# Retrieval of virtual temperature from vertical wind profiles and ground-based microwave radiometer data

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O rd e r N u m b e r 9128674 R etrieval o f v irtu a l tem perature from vertical w ind profiles and ground-based m icrow ave radiom eter data Sienkiewicz, Meta Elizabeth, Ph.D. The University of Oklahoma, 1991 UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. THE UNIVERSITY OF OKLAHOMA GRADUATE COLLEGE RETRIEVAL OF VIRTUAL TEMPERATURE FROM VERTICAL WIND PROFILES AND GROUND-BASED MICROWAVE RADIOMETER DATA A DISSERTATION SUBMITTED TO THE GRADUATE FACULTY in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY By META ELIZABETH SIENKIEWICZ Norman, Oklahoma 1991 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. RETRIEVAL OF VIRTUAL TEMPERATURE FROM VERTICAL WIND PROFILES AND GROUND-BASED MICROWAVE RADIOMETER DATA A DISSERTATION APPROVED FOR THE SCHOOL OF METEOROLOGY Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS I would like to take this opportunity to acknowledge the help I have received from numerous people in the process of pursuing this degree; without such help this disseration would certainly not have taken the form it has. I first recognize Dr. Tzvi Gal-Chen, my advisor. I appreciate the effort he made each semester to arrange tc continue my assistantship. I also am very grateful for the unselfish loan of his office when I needed a quiet place to work, and his wonderful Macintosh is responsible for the quality of the maps in this paper. I next extend my appreciation to the other members of my committee, Drs. Fred Carr, Doug Lilly, Luther White, and Dusan Zrnic, for their review of this dissertation. Last, but certainly not least, I want to thank all of my friends and my family who lent a sympathetic ear to me when I needed it; without their understanding and encouragement I would not have been able to finish this degree. This research was supported in part by NSF Grant ATM-8513364 and by NOAA Department of Commerce Grant NA85RAH05046 awarded to the University of Oklahoma. -Ill- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS....................................................... iii LIST OF TABLES...................................................................vi LIST OF ILLUSTRATIONS................................................... vii ABSTRACT..........................................................................xvi Chapter 1. INTRODUCTION AND OBJECTIVES..............................1 1.1 Introduction.............................................................. 1 1.2 Previous work............................................................3 1.3 Objectives of This Study............................................ 6 2. PHYSICAL THEORY ............................................15 2.1 Relation of Wind to Virtual Temperature.............. 15 2.2 Relation of Radiance to Temperature...................17 3. WIND RETRIEVAL THEORY........................................21 3.1 The Poisson Equation and Compatibility Conditions............................................................... 21 3.2 Dynamic Retrieval as a Variational Problem 22 3.3 Approximations to the Divergence Equation 24 3.4 Dynamic Retrieval in Sigma Coordinates............. 31 4. RADIANCE RETRIEVAL THEORY.............................. 34 4.1 Retrievals Using Scalar Radiances..................... 37 4.2 Retrievals Using Radiance Gradients.................... 60 -iv- R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -V - Chapter 5. COMBINED WIND AND RADIANCE RETRIEVALS ...76 5.1 Temperature and Virtual Temperature................ 76 5.2 Retrievals from Winds and Radiances...................80 6. ANALYSIS METHODS............................................... 83 6.1 Data........................................................................ 83 6.2 Dynamic Retrieval of Height and Temperature From Wind............................................................ 101 6.3 Retrieval of Temperature from Radiance...........113 6.4 Combined Wind and Radiance Retrievals.......... 130 7. DISCUSSION OF RESULTS................................... 133 7.1 Synoptic Overview..............................................133 7.2 Assessment of Retrieval Quality........................ 145 7.3 Dynamic Retrieval from Wind Data....................149 7.4 Retrieval from Radiance Data............................206 7.5 Retrieval of Temperature Using Winds and Radiances........................................................... 243 8. SUMMARY AND CONCLUSIONS................... 281 8.1 Summary of Results........................................... 282 8.2 Possible Improvements......................................285 8.3 Future Applications............................................ 287 REFERENCES...................................................................291 APPENDIX A....................................................................... 299 APPENDIX B....................................................................... 310 APPENDIX C ..................................................................... 314 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES TABLE Page 6.1 AVE-VAS Experiment dates................................... 86 6.2 Estimates of the RMS errors in thermodynamic quantities of AVE/VAS rawinsonde data (After Fuelberg, 1974)....................................................... 86 6.3 Estimates of RMS errors in AVE/VAS rawinsonde wind data (After Fuelberg, 1974).............................86 7.1 Experiments for height retrievals from wind.........153 7.2 Height retrievals from wind data........................... 153 7.3 Experiments for virtual temperature retrievals from wind.................................................................179 7.4 Virtual temperature retrievals from wind data......179 7.5 Experiments for radiance retrievals at stations....210 7.6 Temperature retrievals from radiance data at stations ........................................................... 204 7.7 Experiments for gridpoint radiance retrievals..... 227 7.8 Gridpoint virtual temperature retrievals from radiance ................................................................227 7.9 Experiments for combined retrievals................... 257 7.10 Gridpoint virtual temperature retrievals from radiance ................................................................258 R eproduced with permission of the copyright owner. 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LIST OF ILLUSTRATIONS FIGURE Page 4.1 Contribution function calculated using polynomial basis functions for the four groundbased microwave channels....................................... 43 4.2 Contribution function calculated using polynomial basis functions for both groundbased and satellite microwave channels................. 43 4.3 Contribution function calculated using weighting function basis functions for the four groundbased microwave channels....................................... 45 4.4 Contribution function calculated using weighting function basis functions for both ground-based and satellite microwave channels...........................45 4.5 “Contribution function” for Smith method retrieval using the four ground-based microwave channels.................................................................... 52 4.6 “Contribution function” for Smith method retrieval using both ground-based and satellite microwave channels.................................................52 5.1 Bias (K) and rms difference (K) between “true” virtual temperature fields and virtual temperature fields estimated (a) assuming dry atmosphere and (b) using a correction based on the mean sounding and 50% relative humidity.......................78 5.2 Bias (K) and rms difference (K) between brightness temperatures calculated from “true” virtual temperature fields and those calculated from (a) dry atmosphere and (b) virtual temperature fields estimated using a correction based on the mean sounding and 50% relative humidity.....................................................................79 6.1 Locations of NWS rawinsonde stations in AVE/VAS...................................................................87 6.2 First pass and final response of filter applied to wind data before interpolation to grid..................... 89 6.3 Time cross sections of winds for Oklahoma City. ...90 -VII- R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -V III- FIGURE Page 6.4 Rms error of retrieved heights (m) using raw (solid lines) and smoothed (dashed lines) wind data for the four AVE/VAS experiments..................92 6.5 First pass and final response of Barnes objective analysis.................................................................... 94 6.6 Grid staggering used in objective analysis and dynamic retrievals....................................................94 6.7 Normalized weighting functions for the four ground-based microwave channels and two satellite microwave channels used in this research....................................................................95 6.8 Ground-based weighting functions for 52.85 GHz channel, surface pressure = 850 mb...................... 98 6.9 Ground-based weighting functions for 52.85 GHz channel, surface pressure = 1000 mb.................... 98 6.10 Satellite weighting functions for 54.96 GHz channel...................................................................... 99 6.11 Satellite weighting functions for 54.96 GHz channel......................................................................99 6.12 Response function for second-order centered differences and Lanczos’ derivative formulation..104 6.13 Rms error of temperature retrievals (K) with vertical derivatives calculated by centered differences and by Lanczos’ derivative method. ..104 6.14 Vertical profile of virtual temperature interpolated from gridded data for 1200 UTC, 27 March 1982 at Stephenville....................................................... 105 6.15 Grid points used in dynamic retrieval................... 108 6.16. Grid points at 850 mb level, 1200 UTC 6 March 1982........................................................................ 110 6.17. Rms error (m) for height retrievals in sigma coordinates (two-scale approx. equations) with “true” t v and mean tv fields................................. 112 6.18 Rms error (K) for temperature retrievals in sigma coordinates (two-scale approximation) with ‘true” t v and mean t v fields................................. 113 6.19 Regression retrieval coefficients calculated from OKC “climatology”..................................................116 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -ixFIGURE Page 6.20 Coefficients vwT (WvwT+ ae2i ) _1 for minimum variance retrieval based on OKC 5-year “climatology".......................................................... 117 6.21. Mean squared temperature gradient error ((K/grid division)2) , and the curve-fit estimate for 1/Po .......................................................................... 132 7.1 Surface map for 1200 UTC, 6 March 1982............134 7.2 Surface map for 1200 UTC, 7 March 1982........... 134 7.3 500 mb analysis for 1200 UTC, 6 March 1982....136 7.4 500 mb analysis for 1200 UTC, 7 March 1982.... 136 7.5 Surface map for 1200 UTC, 27 March 1982......... 137 7.6 Surface map for 1200 UTC, 28 March 1982......... 137 7.7 500 mb analysis for 1200 UTC, 27 March 1982...139 7.8 500 mb analysis for 1200 UTC, 28 March 1982. ..139 7.9 Surface map for 1200 UTC, 24 April 1982............ 140 7.10 Surface map for 1200 UTC, 25 April 1982............ 140 7.11 500 mb analysis for 1200 UTC, 24 April 1982.....141 7.12 500 mb analysis for 1200 UTC, 25 April 1982.....141 7.13 Surface map for 1200 UTC, 1 May 1982..............143 7.14 Surface map for 1200 UTC, 2 May 1982..............143 7.15 500 mb analysis for 1200 UTC, 1 May 1982........144 7.16 500 mb analysis for 1200 UTC, 2 May 1982....... 144 7.17 Height analyses for 6 March 1982,1200 UTC, on the 300 mb pressure surface.................................. 151 7.18 RMS error (m) for height fields retrieved from wind using Neumann boundary conditions in pressure coordinates............................................. 154 7.19 Standard deviation of height (m) on pressure levels, averaged over eight observation periods on each VAS experiment day................................156 7.20 Ratio of average rms error to average standard deviation of height on constant pressure levels from retrievals using Neumann boundary condtions and the two-scale approximation to the divergence equation.............................................. 156 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -X - FIGURE Page 7.21 Total number of gridpoints used in calculation of rms error statistics over the 8 observation periods of each VAS day.................................................... 158 7.22 s i score for height fields retrieved from wind using Neumann boundary conditions in pressure coordinates............................................................ 159 7.23 Rms error (m) of retrieved height fields using the two-scale approximation in pressure coordinates............................................................ 160 7.24 S i score for height fields retrieved from wind using the two-scale approximation in pressure coordinates............................................................ 161 7.25 Comparison of rms error(m) for height fields retrieved from winds in pressure coordinates using Dirichlet boundary conditions..................... 162 7.26 Comparison of S i score for height fields retrieved from winds in pressure coordinates using Dirichlet boundary conditions..................... 164 7.27 RMS error (m) for height fields retrieved from wind using Neumann boundary conditions in sigma coordinates................................................. 165 7.28 Rms error (m) of retrieved height fields using the two-scale approximation in sigma coordinates. ...166 7.29 Comparison of rms error(m) for height fields retrieved from winds in sigma coordinates using Dirichlet boundary conditions............................... 167 7.30 Temperature analyses for 7 March 1982, 0000 UTC, on the 700 mb pressure surface................... 170 7.31 Temperature analyses for 7 March 1982, 0000 UTC, on the a = 0.7 surface...................................173 7.32 Rms error (K) for temperatures derived from wind on constant pressure surfaces using the ‘modified’ Neumann boundary conditions and vertical derivatives of various approximate forms of the divergence equation....................................181 7.33 Standard deviation of temperature (K) on pressure levels, averaged over eight observation periods on each VAS experiment day....................185 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -xiFIGURE Page 7.34 Ratio of average rms error to average standard deviation of temperature on constant pressure levels from retrievals using Neumann boundary condtions and the two-scale approximation to the divergence equation.............................................. 185 7.35 s i scores for temperatures derived from wind on constant pressure surfaces using the 'modified' Neumann boundary conditions and vertical derivatives of various approximate forms of the divergence equation.............................................. 186 7.36 Rms error (K) for temperatures derived from wind on constant pressure surfaces using the twoscale approximation for various boundary conditions.................................................................188 7.37 s i scores for temperatures derived from wind on constant pressure surfaces using the two-scale approximation for various boundary conditions. ..190 7.38 Comparison of rms error(K) for temperature fields retrieved from winds in pressure coordinates using Dirichlet boundary conditions..................... 191 7.39 Comparison of S i scores for temperature fields retrieved from winds in pressure coordinates using Dirichlet boundary conditions..................... 192 7.40 Rms error (K) for temperatures derived from wind using Neumann boundary conditions in sigma coordinates and vertical derivatives of various approximate forms of the divergence equation. ...193 7.41 Standard deviation of temperature (K) on constant sigma levels, averaged over eight observation periods on each VAS experiment day........................................................................... 195 7.42 Ratio of average rms error to average standard deviation of temperature on constant sigma levels from retrievals using Neumann boundary conditions and the two-scale approximation to the divergence equation........................................ 195 7.43 s i scores for temperatures derived from wind using Neumann boundary conditions in sigma coordinates and vertical derivatives of various approximate forms of the divergence equation. ...197 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -X II- FIGURE Page 7.44 Rms error (K) for temperatures derived from wind using the two-scale approximation in sigma coordinates for various boundary conditions.198 7.45 s i scores for temperatures derived from wind using the two-scale approximation in sigma coordinates for various boundary conditions.199 7.46 Comparison of rms error(K) for temperature fields retrieved from winds in sigma coordinates using Dirichlet boundary conditions................................200 7.47 Comparison of s i scores for temperature fields retrieved from winds in sigma coordinates using Dirichlet boundary conditions................................201 7.48 Comparison of rms error (K) for retrieved vs. interpolated boundary-layer temperature fields Temperatures are retrieved from wind using the two-scale approximation in sigma coordinates with Neumann boundary conditions..................... 204 7.49 Comparison of s i scores for retrieved vs. interpolated boundary-layer temperature fields Temperatures are retrieved from wind using the two-scale approximation in sigma coordinates with Neumann boundary conditions..................... 205 7.50 Rms error (K) of retrievals at Oklahoma City and Stephenville using regression coefficients............211 7.51 Rms error (K) of retrievals at Oklahoma City and Stephenville using Smith’s retrieval method and the minimum information method..........................213 7.52 Rms error (K) of retrievals at Oklahoma City and Stephenville using regression coefficients, Smith’s method and the minimum information method ........................................................215 7.53 Bias (K) of sounding profiles retrieved with statistical regression , Smith’s method and minimum information using brightness temperatures with (0.5K)2 random error added. ..217 7.54 s i scores of retrievals at Oklahoma City and Stephenville........................................................... 218 7.55 Temperature analyses for 7 March 1982, 0000 UTC, on the 700 mb pressure surface....................223 7.56 Rms error (K) for retrievals on grid from radiance with mean temperature as first guess................... 229 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -xiiiFIGURE Page 7.57 Ratio of average rms error to average standard deviation of virtual temperature on constant pressure levels from retrievals using radiances with mean temperature as first guess................... 229 7.58 Rms error (K) for retrievals on grid from radiance gradients with Neumann boundary conditions and mean temperature as first guess....................230 7.59 Ratio of average rms error to average standard deviation of virtual temperature on constant pressure levels from retrievals using radiance gradients with mean temperature as first guess. ..231 7.60 Temperature analyses for 7 March 1982, 0000 UTC, on the 100 mb pressure surface................... 233 7.61 s i score for retrievals on grid from radiance with mean temperature as first guess...........................234 7.62 S i score for retrievals on grid from radiance gradients with mean temperature as first guess. ..235 7.63 Bias (K) of virtual temperature fields retrieved from radiances with mean field first guess.............235 7.64 Rms error (K) for retrievals on grid from radiance using Smith’s method with mean temperature as first guess............................................................... 237 7.65 s i score for retrievals on grid from radiance using Smith's method with mean temperature as first guess............................................................... 238 7.66 Ratio of average rms error to average standard deviation of virtual temperature on constant pressure levels. Retrievals are with radiance weighting 10 times normal...................................... 240 7.67 Virtual temperature analyses for 7 March 1982, 0000 UTC, on the 700 mbpressure surface..........244 7.68 Analyses of retrieved virtual temperatures for 7 March 1982, 0000 UTC, on the 850 mb pressure surface....................................................................248 7.69 Normalized brightness temperatures for the 53.85 GHz channel calculated from temperature fields at 0000 UTC 7 March 1982......................... 253 7.70 RMS error (K) for retrievals on grid in pressure coordinates from radiance with wind-derived temperature as first guess.......................................259 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -xivFIGURE 7.71 Page RMS error (K) for retrievals on grid in pressure coordinates from radiance gradients with windderived temperature as first guess....................... 260 7.72 Si score for retrievals on grid in pressure coordinates from radiance with wind-derived temperature as first guess...................................... 261 7.73 s i score for retrievals on grid in pressure coordinates from radiance gradients with windderived temperature as first guess....................... 262 7.74 RMS error (K) for retrievals on grid in sigma coordinates from radiance gradients with windderived temperature as first guess....................... 264 7.75 s i score for retrievals on grid in sigma coordinates from radiance gradients with windderived temperature as first guess....................... 265 7.76 Rms error (K) for retrievals on grid in pressure coordinates from radiance with wind-derived temperature as first guess. ‘Observed’ radiances have 0.5K rms error field added.......... 267 7.77 Rms error (K) for retrievals on grid in pressure coordinates from radiance gradients with windderived temperature as first guess. ‘Observed’ radiances have 0.5K rms error field added..........268 7.78 s i score for retrievals on grid in pressure coordinates from radiance with wind-derived temperature as first guess. ‘Observed’ radiances have 0.5K rms error field added..........269 7.79 S i score for retrievals on grid in pressure coordinates from radiance gradients with windderived temperature as first guess. ‘Observed’ radiances have 0.5K rms error field added..........270 7.80 Rms difference (K) between interpolated brightness temperatures and brightness temperatures calculated at gridpoints from constant pressure level analyses of temperature.272 7.81 Rms error (K) for retrievals on grid in pressure coordinates from radiance with wind-derived temperature as first guess. ‘Observed’ radiances are interpolated from station locations.................................................................274 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -XV- FIGURE Page 7.82 Rms error (K) for retrievals on grid in pressure coordinates from radiance gradients with windderived temperature as first guess. ‘Observed’ radiances are interpolated from station locations.................................................................275 7.83 S i score for retrievals on grid in pressure coordinates from radiance with wind-derived temperature as first guess. ‘Observed’ radiances are interpolated from station locations................................................................. 276 7.84 s i score for retrievals on grid in pressure coordinates from radiance gradients with windderived temperature as first guess. ‘Observed’ radiances are interpolated from station locations................................................................. 277 7.85 Rms error (K) for retrievals on grid in sigma coordinates from radiance gradients with windderived temperature as first guess. ‘Observed’ radiances are interpolated from station locations................................................................. 278 7.86 S i score for retrievals on grid in sigma coordinates from radiance gradients with windderived temperature as first guess. ‘Observed’ radiances are interpolated from station locations................................................................. 279 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT Measurements of a wind field and its changes with time (such as may be made by a network of wind Profilers) provide information that may be used, along with the equations of motion and the hydrostatic relation, to retrieve analyses of virtual temperature. Such analyses are poor near the surface where boundary layer sub-grid scale processes dominate, but improve in the mid-layers of the atmosphere. Radiometric measurements such as may be made by ground based (and satellite) microwave radiometers also provide information about temperature fields, and temperature profiles retrieved from the ground-based radiances are fairly good near the surface but become poorer further from the ground. In this study, we want to find whether an analysis derived from a combination of wind and radiometric data can improve on analyses derived from either data source alone. This hypothesis is tested using rawinsonde data taken during a special field experiment (AVE-VAS). Temperature fields are derived from radiosonde wind observations taken at three-hourly intervals. These temperature analyses are combined with ground-based radiances (calculated from radiosonde temperature profiles) and the combined analyses are compared to analyses derived from winds and from radiances alone. Several retrieval methods using radiances and radiance gradients are evaluated. The results show that the temperature analyses from the wind data are improved by the addition of ground-based radiances although there is little effect above 700 mb where the weighting functions of the radiances are small. -xvi- R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. RETRIEVAL OF VIRTUAL TEMPERATURE FROM VERTICAL WIND PROFILES AND GROUND-BASED MICROWAVE RADIOMETER DATA CHAPTER 1 INTRODUCTION AND OBJECTIVES 1.1 Introduction Advances in atmospheric technology have made possible the development of new instruments designed for automated measurement of atmospheric profiles at much more frequent intervals than the conventional rawinsonde network. The twice-daily synoptic rawinsonde ascents do not provide nearly enough temporal resolution (to say nothing of the lack of horizontal resolution) needed to adequately observe mesoscale phenomena such as fronts, squall lines, mesoscale convective systems, mountain waves, land-sea breezes and local circulations, all of which may have a large effect on weather conditions. One such instrument is the wind Profiler (Hogg, e t a i, 1983, van de Kamp, 1988), which produces profiles of wind in the vertical from Doppler radar measurements of energy backscattered by refractive index fluctuations that are advected by the wind. The quality of the wind information is at least equivalent to that observed by the current rawinsonde network. The current design calls for a three-beam (north and east beams at 75° elevation and a vertically- 1- R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. pointing beam) 404 MHz radar that will measure winds every 250 m in the vertical. The measurements start at 500 m above ground level (AGL) and extend to a maximum height of 16.25 km AGL. The wind profiles will be available on an hourly basis, derived from raw radial velocities measured every 6 min. A demonstration network of 31 wind Profilers is in the process of being installed in the central United States, the installation is halfway completed at this time. Hogg, etal., (1983) describe the prototype of a Profiler system designed for unattended operation in almost all weather conditions capable of providing nearly continuous measurements of winds, temperature and humidity. Wind Profilers would be used to obtain winds, and a ground-based six-channel microwave radiometer (also called a thermodynamic profiler) was proposed for temperature and humidity profiling. The ground-based radiometer operates in the microwave absorption bands for oxygen and water vapor. The radiation received at ground level is dependent on the temperature of the atmosphere above and the absorption by water vapor, so measurements from this instrument can be used to produce vertical profiles of temperature and moisture. The Profiler system as described by Hogg etal. (1983) falls short of being able to measure vertical profiles with the accuracy of rawinsondes. The wind observations provided by wind Profilers are good, but the temperature profiles derived from ground-based microwave radiometer measurements show poor resolution of sounding details away from the surface. The wind Profiler data is another potential source of temperature information, however. It is possible to diagnose geopotential and virtual temperature from measurements of winds over an area and their changes with time by use of the equations of motion and thermodynamic equations (Gal- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chen 1978, 1988). These “dynamic retrievals” of height and temperature do not perform equally well at every level, though. The retrievals become very poor in the lower levels of the troposphere where boundary layer sub-grid scale circulations and friction are important (Modica and Warner, 1987). Thus, we have two potential sources of temperature information: the ground-based radiometer, which can be used to retrieve good temperature profiles in the lower troposphere but has poor resolution further aloft; and the Profiler winds, which have problems with obtaining fields near the surface but improve away from the surface. Each of these methods has strengths where the other has weaknesses. This leads us naturally to ask: “Can the two sources of data can be combined to give better temperature analyses, perhaps rivalling rawinsonde accuracy? If so, what is the best way such a combination can be achieved?" These questions are the basis for this dissertation research. 1.2 Previous work Proposals for the specification of atmospheric temperature, profiles from satellite radiance measurements were brought forth as early as the late 1950’s, when King (1956) and Kaplan (1959) suggested that different angular measurements by satellites of outgoing radiance or different spectral measurements at a constant viewing angle could be used to obtain temperature profiles through inversion of the radiative transfer equation. The use of satellite measurements of microwave thermal emission by oxygen to determine atmospheric temperature profiles was discussed by Meeks and Lilley (1963); Westwater (1965) discussed the use of ground-based measurement techniques. Westwater and Strand (1968) discussed information content of ground-based measurements and suggested optimal choices for the frequencies to be used. Westwater (1972) used data from a single frequency Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ground-based radiometer with measurements taken over several different elevation angles to determine temperatures at different locations. Snider (1972) compared results from angular and multispectral ground-based measurements. Decker, et a/., (1978) showed retrievals of temperature and water vapor from measurements taken from a ship, to demonstrate the utility of ground-based measurements from ocean buoys; they also discuss how cloudy profiles can be adjusted so that temperatures can be retrieved. Studies of retrievals from ground-based microwave radiometer measurements show rms errors in temperature determination of about 1 - 3 K in the lower to mid-troposphere, and up to about 6 K near the tropopause; the profile retrievals are improved when satellite radiance measurements (Westwater and Grody, 1980, Westwater, et a!., 1984), radar-determined tropopause heights (Westwater, etai., 1983), or a combination of satellite, radar, and ground-based measurements are used (Westwater, etal., 1985). Several studies have dealt with the use of wind information in retrieval of mass and temperature fields. Saha and Suryanarayana (1971) calculated geopotential fields in the tropics from wind data and several balance type equations. Fankhauser (1974) used winds from a special mesoscale rawinsonde network and the full divergence equation to obtain height and temperature fields in the vicinity of a squall line. Dual-Doppler radar observations of wind have been used to determine characteristics of the planetary boundary layer (Gal-Chen and Kropfli, 1984), thermodynamic structure of a squall line (Roux, etal., 1984), analysis of a tornadic thunderstorm (Hane and Ray, 1985) and a frontal rainband (Parsons etal., 1987). Kuo and Anthes (1985) used the divergence equation in p-coordinates with Dirichlet boundary conditions and model generated winds to calculate geopotential and R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. temperature fields. Kuo et al., (1987b) extended that work in sigma coordinates for several synoptic cases, testing the effect of differing boundary conditions and approximations to the divergence equation; their retrieved temperature fields had rms errors between 1 and 2 K for an observation spacing of 360 km. A few studies have been conducted using actual wind Profiler data; these have necessarily been limited to single station analyses or analyses over a small area, since the larger network of wind Profilers is only now coming on line. Cram, et al., (1988) interpolated wind measurements from four Colorado profilers to a grid and used the full divergence equation to calculate height and temperature fields. They also show how such fields can be blended with output from numerical models through use of variational methods. Neiman and Shapiro (1989) have used the geostrophic thermal wind relation with some success in a single station analysis for diagnosis of horizontal gradients of temperature and of temperature advection using Profiler winds. Their profilerretrieved temperature gradients and advections showed magnitudes and evolution that compared favorably with observed synoptic and mesoscale thermal fields. Hermes (1988, 1991) analyzed wind profiler measurements from PRE-STORM data; she found that overall, the use of the generalized thermal wind equation and profiler wind observations to estimate thermal gradients was not successful; this was partly due to the difference in horizontal scales measured by the profiler triangle and the rawinsondes used for verification. She presented one example where detailed thermal structure was successfully detected using profiler winds. Other authors have proposed ways of combining radiance information and winds to obtain temperature fields. Bleck, etal., (1984), using the balance equation in isentropic coordinates and Profiler winds simulated by radiosonde Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. data, diagnosed mass fields. They proposed that the Profiler winds could be used to modify smooth temperature profiles retrieved from radiances and provide additional vertical detail. Gal-Chen (1988) presented a theory for combining wind and ground-based or satellite radiance measurements to give improved estimates of virtual temperature fields; his proposal is distinguished by using radiance gradient measurements to avoid problems with bias. Lewis, et al., (1989) show a method where model fields are adjusted to match radiances and winds by a conjugate-gradient minimization. 1.3 Objectives of This Study The goal of this research is to determine whether an analysis combining wind information and ground-based radiometer measurements can be made that is better than what can be obtained using each instrument alone. To achieve this aim, the following objectives are to be met: A. Dynamic retrieval from wind i) Produce retrieved temperature and height analyses comparable to those that could be obtained from the proposed Profiler network. ii) Show the effect of using approximate forms of the equations of motion. iii) Compare effects of Dirichlet and Neumann boundary conditions. B. Retrievals from radiance measurements i) Produce retrieved temperature analyses comparable to those that could be obtained from a ground-based radiometer network. a) Show that results from statistical retrievals comparable to previous studies can be obtained using these simulated radiance measurements. b) Compare the statistical retrievals to retrievals by “physically- R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. based" methods. ii) Compare retrievals from radiances on the grid to retrievals using radiance gradient methods. C. Retrievals from combined wind and radiance data i) Produce retrieved temperature analyses using both wind and radiance data and compare to previous analyses. ii) Compare retrievals from radiances on the grid to retrievals using radiance gradient methods. 1.3.1 Dynamic Retrieval From Wind The primary objective for the dynamic retrievals is to use the equations of motion and winds measured by radiosondes to produce height and temperature fields that are comparable to what could be obtained using the planned wind Profiler network. These retrievals will be used to demonstrate the ability of the dynamic retrieval method to obtain useful information about height and temperature fields using only observed wind data. The retrieved temperature fields also serve as a first-guess field for a combined wind-radiometric temperature analysis. Until the demonstration Profiler network is operational it will not be possible to obtain Profiler measurements on a network of sufficient size to test the theory, thus the substitution of other wind measurements for the Profiler winds is necessary. The use of rawinsonde measurements is advantageous in that radiance ‘measurements’ can also be simulated from temperature profiles measured at the same locations, thus the wind and temperature measuring systems are sampling the same scales in the horizontal. One question that arises is how similar are the wind observations used in this study to the ones that will be provided by the Profiler network. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. The radiosonde data to be used was taken during a special field experiment, with observations taken at 3-hourly intervals. Although this temporal resolution is much better than the 12-hourly spacing of conventional observations, it is still somewhat less than the hourly resolution that will be provided by the profiler network. The horizontal station spacing is about 360 km between stations, somewhat larger than in the proposed network. The wind measurements from the Profiler network will be similar to those provided by the current rawinsonde measurements. Weber and Wuertz (1990) presented statistics comparing winds measured twice daily over almost two years by co-located UHF wind profiler and rawinsonde; they showed very good agreement between the measurements with the standard deviation of differences between the horizontal components of 2.5 ms*1, mostly due to meteorological variability in the winds. The correlations between the u- and the v-components of profiler and rawinsonde winds were both better than 0.95. Weber, etal. (1990) compare rawinsonde winds to winds measured by a Unisys profiler (UHF, 404 MHz) of the same design as those to be used in the Profiler demonstration network. Although the profiler and rawinsonde measurements were separated by about 50 km, there was still good agreement between the winds; the differences between measurements had standard deviations of 3.65 ms*1 and 3.06 ms*1 for the u- and v- components, respectively. The correlations between the wind measurements were also slightly lower (.91 and .93 for uand v- components, respectively); the larger differences for these sensors are due to the increased separation between the rawinsonde and profiler measurements. Many of the previous studies which used analyses of wind to produce geopotential and temperature fields had focused on providing initial fields for R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. numerical weather prediction (e.g. Saha and Suryanarayana, 1971, Anthes and Keyser, 1979), their wind analyses were generated using all available data rather than producing analyses derived from wind alone. Other studies (Kuo and Anthes, 1985; Modica and Warner, 1987, Kuo, et al., 1987b) were Observing System Simulation Experiments (OSSE’s) using simulated datasets that were produced from numerical models and so would retain the balance between the wind and mass field (though perhaps somewhat degraded) in the divergence equation that was inherent in the numerical model equations. The methodology of this study is closer to that of an Observing System Experiment (OSE) such as discussed by Ramamurthy and Carr (1987), we are using actual wind and temperature observations rather than model-derived fields. This study differs from the previous ones in that the wind and temperature fields are analyzed separately so that the only relation between the fields is whatever exists in the atmosphere (discounting, of course, the contribution of the measured heights in calculation of radiosonde winds, which should have no influence on the balance between the two fields). A secondary objective is to estimate the error incurred in using approximate forms of the equations of motion to calculate temperature fields, using wind fields analyzed from actual wind observations taken at a meso-a or synoptic scale spacing, rather than data synthesized from numerical model analyses. For various reasons, it may not be desirable or may not be practical to use a divergence equation derived from the full equations of motion when performing dynamic retrievals. For example, we may wish to avoid the use of vertical motion, ro, for instance, since the Profiler measurements of vertical motion may contain motions balanced on a scale much smaller than the horizontal spacing between stations. Also, elimination of some of the other non Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -10- linear terms may prove beneficial if the error in their estimation is large (Kuo, et al., 1987b). This subject will be addressed further in Chapter 3. Kuo et al. (1987b) have presented dynamic retrievals using different approximations to the full divergence equation, this work extends their study. A third area of concern that will be investigated is the effect of boundary conditions on the wind-derived retrievals. Previous researchers (Kuo and Anthes, 1985; Modica and Warner, 1987, Kuo, et al., 1987b) have primarily used Di rich let boundary conditions for the solution of the Poisson equation that is required for the retrievals of heights and temperatures from wind. If Dirichlet boundary conditions are used, the boundary values of height or temperature must be specified from a separate source. Gal-Chen (1986a) suggested the use of Neumann boundary conditions for the solution of the Poisson equation since the lateral boundary conditions can be calculated from the wind measurements. When using Neumann boundary conditions, the only additional information needed is a specification of the mean value of the temperature (or geopotential) field at each level, and (for pressure analyses near the surface) the surface temperature field. For the purposes of this study it is desirable to keep the amount of additional information needed to produce the analyses to a minimum, so Neumann boundary conditions will primarily be used. Kuo, et al. (1987b) found temperature retrievals with Neumann boundary conditions had larger rms errors than similar retrievals using Dirichlet conditions. Errors in specification of Dirichlet boundary conditions increase the error in the retrievals, however (Anthes and Keyser (1979), Kuo and Anthes (1985)); this should be taken into account in comparisons between the retrievals using the differing boundary conditions. The dynamic retrievals will be carried out in both pressure and sigma Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -11- coordinates. The retrievals in pressure coordinates are complicated by the intersection of pressure surfaces with the ground. Gal-Chen (personal communication) suggested that a combination of Neumann and Dirichlet boundary conditions could be used in this case, with Neumann boundary conditions on the ordinary lateral boundaries, and Dirichlet boundary conditions provided by surface observations where the pressure levels intersect with the ground. The effect on the temperature retrievals of these ‘modified’ Neumann boundary conditions will be considered. 1.3.2 Retrievals From Radiance Measurements The primary objective for the radiance retrievals is similar to that for the dynamic retrievals from wind; namely, to obtain temperature fields similar to that that could be obtained by a network of ground-based radiometers, optionally supplemented by microwave satellite measurements. This work can be broken into two tasks: first, show that the data used in this experiment can produce results comparable to the statistical retrievals in previous studies where actual radiance measurements are used; second, compare these statistical retrievals to physical retrieval methods that can also be applied over an area. The first subtask, showing that retrievals can be made that are comparable to previous studies, is necessary because the radiance retrieval part of this study is like an OSSE. Our radiance ‘data’ is simulated by integrating temperature profiles with a somewhat simplified radiance weighting function rather than having actual measurements and using a more complete radiation formulation. The reason for using simulated measurements is the same as for the wind Profiler data given above, there is no network of groundbased radiometers available on which to test the methods presented here. If the results presented agree with previous studies, we can at least say that the R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 2 - data used in this study are not unlike that which would be observed by a network of ground-based radiometers. Previous studies with the ground-based radiometers (Westwater, 1972, Snider, 1972, Westwater,et al., 1975, 1983, 1985, Decker, et al., 1978, Westwater and Grody, 1980, Schroeder, 1990) have used statistical regression retrieval methods applied at a single station location to obtain temperature profiles. Simultaneous radiosonde/radiometer measurements or weighting functions and sounding “climatologies” are used to fit sets of retrieval coefficients needed to produce temperature profiles from radiance measurements. The ground-based retrieval coefficients (unlike those derived for satellite radiance measurements) can be used only at the one station location, for the most part, because of the strong dependence of the groundbased weighting function on surface elevation. A similar set of retrievals will be performed for this research; a 5-year set of spring soundings at one station will be used to derive coefficients to be applied at two stations with similar surface elevations and sounding climatological characteristics. The other task associated with this objective is to compare the statistical retrievals to physically based retrievals performed on the same data. The performance of a statistical retrieval of mid- to upper tropospheric temperature from ground-based radiances depends on the correlation between the temperatures in lower levels (measured by the ground-based radiometer or from the station “climatology”) and behavior of the profile higher in the atmosphere. These covariance values may not be so readily quantified when the retrievals are extended to a large domain and temperature and radiance gradients (as suggested by Gal-Chen (1988)) are used. Thus, it is useful to compare the statistical retrieval to retrievals that are physically oriented and lack R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -13- the covariance information, so that the importance of the covariances in the retrieval process can be estimated. The second objective is to evaluate methods that use radiance gradient information rather than radiances to obtain temperature fields over an area. In his paper on retrieval from combined wind and radiance data, Gal-Chen (1988) presented a retrieval method utilizing a ‘first guess’ or background temperature gradient field and radiance gradient data, so that the effect of bias in radiance measurements could be removed. This concept of using radiance gradient data in retrievals will be expanded in Chapter 4; radiance gradient retrieval methods analogous to regular “one-dimensional” radiance retrieval methods are derived, and these methods are related to the one proposed by Gal-Chen. These retrievals will be compared to retrievals performed using more conventional ‘one-dimensional’ methods to show how the results differ. 1.3.3 Retrievals From Combined Wind and Radiance Data The ultimate goal of this research is to produce temperature fields from a combination of wind and radiance data. The methods to be tested for the combined wind/radiance retrievals are similar to the ones used in the radianceonly retrievals, except that the ‘first guess’ background fields will be windderived temperature or temperature gradient fields rather than mean values of temperature. Since these combined retrievals use the same data and the same methods as the radiance-only retrievals, we can directly compare the resulting fields. Since the initial fields are wind-derived fields from the first section, the combined retrievals can also be directly compared to the dynamic retrievals. As noted in the previous section, the retrieval methods to be used include regular radiance retrievals and radiance gradient retrievals. The reasoning behind using the gradient retrieval formulation (suggested by Gal-Chen, 1988) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -14- is that (a) wind measurements provide estimates of temperature gradient, and (b) bias in radiance measurements make relative measurements of radiance (e.g., gradients) more accurate than absolute measurements, so a good retrieval scheme would be one that produced an analysis as close as possible to the wind-estimated temperature gradients and radiance gradients. The gradient retrieval methods are a new approach, and the consequences of using these methods have not been determined. Thus, it is necessary to compare these retrievals to retrievals by more conventional methods to aid in separating the characteristics of the retrievals due to the retrieval method from the characteristics that result from the input data, independent of the retrieval method. 1.3.4 Dissertation Overview This dissertation is organized into several sections. The next four chapters discuss the theoretical aspects of this study. Chapter 2 reviews the basic physical relationships, the equations of motion and the radiative transfer equation; these equations relate temperatures to wind and radiances and so form the basic foundation that underlies any retrieval method. Chapter 3 covers the theory of retrievals from wind measurements. In Chapter 4, basic radiance retrieval theory is explained and the new gradient retrieval methods are derived. Chapter 5 discusses combining wind and radiance measurements. The last three chapters cover the application of the retrievals of temperature from wind and from radiance measurements. Chapter 6 presents the methodology of this study, it explains how the analysis methods are carried out. Chapter 7 discusses the results of the experiments conducted with the data, and the conclusions and suggestions for future research are given in Chapter 8 . R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2 PHYSICAL THEORY In order to be able to discuss ways to derive a virtual temperature field from physical measurements, it is necessary to first outline the physical theory that relates virtual temperature to wind fields and to radiances. 2J Relation of Wind to Virtual Temperature For scales of motion where vertical accelerations are relatively small, the relation of wind, mass, and temperature in the atmosphere can be modelled using the hydrostatic primitive equations In x , y , p Cartesian coordinates (where p = pressure) the horizontal vector momentum equation is (Haltiner and Williams, 1980): 5? - H r + v • v * - £kXT + <2-1) where v is the horizontal wind (v = u i + v j), <j>is geopotential ( <j> = gz where g is gravity), f is the Coriolis parameter ( f = 2 Q s i n <p, with £2 = Earth’s angular momentum and (p = latitude), and Fr = F j i + f 2j represents turbulent friction forces acting in the x (east) and y (north) directions respectively. The relation may also be formulated in terrain followingsigma coordinates (where o = p / ps; ps = surface pressure) as: -15- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 6 - Dt ' I t + V ' V»V + 6 ^ fk><V + ^ (2'2) where v is directed along a constant sigma surface. Note the added term with ps and virtual temperature t v ( t v = t ( i . + o . 6 iq ) , q = specific humidity). The divergence equation in pressure coordinates is formed by taking the horizontal divergence of the horizontal momentum equation (2 . 1): § * V [<v.V„)v] + Vp.(fkxv) + = -Vpifi + VpFr where (2.3) = V p- v, the two-dimensional divergence of the horizontal wind on a pressure surface. This equation may be written in a more familiar form by d expanding the components and rearranging terms to get 3d 3t + 3d 2 3d f3co 3u 3co 3v u3x + v3y + D + + [ax 3p + 3y 3p 3d - 2 J (U ,V ) - f C + ^ = -V2<> (2.4) where C= ( 3 v / 3 x - 3 u / d y ) is the vertical component ofrelative vorticity and j ( u , v ) = (3u/3x) (3v/3y) - (3u/3y) (3v/3x), is the Jacobian of u and v. This divergence equation is a Poisson equation relating V2<|> with various forcing terms based on horizontal and vertical wind, vorticity, friction, divergence, and the time rate of change of divergence. This relation can be converted to an equation relating V 2t v and the vertical derivatives of the forcing functions through use of the hydrostatic equation: 3<}> 3 In p = -R Tv (2.5) A similar divergence equation may be derived in sigma coordinates. Note that, however, the forcing function includes an additional term involving virtual temperature so a suitably chosen first guess estimate of temperature is R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -17needed before the temperature can be derived. The implications of this will be discussed further in a later section (Chapter 3). 2.2 Relation of Radiance to Temperature Radiometer measurements of the intensity of the microwave radiation received at the Earth’s surface are related to the temperature profile in the vertical above the measuring point by means of radiative transfer theory. Below 50 km molecular scattering of radiation is minimal, and for microwave wavelengths (k = 10'1 to 10-3 m) scattering by most clouds (in the absence of precipitation) is much less than absorption (Snider and Westwater, 1972). Thus, scattering terms need not be considered, and the radiation received at ground level is the sum of the radiation emitted by layers of atmosphere above a point, reduced by some factor to account for absorption by intervening layers. The atmosphere can be considered to be in local thermodynamic equilibrium and itradiates as a blackbody, so the intensity ofradiation offrequency v emitted at level z is related to the temperature at that level by the Planck function, /. e. 1 2 hv3 By (T where t is temperature, c ( z ) ) h is the speed of light. = — c 2--------------------------------- - -- exp (h v /k T ) - 1 is Planck’s constant, k is the Boltzmann constant, and At microwave frequencies (where Rayleigh-Jeans approximation is valid and (2 .6 ) hv « kT ) the (the blackbody radiation at b v frequency v) can be related linearly to temperature as 2 v2 B v (T ) = — k T . (2 .7 ) As radiation passes through the atmosphere, the change of its intensity due to extinction is proportional to the mass of the absorbing gas along the path R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 8 - ds and this relation can be expressed as (Goody and Yung, 1989): d lv (extinction) = - e v v Iv ds (2.8) where e v v is called the volume extinction coefficient. Since scattering at the microwave frequencies used is negligible the extinction is just equal to the absorption (ev v = a v). The absorption in the 50-70 GHz frequency range in which microwave radiometric soundings are taken is due primarily to oxygen, O2. The concentration of oxygen (21% by volume) is nearly constant and so values of the absorption coefficient av can be calculated. In higher levels of the atmosphere the lines in the 50-70 GHz range are more distinct but in the troposphere the pressure (or collision) broadening causes the lines to blend together to form an absorption band centered near 60 GHz. An apparent discrepancy between the linewidth parameters measured at low pressure and those necessary to fit experimental measurements for pressures near 1000 mb was resolved by Rosenkranz (1975), who formulated an expression for the absorption of oxygen by applying the theory of bands consisting of overlapping lines. The expression for the absorption coefficient used in this research (from Ulaby, etal., 1981) is based on Rosenkranz’s work: Oo2 (v) = 1 . 6 1 X 10-2 V2 ( 3 ^ 3 ) F' dB km-1 (2.9) where v is frequency (GHz), p is pressure (mb), and t is temperature (K). f ' is a function incorporating the strengths of the absorption lines; it also depends on v, p, and t . By Kirchoff’s law, the emission coefficient is equal to the absorption coefficient so the total change in intensity can be written as: dXy = ( — (Xy Xv OCy By) dS (2.10) The optical depth t v is the integral of the attenuation along the path s R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. (where ds = - dz) and may be written Tv ( z , z s) = f Zse v v (s) ds = f (Xytz') dz ' . Jz ' Jzs (2.11) The proportion of radiation originating at the level z that is received at the surface level z g is the transmission xv (zs , z ) : xv ( z s - z ) = exp(-Tv (z s, z ) ) = exp(- f Jzs dz) . (2.12) Note 8xv/dz = - a v xv. Multiply (2.10) by xv and then solve the resulting differential equation with proper boundary conditions toget a radiative transfer equation referenced to a sensor located at ground level: »oo ( Z Z) Iv = - J 2 By (T ( Z) ) — V0 ZS/ dz + Bv(z=oo) Tv (zs,oo) . (2.13) A similar expression for the radiative transfer equation may be written for radiation measured by a satellite pointing toward the earth, with the second term including emission from the ground instead of from space. Referring back to Eq. (2.7), the Rayleigh-Jeans approximation can also be used to define a “brightness temperature" t bv, which is the temperature of the blackbody that would produce the radiance i v : Tbv = i k Iv ' (2,U) We can multiply (2.13) by c2/2v2k and use Eqs (2.7) and (2.14) to get the form of the radiative transfer equation that is used in ground based radiometry (Westwater, ef al. ,1985): I^bv = f W(v,z) T (z) dz + Tback Jzs (2.15) w (v,z) is the weighting function (equal to -3tv ( z s, z ) / 3 z ), which takes into account the absorption by the atmosphere and the change in absorption with R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 0 - height. tback is the background microwave brightness temperature received from space (from the “big bang”). It is convenient to express the weighting function in coordinates other than height coordinates. On a constant pressure surface, for example, the pressure dependence of av is eliminated and the dependence on temperature is reduced, thus simplifying some horizontal derivative calculations. Similarly, in sigma coordinates horizontal derivative terms with V z 3 or Vps are avoided. Through the use of the hydrostatic relation and the equation of state we can readily convert from height coordinates to the other coordinate systems. For pressure coordinates, we can define absorption a ttv(p) <Xv RT =— = pg a vr r , py (2.16) and transmission Xv(Ps/P) Anotheruseful (Holton, 1972, = exp (I coordinatesystem p.172), where z* ■'Ps a v (p) dp) . (2.17) is the z* orlog-pressure system = - h i n ( p / p 0) ,h = scaleheight (~ 8 km) and p 0 is a reference pressure (usually 1000 mb). Gal-Chen (1988) uses this coordinate system extensively. In z* coordinates the absorption is _ . .. Po « v ( P ) e xp (-z*/H ) a v (z*)= ------------------------- . The transmission for a ground-based radiometer is [Z* x v ( z 3* , z * ) = exp [ - J a v(z*) d z * ] (2.18) (2.19) and the weighting function is 0xv ( z g*, z * ) w ( v , z s* , z * ) = - ------ ^ ------- = a v ( z * ) t v (zs* , z * ) . (2.20) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3 WIND RETRIEVAL THEORY Now that the physical relations have been presented, an evaluation of solution methods to retrieve the virtual temperature field from the observed fields for wind and radiances using the physical relationships (along with other information as appropriate) can be made. The term “retrieval" was applied early on in meteorology to refer to the calculation of temperature given satellite measurements of radiance; later, the attempts to calculate temperature fields from wind fields were dubbed “dynamic retrievals” in analogy to the satellite retrievals. In Chapter 2 the relationship between the wind and mass fields is described; in this section the methodology of the “dynamic retrieval” for mid-latitude synoptic-scale analyses is considered. 2J The Poisson Equation and Compatibility Conditions The vector momentum equation (2.1) may be rewritten nv VP4> = - — - fk x v + Fr = g (3.1) or in components: so g = G ii + 3x = ~Dt + f v + Fi = Gi (3-2) 9“ = (3.3) - fu + f 2 = g 2 G2j is an estimate of the horizontal gradient ofgeopotential on a pressuresurface. As mentioned in Chapter 2, a divergence equation can be - 21 - R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -22- written (Eq. (2.4)), which is a Poisson equation that, for a given field of v, can be solved for <(>. However, as Gal-Chen (1986a) points out, the solution <(>obtained in this fashion may not satisfy (3.1) (or (3.2),(3.3)) unless the wind data also satisfy the compatibility condition(s) k • V x or, in component form: 9 G i / 9 y = g = 0 (3 .4 ) It can be demonstrated, however, 9 g 2 / 3 x . that the divergence equation gives a solution $ that satisfies the horizontal momentum equation in a least-square sense. 3.2 Dynamic Retrieval as a Variational Problem Suppose the compatibility condition is not satisfied. We may then seek a reasonable estimate of <|> so that the sum of the squares of the differences between grad § and problem: g is as small as possible. This leads to the variational Minimize (3.5) Performing the indicated minimization (see Appendix A) gives the EulerLagrange equation: V • (V<(» — G) = or 0 V 2<J) = V • g (3 .6 ) or, in component form: (3 .7 ) which is equivalent to the divergence equation in full form. The solution of the variational problem also provides the boundary conditions for the problem; they are obtained by setting 8 <(> (V<)> - g) • n = o at each point along the boundary s. Thus the boundary conditions take on two forms: 1) 8 <|> = o, which implies there is no variation of § on the boundary. This corresponds to the fixed or Dirichlet boundary condition, which has values Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -23- of <|> on the boundary specified. Use of this boundary condition requires obtaining <)> on the boundary from another source such as through a retrieval from ground-based or satellite measured radiances or from another analysis or numerical model output. 2) ( V<j> - g) boundary in terms n = o, which specifies the gradient of § on the • of g . These are the natural or Neumann boundary conditions. F o m = nxi + nyj these conditions are written: 3<|> 95 c 3<j> nx + ^ ny = Gi nx + G2 ny (3.8) So, in particular, if the domain is rectangular (3.8) can be written dd> 0^ = Gi on boundary x = c o n s t a n t (3.9) dd> ^ = g 2 on boundary y = c o n s t a n t ^ Gal-Chen (1986a) points out that this choice of boundary conditions can be calculated from wind measurements. The solution of the Poisson equation with Neumann boundary conditions is unique only to a constant, however, and so either the mean value of <|>in the domain must be specified, or the value of <|) at a specific point must be given. As with the Dirichlet conditions, these required values of <|>may be determined from other estimates or from model data. One advantage of looking at the estimation of <> as a minimization problem is that it is then possible to assign a measure of quality to the individual components G i , g 2 as desired. This would be useful to consider when performing retrievals in the lower atmosphere on pressure surfaces. case the estimate of g, In that which is dependent on, among other things, the “friction” parameterization (accounting for the effect of motions that have horizontal or vertical scales unresolved by the wind sensing network), would likely be of poorer quality in areas of the domain that are near the surface and of better R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -24- quality elsewhere. Also, since the wind components in the Profiler measurements are made separately it is possible for the instrumental error to be different in different directions and one may desire to correct for that in the retrieval (although this should probably be taken care of earlier in the processing, in the quality control or objective analysis phase). In that case the equation to be minimized would become: J(<1>) = jj Pi ( ^ - G i ) 2 + p2 (3 ^ ~ G2 ) 2 ds (3.10) where Pi( x , y , p ) , P2 ( x , y , p ) are “precision moduli” that weight the terms relative to their accuracy. The Euler-Lagrange equation for this minimization would be: ^ PGfc ' Gl) + 1 p2 (fr " = G0 0 <3*11) and the boundary conditions would be the same as in (3.9). 3.3 Approximations to the Divergence Equation It is of interest to investigate various approximate forms of the divergence equation for calculation of geopotential and virtual temperature. One objective may be to eliminate need for measurement of co. Although the wind Profiler can measure to to a fair degree of accuracy, the measurements contain variations of co due to internal gravity waves even in areas with flat terrain, away from orographic influences (Gage, et al., 1989, Van Zandt, eta!., 1989, Nastrom, et al., 1989), although perhaps longer averaging times for vertical wind measurement (over 3 to 6 hours or more) would give to values more consistent with the synoptic scale horizontal motions that can be detected by the station spacing of the Profiler network. We also should consider that estimates of the time rate of change of divergence and some of the nonlinear terms may be adversely affected by errors in wind measurement, and it would be desirable to R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -25- see what approximations would have to be made to avoid using those terms. An additional consideration in the choice of divergence equation approximation to use was pointed out by Kuo, et al. (1987b). They made the point that neglecting terms that are small may not be as important as neglecting terms that are poorly estimated. If a larger term (such as d D / d t ) has an estimate that is in error by a large amount it would have a more deleterious effect on the retrieval than if other smaller terms are neglected. One way to begin to look for terms to eliminate in the divergence equation is through scale analysis. Scale analysis allows one to determine relative magnitudes (and hopefully, relative importance) of terms for motions taking place on various scales. However, in choosing an approximate divergence equation the appropriate formulation of boundary conditions should also be considered. The approximate divergence equation would need to be “reverse engineered” in order to derive an approximate horizontal momentum equation corresponding to (3.1) to determine g so that first, one knows whether the approximate equation actually corresponded to the minimum of some functional (and precisely which functional was being minimized) and second, so that Neumann boundary conditions (dependent on our choice of g ) could be formulated. The point is that all approximate horizontal momentum equations yield approximate divergence equations when minimized, but perhaps not all divergence equations correspond to the minimization of a well defined set of momentum equations. Gal-Chen (1988) brings up the point that the choice of an approximate divergence equation (with corresponding momentum equation approximation) is determined only by the size and importance of terms in the full divergence R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -26- equation. it does not imply anything about the relative importance of any of the terms of the momentum equations themselves {e.g., dropping the terms involving 0 in the divergence equation does not imply anything about the relative importance of vertical advection in the horizontal momentum equations). 3.3.1 Quasi-Geostrophic Approximation The simplest approximation to the divergence equation involves eliminating all but the highest order terms from the momentum equations (3.1), leaving only a balance between height gradient (“pressure gradient force”) and Coriolis terms. The divergence equation for such an approximation is V2(J) = -fk-Vxv = - f £ (3.12) Note also that the gradient of f has also been ignored. This is the quasigeostrophic divergence equation (Haltiner and Williams, 1980) that is applicable to synoptic scale mid-latitude disturbances (length scale l ~ 106 m; Rossby number Ro = u / f l~ 0.1, u = typical horizontal velocity scale; time scale ~ 1 day). We may expect geopotential heights calculated using this divergence equation to perform worst in areas of strongly curved flow where parcel accelerations are largest and there is significant deviation from geostrophy. Saha and Suryanarayana (1971) calculated geopotential fields in the tropics from several balance type equations and found that a quasi-geostrophic relation like this one performed worse than the other approximations studied (all of which retained more terms of the divergence equation). The error in calculation of geopotential using that equation was largest near the Equator; the retrieval of <J>was improved when 9 f / 5 y was included. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -27- Calculation of virtual temperature using the vertical derivative of the quasi-geostrophic divergence equation may be more successful than the retrievals of geopotential. The approximate divergence equation then becomes V 2t = - —R oV 2l^n- p^ — R = - k •v x 3 lVn p ’ R 3 l n p— Viv and (3 13) can be obtained by taking the divergence of the thermal wind relation (using the measured wind in place of the geostrophic wind). The error in this approximation would, of course, depend on the vertical shear of the rotational part of the ageostrophic wind v a (where v a = v - v g ) since from the thermal wind relationship V2t - — k • V x ^ dVq— _ 1 k . v x dv— _ L k y x dva— . . v Tv - R K v x d I n p - R * v x a I n p r 3 I n p t05' 1 4 ' Neiman and Shapiro (1989) have used the thermal wind relation with some success in a single station analysis for diagnosis of horizontal gradients of temperature and of temperature advection using Profiler winds. 3.3.2 Balance Equation A pproxim ation Another approximation to the divergence equation is the balance equation proposed by Charney (1962) that is derived by retaining all terms of the divergence equation of order of Ro or greater. The rotational part of the wind is used to calculate the terms and the equation takes the form V - (Vv - W v ) + V2<|> - f£ - kxVf .v v = 0 (3.15) Gal-Chen (1988) (who drops V f for simplicity) retains the frictional forces and rewrites the balance equation as 2 j(u ,v ) ♦ £C + * | f ) = 72$ (3.16) and notes that in this form it may be calculated with the actual winds instead of first having to derive the rotational component of the wind. Equation (3.15) has been used in initialization of numerical models, R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -28- especially in the tropics, or in mesoscale models where the wind field influence is more important than the mass field. Kuo and Anthes (1985), using an observing system simulation experiment (OSSE), compare heights and temperatures calculated using the balance equation to those calculated using the full divergence equation and found that the rms retrieval errors were about doubled in the balance equation retrieval over the retrieval from the full equation. Kuo, etal. (1987b), in another OSSE, also found larger rms error for all cases tested for using the balance equation (versus retrieval from the full divergence equation) with observations measured on a 40 km grid spacing, but rms errors were reduced using the balance equation versus using the full divergence equation for a 300 km spacing of observations. 3.3.3 Steady-state Divergence Equation Approximation The above equations (3.15), (3.16) are valid forms of the balance equation but do not correspond directly to minimizations of an equation approximating (3.1) (unless, of course, the approximation specified that only the rotational wind component should be used). One of the forms of the balance equation used by Saha and Suryanarayana (1971) also included advection of divergence and would be equivalent to calculating (3.15) using observed winds. This equation V 2(j) = tK+ k • Vti x v - V 2 (3.17) (where t| = f + (, is the absolute vorticity) corresponds to a minimization of the approximate form of (3.1) including only the height gradient, Coriolis force, and horizontal advection. When the equation is written in the form (3.17), it may be called the vorticity form of the steady state divergence equation. This equation was the most complete approximation to the divergence equation tested by R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -29- Saha and Suryanarayana and performed the best of the approximations used. Kuo, et al. (1987b) also tested this equation and found that, like the balance equation, the use of this approximation resulted in higher rms error for retrieval with a station spacing of 40 km, but improved retrievals over the full equation when simulated station data was used at a larger (300 km) spacing. 3.3.4 Two-scale Divergence Equation Approximation A scale analysis supporting elimination of only those terms involving <o in the divergence equation was outlined by Gal-Chen (1988). He postulated a flow with two length scales (across and along a front) and two time scales. He showed that when first and second order terms are retained that the local change term remains a part of the equation but the terms involving co drop out, although local time change terms remain. His form of the divergence equation resulting from that scale analysis is: where <j> = <|> + 1/2 (u 2 + v2), a modified geopotential. This divergence equation can be derived by minimizing the horizontal momentum equation (3.1) with terms involving co omitted. 3.3.5 Divergence Equation with Vertical Motion Terms When the terms involving co in the divergence equation are included, the equation may be written: 3d 3d a t + uax + 3d OU 3y + D + < % + - 2 J (u ,v ) - t C, + = -V 2<|) (3.19) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -30- From consideration of the terms that involve co, we see that they could be substantial for cases where the vertical wind shear and horizontal gradient of co is large and/or where co and the horizontal gradient of the vertical wind shear is large. Kuo, et al. (1987b) found that omitting the co terms had little effect on the rms error of the retrieved thermal fields in the synoptic-scale OSSEs evaluated. However, Fankhauser (1974) found that in the mesoscale analysis of a squall line that the term Vco- Ov/9p) was the major contributor to V 2t|>for a grid point just ahead of the squall line near the level of non-divergence (where the horizontal gradient of vertical velocity was the largest). His analyses of heights derived from wind using the full divergence equation including vertical motion were “in qualitative agreement with recognized thunderstorm airflows and pressure distributions’’. Naturally, flow conditions analyzed on small scales in the vicinity of a squall line can be considered to be a special case. Fankhauser found that the winds away from the squall line were essentially in geostrophic balance. Thus, the importance of the vertical motion terms in the divergence equation retrieval is quite situation dependent. 3.3.6 “Friction” and Parameterization of Small-scale Motions One more thing to consider is the effect of small-scale motions that are unresolved by the observing network. The equations of motion are formulated in a continuous form, which means that they include all motions from the smallest scales (near molecular scale) to the largest (planetary scale). The wind observing network will have a station spacing such that only motions with horizontal scales of several hundred kilometers and vertical scales greater than a kilometer or so will be well resolved. The instantaneous time change called for in the equations of motion will also be replaced with a measure of change R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -31- over one or several hours. The unresolved components of motion (as well as the unresolved height and temperature deviations) combine to form a residual error term in the divergence equation. The nonlinear interaction of motions on the resolvable scales with smaller scale motions leads to “frictional effects” as energy from the larger scales is converted to smaller scales and then dissipated. It is necessary to include frictional parameterization (or perhaps a special boundary layer formulation) in numerical weather prediction models to provide a way to dissipate the smallest scale waves generated at each time step. Removal of the smallest scale waves is not as critical for a diagnostic calculation such as a divergence equation retrieval of geopotential. Also, the residual term as mentioned above contains the effect of errors in wind measurement and error due to the geometry of the observing network {i.e., the error in representation of the wind field due to the locations of the stations relative to observed atmospheric features); these will be quite different for the proposed Profiler network as compared to the radiosonde network used for this study. Thus, investigation of optimal methods for dealing with the effects of small scale motions and parameterization of boundary layer effects should be undertaken when the actual Profiler network data is available so that instrument and network dependent effects (that could play a large role in determination of this term) can be taken into account. 3.4 Dynamic Retrieval in Siama Coordinates Recalling the horizontal momentum equation in sigma coordinates §? = H r + v-v«v + ^ = -Va* _ t r Vp- ■ fkxv + P' <3-20) we may decompose the virtual temperature into two parts: t v = t v + t v ‘ , R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -32- where t v is the mean virtual temperature along a sigma level and t v ' a deviation from that mean. Writing r jt * VCT<j> + V (RTV In ps ) = ~ ^7 p m - fkxv + Fr I Vps = G - (3.21) a minimization problem can be formed that is similar to (3.5), except that the objective is to find + r t v i n p s) instead of <|>. One reason that this format is preferred is that the variance of Vc<[> may be fairly large for areas that include significant changes in terrain, and adding r t v i n ps is one way of removing some of the effect of the sloped terrain from VCT<j) and hence reducing truncation error in the Poisson solver. When (3.21) is minimized, following a similar procedure to the one given in Appendix A, the Euler-Lagrange equation is «|> + RTV In ps ) = VCT• G = VCT• ^- — + f k x v + F r Once g RTy Ps L V Ps) (3.22) is determined, (3.22) can be solved for (<j> + r t v i n p s ) with appropriate (Dirichlet or Neumann) boundary conditions. One problem is that an estimate of t v • is needed to calculate g. In sigma coordinates the hydrostatic equation can be written so to solve for virtual temperature we write I R a Va (<j) + d In a rtv In p3) - — + fkxv + Fr R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -33- in solving for virtual temperature, we again have to deal with a factor with t v ' (or actually 3 t v ■ / 3 In cr). We can assign a value to t v ■ to use in calculation of t v and then use this calculated t v to create another estimate of t v • and thus improve the value of t v by iteration. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4 RADIANCE RETRIEVAL THEORY The calculation of radiance that would be received by a ground-based or satellite radiometer resulting from a given atmospheric profile (the “direct” problem) can be accomplished through the use of the radiative transfer equation, Eq. (2.15), provided that the atmospheric transmittance t v can be well estimated. Solution of the inverse problem (determination of the temperature profile that produces a given set of radiance measurements) is not so simply accomplished. One reason for this is because the radiance retrieval problem is ill-posed; it has no mathematically unique solution. It is possible to specify two different vertical temperature profiles that generate the same radiances in each of the frequency channels of a satellite- or ground-based radiometer. In that case the difference between the two profiles would not produce any radiance if integrated with the kernel functions. Considering the problem as a transformation from physical space (temperature profiles) to measurement space (radiance measurements) we see the difference between the two profiles lies in the nullspace of the transformation. Thus, one way of looking at the problem would be to say that the solution of the radiance retrieval problem is not performed with radiances alone but also depends on specifying additional information so that the nullspace component of the temperature profile can be determined. -34- R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -35- However, the problem is more complicated than this, because the factor of instrument error must also be considered. The uncertainty in the measurement due to instrument noise corresponds to an uncertainty in the atmospheric profile that would produce such a measurement. So, our retrieval scheme does not only need to provide information to differentiate between profiles that produce exactly the same radiances; it must be able to further differentiate between profiles that produce radiances ihat aiS. ltl£ same within the measurement error of the instrument. Thompson, et al. (1986) considered the ill-posed nature of the satellite retrieval problem with respect to the limits of retrievabifity of soundings. Using simulated satellite kernels they demonstrate how temperature profiles may be constructed that are significantly different in a mathematical and in a meteorological sense (with temperature differences as much as 10 K at some levels) and that still produce the same radiance values within measurement error. Their further examination of a fairly heterogeneous set of 1600 soundings found 83 pairs of soundings from over a million pairs compared that had radiance differences within noise levels of about 1% of band-averaged radiances, but when smaller "best case” radiance noise levels were used only 8 “dissimilar pairs” of observed soundings were found. These 83 dissimilar pairs had a 10-1000 mb rms deviation of 1.99 K, with the most similar pair having a difference of 0.77 K and the most dissimilar pair having a difference of 4.82 K. They concluded that the modern satellite sounders were capable of distinguishing between naturally occurring thermal fields when considering the forward problem; they also comment that this may not be the case for many retrieval methods for inverse problems that can be dependent on a priori statistics or smoothness constraints. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -36- The results of a similar comparison study (carried out during the course of this research) using weighting functions from the four ground-based radiometer channels are presented in Appendix B. The ground-based radiometer channels did not perform as well as the channels used by Thompson, et al.; from a set of 1470 soundings (5 years of Oklahoma City spring soundings) there were 111 “dissimilar pairs” that had brightness temperatures that differed by less than a “best case” 0.2K. The average rms difference in these soundings for levels from 25 mb to the surface was 2.79 K, and the largest differences between soundings were in the mid- and upper troposphere. The performance was improved when “measurements" from two satellite channels were also considered. In view of the lack of certainty in distinguishing between profiles when considering just the forward problem, it is clear that additional information will have to be supplied in addition to the radiance measurements so that a unique and meteorologically plausible sounding can be obtained by inversion from a limited set of ground-based and satellite measurements. If a number of simultaneous rawinsonde/radiance measurements have been made, these may be used to derive regression coefficients relating radiances to the observed profiles. One may assume atmospheric profiles take on a specified form, such as a linear combination of polynomials or of predetermined empirical orthogonal functions that are typical of profiles in a certain area or season. The constraint may be a “physical” constraint where the radiative transfer equation (2.15) is used and the radiances produced by the retrieved profile are made to match (sometimes only within a specified degree) the observed radiances. Another “physical" constraint would be to eliminate superadiabatic layers in the retrieved sounding as these are less likely to occur in the atmosphere. This R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -37- constraint can be taken a step further by placing a smoothness constraint on the profile so that the deviation of the profile from a mean or “first guess” value or the first or second derivatives in the vertical are minimized. Another “constraint” comes through discretization, representing the continuous temperature profile as values at a finite number of points, though if the spacing between points is sufficiently small this would have little effect on the solution. In the following sections some methods that have been proposed or used in satellite or ground-based temperature retrieval are reviewed, and their utility with respect to retrieving temperatures from ground-based radiances is considered. To simplify the derivations, the boundary terms t BACk e"x(v) are assumed to have been removed from the brightness temperatures. We are also mainly considering the problem from the point of view of radiance retrieval from microwave frequencies. For radiation at infrared (IR) frequencies, the relationship between the blackbody radiation b v and temperature t is non linear, which complicates the retrieval problem. 4.1 Retrievals Using Scalar Radiances The first type of retrievals to be discussed are those that are formulated to use scalar radiance measurements from an individual point. The radiances are used to produce a 1-D vertical profile. 4.1.1 Statistical Regression One method of deriving temperature profiles from radiance measurements is by statistical regression. If a number of simultaneous, collocated measurements of radiances and temperature profiles are available, a set of coefficients relating the radiances to the temperature profile can be obtained. This is done in the standard way by assuming that the temperature at R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -38- a given level can be represented by a linear combination of the radiance measurements and then determining the set of coefficients that minimizes the square of the error of the estimates so obtained. Given M sets of simultaneous measurements of L temperature levels and N radiances the model equation can be written as T where x t = C R (4 .1 ) is the L x M matrix of temperature deviations from the mean, r is the N M matrix of radiance deviations from the mean, and c is the L x N matrix of coefficients to be determined. The coefficients for each level k, c k, are those which minimize the sum of the square of the errors in the retrieved temperatures for level k: ekekT = (T k- c kR ) ( T k- c kR ) T. C = where (t r t ) (T RT) Thus, the coefficient matrix c is: (R RT) - 1 is the matrix of covariances between (4 .2 ) t and r and (r r t ) is the radiance covariance matrix. This formulation, however, may tend to fit the coefficients too closely to the data (yielding large values of coefficients with opposing signs) and so the retrievals become sensitive to small fluctuations in the radiance measurements. This can occur especially when the some of the channels overlap their atmospheric coverage by a large amount. In that case, two or more of the rows of r would be very similar, causing r r t to be ill-conditioned or near singular. If the matrix r in the derivation above is replaced by ( r + e) where e represents the noise in the radiance measurements, and if we can assume that these errors are random and not correlated with the temperature profiles or with the radiances then the expression for c becomes C = where eeT (T RT) (r r t + eeT) - 1 (4 .3 ) is the error covariance matrix. If the measurement errors in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -39different channels are not correlated with one another then eeT is just the diagonal matrix of expected measurement error variances. Using a “noise” term in this way stabilizes or “conditions” the solution, producing coefficients with smaller magnitudes. Lee, et al. (1983) note that this method is similar to the ridge regression method discussed by Marquardt and Snee (1975). This latter study showed that ridge regression produces coefficients that perform better than simple least squares coefficients when the predictor variables (the radiances, in this case) are highly correlated with one another. The statistical regression method has an advantage in that it is not necessary to know the weighting function in order to obtain estimates of the temperature profile. The major disadvantage with this method is the necessity of a number of simultaneous temperature/radiance measurements in order to be able to generate the coefficients. The set of measurements from which the coefficients are calculated must also be selected with some care to insure that their characteristics are consistent with the characteristics of the profile that is to be retrieved. There is an additional disadvantage associated with the use of this method for ground-based radiances over that of satellite measurements. For satellite measurements it may be sufficient to use a sounding data set that was observed within the same latitude band (with possibly a similar climate). However, the ground-based measurements are also highly dependent on the surface elevation, so the coefficients calculated from observations at one station cannot be easily transferred to apply to another area unless the terrain heights are also the same. The ground based weighting functions (e.g. see Fig. 6.7) decrease exponentially with height above ground level. For a given frequency, R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -40- the transmission of radiation from a given pressure level to the ground and hence the weighting function at a given level z* (Eqs. (2.19)-(2.20)) increases with increasing surface elevation. This changes the effect that a change in temperature at that level would have on the brightness temperature at the two stations; so the coefficients for these stations should not be the same. 4.1.2 “Exact” Physical Methods Another group of solution methods includes physical methods that use (2.15) along with appropriate weighting functions wv ( z ) , to obtain temperature profiles that match the observed radiances exactly. This discussion is based on a similar one presented by Rodgers (1976). Note that, of course, the use of the z (height) vertical coordinate is only for convenience; any standard vertical coordinate could be used if w is redefined appropriately. Some of these methods involve a matrix inversion while others are iterative methods. 4.1.2.1 Basis function representation of tem perature. One fairly straightforward way to accomplish this would be to express the temperature profile as a combination of some basis functions tp i (which may be polynomials, trigonometric functions, or empirical orthogonal functions). Supposing there are N observations of radiance; it is simple to solve for temperature as a function of N basis functions. If we take t (z ) = £ c i <Pi. we can express this as: i Tbv = I J Wv( z ) T ( z ) dz = Zg N N Let f wv ( z ) cpi (z ) dz= Av i , so we have a system of N equations to solve for Jzs N unknown coefficients c i. Thus c i = (Avi) -1 t bv and we can solve for the R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. coefficients c i and thus for the temperatures. One problem with this method, however, is that AVi may be an illconditioned matrix, so the solutions c i may be sensitive to errors in radiance measurement or errors in the discretization. Wark and Fleming (1966) show this method can give a reasonable profile when the radiance measurements are exact but when measurement error was added the retrieved profile became useless. Twomey (1977) explains this by noting that the matrix avi has some small eigenvalues (resulting from interdependence or overlapping of the weighting functions wv ( z ) ) and so the inverse ( A y i ) 1 will have some large values making the solution sensitive to errors in t bv. This is an inherent (and nearly unescapable) part of the problem of direct inversion that comes about because of the form of the weighting functions and is not alleviated by double precision arithmetic or higher-order quadrature approximations. Rodgers (1976) defines a “contribution function” d v ( z ) that can be used to demonstrate the sensitivity of the solution to instrument noise. dv (z ) is defined by T (Z ) = £ d v ( z ) T bv . (4 .5 ) V Thus, in this case Dv ( z ) = (AVi) '1 <Pi ( z ) = How does dv(z J wv ( z 1 ) tpi ( z ■) <pi ( z ) d z\ ) show the sensitivity of the solution to noise? The solution is sensitive to noise at a level if d v (z ) has large values that are nearly equal and opposite in sign for different channels. We know that since the radiance weighting functions overlap that the measurements in different channels would be correlated, while the instrument noise would be more uncorrelated or random. So, for equal and opposite values of measurement ( t bv + eBV) the sum of d v ( z ) t bv would dv (z ) , given a tend to cancel out while the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -42- sum of dv ( z ) eBV would not. Figure 4.1 shows the contribution function that results from solving for t ( z ) by (4.4) using polynomial basis functions and the four ground-based channels, while Figure 4.2 includes two microwave satellite channels. Below 700 mb, a 0.2 K instrument error may only be made 5 times larger, to 1.0 K. However, above that level the errors may be multiplied 50 times or more in themid- and upper troposphere (although mid-tropospheric retrievals are somewhat improved by addition of the satellite data). Thus, this method is obviously unacceptable for temperature retrieval. Twomey (1977) also notes that a least-squares solution obtained by solving for fewer than N coefficients c i will still be sensitive to instrument noise. 4.1.2.2 Weighting function representation of temperature. The next question to consider is whether there is a choice of basis functions cpi that are least sensitive to instrumental noise. We want to find a solution that minimizes the contribution function dv ( z ) subject to the constraint that the radiances produced by the profile match the observed radiance. Rodgers (1976) has commented on this and gives a condition to insure the retrieved radiances match the observed radiances: (4.6) where is the Kronecker delta function. If we can assume that the errors in the observations are independent, though perhaps not equal for each channel, the contribution function d v (z ) in the variational problem can be multiplied by a weight pv» which should be directly proportional to the square of the expected value of the error in each channel (so that in channels where the error variance is small the contribution function could be allowed to be bigger than in other R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -43- 200 E 400 2 600 -O— 152.85 GHz r a .- i5 3 .8 5 .G tik 800 -* - i 55.45 GHk Polynomial . 58.80 GHfc 1000 -60 -40 Bajsis Fct. -20 0 20 - 60 40 Contribution Function Figure 4.1. Contribution function calculated using polynomial basis functions for the four ground-based microwave channels. o 200 g 400 5 2 .8 5 G H z S 600 - a — 153.85 GHfc - * — | 5 5 .4 5 G H k .r*-45a.8Q.GJHk. 800 •+ - a 153.74 GHk Polynomial — 154.96 G H z B asis Fct. 1000 -60 -40 -20 0 20 40 60 Contribution Function Figure 4.2. Contribution function calculated using polynomial basis functions for both ground-based and satellite microwave channels. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -44- channels). The resulting variational problem would be to minimize the functional: where Xv is a Lagrange multiplier. Performing this minimization (Appendix A) we can readily show the minimizing d v (z ) would satisfy Y which would be satisfied by choosing basis functions <pi that were the weighting functions or some non-singular linear combination of them. In that case (4.9) Note that pv is not a factor in the minimizing contribution function. This is because of the requirement (4.6) insuring retrieved radiances match observed radiances. Figures 4.3 and 4.4 show the plots of these contribution functions using the ground-based and ground-based plus satellite channels, respectively. It is readily apparent that the values of the contribution function are decreased over those using simple polynomials (Figures 4.1 and 4.2) for levels above 700 mb. Below 700 mb the contribution functions have increased considerably; this comes about because the minimization of depth of the atmosphere. dv(z ) was taken over the entire Once again, the sensitivity is caused by the ill- conditioning of the matrix J wY( z ) wv ( z ) dz that results from the interdependence of the radiance kernels. Another way of writing (4.9) would be to express it as the solution of some deviation from an atmospheric mean or “first guess” profile (designated as R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -45- o -e — 52.85 G H z -a - 53.1 .85' G H z 200 A* 5 5 .4 5 -W « -* • - 58.80 G H z 400 S 600 CL. 8 00 Weighting j F c t Basis . 1000 -50 0 -25 25 50 Contribution Function Figure 4.3. Contribution function calculated using weighting function basis functions for the four ground-based microwave channels. o □ - 200 53.85 G H z -"^•-••55;45GKz... 58.50GH2^ E 400 54.96 G H z £ 600 800 Weighting . £ e t. Basis - 1000 -50 -25 0 25 50 Contribution Function Figure 4.4. Contribution function calculated using weighting function basis functions for both ground-based and satellite microwave channels. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -46- Tf). The equation would then be written T ( z ) - Tf ( z) X X = ( X wy( z) = S X v Wv ( z) dzj Wy(z) | tbv-J z Wv ( z ) T f ( z) ("JJ0WY(z) Wv ( z ) y V 3 dz') 1 Wv ( z) dz j [ T Bv-TBvf ] • (4.10) / where TBvf = f wv ( z ) Tf ( z ) dz is the brightness temperature that would be z3 produced by the first-guess temperature profile. Writing the equation in this fashion uses T f to prescribe the nullspace component of t ( z ) . This solution would have the same sensitivity and the same problems as in (4.9). The solution (4.10) has been suggested for satellite data assimilation in numerical models by Gal-Chen, et al. (1986, their Appendix A). In practice it has not been used with actual radiance weighting functions but rather with derived mean layer temperatures (Gal-Chen, et al., 1986) or thicknesses (Aune, etal., 1987) from mutually exclusive atmospheric layers so that the “weights" in those cases (step functions) do not overlap; thus, they would not experience problems with the error sensitivity that the use of actual radiance kernels and radiances would have produced. 4.1.2.3 Smith's method. Some iterative methods based on adjusting radiances from a first-guess temperature to match observed radiances have also been proposed. These methods avoid the problems associated with inverting an ill-conditioned matrix but would still have problems with noise sensitivity if the iterative process were not stopped soon enough. One iterative method, Smith’s method (Smith 1970, 1983) involves adjusting the temperature at each level by an amount proportional to the sum of the deviations of each step’s calculated radiances from the measurements. For R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -47infrared wavelengths, the blackbody radiation b v(t ) is a non-linear function of temperature and so the derivation given by Smith also involves a linearization of that relation that is not necessary for microwave observations. Thus, a variation of Smith’s method suitable for ground-based calculations may be derived as follows: begin with Eq. (2.14) Tbv = f w v ( 2 ) T ( z ) dz (4.11) JzS where the brightness temperature may be adjusted to remove the contribution of the background microwave radiation t back. This equation can be written in iterative form as TBV- ( Tj+1 ( z) - Tj ( z ) ) dz T3BV= f “ wv ( z) (4.12) Jzs where j denotes the step of the iteration. If it is assumed that thecorrection determined from the brightness temperature with frequency v that is to be applied to the temperature At 3+1 = [Tj+1 - Tj ] is independent of the height z, then N new estimates of temperature at each level can be determined from the N radiance observations: Tbv- TgV= J~Wv (z) dz ATJ+1 (4.13) [T bv - T3V] T3+1( z ) = Tj ( z) + At J+1 = Tj ( z) + — ---------------- (4.14) f Wv (z ) dz •Zo in Smith’s 1970 derivation for satellite weights, the boundary term representing transmittance from the surface was included in the iterative .1 correction, so that f wv ( z ) JZo Tv ( zs, oo) = i. dz + t v ( z s ,° o ) = __ 9Xv(Zs, Z) dz + 3z T v ( Z s ,co) in Smith (1983) it is implicitly assumed that the atmosphere is R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -48opaque so that the transmittance to to the surface t v ( z s / <*>) is zero, and the integral of wv ( z ) over the atmosphere equals one; if that is not the case the observations must be adjusted so that this “opaque" condition is met. For the purpose of ground-based radiance retrieval one would not want to adjust the temperature contribution from space in the iterative procedure, so a small adjustment in the most transparent channels would be necessary. It would also be useful to remove the contribution of the surface temperature to the radiance measurement; this would necessitate a larger adjustment to the weights and to the measured brightness temperatures. For the purposes of the retrieval, we may then define a new normalized weighting function Wv* ( z ) Wv ( z ) = — -------------- , f Wv ( z ) JzS f» [ Wv* ( z ) Zs dz dz = 1 (4.15) and the brightness temperature differences are also to be adjusted as shown in (4.14) so that Tj+1(z) = Tj (z) + At J+1 = Tj (z) + [Tbv* - T^*] (4.16) The final estimate of temperature at a level z for theiteration is calculated as a weighted sum of the temperatureestimatest ^,+ (z ) derived from the N channels. Following Smith (1970), the appropriate weight for the temperature estimate resulting from each radiance measurement is the weight wv* ( z ) at that level and the resulting sum should be divided by the sum of the weights in each of the channels at that level, thus: X = —---= Tj+1( z ) - Tj +1( z ) Wv* ( z ) £ w v* ( z ) (4.17) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. so that at each iteration, the sum of the retrieved profile’s brightness temperatures is equal to the sum of the observed brightness temperatures (though some values will be too high and some too low). The first iteration adjusts the first-guess temperature profile to produce brightness temperatures with the proper sum, subsequent iterations redistribute the brightness temperature error among the different channels. The iterations are continued until the convergence criterion is satisfied. The Smith (1970) convergence criterion is that the change in brightness temperature between iterations for each channel should be no larger than .01% of the observed brightness temperature. The Smith (1983) iterative procedure differs from the earlier one in that it also takes account of measurements with differing error statistics by dividing the temperature estimate resulting from each radiance measurement by a factor proportional to the expected error of the measurement. This causes the less reliable measurements to be weighted by a smaller amount. Thus, for the Smith (1983) method w v * (z) from Eq. (4.15) is redefined as R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. where ev is the expected error of the brightness temperature measurement. This also means that at each step the sum of brightness temperatures weighted by i / E v is set equal to the weighted sum of observed brightness temperatures. The convergence criteria for Smith (1983) is X £t b v * - T b v 1* v=l }2 * ^ £ £v2 (4.20) v=l where the factor V 10 was found empirically from experience to insure that the temperature profiles did not change significantly from one iteration to another. The Smith (1983) procedure also added another step after the iterative retrieval where a direct least squares solution is made to attempt to determine perturbations in the smoothed profile based on the differences of the measured radiances between channels. This additional step does not seem feasible to implement for the ground based measurements since the levels it used were chosen based on the vertical distribution of the weighting functions for the VAS sounder; the ground-based weighting functions are much less independent than the VAS channels so the differences between the measurements would be more sensitive to measurement error. The adjustments to the “first guess” temperature profiles would, at first glance, appear to be similar to those described earlier in Section 4.1.2.2, but they are not the same. Although the adjustments to the temperature profile at each level are multiplied by the radiance weighting functions, there is also a division by Swv* ( z ) that changes for different levels; thus, the profile of the difference between the retrieved temperature and the first guess temperature profiles is not a linear combination of the weighting functions. From (4.16), Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -51- (4.17) we can see that Wv* ( z ) t J +1( z ) = t U z) ATJ+1 + v v ------2 ,Wy* ( z ) (4.21) y so the final retrieved profile will be equal to the initial profile plus some linear combination of wv* (z ) / (Ewy* (z ) ) . This function plays the role for the Smith physical retrieval of the “contribution function” mentioned earlier, but because this method is an iterative procedure these curves represent the effect of measurement error for only a single iteration and so the total effect would be more complicated. Values of this “contribution function” that correspond to the ground-based channels only and the ground-based plus satellite channels are plotted in Figures 4.5 and 4.6, respectively. Notice in Figure 4.5 that the influence of the 52.85 GHz channel (circles) is greatest above 700 mb and is large and nearly constant above 300 mb. Even though the 52.85 GHz channel has a small weighting function at those levels, the other three ground-based channels have much smaller weighting functions at those high levels so when the weighting functions are divided by the sum of the weights on a level the values are amplified. This means that a large correction could be applied at levels that have little effect on the ground-based radiance measurements, so one must be cautious in applying the Smith method retrieval in such a case. (The large values of contribution function in Figure 4.6 are not as much of a problem as those discussed in Figure 4.5 since the 54.96 GHz satellite channel has most of its weight at the levels where its contribution function is large.) Some advantages of the Smith method over the “exact" methods mentioned thus far are that the calculations are simpler (no matrix inversion is R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -52- o 200 400 600 ■& — 800 5 2 .8 5 G H z a -5 9 .8 5 GHz ♦ -5 5 .4 5 GHz * - - 5 8 .8 0 G H z 1000 0 0.2 0.4 0.6 0.8 ’ Contribution function* Figure 4.5. “Contribution function” for Smith method retrieval using the four ground-based microwave channels. 200 p 400 2 600 a --5 3 ;8 5 G H z -~ 800 -♦ -5 f4 5 G H z ! - • - 5 8 .8 0 G H z - — Or - 5 4 . 9 6 G H z - 1000 0 0.2 0 .4 0.6 0.8 "Contribution function* Figure 4.6. "Contribution function” for Smith method retrieval using both ground-based and satellite microwave channels. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -53- necessary) and that the iteration process can be stopped before the retrieved radiances have been adjusted to exactly match the observed radiances so that it can accommodate errors in the measurements. Naturally, this retrieval method cannot be applied to levels where Zwv* ( z ) is zero; however, as mentioned above, it would also be better to not apply corrections at levels where Zwv* ( z ) is small. If more information is available about the error of the “first guess” profile it would be advantageous to incorporate that into the inversion method, rather than being limited by the assumption that the errors in the brightness temperatures are due to errors in the guessed profile that are independent of height. This can be done by adding additional constraints to the physical solution. One form of constraint minimizes the deviation of the profile from the first guess profile, or minimizes the magnitude of the first or second derivatives in the vertical. Additional information such as covariance of the error of the first guess between the different sounding levels or covariance of the measurement error can also be included if it is available. Some of these approaches will be discussed in the next section. 4.1.3 4.1.3.1 Constrained Physical Retrievals Smoothness constraints. Twomey (1963,1977) presents a general method whereby retrieved profiles are found that have radiances that are close as possible to the measured radiances, while the profiles satisfy a smoothness constraint. One way in which the instabilities of the “exact” solution method manifest themselves is in creation of unrealistic “wiggles” in the atmospheric profile. (These "wiggles" occur because we are trying to fit to a noise increment in one R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -54- channel without changing the brightness temperatures of the other channels, the positive and negative deviations of the profile cancel out when convolved with the weighting functions of the other channels.) Thus, a smoothness constraint (if not applied so severely as to eliminate legitimate atmospheric profile changes) would help to stabilize the solution. Twomey’s approach was to first specify that the error of the radiances produced by the retrieved profile should be less than or equal to the measurement error of the instruments. For convenience the temperature profile and other terms in the radiative transfer equation are represented in discretized form as vectors, thus temperature profile at L levels in the vertical, and t b t is the are the brightness temperatures for the N measured channels, and w is the N x L matrix of weighting functions. The integrations in the vertical are replaced by matrix multiplications; e.g., the set of N equations replaced by the matrix equation t b = w t. t bv = f°°wv ( z ) (z ) dz t is (Note that w therefore contains integration weights as well as the radiance function.) Twomey’s specification for the temperature profile t with the desired squared error of theradiances is written: (t b - w t )t (t b - w t ) < 2 a 2 , where a 2 are the expectedinstrument errors in each of the channels. Twomey then attempted to find the profile that best satisfied a smoothness constraint problem; i.e. h = i q (T ) = tt h t . The h matrix is chosen to suit the would serve to minimize the variance of formulated to calculate the first or second derivative of t t , or could be h in the vertical. The functional to be minimized is J = and the minimizing (t b t - w t )t (t b - w T) + y (t t h t ) (4.22) is (by inspection) t = (wr w + yH ) - 1 wr t b (4.23) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -55Twomey recommends that the multiplier y should be determined by trial calculations to find a value that gives adequate smoothing and still keeps the error (t b - w t )t (t b - w t ) at a reasonable level. This method would be fine for a situation where little was known about the desired temperature profile characteristics beyond smoothness and the expected error of the measurements; however, if there is other information available about the expected statistical characteristics of the temperature profile, it should also be incorporated. This will be discussed in the next section. Tikhonov (1963) independently developed a similar method to this, thus the method is known as the Twomey-Tikhonov method (Rodgers, 1976). If t and t b are taken to be deviations from some mean or first guess values (as in Eq. (4.10)) then the choice of h = i would be appropriate as it minimizes the mean square difference of the retrieved profile from the first guess. In that case (4.23) could be written t = (W1 w + 7 1 ) - 1 w1, t b = W1 {vnf + y i)-1 t b (4.24) This form has the advantage that the number of brightness temperature measurements (and hence the dimension of wwT) generally is smaller than the number of quadrature points (and hence the dimension of wT w) so that the matrix to be inverted would be of a smaller dimension. We can compare this method with the “exact” retrieval using weighting functions as a basis. If we write (4.10) in a discretized form T = WT (WWT) - 1 t b (4.9') we can see that it is of the same form as (4.24) except for the addition of the factor yi to the matrix that was to be inverted. This factor acts to stabilize or condition the matrix (W1 w + y i ) -1. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -56- 4.1.3.2 Minimum variance estimate. If there is a priori statistical information about the mean value and covariance of a representative sample of atmospheric profiles to which we can expect the profile we wish to retrieve to belong, that information can be used to determine a minimum variance estimate of the temperature profile (Strand and Westwater, 1968, Westwater and Strand, 1968, Luenberger, 1969). The method is similar to that of the statistical regression in section 4.1.1, except that the knowledge of the physical relationship between the radiances and the temperature profile substitutes for the need to have simultaneous measurements of temperature and radiance. In this case the known quantities are the mean temperature profile, t the , matrix of the covariance of temperature between levels, s, the weighting functions, w, and the error covariance matrix of the brightness temperatures, n . Let T t be a discrete representation of the “true" vertical temperature profile with the mean removed, and let T B=wTt represent the “true” brightness temperature measurements with the contribution from the mean temperature (i.e., removed. We take e [Tt ] to denote the expected value of x in a population, thus, e [x] = t , the mean, and e [ T cT tT] = s, the covariance matrix. The objective is to find a temperature profile value of the squared error of the profile c w t) such that the expected t ( t - t c>T ( t - t c) is minimized. Let t = again the temperature profile is a linear combination of the ( t b+ e ) , so measured brightness temperatures. Thus (assuming there is no correlation between e e and t b or T t), the expected value of the squared error is: = E [(T -T t ) T(T -T t ) ] = = E [ ( C T B+ C e -T t ) T (CTB+C E-Tt ) ] E [T bt CtCTb] - E [T bt CtT c] - E [ T ctCTb] +E [ T t TT c] +E [£TCTC£] = EtTtTwTCTCWTj -E [TtTwTCTTt] - E [ T c'rCWTc] +E [ T CTT C] + E [ e TCTC£] = e i v k # . p i - 2 e i t c * w K t c j + E t T t , T . „ > + e ' <4 - 2 5 > Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -57where we have switched to tensor notation (summing over y,v, and m) for convenience. To find the coefficients c that minimize the error e, we set the derivative of (4.25) with respect to each equal to zero: 0 e /8 c 0 = C(WSWT + N) - SVF; C = SWT (WSWT+ N ) ' 1 (4.26) Thus, the retrieved temperature profile deviation from the mean is: T = SWT (WSWT+ N) -1 (T b+E) or, letting r = t b+ e +w t (4.27) be the measured brightness temperature, the retrieved tem perature profile would be T = T + SWT (WSWT+ N ) ' 1 (R-W T) (4.28) This solution is actually very similar to the solution from statistical regression, except that the requirement for simultaneous measurements of brightness temperature has been replaced by specifying the weighting function. Note that Eq. (4.27) can be derived from (4.1), (4.3) by replacing r by w t . This solution is sometimes referred to by another name, the “maximum likelihood" solution: Rodgers (1970,1976) has shown that solution is equivalent to (4.28) if the error statistics are Gaussian. In addition, this solution (4.28) can also be derived by taking a least-squares minimization of a combination of first guess and radiance measurements weighted by the inverse error covariance matrices, so that t minimizes J = ( T - T ) t S' 1 ( T - T ) + (WT—R ) TN ” 1 (WT—R ) (4.29) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -58- 4.1.3.3 “ Minimum information” method. The coefficients for the minimum variance method just described can be calculated when temperature covariance matrices and error covariance matrices are available. This may not always be the case; for example we may want to use some other “first guess” temperature profile such as a forecast field or another temperature analysis, rather than calculating the profile as deviations from the mean profile t . A full temperature covariance matrix may not be available, so we would have to make do with only some estimate of the expected variance of the forecast profile from the true temperature. When the information available consists only of the various error variances of the first guess temperature profile at each vertical level, then the full covariance matrix s may be substituted by a diagonal matrix consisting of only the different error variances, with off-diagonal elements equal to zero (in effect assuming no correlation of errors between levels or between measurement errors in different channels). This approximation could be limited even further. Suppose that our information about the validity of the first-guess temperature is limited to one error value a T2 representing the error for the entire first guess profile. If we substitute s = ctt 2i (where a T2 is the error variance of first-guess temperature) and n = <re2i (where ae2 is the measurement error variance) into (4.28) we get T = T + (Jt 2Wt (WO^WTh- <Te2I ) - _1 (R-WT) Oc2 = T + WT (WWT+ ------ I ctt 2 )"1 _ (R-WT) (4.30) This solution is known as the “minimum information” method (Foster, 1961, Smith, et a i, 1972, Fritz, 1977). Note, however, that if we take y = ae2 /o T2 then (4.30) is essentially the same as (4.24). This shows a statistical basis for the choice of y in the Twomey-Tikhanov method. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -59- 4.1.4 Other Retrieval Methods We have examined several simple methods for retrieving temperature profiles from radiance data. This listing of methods is not intended to be comprehensive. Rather, these methods represent the basic building blocks, from which other, more sophisticated retrieval methods are formed. For example, the statistical regression method discussed in Section 4.1.1 can be improved, if we can insure that the soundings in the data set used to form the regression coefficients have properties similar to those of the profile that we are trying to obtain. This can be accomplished by stratifying the soundings used to form coefficients, subdividing them into sets of soundings that share similar characteristics. This is the strategy used by Westwater, et at. (1983, 1985); they stratify the soundings based on tropopause height, which can be measured independently from VHF radar measurements (Gage and Green, 1979). For satellite retrievals, various investigators {e.g., Uddstrom and Wark, 1985, Thompson, et al., 1985, Chedin, et at., 1985) have investigated using the radiance measurements themselves to help select the best possible first-guess profile and to aid in the radiance retrieval. Additions and improvements to other methods discussed in this chapter have also been implemented for satellite radiance retrieval. Methods for retrieval from infrared radiances are complicated by the non-linear relationship between blackbody radiation and temperature (Eq. (2.6)), and the influence of water vapor and cloud-contamination of the radiances. An example is Eyre’s (1989) method, which uses minimum variance/maximum likelihood estimation with non-linear Newtonian iteration to obtain profiles from cloud-contaminated radiances with a first-guess derived from a numerical weather prediction (NWP) model (or to incorporate radiances into an NWP analysis routine). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -60- 4.2 Retrievals Using Radiance Gradients The discussion of retrieval methods up to this point has concerned applying the radiance or brightness temperature measurements to obtain a one-dimensional profile of the variation of temperature in the vertical. This next section presents possible methods for using measurements of radiance gradients to obtain three-dimensional temperature analyses. We first consider why one would want to deviate from the more “traditional” methods of 1-D radiance retrievals. Measurements of radiance from a network of ground-based radiometers (or from satellites) provide information also about the gradient of radiance that is only accounted for implicitly, through the gradient of the retrieved temperature profiles. However, because of calibration error and uncertainties about the absorption coefficients that determine the weighting functions, there may be substantial biases in the radiance measurements and the absolute measurement of radiances (required for the 1-D retrievals) is less accurate than a relative measure (Menzel, etal., 1981, Westwater, e ta l., 1985). Thus, it is useful to consider methods for incorporation of radiance gradient measurements to retrieve temperature gradients and hence three-dimensional temperature analyses. Another motivation to look for radiance retrieval methods that incorporate gradient methods is the desire to incorporate temperature information derived from wind data (as discussed in section 3.1) together with the radiances to obtain temperature analyses. Since the momentum equations (3.2), (3.3) through use of the hydrostatic equation relate the wind and vertical wind shear to the horizontal temperature gradients it is a natural extension of this to seek methods that incorporate the gradient information directly without having to R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -61- previously solve for the temperature profiles. Chapter 5 contains further discussion of the combined retrievals. Before we discuss the various methods that can be used to obtain temperature fields from radiance gradients, we need to devise some way of determining Jzs„wv (z*) Vht dz* from the radiance gradients V ht bv. We will work in the z* coordinate system since the horizontal derivative of the weighting function is somewhat simpler than in height coordinates. For the purposes of these derivations we are assuming that the boundary terms (t back e'T(v) for ground-based measurements and Tsfc e'T(v) for satellite measurements) have been subtracted from the brightness temperature; these terms would also make a contribution to the radiance gradient (especially for the satellite measurements) but we assume these terms can be specified. We may write (using Leibnitz’ rule): V hT bv=V h [ / z ,Wv ( z *) T ( z * ) d z * j = Jz ,Wv (z*) VhT ( z *) dz* + f • 'Z s (VHWv ( z * ) ) T ( z * ) d z * - Wv ( z s*) T ( z s* ) V Hz s* (4 .3 1 ) and then solve for f ,wv (z*) V ht dz* . The last term on the right hand side, VZS wv ( zs* ) t ( zs* ) VHzs*, can be specified if we know the surface temperature and pressure. For ground based radiance measurements, Gal-Chen (1988) showed that the second term on the right hand side f . (VHwv ( z*) ) t ( z *) d z * , can be zs expressed in terms of the brightness temperature. Recall that wv ( z * ) = 3xv ( zs* , z * ) /9z* (where xv, the transmission, is given by Eq. (2.19)), so V HWv ( z * ) = - 0 f r ( V HXv(zs* , z * ) ) = VH(exp [ JZs, a v ( z * ) d z * ] ) } = “ a f r j e x p [JZgtav ( z*) dz* ] a v ( z s* ) V Hzs* R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -62- = a v ( z s*) V Hz s* Wv ( z * ) (4.32) (assuming that V Ha v ( z * ) , which results from temperature changes on a pressure surface or horizontal gradients of the absorbing gas, is negligible.) So, for ground based radiance measurements, = av ( z s* ) V Hz s* T bv (4.33) Thus, we may write an expression for the weighted temperature gradient for ground based measurements: V ht bv - a v ( z s * ) V HZS* T bv + WV ( ZS* ) T ( z s* ) V HZ g *. (4 .34 ) It is evident that hv defined in this manner is only approximately known since the measurements of z s* , t bv and t ( z s* ), as well as wv ( zs* ) and av ( z s* ), are all subject to errors. For microwave radiance measurements from satellites, V Hwv ( z * ) is approximately zero since the weighting function change along a constant pressure or z * surface is due only to the weak dependence of the weighting function on temperature. Thus, the equivalent expression for (4.34) in z * coordinates for satellite measurements does not include the V Hwv ( z * ) term and is written: V HT BV + WV ( Z g * ) T(Zg*) V HZ g * . (4 .3 5 ) When working in z * or in p coordinates, solution of the Poisson equation could be complicated by the intersection of the coordinate surfaces with the R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -63ground. If the Poisson equation were solved in the terrain-following a coordinates, this would not be a problem. However, in a coordinates, it is not possible to write J V CTwv (c) t (c ) da only in terms of the brightness temperature. In order to eliminate this term we would have to use some first guess temperature such as may be retrieved from radiances or determined from wind measurements. Then, in a coordinates hv is redefined as: o (4.36) In a coordinates the magnitude of VCTwv (c) for ground-based weighting functions is not very large so the error incurred in using a first-guess temperature Tf to approximate this term may also not be very large. In fact, test calculations show an average error of less than ± 6 % in calculations of hv using a mean temperature field for Tf for one case. For measurements from satellites, however, the error in the term Vcwv (a) could be substantial (~20%) in an area where there is a large gradient of surface pressure. The estimates of hv can be improved through iteration with retrieved temperature fields substituted for the first guess fields. How should one choose to use radiance gradient information to retrieve temperature gradients? The answer to this depends on what data is available and what kind of problem is desired to be solved. A useful way to investigate the possibility of radiance gradient retrieval is to look at it as an extension of the simpler one-dimensional retrieval problem and apply similar solution methods to deal with the similar kinds of problems that may arise. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -64- 4.2.1 Regression Methods The first method discussed above for the one-dimensional retrievals was the statistical regression method. For gradient temperature retrieval, we could postulate a linear relationship between the radiance gradient and the temperature gradient and fit regression coefficients just as in the one dimensional case. From (4.3), a possible set of coefficients could be c = ( V t Vr t ) • ( V r V rt + eeT) - 1 (4.37) and thetemperature field that has the minimum mean squaredifference from the retrieved temperature gradient estimates could be determined from the Poisson equation: V2t = V • (c • V r ) , (4.38) (where the dot products of matrices also imply a matrix multiplication). Another possible approach would be to postulate a linear relation between temperature gradient and the adjusted radiance gradient h v discussed in the previous section; the derivation of coefficients would be similar to (4.37), (4.38). A difficulty with this regression approach is that a large number of simultaneous temperature and radiance gradient measurements over an area would be needed in order to be able to calculate the coefficients. In addition, for ground-based remote sensing, the coefficients c would be very dependent on terrain height and so must be expressed as a function of horizontal as well as vertical position. These considerations make the use of this method impractical for use with ground-based radiometers. 4.2.2 “Exact” Physical Methods Section 4.1.2 dealt with radiance retrieval methods that used weighting R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -65functions wv ( z * ) to obtain temperature profiles that match the observed radiances exactly. These methods in general suffered from sensitivity to noise in the measurements and instability, and similar methods using radiance gradient measurements would suffer from the same faults. 4.2.2.1 Sm ith's method. Smith’s method did not exhibit quite as much error sensitivity as the matrix inversion methods so development of a “gradient Smith’s method" solution may be useful. Recalling (4.34) and (4.35), we note that h v can play the same role in estimating V ht that t bv played in (4.14) to estimate t . (Note that hv already takes into account the variations in surface height.) We define an adjusted hv* as hv (z*) (4.39) then the iteration procedure can be written as V ht ^ +1 ( z * ) = V hT 3 ( z * ) + [hv* - h ^ * ] (4.40) where (4.41) and then Vht ^+1 = V £ w v* ( z * ) (4.42) V The iterations can be continued until some convergence criterion is achieved, giving us an estimate of V ht . This estimate can then be used in a Poisson equation to solve for t . The boundary conditions are specified as for the wind retrieval, fixed Dirichlet conditions or gradient Neumann conditions from (4.42). The response to errors in the measurements would be similar for this gradient method as in the Smith radiance retrieval method; however, the solution of the R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -66Poisson problem may reduce some of the error in the Vht estimate just as in the case of the wind retrievals discussed in Chapter 3. 4.2.3 Constrained Physical Retrievals The next approach to be considered is that of applying statistical constraints to the retrieval with radiance gradients. The methods discussed for radiance retrieval were the use of smoothness constraints (Twomey-Tikhonov), the minimum variance estimate, and the “minimum information” method. For gradient retrievals, these methods can be adapted just as with the Smith method above; using (4.34) or (4.35) to convert from retrieval of temperature to the retrieval of temperature gradient, followed by solution of a Poisson equation to derive a temperature field from the estimates of the temperature gradient. An extension of this concept is to use the same statistical constraints on the gradients but to solve directly for temperature fields rather than for temperature gradient fields, such as in Gal-Chen's method (Gal-Chen, 1988; Sienkiewicz, 1990). 4.2.3.1 Smoothness constraints. vertical temperature profile as V ht = 3 t Let us denote the gradient of the / 3 x i + di/dy j . For the smoothness constraint method, Eq. (4.23) can be revised as: Vht = (wtw + yh ) - 1 wT h (4.43) hence the Poisson equation would be: V h2t If we choose h = Vh- ( (vTw + YH ) - 1 wT h) . = i we can write (4.45) in a form similar to (4.24) V h2t = VH- ( (wtw + y i ) ' 1 wT h) = VH- (w’Mww'1’ + y i ) - 1 h) . 4.2.3.2 (4.44) Minimum variance method. (4.45) In a similar fashion, we can obtain a gradient minimum variance method by following a derivation much like Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -67- that in section 4.1.3.2; except we use the temperature and radiance gradients rather than the scalar values. Since VTt is an array of vectors (with elements being the temperature gradients at each level) then the covariance matrix Sy = e [ ( V t c) (VTt ) T] has elements that are dyads (e.g. Symn = 3Ttm3Tt n a T ^ d T t 11 EI‘ & T 1 T 'U + I T 3T t m3Tt n i r 13 + I T 3 T t m dTt n i r 33 + I T e [ V ^ V t , ; 11] = , I T 331, Where m and n are different vertical levels). The derivation can be carried out in the same fashion as in Eqs. (4.25)-(4.27). Using the measured brightness temperatures we estimate the array h = [hvl) hV2, ... hvN] . Our objective is to find the coefficient dyad matrix c that produces the retrieved profile of temperature gradient V t = H ——— " c • (h - w V t c + ey) + VTt that minimizes the expected value of the gradient profile squared error (V t - V t j T (Vt - V t c) . Following the same procedure as ^ ^ y in section 4.1.3.2, we find that the appropriate value for c is c = +N y ) ^ y Sy ^ • wT ( w s y w 7 and so the retrieved temperature gradient profile would be: VhT = VHTt + Sy • WT(WSyWT + Ny) _1 • (h - W VHTt +ey)(4.46) These gradient estimates then can be used in a Poisson equation to produce a temperature field V h2T = V h - V HT t + V H* ( S y - W T {WSyWT + N y ) ’ 1 * (h - W V HT t + Ey) ) (4.47) The boundary conditions, as mentioned above, could be fixed Dirichlet conditions or Neumann conditions specified by (4.46). 4.2.3.3 “ M inim um in fo rm a tio n ” m ethod. Estimates of the full covariance matrix s y may be perhaps even more difficult to obtain for the gradient minimum variance method than for the scalar case (since it would involve finding cross-correlations between gradients in the x- and y- direction at many levels from analyses over an area, rather than involving measurements at perhaps only one station.) Thus, it would be desirable to make a similar adaptation for gradient solution to the “minimum information” method of section Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -68- 4.1.3.3. The covariance dyad matrix s v in (4.46) would be replaced by a diagonal matrix or a single value. Thus Eq. (4.47) may be written in the simplest “minimum information” form if the variances are assumed to be the same for all levels of t (Sv = <jt 2 i ), and if the noise variance is the same for all channels ( n v = o e2i ) . Vh2t = VH- ( Vht ) + VH• (ot 2^ewt • ( a T2wwT?+ ae2? ) - 1- (h-w V H - ( V ht ) + V h - | w t ( w w t :e + — ? ) - i ■ ( h - w V ht ) V V H - ( V ht ) ° T + V H - f wT ( ww T+ I 5 s _ ) - i ( h - w V ht ) 1 c T2 I (4.48) This equation is similar to (4.45), the equation for the smoothness constraint. As with the gradient Smith’s method, the boundary conditions could be fixed Dirichlet conditions or Neumann conditions. 4.2.3.4 G al-C hen’s m e th o d . The gradient retrieval methods mentioned above were, in essence, derived by finding the temperature gradient V ht that had the minimum expected squared error from the true temperature gradient, then finding the temperature profile that satisfied the retrieved gradients most closely. However, we may want to define our optimal field in a different fashion, since in the above methods there are no guarantees that the squared error of the temperature field (rather than the gradient field) would be minimized. In order to obtain the temperature field with the minimum squared error it is necessary to write an equation analogous to (4.29) where gradients of temperature and radiance replace the scalars, and find the temperature field that minimizes the functional. The equation is written as continuous integrals rather than in a matrix form in order to better demonstrate the effect of terrain variations on the minimization. The functional to be minimized is written: R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. where l (z * , z * ') is a dyadic weighting function that corresponds to an inverse ^ y ^ of the error covariance matrix Sv, and mV7 corresponds to an inverse of the <-> observation error covariance Nv- The details of this minimization are given in Appendix A, section A.3; the resulting Euler-Lagrange equation is L ( z * , z * ‘ ) • V ht ( z * ' ) d z * 1 + X 2 wv ( z * ) v y Mvy * (4.50) As discussed before, the calculation of this is simplified if the covariance dyad matrices S y and N v (and/or the weighting matrices l approximated by diagonal matrices. and m ) are Gal-Chen (1988) has presented a derivation equivalent to (4.49), (4.50) using that simplification and he describes a method by which the resulting three-dimensional Euler-Lagrange equation may be solved iteratively as a series of two-dimensional Poisson equations at each horizontal level. In his notation the weighting dyad function l is replaced by the function (30 ( z * ); and m is replaced by the column vector pv- Then (4.49) in integral form is written oo J S J zs * P oV h [ ( T - T ) ] 2 d z * JS Pv dS + [fZs.wv<z *> V «T dz* - hv] dS (4.51) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -70- and the optimal solution t must satisfy the following partial differential equation if we assume the surface values are fixed (Sienkiewicz, 1990): VH • PoVhT + Z v p J #VH- [Wv (z*)Wv ( z * , ) VHT ( z * , ) ] d z * ' = V h *P oVhT Zs + Zy PvfVH • wv (z*) hv + VHzs*-Wv (z*) Wv ( z s *) VHT ( z s *) ] . (4.52) A derivation of thisis also given in Appendix A. Note that all the terms on the right-hand side of the equation are known or can be estimated from measurements. The boundary conditions for (4.52) may be either Dirichlet conditions or Neumann boundary conditions. The Dirichlet boundary conditions consist of setting the values on the boundary r equal to some previous estimate of temperature (perhaps from a numerical model or derived from the radiance measurements): T (I\z*) = f(I\z*) (4.53) The Neumann boundary conditions have the following form: Po(VHT • n) + Sy pv Wv ( z *) J ^wv (z* • ) ( V hT • n ) ] d z * 1 = 2S Po(VHT - n ) + ZvPv wv (z*) hv * n (4.54) where all the functions in the above equation are evaluated on the lateral boundary of the region, r, and n = ( n x , n y ) is a unit vector normal to the boundary. Since the Neumann solution is unique only to a constant, the mean value of the field or the value at some particular point must also be specified. Gal-Chen (1988) details how these boundary conditions can be simplified so that they may be solved for directly (without resorting to an iterative solution). If (4.54) is multiplied by wY( z * ), divided by p0, and integrated in the vertical, then a set of linear equations results (one for each channel number y), which may be solved for bv = f wv ( z * 1 ) ( V ht • n) dz * ■ through a matrix Jzs * inversion. This set of linear equations is written: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -71f Jz3* WY( z * ) ( V HT ( z * ) t i) dz* + Pv « p0 (z*) wy(z * ) wv( z*) ,Wv (z* 1 ) (VHT (z* ' ) • n ) ] d z * ' dz* = Jz Ev zs* <00 Syv + 2v JZ_*^Z * 0 s O 1 V Wy(z*)Wv (z*) f wWv (z* ' ) VhT • n d z * ' = Po(z*) JZs* Pv p0 (z*) Wy(z*)Wv ( z * ) h v • n d z*, ^Wy(z*)VHT ( z * ) • n dz*+Zy J (4.55) ZS* where 8yv is the Kronecker delta. Thus, the gradient on the boundaries can be pre-specified before the iterative solution is performed. It is interesting to note that the matrix to be inverted has a form similar to those described earlier in the discussion of the minimum variance method. Let { y { y ^ y ^ y s v correspond to Po_1i and Nv correspond to P v ^ i ; the matrix equation to be solved can be written symbolically as (wsvwT + Ny) • b = g, where b is the integral bv and g is derived from the right-hand side of (4.54). Since a matrix equation of this form is also used for solution of the interior, this implies that the inversion method will have some of the same qualities as the other inversion methods; the values of pv and p 0 relative to one another will determine the conditioning of the matrix (wsvWT + Nv). Increasing the weight on the first guess, p0 or decreasing pv will decrease the sensitivity of the solution to noise in the radiance measurements. The effect of this matrix and its sensitivity to noise is not as straightforward for this retrieval scheme as for the others mentioned above since the matrix inversion is used to estimate only one of several terms that are involved in the retrieval. Gal-Chen shows that in Eq (4.52), the operator b (x , y , z * ) acting on the temperature field, which is defined as: —B (x , y , z *) = V H • PoVH + Ev Pv J s d z * 1VH • Wv ( z * ) Wv ( z * 1 ) VH (4.56) is positive semi-definite so (4.52) is an equation of elliptic type and well posed. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -72- In order to solve (4.52) with the boundary conditions (4.53) or (4.54), the operator b is separated into two parts, i.e., - b = m + n where m is an operator whose inverse m+ is easily found, while the correction term for the iterations m+n t is not too large so that convergence is readily reached. His choice of the terms mt and n t is: MT = p 0VH2T + ZvpvWv ( z * ) J ,Wv ( z * ' ) V h2T d z * 1 (4.57) NT = V Hpo • VhT + Zv Pvf jtV H[ w v ( z * ) W v ( z * 1 ) 3 - V HT ( Z * ' ) d z * 1 (4.58) zs Then, an iterative solution method can be written: Tk+1 = (1—CO) Tk + G)M+ [NTk + / ] = [ (1 -0 )) I-(0M+N ] Tk + OM+f (4.59) where / denotes the terms on the right-hand side of (4.52), all of which are known or can be estimated; co is a relaxation parameter chosen to accelerate convergence of the solution, and k is an index for the iteration count. At each step of the iteration, a forcing function [NTk + / ] is calculated using the temperature field from the previous step. Then, the Poisson equations M T k + 1 * = [N T k + / ] ( 4 .6 0 ) or, in expanded form: 0oVH2Tk+1* + ZvPvWvU*)/ .Wv ( z * ' ) V H2Tk+1* dz* 1= - VHpo • VHTk ¥ZS - E v Pvf tVH[Wv ( z * ) W v ( z * 1 ) 3 • V HT k ( z * 1 ) d z * 1 + V H * 0 OV HT S + Ev 0v[VH-Wv(z*) hv + VHzs* • Wv (z*) Wv (zs* ) VhT ( zs* ) ] (4.61) are calculated for each horizontal level by a method similar to thatdescribed for solving for the boundaries; the equation at each level is multiplied by wY( z * ) and integrated in the vertical to create a set of linear equations that can be f°° solved for JZs*wY( z * 1 ) V h2t ( z * 1 ) d z * 1. These values can be substituted into (4.61) and the 2-D Poisson equation for Tk+1* can be solved at each level. Then Tk+1 = (l-co) Tk + «)Tk+1* is calculated to complete the iteration step. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -73- Sienkiewicz and Gal-Chen (1988, 1989) posed the minimization problem (4.51) in a slightly different fashion. The approximation hv was not used so the functional to be minimized was written J = J •'S J PoVH[ (T —T ) ] 2 d z * dS + zs * is | iz ^ d z * — Tbv j J dS (4.62) and this shows more clearly that the functional to be minimized is a weighted sum of the first-guess gradients and the observed brightness temperature gradients. When this minimization is performed (see Appendix A) we find that the optimal t in this case must satisfy V H * PoVhT + EvPv Wv ( z * ) V H2r Wv ( z * 1) T ( z * 1 ) dz * 1 = s* Vh ' PoVhT+ Ev py Wv (z *) V h2 T bv (4.63) The solution method for this equation is similar to that detailed for (4.52) above. The Neumann boundary conditions would be Po(VHT * n ) + Ev Pv Wv ( z * ) V H •{£ ^Wv ( z * 1) T d z * ' J - n = Po (VHT ■n) + Ey Pv Wy ( z * ) (V^Tbv ' n) (4.64) The above equation (4.64) can be solved for ( V ht • n ) just as wasdone for Eq. (4.54) if the equation is first rewritten as: Po ( VhT • n) + Ey pv Wv ( z * ) JJz3 Wv (z* ' ) (VhT •n ) ]d z* ' Po(VHT - n ) + Ev Pv Wv ( z * ) [ l z *VhWv ( z * ' ) T ( z*) = (VhTbv * n) d z * ' + W v (zs*) T ( z s*) VHzg*J -n (4.65) Eqs. (4.33) and (4.34) can be used to put (4.65) into the form of (4.54), or one could use the first guess temperature to specify I s* VHwv (z * •) t (z *) d z * ' . It may also be possible to improve the estimate of this term by re-evaluating R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -74- (4.65) after each iteration. Eq. (4.63) also can be written in terms of an operator b as was (4.56): -B (x,y,z) = VH • PoVH + EvPv Wv ( z * ) V h2 [ d z * ' Wv ( z * ' ) 'Z s * (4.66) and can be broken into two parts in a similar fashion. Here we take MT = p 0V H2T + ZvPvWv(z*)f wv ( z * ' ) V (4.67) h2T d z * 1 *Zo NT = V Hpo - V HT +2y Pv Wv ( z * ) VH2 Wv ( z * 1 ) T ( z * ' ) d z * 1 - JZg* (4.68) The solution of the system of equations (4.63) then proceeds as described above according to the method presented by Gal-Chen (1988). It is not clear how much the solution obtained by this method would differ from that of (4.52). The major difference between the two equations is that I * (VHwv ( z * ) ) *z9 t ( z *) d z * was approximated using t bv (or possibly t ) in the Gal-Chen (1988) method while in the Sienkiewicz and Gal-Chen (1988, 1989) method it is in effect recalculated at each iteration. The greatest effect would be if there were some large error or bias in 1 „ (VHwv ( z * ) ) t (z*) dz*. t bv, which could change the value of However, we could expect the effect to be, at •' Z s most, very small, since typical errors in radiance measurements are ~ 1 % or less. (The effect of bias in t bv on VH • f (VHwv ( z * ) ) t ( z *) d z * would be J ZS * small except in areas where z s * has a large second derivative.) If there were no gradient of wv ( z * ) in the horizontal, then the systems (4.52) and (4.63) should give the same solution. If (3o was also taken to be constant, it can be shown that the solution reduces to that of the gradient “minimum information” method (4.48). Furthermore, when there is no horizontal gradient of wv ( z * ) , the solution of the gradient “minimum information” method R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -75- would be the same (to within a constant term on each level) as the scalar radiance “minimum information” solution (4.30). Note that, because we are writing Eqs. (4.51) and (4.62) in terms of continuous integrals instead of sums, the units of po should be K^m - 1 or inverse squared error per unit height. Similarly wv ( z * ) also contains a factor of m-1. When the equations are expressed in discrete form, the factor Az in each term is incorporated into Po and wv. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5 COMBINED WIND AND RADIANCE RETRIEVALS Before we can perform retrievals combining wind and radiance data, we must first examine the difference between temperature, which enters into the radiance calculations; and virtual temperature, which the wind retrievals provide. 5.1 Temperature and Virtual Temperature The virtual temperature is an adjusted temperature that takes into account the differing densities of dry and moist air (rather than adjusting the gas constant r by the changing proportion of these atmospheric constituents). Virtual temperature tv is approximately related to temperature t by the equation Tv = T (1 + 0 . 6 1 g) (5.1) where q is the specific humidity (ratio of the mass of water vapor to the mass of air) measured in 9 (H2°) / g(air). The difference between temperature and virtual temperature is generally small but can be substantial near the surface where temperatures and specific humidities can be large. The difference between temperature gradients and virtual temperature gradients can also be substantial especially in areas where the specific humidity changes rapidly (such as in the vicinity of a dryline). At 1000 mb, given a temperature of 300K, the specific humidity could be greater than 20 g kg-1; this could result in differences of close to 4 K in virtual temperature between areas of high and low humidities. -76- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -77- Eq. (2.15) shows the relationship between temperature and brightness temperature. To perform retrievals of virtual temperature from radiances, we need a formula that expresses a relationship between the brightness temperature and a weighted integral of virtual temperature; so we rewrite (2.15) in this form: Tbv ■T Ba c k e~x(v) + [°° W( v, z ) 0 . 6 1 T ( z ) q ( z ) d z Jzs (5.2) The first two quantities on the right hand side are known; the brightness temperature is measured and presumably the background radiation from space is also known or can be well estimated. The third term is a virtual temperature correction term for the brightness temperature; it may be estimated from radiometric measurements or from a prior model run. How much of an error are we making with this estimate? If we suppose that the radiometric measurements can determine the temperature to within ± 2 K through the atmosphere, and the specific humidity to within ± 5 g kg-1 in the lower troposphere, then the correction factor would be in error no more than about ± 1 K. In practice the error is probably less. (This question was also discussed by Gal-Chen (1986b) who came to a similar conclusion for the more general case of radiances not linearly related to temperature.) Figure 5.1 shows the mean difference (bias) and rms difference between analyses of virtual temperature and estimated (q = 0, tv = t ), tv assuming (a) dry atmosphere and (b) corrections using a mean temperature profile with 50% relative humidity (t v = t + o. 61 t (0 . 5 qs ( t ) )), for each of the four VAS experiment days. The bias is fairly large when a dry atmosphere is assumed, but it is reduced when some reasonable values of temperature and humidity are assumed in order to calculate a correction. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -78- —0— VAS2 —a - VAS3 — VAG4VAS5 200 -O— -a — -» - 200 400 400 600 600 600 600 VAS2 . VAS3 . VASA-..VASS ♦ 1000 1000 -1.5 •0.5 0 0.5 0.05 0.1 0.15 0.2 0.25 0.3 0.35 RM S difference (K) Bias est - obs (K) tt - e —t VAS2 -a VAS3 4 vr*4*VAS4-f 200 - * - j VAS5 f S I - e — YAS2 . - a - VAS3 • '-V A S 4 -YAS5 ’ 200 > ! 400 400 600 600 600 600 r 7?_rrrr f t r.-rrr ,^i>60%tRH porTBCtioo- 1000 1000 •0.5 •0.4 •0.3 • 0.2 • 0.1 0 0 0.1 Bias est - obs (K) Figure 5.1. 0.05 0.1 0.15 0.2 0.25 0.3 0.35 RM S difference (K) Bias (K) and rms difference (K) between “true” virtual temperature fields and virtual temperature fields estimated (a) assuming dry atmosphere and (b) using a correction based on the mean sounding and 50% relative humidity. Figure 5.2 shows the bias and the rms difference between the “virtual” brightness temperature (f w ( v , z ) Jz3 tv( z ) dz ) and (a) the dry brightness temperature and (b) “virtual" brightness temperature using the 50% relative humidity correction 0 . 6 1 T ( 0 . 5 q s ( t o . The averaging in the integration causes the biases to be smaller for these brightness temperature fields than for the temperature fields in Figure 5.1. The bias can be reduced to less than 0.3 K using the simple 50% RH correction; it is likely that radiometric estimates of humidity could do better if only because they would include some humidity gradient information. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -79- 5285GHz 53.85 GHz 55.45 GHz 55.45 GHz 58.80 GHz 58.80 GHz 53.74 GHz 53.74 GHz 54.96GHz 54.96 GHz • 0.6 •0.4 Bias ast • obs (K) - 0.8 •1 0 0 • 0.2 5285 GHz 5285 GHz 53.85 GHz 53.85 GHz 55.45 GHz 55.45 GHz 0.05 0.1 0.15 0.2 0.25 RMS differeno® (K) 0.3 0.35 56.80 GHz 53.74 GHz 53.74 GHz 54.96 GHz £o% Ri-| correction 54.96 GHz WMfW •0.3 -0.25 Figure 5.2. -0.2 -0.15 -0.1 Bias est • obs <K) -0.05 0 0.05 0 0.05 50% RH Correction 0.1 0.15 0.2 RMS differenoe (K) 0.25 0.3 Bias (K) and rms difference (K) between brightness temperatures calculated from “true” virtual temperature fields and those calculated from (a) dry atmosphere and (b) virtual temperature fields estimated using a correction based on the mean sounding and 50% relative humidity. Thus, it is possible to estimate a “virtual” brightness temperature using only simple assumptions (such as using only a mean relative humidity) that is in error by less than ± 0.5K. Since the instrument error of the microwave radiometers is about ± 0.5 K, we could expect our “virtual” brightness temperature estimates to be in error by less than ± 1K. Using such estimates in a combined wind-radiance retrieval would be feasible. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 80 - 5.2 Retrievals from Winds and Radiances There are many possible ways to combine wind and radiance data to retrieve temperature or virtual temperature. The radiance retrieval could be carried out separately from the wind retrieval and the results combined as some weighted average. This would be the simplest method in concept, but would not necessarily yield the best results. For the radiance retrievals, the first guess can make a substantial contribution to the final profile if it has a large component in the radiance nullspace. If this first guess profile were chosen from inappropriate statistics it could have a negative influence on the solution. The co-analysis of radiance and wind derived temperature fields would have to be carefully done to insure that the contributions of the radiance derived fields were significant only at levels where the radiance weighting function is substantial. The analysis performed by Bleck, et al., (1984) seems to fall in this category. They performed an analysis in isentropic coordinates where a “radiance-derived" Montgomery potential is variationally adjusted so that the second derivative in 0 is as close as possible to the second derivative in 0 of a Montgomery potential derived from the winds using a balance equation. Their analysis did not include any vertical variation of relative weights of the radiance derived and wind correction fields. The first guess Montgomery potential field was simulated by using smoothed analyses of radiosonde data, rather than using profiles actually derived from radiance data. Their study showed that some vertical detail that had been smoothed from the Montgomery potential field (to simulate the poorer vertical resolution of the radiance retrievals) was restored through the use of the wind field analyses. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -81- An alternative strategy would be to perform the wind retrieval first and then the radiance data could be used to adjust the initial analysis. If the wind retrieval were performed first it could be used as a first guess profile for the radiances and one of the methods described in Chapter 4 (that take into account the shape of the radiance weighting functions) could be employed to obtain a final retrieved virtual temperature field. A third possibility is to perform one analysis utilizing both forms of data. This strategy would allow us to explicitly include the relationships between the winds and radiances and the virtual temperature (in the form of the equations of motion and the radiative transfer equation) and estimates of observation error. The study by Lewis, etal., (1989) tests a retrieval scheme that uses wind shear and radiance directly. Their retrieval method is a constrained minimization problem that seeks to adjust a first guess background temperature so that it approximately satisfies constraints relating the temperature to the radiance and wind shear. The wind shear constraint is based on a gradient wind balance; the radiance constraint is derived from a regression equation between the radiances and the temperatures at the surface and 9 mandatory reporting levels, rather than being in the form of the full radiative transfer equation. The solution of the minimization was found through the conjugate gradient method. Their results for experiments using simulated VAS radiances showed that the inclusion of wind reduced the dependence on the first guess field from that of a retrieval using only radiances. These combined wind-radiance retrievals could also be accomplished using methods similar to the gradient radiance retrieval methods presented in section 4.2. The gradient retrieval methods require first-guess temperature gradient fields; they may be modified by substituting the wind derived gradient R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -82- temperature estimates 1 06 lng>) (where g is defined in Eq. (3.1)) in place of the first-guess mean gradient V ht c . In this research we use the temperature fields obtained by solving the Poisson equation V2t = - V *| ^ 3a I „ can be shown that these solutions should be the same as those obtained by using the wind-derived gradients directly, since we are not varying j} 0 in the horizontal. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6 ANALYSIS METHODS The aims of this research were stated in the first chapter: to evaluate various retrieval methods used to derive temperature fields from Profiler winds, ground-based radiance measurements, and a combination of the two types of observations, and to determine if the retrievals using both forms of information improve on the ones where only one type of data is used. This chapter describes how the analyses and retrievals are performed. The first section is a discussion of the data used in the research and how it is processed before being used in the retrieval schemes. The second and third sections describe the application of the retrieval methods of wind and radiances that have been discussed in the previous chapters. The final section covers the methods used to produce retrievals from a combination of the two kinds of data. 6.1 Data To evaluate retrieval methods that can be applied to the wind Profiler network requires the use of data or simulated data from an observation network of a similar scale and observation frequency. Such Profiler wind and radiometer data have been not available, though Profiler observations have been made on small networks of three or four sites {e.g., PRE-STORM, Colorado Profiler network). As mentioned previously, prior research on dynamic retrieval from -83- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -84Profiler data primarily followed the OSSE (Observing System Simulation Experiment) methodology, with wind and temperature fields simulated from numerical model forecasts (and subsequent artificial forcing of compliance with the forms of the equations of motion used in the model). This research follows the OSE (Observing System Experiment) methodology; the actual wind observations from a special rawinsonde network (with time and space resolution approaching that of the Profiler network) are used to substitute for the Profiler data. For the radiance retrievals, the OSSE approach is used. The radiances used are calculated from the observed vertical temperature profiles, so the ‘observed’ radiances satisfy the form of the radiative transfer equation and the radiance weighting functions used in the research. It is necessary to use this approach since there is no other substitute for the radiance observations. The effect of random error in the data is simulated, and the sensitivity of the radiance retrieval methods to random error is evaluated. 6.1.1 Sounding Data The rawinsonde observations used in this study were taken during NASA's AVE/VAS Ground Truth Field Experiment conducted in the spring of 1982 (Hill and Turner, 1982). The 1982 Atmospheric Variability Experiment (AVE) was conducted as a part of NASA's Visible and Infrared Spin-Scan Radiometer (VISSR) Atmospheric Sounder (VAS) demonstration. Soundings at National Weather Service (NWS) and special rawinsonde stations were taken at 3 h intervals between 1200 UTC and 0600 UTC on each experiment day, with an additional sounding at the normal 1200 UTC observation time at the NWS rawinsonde stations. The data from this field experiment are a good choice for this retrieval study since (1) the four experiment days offer a range of R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -85weather from the strong gradients during VAS 2 to the weak anticyclonic flow of VAS 5; (2) the data coverage is similar to that of the proposed Profiler network, the 3 h observation interval approaches the Profiler’s hourly observations; and (3) other studies of VAS retrievals for this case are available for comparison. The experiment dates and times are summarized in Table 6 .1 . Only data from the 24 NWS rawinsonde stations are used in this study. Their locations are shown in Figure 6 .1. The reduction and error analysis of the data followed a procedure described by Fuelberg (1974). Thermodynamic variables were calculated from measurements extracted at each pressure contact level, and were interpolated to even 25-mb levels. Winds were calculated from angle measurements taken every 30 or 60 s by centered finite differences, filtered, and interpolated to even 25-mb levels (Sienkiewicz, 1982 a,b, 1983 a,b). The error estimates for the thermodynamic data and wind data are given in Tables 6.2 and 6.3. 6.1.2 Wind Data Processing It was necessary to perform some pre-processing on the 25-mb wind data before they were interpolated to the grid. The first step was to fill in some of the missing wind levels. The u- and v- wind component data from the eight sounding times for all pressure levels at a single station were each copied into a two-dimensional array and interpolated using the IMSL routine SURF (a twodimensional spline-fitting routine) (IMSL, 1987a). Cross-section plots of the wind data from each station were then inspected and compared with the raw data to find and remove levels where the interpolation was bad (i.e. levels in soundings where the winds were completely missing or where there were too few neighboring points to justify interpolation). The resulting data arrays were generally of good quality. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -86- Table 6.1. AVE-VAS Experiment dates Experiment Dates Observation times, UTC AVE/VAS II 6-7 March 1982 12,15,18,21,00,03,06,12 AVE/VAS III AVE/VAS IV 27-28 March 1982 24-25 April 1982 12,15,18,21,00,03,06,12 12,15,18,21,00,03,06,12 AVE/VAS V 1-2 May 1982 12,15,18,21,00,03,06,12 Table 6.2. Estimates of the RMS errors in thermodynamic quantities of AVE/VAS rawinsonde data (After Fuelberg, 1974). Parameter Approximate RMS error Temperature 0.5 K Pressure 1.3 mb from surface to 400 mb; 1.1 mb between 400 and 100 mb; 0.7 mb between 100 and 10 mb. 10 percent 10 gpm at 500 mb; 20 gpm at 300 mb 50 gpm at 50 mb. Humidity Pressure Altitude Table 6.3. Pressure 700 500 300 Estimates of RMS errors in AVE/VAS rawinsonde wind data (After Fuelberg, 1974). RMS errors (m s*1) in speed RMS errors (deg) in direction 10 deg elev. 10 deg. elev. 2.5 4.5 7.8 40 deg elev. 0.5 40 deg elev. 0.8 9.5 13.4 1.8 1.0 18.0 2.5 1.3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -87r r r r t t l> •• 4 - « — 4 >—4 I* _ [A- « t X. - _ iB F i i » t i i 5 »<wM 4 ■ »4 * fv i ~"£J\ i r i — — I —— }• ■ 'J ■ »— I—— I—~ I i t -*^ 4 i i L _ J. _ I I I I I I___ L — L _ .L _ X I 1 I I I I I 1___ I___ L - J I I Ll. | J 1 _L*. > 1 _1 _ i _ I I < I +QP1 I I I I I I 1 . J ____I ___ t __ l_ I “ T " i ------ 1 - I — r - i —n ~ r "i— i— r " A t i— | !★ | I I I | I I ' “ i— vrct - t - t i n — i “ iF r | I LZV r<aisu mF i ki i i i - -i--i— i i i t i L . - + - - 4*— + — 4 iV i i i i L - ^ i . » J. _ J___ I Figure 6.1 Locations of NWS rawinsonde stations in AVE/VAS. The grid is the 1° latitude-longitude grid used for height and temperature fields. The next step in wind preparation is to filter the wind component data in each sounding. There are a couple of things to consider when choosing a filter for the winds. We want to reduce the effects of random error in the wind on the final analysis. This means that some filtering is required to remove the smallest wavelengths ( X , = 2 A p - 4 A p or 50-100 mb). On the other hand, one of the motivations for the dynamic retrievals from wind is to provide profiles that have better vertical resolution than profiles derived from radiance measurements. The vertical resolution of the VAS sounder, for example, ranges between 2 km at the surface and 10 km at 100 mb (Smith, 1986); which corresponds in pressure coordinates to a resolution of about 200 mb. Thus, any filter that we apply should remove or reduce wavelengths of less than 100 mb, to reduce observation noise; but pass wavelengths greater than 100-200 mb, or frequencies 0.125-0.25 cycles/25 mb. These guidelines also apply in the choice of a filtering differentiation method for calculation of temperature (which R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -8 8 - will be discussed in section 6 .2 ). A 9-point Gaussian weighting function is applied to lightly smooth the data. The method used is similar to that employed in the Barnes interpolation scheme (Barnes, 1964, 1973), except that this was only employed in one dimension and no interpolation was involved. For a given pressure level p, the Gaussian weight has the form w (p+nAp) = e x p (- n 2/K), n = -4,..., 0,..., 4 (6.1) with the sum of the weights adjusted to be equal to one. Two passes of the filter are applied; on the first pass (with k = l .2 ) the weights are applied to the data values, while on the second (correction) pass (with k = 2 .4) a weighted sum of the difference of the first pass values from the observations is added at each point. The response for this truncated set of Gaussian weights is the discrete Fourier transform: 4 R(f) = 51 n = w (p+nAp) cos (27tfnAp), (6.2) -4 where f is frequency (cycles / 25 mb). If we take Ri ( f ) to be the response of the first-pass weights and r 2 ( f ) to be the response of the second-pass weights then the final response function after the correction pass would be Rf( f ) = R i( f ) + (1 - R i ( f ) ) R2 ( f ) (6.3) Figure 6.2 is a plot of the first pass and final response function. Time cross-sections of raw sounding winds, filtered winds, and the difference between the two analyses for the 6-7 March 1982 Oklahoma City soundings are depicted in Figure 6.3. Note that some of the missing winds in the lower levels have been filled in; otherwise, the wind data shows very little change. The wind field shows the passage of a short-wave trough at around R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -89400 mb in the 2100 UTC sounding; such a feature would have been completely missed in the conventional synoptic sounding standard level data. Figure 6.4 shows some examples of the rms error of height retrievals in pressure coordinates using wind data before and after the smoothing filter was applied. (The missing data gaps had already been filled in the “raw” wind data.) The figure shows that in each case there is at least a small improvement in the retrievals after the filter was applied. The use of the filtered winds also gives a corresponding improvement in retrieved temperature fields. 1 -4 First pass response ^ R n a lr e s p a n s e 0.8 ‘ J 0.6 Q. </> cr 0.4 0.2 0 0 Figure 6.2 0.1 0.2 0.3 Frequency (cycles/25 mb) 0.4 0 .5 First pass and final response of filter applied to wind data before interpolation to grid. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -90- 100 200 300 500 500 700 300 900 1000 1200 1500 1800 2100 0000 0300 3 / 6/ 8 2 T im e Figure 6.3 0600 1200 3 / 7/8 2 (U T C ) (a) Time cross section of raw sounding winds for Oklahoma City. 100 200 300 500 600 700 300 900 1000 1200 1500 1800 2100 0000 T im e Figure 6.3 0300 0600 1200 3 / 7 /8 2 3/ 6/82 (U T C ) (b) Time cross section of filtered winds for Oklahoma City. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -91- J : ; lOOr ; v c ' -T r ’■ - 200 : 300 j 3 4oo" - sooF- : * . : . . . : . - : : - q j " O 3 S ^ 600 - £ [ • • • • • • • ■ J 700: j soo i ; 900 - • : 1 0 0 0 t --------------1--------------- 1---------------1---------------1__________I__________ I_________ !__________ I_________J 1200 1500 1800 2100 0000 0300 0600 1200 3/ 6/82 3 / 7/82 T im e Figure 6.3 (U T C ) (c) Difference between raw and filtered winds. 6.1.3 Objective analysis The height, virtual temperature and the u- and v components of the wind are interpolated to a 1° latitude-longitude grid by the Barnes method (Barnes, 1964,1973; Koch, etal., 1981,1983). The average minimum distance between the stations in the AVE-VAS network was An = 2.95°. Following the guidelines suggested by Koch, et al., (1981) (V3 < Ax/An < V 2) the 1° latitude-longitude grid spacing is chosen to adequately represent the wavelengths resolvable by the observing network while at the same time not being so fine as to cause problems in calculation of divergence. Sounding parameters are interpolated to all points on the 25x15 grid, but only the grid points lying within an area bounded by the station locations are retained to be used in the retrieval. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -92- b) VAS a ) VAS 200 200 s, 400 S 600 S 600 600 800 ■" a 1 Raw winds — 6 — Smoothed winds 1000 0 5 1000 10 15 20 5 0 25 RMS error (m) 0 20 25 d) VAS 5 c) V A S 4 200 10 15 RMS error (m) 200 400 S 600 600 600 600 — RawwW)ds — Smoothed winds Rawwiftds Smoothbd winds 1000 0 Figure 6.4. 1000 5 10 15 RM S error (m) 20 25 0 5 15 RMS error (m) 10 20 25 Rms error of retrieved heights (m) using raw (solid lines) and smoothed (dashed lines) wind data for the four AVE/VAS experiments. In the Barnes interpolation method, two passes through the observations are made: for the first pass, the value at a grid point is the weighted sum of nearby observations; on the second pass, a correction is added to each gridpoint value that is a weighted sum of the difference between the observations and the first pass grid analysis. Gaussian weights are used that are dependent on the square of the distance of the observation from the gridpoint. The weights have the form wm = e x p Lm (6.4) K R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -93The parameter k determines the closeness of fit of the analysis to the observations. To speed the convergence of the analysis to the observations, the value of k on the second pass is decreased from its first pass value (k0) by a convergence factor y (with 0 < y < 1), such that Ki = y k 0 . Koch, (6.5) etal. (1981,1983) recommend that k 0 be chosen asa function of the station spacing, to limit the response of the analysis for wavelengths less than or equal to 2 An. They recommend K0 = 5.052 • Thus, for the analyses in this research, the value (6 .6 ) k0 = 17.82 is used. The convergence factor y was chosen to be 0.3. Figure 6.5 is a plot of the theoretical response function for the analysis after the first and second passes. Barnes (1973) shows that the response for the analysis after the first pass is D0 (a,Ko) = exp(-a 2KQ/4) = exp (-Ko7i2/A,2) (6.7) where the wave number a = 2k/X, and X is the wavelength (in degrees). The final response, after the correction pass, is given by D’ = D0 (1 + Do?'1 - D0?) . The heights and virtual temperatures are interpolated to (6 .8 ) the gridat even 1° latitude-longitude points This grid array was depicted in Figure 6.1. The uand v-grid points were displaced from the height/temperature grid by 1/2 grid length to the west and south, respectively, which corresponds to the Arakawa 'C' grid (Haltiner and Williams, 1980). Figure 6.6 shows the relative positions of the staggered grid points. There is no vertical staggering, the interpolations for all variables were carried out at each of the even 25-mb levels. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -94- 1 0.8 0 .4 0.2 — DQti 1st p a ss — D \ 2nd pass 0 0 5 10 20 15 25 Wavelength (°) Figure 6.5 First pass and final response of Barnes objective analysis. v. . 1 1 ' 3 +I u . . Ti , : i Figure 6.6 u. 6 / i . 3~ ~ Grid staggering used in objective analysis and dynamic retrievals. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -95Interpolation in sigma coordinates is handled in a similar fashion to that in pressure coordinates. The sounding values at the stations are interpolated to levels spaced 0.025 a apart, assuming that the quantities vary linearly with in p ; these are interpolated to the grid in the same way that has been described for interpolation on pressure surfaces. 6.1.4 Radiance Calculations Microwave brightness temperature measurements were simulated by integrating the temperature profiles measured by the radiosondes with weighting functions for the four channels of the ground-based radiometer (with frequencies of 52.85 GHz, 53.85 GHz, 55.45 GHz, and 58.80 GHz), and two of the three channels (frequencies of 53.74 GHz and 54.96 GHz) of the satellite 30 — ■a -+ -* •+ ■ ■e 25 i> + 20 * 52185 GHz 53I85 GHz 55I45 GHz 58180 GHz 53I74 GHz "A-StflSeGHz 15 10 —4 5 0 -I - 0 Figure 6.7 +- 0.2 0.4 0.6 0.8 Normalized weighting function 1 Normalized weighting functions for the four groundbased microwave channels and two satellite microwave channels used in this research. These were calculated using a standard atmosphere temperature profile, with surface pressure = 1000 mb. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -96Microwave Sounding Unit (MSU) described by Westwater, etal. (1985). Figure 6.7 shows an example of these weighting functions, normalized so that the largest value is equal to 1 . The absorption coefficients for radiance calculations are computed using the formula (2.9) (Ulaby, et al., 1981). The calculations are made at a single frequency for each channel instead of being integrated over a frequency window. The units are converted to km*1 by multiplication by the factor (0.1 In 10). For convenience, the absorption is converted to z* coordinates using (2.16) and (2.18): prn a v (z*)= av ^ . (6.9) The transmission xv is calculated by integrating a v using the IMSL high accuracy quadrature routine QDAGS (IMSL, 1987b) with standard atmosphere values for t . The integration of a v for the ground based transmission functions begins at z* = -0.5 km (which corresponds to p = 1064 mb). The values of Ov and xv are calculated at 0.05 km intervals for -0.5 km < z* < 2 km, and thereafter at 0.4 km intervals from z* = 2 km to z* = 46 km. The finer discretization is used in the lower levels to better represent the variation of transmission function due to surface pressure changes. We obtain values of transmission function for different z g* values by dividing by xv ( - o . 5 , z 3* ), since: f Z* Tv (zs * , z * ) = exp [ - Jj*zzs*av ~ (z*) dz * ] z* = exp [ - J av(z*) dz* + J &v(z*) dz*] -0.5 exp [ - J a v (z*) -0.5 dz* ] _______________- Q . 5 exp [ - J a v (z*) xv ( - 0 . 5 , z * ) Xv ( - 0 . 5 , z s* ) (6.10) dz*] -0.5 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -97For the satellite radiances, the finer resolution of xv near the surface is also needed since the surface transmission function is used to determine the contribution of surface temperature to the radiance measurement. Even though use of the standard atmosphere t can be thought of as linearizing (and hence simplifying) the radiance calculation, it is also true that the temperature dependence given the variation of temperature on constant pressure surfaces is not very great. This is demonstrated in Figures 6.8 - 6.11, which show the ground-based and satellite channels with weighting functions that were affected most by variations in t . Figure 6.8 shows the ground-based weighting function (-5xv/9z* = <xvxv) for a frequency of 52.85 GHz, calculated for a surface pressure of 850 mb, using the standard atmosphere profiles and the temperature profiles from Boothville (BVE) and Denver (DEN) at 0000 UTC 7 Mar 1982. Figure 6.9 shows the weighting function for the same frequency, for a surface pressure of 1000 mb, using the standard atmosphere temperature profiles and temperature profiles from Boothville at 0000 UTC 7 March 1982 (AVE/VAS 2) and 0000 UTC 2 May 1982 (AVE/VAS 5). There is little difference between the curves in each figure; thus we can see that the temperature variations that might be expected across this network or between experiment dates had little effect on the ground-based weighting functions, all other things being equal. Figures 6.10 and 6.11 show satellite weighting functions (dxv/5 z* = avxv) for the 54.96 GHz channel calculated using the same sounding profiles as in Figures 6.8 and 6.9 respectively. These satellite based weighting functions are not a function of surface pressure. The figures show that, as with the ground-based radiance weighting functions, the temperature variations between soundings at two stations at a single time or temperature variations at R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -98- o Std. Atm. 5 4 8 5 G H z B Y i. 5 2 .8 5 .4.H.?.......... 200 D E N 5 2 .85 dlHz 400 £ 600 800 1000 0 Figure 6.8 0.0 5 0.1 0 .1 5 W eighting function < k/3z* 0.2 0 .2 5 Ground-based weighting functions for 52.85 GHz channel, surface pressure = 850 mb, using standard atmosphere temperatures and two temperature profiles measured during AVE/VAS 2. o Std. Atm. 52 .8 5 G H z yAS2 52.85.isHz..... 200 V A S 5 5 2 .8 5 |G H z •§ 400 £ 600 800 1000 0 Figure 6.9 0.0 5 0.1 0.1 5 W eighting function 3 t/3 z * 0.2 0 .2 5 Ground-based weighting functions for 52.85 GHz channel, surface pressure = 1000 mb, using standard atmosphere temperatures and temperature profiles measured at Boothville during AVE/VAS 2 and AVE/VAS 5. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -99- Pressure(mb) 200 400 600 S td . Atmi 54 .96 © H z 800 ^'-|VE54^6GHz'|... ♦ -6EN 54.46G H z I 1000 0 0.01 0 .0 2 0 .0 3 0 .0 4 0 .0 5 0 .0 6 0 .0 7 0 .0 8 © W eighting tunction < h/3z* 6.10 Satellite weighting functions for 54.96 GHz channel, using standard atmosphere temperatures and two temperature profiles measured during AVE/VAS 2. Pressure(mb) 200 400 600 -e— S td. Atm 54 .9 6 © H z 80 0 -j—0 -V AS2 54 9 6 G H z t ♦ -\iA S 5 54 9 6 GHzi 1000 0 0.01 0 .0 2 0 .0 3 0 .0 4 0 .0 5 0 .0 6 0 .0 7 0 .0 8 W eighting function < k/3z* Figure 6.11. Satellite weighting functions for 54.96 GHz channel, using standard atmosphere temperatures and temperature profiles measured at Boothville during AVE/VAS 2 and AVE/VAS 5. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 0 0 - one location on different dates do not cause a large variation in the satellite weighting function. For the purposes of this investigation, we do not make a large error by using standard atmosphere temperature profiles to calculate the weighting functions. The brightness temperature is calculated as the integral of the weighting function with temperature. To perform this integral, we must choose a way to represent the temperature profile between the m data levels. Usually, the temperature at a data level is assumed to represent the mean temperature for a layer; this is not a bad assumption if the layer is not too deep and the weighting function does not change too quickly in the layer. However, it is quite possible for Jw t dz* to be a poor estimate of Jwt dz* in calculation of ground- based radiances, since the weighting function is so steep near the surface. We can improve the estimate by assuming temperature changes linearly between coordinate levels. If we take temperature to vary linearly with z * (or i n p) we can use integration by parts to derive (l is 1st level above surface): dz* + T back xv ( z s*,°°) ,zk+l R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -101This expression is expanded in order to find the appropriate integration weights to multiply t ( z * ) at each level. For instance, the weight w^forT(zk*) is f zk* J7, 1. x v ( z s * , z * ) K = f zk+l* dz* - J Tv ( z s * , z * ) d z * . AzJ*------------------- * (6‘12> The calculation of the integration weights requires integration of xv over each of the sublayers; this was accomplished by fitting a cubic spline to the previously calculated values of xv using the IMSL routine CSINT, and evaluating the integral of the spline using the routine CSITG (IMSL, 1987a). 6.2 Dynamic Retrieval of Height and Temperature From Wind Retrievals of height and temperature on constant pressure and constant sigma surfaces are performed using Dirichlet and Neumann boundary conditions. Several approximations to the full momentum equations (discussed in Chapter 3) are used to obtain the estimates of gradients of geopotential. This is motivated by the desire to avoid the use of terms with co, since the vertical motion is poorly estimated from radiosonde measurements, and since the Profiler co measurements may also be inconsistent with the horizontal scales measured in the network. It is also useful to examine the importance of the time derivative and the non-linear terms in the retrieval since these are more adversely affected by observation error than the Coriolis terms. Geopotential gradient estimates are obtained using finite-difference forms of the horizontal momentum equations (3.1) and (3.21); with space derivatives approximated by second order centered differences. Time deriva tives are also approximated by second-order centered differences calculated over 2 At = 6 hr. A finite-difference analogue of the least-squares minimization is used to find height and temperature fields that best fit the gradient estimates. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -102- 6.2.1 Pressure Coordinates The approximate forms of the momentum equations used for retrievals in pressure coordinates are: Geostrophic: g ' 1' = f 0v , Balance_appx.: g«d = - u§£ - Im-scale appx.: g<d + f 0v, g<2> g<2> ~ = - = - ^ Gl2) = - I f where g ' 1’ ~ 3(j>/8x, (6.13) g<2> = - f 0u + ^ u ^ -v ^ f 0v (6.14) - f oU , - f °u (6.15) 3<j>/3y. The non-linear terms in these equations are written in advection form. The retrievals were originally tried by omitting the vertical motion terms from the equations written in flux form; those retrievals were less successful before the addition of u |^ = - ud and v |^ = - v d (thus changing the equations to advection form). Note that the ‘balance approximation’ of (6.14) is not the same as the balance equation discussed in section 3.3.2; rather, it yields (after the minimization is performed) the steady state divergence equation approximation discussed in section 3.3.3. This would be equivalent to the ‘balance equation’ if the winds used in the calculation were the rotational part of the wind only. Virtual temperature gradient estimates are calculated from the geopotential gradient estimates using the hydrostatic equation (2.5). This calculation involves taking a vertical derivative of the height gradient estimates, (3V<(> /<?inp). This differentiation amplifies small-scale noise in the vertical. Anthes and Keyser (1979) mentioned this problem in connection with their use of winds in a balance equation numerical model initialization; the errors in geopotential differenced in the vertical produced errors in temperature and R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -103hence errors in static stability. Kuo and Anthes (1985) also give an example of a retrieved sounding with short wavelength noise, produced from modelderived winds that did not have any added observational error. They noted that vertical smoothing reduced the rms error of their retrievals. To reduce the effect of small-scale vertical errors on the temperature retrievals, we would like to use a filtering differentiator. Thus, the vertical derivative (3V<|> / 3 inp ) is calculated using a derivative formulation suggested by Lanczos (1956) for differentiation of an empirical function where the observations are not error free. If we assume that the curvature (32V<j>/ (3 in p )2) varies only by a small amount over 5 levels, then we can fit a parabola to the five points by a least-square minimization and evaluate the derivative from that curve. The formula for the derivative is then „ , , f(x ) - 2 f ( x - 2 A x ) - f(x-A x ) + f(x+Ax) + 2f(x+2A x) = ------------------------------- ^ -------------------------------- (6.16) Figure 6.12 shows the response function for Lanczos’ derivative formulation compared to simple second-order centered differences. (Note that the analytic response for a derivative is a line with 2 n slope and zero response at zero frequency.) The Lanczos derivative reduces the response for wavelengths smaller than 200 mb. (The negative response of the Lanczos derivative is for wavelengths already reduced by the vertical smoothing of the wind.) The derivatives for levels nearest the surface and the top are calculated using formulae obtained in a similar fashion as (6.16), from 4-point parabolic curve fits. Figure 6.13 compares the rms temperature errors for retrievals in pressure coordinates using Gal-Chen’s two-scale approximation for the centered difference and Lanczos derivative calculations. The Lanczos R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -104- 0.8 0.6 o 04 (O S . 0.2 c GC n CD - GentefedditterenceLanczos' derivative 0.2 -0.4 0 0.1 0.3 0.2 0.4 0.5 Frequency (cycles/data interval) Figure 6.12 Response function for second-order centered differences and Lanczos’ derivative formulation. [Centered difference E 200 200 400 E 400 600 600 800 600 a) VAS b) V A S I3 1000 J 0 1 2 3 RMS error (K) 4 1000 L 5 0 1 2 3 5 4 RMS error (K) jCentered difference E 200 200 400 E 400 600 600 800 800 c) VAS VAS 5 1000 0 1 2 3 RMS error (K) 1000 4 5 0 1 2 3 4 5 RMS error (K) Figure 6.13. Rms error of temperature retrievals (K) with vertical derivatives calculated by centered differences (solid lines) and by Lanczos’ derivative method (dashed lines). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -105- 100 -90 200 -80 -70 300 -60 400 500 -50 600 700 -40 800 900 1000 -30 -20 20 30 Figure 6.14 Vertical profile of virtual temperature interpolated from gridded data for 1200 UTC, 27 March 1982 at Stephenville. Dashed line: centered difference retrieval. Thin solid line: Retrieval using Lanczos’ derivative. Thick solid line: Observed sounding profile. derivative clearly improves the retrievals, the rms error decreases by more than 0.5K at several levels for some of the experiments. Figure 6.14 shows an example of a vertical profile from wind-derived temperature fields interpolated to a station location. This is an example of one of the worst retrievals; the sounding is from 27 March 1982, 1200 UTC, at Stephenville, TX. The dashed line is interpolated from a retrieval carried out Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -106using simple centered differences. The thin solid line is the sounding interpolated from a retrieval performed using Lanczos’ derivative. The thicker solid line is the observed virtual temperature profile. For the derivative calculations using centered differences over 50 mb layers, we can expect some difficulty with separation of solutions; taking the difference of height gradient estimates at odd levels produces the temperature gradient estimates at even levels and vice versa, without any common points used in the calculations. In the centered-difference sounding this shows up in the 2 Ap ‘kinkiness’ with abrupt changes in the profile between levels. We can readily see that the Lanczos’ derivative filtering does remove the kinks that are present in the centered-difference sounding. However, there is still a substantial component with wavelength of about 200 mb (8 Ap) in the Lanczos derivative sounding. The response of the Lanczos derivative (Figure 6.12) at the frequency corresponding to 8 Ap (.125 cycles/data division) is .54142, which is almost the maximum response for any wavelength; thus the filtering of the Lanczos derivative is not as effective at removing those wavelengths as at shorter wavelengths. (Note, however, a continuous derivative has response 2nf = .785 for a wavelength of 8 Ap.) We would not want to filter out this wavelength, however, since there could be meteorologically significant variations with a wavelength of 200 mb, or shorter. To reduce noise in small-scale horizontal wavelengths, the estimates of geopotential gradients G (G = G<1 ) i + G < 2> j ) and virtual tem p e ra tu re gradients X 06 - - are smoothed with a 1-2-1 smoother applied in the x- and y- directions. The least-squares minimization (discussed below) is then applied to obtain the final fields of height and temperature that best fit the gradient estimates. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 0 7 - The discretized analogue of the minimization problem (3.5) is written in finite-difference form, with second-order centered differences to approximate horizontal derivatives, and a simple summation over all grid points for the integration over the domain. This gives the discretized form: Minimize J(<j>) i j f - <|>*j _i ) - (G.jj (6.17) Ay V where the latitudinal dependence of Ax = a cos <p A K (a = Earth’s radius, <p= latitude, ^longitude) is taken into account. A similar expression can be written using the virtual temperature gradients. minimizes j An expression for the <l> field that (<>) can be found through variational calculus, but the minimizing <f> may also be found by simply setting the derivative of j (<J>) with respect to each <J>ij equal to zero and solving the resulting set of simultaneous equations. The field of <|>ij that minimizes j (<>) (determined by either method) must satisfy (2< j> ij-<}> i+i j - - G i + 1j - G [ . j j ) Axj2 - <|>ij + i - <t>i j - 1 ) - Ay2 0 (6.18) at each interior point (i , j ) on the domain. (The interior points are defined as those with gradient estimates available on all four sides of the grid point.) Eq. (6.18) can readily be seen to be the finite-difference equivalent to the divergence equation (3.6), which was the solution of the continuous problem. Figure 6.15 illustrates the solution domain of the integration on the 25x15 grid area, for a constant pressure level in the mid-troposphere. The points labeled T are interior points; points ‘ a ’ .' b ’ .' c ’/ d ’ are edge points on the west, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -108- A 2 I I I I C D I I I I I I 1 D I I I I I I I . • . • • • • • • • • • A A A A D I I I I I I I C 2 I I I I I I I C D I I I I I I I I 1 D I I I I I I I I I C 2 I I I I I I I I I C D I I I I I I I I I I C D I I I I I I I I I I C D I I I I I I I I I I 1 D I I I I I I I I I I I D I I I I I I I I I I I C D I I I I I I I I I I I C D I I I I I I I I I I I C D I I I I I I I I I I I D I I I I I I I I I I I 4 I I I I I I I I I 3 D I I I I I I I I I C . . . B B B B B B B B B • • • • • • • • • • • • • • • • • • . . . Figure 6.15. Grid points used in dynamic retrieval. The points T ’ are interior points; ‘a ’.'b ’.'c ’/ d ’ are edge points;T, ‘2 ’, ‘3 ’, ‘4 ’ are corner points. east, south and north sides of the domain, respectively. Points T , ‘2 ’, ‘3 ’, ‘4 ’ are corner points that are, in essence, edge points that belong to more than one edge. The edge and corner points are held fixed for Dirichlet boundary conditions; for the Neumann conditions the required equations come from the minimization of (6.17). The equations for Neumann boundary conditions are: (<j)ij - 4>i+ij) - (Gij* ‘ G 'i+lj ) West edge (‘a’): ------------------ ^ 7 2 ------------------ = 0 East edge (‘b ’): ------------------ ^ 7 3 = 0 (6.19b) South edge (‘c ’): ------------------ ttz------------------- = 0 (6.19c) North edge (‘d ’): ( ^i j ■ " (Gij* ■ G 'ij +l ) ---------------------- -r ~ 2 -------------------------= 0 (6.19d) (<t»ij - (<j>ij - <|>ij + i ) (6.19a) *G ^ ) - (G ij “ G ij+ i) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -109j " ^ i+ lj) ~ (G-j’ - G 'i+|j ) + Ax-j2 ((j>ij - <j>ij +i ) - (Gj^j rG (2) i j +l ') Ay2 SW corner (‘l ’): NW corner (‘2 ’): («>i:j • ‘t’i+ ij) - (G ij 1 ■ G(i+ij ) + Axj2 ■*t*ij-l) - (G ij 1 ■ G ‘ij- i) Ay2 SE corner (‘ 3 ’): (6.19f) - (Gij -G i.ij) + Axj2 + “ (Gj.j rG (2) ij +i)t <<>ij (6-19g) Ay2 ‘ NE corner (‘4’): (6.19e) _ fG^j* - G i - i j ) ‘ + Axj2 (<t>ij " 01j - 1) ~ t G j Ay2 rG (2) i j - 1t' (6.19h) These equations are clearly the tinite-difference analogues of Eq. (3.9). Their form is dependent on the limits of the summation over i and j ; they change if the summation limits are set to include gradients along the edge of the domain. The system of equations (6.18), (6.19a-h) is solved using sequential over-relaxation (SOR) with a zero initial guess field. For Dirichlet boundary conditions, the fixed values on the edges are set equal to the “true” height or temperature field with mean value removed. For the Neumann retrievals, the mean value of the final solution fields is adjusted to match the mean value of the True” fields (which is assumed to be known). The solution surfaces in pressure coordinates intersect with the ground at the lowest levels. We assume that the surface height and temperature is known, so it is possible to have levels using both Dirichlet and Neumann R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -110- boundary conditions near the surface. This is illustrated in Figure 6.16, which shows grid points on the 850 mb level at 1200 UTC, 6 March 1982. For the purpose of this research, a gridpoint was considered to intersect with the surface if the surface pressure was less than 50 mb greater than that level’s pressure. These points, denoted by '#’ have the temperature or height value set equal to the “true" value. The 50-mb value was chosen because it seemed to work best with the available wind data (i.e. if the wind gridpoint 1/2° west of a <t> point is below the surface, it does not have a wind value that complicates the calculation of the derivatives). Use of this 50-mb value was sufficient to show how these retrievals with ‘modified’ Neumann boundary conditions (Neumann with Dirichlet at surface intersection) would compare with retrievals using only Neumann or only Dirichlet boundary conditions. • • . # # # # # # • # # • . . . # I • A I # • • # I I . . C • • # I I I • • c 1 I 2 I I I I I I I I I c c D D D I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I C C 1 D D I I I I I I I I I I I I I I I I I I I I I I C D D D I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I C C D I I I I I I I I I I I 4 I I I I I I I I I 3 D . . . I B • • I I I I I I I I C B B B B B B B B • • • • • • • • • • • • . • • • • . . Figure 6.16. Grid points at 850 mb level, 1200 UTC 6 March 1982. The points '# ’ are fixed values where surface “intersects" the ground. Other points are as in Figure 6.15. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -111- 6.2.2 Sigma Coordinates The retrievals in sigma coordinates are performed in a fashion similar to RT retrievals in pressure coordinates. There is an extra term, —^Vpg, that must be Ps dealt with in sigma coordinates. The height retrievals followed the form of (3.21), where the gradient estimate g = Va (<S> + r t v of Va<)>. The term in is calculated instead p s) RT 1 Ps -Vps was not included in the gradient approximations. Thus, the equations for calculation of g are the same as (6.13)-(6.15), except that the horizontal derivatives are taken along the constant sigma surface. These gradient approximations were used to derive a field of (<|) + r t v in p s) the same least-squares minimization described in Eq. (6.17)-(6.19). resulting field was adjusted by subtracting rtv in Figure 6.17 shows the effect of ignoring t by The p s. v' on the rms error of height retrievals. These retrievals were made using Gal-Chen’s two-scale approxima tion (6.15) and Neumann boundary conditions. Note that the largest change in rms error between the “true" tv retrieval and the retrieval using only 2 m. In fact, some levels showed slightly improved retrievals when tv tv is about was used. The calculation of approximate (virtual) temperature gradients from the momentum equations in sigma coordinates was performed a bit differently than the calculation of height gradients. The temperature retrievals are not as 3t affected by sloped terrain so it was not necessary to use ■ the terrain effect from V ctt v. Instead, R T vV ( i n p s) ln Ps t0 remove was subtracted from the gradient estimates used above prior to calculation of the derivative by Lanczos’ method. As in the height retrieval, the contribution by t v ' was not included in the retrieval, and the retrievals were not “improved” by iteration. Figure 6.18 shows the difference in rms error between retrievals performed using those using the “true” t v field to calculate the RTvV ( i n p s) tv versus term. (The two-scale Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -11 2- a) VA S 2 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 ,S> 0 5 15 20 25 0 30 5 15 20 25 30 20 25 30 RMS error (m) RMS error (m) 0 C) VAS:4 d) VAS 5 0.2 0.2 0.4 0.4 cd a 6 CD E CD * 0.6 55 0.6 0.8 0 5 10 1 15 20 25 30 0 5 RMS error (m) 10 15 RMS error (m) Figure 6.17. Rms error (m) for height retrievals in sigma coordinates (two-scale approx. equations) with “true” t v and mean t v fields. approximation and Lanczos’ derivative formulation were used.) The differences were generally less than 0.1 K except for a few levels where the differences in the RMS error of the analyses was about 0.2-0.3 K. Thus, it is unlikely that iterative adjustment of t v would have much effect on these retrievals. Naturally, once the estimates of V 0 (<J> + rtv in ps ) and V ctt v are calculated, the height and temperature fields can be derived in sigma coordinates through the same least-squares minimization Poisson equation that is used in the pressure coordinate dynamic retrieval. The same solution domain (see Figure 6.15) is used at all levels, since sigma surfaces do not intersect with the ground. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 1 3 - 0.2 0.2 0.4 0.4 CO E g> « 0.6 0.6 Mean Tv 0.8 b) V A S j3 a) V A S |2 0 1 2 3 4 o 5 1 RMS error (K) 0 2 3 4 5 RMS error (K) 0.2 0.2 0.4 0.4 co ra “ 0.6 E a> 0.6 -Q — Mean Tv - s — Mean Tv 0.8 1 0 1 2 3 RMS error (K) 4 5 0 1 2 3 4 5 RMS error (K) Figure 6.18. Rms error (K) for temperature retrievals in sigma coordinates (twoscale approximation) with “true” t v and mean t v fields. 6.3 Retrieval of Temperature from Radiance Several of the methods discussed in Chapter 4 are used to retrieve temperature profiles from the brightness temperatures calculated from soundings: statistical regression, Smith’s method, and a minimum information method. Brightness temperatures are also calculated on the grid depicted in Figure 6.1, and retrievals using Smith’s method and minimum information are performed using the gridded brightness temperatures. Additionally, brightness temperature gradient retrievals with a mean field first guess are tried using the gradient version of Smith’s method, the minimum information method and GalChen's method. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -114- 6.3.1 Retrievals from Radiances at Station Locations Previous studies of ground-based radiometer retrievals (Westwater, 1972; Decker, eta!., 1978; Westwater and Grody, 1980; Westwater, etal., 1975, 1983, 1984, 1985) were mainly regression-type retrievals using data measured at one station. A regression retrieval is performed using AVE/VAS data to provide retrievals comparable in nature to the previous studies. Oklahoma City (OKC) is a natural choice for the regression retrieval study; it is centrally located in the network so it is assured of having wind-derived soundings available for comparison; also, a 5 year set of OKC spring sounding data was available for the “climatology” to create the regression coefficients. It was necessary, however, to determine if soundings from any other stations in the network could also be used to evaluate the regression retrieval, since the 29 useful OKC soundings in AVE/VAS is too small a set to use in such a test. Since the ground-based weighting functions are extremely dependent on surface pressure and the terrain elevation, the regression coefficients developed at one station can only be used for locations with similar surface pressures and elevation as well as a similar climatology. Fortunately, Stephenville, TX (SEP) has a station elevation very close to OKC (399 m as compared to OKC’s 392 m elevation) and does have a similar climatology to OKC so is a good choice to supplement the OKC soundings. The calculation of the regression coefficients followed the steps outlined in section 4.1.1. The OKC “climatology” data is interpolated from the significant level points to even 25-mb levels to match the AVE/VAS data spacing. Brightness temperatures are calculated from the significant level data assuming the temperature varies linearly with i n p , as discussed in section 6.1.4. The mean temperatures at each level and the mean brightness temperatures for R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -115each channel are calculated and subtracted from the data. The brightness temperature covariance matrix (r r t ) is calculated and an error variance of (0.5 K)2 was added to the diagonal before taking the inverse in order to reduce the sensitivity of the solution to errors in the measured brightness temperatures. The regression coefficients are then calculated according to Eq. (4.3), and used in Eq. (4.1) with brightness temperatures to obtain temperature profiles. Figure 6.19 shows regression coefficients calculated using the four ground based channels and using ground-based and satellite channels together. Figures 6.19(a) and (b) were calculated without using the (0.5 K)2 conditioning factor. These coefficients show a sensitivity to noise that is similar to some methods demonstrated in Chapter 4; the coefficients are moderately large in magnitude and of opposing sign. The sensitivity is not as great as that demonstrated in Figures 4.1 - 4.4. The coefficients in Figure 6.19(c) and (d) were calculated using the conditioning factor, and their magnitude is reduced somewhat over the coefficients in (a) and (b). The disadvantage of adding the conditioning factor is that the retrieved soundings' detail is reduced since the coefficients are less sensitive to small changes in brightness temperature.The regression coefficients in 6.19 (c) and (d) are used to retrieve temperature profiles from brightness temperature calculated from 55 soundings taken at OKC and SEP during the AVE/VAS experiment. The same 55 soundings are used in two other retrieval methods to provide a comparison against the regression retrieval method. One of the methods used is Smith’s retrieval method implemented as described in section 4.1.2.4, using the integration weights from (6.12) normalized via Eq. (4.15) and applying retrieval equations (4.16) and (4.17). The first guess used here was the mean 5-yr sounding, and the convergence R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -116- j..................... - 200 § 200 ' 1 7 ] ...... “ p - .................. 400 s I 600 600 800 800 • Q_ ill 1».__ — *e — 5 3 .8 ^ w = -e * -a - 53.85 GHz,. -♦ - 55.45 GH l , 58.80 GHz. - - • — 55.45GHz -;*" v*”—5ff.B O jQ ’ i f ' •'"*£; • • • + • 5X754GHz v .* — 4 — 54.96 GH2r~ _z^ ' % " 0 •5 10 5 » ir . i 10 Coefficient ■d> i 200 200 400 E. 400 - ;— . | i 1 if 600 ! i r r | ' i 11 V i iT V H i v i - v > I 600 Lv? . •, ^ * • ■ ■ + • : 53.74 GHz*— A —j 54.96 GHz •2 >1 0 Coefficient 1 2 3 | i -:"."«-;-V5aB0Xit*"” 'ii'"ii!+v" '^ " 1 " " 1000 i ' •'.............................. I ■---- 9 -4 52.85GHz A d i ^ • — - e —: 53.85 GHz. V • ■ - - ♦ - 1 55.45GH* K : 800 . r 'lV 'U .1 I •3 > i-i •5 •10 Coefficient o L 9 ;> ---------- 1000 1000 •10 , I . .. . I ■1 o S "t ’ .. 1 Coefficient Figure 6.19. Regression retrieval coefficients calculated from OKC “climatology”. The coefficients are: (a) ground-based channels, no conditioning factor; (b) ground-based and satellite channels, no conditioning factor; (c) ground-based channels, (0.5K)2 variance added; (d) ground-based and satellite channels, (0.5K)2 variance added. Note the change in scale between graphs. criterion for this solution was that the sum of the square of the change of brightness temperatures between steps should be less than 0.016 K. The other method used on sounding data at stations is the “minimum information" method described in section 4.1.3.3, but using the diagonal of the covariance matrix of temperature profiles from the “climatology" data (with a different variance value at each level) rather than using a single value of oT2. If we take v to be a diagonal matrix of temperature variances at each of the 25-mb levels, the retrieval equation for this “minimum information” version is R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. T = T + VWT (WVWT+ Oe2! ) '1 (6.20) (R-W T) The error variance ore2 is assumed to be (0.5 K)2 for all channels. IMSL routines DLFCRG/DLFIRG are used to factor and solve the matrix equation. We can regard vwT(wvwT+ ae2i ) - 1 as the “coefficient” or contribution function for this method; these values calculated with surface pressure = 1000 mb are shown in Figure 6.20. The curves are similar in shape to the weighting functions, modified by differing v at each level. The changing variances at different levels produce the zigzag pattern; the mandatory reporting levels in the 5-yr data generally had different variances than the other levels. Note that these coefficients do not exhibit as much error sensitivity as the regression retrievals. On the other hand, these retrievals do not use covariances between levels; for the retrievals such as the ground-based only case (Figure 6.20a) there is little change from the first guess field above 600 mb. Additional retrievals are performed using brightness temperatures with added random errors (Gaussian distribution, zero mean, 0.5 K standard deviation) to assess the sensitivity of each of these methods to errors in the b) 200 200 & 52.85 GHz -a - 53.85 GHz * • - - - 5 * * 5 -GHz 58.80 GHz Z3 1 600 Q. 800 800 52*5 GHz 53.55 GHz 55.45. GHz.......... | 56.80 GHz K ill if T J) ' • ••+ • 53.74 GHz /{ * , - 6 - 54.96GHz vil-"*. 1000 1000 -6 •4 •2 0 2 4 Coefficient Figure 6.20. Coefficients vwT(wvwT+ cte2! ) - 1 for minimum variance retrieval based on OKC 5-year “climatology", (a) Ground-based channels; (b) Ground-based and satellite channels. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -118brightness temperature. Each sounding is used 20 times with different random errors, which gives over 1000 data points per level above the surface. 6.3.2 Retrievals from Radiances at Gridpoints Retrievals using scalar brightness temperatures at gridpoints are performed using Smith’s method and the “minimum information” method in a similar fashion as the retrievals at station locations. These calculations are made in pressure coordinates. The radiance values for these tests are calculated from gridded values of virtual temperature. The weighting functions used are the integration weights defined in Eqs. (6.11) and (6.12), with a slight modification (which is discussed in the next section). The first guess profile is the mean virtual temperature profile for the area. For the Smith retrievals, the surface and space contributions are removed from the brightness temperatures. The weighting functions are then normalized (Eq. (4.15)), and the brightness temperatures scaled by the same factor. As with the retrievals at station locations, the convergence criterion for the solution is that the sum of the square of the change of brightness temperatures between steps should be less than 0.016 K2. The variances v used in the “minimum information” retrievals are variance of temperature on each level, averaged over the 8 time periods of each observation day. The values at the surface and the space contribution are specified, and not adjusted in the retrieval. Also, the first guess mean profile and an estimate of the variance of the temperature field about the mean are regarded as given; in practice they might be obtained from prior sounding observations or from a forecast model. As with the retrievals at station locations, additional retrievals are performed to assess the sensitivity of the methods to error in the brightness Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -119temperature. The error is not added as random errors at gridpoints, however. Rather, an error field is generated by interpolating random error “observations” from gridpoint locations in the same fashion as the temperature fields are interpolated. The retrievals with added random error are performed twice for each observation time; these also provide more than 2000 data points at each level above the surface levels for each experiment. 6.3.3 Retrievals from Radiance Gradients at G ridpoints Retrievals from brightness temperature gradients are performed using Smith’s method, the “minimum information” method and the two variations of Gal-Chen’s method that are discussed in section 4.2. The finite-difference approximations of the horizontal gradients are second-order centered differences. The first-guess fields are the same as used in the scalar retrievals, i.e. the mean virtual temperature on each of the analysis levels. (This gives a zero first-guess gradient field.) The radiance values and weighting functions used are the same as for the scalar retrievals. The sensitivity tests for each method are performed in the same way as in the scalar retrievals. 6.3.3.1 Calculation of hv. The calculation of h v = [ wv ( z * ) V ht d z * Z S* from the brightness temperature gradient measurements in pressure coordinates is carried out in a slightly different fashion than indicated in Eq. (4.34), in order to make the formulation compatible with the centered difference scheme. It is necessary to find a relationship between the finite difference form of h v and the finite difference form of V ht bv. and this requires a slight modification of the weighting function formulation. We want to separate the terms of (4.34) that result from terrain effects into terms that can be calculated from “known” quantities. For levels above the surface, the integration weighting Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -120function given by (6 .12 ) can be expressed in terms of a weighting function calculated from a reference level, divided by the transmission function from the reference level to the surface. We can substitute (6.10) into (6.12) to get: wv„ = ijk fr z k* k r z2k k+ +l * J2k_i*?v(zs * ' z *> d z * ” Jzk* ^vfZs*, z*) dz*. A Z j* f zk * _ f zk + l * _ Jzk_1*Tv f - 0 . 5 , z * ) d z * - Jzk. T v ( 0 , z g*) Xv ( - 0 . 5 , z * ) dz< Azi* W^°)v (6 .21 ) TV ( 0 , Z g * ) where w^0)v is the integration weight for frequency v at level k for a radiometer located at the base level of radiance calculations, z s* = -0 .5 km. Thus, in pressure coordinates, above the surface, the vertical variation of the weights can be separated from the horizontal variation. (This will not work on a sigma surface as the values of z k* = in h (pk/ p 0) change in the horizontal on a sigma surface.) We define d^ = ( t v (o, z s * ) ) - 1 for convenience. What about at level l , the first coordinate level above the surface? For that level, the weighting function is written: f ZL*_ f zL*l*_ J, .'Cv(Zg*,z*) dz* - J Tv ( z s * , z * ) dz* -------------------------- • WI jL = (6.22) This cannot be separated into horizontal and vertical factors as the other weights are, because of the first integral’s dependence on z s* (x , y ). We want to remove this horizontal dependence from the weight so it can be handled in the same fashion as the other levels. Recall that, as in the retrievals from the wind, we do not begin calculation of the temperature fields until 50 mb above the surface. This means that there is at least one coordinate level between the beginning level l and the surface. Suppose that we take the surface temperature to represent a mean value in the (thin) layer between the R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -121surface and level l - i , and let the temperature vary linearly with i n p from that point onward. In that case (6.11) is rewritten: s ^ a rtz * )! T bv = T ( z s *) + X "~"3z* dz* rZk+i* I •'zjt* L* L xv ( z s * , z * ) d z * k= L -l /T’ (( Zz lL **)) -- TT(( Zz g3** )) \n fZ |-zlL ** = T ( z s *) + I---------- 1— ---------[ I A zs * + , !?v(Zs*,z*) d z * M -l $=.YT(Zk+i* ) - T ( Zk* ) A f zk+l* ------------JJzk* ^v(zs*.z*) dz* (6.23) k=L and the weight at level l becomes: f ZL* „ f zL+l*_ .Xv(Zs*, z * ) d z * - J xv ( z s *, z * ) d z * J wIjL " — ------------------------------- = dL < ° )V- We may easily verify this expression, which ^ ^ I j k = ^ k °>VV„<^ij = ^k°)Vd^j (6.24) is analogous to (4.32): a v ( z s * ) V Hz s * = a v ( z 3* ) V „ z 3*w^jk (6.25) but it is more useful in a finite difference form {e.g., for x direction): _ _V _ _V ijk - v i-ljk Ax, - - V d i j ‘ d i-lj wvijk ----"= "vijk d . .Axj ID (6.26) J Define TBVijk s X . ” ^ - ^ i i k as the brightness temperature with the surface and space contribution removed. Suppose that, at two points ( i , j ) and ( i - i , j ) , the brightness temperature summation begins at the same lowest level l . The x-component of hv between these two points is: M -l v y V V i-lD k 1 1' 1/2j = £ - y i 1 Wi j k T iP k 2u k=L T iT k -T j-j-ik 2 Wi-ijk T i-ljk Axj ^ ^ - 2mi Wj j k Wi- l j k Axj T ijk + T i 2 k= L Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -122m T s v ijk m Ax-j ~ ^ B v i-ljk ^^ v i - l i ^MT_1 X * M~1 v V v 1 Q v i 1~ Q v i - l j dvijAxj 2LiWijkTi3k*2 dvi-ijAxj Z^WijkTi3k 1 C ivil '2 k= L T B V ijk ^ B V l-ljk Axj 1 d y jj-d y j-l-j "2 dv ijAxj k=L /V 1 d y jj-d y j-lj Tevijk“ 2 dvi-ijAx-j /\ TBvi-ijk • (6.25) A similar formulation can be derived for the y direction. Note that the hxv and hyv values are defined at gridpoints displaced by V 2 grid distance from (i , j ). The other term in (4.34), w ( v , z g* ) T ( z s *) V Hz g\ is taken into account in part by the surface terms, through the variation of the surface level’s weighting function. Another place where the terrain effects enter is in the situation where the lowest coordinate level of the integration changes between two gridpoints. If two adjacent points have different starting levels l , the lower starting level point is designated as a “Dirichlet” point such as is used in the wind-only retrieval, and is assumed known (or at least, able to be approximated rather than being adjusted by the scheme). The gradient retrievals are also performed in sigma coordinates. The motivation for this is to avoid dealing with coordinate surfaces intersecting with the ground. For the sigma coordinate retrievals, the hv field is calculated using the temperature first guess field as well as the radiance field, according to (4.36). In discretized form the x component of hv is : , v h x i - 1 / 2j TBi j k = Tsi-ljk A xj 3 ^ ~ f * k= L Wijk Wi - l j k 3 T ijk + T i-i-jk 2 (6 > 26) where, as in the pressure coordinate retrievals, the contribution of the surface and transmission from space have been removed. 6.3.3.2 Smith’s method and Minimum inform ation m ethod. The retrievals of temperature gradients using Smith’s method and the “minimum information" method are carried out in the same fashion as the scalar retrievals Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -123of temperature. The gradient variances on the grid, averaged over the 8 time periods of each experiment, are used in the variance matrix v for the "minimum information” retrieval. (It is assumed that such estimates of variance of temperature gradient could be estimated from “climatology", or taken from a previous analysis or forecast field.) Once the temperature gradient estimates are obtained, a 2 -D Poisson equation of the same form as used in the wind retrievals is solved to obtain the final temperature fields at each level. The x- and y- temperature gradient estimates take the place of G ^ /2j and G {^/2j used in the wind retrievals. 6.3.3.3 Gal-Chen’s m ethod. For Gal-Chen's method, the solution equations are found by minimizing the finite-difference form of the functional j (which was defined in (4.51)): M (l^>ijk ~ T i-ij)c) ~ ( T i j k - T i - i j k) j = X X Pok Axj L .i k= L M +X X Pok i , j k= L V N m + X XPv i.jv = l (r^ijk~1>Lj-lk) ~ ( T ijk Ay <, V Wi j k + W ._ l j k 2 ^ k= L T i - j k - T i - 1 -ik V Axj * h x i - V 2j ~ J V V Wijk+Wij.ik T iik -T ii_ ik 2, 2 A y '1 m - X X M i, j v=l T jj-ik ) v (6.27) k=L Note that the surface contributions have again been removed, so that (for retrieval in pressure coordinates) some of the terms dealing with terrain effects will not appear. At each location where a horizontal derivative is calculated, the lowest level l of the summation/integration is taken to be the higher of the bottom levels of the two points involved in the derivative; if the two points’ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -124radiance integration does not begin at the same level, then the extra level in the calculation is a ‘Dirichlet’ level and must be estimated by other means. The functional j expressed in the sigma coordinate system has the same form as (6.27), except that vertical summations always begin at the first level above the surface, instead of beginning at different levels l depending on the surface pressure. When j in (6.27) is minimized by taking the derivatives with respect to Tijlc and setting them all equal to zero, a set of (M-L+1) simultaneous equations is generated for each point ( i , j , k) in the interior. These equations are of the form: ii+ ik j Axj2 X' V WV.,+WV ijk i+ ljk N + A y2 p Uv 2 A x;j v=l A x■j ,m=L wv.. +wv , N V ft ijk M i-ijk 2A WV . +W V , . ijm l-ljm „ m T jim -T j-i-ir Axi xh ,m=L v= 1 N V Wl.j m. +Wl.+ l,j m. Tmj + i - i m - Tmi - i m M WV. ■ +WV. . 2■A yl j J Ay ,m=L V=1 V V W +W N 2A y Ay ,m=L V=1 P Ok T T jjk + T j+ ijk i- ljk ~ 2 A x j2 V -X P X T I J A. v ij- ik ~ 2 2 A x j . jjk + W...+W. ijk i-ijk v V h x i + 1/ 2j T T ij jj + l k j A y2 V wi j ..+w , k i+ ljk N T + J.JJI ' j - l j l i 2 A x j . v V h x i - 1 / 2j v=l tv X p. V V V V w;..+w . .. ijk lj+ lk w:..+w:. ijk ij - l k 2 Ay 2Ay v ^ ij-V z (6.28) v=l R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 2 5 - The set of equations comes directly from taking derivatives of (6.27); there is no need to perform any “summation by parts" analogous to the integration by parts in the continuous case. The Neumann boundary conditions are also derived from the minimization of j in (6.27). With the Neumann boundary conditions, if ( i , j ) is, e.g., a point on the western edge of the domain the temperature profile at (i , j ) must satisfy: .V rm r *M m T i+ lik Axj 2 W W .. ,, -NN 1 T j-jk Y R J + 2 , Pv ..V +W.13k Axj .. +V 1 + 1 : ik 2 v= 1 _ ■ Po4 (" T i +i j k to ? T ijk ^ f M i m=L . Tv . _v . . +W . . m i+ljm ijm W. 2 Wi + l j k + W i j k m Ax-i v _ + 1 P » — 2 ^ ] ----- h x i .V !j _ Q. (6.29) V—1 and similar expressions hold for the other edges of the domain and for the corners. This method for solving this system of equations is similar to the one outlined in Gal-Chen (1988) for the continuous case. In the case of Neumann boundary conditions, the temperature gradients on the edges are solved first. At the corners, the gradients in the x- and y- directions are solved separately. Thus, each of the simultaneous equations to be solved on the edges has a form similar to (6.29). The solution at an edge point is performed by first dividing each equation by p0k. multiplying by the weighting functions for each of the N channels and summing over the M levels. This yields a set of simultaneous linear equations that can easily be solved for £ w V t •n. For a point on the western edge, these N simultaneous equations generated have the form (where 8vy is the Kronecker delta): R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -126wY , +wY i+ ljk M k=L M X k=L NR Y»_Pv_ L Ax-j ’ilk j wv . +wv .. wT ,.,+wT.. l+ ljk 13k l+ ljk 13k 2 2 f^M ^ v = 1 P0k wV - 1 V k=L l t Ok M i+ ljm S i+ l]k 13k T i +ljk 2 A + +w w '’ ijm T i + lim - T i i m Axj2 2 m=L y r TV m w ! ,, 2 ............ , W Y . . . + W Y .. Axj2 ■V V =1 T j+ i-jm T j-jm 2 m=L NRv Wvi + ,l j k +wv., WY , +WY. ijk i+ ljk ijk n . +WY. ijm i+ ljm y T ijk Axj 2 k=L m 'V 1 n r w .,.. V '1 Pv l+ lik 2 * " R n. k=l v=r +w :.„ wY , 13k i+ ljk 2Axj +w:., ijk 2 v (6.31) h3Ci+1/ 2j ‘ The quantities on the right hand side are known, thus the solution for £ M WY . +WY.mTm m “I y-. i+ljm Ti+lim-Tiim| ^ 2 L Ax-)2 J - ^ 13 5vy+ k=L m=L w ,,,+ w i+ ljk NR % wVt -n: WV +WV WY +WY i+ ljk ijk i+ ljk ijk - X v=l Pok \ ' T 13k i+ ljk " T ijk v 'P v ^ i+ ljk + ^ ijk + -2pt 0k- 2Ax-i v hx.i+ 1/,2O. (6.32) V is calculated by means of the IMSL linear system solution routines DLFCRG and DLFIRG. This solution is then substituted into (6.29), which is solved to get the temperature gradient needed for the boundary conditions. Once the boundary conditions for T ijk are determined the set of simultaneous equations for the interior points can be solved iteratively. The set of equations (6.28) is broken up into finite-difference analogues of mt and nt (which were defined in (4.57) and (4.58)): -l~ik~2 Ti-jk+Tj+i-jk Tj-j_ik-2 Ti i k + T j i + i k l MT Axj2 Ay/2 2 J + XPv w*j.k v=l X wVm=L /T l-ijk ~ 2 Tj jk+Tl+iik Ti-j-ik-2Ti-jk+Ti-i4.ik A x j2 Ay2 F v (6.32) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. It is easy to see that these equations (6.32)-(6.33) are the finite-difference analogues of (4.57)-(4.58); however these equations are not in the form that a finite-difference representation of (4.57)-(4.58) would usually take, if some kind of centered-difference approximation were assumed. This shows the importance of performing the functional minimization on the discrete forms as well as the continuous forms of the equations. The minimization of the continuous form yields expressions that can be used as guidelines for the design of the experiment; the finite-difference minimization provides expressions that are consistent with the form of the functional being minimized. The iterative solution method proceeds as follows: on the first iteration, the first guess field is used to provide the values for T ijk in nt; otherwise the previous step’s temperature field is used. The terms on the right-hand side (/) of (6.28) are then added to nt. This set of equations mt = nt +/ is then solved for ] T wV 2t using the same methods as used in the edge solutions; the equations are multiplied by wj and summed in the vertical, and the resulting set of N simultaneous equations is solved using the IMSL routines. (The factorization of this system is performed only once at each point, and then used R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -128at each iteration with a new right-hand side vector.) When the field of ]T wV2t is obtained, it is substituted back into (6.28), which is then solved for the temperature field using the same 2-D Poisson solver as was used for the retrievals from wind data. The boundary conditions are the fixed Dirichlet conditions or the Neumann gradient conditions that were previously obtained. When using the Neumann boundary conditions, special care is taken to insure that the mean value of the temperature field at each coordinate level is held fixed between iterations. If the new estimate of the temperature field differs from the previous guess by less than 0.1 K at each gridpoint the iteration process is stopped, otherwise the new field is used to calculate a new estimate of nt . The retrievals in sigma coordinates required less than three iterations through this method to converge to a solution; the retrievals in pressure coordinates required more iterations. To speed the convergence for pressure coordinate retrievals, an under-relaxation factor of 0.6 was used (i.e., at iteration n, the new temperature field is calculated as 0.4 t 11- 1 + 0.6 Tn). The same solution procedure is used for the version of this method presented by Sienkiewicz and Gal-Chen (1988,1989). It is slightly easier to write the finite-difference analogues of (4.62)-(4.68) since the average value of the weighting function between grid points does not enter into the formulation. The functional j to be minimized is written: 2 J ~ X X Pok Axj i,j k (T jjk + X X Po* T jj-lk ) ( T ijk T ij-lk ) Ay i, j k R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -129- I S i pV k v ,3 Axh <3kTi3 ^ X, r 2* X - i* TiJ--iic • ( t »«- tb« - i ) ♦ S lP v X i, j v ^ V (6.34) J Ay and the T ijk field that minimizes (6.34) must satisfy, for each interior point ( i , j ): T ii +lk~2 Ti-j)c+Ti-i-i)c*| i+ lik S PoKp1 Axj 2 ,v X X P v w'ijk Ay2 J y fX y Wv . . T. , . -22 /> j Wv . T T.13m . . + /> / V Wv , . T. , . n-l3m l+lnm 13m 1 3 m ljm i - l 3 m i - l3 m Axj2 V \ / A + SEPv i. j = X P°k 7 wv. . tT 1 ..3+lm , - 2 7 wv, T .. + > WT. . T.13. -lm . 13+lH 13-110 ,v wijk Ay2__ _V v T i +l jk~2 T jjk+ T j-ijk __ A xj2 Ay2 V XXPv w* y T i j+i k~2 T j j k+ T jj-ik^ ~ V V Tbj j . 1” 2 TBj_.+Tb. j jk Axj2 + Ay2 A (6.35) Th e N eum ann boundary conditions are also determ ined from the minimization of (6.34), but are not in a form to be so readily solved as those from the G al-Chen (1988) problem. The solution can be facilitated by substitution of the expressions for h v as derived above, and then the boundary conditions have the sam e form as (6.29) above and can be solved in the sam e fashion. Eq. (6.35) can be separated into two parts as before: MT - P *p “ ^XPv wjijk - lik ^Tj-jk+Tj+i-jk T i i - i k-2 T iik + T ii +lk] /2 Axj 2 Ay2 ■lik~^Tiik+1’i+i-|)t Axj2 J Tj-j-i)c- 2 Ti-ji<+Ti-i +i)c> Ay2 (6.36) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -130- i .. jj AXj2 v J ’) (6.37) and the solution method is the same as for the Gal-Chen (1988) method described above. 6,4 .Combined Wind and Radiance Retrievals The choice of combined wind and radiance retrieval methods is guided by the aim of this research to determine whether retrievals using a combination of wind and radiance data can outperform retrievals using only wind data or only radiance data. We take the temperature fields retrieved from wind as described in section 6.2 and use them as first-guess fields in radiance retrievals similar to those described in section 6.3. In this way we can see if the wind derived fields are improved by adding radiance data. We also can compare the temperature fields retrieved from radiance using wind-derived temperatures as first guess and the retrievals where only a first guess mean sounding is used. The combined wind and radiance retrievals are performed in the same fashion as the retrievals at gridpoints where the radiances alone were used. Retrievals with wind-derived first guess temperature fields using Smith’s method and minimum information are performed on the gridded brightness temperatures. Additionally, brightness temperature gradient retrievals with a wind-derived first guess are done using the gradient version of Smith’s method, the minimum information method and Gal-Chen’s method. In each case, a test Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -131of sensitivity to random error is carried out in the same fashion as for the radiance-only retrievals. The weighting factor Po used in Gal-Chen’s method and the weighting diagonal matrix v used for the “minimum information” method are chosen based on the expected value of the error of the first guess field so that, for example, 1/po = e [ ( V ht ( z * ) - V HT f ( z *) ) 2] . If these methods were to be applied operationally we would not have a ‘true’ temperature field to compare the wind retrievals against and so we would have to estimate the error of the first guess field. Figure 6.21 shows plots of e [ ( V ht ( z * ) - V HT f ( z * ) ) 2] averaged over the 8 observation times for each experiment day. We can see that the error curves are somewhat similar in shape, with the largest values near the surface. In addition, the magnitudes of the errors when compared between the experiments were proportional to the mean squared value of the temperature gradient field. The mean squared value of the temperature gradient field could be estimated from a previous analysis, forecast model, or even perhaps the wind-derived analyses themselves. Thus, it seemed reasonable to divide the mean squared error values for each experiment day by the mean squared temperature gradient, take the average of these “normalized errors" and find a best-fit curve to represent the “structure" of the gradient error of the wind-derived retrievals. The curve fitted to the averaged ‘normalized’ errors is of the form e2= a + b e x p (C p ). This best-fit curve is then multiplied by the mean squared temperature gradient for each experiment, and used to estimate 1/Po. The results of this curve fit are also depicted in Figure 6.21. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 3 2 - — Mearjsq.error ! 200 200 E 400 E 400 600 600 800 800 a ) VAS 2 1000 0 1 1000 2 3 4 5 0 1/3, !-Q — Mearjsq. error 2 i/R ! ... 200 1 E 400 600 600 800 800 rnnrrr.. c j VAS 4 o 1 2 3 4 5 Fstimate— d)i V A S 5 1000 1'P, 4 -9 — Meari sq. error • 200 E 400 1000 3 5 0 1/P, Figure 6.21. Mean squared temperature gradient error ((K/grid division)2) for the four VAS experiment days, and the curve-fit estimate for 1/Po used in the gradient temperature combined retrievals. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 7 DISCUSSION OF RESULTS ZJ Synoptic Overview Before discussing the results of the various retrieval methods that were presented in previous chapters, we need to first review the weather conditions that were present during each of the different AVE/VAS days. This review of the conditions that prevailed during each of the experiment periods will aid in the understanding of the results. 7.1.1 AVE/VAS II: 6-7 March 1982 The 6-7 March case (AVE-VAS II, or VAS 2 ) is the best-known of the four regular AVE/VAS experiment days, and has been investigated in a number of studies e.g. Jedlovec, 1985; Fuelberg and Meyer, 1985; Chesters, et al., 1988; Doyle and Warner, 1988). Figures 7.1 and 7.2 show surface analyses for 1200 UTC on the 6th and 7th of March, respectively. At the beginning of the experiment (Figure 7.1) the major surface features are a cold front (marked as stationary at that time) in the Gulf of Mexico, just off the coast of Texas. The cold air behind this front had spread throughout the experiment area. However, a secondary front with an even colder high pressure area behind it, which at this time extended from Montana to the northern part of Nebraska to the Great Lakes, was starting to move into the northern part of the experiment area. -133- R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -134- \024 )fn 6 10081 Y008 R iki 6 March 1982 1200 UTC Surface Figure 7.1 % » I *''■ ^ 7^ Surface map for 1200 UTC, 6 March 1982. Isobars are at 8 mb intervals. (After NOAA, 1982.) & (7 M arch 1)982 1 2 0 0 UT S u rfa c e Figure 7.2 Surface map for 1200 UTC, 7 March 1982. Isobars are at 8 mb intervals. (After NOAA, 1982.) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 3 5 - By 1200 UTC on 7 March (Figure 7.2) the first front had swept across the Gulf of Mexico, while the secondary front moved in behind it, reaching the Texas-Oklahoma border, and extending up to Ohio. The 500 mb analyses for 1200 UTC on the 6th and 7th are depicted in Figures 7.3 and 7.4. The upper level flow was dominated by a large trough that extended across the country, from the west to the east coast. At 1200 UTC on the 6th, the major axis of this trough extended from a low in Hudson’s Bay down through the central United States into Texas. There was also a short wave trough with an associated cold pool moving through the system; at 1200 UTC the axis of the short-wave trough extended from Iowa into Kansas, then across into New Mexico. This is clearly depicted in Figure 7.3. The short wave moved through the base of the trough and had reached the eastern edge of the VAS experiment area by 1200 UTC on the 7th. (The wave in the height and temperature pattern depicted in Figure 7.4 is the location of the main trough.) The major features to note in the VAS 2 analyses are the strong baroclinic zone in the eastern part of the region (associated with the surface front and the eastern side of the upper level trough) and the smaller scale features such as short-wave trough and the associated cold pool that move through the region. 7.1.2 AVE/VAS III: 27-28 March 1982 The second regular experiment day in the AVE/VAS experiment was AVE/VAS III (VAS 3), which was held on 27-28 March 1982. Surface analyses for 1200 UTC on the 27th and the 28th are depicted in Figures 7.5 and 7.6 respectively. On 26 March, the surface map is dominated by the large high pressure area extending from north of the Great Lakes down through the Mississippi valley. There is also some indication of a weak lee trough on the R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 M arch 1 9 8 2 1200 U TC 50 0 m b r Figure 7.3 500 mb analysis for 1200 UTC, 6 March 1982. Height contours (solid) at 60 m intervals, temperature contours (dashed) at 5 K intervals, winds in knots (full barb = 5 ms-1, flag = 25 ms-1). \ \ 7 M a rch 19 82 1200 U TC 500 mb Figure 7.4 500 mb analysis for 1200 UTC, 7 March 1982. Height contours (solid) at 60 m intervals, temperature contours (dashed) at 5 K intervals, winds in knots (full barb = 5 ms-1, flag = 25 ms-1). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -137- 0 2 4 0 1 6 OOfOl 008 27 March 1982 1200 UTC .Surface 016) Figure 7.5 Surface map for 1200 UTC, 27 March 1982. Isobars are at 8 mb intervals. (After NOAA, 1982.) v 008 * 0 1 6 016V 00 28 March 1982 1200 UTC Surfa Figure 7.6 Surface map for 1200 UTC, 28 March 1982. Isobars are at 8 mb intervals. (After NOAA, 1982.) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -138eastern side of the Rockies. By 1200 UTC on 28 March, the high pressure system has moved southeastward and is centered in Ohio. The 500 mb analysis at 1200 UTC on 27 March shows an upper level ridge located over the analysis area, with a small cut-off low centered in the southeast corner of Colorado (Figure 7.7). A large trough is located to the east of the VAS analysis area; the western edge of this trough is over Illinois. As the day progresses the cut-off low moves slowly eastward (through the large-scale ridge) and weakens. At 1200 UTC on 28 March (Figure 7.8) the low appears as a weak wave in an almost zonal flow. 7.1.3 AVE/VAS IV: 24-25 April 1982 Figures 7.9 and 7.10 show surface conditions for 1200 UTC on 24 and 25 April, the first and last observation times for AVE/VAS IV (VAS 4). At 1200 UTC on the 24th (Figure 7.9), the main weather feature in the VAS experiment area is the trough developing in the lee of the Rockies. A warm/stationary front stretches across the Gulf. There is also a cold front stretching down from a low over Manitoba across the western states of Montana, Idaho, and Nevada; this front moves south during VAS 4 but only reaches the northernmost part of the VAS analysis area by the end of the experiment. At 1200 UTC on the 25th (Figure 7.10), the surface low that had been in Colorado has moved down into the Oklahoma panhandle, and a trough line is indicated through western Texas. A wave has formed on the front in the Gulf of Mexico, with a low pressure center in Mississippi, just to the east of the VAS analysis area. Figures 7.11 and 7.12 show the 500 mb analyses for 1200 UTC on the 24th and 25th, respectively. The major feature is a cut-off low that moves northeastward from the Colorado-New Mexico border (at 1200 UTC on the 24th) to western Kansas (at 1200 UTC on the 25th) and deepens. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 7 M a rc h 1 9 8 2 1200 U TC 500 mb Figure 7.7 500 mb analysis for 1200 UTC, 27 March 1982. Height contours (solid) at 60 m intervals, temperature contours (dashed) at 5 K intervals, winds in knots (full barb = 5 ms*1, flag = 25 ms*1). M a rch Figure 7.8 500 mb analysis for 1200 UTC, 28 March 1982. Height contours (solid) at 60 m intervals, temperature contours (dashed) at 5 K intervals, winds in knots (full barb = 5 ms'1, flag = 25 ms-1). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -140- 24 April 1982 1200 UTC Surface . Figure 7.9 Surface map for 1200 UTC, 24 April 1982. Isobars are at 8 mb intervals. (After NOAA, 1982.) 25 April 198! 1200 UTC Surface Figure 7.10 Surface map for 1200 UTC, 25 April 1982. Isobars are at 8 mb intervals. (After NOAA, 1982.) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -141- -20 570 576 2 4 April 1 9 8 2 1200 U TC 500 mb Figure 7.11 500 mb analysis for 1200 UTC, 24 April 1982. Height contours (solid) at 60 m intervals, temperature contours (dashed) at 5 K intervals, winds in knots (full barb = 5 ms-1, flag =25 ms-1). 2 5 April 1 9 8 2 1200 UTC 500 mb Figure 7.12 500 mb analysis for 1200 UTC, 25 April 1982. Height contours (solid) at 60 m intervals, temperature contours (dashed) at 5 K intervals, winds in knots (full barb = 5 ms*1, flag = 25 ms-1). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -142- 7.1.4 AVE/VAS V: 1-2 May 1982 The last experiment day was AVE/VAS V (VAS 5) on 1-2 May 1982. Figures 7.13 and 7.14 depict surface features at 1200 UTC on each day. The dominant surface feature in the VAS experiment area was a large area of high pressure, centered in Iowa at 1200 UTC on 1 May. This high pressure center moved southeastward and weakened by 1200 UTC on 2 May. The pressure gradients were quite weak over the VAS area and the surface winds were generally light and variable. Figures 7.15 and 7.16 show the 500 mb analyses for 1200 UTC on the 1st and 2nd. A high pressure ridge stretched across the area at 1200 UTC on the 1st (Figure 7.15). This ridge built toward the northwest and strengthened slightly by 1200 UTC on 2 May (Figure 7.16). As at the surface, the gradients of height and temperature are weak, and the winds are light. Note that the 500 mb height contour lines in these analyses are drawn at 30 m intervals, while those for the other experiments were drawn with contour intervals of 60 m. In summary, then, the four VAS experiment days provide a few different weather situations in which to test the retrieval schemes. VAS 2 is a winter-type case characterized by strong height and temperature gradients. VAS 3 and VAS 4 are spring cases that do not have as strong gradients as VAS 2 but include small scale features that move and develop through the periods. VAS 5 is more like a summer case where the gradients are weak and the wind flows are not well defined. Although none of these cases have the strong convective activity and storm-scale circulations that are found in cases such as SESAME ‘79, this means that there is less of a problem dealing with unresolved small scale motions when evaluating the results of the retrievals. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -143- 1 May 1982 1200 UTC Surface Figure 7.13 Surface map for 1200 UTC, 1 May 1982. Isobars are at 8 mb intervals. (After NOAA, 1982.) 2 May 1982 1200 UTC Surface Figure 7.14 Surface map for 1200 UTC, 2 May 1982. Isobars are at 8 mb intervals. (After NOAA, 1982.) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -14457: -15 >76 579 J 1 May 1982 1200 UTC 500 mb Figure 7.15 500 mb analysis for 1200 UTC, 1 May 1982. Height contours (solid) at 30 m intervals, temperature contours (dashed) at 5 K intervals, winds in knots (full barb = 5 ms*1, flag = 25 ms-1). -15 B76 576 2 May 1982 1200 UTC 500 mb Figure 7.16 500 mb analysis for 1200 UTC, 2 May 1982. Height contours (solid) at 30 m intervals, temperature contours (dashed) at 5 K intervals, winds in knots (full barb = 5 ms-1, flag = 25 ms*1). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -145- 7.2 Assessment of Retrieval Quality The next topic to consider is the assessment of the quality of the retrieved fields. How can we determine whether a retrieval is "good”, and how can we compare the fields that are derived using different retrieval methods? One way would be to look at retrievals and somehow visually compare them to decide which fields and/or profiles are the “best”. This method is unsatisfactory for several reasons: (a) it is subjective, it depends too much on our opinion of how the retrieved field should look, (b) it is not quantitative, it does not necessarily assign a value to each of the retrievals so it is difficult to assess the “goodness" of several different retrievals except by direct comparison of the different fields, (c) it is not practical to present every field and every profile as there are thousands of fields generated from the dynamic retrieval methods alone, let alone the different radiance retrieval methods. However, this is not to say that it is not useful to look at and compare the analyses produced by different methods; thus, there will be some examples of each retrieval method presented that can serve to demonstrate how typical retrieved fields and profiles look. There are some objective measures of the quality of the retrieved fields that can be used. We can calculate the error of a retrieval by subtracting the “true” value of the field or profile from the retrieved value. The average error of a set of retrievals is the bias, which tells us if a retrieval method has a tendency to produce retrieved fields that are too high or too low. The root-mean-square (rms) error is the square root of the average of the square of the error (where the error has had the bias removed). These are measures commonly used to determine retrieval quality. In the case of the dynamic retrievals from wind data, each retrieval R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -146produces a two-dimensional field of grid points either on a constant pressure or sigma level. If the retrieval is made using Neumann boundary conditions, then we obtain deviations from the mean at the level and so we cannot find a bias for that retrieval (or rather, the bias of the retrieval would be the amount that our estimate of the average field deviated from the true average). For these retrievals, the rms difference between the retrieved field and the ‘true’ field for all the points on a single level at a single time can be used as a measure of the error of the retrieval. These rms values can be averaged over the eight observation times to get a final measure of the quality of the retrieval for each experiment day. Since the wind-derived temperature and height fields are calculated from gradient fields, we would also like to evaluate the height and temperature gradients. One measure of skill that is commonly used in the verification of numerical forecasts is the s 1 score (Teweles and Wobus (1954)). This score was originally used in verification of surface pressure forecasts and was originally defined in terms of differences between the analyses at pairs of station locations: Si - 100ffr ■ where eG (7-1) is the error in the forecast pressure difference and gl is either the observed or forecast pressure difference between stations, whichever is largest. In this research, the definition is modified so that e G is the error of the difference between height or temperature values at two adjacent grid points in a constant level analysis and gl is either the retrieved or ‘true’ difference in the fields between the two grid points, whichever is largest. (This is consistent with more recent application of the s 1 score for numerical weather prediction.) The s x score is graded in the following fashion: forecasts with s x values R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -147- of 2 0 are considered near perfect (since this corresponds to the difference in subjective analyses from two human analysts); those with s 1 values of 70 are considered worthless. The other retrievals performed on the grid can also have the rms error and s x score determined in the same fashion, so that these retrievals can be directly compared with the wind-derived temperature retrievals. We should note that both the rms error and s 1 score tend to be case dependent. The rms errors tend to be larger for cases when the variance of the retrieved field is large. The s x score shows the opposite effect, the values are smaller when the gradients are large. Bearing this in mind, when comparing the retrieval results we need to make the comparisons on a case-by-case basis if possible. What about the methods where individual soundings at stations are retrieved? Consider the regression retrievals at OKC and SEP, which are performed for comparison with microwave retrieval studies such as the one by Westwater, e ta i, (1985). To do the comparison, we need to calculate the bias and rms error for these soundings. A single experiment day has less than 16 soundings (because some soundings were missed or were terminated at too low an altitude to be used) and so may not represent the variety of weather types in the sounding sets used in the previous research. Thus, it seems preferable to use soundings from all the experiment days to calculate the statistics for this method. Unlike the other studies of retrievals from ground-based measurements, there are simultaneous ‘soundings’ from two stations available in this study. This offers an opportunity to study how well the ground-based channels can determine temperature gradients when a regression method is used for retrieval. The retrieval coefficients and first guess profile used are the same for R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -148- both stations, thus the difference between the soundings is solely due to the difference in the ‘measured’ brightness temperatures between the two stations. Thus, this OKC-SEP score using only these two stations can be calculated from all the pairs of soundings and can be used to evaluate the gradient retrieval. These statistical regression retrievals can be directly compared to retrievals using Smith’s method and the “minimum information” methods that are derived from the same radiance data. The rms errors and s x scores for retrievals at sounding locations should not be directly compared to the ones from the gridded data, however, as they involve somewhat different sets of data. (Also, the gridpoint retrievals are of virtual temperature rather than temperature.) As we can see, choosing an assessment method for retrievals requires careful consideration. Based on the above discussion, the quality assessment of the retrieval methods will be: a) Gridpoint retrievals: Bias (where applicable), rms error, and Si scores will be calculated for each observation day (combining 8 observation times). Examples of constant level analyses and vertical temperature profiles will also be presented. b) Retrievals at the Oklahoma City and Stephenville sounding locations: Bias and rms error for the entire VAS experiment will be calculated (i.e., including all 4 experiment days). Si scores will be calculated based on differences between the profiles at Oklahoma City and Stephenville, for observation times where soundings from both locations are available. Examples of vertical temperature profiles will also be presented. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -149- 7.3 Dynamic Retrieval from Wind Data We have reviewed the synoptic conditions that prevailed on the experiment days used in this research, and have considered methods of evaluating retrievals; we are now ready to examine the results of the retrievals. The first objective of this research is to produce analyses of mass fields using wind data that are comparable to what can be derived from wind Profiler measurements. Height and virtual temperature fields are derived from the wind data using the dynamic retrieval method described in section 6.2. The retrievals are performed both in sigma and pressure coordinates, and with Neumann and Dirichlet boundary conditions. The first set of retrievals to be discussed are the height retrievals, then the temperature retrievals will be presented. 7.3.1 Dynamic Retrieval of Height The first objective to be carried out for the dynamic retrievals is to produce retrieved analyses like those that can be obtained using Profiler winds. Figure 7.17 shows some examples of height fields on constant pressure surfaces obtained from wind data by the dynamic retrieval method. The analyses are for the 300 mb level on 6 March at 1200 UTC. The solid lines are height contours with a contour interval of 60 m. The dashed lines are contours for the difference field between the retrieved height and observed height; the contour interval is 15 m. Figure 7.17(a) is the verification field of 'true' heights analyzed from the rawinsonde measurements. The field shows a large trough over the VAS analysis area with a short-wave trough superimposed. The short wave trough axis extends from Kansas to the Oklahoma and Texas panhandles and into northern New Mexico. (This short-wave trough also can be seen in the 500 mb analysis in Figure 7.3.) Figures 7.17 (b)-(d) show, respectively, height fields retrieved using the R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 5 0 - geostrophic, balance, and two-scale approximations (Eqs. (6.13)-(6.15)). These analyses demonstrate some typical characteristics common to most of the retrieved height fields. The largest scale horizontal features, such as the large trough across the area, are quite well represented in the retrievals. The smaller scale features, such as the short-wave trough, are not as well represented, however. The analyses are, in general, smoother than the ‘true’ height analyses. The analysis derived using the geostrophic approximation (Figure 7.17 (b)) is the smoothest of the three analyses; there is almost no indication of the presence of the short-wave trough. The balance and two-scale approximations (Figures 7.17 (c) and (d)) show a little more indication of the trough. In these analyses, the retrieved height gradient in the northeast part of the VAS area is worse than for the geostrophic approximation, however. Dynamic retrievals of heights on constant sigma surfaces have also been performed. The results of these height retrievals are similar to the ones performed in pressure coordinates; the derived fields were in general somewhat smoother than the analyzed fields of height. Examples of height retrievals on sigma surfaces will not be presented since the fields are dominated by the variation of surface pressure and so are not as easily interpreted as the analyses in height coordinates. The other objectives for the dynamic retrievals are to show the effect of using approximate forms of the equations of motion (i.e., Eqs. (6.13) - (6.15)) and to compare the effects of using Dirichlet and Neumann boundary conditions in the retrievals. Table 7.1 summarizes the experiments performed to carry out these objectives; Table 7.2 shows the analyses that are used in each of the experiments. The remainder of this section shows the statistics (rms error and Si scores) that are used in making the comparisons between the retrievals. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 5 1 - 894 900 '9 0 0 906 906, 918 .918 918 93Q. a) 'True' heights 894 -3 0 •so -0- b) Geostrophic approximation Figure 7.17 Height analyses for 6 March 1982, 1200 UTC, on the 300 mb pressure surface. Solid lines: height contours (interval = 60 m). Dashed lines: difference between retrieved field and 'true' height field (contour interval = 15 m). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -152- ^^90 -i2jT\§82 -60 -60 V /----V924 I *o *60 c) Balance equation approximation .f°8?4. (d) Two-scale approximation (Eq. (6.15)) Figure 7.17 (continued) Height analyses for 6 March 1982, 1200 UTC, on the 300 mb pressure surface. Solid lines: height contours (interval = 60 m). Dashed lines: difference between retrieved field and ’true1 height field (contour interval = 15 m). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -153- Table 7.1 Exp A B C D E F Experiments for height retrievals from wind Purpose of experiment Vertical Remarks Coordinate show effects of using Pressure Differences between the different approximations to retrievals are small the divergence equation show effect of different Pressure Dirichlet B.C.’s have boundary conditions smaller error than Neumann B.C.’s show sensitivity of Dirichlet Pressure Boundary error increases boundary condition to error error of retrieval to near on boundary that of Neumann B.C.’s. show effects of using Sigma Differences between the different approximations to retrievals are small the divergence equation show effect of different Sigma Less difference between boundary conditions retrievals than in pressure coordinates show sensitivity of Dirichlet Sigma not as sensitive as boundary condition to error retrievals in pressure on boundary coordinates Table 7.2 Height retrievals from wind data Vertical Approximation to Coordinate divergence equation Pressure Geostrophic (6.13) Pressure (6.14) Balance Pressure Two-scale (6.15) Pressure Two-scale (6.15) Pressure Two-scale (6.15) Sigma Geostrophic (6.13) Sigma (6.14) Balance Sigma Two-scale (6.15) Sigma Full inviscid Sigma Two-scale (6.15) Sigma Two-scale (6.15) Boundary conditions Neumann Neumann Neumann Dirichlet Dirichlet Neumann Neumann Neumann Neumann Dirichlet Dirichlet Error on Used in boundary experiment — A — A — A,B no B.C 10 m RMS C — D — D — D.E — D no E,F F 10 m RMS R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -154- i>) vAd 3 ' i- .v 200 200 g g 400 fx .. 400 £ i 600 a. 600 -O-T- Geostrophic 800 600 *e -*• BaJanceieqn. 1000 -*-< Two scata eppx. 1000 0 5 10 20 15 25 0 30 10 5 RMS error (m) 15 20 25 30 RMS error (m) ;d ) V /\£ 5 c u j ' ’ ............j .......... ....................[.............. iy ” E 400 ¥ ! 1 600 r Geostrophic - Geostrophic -a -j- Balance! eqn. - Balance! eqn. -o>*j Two scale appx. ■ ■* I • ■ 1S RMS error (m) 20 25 Twoscafoappx. • 1000 30 0 5 10 15 20 25 30 RMS error (m) Figure 7.18 RMS error (m) for height fields retrieved from wind using Neumann boundary conditions in pressure coordinates. Circles: Geostrophic approximation (Eq. (6.13)). Squares: Balance equation approximation (Eq. (6.14)). Diamonds: Two-scale approximation (Eq. (6.15)). The first experiment involves showing the effect of using different approximations to the divergence equation on pressure coordinate retrievals. Figure 7.18 shows the rms error for dynamic retrievals of geopotential height in pressure coordinates, with Neumann boundary conditions. Generally, the rms error of the retrievals is small near the surface, and increases with height. The errors in the lower troposphere (1000 - 500 mb) are around 5 - 10 m. Between 500 and 200 mb the errors are generally between 10 and 15 m. These error values compare favorably with the measurement error of the radiosondes themselves (see Table 6 .2 ). Above 100 mb the rms errors are R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 5 5 - quite large, sometimes in excess of 30 m. This is due, in part, to error in the ‘observed’ analysis of radiosonde height measurements at those levels. The rms errors of the retrievals produced from the different approximations are indicated by different symbols; the geostrophic approximation by circles, the steady-state balance equation by squares, and Gal-Chen’s two-scale approximation (with time derivatives but no vertical motion) by diamonds. For the most part there is little difference between the rms errors for the retrievals between the different approximations, except for the case in VAS 2 where the geostrophic approximation rms errors for the upper troposphere are much worse than those of the other approximations. The VAS 5 analyses (Figure 7.18 d) have the smallest rms error; this is because the gradients of height are smaller for this case than for the other cases. This shows the tendency for skill in rms error to be case dependent, with larger rms errors in cases where the standard deviations are larger. The average standard deviation of height on the constant pressure levels for each VAS day is presented in Figure 7.19. (Note the difference in scale between this figure and Figure 7.18 above.) It is quite clear the VAS 2 has by far the largest standard deviation of height, typical of such a wintertime case. Figure 7.20 is a plot of the ratio of the rms error from dynamic retrievals (two-scale approximation, Neumann boundary condition) averaged over the 8 observation times to the averaged standard deviation of the ‘true’ height field. Dividing the rms error by the standard deviation at a level normalizes the rms error so that some of the case-dependent effects (larger rms errors in cases with larger variances) are removed. In addition, this plot can be regarded as a kind of signal-to-noise ratio. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 5 6 - o 200 400 CO <2 600 w CL -4- VAS2 -L..VAS-3•f VAS4 -! VAS 5 800 1000 0 20 40 60 100 80 Standard deviation (m) Figure 7.19 Standard deviation of height (m) on pressure levels, averaged over eight observation periods on each VAS experiment day. 200 VAS 2 - VA S 3 ■ 'VA ST"'" Pressure (mb) VAS 5 ! 400 600 800 1000 0 0.5 1 1.5 2 Figure 7.20 Ratio of average rms error to average standard deviation of height on constant pressure levels from retrievals using Neumann boundary conditions and the two-scale approximation to the divergence equation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -157We begin the retrieval process with only one piece of information, the mean at each level, so this ratio is a measure of how much our lack of knowledge about the field has been reduced by the addition of wind information in the dynamic retrieval. If the ratio is small, we have removed a large amount of uncertainty about the field. If the ratio is near one, we have gained practically no additional information, or, perhaps worse, wrong information. Judging by the ratio of rms error to standard deviation, the VAS 2 retrievals have by far the best score. The rms error in the VAS 2 retrievals is only about 20% of the total standard deviation of the field, above 800 mb. The VAS 4 retrievals have done nearly as well, but the VAS 3 and VAS 5 retrievals both have some levels where is ratio is 70% or more (so that the error variance would be about 50% of the total height variance). An important thing to note about the retrievals on constant pressure surfaces is that the size of the domain changes for the lowest pressure levels. In Figure 6.16, the northwest part of the grid is ‘underground’ at the 850 mb level. Thus, fewer grid points enter into the rms calculations for levels below 800 mb than for the other pressure levels. Figure 7.21 shows the number of gridpoints that enter the rms computation at each level. We can see that the error statistics for the first level, 975 mb, are calculated from a total of less than 100 gridpoints from the 8 observation times of each VAS day; this contrasts with ~ 1 100 grid points used in calculations for the 800 mb level and above. Since the statistics for the lowest few levels are taken over a much smaller area and calculated from a much smaller number of points, we need to be careful when comparing these levels with other levels that represent a larger sample area. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -158- ®...4VAS2. E l ! V A S 3 u ! V A S 4 '0 '" 'r V A S 5 i 0 i l i i 200 i I i 400 ■ i i 600 i i i i 800 i 1 < i ■ i 1000 i ~ 1200 Total no. of gridpoints Figure 7.21 Total number of gridpoints used in calculation of rms error statistics over the 8 observation periods of each VAS day. Figure 7.22 shows the Si score for each level, averaged over the 8 observation times of each experiment day. As with the rms error, the large values above 200 mb are probably due to problems at those levels with specifying the ‘true’ field against which the verification is done. The lowest Si scores for each experiment day are in the upper troposphere near 200 mb. The Si scores increase nearer the surface. By the standards given earlier (20 = very good, 70 = useless) the retrievals in the upper troposphere between 200 and 500 mb are good for ail the experiments. For two experiments (VAS 2 and VAS 4) the s i scores are less than 30 through the mid-troposphere to 800 mb. Below the 800 mb level, the Si scores increase, probably because these levels intersect with the surface in the western part of the domain. The poorer (larger) values of s i score for the VAS 5 retrievals come R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -159about because the gradients are much weaker for that experiment. Recall from Eq. 7.1 that the Si score is defined in terms of the absolute value of the error of the gradient divided by the absolute value of the gradient, thus the same error has a larger effect on the Si score when gradients are weak than when they are strong. The Si score is also fairly large near 800 mb in VAS 3, this also corresponds to levels where the standard deviation of height (and hence also the height gradient) is small. There is little difference between the Si scores for the different approximations to the divergence equation. The non-linear terms included in b ) V A S |3 200 200 E 400 E 400 600 600 600 600 ; ^Q-"Balanc9eqn ! - « — Twosc&leappx. . 1000 1000 0 20 60 40 80 0 100 20 40 60 80 100 S .sco re c) VAS A d) VASp -e j— Geostrdphic Kip—"BafarSwi’oqn...... •♦ j— Two scale appx. 200 •4— Geostrdphic 200 e 400 E, 400 600 600 600 600 1000 0 1000 20 40 60 S ,sco re 60 100 0 20 40 60 80 100 S , score Figure 7.22 Si score for height fields retrieved from wind using Neumann boundary conditions in pressure coordinates. Circles: Geostrophic approximation (Eq. (6.13)). Squares: Balance equation approximation (Eq. (6.14)). Diamonds: Two-scale approximation (Eq. (6.15)). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 6 0 - ; bj VA^ 3 200 200 f f 400 Q. 400 e 3 s 600 600 600 800 1000 1000 0 5 10 15 20 25 0 30 5 10 RMS error (m) 15 20 25 30 20 25 30 RMS error (m) 200 200 g 400 ¥3 0. 600 600 800 800 1000 1000 0 5 10 15 RMS error (m) 20 25 30 0 5 10 15 RMS error (m) Figure 7.23 Rms error (m) of retrieved height fields using the two-scale approximation in pressure coordinates. Circles: Dirichlet boundary conditions. Squares: Neumann boundary conditions. the balance equation approximation and two-scale approximation are small. Also, these terms may not be very well determined because the wind observations (at rawinsonde station spacing) do not resolve the smaller scale components very well. The next two experiments involve the comparison of the use of Dirichlet and Neumann boundary conditions, and the sensitivity of the retrievals using Dirichlet boundary conditions to error on the boundary. Figure 7.23 shows a comparison between the rms error of retrievals using Dirichlet and Neumann boundary conditions. The retrievals using Dirichlet boundary conditions (with Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 6 1 - a ) VAS 200 200 f 400 600 600 800 600 ■ ■»■■■ Dirichlet — -e — ftoumann 7 Tr 1000 0 20 40 60 Sj score 1000 80 0 100 c) VAS 4 40 S 1 score 80 100 d) VA S 200 200 f 400 600 6 00 800 8 00 — (Dirichlet — Neumann 1000 0 20 40 60 S j score 80 100 1000 0 40 S 1 score 80 100 Figure 7.24 s i score for height fields retrieved from wind using the two-scale approximation in pressure coordinates. Circles: Dirichlet boundary conditions. Squares: Neumann boundary conditions. the “true” height specified on the boundary) have rms error generally about 5 m less than that of the retrievals performed with Neumann boundary conditions. This comes about because using Dirichlet boundary conditions adds information to the retrievals. Figure 7.24 compares the Si score for height fields retrieved using Dirichlet and Neumann boundary conditions. Again, the retrievals using Dirichlet boundary conditions performed better than the retrievals with Neumann boundary conditions. By specifying the boundary conditions using the ‘true’ height field, we force the retrieved fields to have the correct mean gradient across the area and in doing so reduce the gradient error. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -162- ;b) V A 4 3 200 200 i ¥ 400 Z» g 600 600 800 800 1000 0 10 5 ;c ) VAS 1000 \2 2 T ' ■U ' s. .......s■V'^' 200 20 15 RMS error (m) 25 30 '"■■■-a- 0 / V) 600 I 600 if * (I * . 15 RMS error (m) - f- O — Nc error —i -o - 10 iRMS : i 11 25 30 r - i- i- r - ' 30 x 'n T -f :__ ............................... f i " ■f ■ ( i '1 ............. ■ / j- o 1' I—L t . 1 1 1. 10 No —10 —i - e 1 L l. 1 1 I 15 20 RMS error (m) i Ill: 1000 *...... i... 10 25 /] / 400 ¥a A *N 20 1........ l j C®.., / \ ........... - d ’ J 15 RMS error (m) ■ i" l- j" r - r - r - n ;d) V A S 5 ! \ 10 5 25 30 Figure 7.25 Comparison of rms error(m) for height fields retrieved from winds in pressure coordinates using Dirichlet boundary conditions. Circles: retrievals using "true" height field as boundary condition. Squares: retrievals using height field with added error (oe = 10 m). Dirichlet retrievals with perfect boundary conditions are compared to retrievals where the boundary conditions have a 10 m rms error field added in Figure 7.25. The error field used is the interpolated observation error field described in Section 6.3.2, scaled so that the mean of the boundary points’ error is zero and the standard deviation is 10 m. This is done to simulate a correlated error that might be found in estimates derived from a numerical model forecast field. Note that the effect of this added error removes much of the advantage in using Dirichlet boundary conditions in height retrievals. Anthes and Keyser (1979) give an analysis of the effect of boundary errors on solutions to Poisson’s equation. They show that the influence of R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -163lateral boundary conditions damps exponentially with the distance from the boundary at a rate determined by the wavenumber of the harmonic being considered. Their analysis is modified somewhat for a discrete domain; the influence of shorter waves is increased somewhat over the continuous case. Anthes and Keyser’s (1979) analysis shows that it is important to specify the large-scale variation of the boundary conditions accurately since the amplitude errors for long wavelengths damp less rapidly than short wavelengths. The effect on the Si score of using the boundary conditions with error added is illustrated in Figure 7.26. As with the rms error, we see that the use of boundary conditions that are in error reduces the advantage of using the Dirichlet boundary conditions. The change in Si score is smaller at higher levels because, although the added error is the same at all levels, in a relative sense the error is smaller at higher levels since the gradients there are larger than in the lower troposphere. The next set of experiments involves performing the same analyses in sigma coordinates. The first set of retrievals to be compared are the ones using various forms of the divergence equation. Figure 7.27 shows the rms error of the heights retrieved using Neumann boundary conditions on sigma surfaces. The rms error is about 10-15 m near the surface and decreases to 5-8 m at about the a = 0.8 level. Modica and Warner (1987) showed that the error near the surface increases substantially when frictional and sub-grid scale effects are not included in the retrievals; that increase is also demonstrated here. The error increases through mid-levels at a moderate rate, up to the a = 0.2 level. The sharp increase above that level again is probably the result of problems with the height analysis used to verify the retrievals, as well as increased error in the wind analysis (due in part to missing data). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -164- b) VAS b ; a) VA S 8 200 200 E, 400 600 600 600 800 jNo error 1000 :10m RMS ■10mRMS 1000 0 20 40 60 S , score 60 0 100 20 60 80 100 40 60 S , score 80 100 40 S 1score c) V A S 4 200 200 E, 400 E 400 600 600 800 800 •10m RMS . 1000 1000 0 20 40 60 80 S 1score 100 0 20 Figure 7.26 Comparison of Si score for height fields retrieved from winds in pressure coordinates using Dirichlet boundary conditions. Circles: retrievals using "true" height field as boundary condition. Squares: retrievals using height field with added error (ae = 10 m). Comparison with Figure 7.18 shows that the error of the retrievals on sigma surfaces is quite similar to the retrievals on pressure surfaces, though a careful level-by-level comparison shows the error to be somewhat larger for the sigma retrievals. The larger error in sigma retrievals near the surface comes about in part because the pressure retrievals are calculated over a smaller area near the surface (discussed above). When the rms errors of the retrievals using different approximate forms of the divergence equation are compared, it is evident (as with the retrievals in pressure coordinates) that there is little difference between the retrievals. There R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 6 5 - T T 0.2 0.2 0 .4 0 .4 CO 0.6 ■©— •Geostrophic ■a —{Balance «qn. 0.8 0.8 Two-scale appx. -» - iFuileqn. i 25 Fulloqn. j 0 30 5 10 RMS error (m) 15 20 25 30 RMS error (m) 0 c) VAS|4 d )V A s j 0.2 0 .4 E CD 05 0.6 S? 0.6 Geostrophic Balance fjqn. 0.6 —|Two-scaJ0 appx. Two*scal4 appx. iFuileqn. : Fulleqn. : 1 30 0 5 RMS error (m) 10 15 20 25 30 RMS error (m) Figure 7.27 RMS error (m) for height fields retrieved from wind using Neumann boundary conditions in sigma coordinates. Circles: Geostrophic approximation. Squares: Balance equation approximation. Diamonds: Two-scale approximation. X’s: Full inviscid equation including a terms. are variations, such as the larger rms error for the geostrophic approximation in VAS 2. However, even the addition of vertical motion terms using a (retrievals denoted by X) made little difference in the rms error. The next experiments compare the effect of using fixed (Dirichlet) versus gradient (Neumann) boundary conditions in sigma coordinates. The rms errors for retrievals using Dirichlet and Neumann boundary conditions with GalChen’s two-scale approximation are compared in Figure 7.28. The advantage in having error-free Dirichlet boundary conditions over Neumann conditions is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -166- 0 B) VAS 2 p ) VA S|3 0.2 0.2 0.4 0.4 <o £ SP S» « 0.6 CO 0.6 0.B — DiHchlet — Neumann 1 0 5 10 15 20 25 30 RMS error (m) 0 s 10 15 20 25 30 RMS error (m) o VAS 5 c) V AS 0.2 0.2 0.4 0.4 cd £> & s* « 0.e w 0.6 0.8 — Dirichlet — Neumann 0 5 10 15 20 1 25 30 RMS error (m) 0 5 10 15 20 25 30 RMSerror(m) Figure 7.28 Rms error (m) of retrieved height fields using the two-scale approximation in sigma coordinates. Circles: Dirichlet boundary conditions. Squares: Neumann boundary conditions. not as clear in this case as in the retrievals in pressure coordinates. The reason for this is not clear; it is not unprecedented, however. Kuo, et ai, (1987b) reported rms geopotential height errors from retrievals on sigma surfaces that showed very little difference between the Dirichlet and Neumann retrievals. Their mean rms error over 9 a levels (from a = 0.865 to cr = 0.15) for all their experiments was 9.4 m for the Dirichlet retrievals and 9.6 m for the retrievals using Neumann boundary conditions. Figure 7.29 compares the rms error of the height fields retrieved using error-free Dirichlet boundary conditions to boundary conditions with rms error of R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -167T“"T b) VAS 3 0.2 0.2 0.4 0 .4 0.6 0.6 0 5 10 15 20 25 30 0 5 10 15 RMS error (m) 20 25 30 0 5 10 15 RMS error (m) 20 25 30 RMS error (m) c) V A ^ 4 0.2 0.2 0.4 0 .4 <8 E CD E CD * 0.6 0.8 0 5 10 15 20 25 30 RMS error (m) Figure 7.29 Comparison of rms error(m) for height fields retrieved from winds in sigma coordinates using Dirichlet boundary conditions. Circles: retrievals using "true" height field as boundary condition. Squares: retrievals using height field with added error (ae = 10 m). 10 m. The change in rms error for these retrievals is not as large as with the retrievals in pressure coordinates (Figure 7.25). S i scores were not calculated for the height retrievals in sigma coordinates because the gradients of height on sigma surfaces are very large. These height gradients are primarily a function of variation in surface pressure, the error in the retrievals is a small fraction of that large gradient value; hence the calculation of S i scores produces extremely low values that are unrepresentative. Overall, the results of the experiments in dynamic retrieval of height can Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -168be summarized as follows: (a) It is possible to retrieve height fields using wind measurements and the divergence equation; these fields have rms error of ~ 5 - 20 m, and fair to good (20-40) Si scores. (b) Different forms of the divergence equation (derived from different approximations to the equations of motion) give retrievals that have similar overall rms error and S i scores. The individual analyses show that the geostrophic approximation gives fields that are slightly smoother than the other retrievals that retain the non-linear advection terms. (c) Using correct Dirichlet (fixed) boundary conditions in pressure coordinate retrievals can reduce the error of the retrievals substantially, as shown by the decrease in rms error and Si score when compared to the Neumann (gradient) boundary conditions. There is less difference between the retrievals when performed in sigma coordinates. The estimates of height on the boundary for Dirichlet retrievals must be obtained from an external source such as radiance retrievals or a numerical model; if these boundary conditions have errors it may increase the retrieval error to a level comparable to the Neumann boundary condition retrievals. (d) The rms error of the retrievals in pressure and sigma coordinates is similar through the mid- to upper troposphere. In the lowest levels of the atmosphere, the rms error of the pressure coordinate retrievals is much smaller than for the sigma coordinate retrievals because the retrieval area, and hence the temperature variance, in the pressure coordinate retrievals decreases in the lowest levels where the pressure surfaces intersect with the ground. The sigma coordinate retrievals show an increase in rms error for the levels nearest the surface in agreement with the results of Modica and Warner (1987). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -169- 7.3.2 Dynamic Retrieval of Temperature The next task is to evaluate the dynamic retrievals of virtual temperature. The objectives for dynamic retrieval of temperature are the same as those for the height retrievals: to show examples of temperature fields that can be derived from wind measurements, to determine the effect of using approximate forms of the divergence equation in the retrieval, and to compare retrievals using different boundary conditions. We will first examine some examples of retrieved fields and vertical temperature profiles, and then use statistics to compare the results from different dynamic retrieval runs. The virtual temperature fields retrieved from wind using the dynamic retrieval methods are of poorer quality than the height retrievals discussed in previous sections. The forcing functions - - used for the temperature retrieval in pressure coordinates involve taking differences between the forcing functions at different levels. When the error in the height gradient forcing functions 6 is not vertically correlated, it is possible for the errors between levels to add together in the calculation of temperature. As noted in the discussion in section 6 .2 .1 , the process of taking a derivative amplifies components that have smaller vertical wavelengths,such as the noise components. Figure 7.30 shows an example of temperature fields retrieved from winds from 6 March 1982 at 1200 UTC on the 700 mb surface. The ‘true’ field analyzed from rawinsonde data shows cold air centered in the north central part of the VAS area, with strong gradients in the west and east. The temperature field retrieved from wind using a geostrophic approximation exhibits the strong, near east-west temperature gradient in the southeast part of the area. The north-south temperature gradient in the mid-section was somewhat lacking. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -170- a) ‘True’ field s' b) Geostrophic approximation Figure 7.30 Temperature analyses for 7 March 1982, 0000 UTC, on the 700 mb pressure surface. Solid lines: temperature contours (interval = 2 K). Dashed lines: difference between retrieved field and 'true' virtual temperature field (contour interval * 2 K). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. c) Balance equation approximation -12 \-8 0 -- d) Two-scale approximation Figure 7.30 (continued) Temperature analyses for 7 March 1982, 0000 UTC, on the 700 mb pressure surface. Solid lines: temperature contours (interval = 2 K). Dashed lines: difference between retrieved field and 'true' virtual temperature field (contour interval = 2 K). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -172The -12 °C isotherm, which in the ‘true’ analysis was north of Oklahoma City, extended down into Texas for each of the retrievals. For this particular level and time, the balance equation approximation produced an analysis closer to that of the ‘true’ field than the one derived using the two-scale approximation (with time derivative terms). In general, for the levels closer to the surface, the strong features (such as the gradient in the southeast in this analysis) are captured in the retrievals, but weaker gradients are usually not as well depicted. Figure 7.31 shows examples of temperature fields retrieved from winds from 6 March 1982 at 1200 UTC on the a = 0.7 surface. This “true” analysis differs somewhat from the analysis on the 700 mb level presented in Figure 7.30. The cold air center on this surface is displaced toward the west. This is because temperatures are decreasing with decreasing pressure through this layer and since the surface pressures are lower in the west the constant sigma surface is at a lower pressure (approximately 600 mb). Thus, the temperatures on a constant sigma surface are, in part, a function of surface pressure. The retrieved analyses of temperature in sigma coordinates have some of the same problems as the corresponding analyses in pressure coordinates. The 12 °C isotherm extends too far south in all of the analyses, just as in the retrievals on constant pressure levels. Following the 12 °C isotherm to the east, we see that in the ‘true’ analysis there is a colder area in Missouri with a relatively flat temperature gradient (no doubt associated with the with the increase in surface elevation over the Ozarks) but this feature is poorly depicted in the retrieved fields. The field retrieved using a geostrophic approximation has an even temperature gradient across the east with no indication of the Missouri temperature ‘trough’. In Figures 7.31 (c)-(e), this small-scale wave in the temperature field is displaced southward, over Arkansas. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 7 3 - The gradients in the southeast part of the analysis are stronger in the retrieved fields than in the 'true' analysis. This leads to fairly large temperature errors in the southeast, as much as 8 °C for the retrievals using the two-scale approximation and the full inviscid equation. a) ‘True’ field Figure 7.31 Temperature analyses for 7 March 1982, 0000 UTC, on the o = 0.7 surface. Solid lines: temperature contours (interval = 2 K). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. b) Geostrophic approximation -16 c) Balance equation approximation Figure 7.31 (continued) Temperature analyses for 7 March 1982, 0000 UTC, on the a = 0.7 surface. Solid lines: temperature contours (interval = 2 K). Dashed lines: difference between retrieved field and 'true' height field (contour interval = 2 K). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. d) Two-scale approximation -16 -16 --G -, e) Full inviscid equation (including a terms) Figure 7.31 (continued) Temperature analyses for 7 March 1982, 0000 UTC, on the a = 0.7 surface. Solid lines: temperature contours (interval = 2 K). Dashed lines: difference between retrieved field and 'true' height field (contour interval = 2 K). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -176A set of virtual temperature ‘soundings’ for Stephenville and Oklahoma City at each observation time has been produced by bilinear interpolation to station locations from the gridded retrieved fields. Some examples of these ‘soundings’ are shown in Appendix C. Besides allowing the evaluation of the retrieval of temperature profiles from wind data, this set also serves as a basis for comparison with the radiance-only retrievals of the Oklahoma City and Stephenville soundings at the station locations (also presented in that appendix). Figures C.1 and C.2 show examples of soundings interpolated from fields obtained from winds by dynamic retrievals in pressure and sigma coordinates. In these figures, the heavy solid line is a profile interpolated from the ‘true’ gridded virtual temperature analysis, a profile quite close to the virtual temperature profile that was actually measured at that time. The thin solid line is interpolated from a two-scale approximation retrieval with Lanczos vertical derivatives. The dashed line is the mean of the ‘true’ gridded fields, or, in a way, the first guess field for the dynamic retrievals. There are a couple of problems that can occur with the wind-derived soundings. As mentioned before, one difficulty with the method arises because of the amplification of noise through the vertical differentiation necessary to obtain the estimates of temperature gradient from the gradients of geopotential. The measures used to control small scale noise are vertical filtering of winds (section 6.1.2) and the use of Lanczos’ derivative formula to calculate temperature gradient estimates (section 6.2.1). The use of these filters removed most of the small-scale noise. Unfortunately, there can be error in larger scale components as well as in the small scale components; this can be detrimental to sounding quality. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -177The profile on the right of each figure is from 27 March 1200 UTC at Stephenville. (This sounding is also shown in Figure 6.14.) This observation time produced some of the worst of the retrieved profiles, and this profile is far from typical. The retrieved profile (thin line) is quite unlike the ‘true’ profile (thick line) at this observation time, it has a substantial component with wavelength of about 200 mb, or 8 Ap. (Recall Ap = 25 mb in the pressure coordinate retrievals.) Since the mean profile is quite smooth, it must not have directly contributed to the 200 mb wavelength noise in the retrieved profile; thus the 8 Ap wave must come about through the mis-specification of the temperature gradient in the estimates derived from the wind analyses. The profile on the left of these figures is a better example of the sort of vertical profile that can be obtained from the dynamic retrievals. This sounding is for Oklahoma City at 1200 UTC, 7 March 1982. This retrieved sounding shows a more realistic structure, without the noise problem of the sounding described previously. There is still some indication of an 8 Ap wave in the retrieved sounding but the amplitude is much smaller. Note the difference in tropopause level between the retrieved and mean sounding; this demonstrates that information has been added to the mean first guess by use of the wind data. Retrieved profiles for other station locations show a similar improvement over the mean first guess profile. The large deviation between the ‘true’ and retrieved soundings just below 800 mb in both of the soundings in Figure C.1 is due to another problem in the pressure retrieval formulation. In many of these retrieved soundings, there are discontinuities between the 850 and 800 mb level that come about because of the use of ‘modified’ Neumann boundary conditions suggested by Gal-Chen: Dirichlet boundary conditions for grid points ‘intersecting’ with the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -178surface for levels up to 800 mb, and Neumann boundary conditions with a specified mean value for all levels above 800 mb. This could probably be remedied, at the cost of more error in the near-surface ‘Dirichlet’ points, by specifying the mean of the lower layers as well as the layers above the surface. This discontinuity is absent in the retrieved profiles in sigma coordinates (Figure C.2), but the lowest levels of the soundings are still very poor. The mean or ‘first guess’ profile (dashed line) is quite different for the sigma coordinate retrievals than for the pressure coordinate retrievals. We can certainly expect this to be the case, since the temperature values on each sigma surface represent a range of pressure levels. However, if we compare the retrieved profiles in Figures C.1 and C.2, we see that (aside from the discontinuities between 800 and 900 mb) they are quite similar. From these examples of virtual temperature retrievals, we can see that it is possible to obtain temperature fields using the equations of motion and wind data. The retrievals can show major features of the temperature fields in the mid- and upper troposphere. The vertical temperature profiles (soundings) demonstrate that the wind data is providing additional information about major features such as the tropopause level. There are some problems with noise in the vertical profiles, this is an inherent part of the retrieval method because the vertical profiles are derived separately at each level. (The error minimization is done globally on each level without any interaction between levels). The next task is to use statistics (rms error and Si scores) to make comparisons between retrievals using different approximate forms of the divergence equation and different boundary conditions. Table 7.3 shows the experiments that are performed to carry out these objectives. The analyses used in the experiments listed in Table 7.4. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -179- Table 7.3 Experiments for virtual temperature retrievals from wind Exp Purpose of experiment Vertical Remarks Coordinate ” show effects of using Pressure Differences between the different approximations to retrievals are small. the divergence equation H Pressure show effect of different Dirichlet B.C.’s have boundary conditions smaller error than Neumann B.C.’s. I show sensitivity of Dirichlet Pressure Effect of boundary errors boundary condition to error is small. on boundary J show effects of using Sigma Differences between the different approximations to retrievals are small. the divergence equation K show effect of different Sigma Dirichlet B.C.'s have boundary conditions smaller error than Neumann B.C.’s. L show sensitivity of Dirichlet Sigma Effect of boundary errors boundary condition to error is small. on boundary Table 7.4 Virtual temperature retrievals from wind data Vertical Coordinate Pressure Pressure Pressure Pressure Pressure Pressure Sigma Sigma Sigma Sigma Sigma Sigma Boundary Error on Approximation to Used in divergence equation conditions boundary experiment — Geostrophic (6.13) mod.Neumann G — (6.14) mod.Neumann Balance G — Two-scale (6.15) mod.Neumann G,H — Neumann Two-scale H (6.15) Two-scale Dirichlet no (6.15) H,l Two-scale Dirichlet I (6.15) 10 m RMS — Neumann Geostrophic (6.13) J — (6.14) Neumann Balance J — Two-scale Neumann (6.15) J,K — Neumann Full inviscid K Two-scale (6.15) Dirichlet no K,L Two-scale (6.15) Dirichlet F 10 m RMS R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 8 0 - The first experiment compares retrievals in pressure coordinates using different approximations to the divergence equation. Figure 7.32 shows the rms error for the dynamic retrievals of temperature on constant pressure surfaces using the 'modified' Neumann boundary conditions (Neumann boundary conditions except where pressure surfaces intersect with the ground). The rms errors for the lowest levels are small but, as discussed before, this is due to the reduced area of calculation and the influence of the Dirichlet boundary conditions on the western edge of the grid. Above the surface, the rms values generally range from about 2.5 K at 900 mb to close to 1 K at the 200 mb level. The values for VAS 5 were somewhat smaller but the temperature gradients (and hence the temperature variance) were somewhat weaker for that case than for the others. Note the wave-like pattern in the rms error. This pattern is the clearest in the VAS 5 rms error although it is also evident in the rms error for the other cases; it seems to have a wavelength of 200 mb (similar to that of the sounding in Figure 7.30). Referring back to Figure 6.13, which compares rms errors of temperature for centered differences versus Lanczos’ derivative, we see that the vertical variation of the rms error for the centered differences is much noisier than for Lanczos' method. This makes sense if we consider that in the centered difference method errors in height map into temperature errors on alternate levels, while in the Lanczos method the temperature gradient estimate is derived from height gradient estimates of both odd and even levels, and so can contain errors from many levels. As with the height retrievals, there does not seem to be a clear advantage for one approximate method over another. In some cases (VAS 3, for instance) the geostrophic approximation seems to do a better job than the R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -181- a) VASL 200 200 £400 £ 4 0 0 ® 600 600 600 600 1000 0 1 2 3 1000 4 5 0 1 3 2 4 S RMS error (K) RMS error (K) Geostrophic fe ■ Geostrophic 200 200 — -> — Two scfclo appx. — — Two sckJe appx. £ 4 0 0 £ 600 © 600 600 800 1000 0 1 2 3 RMS error (K) 4 5 1000 0 1 2 3 4 S RMS error (K) Figure 7.32 Rms error (K) for temperatures derived from wind on constant pressure surfaces using the ‘modified’ Neumann boundary conditions and vertical derivatives of various approximate forms of the divergence equation. Circles: Geostrophic approximation. Squares: Balance equation approximation. Diamonds: Twoscale approximation. retrievals where the non-linear terms are retained; this may be due to some correlation of error in the vertical for the geostrophic method. In VAS 2, the twoscale approximation has somewhat larger rms errors at several levels. It is useful to consider what sort of error should be expected from the temperature retrievals based on the expected error of the winds. We can derive an estimate of the order of magnitude of the error that we may expect based on the error in wind measurements. (Naturally, this analysis does not take into account interpolation error, or the error resulting from modelling assumptions.) Table 6.3 gives the expected error estimates for the winds in AVE-VAS, from R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 8 2 - which we see that the largest rms errors we would expect in the winds would be less than 10 m s_1. Suppose we assume a value for the error 8u = 10 m s-1. The error in the wind gradient between stations would be approximately 5u/As where As is the station spacing, thus 8 u /A s ~ .33 x 10' 4 if we assume a station spacing of around 300 km. We can take a fairly generous value for the wind gradient, say about 10 m s_1 /1 0 0 km = 1 x 10 -4; we would not expect a larger value since this value makes the advection terms in the equations of motion to be of about the same order as the Coriolis term. We would expect the error in the height calculation to be of the order Ay 0 ( 8 z ) = “ x max (f 0 ( 8 u ) , 0 ( 8 u | V v | ) , 0 ( u 8u/As) ) ; (7.2) if we assume u ~ 30 m S'1 , f ~ 10*4 s *1 and Ax ~ 100 km, then o ( 8 z ) ~ 10 m. This agrees well with the results previously presented. Now, if we assume the height error is 10 m and that the errors are nol vertically correlated, then, by the hydrostatic equation the error in temperature should be of the order 0 (8t) . M 0 ^ 1 . (7.3) For a difference Ap = 50 mb, p = 500 mb, o ( 8 t ) ~ 3.5 K. Fortunately the rms errors in the temperature retrievals at 500 mb are less than half that, even for the centered difference method. If the error in height is vertically correlated, then the error in the calculated thickness of a layer would not be as great as the error of the height fields used to calculate it. One thing to note is that the effect of the wind errors on the retrieved temperatures, according to Eq. 7.3, should be directly proportional to pressure. In fact, this effect is could be partly responsible for the decrease in rms error from the 800 mb to 200 mb level in ail the experiments. Naturally the slope of the rms error decrease is not quite linear with height. This is because the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 183 - magnitudes of the winds and the wind errors increase with height, so 5 z , the error of the height retrieval, increases with height, thus changing S t . This analysis does bring out another point, however, that if the height derivatives are taken over deeper layers (where we attempt to find layer mean temperatures, say, rather than to estimate temperatures at a level), we can expect that the contribution of wind measurement error to the error of the retrieval will decrease. However, since the error is proportional to 1/Ap, the improvement in error for calculations over deeper layers decreases with increasing Ap and begins to level off at a layer depth of 200 mb (with o ( S t ) ~ 0.85 K for that layer thickness). This analysis also helps to explain why there is little difference between the retrievals using various approximations to the divergence equation. Suppose that the advection terms are not the same order of magnitude as the Coriolis term, but about one order of magnitude less. In that case, the error in the analysis resulting from error in the wind observation (calculated as error in the Coriolis term) would be about the same size or larger than the contribution of the advection terms. Also, the advection terms themselves could be in error by nearly 100% because of error in the observed wind gradient. Naturally, the numbers given above for wind observation error represent ‘worst-case’ scenarios, but this still gives some insight into the question of why the addition of more terms did not improve the analysis. A plot of the standard deviation of the 'true' temperature field is shown in Figure 7.33. These standard deviations (as well as the rms error statistics presented above) are calculated using only interior points on the grid. The standard deviation of temperature on the constant pressure surfaces is quite large at the 800 mb level for VAS 3, 4, and 5. In those cases, a considerable R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -184portion of the grid in the western part of the area has its first ‘non-Dirichlet’ level at around 800 mb. As the day progresses, the levels nearest the surface warm in response to solar heating. Thus, the temperatures in the western part of the analysis area at 800 mb became considerably higher than those in the eastern part of the grid. The lapse rate in the lowest few levels of the western soundings is steeper than that in the east, nearly dry adiabatic, so the temperature difference relative to the eastern half of the grid decreases with height. This effect was particularly strong in VAS 3. In VAS 2, the temperature gradients were quite strong up to the tropopause, so the variance of the temperature fields stayed fairly large through the mid- to upper troposphere. As discussed in Section 7.3.1, the decrease in the standard deviation in the lowest levels is due to the decreased area of those constant pressure surfaces. The ratio of the average rms error to average standard deviation of temperature on the constant pressure levels for each of the AVE-VAS cases is shown in Figure 7.34. These retrievals use the ‘modified’ Neumann boundary condition with the two-scale approximation to the divergence equation. When compared to their respective standard deviations, the temperature retrievals have indeed performed more poorly than the height retrievals (Figure 7.20). The VAS 2 retrievals perform better than the others, with rms errors of the retrieved fields being generally only 50 - 70% of the standard deviation of the fields. The retrievals from the other VAS experiment days are worse, with the rms error greater than the standard deviation at some levels. For VAS 5, this is because the standard deviation of the field is less than 1.5 K through a large portion of the troposphere. The largest values of this ratio for VAS 3 and VAS 4 are also at levels where the standard deviation is relatively small (less than 2 K). Below 800 mb, all the retrieved fields are poor. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -185- VAS 2 VAS 3 200 VAS 4 !§. P> 3 co in £ Q. 400 600 800 ^ 1000 0 1 2 3 Standard deviation (K) 4 5 Figure 7.33 Standard deviation of temperature (K) on pressure levels, averaged over eight observation periods on each VAS experiment day. o 4 — VAS 2 ■d - VAS 3 200 . ' T - ' v a s 4 " '“ -i- VAS 5 ' 400 600 800 1000 0 0.5 1 1.5 2 V °T Figure 7.34 Ratio of average rms error to average standard deviation of temperature on constant pressure levels from retrievals using Neumann boundary conditions and the two-scale approximation to the divergence equation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 8 6 - b) VAS a) VAS |2 200 B 400 2 600 200 E 400 600 800 1000 600 %uu*9uu»4eo9trophiei..... — a — Balance e q ri - *+ — Two scale ajipx. 0 20 40 60 80 1tt 1000 0 20 40 c) VAS 60 80 100 60 80 100 S . score S . score d) VA S|5 A 200 200 E 400 600 800 600 -nTnQrm»<|eo9trophlcj..... — o — Balance e q ri — » — Two scale ajSpx. 800 1000 0 20 40 60 S 4score 60 100 M m i^m r-^e o s trophtg.... — □ - Balance eqri. — Two scale appx. 1000 0 20 40 S . score Figure 7.35 s i scores for temperatures derived from wind on constant pressure surfaces using the 'modified' Neumann boundary conditions and vertical derivatives of various approximate forms of the divergence equation. Circles: Geostrophic approximation. Squares: Balance equation approximation. Diamonds: Twoscale approximation. Figure 7.35 shows the Si scores for the retrieved temperature fields for each of the various approximate divergence equation retrievals. The largest Si scores are close to the surface, where presumably the error is due to neglect of frictional terms or poor resolution of smalier-scale circulations. In mid-levels, the Si scores go from poor to only moderately good. The larger Si scores for VAS 5 are related (as with the height retrievals) to the small magnitude of the temperature gradients in that summer case. The large values of Si score at the highest levels are again probably due to a poor analysis of the ‘true’ R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1 8 7 - temperature field or problems with the wind analyses at those levels. The lowest Si scores are on the VAS 2 experiment day where the gradients are the largest. The VAS 2 and VAS 3 retrievals have fair to mediocre Si scores, but for both VAS 4 and VAS 5 the Si scores are in the near useless range for retrievals below 500 mb. As noted before for the height retrievals, the differences in Si scores between the retrievals using different approximations to the divergence equation are small, except for a few layers where the geostrophic approximation has slightly smaller s x scores. Overall, the Si scores for the temperature retrievals are much poorer than for the height retrievals. In Figure 7.22, the Si scores were generally less than 40 for the height retrievals. For the temperature retrievals, nearly all the levels have average Si scores greater than 40. For the VAS 2 and VAS 4 retrievals, the temperature Si scores are generally about twice the Si scores of the height retrievals, while for VAS 3 and VAS 5 the Si scores of the temperature retrievals are about 1 V2 times the size of the Si scores for the height retrievals. The relative size of the Si scores of height retrievals and temperature retrievals from the same case would depend on how well correlated the height fields are in the vertical (hence, the intensity of the temperature gradient) as compared to the correlation of the error fields. If the error fields are well correlated in the vertical, then the temperature gradient error would be small. Conversely, we would have much worse Si scores given the same error field for a barotropic case where the height fields are better correlated and temperature gradients are small than for a baroclinic case with large temperature gradients. The next experiment involves comparing the retrievals performed in pressure coordinates with different boundary conditions. Figure 7.36 is a comparison of the rms error for retrievals of virtual temperature from wind using R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -188- o ; -o 200 Dirichiet | Modified Neumann 200 ^VP'toumanft....... 400 400 800 600 800 600 1000 0 1 2 3 JL 4 1000 5 0 1 2 3 4 5 RMS error (K) 0 4 - Dirichiet i t - Modified Neumann 200 -f- Dirichiet j * 400 400 600 600 800 800 1000 0 1 2 3 RMS error (K) 4 5 t- Modified Neumann 200 1000 RMS error (K) Figure 7.36 Rms error (K) for temperatures derived from wind on constant pressure surfaces using the two-scale approximation for various boundary conditions. Circles: Dirichiet boundary conditions. Squares: ‘modified’ Neumann boundary conditions. Diamonds: Neumann boundary conditions. the two-scale approximation with Dirichiet, modified Neumann, and Neumann boundary conditions. Just as in the height retrievals, the rms error for retrievals using Dirichiet boundary conditions is much smaller than that for retrievals using either the Neumann or ‘modified’ Neumann boundary conditions. The use of modified Neumann boundary conditions, where the points intersecting the surface are specified, reduces the rms error in the lowest levels for these retrievals over the retrievals where only Neumann boundary conditions are used. The reduction in rms error can be fairly large; it was over Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1891 K in the VAS 2 and VAS 3 cases. From Figure C.1, we have seen that the modified Neumann boundary condition can produce discontinuities in the vertical profiles; if this is corrected by adjusting the mean values at these levels it will not change the rms error, however. There was little change between the Neumann and modified Neumann boundary conditions in VAS 5, but the rms error is fairly small in that case. Figure 7.37 compares the S i scores for retrievals using the two-scale approximation with the Dirichiet and ‘modified’ Neumann boundary conditions. Use of Dirichiet boundary conditions again produces a considerable improvement in the Si scores. The Si scores of the temperature retrievals are still about twice those of the corresponding height retrievals, however. The effect of error in Dirichiet boundary conditions is demonstrated in Figure 7.38. The rms error for a two-scale approximation temperature retrieval using perfect Dirichiet boundary conditions is compared with a retrieval where a 1 K rms error field has been added. The error field used is the interpolated observation error field adjusted so that the mean boundary point error is zero and the standard deviation of the error is 1 K. The introduction of this spatially correlated error field on the boundary conditions does not have as much effect on the rms error as the addition of 10 m rms error to the boundary has on the height field. (See Figure 7.24.) The average difference between the rms error of the analyses was about 0.15 K, except for the VAS 5 case where the average difference was 0.2K. The difference between the rms errors remains fairly constant with height, whereas for the height retrievals the difference between rms errors decreased away from the surface. Figure 7.39, which compares Si scores of temperature fields retrieved with and without error in the boundary conditions shows a similar result. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a ) V A S 12 b ) V A S |3 200 200 £ E 400 400 600 600 60 0 800 — Dirichiet — Modified NejumanrT — O iric h le P ^L — Modified M^umann 1000 1000 20 40 80 100 S 1score o c ) VAS 4 60 80 100 60 80 100 40 S 1score d) VA SI5 200 E 20 200 400 E 400 600 600 800 800 — Dirichiet j - Modified Nqi 1000 1000 20 80 S j score 100 0 20 40 S 1score Figure 7.37 Si scores for temperatures derived from wind on constant pressure surfaces using the two-scale approximation for various boundary conditions. Circles: Dirichiet boundary conditions. Squares: 'modified' Neumann boundary conditions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -191- — No error - 200 ■o— No error ■a — 1-OKRMS 1|0K RMS 200 400 400 600 600 800 800 1000 0 1 2 1000 3 4 5 0 1 2 3 4 5 4 5 RMS error (K) RMS error (K) — No error - 200 1i:0K RMS 200 400 400 600 600 800 800 1000 0 1 2 3 RMS error (K) 4 S 1000 0 1 2 3 RMS error (K) Figure 7.38 Comparison of rms error(K) for temperature fields retrieved from winds in pressure coordinates using Dirichiet boundary conditions. Circles: retrievals using "true" temperature field as boundary condition. Squares: retrievals using temperature field with added error (ae = 1 K). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -192- — I' Pressure (mb) -»---- 1-----1-----(— 1 ; b ) V A S |3 a ) VAS 200 200 400 E 400 600 600 800 600 20 40 60 60 20 100 - i- 'i—r ; d) VAS c) VAS 200 Pressure (mb) Ijlo error __ 1.0K RMS | “is i . .- T * 7 ■ \ 40 200 400 E, 400 600 600 600 600 No error 21.OK RMS 0 20 1000 60 40 S 1score 80 p-T" r - r - , , 60 —1 —1 —1 — 5' ............................................................ ............... fS i .................... .............. ......................................................... . . 1000 '-J S 1score S . score o — i 1000 0 r-r1 ' 17 - ^ : M i . A ........ ■— ■— o 1000 1»‘ >■ 0 No error ---- a - il.OK RMS : : ^ • ..........i .............4 < T , . p -.r . 100 . . 80 S 1score Figure 7.39 Comparison of Si scores for temperature fields retrieved from winds in pressure coordinates using Dirichiet boundary conditions. Circles: retrievals using "true" temperature field as boundary condition. Squares: retrievals using temperature field with added error (ae = 1 K). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -193- o ; o | Geostropftjc - a I- Balance eqn. .-♦■■i— Two*eoa*4-appjfc-**• Futf Invis4*d Geostrophic -a j - Balance •qn. *e.j»».TwftscaJ4 appy.. • * j- Full invts^id 0.2 0.2 0.4 0.4 <s E 0.6 S> (0 0.6 o.e 0.8 .s> b) VAS;3 a) V A S 0 1 2 3 4 0 5 1 RMS error (K) 2 3 4 5 4 5 RMS error (K) 0.2 0.2 0.4 0.4 <8 E 8 0.6 0.8 d) V A S I5 o 1 2 3 RMS error (K) 4 5 0 1 2 3 RMS error (K) Figure 7.40 Rms error (K) for temperatures derived from wind using Neumann boundary conditions in sigma coordinates and vertical derivatives of various approximate forms of the divergence equation. Circles: Geostrophic approximation. Squares: Balance equation approximation. Diamonds: Two-scale approximation. X’s: Full inviscid equation including a terms, difference in Si scores is much less than for the height retrievals near the surface but it does not change much with height. This is because temperature gradients do not increase with height the in the way that height gradients do. The next set of experiments involves the comparison of retrievals performed in sigma coordinates. Figure 7.40 shows plots of the rms error of temperature retrievals on sigma coordinate surfaces with Neumann boundary conditions using the different approximations to the divergence equation. Below the a = 0.8 level the rms error of these retrievals increases sharply R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -194with increasing a (or decreasing height above ground) with extremely large values near the surface. As with the height retrievals in sigma coordinates, this increase in error can be attributed to the neglect of frictional and sub-grid scale effects (Modica and Warner, 1987). Above the a = 0.8 level the rms errors are generally between 1 - 2 K. A wave pattern with wavelength ~ 200 mb is present in the rms errors; the VAS 2 rms error (Figure 7.40 (a)) shows the pattern most clearly. The sharp peaks in rms error around the 0.1 a level are related to the large error in the wind analysis just above that level due to missing data; the analysis error is well correlated in the vertical for the levels above this and so the difference of the height gradient error between levels (and thus the temperature error) would peak and then drop off. The rms errors above the surface are the largest for the VAS 2 (strong gradient) case and smallest for VAS 5 (weak gradient). As with the retrievals in pressure coordinates, there is not a clear advantage for one approximate method over another, although the simple geostrophic approximation had the lowest rms error in some layers. Using vertical motion terms (the ‘full inviscid’ retrieval) did not appreciably improve the analyses; the rms error is the largest for these retrievals on a few of the levels. The rms error values above a = 0.8 do not differ substantially between the sigma and pressure coordinate retrievals. Figure 7.41 shows the standard deviation of temperature from the ‘true’ analyses on constant sigma levels. The standard deviation of temperature on sigma surfaces is generally higher than the standard deviation on pressure surfaces (Figure 7.33). Near the surface, the standard deviation is between 2 3 K. The standard deviations decrease slightly up to about the a = 0.85 level, except in VAS 5 where the decrease continues to the 0.75 a level. Through the mid-troposphere, standard deviations increase to a maximum at or just above Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -195- o VAS 2 : VAS 3 -VAS’4'-— VAS 5 • 0.2 0 .4 0.6 0.8 1 0 1 2 4 3 5 Standard deviation (K) Figure 7.41 Standard deviation of temperature (K) on constant sigma levels, averaged over eight observation periods on each VAS experiment day. o 0.2 .YAS..2. 0 .4 VAS 3 VAS 4 VAS-5- 0.6 0.8 1 0 0.5 1 1.5 2 °E/ffr Figure 7.42 Ratio of average rms error to average standard deviation of temperature on constant sigma levels from retrievals using Neumann boundary conditions and the two-scale approximation to the divergence equation. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -196the ct = 0.4 level, except in VAS 4 where the largest standard deviation is at a = 0.425. Above this level standard deviations decrease, to 1-2K at a = 0.2. The ratio of the average rms error to average standard deviation of temperature on the sigma surfaces for each of the AVE-VAS cases is illustrated in Figure 7.42. The rms errors used are from the retrievals with Neumann boundary conditions and the two-scale approximation to the divergence equation. The retrievals at levels near the surface are extremely poor. The cut off on the graph is at a ratio of 2 , to match the scale in Figures 7.20 and 7.34. For VAS 4, the ratio of rms error to standard deviation is 6 to 1 at the surface. The ratio is 0.6 or greater below the o = 0.8 level for all cases. Above the a = 0.8 level , the rms error is between 20 % and 60% of the standard deviation of the sigma surface, up to the 0.25 a surface. The ratio increases to above 1 at the a = 0.1 level for all the observation days, this again is because of missing data leading to a poor wind analysis at and above this level. Between the o = 0.8 and o = 0.2 level the ratios are generally less than for retrievals on pressure surfaces in the corresponding 800 - 200 mb layer. This is because the standard deviation of temperature on sigma surfaces is larger than on pressure surfaces in the mid- to upper troposphere. Figure 7.43 compares the Si scores for retrievals using the various approximations to the divergence equation and Neumann boundary conditions. The Si scores are large through the surface layer up to a = 0.8, the retrievals in the lowest levels are very poor. In middle levels the Si scores are generally around 40 - 50, somewhat better than the retrievals at corresponding levels in pressure coordinates. This is probably due to the larger values of gradient on sigma surfaces rather than any real decrease in the gradient error. Again, the large Si scores in the upper levels result from the lack of data at those levels. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1970 a) VAS 0.2 0.2 cd E 0.4 to E a> a C/5 CO 0.6 Q (jlO M tropm o^ -Bafanca aqn-ft^ — — Two*cale«ppx » - tfull inviscid • 0 20 40 0.4 0.6 ■o— jGoottrophic 0 ‘w.}Bafanwiqii:**” -• — >Two scale agpx • » - jfu ll invaod: 0.8 1 60 80 0 100 20 40 60 80 100 80 100 S .sco re S.score o c) V A S 4 d) VAS S 0.2 0.2 0.4 a E S> co SP 0.6 0.4 0.6 0.8 0 20 40 60 80 100 1 0 20 40 60 S . score Figure 7.43 s i scores for temperatures derived from wind using Neumann boundary conditions in sigma coordinates and vertical derivatives of various approximate forms of the divergence equation. Circles: Geostrophic approximation. Squares: Balance equation approximation. Diamonds: Two-scale approximation. X ’s: Full inviscid equation including 6 terms. The differences in Si scores between these retrievals using different approximations to the divergence equation are generally small, except for some levels in the VAS 2 and VAS 3 cases where the geostrophic approximation has slightly smaller Si scores than the other retrieval methods. As noted in the discussion of rms error, the addition of terms involving vertical motion did not significantly improve the retrieval; in some levels the ‘Full inviscid’ retrieval has the highest s x score of all the approximations compared. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -198The differences between retrievals using different boundary conditions are demonstrated in the next set of graphs. Figure 7.44 compares the rms error of virtual temperature retrievals in sigma coordinates using the two-scale approximation with Dirichiet and Neumann boundary conditions. Once again, using Dirichiet boundary conditions reduces the rms error of the retrievals considerably. This decrease is generally between 0.5 and 1 K, except near the surface where the improvement is larger. The rms error above a = 0.8 is 1 K or less for nearly all levels; in VAS 5 the error is only about 0.5 K. In the lowest levels (below a = 0.8) the Dirichiet retrievals are much poorer than in the higher layers, though the values are smaller than in the Neumann retrievals. Unlike 0 Q irichlet tysumann 0.2 0.2 0.4 0.4 cd E a> to E .S> CO <0 0 .6 0.6 0.8 0.6 a) VAS 0 1 1 2 3 RMS error (K) 5 b) VAS 3 o 1 2 3 4 5 0 — dinchlet 0.2 a — Neumann 0.2 0.4 a E O) 0.4 0.6 W0.6 0.8 0.6 c) VAS 4 o i .1. .1-1, i 1 2 3 RMS error (K) 1 4 5 d) VAS 0 1 2 3 RMS error (K) 4 5 Figure 7.44 Rms error (K) for temperatures derived from wind using the twoscale approximation in sigma coordinates for various boundary conditions. Circles: Dirichiet boundary conditions. Squares: Neumann boundary conditions. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -199- a) VAS a 0.2 es 0 .4 E a 0) 0.6 Dirichiet — Neumann 0 20 40 60 S 1score 80 100 S .sco re c) V A S 4 d) VAS 5 0.2 0 .4 a> 0.6 0.6 — Dirichiet — Neumann 0 20 Dirichiet Neumann 40 60 S.sco re 60 100 S.score Figure 7.45 Si scores for temperatures derived from wind using the two-scale approximation in sigma coordinates for various boundary conditions. Circles: Dirichiet boundary conditions. Squares: Neumann boundary conditions. the height retrieval in sigma coordinates (Figure 7.28), there are no levels where the Neumann retrievals have lower rms error than the retrievals using the Dirichiet conditions. The Si scores for the retrievals in sigma coordinates using the two-scale approximation with Dirichiet and Neumann boundary conditions are shown in Figure 7.45. Once again, the retrievals performed using Dirichiet boundary conditions have lower S i scores than the Neumann boundary condition retrievals; this reflects the information added by use of the Dirichiet boundary conditions. The difference between the Si scores near the surface is smaller for R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 0 0 - a) M s 2 0.2 0.2 0.4 0.4 .S’ .S’ 0.6 0.6 Dirichiet jboundary conditions Dirichiet boundary conditions No error U0KRMS 0 1 2 3 RMS error (K) i . . . , i . . \ : c) v a 4 5 1 Q 3 . . . . . . . . . ! . . . . 0.2 ■ ?S : ........ € : f t L I .. . ................*|................. ................. 1............... - 5 i . i . 1 . . . . 0.8 ! ........... -------------- - Dirichiet jboundary . conditions i 0.8 l L ............ 1 i ... 0.6 Dirichiet jboundary . conditions Q ----- - — tO K RMS ■ .,.. 4 |; 0.2 E (S’O 2 RMS error (K) — o - No error 1;0 K RMS • ■. . . . i T T " : — • < RMS error (K) RMS error (K) Figure 7.46 Comparison of rms error(K) for temperature fields retrieved from winds in sigma coordinates using Dirichiet boundary conditions. Circles: retrievals using "true" temperature field as boundary condition. Squares: retrievals using temperature field with added error (cre= 1 K). the sigma coordinate retrievals than for the retrievals performed in pressure coordinates; we would expect this since the area over which the retrievals are performed is smaller for the lower levels in pressure coordinates. Figures 7.46 and 7.47 show the effects of error in the Dirichiet boundary conditions for the sigma coordinate retrievals. The retrievals with perfect boundary conditions are compared to retrievals where a 1 K rms error field has been added to the boundary. The results of this experiment are quite similar to the results in pressure coordinates (Figures 7.38 and 7.39), the error in the boundary conditions has little effect on the rms error. The changes in Si scores R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -201- a) VAS I 0.2 0.2 0.4 0.4 0.6 0.6 Dirichiet jboundary. conations Dirichiet jboundary • concfitions 0.8 o No error -a - ijjK R M S 0 20 40 60 S . score 80 0 100 c) VAS 4 60 S . score 80 too d) V A S 5 0.2 0.2 0.4 0.4 ot CO 0.6 0.6 Dirichiet jboundary conditions Dirichiet jboundary. concfitions 0.8 ■■ e Nb error - -a - 1.0 K RMS 0 20 40 60 S 1score 80 —©— Nb error . -a - 1.0 K RMS 100 0 - 100 S j score Figure 7.47 Comparison of s i scores for temperature fields retrieved from winds in sigma coordinates using Dirichiet boundary conditions. Circles: retrievals using "true" temperature field as boundary condition. Squares: retrievals using temperature field with added error (ae = 1 K). are relatively larger than the changes in rms error, but neither the rms errors nor the Si scores become as large as those for the retrievals with Neumann boundary conditions. The large values of rms error and Si score near the surface are a problem with the sigma coordinate retrievals. The fields are very poor, even when Dirichiet boundary conditions are used. Modica and Warner’s (1987) analysis in sigma coordinates using data from the PSU/NCAR model (Anthes and Warner, 1978) showed large errors in height fields in the boundary layer, when calculations were made with the frictional terms omitted from the R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 0 2 - divergence equation. Their analysis confirmed that the frictional terms play an important role in the balance of height and wind fields in the boundary layer of the PSU/NCAR model. We can infer a similar effect in the calculations here, that the height and wind perturbations unresolved by our observing network play an important role in the balance between wind and height (and hence temperature) in the boundary layer. Unfortunately, the error cannot be removed by simply adding another term to the divergence equation here, as it could be from the model generated data. It could be necessary to do something on the order of creating a boundary layer numerical model if we wanted to account adequately for the unresolved motions. (Naturally this is beyond the scope of this research.) Another factor to take into account is that the Profiler measurements in the boundary layer will be limited, the first wind level available will be 500 m above the surface (at approximately the 0.95 a level) so there will be a lack of detailed wind information near the ground in any case. Considering this lack of observations, then, and the lack of quality of retrievals even if observations were available, it is advisable to look into other methods of obtaining estimates of temperature fields in the lowest levels that do not require divergence equation retrievals. In other words, maybe we should just not bother trying to use the wind information in the lowest few layers, but replace the wind-derived fields with better estimates of temperature. Our eventual goal for this section is to obtain first guess temperature gradients for use with ground-based radiances in the combined retrievals. We want to find better temperature estimates than can be obtained with low level winds. We assume that we have observations of temperature on the surface that will give us fairly good quality estimates of temperature gradients there. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 0 3 - The statistics above show the sharpest increase in error in the wind derived fields occurs below the 0.825 a level. If we assume that the meso-oc scale or synoptic scale horizontal temperature gradients (the scales observed on this network) vary fairly smoothly with height, then our best estimate of temperature gradient is a linear interpolation between o = 1 (surface) and o = 0.825. Since the intent is to obtain estimates of the temperature gradients, the we interpolate deviations from the mean value of the fields at the surface and a = 0.825, and adjust the mean values of the interpolated fields to match the ‘true’ mean. Kuo, et al., (1987) also used interpolation between the surface temperature field and a retrieved field at a higher level in their study, their interpolation was between the surface and a = 0.865. The motivation in their study was to improve error statistics, rather than to obtain optimal temperature gradient estimates. Figures 7.48 and 7.49 compare the rms error and S i scores for temperature fields retrieved from wind with those derived by interpolation between the measured surface field and the retrieved field at a = 0.825. GalChen's two-scale approximation and Neumann boundary conditions are used to produce the retrieved temperature fields shown. The improvement in the error statistics is dramatic, the decrease in rms error and Si score is quite large. The interpolation produces lower level fields with error statistics comparable to the wind retrievals at other levels. These improved fields, when interpolated to constant pressure surfaces, also have smaller rms errors than the retrievals from winds in pressure coordinates. The results of the dynamic retrievals of temperature can be summarized as follows: Virtual temperature fields can be derived using observed winds and the equations of motion. The retrieved temperature fields using Neumann R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -204' ' Mis1- - t-T -r-r l) VAS b) 1 1 1 1 i -r-.-r-T - | i- r - r - T - — 1.......... o i 0.2 - i- r - 7 VAS S 4- F: 0.4 E in i i .9 0.6 0.6 < *-" > - No interp 4- No if — -g . -S fC ;0 .8 2 5a interp. Sfc - 0.825a interp. : ...............i ........ ^ i i • : . . . . 0 1 2 3 RMS error (K) 4 T . . . , L_«_ i—i_i 2 5 0.2 : ■ 0.4 L ....... £ J ................. .S> CO 0.6 -o 4- No interp j £ > 2 p - . 4 H :.............i> .1. i ................. . ^ - No interp — -a • - Sfc - 0.82 5a interp. 5 2 • ----- ~ ~ \ _____ ■ «, 7 7 ■. . . / i . . . . 3 RMS error (K) . . . . . , . ‘ T i •a t- sfc - 0.825a interp. 1 L. RMS error (K) • 1 ■ ■ ' d ) VASnS-----------3 ► 0 .... 3 3 RMS error (K) Figure 7.48 Comparison of rms error (K) for retrieved vs. interpolated boundary-layer temperature fields Temperatures are retrieved from wind using the two-scale approximation in sigma coordinates with Neumann boundary conditions. Circles: No interpolation, lower levels retrieved from wind. Squares: Lower levels interpolated between surface and a = 0.825. boundary conditions have somewhat higher error than the retrieved height fields; the rms error of the height fields was equal or better than that of the radiosonde measurements while the rms temperature error is somewhat larger. The Si scores (generally ~40-50) are also only fair. There was little difference in errors when different approximate divergence equations are used; this may be because the lower order terms are about the same size as the error in the retrievals due to error in the wind observations. Using Dirichiet boundary conditions improves the retrievals; the addition of a 1 K rms error to the boundary conditions had only a small effect. The sigma and pressure R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 0 5 - coordinate retrievals had similar errors, except near the surface. The large surface error values for sigma coordinate retrievals can be reduced by using surface temperature fields to get a better estimate of the temperature gradients. T b) VAS h a) VAS 2 0.2 0.4 No Interp Sfc *0.9259 interp. - 9 — No interp *4 — Sfc *0.0259 interp. 0.6 0.8 40 0 100 60 20 40 60 60 100 S ,score 0 C) VAS 4 ; d) VAS 5 0.2 0.2 0.4 0.4 I cd E CO -Q — No interp * 4 — Sfc*0.625ainterp. 0.6 i ^ \ 0.6 V 0.8 1 : .............. I........C > 0.8 -t-* . . 0 20 40 60 S 1 score 80 — cj — No Interp — *4 — SfC'0.625ointerp.’ ...... 1.................j ............... ' I i i- i^ . i I _ j_ i ............~ . . -7 -* 100 S j score Figure 7.49 Comparison of s i scores for retrieved vs. interpolated boundarylayer temperature fields Temperatures are retrieved from wind using the two-scale approximation in sigma coordinates with Neumann boundary conditions. Circles: No interpolation, lower levels retrieved from wind. Squares: Lower levels interpolated between surface and a = 0.825. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 0 6 - 7.4 Retrieval from Radiance Data The next set of objectives involves the use of the simulated groundbased radiometer data to produce temperature profiles. The radiance retrieval theory is discussed in Chapter 4. The methodology used in these retrieval calculations is discussed in sections 6.3.1-6.3.3. One objective is to show that our simulated radiance data can produce results similar to those of previous studies; this is demonstrated with a set of retrievals using regression coefficients of soundings at Oklahoma City and Stephenville. The same set of Oklahoma City and Stephenville soundings is used in another task, to compare the statistical regression retrievals to physically based retrievals carried out using Smith’s method, and the “minimum information” method, to show how the choice of retrieval method can influence the results. Since we are using the OSSE methodology, where our ‘simulated’ radiances exactly satisfy our radiative transfer equation, we also need to simulate the effect of observational error and show the sensitivity of the retrievals to error in the observations. The second objective is to obtain radiance retrievals of virtual temperatures that can be compared to the retrievals from wind information and the combined retrievals. Smith’s method and the “minimum information" method are used to retrieve vertical profiles of virtual temperature at gridpoints. These ’one-dimensional’ retrievals are compared to retrievals using gradient versions of Smith’s method and the “minimum information" method, and GalChen’s method, so that the concept of using radiance gradients to retrieve temperature gradients can be investigated. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -207- 7.4.1 Temperature Retrievals at Oklahoma City and Stephenville In our study of radiance retrievals at Oklahoma City and Stephenville, the first thing we would like to examine are examples of soundings that are retrieved using statistical regression, Smith’s method and the ‘minimum information’ method. These soundings can be found in Appendix C, Figures C.3 - C.8 . In these figures, the thick solid line is the observed temperature profile at that time and location, the thinner solid line is the retrieved temperature profile obtained from the radiances, and the dashed line is the mean sounding from the 5 years of Oklahoma City data that served as a ‘first guess’ profile in the retrievals. Both soundings from Oklahoma City; the sounding on the left in each figure is from 1200 UTC on 7 March 1982, the one on the right is from 1200 UTC on 27 March 1982. These soundings correspond to the ones presented as examples for dynamic retrievals. (The Stephenville sounding on 27 March at 1200 UTC was not used since it was incomplete.) Figures C.3 and C.4 show retrievals from statistical regression coefficients, C.5 and C.6 are Smith’s method retrievals, and C.7 and C.8 have soundings retrieved by the ‘minimum information’ method. The odd figures (C.3, C.5, and C.7) are derived using only ground-based measurements, the even figures (C.4, C.6 , and C.8 ) also have data from two satellite channels entering into the retrieval. Note that the 27 March observed profile (thick line) is much closer to the first guess mean sounding profile (dashed line) than the observed profile from 7 March. We can see that the quality of the first guess profile has an effect on the retrievals; in each figure the 27 March retrieval is closer to the observed profile than the retrieval from 7 March. This is something that we can expect for retrievals from ground based radiances; most of the information contained in the R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -208observations is from layers near the surface, so the first guess profile must provide information about other levels in the sounding. When that information provided by the first guess is good, the retrievals are of better quality. Another characteristic of these retrieved soundings to note is that the retrievals performed using both ground-based and satellite radiance measurements tend to be better than the retrievals performed with groundbased measurements alone. The improvement in the profiles is primarily in the mid-troposphere (700 to 400 mb). The addition of satellite data is not sufficient to make a poor retrieval become a good one; this further emphasizes the importance of a good first guess profile. In these examples, the statistical regression retrievals are the best overall. The retrievals using both ground-based and satellite radiances (Figure C.4) are fairly close to the observed sounding, but both sets of retrieved profiles are smooth and do not contain much detail in the vertical. A typical feature of the Smith’s method retrievals from ground-based radiances is demonstrated in Figure C.5: the difference between the mean profile and the retrieved profile is nearly constant with height. This characteristic should be expected, considering the shape of the ‘contribution function’ depicted in Figure 4.5; retrieved profiles above 500 mb contain near constant contributions with height from two of the four channels. While this characteristic may yield fairly good retrievals in the lower to mid-troposphere (as in the 7 March retrieval), the retrievals above 300 mb are nearly useless. Figure C .6 shows that the addition of satellite data causes the adjustment in upper levels to be smaller; however, this gives little improvement in the retrievals. The 7 March retrieval is much worse below 300 mb when the satellite data is added. The minimum information retrievals in Figures C.7 and C.8 are somewhat Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -209noisy; this is because the profiles of rms error from the 5 years of Oklahoma City data are not smooth, and the weights given to the first guess profile are inversely proportional to the square of the rms error. (The coefficients for these retrievals were shown in Figure 6.20.) The retrievals for 7 March are not as good as those for the statistical regression, which shows that with a poor first guess the statistical regression method can perform better than minimum information. The retrievals for 27 March using minimum information retrievals are slightly better than the statistical regression retrievals. In summary, these retrieval examples show us that it is possible to obtain fairly good sounding profiles provided that the mean profile used as first guess is not too far from the actual profile we wish to retrieve. The retrieved profiles tend to smooth out smaller scale features such as inversions. Adding satellite radiances to the retrievals can improve the retrieved soundings. We want to look at the statistics now, to determine whether the results of the statistical regression retrievals are comparable to the ones previously reported by Westwater, et al., (1985), and to compare these retrievals with retrievals performed using Smith’s method and the minimum information method. The experiments using radiance retrievals at Oklahoma City and Stephenville are summarized in Table 7.5, and the retrievals performed are listed in Table 7.6. In the discussion of the statistical regression method (Section 4.1.1) it was stated that coefficients calculated using the inverse of the radiance covariance matrix (r r t ) may fit coefficients too closely to the data, and that a noise term eeT can be added to condition the solution. The use of conditioned coefficients is advisable in this study since the radiance ‘measurements’ used to generate the coefficients are calculated from the sounding data and thus satisfy R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. •2 1 0 - Table 7.5 Experiments for radiance retrievals at stations Exp Purpose of experiment Remarks M compare regression retrievals with Conditioned coeff. retrievals conditioned and unconditioned have more error but are less sensitive to observation noise coefficients N show sensitivity of physical retrieval Relatively insensitive to methods to obs. noise observation noise 0 compare ground-based retrievals satellite data improves retrievals with and without satellite data in mid- to upper levels Table 7.6 Temperature retrievals from radiance data at stations Radiance data used Retrieval method Ground Ground + satellite Ground Ground + satellite Ground Ground + satellite Ground Ground + satellite Ground Ground + satellite Ground Ground + satellite Ground Ground + satellite Ground Ground + satellite regression, unconditioned regression, unconditioned regression, conditioned regression, conditioned regression, unconditioned regression, unconditioned regression, conditioned regression, conditioned Smith’s method Smith’s method Smith’s method Smith’s method minimum information minimum information minimum information minimum information Instrument Used in error experiment no M no M no M M no 0.5 K M 0.5 K M 0.5 K M,0 0.5 K M,0 no N no N 0.5 K N,0 0.5 K N,0 no N N no 0.5 K N,0 0.5 K N,0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 1 1 - No error V i 0.5KRWS error 200 200 E, 400 E, 400 r " 2 600 600 800 800 : Ground^jsQTSnna UnronditioAed coeff. Ground: based iUnconditiohed coeff. 1000 1000 o 1 2 3 4 o 5 1 RMS error (K) Stoerror Tgi»: _ 4 5 No error T» I O.SKRMS errbr 200 2 3 RMS error (K) 0.5K RMS error 200 E, 400 E, 400 600 600 800 800 Ground based Conditioned coeff. Ground + satellite Conditioned coeff. 1000 1000 o 1 2 3 RMS error (K) 4 5 0 1 2 3 RMS error (K) 4 5 Figure 7.50 Rms error (K) of retrievals at Oklahoma City and Stephenville using regression coefficients. Top panels: retrieval from coefficients fitted without conditioning factor. Bottom panels: retrieval from coefficients with (0.5K)2 conditioning factor added to diagonal of covariance matrix. the radiative transfer equation exactly, without any measurement errors. The purpose of the first experiment is to determine if we will need to use this conditioning factor to control the sensitivity of the solutions to observational noise, and to find out if decreasing sensitivity also results in a degradation of the retrievals. The rms error of retrievals from conditioned and unconditioned coefficients are compared in Figure 7.50. The two graphs on the left show the rms error of retrievals derived without the (0.5K)2 conditioning factor; the graphs on the right show retrievals from coefficients that had the conditioning factor Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 1 2 - added to to the radiance covariance matrix. The graphs on the top of the figure clearly demonstrate the sensitivity to error of coefficients derived without the error factor, while the graphs on the bottom demonstrate the effectiveness of the error factor in reducing the sensitivity to error in the brightness temperature measurements. The unconditioned coefficients have retrievals with rms errors as much as 4 K greater if 0.5K Gaussian noise is added to the brightness temperatures. The rms errors of the retrievals from conditioned coefficients increases by only about 0.5K, or less, given the same noise-contaminated brightness temperatures. The retrievals have the smallest error near the surface, about 0.6 K. The rms errors increase to 1.5 K by mid-troposphere, and reach a maximum of about 3 K near the tropopause level (200 mb). These results are in good agreement with the results presented by Westwater, et al. (1985) for retrievals of Denver soundings. The use of a conditioning factor increases the rms error of the groundbased retrievals; the difference is about 1.5 K at 500 mb, if the observations are without error. The unconditioned coefficients produce large rms errors for observations that have error, though, so omitting the conditioning is not a viable alternative. The difference between rms errors for conditioned and unconditioned coefficients is much smaller for retrievals with both ground-based and satellite observations, the maximum difference is less than 0.5 K. The disadvantage of the increase in rms error when using conditioned coefficients is outweighed by the necessity for decreasing the sensitivity of the retrievals; thus, use of conditioned coefficients is best. Our next area of concern involves the physical retrieval methods used in this research, Smith’s method and the minimum information method. How do these retrieval methods compare to the statistical regression retrievals, and are R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 1 3 - 200 £ 400 £ 600 200 £400 600 800 800 error 1000 1000 8 0 10 1 RMS error (K) Ground-based “ Min. information 2 3 RMS error (K) 4 5 GroundfSftellf Min. information 200 £400 e 400 600 ® 600 800 No — -e — 9£K error •e — 9.5K error 1000 0 1 2 3 4 5 RMS error (K) RMS error (K) Figure 7.51 Rms error (K) of retrievals at Oklahoma City and Stephenville using Smith’s retrieval method (top) and the minimum information method (bottom). Left panels: retrievals using ground-based channels only (note scale change on axis for Smith’s method). Right panels: retrievals using both ground-based and satellite channels. either of these methods sensitive to observation noise? Figure 7.51 shows the rms error for retrievals by Smith’s method (top) and the minimum information method (bottom) using the same radiances and the same observation error as for the regression retrievals. The plots demonstrate that these retrieval methods are less sensitive to observation noise than the retrievals by statistical regression. The increase in rms error due to the 0.5K rms observational noise was generally about 0.2K or less for the Smith’s method retrievals and 0.5K at most for the minimum information retrievals. The observational noise produced little change in rms error above 500 mb for R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -214minimum information retrievals. Note that the plot on the upper left has a horizontal scale twice that of the other plots; the Smith method retrievals have very large rms error in the highest levels when only ground-based channels are used. From Figure 4.5, which shows the ‘contribution function’ for a Smith method retrieval when only the ground-based channels are used, we see that the measurements from the most transparent channel (52.85 GHz) produce large changes in temperature in the highest levels even though the weighting function is small at those levels. This means that unless the radiance of the “first guess” profile for that channel is close to the observed radiance, the Smith retrieval method will produce a large, probably erroneous, adjustment to temperatures at higher levels. Figure 7.52 shows the rms errors for the three retrieval methods replotted so that the retrievals using only ground-based data can easily be compared to the retrievals using both ground-based and satellite data. The standard deviation of the sounding data that are used in the retrievals is plotted for reference (solid line). Use of satellite data produced much lower rms error for levels above 800 mb in the regression retrievals from conditioned coefficients (top panel). The largest difference, about 1 K, was at 500 mb. The satellite data did not improve the Smith’s method retrievals (middle panel) below 300 mb. In fact, the retrievals using satellite data had rms errors about 1 K greater at 350 mb. Above 300 mb the retrievals using ground-based radiances only have extremely large rms errors, the retrievals with satellite data perform much better in those levels. This is not to say that the Smith’s method retrievals with satellite data are good at those levels; on the contrary, the rms errors are almost as large as the standard deviations of the sounding data so that the retrieved profiles show little improvement over the first guess. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 1 5 - 200 I 400 s 3 600 Q. 800 1000 0 2 1 3 5 4 6 7 T ^ RMS error (K) . 1' ' ' ! ‘ 1 1! 1 1‘ v2 ~ - r - r f . 200 ------- Std. deviation Grotfid based * E 400 : I 1 . 3 ' ' ! ! ............ S 'S 600 : ............... P : ; / H V j r ................. !.................. : : : a ’Sij T ! ] Sm iths mqthbtl^_ ).5K rms e m x To ' — ■ \ j c 1000 3 4 RMS error (K) ............... j 1 ’ 1 1 | 1 ' " ’" H t 200 E : ^ 1 I 400 • ........ | .............. .. ' ' ' '' j ' ■ " > “‘j r 0 \ ------- Std. deviation ■ a Ground-based ----- ;— .__ “ \ I N * - o - Ground+satellite dS.^ | / 600 1............... j ................. .................................. i T i 800 \ 1 \ \ i i...... i \ i ! 1000 Min. inform iatibv. ).5K rms ertor TD . . . , . i . p , , 3 4 RMS error (K) Figure 7.52 Rms error (K) of retrievals at Oklahoma City and Stephenville using regression coefficients (top), Smith’s method (center) and the minimum information method (bottom). The solid line in each graph is the standard deviation of the temperature profiles used in the retrievals. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -216For the minimum information retrievals, the rms errors are lower for levels above 500 mb when satellite data are used, and the improvement is not as large as for the regression retrievals. The largest differences (0.8 K - 1.2 K) between the retrievals with and without satellite data are in layers centered on the 500 mb level and the 200 mb levels; these correspond to the levels of the maximum weighting functions for the satellite channels (see Figure 6.7): z * = 4.8 and z * = 10.8 (~ 550 mb and 250 mb). The statistical regression retrievals have the lowest rms error overall of the three retrieval methods, particularly when satellite data is used. The Smith’s method retrievals have lower rms below 300 mb when only ground-based channels are used, but the extremely large errors above 300 mb make this method unsuitable. The minimum information method rms error for groundbased only retrievals is close to that of the statistical regression, but this method does not show as much improvement when satellite data is added. Let’s look at some other statistics to compare these retrieval methods. Figure 7.53 shows the average of (retrieved - observed) temperatures, i.e., the bias of the retrieved soundings. The solid line is the difference between the mean value of the 5 years of spring data used in calculating the coefficients and the mean of the soundings used in the retrieval. The 5-yr “climatological” mean temperature in the lower troposphere was much higher than the mean of the AVE/VAS sounding sample used in the retrievals. However, this did not translate into a large bias in the retrievals; in fact, the bias of the retrieved soundings was only about ±0.5 - 1 K through most of the troposphere. At tropopause level (above 300 mb) the “climatological” bias becomes negative. The retrievals by Smith’s method using only ground-based channels has a much larger bias above 300 mb, reflecting the large error in the retrievals at R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 217 - 200 200 E, 400 £ £ 600 600 800 800 jCorxJitionegcoaffe. j 0.5K B 1000 1000 -2 6 Bias (retrieved • obs) (K) 0 2 Bias (retrieved * obs) (K) 4 6 Syr mean • sample mean ■9 — Ground-based 200 E, 400 S 600 800 Minimum irffohnation j O.SKrmsjBrm^Tp 1000 -4 •2 0 2 4 6 Bias (retrieved - obs) (K) Figure 7.53 Bias (K) of sounding profiles retrieved with statistical regression (top left), Smith’s method (top right) and minimum information (bottom) using brightness temperatures with (0.5K)2 random error added. these levels. The bias of the other retrievals is close to the “climatological” bias at these levels, reaching -2 K for statistical regression and minimum information retrievals. The bias values for the statistical regression agree with Westwater, et al., (1985), who reported bias values ranging from less than 0.5 K near the surface to 2 K at 100 mb. Figure 7.54 shows the s x scores for retrievals performed at Oklahoma City and Stephenville. These Si scores are calculated using differences between the OKC and SEP soundings at each pressure level. The regression retrievals did the best of the three methods in the lowest levels, below 800 mb. The Si scores above 800 mb all tended to increase with height, though the R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -218- | . I I | i— «— j Statistical regression . Conditioned coefficients - .......... +W « T 8 fT g —f™ ........... - ■ - T - r 200 . S, 400 ¥3 a S 600 ................. i ............. 1000 - ............... i ..............a * . . : i _ r> : S : i—1 1 . . i l . J................ T • • 1 I •' ; Smith's riethod N o e m *T „ < b ! v I .......... |...............: : Ts : **Q - - JiJu _ I ; *^ -C 800 r—r—r : ..... j .... 1 | £ ■ •—J » « ■ 1 ■ . ■ ' ' I j--. J I |........ ! I F-. T........ : ■ <db 800 - .............. T ! *r■a* ’— — Ground-based • - • o - Ground+saieHrte 1000 i i i i,> t i i i — ie — Ground-baaed -- ■ jo - Ground♦ satellite. I ' i..........\s(S ... 400 e e 6oo EL . • 20 40 6*} -I 40 60 S -sco re I -J la- 80 100 Minimumiinformationf No eirrorT, B 200 £ 400 2 a. 600 s<.' 800 i t*** : am — -o — Ground-based 1> - Ground+saleilite 1000 .... . _i_,_ . , L i . . , 20 . j_ ._ T ^ r-L 40 60 100 S 1 scdre Figure 7.54 Si scores of retrievals at Oklahoma City and Stephenville. Top left panel: statistical regression retrievals. Top right panel: Smith’s method. Bottom panel: Minimum information retrievals. statistical regression and Smith’s method retrievals both showed a decrease in Si score at 400 mb. The minimum information Si scores are higher than those of the other retrievals. Above 300 mb, ground-based Smith retrievals have Si scores off the chart, again demonstrating the extreme error in the retrievals at those levels. Overall, the Si scores are comparable to ones that would be produced by having an error of 1.0K in the gradient calculation. (For the radiosonde accuracy of 0.5K, the Si score would range between 20 and 40.) The profiles at Oklahoma City and Stephenville are derived with the same retrieval methods and ‘first guess’ sounding, so any difference between the profiles at these stations must be entirely the result of the difference R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -219between the brightness temperatures. Thus, the gradient error statistics should be a good measure of how well the brightness temperatures can be used to distinguish one profile from another. The Si scores show the same things as the rms error statistics: the ground-based radiance measurements can do a fair job of determining the temperature gradients near the surface (below 700 mb) but the retrieval quality decreases with height (though use of additional information in the form of two microwave satellite channels can be used to improve the upper level retrievals to some extent). In summary, all the retrievals perform best near the surface where the ground-based weighting functions are the largest; both the bias and the rms error are larger at higher levels. This effect is more pronounced for retrievals using ground-based measurements only, but it is still present when measurements from the two satellite channels are added. Examination of the statistics does not reveal any one ‘best’ method. The statistical regression method appears to have smaller errors overall than the minimum information method. The advantage of the regression method over the minimum information method is that it includes information about covariances of temperature between levels as well as the error variances. The Smith method retrieval statistics are comparable to those from the statistical regression retrievals, except for the highest levels in the “ground-based channels only” retrievals. The ability of the Smith method to match the the regression retrievals may be due to its fitting the profile somewhat closer to the radiance measurements than the conditioned regression coefficients. The unconditioned regression coefficients had the smallest rms errors but also demonstrated the greatest sensitivity to measurement noise, so they should not be used in a practical retrieval scheme. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 2 0 - 7.4.2 Virtual Temperature Retrievals at Gridpoints The next objective is the evaluation of virtual temperature retrievals from brightness temperatures at gridpoints. The two physical retrieval methods used at station locations, Smith’s method and the minimum information method, are used to retrieve virtual temperature profiles at gridpoint locations. These retrievals using brightness temperatures are compared to retrievals by Smith’s method and the minimum information method using brightness temperature gradients as derived in Section 4.2 and described in Section 6.3.3. Retrievals from brightness temperature gradients are also performed using variations of Gal-Chen’s (1988) method. In this section, our purpose is to demonstrate how the radiance retrieval methods perform and to compare the conventional “one dimensional" radiance retrievals to the retrievals using radiance gradients. These retrievals, performed using radiance data only, will be compared to combined wind/radiance retrievals in the next section. The conventional “one dimensional" radiance retrievals are performed in the same fashion as the retrievals at station locations. The mean temperature field on the grid is used as the ‘first guess’ for the retrievals. The gradient radiance retrievals are performed by using brightness temperature gradients and either Smith’s method or the minimum information method to derive temperature gradient estimates. The temperature field is then calculated to have the best fit to these gradient estimates in a least-squares sense. A third gradient radiance retrieval is performed using Gal-Chen’s method with the mean temperature field as first guess. Because the mean temperature field is used as first guess, the first guess temperature gradient for all of these retrievals is zero. This is a very poor first guess, except perhaps for the VAS 5 case R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 2 1 - where temperature gradients are small. We will first look at some examples of retrieved fields on constant pressure surfaces. Figures 7.55 (a) - (d) show the ‘true’ and retrieved virtual temperatures analyzed on the grid at the 700 mb level for 7 March 1982, 0000 UTC. (Retrievals for this level and date from wind data are presented in Figure 7.30.) The true field shows the cold air centered in the north-central part of the analysis domain, spreading southward into Texas. The retrieval using Smith’s method in Figure 7.55 (b) shows the strength of the cold trough fairly well, but the analysis seems to be quite noisy, because the vertical temperature profile at each gridpoint is determined separately. The retrieved profile has a large error in the southeast corner of the domain. The problem with the southeast corner may be in part due to the vertical discretization; the lowest gridpoint pressure level is at 975 mb, and the surface pressures in this corner are greater than 1000 mb. The minimum information retrieval (Figure 7.55 (c)) is much more smooth than the Smith’s method retrieval but the temperature gradients are also underestimated in this retrieval. Note how the -12 isotherm remains to the north, in Kansas, rather than extending down into Oklahoma. Figure 7.55 (d) shows the retrieval produced by Smith’s method and the radiance gradient field. This retrieval is much more smooth than the Smith’s method retrieval in Figure 7.55 (b). The temperature gradients in this retrieval are stronger than in the minimum information retrieval in Figure 7.55 (c). The strong gradient in the southeast corner is not represented. Figure 7.55 (e) and (f) are the minimum information gradient retrieval and Gal-Chen’s method retrieval for this time. The two analyses are quite similar. The retrieved field is almost flat; there is no -12 isotherm at all on this level. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -222- -12 a) ‘True’ field J2 (b) Smith’s method, retrieval from radiances Figure 7.55 Temperature analyses for 7 March 1982, 0000 UTC, on the 700 mb pressure surface. Solid lines: temperature contours (interval = 2 K). Dashed lines: difference between retrieved field and 'true' virtual temperature field (contour interval = 2 K). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 2 3 - (c) Minimum information, retrieval from radiances J2 (d) Smith’s method, retrieval from radiance gradients Figure 7.55 (continued) Temperature analyses for 7 March 1982, 0000 UTC, on the 700 mb pressure surface. Solid lines: temperature contours (interval = 2 K). Dashed lines: difference between retrieved field and 'true' virtual temperature field (contour interval = 2 K). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -224- (e) Minimum information, retrieval from radiance gradients -10 •4----- (f) Gal-Chen’s method, retrieval from radiance gradients Figure 7.55 (continued) Temperature analyses for 7 March 1982, 0000 UTC, on the 700 mb pressure surface. Solid lines: temperature contours (interval = 2 K). Dashed lines: difference between retrieved field and 'true' virtual temperature field (contour interval =2 K). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 2 5 - These retrieved fields are much closer to the flat ‘first guess’ field than the fields from other retrieval methods presented here. The retrievals at the 500 mb level show even less of a temperature gradient than at the 700 mb level. The flat ‘first guess’ field receives more weight in these analyses than in the gradient Smith method analyses. Figures C.9 - C.13 in Appendix C show examples of vertical profiles of brightness temperature derived using the methods listed above. These profiles have been interpolated from the grid in the same fashion as the example profiles for the wind-derived retrievals and are for the same times and stations. The profiles on the left of each figure are from Oklahoma City, 1200 UTC 7 March 1982; the profiles on the right are from Stephenville, 1200 UTC 27 March 1982. The line convention is the same as in the previous figures: the thick solid line is the observed temperature profile at that time and location, the thinner solid line is the retrieved temperature profile obtained from the radiances, and the dashed line is the ‘first guess’ profile, which in this case is the mean temperature of the gridded field for the particular observation time. The Smith’s method retrievals in Figure C.9 follow the true sounding fairly closely, and are indeed much better than the Smith’s retrievals shown in Figure C.6 . The improvement in the retrievals is undoubtedly due to the use of 'first guess’ mean profiles that are closer to the true soundings. Both the soundings capture the major low level features, though they are somewhat smoothed when compared to the true profile. The correction to the ‘first guess’ mean profile is small in the upper levels, as is characteristic of the Smith’s method retrievals in this study from ground-based and satellite data. Retrievals at gridpoints from the minimum information method are shown in Figure C.10. These retrievals also show the value of having a good ‘first R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 2 6 - guess' to work with. The 7 March sounding is a major improvement over the retrieved sounding in Figure C.7, but the radiances appear to have made little contribution to the information in the retrieved sounding since the correction to the ‘first guess’ in this retrieval is quite small. The 27 March profile was not retrieved as well; the retrieved profile picks up the strong inversion at 800 mb but then follows the ‘first guess’ above 700 mb. The ‘first guess’ profile for these retrievals have a greater relative weight than the ‘first guess’ for the minimum information retrievals at station locations; the error standard deviation is much larger for the station retrievals than for the gridpoint retrievals and the weighting of the first guess relative to the radiance measurement for the minimum information method is inversely proportional to the standard deviation of the error of the first guess. Figure C.11 shows virtual temperature profiles retrieved using Smith’s method retrieval and brightness temperature gradients. Neumann boundary conditions are used in estimating the temperature field from the gradients. The difference between the retrieved profile and the first guess here is nearly constant for levels away from the surface. This leads to a fairly good fit with the ‘true’ profile between the surface and 300 mb, but the retrievals are poor above that level. The low level inversion is reproduced in the 27 March retrieval but the retrieved inversion is too weak. The profiles in Figures C.12 and C.13 from the gradient minimum information retrievals and Gal-Chen’s retrieval method are quite similar. Both sets of retrievals do a fair job of correcting the first guess below 800 mb, but by the 700 mb level the retrieved profiles follow the first guess. These profiles are not as good as the ones from the one-dimensional methods, which demonstrates the poorness of a zero-gradient first guess for these retrievals. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -227The above examples demonstrate the characteristics of the retrievals from radiances and from radiance gradients at gridpoints. We turn our attention now to examination of the error statistics, so that we can fulfill our objective of comparing the "one-dimensional” retrievals from radiances to the new methods for retrieval from radiance gradients. Table 7.7 summarizes the experiments that are performed with the radiance retrievals; the retrievals that have been performed for these experiments are listed in Table 7.8. Table 7.7 Experiments for gridpoint radiance retrievals Exp P Purpose of experiment Remarks compare radiance retrievals with rms error similar; radiance retrievals from radiance gradients retrievals affected by bias Q show sensitivity of retrieval methods some effect with Smith’s method, to obs. noise others insensitive to noise R determine effect of increasing the reduces error of retrieval, relative weighting on radiance increases sensitivity to noise Table 7.8 Gridpoint virtual temperature retrievals from radiance Data used in retrieval radiance radiance radiance gradient radiance gradient radiance gradient radiance radiance radiance gradient radiance gradient radiance gradient radiance radiance gradient radiance gradient Retrieval method tb wgt. Smith’s method minimum information gradient Smith’s method gradient min. information Gal-Chen’s method Smith’s method minimum information gradient Smith’s method gradient min. information Gal-Chen’s method minimum information gradient min. information Gal-Chen’s method (4.21) (6 .20 ) (4.42) (4.48) (6.28) (4.21) (6 .20 ) (4.42) (4.48) (6.28) (6 .20 ) (4.48) (6.28) 1 — 1 1 — 1 — 1 1 10 10 10 obs. Used in error experiment no P.Q no P.Q no P,Q no P,Q no P.Q 0.5 K Q 0.5 K Q 0.5 K Q 0.5 K Q 0.5 K Q no R no R no R R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 2 8 - In the first experiment, we will use the rms error and sx score to compare the results of the radiance retrievals with those from radiance gradients. Figure 7.56 shows the rms error of the “one-dimensional” (1-D) Smith’s method and minimum information method retrievals. The solid line is the standard deviation of virtual temperature on the grid. The rms error tends to be small near the surface and increases with height; the error from Smith’s method (circles) is usually smaller than that of the minimum information method (squares). Figure 7.57 shows the ratio of the rms error to the standard deviation of virtual temperature for the Smith’s method and minimum information retrievals just discussed. This ratio is fairly low near the surface, around 0.3 for the Smith’s method retrievals, but somewhat larger, around 0.5 for the minimum information retrievals. There is more variability in the curves for the Smith’s method retrievals; the minimum information curves are quite similar for VAS 3, 4, and 5. Both methods show relatively lower values for the VAS 2 case, which had larger standard deviations than the other three cases. These ratio values can be directly compared to Figure 7.34, the ratio of rms error to standard deviation for the retrievals from wind data. The wind-derived retrievals tend to be in the 0.5 - 0.7 range in the mid- to upper levels. The wind retrievals generally do better above 400 mb than the Smith method retrievals, and they are better than the minimum information retrievals above about 700 mb. The rms errors for the gradient radiance retrievals are shown in Figure 7.58. The rms errors from the minimum information method and Gal-Chen’s method are quite similar to each other, they are small near the surface and approach the standard deviation value fairly quickly with height. The rms errors from the gradient retrievals using Smith's method are much lower than the other gradient retrievals below 300 mb, but above that level they become quite large. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -229- 1 1 1 i ■1 1 1 a) VA! b) 1-D retrieval Moan T first guess 200 200 E 400 E 400 o 600 600 800 800 VAl 1-D retrieval Mean T first guess a. • ‘Ttnnrtfiirtflsirielinn— ^ • ( Smart's method — 0 4 Minimum information 1000 1000 0 1 2 3 RMS error (K) C) VAi 4 0 5 200 200 E 400 E 400 o 600 9 600 1000 1 3 2 S 4 4 5 1-D retrieval Mean T first guess 800 Standard deviation - 4 4 - Smith's method - - - o 4 Minimum ^formation 0 2 3 RMS error (K) d) VAl 1-D retrieval Mean T first guess 800 1 ■j ■ Standard {deviation — j - Smith's rrfethod - - -o- f Minimum Information 1000 0 1 RMS error (K) 2 3 RMS error (K) 5 4 Figure 7.56 Rms error (K) for retrievals on grid from radiance with mean temperature as first guess. Solid line: Standard deviation of temperature on the grid (error of first guess). Circles: Smith’s method. Squares: minimum information. ‘ Smith's method Minimum information 200 200 A E 400 2 3 CO £ Q_ 600 600 ■O— VAS 2 ' 800 h o- ’V A S 3 “ VAS 2 VAS3" VAS 4 VAS 5 800 •e - VAS 4 ‘ - » • VAS5 ! 1000 1000 0 0.5 1 1.5 2 0 0.5 1 1.5 2 oJa. Figure 7.57 Ratio of average rms error to average standard deviation of virtual temperature on constant pressure levels from retrievals using radiances with mean temperature as first guess. Left panel: Smith’s method. Right panel: minimum information. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 3 0 - b ) VAl a) VAJ 200 200 1 400 E ,400 <8 600 600 600 600 1000 0 1 2 3 1000 4 5 0 1 RMS error (K) 3 2 4 5 RMS error (K) c) VAS* 200 200 E, 400 600 600 600 600 -- 1000 0 1 2 3 RMS error (K) 1000 4 5 0 1 2 Standard deviation matHcwT....... *< Minimum information w i GaFChen> method 3 4 5 RMS error (K) Figure 7.58 Rms error (K) for retrievals on grid from radiance gradients with Neumann boundary conditions and mean temperature as first guess. Solid line: Standard deviation of temperature on the grid (error of first guess). Circles: Smith’s method. Squares: minimum information. ‘X’s: Gal-Chen’s method. Figure 7.59 shows the ratio of the rms error of the gradient retrieval methods to standard deviation of temperature for each of the VAS experiment days. The left top panel is from the Smith’s method retrievals, the minimum information retrievals are on the right, and the retrievals from Gal-Chen’s method are on the bottom. The gradient retrievals for Smith’s method have ratios that are similar to the 1-D retrievals (Figure 7.57) below 400 mb. Above the tropopause, the retrievals are poor with a large retrieval ratio. The gradient Smith method corrects the gradient approximately the same way all through the atmosphere. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 3 1 - I Smith's method 1 Minimum information E 400 E 400 a) 600 Graqient retrieval M eariT first guess VAS2 - VAS3 - r + - VAS4 Gracfient retrieval . MeariT first gue 68- VAS2 'r* - VAS4 . • VAS5 • Gal-Chen’s method 200 E 400 ® 600 Gradient retrieval Mearj T first guess Ip — VAS2 t O - VAS3 U - VAS4 600 1000 0 0.5 1 1.5 2 Ceo/0° T Figure 7.59 Ratio of average rms error to average standard deviation of virtual temperature on constant pressure levels from retrievals using radiance gradients with mean temperature as first guess. Top left panel: Smith’s method. Top right panel: minimum information method. Bottom panel: Gal-Chen’s method. This is not a bad strategy to pursue in the lower to mid-troposphere where there may be considerable correlation between the temperature gradients at successive levels. However, the method fails at higher levels at and above the tropopause because the stratospheric temperature gradients are quite different from the temperature gradients lower in the atmosphere. The ground-based channels contain little information about the temperature gradients higher in the atmosphere. The satellite channels contain information mainly about the average gradients through a fairly deep layer since their weighting function R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 3 2 - peaks are so broad. (See Figure 6.7.) The 54.96 GHz channel has weighting function values that are half the peak value at the levels z * ~ 5 km (535 mb) and z* = 18 km (105 mb); this includes a fair amount of both troposphere and stratosphere. Thus, the effect of the satellite channels on the retrieved profile is small and is spread out over many levels. Figure 7.60 is an example of a stratospheric gradient Smith's method retrieval from the 100 mb, 0000 UTC, 7 March 1982; the same observation time as the analyses in Figure 7.57. The ‘true’ analysis (Figure 7.60 (a)) has a south to north gradient with the warmest air over Kansas, the temperature gradient is the reverse of that found in lower levels; but the Smith’s method gradient retrieval has temperatures increasing to the south, just as in the lower levels. The ratios for the minimum information gradient retrievals and GalChen’s method are similar to the 1-D minimum information retrievals, except that the values of the ratio approach 1 more closely and at a lower level for the gradient retrievals than for the 1-D radiance retrievals. This shows the tendency for the corrections to the mean field ‘first guess’ to be smaller for the gradient minimum variance retrievals than for the 1-D retrievals, although neither method gives very much correction above the 700 mb level. Figure 7.61 shows the Si scores for the 1-D retrievals at gridpoints. As with the rms errors, the Smith’s method retrievals have consistently lower Si scores than the minimum information retrievals. The Si scores are fairly low near the surface for the Smith’s method retrievals. The minimum information retrievals have some rather large Si scores near the surface. Note that the standard deviation of the virtual temperature is small near the surface; this was explained in Section 7.3.1 as due to the decreased area of the constant pressure surfaces. The weighting of the ‘first guess’ field relative to the R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 3 3 - -54 ,-56 -56 (a) True’ field -60 -58 &• -56 (b) Smith’s method, retrieval from radiance gradients Figure 7.60 Temperature analyses for 7 March 1982, 0000 UTC, on the 100 mb pressure surface. Solid lines: Temperature contours (interval = 2K). Dashed lines: difference between retrieved field and ‘true’ virtual temperature field (contour interval = 2 K). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -234- ajVASfc i - r - T- i - r ; 1-0rotodval i ;b j V A S k ' | : M e a/rl tjret guess I I \ S. 400 | - ........... | ..............}............... I i 800 \ , , —i \ ' I I 200 i s* B 400 — + Smith's method Minimum information . 4? -ms£Zi . i- .- . , t- r - r - r - i . . . 4 T i . , i . ., 100 60 i i J * 1,1 1 1 1_*VDre*n4a*l J M«pi T fjrst gues* i CL J Q I I' ■ I i 40 60 S., score I ■1-D‘ retn$vai 1I , d) VAS k — Mean T fjrst guess " 4.........4-«.9—s.: 200 i i * - * j : r ' 9 ■ £, 400 S 3 8 s CL , 111 40 I r.........r kT ! 1 -......... ;•* ..............i ...... ............................. U ............. .. i , 2 S ...... i» : — 4 Smith’s method - - -o -i Minimum information , ;c j VA S 4 ' - , . - k * - “ ’ "j £ 20 r r - r - i -t— | » I , 1 1-D retrieval Mean T firstgGess i I ..............S.................................... ................. _ S 600 Q. ? ,? j 1000 i i : J4 **• S 2 600 a. . i 600 \ * jf ? \ !- i & \ : D . -r" ____ * j ........... " ‘jjSET'— 'Je'ir"Sfmrfffs Minimurflinformation J -* .-* < -• : 1 . . -r. t . .» . i , . . i . . . b. - S 600 Q. i ^ 800 1000 , _i 1 i 20 — -e 4 ^ H f r s method - - - o - j Minimum ififormatfcn-: t i 20 i . 40 i 60 . . . I . ■ 80 Figure 7.61 s i scores for retrievals on grid from radiance with mean temperature as first guess. Circles: Smith’s method. Squares: minimum information. radiance measurements is inversely proportional to the standard deviation of the temperature field, thus the ‘first guess’ is weighted somewhat more at the lowest levels than at levels just above, thus contributing to a larger gradient error in the lowest levels of the analysis. Figure 7.62 shows the s x scores for the radiance gradient retrievals. The variation of Sj. score in the vertical is quite similar to the variation in rms error (Figures 7.58, 7.59). The Si scores for the Smith’s method retrieval start fairly low, lower than the Sj. scores for the 1-D Smith’s retrieval. These Si scores have a moderate rate of increase with height up to the tropopause, then become extremely large. Naturally, the largest Smith’s method Si scores are for R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 3 5 - ;a j V A S fe ’T~ , T“ 'l ' ' ' I « "r“ i — -® —I Smith's method - - -o - i Minimum information ..............|..... • • -x • : Gj^hen'simethod b jV A S i ' 200 .. ...............?** 1 i*~ 1 » ................................ \................. 1..... ! I 1 I J ‘ m* /» : : • .• E, 400 s a 6oo Q. ? ! i I * ! 3E,* 400 * ■ s s £ . . . > | X £ * <^>| 600 \ i jf? ! r i i • •*«-* .. ...............T” \ .........[................................... | ..............** 4- “ .................. i • •** * ’ 7 i i i j r - 'i i . — -9 —i Smith's mefhod - - -o- - j Minimum information * ......... i..„;.-*.;jG a t£ h e n 'S :m e rth o d ■ > 800 • . ■ ............... .. . . . 1000 i a i - * ' eoo Gradwit retrieval . Mean Tjtimt guess - ?■ — \f I 7 7 7 ^ t- i | J: m Gradierit retrieval . i ■; MeanTjfiratguess • S t score T cj VAS 4 i <*r E 400 6 ! ! i if «-*: n ✓j ! : ; " : 200 j , r -p -r-.-r-, ' ■_ — -© —• Smith's method - - * o -M in im u m information - • -x * : GaFChen'simetito^. « :r : ** T i ,*I i $ s i 3 : ftS Q_ !i . V** 1 s .- * w-*.Tx . f* ^ 600 ^ ^ I BOO - ...............I...... : — - -j 1000 ^ r ............... j..... y7%j~»^Vjriisrjt retrieval ■1« « » 1 «--» ■ 1Tifirst ■'■guess » .« J 1000 , ;d ) V A S & ' S ' E, 400 i..............ri............... r .......... S 600 CL 0 • • ■ i ■ ■ ■ i ■. t- L— -o —j Smith's method - - -o- - i Minimum information Gal-Chen'simethod £ •, , .\T , .....^ I - • ..................................................... .. _ j . . •Gr&dieflL.cetheval . ..!•••** I Mean Tjfirst guess * 20 40 100 Figure 7.62 Si scores for retrievals on grid from radiance gradients with mean temperature as first guess. Circles: Smith’s method. Squares: minimum information. ‘X’s: Gal-Chen’s method. i Smith's method 1-D retrievil VAS 2 . — VAS 9 . .*.m..VAS.4..._ - » • VAS 5 . 200 400 ’ Minimum inlocmation 1-D retrieval . J 4 i i - - -K • ■ : ; 1 v# !__________ i...................... 2 600 a. . . ...................... ................... T ■ . -----d — VAS 2 — -g — VAS 3 VAS 5 - 600 ...................... l—i_.i 1000 1 •0.5 0 0.5 Bias (retrieved • obs) (K) I - I . . T T T * "t ^ T T t -fr i ■ .i.. !,. ' 1 B ias (retrieved - obs) (K) Figure 7.63 Bias (K) of virtual temperature fields retrieved from radiances with mean field first guess. Left panel: Smith’s method retrieval. Right panel: minimum information. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 3 6 - the weak gradient case of VAS 5. The Si scores for the gradient minimum information method and Gal-Chen’s method are low to moderate near the surface, but reach the “worthless" value of 70 by the 800 mb level. The s x scores approach 1 by the 500 mb level, where the retrieved fields have almost a zero gradient. The formulation of the gradient retrievals is such that, when Neumann boundary conditions are used, the fields are retrieved as deviations from an adjustable mean value. As demonstrated in the last section, it is possible for the fields retrieved using the conventional methods to have bias. Figure 7.63 shows the bias of the virtual temperature fields retrieved from radiances using Smith’s method and the minimum information method. These bias values are smaller than those from the station retrievals since the “first guess” field has no bias error. These biases have a similar pattern in both sets of retrievals, the 975 mb values are too high, then the temperatures between 950 mb and 850 mb are too low, overall. At 800 mb, the average temperature on the grid is again too high, but by 700 mb the biases are near zero, with average temperatures on the grid through the mid troposphere 0.1 K too low for the minimum information method and between 0.1 K and 0.3 K for Smith’s method. Note that the largest biases are near 800 mb, a level where the variance of temperature is also large and so the ‘first guess’ temperature field is relatively poor. The 1-D retrievals are also subject to the effects of bias in the radiance measurements. When the ‘observed’ radiances are given a 0.5K bias it caused a 0.5K increase in the bias for both sets of 1-D retrievals, though the effects decreased with height for the minimum information retrievals. Another consideration in the evaluation of retrieval methods is the sensitivity to observation error. With the retrievals at Oklahoma City and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 3 7 - b) V A S 200 200 400 ■g 400 600 600 800 800 -f -f 0.514RMS error 1000 1000 J— 0 1 4 0 5 1 2 9 -a 3 No^rror — 0.5K RMS error 4 5 RMS error (K) 0 :j «»>> Smith's method 1-d retrieval Mean.£ fjrpt flue* C ) V A s i 1-D retrieval 200 200 400 400 600 600 800 000 -e— Noerror -a — 0.5^ RMS error 1000 1000 0 1 2 3 RMS error (K) 4 5 ■o No error •e — 0.5*4 RMS error o 1 2 3 4 5 RMS error (K) Figure 7.64 Rms error (K) for retrievals on grid from radiance using Smith’s method with mean temperature as first guess. Circles: Error-free brightness temperatures. Squares: Brightness temperature field with 0.5 K rms error added at station locations. Stephenville, the methods that produced the lowest rms errors also had the highest sensitivity to observation error. This is also the case for the retrievals performed on the grid. The next experiment tests the effect on the retrievals of adding random errors to the ‘observed’ radiances. Figure 7.64 shows the rms error for the 1-D Smith’s method using brightness temperature with and without an added error field. The standard deviation of the added error is 0.5 K at the station locations, the interpolated error field has a standard deviation of about 0.35 K. The added error produces changes in the rms error of the Smith’s method retrieval of 0.5 K near the surface, decreasing with height; these rms errors are still less than or equal to R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 3 8 - a) VAS 2 ; b ) V A S j3 ' ! ! ' 200 200 ........... j........ ....... E 400 ! 600 2 600 I j /T eoo ii 800 ’ - 0|S K 1000 1000 0 20 40 60 80 q . No error -a — 0;5 K error -,.i i —i ..1, t i i 1 X ^ ^ ^ 1 i Tiu. i ... Smith’s m tf hod . 1-D| retrieval Mean j* first guesr 100 Sjscore S j score c) VAS j4 d) V A S j5 200 200 E 400 E 400 2 600 O. 600 800 800 1000 1000 20 40 100 0 S1 score 20 40 60 80 100 S j score Figure 7.65 Si score for retrievals on grid from radiance using Smith's method with mean temperature as first guess. Circles: Error-free brightness temperatures. Squares: Brightness temperature field with (0.5 K)2 error added at station locations. the rms errors of the 1-D minimum information retrievals (Figure 7.56). Figure 7.65 compares the sx scores for the retrievals using Smith’s method with and without the added error field. These values also show an increase in the lower levels, with smaller increases higher up. The increase in Si score is enough to change the analysis from good to fair (VAS 2) or from fair to poor (VAS 4, VAS 5). Again, this change does not increase the error beyond that of the 1-D minimum information retrievals (Figure 7.61). The gradient Smith’s method retrievals shared a similar sensitivity to observational error; the error in the analysis increased below 300 mb, but not so much as to exceed the error of the other gradient methods. We must remember, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -239however, (Figures 7.58 - 7.61) that above 300 mb the fields produced by this retrieval method are useless. The closeness of fit to the brightness temperature measurements of the Smith’s method retrievals is controlled by how closely the iteration process is permitted to converge to the observed brightness temperature values. The convergence criterion may be a little too small for these retrievals; the sensitivity to observation error can be reduced if this criterion were adjusted. The minimum information method retrievals and the retrievals by GalChen’s method are less affected by added observation error than the Smith’s method retrievals. (These statistics are not shown.) The rms error increased by at most 0.1 to 0.2 K near the surface for the minimum information retrievals from brightness temperatures. There was practically no change in rms error and Si score for the gradient minimum information retrievals and Gal-Chen’s method. The poor error scores and lack of sensitivity to error in the observations indicates that the ‘first guess’ field is probably being given too much weight in the minimum information method and Gal-Chen’s method retrievals. The relative weights of the ‘first guess’ field and the radiance measurements are chosen under the assumption that each 'first guess’ level adds independent information to the retrieval and so each level should be treated equally as a separate observation. Naturally, this is not the case with these retrievals, the ‘flat field first guess’ does not represent an independent observation at each level. The retrievals fit much too closely to the mean field. The error would be reduced if more weighting is given to the radiance measurements (though the sensitivity to observational error would also be increased). To test this theory, a set of retrievals has been performed where the weighting on the radiance observations is set to be 10 times larger than what is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -240- 1'* ' r-'-r-r ~ | k% ’ • 1 */ > w* 200 ' ■j • • • • ' J ,,, : j E400 Miremtrn information 1-6 retrieval X / Z • ................ L-*—i . . I' . . , a — b— VAS 2 ! — -fr — VAS 4 VAS5 . - , . i . , . Ji T Minimum information Grad ipnt retrieval ^...........jV.................... 10 X njdianoe weight. ■ 1000 { ....... MjwiT.ftf5!.gy<wa..J ut'* > c r ' v y-i: ■ -/* r.A .Z 10 Xrcjdianoe weight — <»— VAS 2 :................_______________ k _ VAS4 VAS5 , \= / T\ E 400 GaFGhenfc method " ........MflM.Lttol.aYess.J Bf 4fal 10 Xritdianoe weight. S - U— VAS2 — -vas a-— H> - VAS 4 VAS 5 Figure 7.66 Ratio of average rms error to average standard deviation of virtual temperature on constant pressure levels. Retrievals are with radiance weighting 10 times normal using radiance or radiance gradients with mean temperature as first guess. Top left panel: 1D minimum information method. Top right panel: gradient minimum information method. Bottom panel: Gal-Chen’s method. used in the retrievals above. Figure 7.66 shows the ratio of the rms error of these retrievals to the standard deviation on the pressure surfaces. Comparing with Figures 7.57 and 7.59, we see that this ratio of error to standard deviation has decreased; the greatest improvement in the retrievals is in the 800 - 400 mb layer. This adjustment in radiance weight brings the error ratio for the 1-D minimum information retrievals close to that of Smith's method, with a corresponding increase in error sensitivity. The error in the gradient retrieval methods is still larger than the 1-D retrievals above 700 mb. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 4 1 - All the gridpoint retrievals presented above have been performed using both ground-based and satellite data. When satellite data is not used, the 1-D Smith’s method retrievals have extremely bad retrievals in the stratosphere, just as for the retrievals at station locations. The minimum information method and Gal-Chen’s method retrievals are not as dependent on the satellite data for their retrievals (although the retrievals above 300 mb are useless for these methods in any case), the rms error in mid-levels (700 - 300 mb) only increases about 0.2 - 0.3K at most when the satellite data is not used. What do these radiance retrieval tests tell us? The retrievals from the ground-based radiance measurements do a fair job of retrieving fields near the surface, but retrievals become much poorer above 700 mb especially for the minimum information and Gal-Chen’s method retrievals. The Smith’s method retrievals carry their improvement from the first guess profile over a deeper layer than the other retrievals because of the hypothesis upon which the method is based, that the contribution to the error in the brightness temperatures is nearly equal for all levels. The other methods are limited to making corrections near the surface because the contributions to the brightness temperature (gradient) of the vertical temperature (gradient) profiles is the largest for levels near the surface where the ground-based weighting functions are large. The ground-based channels, even when supplemented by two satellite channels, do not provide enough information to retrieve good fields above the 700 mb level. The retrievals perform better when more information is available. The best retrievals are the regression retrievals that have information about covariances between brightness temperature and the temperature profiles through the atmosphere. The retrievals with satellite data do better than those without because the satellite data adds information about the mid- and upper R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 4 2 - tropospheric mean temperatures. Some of the retrievals at gridpoints had less error than the corresponding retrievals at station locations when the mean level temperature was a better first guess than the climatological mean. The gradient retrievals had larger errors than the corresponding 1-D retrievals, because using a zero gradient first guess field is much more detrimental to the gradient retrievals than to the 1-D retrievals. The first guess gradient field is extremely biased, the gradients are much too small at all levels. In the gradient Smith’s method retrievals, the assumption that all levels contribute the same amount to the brightness temperature error causes the correction of the first guess brightness temperature gradient, which is dominated by low level gradient error, to also be applied to stratospheric levels where the gradient first guess error is of the opposite sign, which results in extremely bad retrievals. The strong correlation in the gradient error between tropospheric levels led to poor retrievals in the mid-troposphere from the gradient minimum information method and Gal-Chen’s method because those method assume that the covariance between the first guess error at different levels is zero. Despite these problems, it is clear that these gradient retrieval methods do work as designed, they produce temperature fields with gradients that are a weighted combination of first guess gradients and gradients inferred from radiance measurements. These results reinforce the motivation for this study, they demonstrate that the ground-based radiance measurements provide information near the surface where wind retrievals are poor, but additional information such as can be provided by the wind observations is needed to produce useful temperature fields above 700 mb. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 4 3 - 7J5- Betrievat of Temperature Using W inds and Radiances We now approach the ultimate goal of this research, to produce temperature fields from a combination of wind and radiance data and determine if these analyses are better than the retrievals from wind or from radiance alone. The combined wind/radiance retrievals are performed in the same way as the radiance-only retrievals, except that the “first guess” fields are wind-derived temperature or temperature gradient fields rather than mean values of temperature. These combined wind-radiance retrievals can be directly compared to the radiance retrievals in the last section and the dynamic retrievals from wind data presented in Section 7.3.2. The objectives in this section are to produce retrievals using both wind and radiance information, and compare these with the retrievals performed in previous sections from either wind or radiance data alone. Our hypothesis is that the radiance data will provide the information needed for successful retrievals near the surface, and the wind data will provide the information needed for retrievals in higher levels that is lacking in the radiance data. As in the other sections, we first look at examples of the temperature fields produced by the retrievals. The temperature fields shown in Figures 7.67 and 7.68 are from the 7 March 1982, 0000 UTC. Figure 7.67 shows the 700 mb level of the retrievals, which is the same analysis that has been used to demonstrate the wind-only and radiance only retrievals. We see from these figures that the combined retrievals at this level do not change very much from the wind-derived first guess, though the -12 isotherm in Texas is moved further north in each of the combined retrieval fields. The gradient in the southeast appears weaker also, becoming more like the ‘true’ field. The largest changes appear to be in the Smith’s method (radiance and gradient radiance) retrievals. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -244- a) ‘True’ field -12 (b) ‘First guess’ retrieval using only wind information Figure 7.67 Virtual temperature analyses for 7 March 1982, 0000 UTC, on the 700 mb pressure surface. Solid lines: temperature contours (interval = 2 K). Dashed lines: difference between retrieved field and ’true’ virtual temperature field (contour interval = 2 K). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (c) Smith’s method, radiance retrieval (d) Minimum information, radiance retrieval Figure 7.67 (continued) Virtual temperature analyses for 7 March 1982, 0000 UTC, on the 700 mb pressure surface. Solid lines: temperature contours (interval = 2 K). Dashed lines: difference between retrieved field and true' virtual temperature field (contour interval = 2 K). / Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (e) Smith’s method, radiance gradient retrieval -12 (f) Minimum information, radiance gradient retrieval Figure 7.67 (continued) Virtual temperature analyses for 7 March 1982, 0000 UTC, on the 700 mb pressure surface. Solid lines: temperature contours (interval = 2 K). Dashed lines: difference between retrieved field and true' virtual temperature field (contour interval = 2 K). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -247- '0.-4 (g) Gal-Chen’s method Figure 7.67 (continued) Virtual temperature analyses for 7 March 1982, 0000 UTC, on the 700 mb pressure surface. Solid lines: temperature contours (interval = 2 K). Dashed lines: difference between retrieved field and ’true’ virtual temperature field (contour interval = 2 K). In Figure 7.68, the retrieved fields from the 850 mb level are presented. This series of figures is intended to show how the combined retrievals modify the first guess field at a lower level where the corrections are larger. Figure 7.68 (a) shows the ‘true’ field analyzed from radiosonde observations. The other fields shown in Figure 7.68 are various retrievals at that level. The first guess temperature field is shown in Figure 7.68 (b). The field was retrieved from wind data using the two-scale approximation to the divergence equation with modified Neumann boundary conditions. The dashed lines in Figures 7.68 (c) - (g) are difference fields between this first guess field and each of the combined retrieval fields. The difference between the first guess and the retrievals is zero on the western part of the grid because Dirichlet (fixed) boundary conditions are used where the pressure surface intersects with the R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -248- ground. In the ‘true’ analysis cold air is centered over northern Arkansas and southwest Missouri, with a strong temperature gradient to the south and west; in the retrieval from wind data, the cold center is displaced further to the southeast. In each of the combined retrievals in Figures 7.68 (b) - (e), there is some attempt to bring the cold center further to the north, and to enhance the northsouth temperature gradient in Kansas. The Smith’s method retrievals (b) and (d) are the most successful at bringing the cold center further northwest and adjusting the temperature gradient in Kansas. We noted in the last section that the Smith’s method retrievals tend to be more successful in the lower levels because these methods make the retrieved fields fit the observed brightness temperatures more closely than the minimum information methods (and consequently also (a) True’ field Figure 7.68 Analyses of retrieved virtual temperatures for 7 March 1982, 0000 UTC, on the 850 mb pressure surface. Solid lines: temperature contours (interval = 2 K). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -249- (b) ‘First guess’ retrieval using only wind information /■ 2-C" (c) Smith’s method, radiance retrieval Figure 7.68 (continued) Analyses of retrieved virtual temperatures for 7 March 1982, 0000 UTC, on the 850 mb pressure surface. Solid lines: temperature contours (interval = 2 K). Dashed lines: difference between retrieved field and first guess virtual temperature field (contour interval = 2 K). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. (d) Minimum information, radiance retrieval (e) Smith’s method, radiance gradient retrieval Figure 7.68 (continued) Analyses of retrieved virtual temperatures for 7 March 1982, 0000 UTC, on the 850 mb pressure surface. Solid lines: temperature contours (interval = 2 K). Dashed lines: difference between retrieved field and first guess virtual temperature field (contour interval = 2 K). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 5 1 - (f) Minimum information, gradient radiance retrieval 0-' (g) Gal-Chen's method Figure 7.68 (continued) Analyses of retrieved virtual temperatures for 7 March 1982, 0000 UTC, on the 850 mb pressure surface. Solid lines: temperature contours (interval = 2 K). Dashed lines: difference between retrieved field and first guess virtual temperature field (contour interval = 2 K). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 5 2 - have greater sensitivity to observation error). The minimum information methods and Gal-Chen’s method did not make as large a correction to the wind-derived field, but these methods are less sensitive to observation error. There appears to be very little difference between the gradient minimum information retrievals and the retrievals by Gal-Chen’s method. Recall that the difference in sensitivity to small changes in brightness temperature is part of the design of these methods. The Smith’s method retrieval is performed by iteratively correcting the temperature (gradient) profile until the brightness temperature (gradient) matches the observations within a specified degree of error. Thus, there is no weight given to the brightness temperature field of the first guess. In the minimum information method and Gal-Chen’s method, the first guess temperature (gradient) field and brightness temperature (gradient) field are weighted relative to the expected error in the fields; thus the brightness temperature (gradient) of the retrieval will be a combination of the observed brightness temperature (gradient) and a contribution from the first guess field. Naturally, the ability of the radiance retrieval to improve a first guess field depends on how large the error of the first guess brightness temperature is. If the brightness temperatures of the first guess field matches those of the observations then it is not possible to correct the first guess. If we think of the process of observing brightness temperatures as a transformation from physical space into radiance space, the component of the error of the first guess that can be corrected is the projection of the error in radiance space. The part of the error that cannot be corrected is the projection of the error into the nullspace of the radiance transformation, i.e., the portion of the error that cancels when integrated in the vertical with the radiance weighting functions. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 5 3 - 258 262 •266 266" " (a) Brightness temperature from “true” temperature field 258 ,262 262 266 c - (b) Brightness temperature from wind-derived temperature field Figure 7.69 Normalized brightness temperatures for the 53.85 GHz channel calculated from temperature fields at 0000 UTC 7 March 1982. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 5 4 - For this example (7 March 1982, 0000 UTC) there is only a small difference between the brightness temperatures calculated from the “true” temperature field and those that would be produced by the wind-derived first guess temperature field. The largest differences are generally about 2 K. Figure 7.69 shows the ‘true’ and first guess brightness fields for the ground-based 53.85 GHz channel. These brightness temperatures shown in the figure have been adjusted as described in Section 4.1.2.4 so that the optical depth (integral of the weighting function from the surface to the top of the atmosphere) is equal to one. (The variation of brightness temperature due to changes in optical depth because of changes in surface pressure across the region would otherwise outweigh the variations due to changes in temperature.) This figure gives us some idea of how small the difference is between the ‘true’ and first guess brightness temperatures, and thus how little the retrieval methods have to work with for improvement of the first guess. The largest differences between the fields are in the north in Kansas, where the combined retrievals also show large changes from the first guess (Figure 7.68). Some examples of vertical temperature profiles from retrievals in pressure coordinates are shown in Appendix C, Figures C.14 - C.18. In these figures, the dashed line is the first guess profile from the wind, the thin solid line is the retrieved profile and the thick solid line is the ‘true’ profile. It is apparent that, while the addition of radiance information is able to improve the horizontal temperature fields to some extent, it is not able to correct some of the grosser errors of the first guess fields in the vertical profiles. This is not surprising, since the 1-D radiance retrievals only constrain the vertical profiles so that the brightness temperature obtained by integrating the profile with the radiance weighting functions is close to the observed profile. It appears that a large Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 5 5 - portion of the error of the first guess temperature cancels in the vertical integration with weighting functions, since the differences in brightness temperatures are small. As for the gradient retrievals, there is no control over the smoothness of the vertical profile except perhaps through adjustment of the mean values of each level. Figures C.19 - C.21 show soundings from combined retrievals in sigma coordinates using a wind-derived first guess and gradient retrieval methods. The first guess wind retrieval was performed using Neumann boundary conditions and the two-scale approximation to the divergence equation, with temperature gradients interpolated between the surface and a = 0.825. The addition of radiance data improved the profiles from the surface to 850 mb, however, as in the pressure coordinate retrievals, some of the larger deviations of the first guess profile could not be corrected by the radiances. These examples show that the addition of radiance data to a windderived first guess can improve the retrieved temperature fields and profiles, but the effect of adding the radiance information is mainly confined to the lowest levels where the ground-based weighting functions are large. The utility of adding wind data to radiance retrievals is not as clearly demonstrated in these examples. The 700 mb analyses in Figure 7.67 do not show a definite “improvement" over the radiance-only analyses in Figure 7.55. Some radiance-only retrievals are already beginning to show the effect of the zero-gradient first guess at this level, though, and a comparison between 500 mb analyses will show that the radiance-only retrievals by the minimum information and Gal-Chen’s methods become completely useless. The Smith’s method retrievals are sometimes not too bad at 500 mb, but they become useless at higher levels that have gradients that are far different from the ones Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 5 6 - in lower levels that are measured by the ground-based channels. The radiance-only retrievals really have little information being input about the temperature gradients above 700 mb, and the retrievals such as Smith’s method work only insofar as the constraints under which they were derived (i.e. equal contribution to error from each level) are valid. In contrast, the combined-wind radiance retrievals have considerable information about the temperature gradient field above 700 mb through use of the wind observations and the equations of motion that go into obtaining the wind-derived first guess. The constraint on the temperature gradients imposed by the equations of motion is valid over a much wider range of circumstances than, e.g., the assumptions made by Smith’s method, or the constraints imposed by some form of statistical relationship that may not hold in a different location or at a different time of year. The fact that the combined retrievals are not modifying the wind-derived fields to any extent above the 700 mb level shows that the methods are behaving correctly in situations where knowledge of the ground-based radiances does not add new information. Looking at examples only tells us how specific times or specific levels of the retrievals compare to one another. If we want to know how the combined retrievals compare to the wind-only and radiance-only retrievals over all levels, locations, and times, we need to look at error statistics calculated from the retrievals. This next section presents the error statistics for several experiments using combined retrievals. Table 7.9 lists the experiments that have been performed for the combined wind-radiance retrievals. Table 7.10 lists the analyses that are performed for these experiments. We are using Neumann boundary conditions for these retrievals (except where Dirichlet conditions are required to be used Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 5 7 - when pressure surfaces intersect the ground), since we want to perform these analyses with a minimal amount of externally supplied information. The ‘first guess’ temperature field used for these analyses is the ‘two-scale’ approximation; if a choice were to be made based on the error of the windderived retrievals it would not matter very much which approximation was used, so it seemed best to use the approximation where most of the terms of the equations of motion were retained. The retrievals are performed using groundbased and satellite channels; recall in the last section that, for the gridpoint retrievals, the satellite channels either had a small effect on the retrieval (minimum information and Gal-Chen’s method) or were necessary for the success of the retrieval (Smith’s method). Thus, presenting statistics for retrievals using only ground-based channels would add very little to this study. Table 7.9 Experiments for combined retrievals Exp Purpose of experiment S compare combined wind/radiance retrievals to wind-derived first guess T compare radiance retrievals with retrievals from radiance gradients show sensitivity of retrieval methods to obs. noise Pressure use brightness temperatures interpolated from stations Pressure, Sigma U V Vertical Coordinate Pressure, Sigma Pressure Remarks Definite reduction in error below the 700 mb / a = 0.7 level Error statistics are close, gradient methods slightly better, no bias. Smith’s method retrievals sensitive, other methods have slight error increase Still some improvement over first guess, Smith’s methods added error. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 5 8 - Table 7.10 Gridpoint virtual temperature retrievals from radiance Data used in retrieval radiance radiance radiance gradient radiance gradient radiance gradient radiance gradient radiance radiance radiance gradient radiance gradient radiance gradient radiance radiance radiance gradient radiance gradient radiance gradient radiance gradient radiance gradient radiance gradient radiance gradient radiance gradient radiance gradient Retrieval method Smith’s method minimum information gradient Smith’s method gradient min. information Gal-Chen’s method Gal-Chen’s method Smith’s method minimum information gradient Smith’s method gradient min. information Gai-Chen’s method Smith’s method minimum information gradient Smith’s method gradient min. information Gal-Chen’s method gradient Smith’s method gradient min. information Gal-Chen’s method gradient Smith’s method gradient min. information Gal-Chen’s method (4.21) (6 .20 ) (4.42) (4.48) (6.28) (6.35) (4.21) (6 .20 ) (4.42) (4.48) (6.28) (4.21) (6 .20 ) (4.42) (4.48) (6.35) (4.42) (4.48) (6.28) (4.42) (4.48) (6.28) Exp. obs. Vert. error coord. no P S.T.U.V no P S.T.U.V no P S,T,U,V no P S.T.U.V no S,T,U P no S,T.V P 0.5 K U P 0.5 K U P 0.5 K U P 0.5 K U P 0.5 K U P interp. V P interp. V P interp. V P interp. V P interp. V P no S a no S a no S a V interp. a V interp. a interp. V a R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 5 9 - b) VAS, a) VAS Wind T first guess 200 200 £400 £400 600 8 600 I Wincj only 4—o — Smith’s method i Wind only - j—o — Smith’s method - Min. information BOO 800 1000 1000 0 1 2 3 RMS error (K) 4 0 5 1 2 3 4 5 RMS error (K) 1-d retneval Wind T first guees 200 200 £400 600 600 Wind only Smith's method Min. information 800 800 1000 j ■ Wind only -4—o — Smith's method - i -e — Min. information - 1000 0 1 2 3 4 5 0 RMS error (K) 1 2 3 RMS error (K) 4 5 Figure 7.70 RMS error (K) for retrievals on grid in pressure coordinates from radiance with wind-derived temperature as first guess. Solid line: Wind only. Circles: Smith’s method. Squares: minimum information. In the first experiment, we compare the statistics from the combined windradiance retrievals with the ‘first guess’ wind only retrieval. Figures 7.70 and 7.71 show the rms errors for the combined retrievals in pressure coordinates. These statistics show that the use of radiance data improve the retrievals between the surface and 700 mb, but there is little improvement (if any) above that level. The decrease in rms error is greatest for the levels where the rms error for the wind-derived field is the largest. The reductions in rms error in the lower levels range from more than 1.5 K in VAS 2 to the very small improvement in VAS 5. The rms errors for the combined retrievals are generally less than 2 K except for near the 800 mb level in VAS 3. (Recall, however, that the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 6 0 - Gradient retrieval ■ WindTlinftgueet • a) VAS 200 200 £400 E 400 600 s 600 800 Gradient retrieval Wind Tjiret guess b) VAJ 1 ; Wind ofify — 0 — Smith's: method -e — Min. information ; Wind only — <> — Smith's: method 800 ■“ "-■it.- 1000 1X0 0 5 1 RMS error (K) 3 2 Gai-Oien's method. S 4 RMS error (K) i nr1t i I i i i i Gradient retrieval Wind Tfirst guess Gradierjt retrieval ■ Wind T first guess * 200 200 £400 E 400 <? 600 600 i— Wind oftly _ 6 — Smith’s method 800 800 1000 --■ X - 1000 5 RMS error (K) 0 1 2 3 GaJ-Chpn*method. 4 5 RMS error (K) Figure 7.71 RMS error (K) for retrievals on grid in pressure coordinates from radiance gradients with wind-derived temperature as first guess. Solid line: Wind only. Circles: Smith’s method. Squares: minimum information. ‘X’s: Gal-Chen’s method. temperature analysis on that level has an extremely large standard deviation: see Figure 7.33.) The rms error for the Smith’s method retrievals is sometimes smaller than that of the other retrieval methods, but the Smith’s method retrievals also produced changes in higher levels that increased the error of the analyses. The gradient minimum information method and Gal-Chen’s method had practically identical rms errors. The Si scores for these combined retrievals are shown in Figures 7.72 and 7.73. These figures show much the same thing as the plots of rms error; a marked decrease in error in the surface levels, but no improvement above 700 mb. The 1-D methods (Figure 7.72) have somewhat larger s x scores in the R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 6 1 - i i i I b) VAS 9 200 £ 400 2 600 200 £ 400 600 1*dretrie\jal Wind T firit guess 1-d retrieval Wind T f r i t guess Jp ....... ytftn&orfy''’ *:...dtp’ — -e — Smith's method ( * 600 --< > • 600 — -e — Smith's method ^ — o - Minimum information Minimum intofmati^1000 1000 0 100 S 1score 20 40 60 80 100 S t score —i— i— r - r - d) VAS 5 200 200 E, 400 2 E, 400 600 600 1-d retrieval Wind Tfifct guess *^{n d o n ly""l.......... — -g — Smith's method - - - o - Minimuminformation 800 1-d re tried Wind T ffe t guess 600 **■ * ^ * lfen3o*n|y"*‘*t.... — — Smith's method 'j - - - o - MinimuminfoijnatiorP-.*' 1000 1000 0 20 40 60 60 100 S . score 60 S 1score 100 Figure 7.72 Si score for retrievals on grid in pressure coordinates from radiance with wind-derived temperature as first guess. Solid line: Wind only. Circles: Smith’s method. Squares: minimum information. lowest levels than the gradient methods, but there are far fewer gridpoints entering into the calculations in the low levels so it is not certain how reliable these statistics are. The Si scores for the gradient retrievals (Figure 7.73) show a great deal of improvement over the wind-only retrieval. The addition of radiance data brings the Si scores for VAS 2 to 50 or less for all the tropospheric levels. Once again, we see that the statistics for the gradient minimum information retrievals and Gal-Chen's method are practically identical. The Smith’s method retrievals are again somewhat better in the lowest levels but that is offset by the degradation of the analyses above 400 mb. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 6 2 - • n\ 200 1 1 • ; b) VAS 13 ■ Gradient retrieval “ Wifa*T'TinJf'gue4s Wjnd only — Smith's method ’ Gradient retrieval....... ...... _____________ . Wind T first guess ] ■— —- O — r. - -x S • — — - Wjnd only Srjifth'a method Minimum information^ Gal-Cherf* m ethod^ ! WpimumintorniaUQD..^^ Gad-Cherfs method 600 ...............J : -------------i ................ b f ? Y < <! . . . i . . . ' K ' 1. , j .............. I------------i . , . i . . . S . score S« score c) V A S d) VAS |5 4 firflriwnt ratnfttal Wind T first guess Wfind only — Sfnith's method Wind T fir'it guess ’ MTriinium ih T o n ^ ion —' -X - GaJ-Cherfa method S 4score Wind only — Smith's method —’ '•£ —"' WTriimum’ mTormaiioo - - - X - AaJ-Chetfs method S j score Figure 7.73 s i score for retrievals on grid in pressure coordinates from radiance gradients with wind-derived temperature as first guess. Solid line: Wind only. Circles: Smith’s method. Squares: minimum information. ‘X’s: Gal-Chen’s method. Overall, when ‘perfect’ brightness temperature are used, with no added observational errors, the statistics for the 1-D and radiance gradient retrievals are comparable. The gradient retrievals had better Si scores near the surface; recall also that the gradient retrievals are not affected by bias in the radiance measurements. The rms error and Si scores for the combined retrievals in pressure coordinates can also be compared also with the error statistics of the radiance only retrievals. Figures 7.56 and 7.58 showed the rms error of the 1-D and gradient retrievals, respectively. The Si scores are shown in Figures 7.61 and 7.62. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 6 3 - The rms errors (Figure 7.56) and the Si scores (Figure 7.61) of the radiance-only 1-D retrievals are a little bit lower than the errors of the combined retrievals for the levels below 700 mb. Above 700 mb the combined retrievals are better than ttee radiance-only minimum information retrievals; the radianceonly 1-D Smith’s method retrievals did somewhat better than the combined retrievals through at least 500 mb. The gradient retrievals behaved in a similar fashion. The radiance-only and combined retrievals for minimum information and Gal-Chen’s method are quite similar below 700 mb, and the radiance-only retrievals become much worse above that level. The Smith’s method radiance-only retrievals did better through the troposphere but are useless above 200 mb. Although the radiance-only Smith’s method retrievals had better error statistics through much of the troposphere than any of the combined retrieval methods, the behavior of the retrievals at and above the tropopause makes the use of Smith’s method somewhat less than desirable. We commented in the last section that the performance of the Smith’s method retrieval depends on the validity of the underlying assumption of the method, that all the levels of the atmosphere make equal contribution to the total error in the first guess radiance. If we only consider the differences between the minimum information and Gal-Chen’s method retrievals, we can definitely say that the addition of wind information is beneficial to the retrievals. (The addition of any realistic radiance first guess is necessary for the gradient minimum information and Gal-Chen’s method retrievals.) Figures 7.74 and 7.75 show the rms error and Si score for gradient retrievals in sigma coordinates. The first-guess wind derived field here is the two-scale approximation retrieval with the surface levels replaced by R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 6 4 - ■b) V A S 4 •a) VA S 0.2 0.2 0.4 0 E £ 0.6 0.4 0.6 gradient retrieval Wind derived tin t guest Gradient retrieval WirjJ derived fir^t guess 0.8 ^ 'S ^ H S T .... .... -o Minimum information •X •! Gal-Chen* method - o - j Minimum irtormeoon •X ■: Gal-Chen't method 0 1 2 3 4 0 5 1 RMS error (K) 2 3 RMS error (K) 4 5 0 ;C) V A S f 0.2 0.2 0.4 a E 0.6 0.4 0.6 Gradient retrieval Wirid derived firv guess 0.6 Gradient retrieval Wirid derived firjt guess ... ... 0.8 -o -• Minimum information •X *: Gal*Chen'» method 1 0 1 2 3 RMS error (K) 4 •o -j Minimum iiformabon •X *: Gal-Chen‘8 method 5 0 1 2 3 4 5 Figure 7.74 RMS error (K) for retrievals on grid in sigma coordinates from radiance gradients with wind-derived temperature as first guess. Solid line: Wind only, lower levels interpolated between surface and o = 0.825. Circles: Smith’s method. Squares: minimum information. ‘X’s: Gal-Chen’s method. interpolated gradients from the surface to a = 0.825. These plots show that even though the first guess fields are improved over the original sigma coordinate retrievals (see Figures 7.48 and 7.49) by replacing the wind derived gradient estimates with interpolated gradients in the lowest layers, the error can be reduced even more if good ground-based radiance data is available. The improvement in error statistics is confined to below the a = 0.7 level, however. Once again, the Smith’s methods retrievals have smaller errors than the minimum information and Gal-Chen’s method retrievals. Again, the VAS 2 case R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 6 5 - - V I' I IT - a) V A S 2 I ■ I I I- ' 1 1 * | b) VAS 3 I" I | ________ i Gradient retrieval [Wind T first gu Win'd’oniy: Smith's method — -« f* Minimum information K.i...GAHChcnkmeUjoeL : Gradient retrieval Wind T fir* guess Wind’orfy — Smith's method —- Minimurft informatiOQ 4ft.T...Gai£h4Q&.mail)oeL 40 60 S . score S - score c) V A S ' d VAS S 4 0.2 0.2 0.4 0.4 . . 0.6 I Gradient retrieval I Wind T fins guess . *‘t‘NrSr*t .S> 0.6 . j WTncforiy ’ — Smith's method — -q — Minimum information r.r.^.r...Gat£thdn'sjsflttiod_ 0.8 0 20 40 60 60 100 S . score Figure 7.75 Si score for retrievals on grid in sigma coordinates from radiance gradients with wind-derived temperature as first guess. Solid line: Wind only, lower levels interpolated between surface and a = 0.825. Circles: Smith's method. Squares: minimum information. ‘X’s: Gal-Chen’s method. shows the largest improvements, and the VAS 5 case (where errors and gradients were small) showed the least improvement. The rms error of the sigma coordinate retrievals is comparable to the error of the retrievals in pressure coordinates shown in Figure 7.71. The rms errors near the surface are smaller in the pressure coordinate combined retrievals, but the pressure coordinate statistics are calculated over a smaller area. If the sigma coordinate combined retrievals are interpolated to pressure coordinates their rms error is equal to or smaller than that of the pressure coordinate combined retrievals. The Si scores for the sigma coordinate R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 6 6 - retrievals are somewhat better than for the pressure coordinate retrievals, but this is in part because the gradients on sigma surfaces are also larger than for pressure surfaces. The combined retrievals considered so far have been best-case scenarios; the radiances used in these retrievals are calculated from the ‘true’ temperature analysis; thus, they exactly satisfy the radiative transfer equation used in the retrieval methods. We now want to do some retrievals that are more realistic. The first set of retrievals will assess the sensitivity of the combined retrievals to errors in the brightness temperature measurements, in the same fashion as for the radiance-only retrievals. The error added to the brightness temperature fields has a standard deviation of 0.5 K at the station locations, so the interpolated error field has a standard deviation of about 0.35 K. A similar error field is added to the surface temperature field. Figures 7.76 and 7.77 show the rms error for these combined retrievals using the brightness temperatures with simulated observation error. The minimum information retrievals and the retrievals by Gal-Chen’s method were only slightly affected by the added error. The Smith’s method retrievals both showed an increase in error; the gradient Smith’s method retrievals had a great deal more error in the upper levels, more than the wind-derived retrieval. The results are also similar for the Si scores shown in Figures 7.78 and 7.70. The combined retrievals still show some improvement over the wind-only retrievals, though not as much as when ‘perfect’ observations are used. Because the brightness temperatures used in retrievals up to this point are calculated from gridpoint temperature profiles, these retrievals are effectively from a network of ground-based profilers with station spacing of 1 ° in R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 6 7 - 0 1-d retrieval Wind T first guess 0.5 K RMS error T_ a ) VAS 200 200 E 400 600 600 -r . — Wind only — - I—o — Smith's method ^•^•ui-Mtn/Snformaihyi 600 Wind only —o — Smith's method 800 ca — Min. information 1000 1000 0 1 2 3 RMS error (K) 4 0 5 1 2 3 RMS error (K) 4 S 0 Wind T first guess 0.5 K RMS error T_ 200 1-d retrieval Wind T first guess 0.5K RMS error T■B— 200 E 400 600 600 — Wind only ■■ Wind only — Smith's method 800 r? •a 800 — Min. information 1000 0 - o — Smith’s method -a — Min. {information 1000 1 2 3 RM Seiror(K) 4 5 0 1 2 3 4 5 RMS error (K) Figure 7.76 Rms error (K) for retrievals on grid in pressure coordinates from radiance with wind-derived temperature as first guess. 'Observed' radiances have 0.5K rms error field added. Solid line: Wind only. Circles: Smith’s method. Squares: minimum information. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 6 8 - I I I I T I I 0 I Gradient; retrieval Wind T fjrst guees 0.5 K RMSerror T_ 200 Gradient retrieval Wind T first guees 0.5K RM8efror Tc •B*— b) V A a 200 E 400 E 400 600 600 — ■4 - Wind ojily -—9 — Smiths method ^$ ..rr..Mw..rofenrnii9n ! — }*■- GaFCh&n1* method. 800 0 1 2 3 RMS error (K) ;c) V A S i E 400 77 600 rr:..dB.rT:.MiCLinfclJ » lJ £ in .-_ GaKChen's method1000 1000 200 ■■ Wind only — p — Smith’s method ‘S ; 1.......... V . . . ^ .....j.... | j i 4 Gradient retrieval * Wind T fjrst guess * 0.5KRMSerror T„ ' M 1 0 5 -d ) va £ £ ' : s ■ V 2 3 RMS error (K) : ' u i 4 5 Gradient retrieval * Wind T f|rst guees 0.5 K RMSerror T„ * £400 ¥ 2 300 Q_ 800 : ............... h y - r ir ■ ‘ » ■ 1 ■ ■ ■ ‘ -1 » ■-■j — Windofily — O — Smiths! method .... ^...-^.Tr..Min..inh3timlion._ - - • i f - GaJ'Chhn’s method j—i—i. 1 i — i—i_i RMS error (K) 1 i ■t • v — 9 — Smiths- method ...........M in, inform ation : — («■- Gal-Chhn's method i—»—«—i - i i . -i . . i . . . . ‘ RMS error (K) Figure 7.77 Rms error (K) for retrievals on grid in pressure coordinates from radiance gradients with wind-derived temperature as first guess. ‘Observed’ radiances have 0.5K rms error field added. Solid line: Wind only. Circles: Smith’s method. Squares: minimum information. ‘X’s: Gal-Chen’s method. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 6 9 - b) V A S ^ 200 Pressure (mb) 200 E 400 600 600 i'^'ritrwva) T" Wind T first guess 0 5KRMSerrTB j 800 400 800 ■■■■ ^indonfy*'* *t** — Smith's method -o - Min. information 7<frijir'Nrraf........ Wind T first gusss j 0.5KFMS«rrTQ I ~ ■“W tn r jo n iy 1000 1000 0 100 40 20 score c ) VAS 4 200 Pressure (mb) 40 60 S 1score 80 100 80 100 , ■;—T—T~ d) VAS 5 —I— I— I— r 200 E 600 400 600 Wind T first guess 0.5 KR^SerrTg 800 j" ‘ — — Sfreth's methqd - • ■ o - Min. information -9 : j Wind T first guess O.SKRMSerrTg 800 -—’Windonly...... — -q — Smith's method o - Mi». information — ' Wrndoniy..... :* — Smith's method - - -o - M>n. information 1000 0 : I 1000 20 40 60 S 1score 60 100 S 1score Figure 7.78 s x score for retrievals on grid in pressure coordinates from radiance with wind-derived temperature as first guess. 'Observed' radiances have 0.5K rms error field added. Solid line: Wind only. Circles: Smith’s method. Squares: minimum information. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 7 0 - -r-,-r-| , 1 * 1 I * ■■ » ^ Gradient retrieval Wind T first guess !K5KFWSiwrBr.:% ’ Gradient retrieval ' Wind T fir# guess 200 ~ E 400 j r ----------- wjnd only ■— o — Smith’s mathoc wjmnfttmane ; - - X - Git-Chen’sme E ; .... t ............... i ................. Wind only E & 400 i ...... : ........ M — Min. information - X - Qal-Cherfs njethod 600 (& ' I ,......5 X . . i ..................i ...............- 800 • a) V A S ji * —i— i ,.?*4— i_ .i —i— 40 60 S . score i... i b) V A S 3 i —l.. 80 s .s c o re -T r r- . " j - r r"T ' i ■ .............................. Gradient retrieval Wind T firtf guess i 1 i To lQ.5KBMS.jsfTOr.TB.....I. ... J l. E 400 • — — Wind only ■ — © — Smith's method 200 Z & -* ” 1 sf Gal-Chen's method 600 .......... ..... 800 • ^ • ■ ■ I< r . . 4 20 L . i 40 60 S-score # ....... [ . . . S . I i I I— I— [— I Gradient retrieval Wind T fir^t guess 6. 400 | - • -X - 2 - I — I— I— I ® Q. • _ ■ Wind only — o — Smith's method —* 'Wfri!Trir6rrraibn..... . . -X - Gal-Chen’s method 600 800 ; d) VAS j5 1000 40 60 80 100 S . score Figure 7.79 Si score for retrievals on grid in pressure coordinates from radiance gradients with wind-derived temperature as first guess. ‘Observed’ radiances have 0.5K rms error field added. Solid line: Wind only. Circles: Smith’s method. Squares: minimum information. ‘X’s: Gal-Chen’s method. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 7 1 - latitude and longitude. The second set of retrievals to be presented in this sensitivity study uses a more realistic station spacing, the spacing of the rawinsonde network. The brightness temperatures used in the next set of retrievals to be presented are calculated from observed profiles at rawinsonde stations and then interpolated to the grid. This has more of a detrimental effect on the retrievals than just adding error to the brightness temperature because the interpolated radiances do not have as much horizontal detail as the radiances calculated at gridpoints. The brightness temperatures for at least two of the ground-based channels have a strong dependence on the optical depth (and hence the surface pressure). The interpolation to gridpoints has to be designed to accommodate this dependence; the brightness temperatures are normalized (adjusted so the sum of the weighting functions would be equal to one) to remove the effect of changing optical depth, these normalized brightness temperatures are interpolated to gridpoints using the same Barnes (1973) interpolation scheme as was used for the temperature interpolations, and then the interpolated gridpoint normalized brightness temperatures are adjusted by multiplying by the sum of the weighting functions at each gridpoint. This interpolation method gives a radiance field similar to the radiance field calculated at the gridpoints. Figure 7.80 shows the rms difference between the gridpoint brightness temperature analyses and the interpolated brightness temperatures. The interpolated brightness temperatures are not too far from the gridpoint values, except for the 58.80 GHz channel. The problem with the 58.80 GHz channel lies more with the discretization of temperature in the vertical on constant pressure levels and the formulation of the weights to accommodate this. (Recall that in Eqs (6.23), (6.24) that the weight for the first level above the R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 7 2 - t--- 1—i-- 1—i-- 1-- 1-- 1-- r 52.85 VAS 2 - ■il--i~~VAs-3-— : 5 3 .8 5 *....................... a □ m Nl c ® 5 8 .8 0 □ I VAS 4 : □ j vas s : a * cr £ 53 .7 4 5 4 .9 6 I ... 0 0.2 0 .4 I 0.6 0.8 RMS error (K) Figure 7.80 Rms difference (K) between interpolated brightness temperatures and brightness temperatures calculated at gridpoints from constant pressure level analyses of temperature. surface required special handling so that the horizontal and vertical dependence of weighting function could be separated.) The 58.80 GHz channel is the most opaque of the ground-based channels, thus the weighting function is non-zero only for a very few levels above the surface. This means that this channel is extremely sensitive to variations in surface pressure; the gridpoint analyses of brightness temperature show some discontinuities or noise where the lowest constant pressure level used in the calculation changes between gridpoints. (The brightness temperatures calculated using constant sigma temperature analyses are much more continuous.) This noise in the gridpoint brightness temperature analysis is matched by similar characteristics in the brightness temperatures, which are calculated while performing the retrievals; so the noise has not been a problem with the retrievals performed thus far. However, the mismatch between the input Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 7 3 - brightness temperature gradient from the interpolated brightness temperatures and the brightness temperature field expected by the Gal-Chen’s method retrieval program is enough to cause problems with those retrievals. Thus, for the purposes of these retrievals, the alternate formulation of Eq. (6.35) is used instead of Eq. (6.28). Eq. (6.35) is based on the Sienkiewicz and Gal-Chen (1988,1989) formulation of the retrieval (Eq. (4.64)). In Chapter 4 we said this formulation would be more appropriate than Gal-Chen (1988) if there were a large error or bias in t bv; this also applies when the finite difference V t bv in the retrieval is inconsistent with the measured gradient of brightness temperature. Figures 7.81 - 7.84 show the rms error and Si scores for these retrievals from brightness temperatures interpolated to the grid from radiosonde locations. The results are similar to results from the retrievals where an error field had been added to the observations. The minimum information retrievals and the retrievals by Gal-Chen’s method still show a little improvement over the first guess. The Smith’s method retrievals are worse in some levels. The gradient retrievals are a little better than the 1-D retrievals. The retrievals for VAS 2, the strong gradient case, show the largest improvement. The weak gradient case of VAS 5 shows almost no change, or worse, even greater error after the retrieval. The weak temperature gradients give rise to weak radiance gradients, especially for the 58.80 GHz channel. The rms difference between the VAS 5 interpolated brightness temperatures and those calculated from the weighting functions used in the retrieval program is about the same as for the other VAS days, but the effect of the difference is larger when the gradients are weak. Figures 7.85 and 7.86 show the rms error and Si score for retrievals in sigma coordinates using brightness temperatures interpolated from station locations. The effect of using interpolated values appears to be limited to the R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 7 4 - a) V A S b g . 200 200 E 400 600 600 — Wirxjonfy Wind only - o — Smith’s method 800 Smith's method 800 -a — Min. jjrfomiolion Min.jhiformaiion 1000 1000 0 5 1 RMS error (K) 2 3 5 4 RMS error (K) 0 Wind T jiret guees ‘ 200 ..IP A ® . •?!..Tg..J 200 E 400 E 400 600 600 Wimj only Wirxjonly -O — Smith's method 600 Smith's method ' 800 •e — Min. information Min. information - 1000 0 1 1000 2 3 RMS error (K) 4 5 0 1 2 3 4 5 RMS error (K) Figure 7.81 Rms error (K) for retrievals on grid in pressure coordinates from radiance with wind-derived temperature as first guess. ‘Observed’ radiances are interpolated from station locations. Solid line: Wind only. Circles: Smith’s method. Squares: minimum information. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 7 5 - a )V A S £ s » _ > . 1 1 1 1" | 1 1 1 1 I r—t " i i j i i i i Gradient! retrieval * Wind T first guest ' Interpolated T_ j L ...................................................... t>) V A S * 200 ¥ s Q_ — Wndofily > — Smith1*; method ....... goo H 4 ^ * ' - - - Gal-Chin1* msthod1— * -------1 . i . . . . 1 .T T . i . . . . S iZ ■ , L ..r-rr ' P T . — Wind only >— Smiths: method r——Min.,information— F- GaFChenS method *■, , . , i , . , , * 3 RMS error (K) Gradient retrieval * Wind T ffrst gueee ‘ ; ;d) V M 5 / 200 ............. - i r - n j 'i—i - i i Gradient* retrieval * Wind T fjrst guees ' 200 : £400 — 2 r^T i | »i i i | i i i i ■n *i ■t i i i i i \> \ 4 r - r ' :‘^ C 2 3 RMS error (K) ;c )V A ^ 4 i i i i—f i i i i Gradient retrieval * Wind T first guees * Interposed i___ £400 ¥ £400 | - 2 ....... £ H 400 ¥ : 55 600 Q, — Wind oftly >— Smiths; method jd rfS ’ rC : 1 t it 1 Li h—Mif^Hofbrmalion— H t- • .1,,. 2 3 RMS error (K) — Wind oftly — i — Smith's; method ........ —— i — Min^infermatioo— - - - h - Gal-Chpn’s method 800 Gal-Chpn*# method' . . . . i ' . . . . . . . . 2 i . . . . * 3 RMS error (K) Figure 7.82 Rms error (K) for retrievals on grid in pressure coordinates from radiance gradients with wind-derived temperature as first guess. ‘Observed’ radiances are interpolated from station locations. Solid line: Wind only. Circles: Smith’s method. Squares: minimum information. ‘X's: Gal-Chen’s method. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 7 6 - i i i l b) V A S 3 200 200 .o E 400 E 400 s 3 1 CL 600 600 f-y'r&Vwval..... Wind T first guess Interpolated T_ 800 600 rrma oniy -o - Min. information 40 20 c) VAS 4 600 Min. information 1000 60 80 0 100 S . score S ffin d o n ly.... •q — Sfnith's methcjd o - 0 400 p -q — Smith’s method 1000 E T<Tf<j(ntwaf Wind T{lrst guess j Interpolated Tg 40 60 S . score 80 100 | E T^'r&'rieva] Wind T first guess Interpolated T 400 1-d retrieval Wind T first guess : Interpolated T WindoKly -e — Smith’s method -o- - Min. information *-1.-*-. ■_ 20 . i , , ■I S . score Wtndoniy.... — Smith’s method - -o - Min. information I . . ■l 40 60 S.. score Figure 7.83 s i score for retrievals on grid in pressure coordinates from radiance with wind-derived temperature as first guess. ‘Observed’ radiances are interpolated from station locations. Solid line: Wind only. Circles: Smith’s method. Squares: minimum information. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 7 7 - 1 1 1 1 * Gradient retrieval j * Wind T first guess j ........ j...... I I i | ■p- I Gradient retrieval Wind T firjt guess JlttWBOteHKLT.0 E Wjnd oniy — Smith's met •'Mrf.Mrfformanoff - - X - GaVCherfs meih 400 . ---------- Wind only E 2 i—~-* —^ i . E, 400 ; 1 ’ ' ! 1 1 * » "•" ! _ L _ L = | ; 1 ■ . ... , r* |— r - r - r Gradient retrieval Wind T M t guess .IfllKEQlflWd.Ig ----------- Wjnd only •— 0 — Smith's methoc :... Wjnd only — Smith's method -MfrTnfwrreutoh - Gal-Chen’s method wirrinfemwn& ¥ 2 Q. ■ ■ ' ” S . score S - score ■ Gradient retrieval ’ Wind T fir<t guess ' . — -a — Uin. information ■ - - - X - fcaWherfs njiethod^M :b)VASfc / . , i . a) V A S 2 -i—» i 400 . — - -X - Gal-Cherfe me hod 600 ■ | :c)VAsk ! ji . . > i . > . 1 1i . . . : d) V A S p 60 S - score Figure 7.84 Si score for retrievals on grid in pressure coordinates from radiance gradients with wind-derived temperature as first guess. ‘Observed’ radiances are interpolated from station locations. Solid line: Wind only. Circles: Smith’s method. Squares: minimum information. ‘X’s: Gal-Chen’s method. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 7 8 - a) VAS 0.2 '"T j 1Gradient l~r ' retrieval I 11 1 b) V A S 4 Gradient retrieval Wind T firpt guess ... Interpolated .Tp... I Wind T fir it guess L.Jnlarpddted.Tg.... 0.2 0.4 E S> s> V> 0.6 • Wind only! »*,9rnittf*,m*thod...... Min. information -X - Gal-Chen1# method 0.8 0 1 3 2 4 — o -4 Wind only: '^'H »^*Sm ithfrrn#thod,’' ” **‘ Min. inforreetton - - -X -i Gal-Chen** method 0.8 5 3 1 2 X ) VAS 0.4 •d) V A 9 $ — -----0.2 i >( V) f ................ x | i ■• • * i - ............................... : i- ......... E 0.8 ■ . . . i i . i i Gradient retrieval Wind T first guess ' ...Jnterpolated.Xg.— ' 3 4 RM S error (K) RMS error (K) Gradient retrieval Wind T first guess ] ...intarpnlatBd.Tg___ v E S? (0 0.6 _ o _ Wind only; •■SmithfSTnithod........ .. Min. information - - - x - Gal-Chen*# method * — oi | ■. : . .V i, . . ) RMS error (K) 1 Wind only; Min. information - - -X -I Gal-Chen*# method 2 3 RMS error (K) 4 * 5 Figure 7.85 Rms error (K) for retrievals on grid in sigma coordinates from radiance gradients with wind-derived temperature as first guess. 'Observed' radiances are interpolated from station locations. Solid line: Wind only, lower levels interpolated between surface and o = 0.825. Circles: Smith’s method. Squares: minimum information. ‘X’s: Gal-Chen’s method. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 7 9 - ;a) V A S T T - '—r ' I I ; b) V A S & . . t e l:“ fSraflfenf[eBiSi5r' " : Sigma i Wind T lirit guns : Interpolated T_ — jo — Wndqnfy — -js — Smith1* method Min. Information '- - 'f y ’’G'if.'CiSh’i' mefiw «•?......... -3gL~ 0.4 4 .>£ X - ............... - .......... 0.8 - ................... i ■ ! " -v . 0.6 I..... Gradient Retrieval j Wind T fifjst guess • —p — Wlndofity — -ja — Smfth’i method Ged-Clien*»method i .S T *r?t~* i i . i i _i . _i_ i ■ i 80 100 S1 score S 1score ;c) V A S 4 d ) V A S fe : N' i ar Sigma yjP ' 3 : Gradient retrieval I Wind T fiijst guees Interpolated - 0 — \Mnd only -e — Smfttfs method ...L..........r..-.^>..-...A4n.intonnfltton..- • r* - " “GaJ-Ctfen'smethod " T ...... 0.2 . Gradient retrieval i .... I •Srt-I ^ j. j Wind T firjst guees i . ,nte»pol4ted a T . '.v . < \ ........Tv. 0.6 — to — Wnddnly — <jfl — Smith** method ™t iP.-..M o,j/go.rmoftcfi._ Gal-Chen**methoe £ r * * " t ......... ■ ■_! i .^ * r v . > i . . . \ 100 S . score S1 score Figure 7.86 Si score for retrievals on grid in sigma coordinates from radiance gradients with wind-derived temperature as first guess. ‘Observed’ radiances are interpolated from station locations. Solid line: Wind only, lower levels interpolated between surface and o = 0.825. Circles: Smith’s method. Squares: minimum information. ‘X’s: Gal-Chen’s method. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 8 0 - lowest layer for the minimum information and Gal-Chen’s method retrievals. The Smith’s method retrieval does not appear to be adversely affected as in the corresponding retrieval in pressure coordinates. We can summarize the results of the combined retrievals in the following fashion: Addition of ground-based radiance data to a wind-derived first guess temperature field reduces the error of the retrieved fields, but only up to about the 700 mb level. Above 700 mb, the temperature field is changed very little. The amount of adjustment is affected by the difference between the observed brightness temperatures and those that can be calculated from the wind-derived first guess field. The addition of radiance data has only a small effect on the discontinuities in the vertical temperature profiles from the wind derived first guess in pressure coordinates. The combined retrievals show improvement over the radiance-only retrievals as well. The minimum information retrievals and retrievals by GalChen’s method benefit greatly above the 700 mb level where the radiance data does not improve on the first guess field. The Smith’s method retrievals do not show as much improvement except in the upper troposphere and stratosphere where the temperature gradient fields are substantially different than the gradients at lower levels. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 8 SUMMARY AND CONCLUSIONS The goal of this research has been to determine whether it is possible to combine Profiler wind observations and ground-based measurements of microwave brightness temperatures to obtain analyses of temperature that are better than what can be determined by just one of the sources of information. In pursuing this goal, we first reviewed the physical relationships that form a basis for the dynamic retrievals from winds and the radiance retrievals the equations of motion and the radiative transfer equation. Then, the retrieval equations for the dynamic and radiance retrievals were developed. For the dynamic retrievals, we showed that the solution of the divergence equation for $ produces a height field that is closest in a least-square sense to the height gradient estimates obtained using the equations of motion and the wind observations, and the boundary conditions for solution of the Poisson equation are specified from the minimization. Some justification was given for using various approximate forms of the equations of motion to estimate the height and temperature gradients for the retrievals. The section on radiance retrievals reviewed several ways that radiance retrievals can be performed. Smith’s (1970, 1983) retrieval method can be used to obtain temperature profiles that satisfy the observed radiances nearly exactly. The regression retrievals fit coefficients based on the covariance between radiance measurements and sounding profiles derived from a station -281- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 8 2 - climatology. The minimum information method uses a simple model of error covariance to derive profiles that are a weighted combination of the first-guess information and the radiance measurements. We also considered the retrieval method proposed by Gal-Chen (1988) that uses first-guess and radiance gradients to obtain temperature fields. We show how his method is related to a more general form using inverse gradient covariance matrices to weight the different terms in the minimization. Other gradient retrieval methods based one-dimensional Smith's method and minimum information method are derived. These gradient retrieval methods have the advantage of not being influenced by radiance observation bias. 8.1 Summary of Results A. Dynamic retrieval from wind The height fields obtained from winds were fairly good. Some of the smaller scale height features are not captured well by the dynamic retrievals but this may be in part because they are not well resolved by the wind observations. The rms errors of the retrieved heights are close to or smaller than the rms error expected from rawinsonde measurements. The retrieved temperature fields do not compare as well to the rawinsonde measurements. Calculating the temperature gradient estimate requires taking a vertical derivative; the process of differentiation amplifies small scale errors. The errors are quite large near the surface where the vertical differences for the hydrostatic equation are taken over thinner layers, and where unresolved small scale circulations and frictional processes are important. The sigma coordinate retrievals show larger rms errors than the pressure coordinate retrievals near the surface; these rms errors can be reduced by replacing the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 8 3 - wind-derived gradients with temperature gradients interpolated between the surface (known) and c = 0.875 (derived from wind). There was very little difference between the retrievals performed using different approximations to the divergence equations. This may come about if the size of the error in the geostrophic terms resulting from wind error is the same order or greater than the size of the non-linear terms. If the retrieved fields were to be used in a numerical forecast model, however, we would still want to include the non-linear terms in the retrieval. Retrievals using Dirichlet boundary conditions had lower rms errors than the corresponding retrievals using Neumann boundary conditions. Use of Dirichlet boundary conditions eliminates the bias in the estimated height or temperature gradients. However, if the Dirichlet boundary conditions have errors this can have a detrimental effect on the retrievals. The vertical temperature profiles derived from retrievals in pressure coordinates have discontinuities in levels that use the ‘modified’ Neumann boundary conditions (where intersections with the Earth’s surface are given Dirichlet boundary conditions). Retrievals in sigma coordinates using Neumann conditions for all boundaries, and retrievals in both coordinate systems using Dirichlet boundary conditions, do not produce such discontinuities. B. Retrievals from radiance measurements The retrievals at Oklahoma City and Stephenville using the statistical regression produced results comparable to those of previous studies such as Westwater et a i, (1985). The statistical retrievals generally performed better than the retrievals by physically based methods since the coefficients incorporate additional information in the form of radiance-temperature covariances. The physically based Smith’s method retrievals do almost as well Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 8 4 - as the regression retrievals but are also sensitive to observation error. The temperature fields retrieved using radiance gradient methods have rms errors similar to the conventional radiance retrievals performed at gridpoints. The retrievals by Smith’s method have lower rms errors than the other methods in lower levels. However, Smith’s method continues to apply the same adjustment to levels in the upper troposphere and stratosphere where the ground-based channels supply no information, so the retrievals are useless at those levels. Overall, the retrievals from ground-based radiance measurements performed best near the surface but were poorer above 700 mb where the weighting functions are small. This reinforces the original idea that there is a need for additional information to be added to the radiance retrievals away from the surface. We used a mean temperature field as first guess for the gradient retrievals so that they would begin with the same information as the one dimensional retrievals. The use of this zero-gradient first guess was detrimental to the gradient retrievals. The minimum information method and Gal-Chen’s method include an assumption that the covariance of the errors between levels of the first guess is small, which is not the case with a zero gradient first guess. If the weights on the radiance pv (Eq. 4.51) are increased by a factor of 10, the minimum information and Gal-Chen’s method retrievals have rms errors reduced to being close to those of the Smith’s method retrievals in the lower to mid-troposphere. C. Retrievals from combined wind and radiance data Adding radiance data to the wind-derived first guess fields reduces the error of the retrievals. The changes are confined mostly to below 700 mb. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 8 5 - combined retrievals are limited in the amount of adjustment they can make, because the difference between the first-guess and observed brightness temperatures is not very large. The radiance-only retrievals have smaller errors at some levels; however the combined retrievals are better when all levels are considered, because the combined retrievals have more information about levels above 700 mb. The error in the retrievals from unbiased radiances by gradient methods and the one-dimensional methods were very close; the gradient methods had slightly smaller Si scores near the surface. The main advantage of the gradient methods is that they are unaffected by bias in the radiance observations. The tests with more realistic radiance observations show that these retrievals have small sensitivity to random observation error. There is a larger effect for the pressure coordinate retrievals if the radiances are interpolated from station locations, because the interpolated radiances do not have the horizontal detail (or noise) that radiances calculated at gridpoints contain. The sigma coordinate retrievals are not so adversely affected since they do not have as abrupt changes in weighting function in the horizontal. 8.2 Possible Improvements The combined wind-radiance retrievals do show improvements over the wind-only and radiance-only retrievals, but there are still problems with the retrievals. We had hoped that the dynamic retrievals would provide greater detail away from the surface than the retrievals from ground-based radiances; instead, they provided greater noise. Also, the error of the retrieved temperatures calculated on pressure or sigma surfaces is larger than that of the radiosondes. The greatest problems are with the pressure coordinate retrievals; these Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 8 6 - problems are caused by the way we have to deal with changes in terrain height. The discontinuities in the weighting functions led to problems with the combined retrievals when radiances interpolated from station locations were used; the problems could perhaps be reduced by adjusting or horizontally smoothing the weighting functions used at each gridpoint so that they do not produce as much small scale detail in the calculated radiances. The retrieval method must be adjusted so it will work with the type of input one is able to give it. Smoothing the weights would probably also speed the convergence of Gal-Chen’s method; at present it takes 10-15 iterations to converge to a solution in pressure coordinates while in sigma coordinates it only requires at most 3 iterations. The discontinuities near the surface in the temperature profiles from the pressure coordinate retrievals can be reduced if the mean values of those levels are adjusted to match the ‘true’ mean; the error may also be reduced for retrievals using Dirichlet boundary conditions on all boundaries, provided that these boundary conditions do not contain too much error. The sigma coordinate retrievals did not have problems with discontinuities but shared the problem of the small vertical scale noise in the temperature profiles. This noise in the vertical profiles causes noise in the horizontal gradients of temperature if we have to interpolate the sigma coordinate retrievals to constant pressure surfaces (or vice versa). In view of this, it would be best to try to perform the retrievals in the coordinate system in which they are meant to be used. The noise in the vertical profiles comes about because in the dynamic retrievals one minimizes the error on horizontal surfaces without considering the error in vertical profiles. If we have bad gradient estimates in one area on one level, they affect the values at all the gridpoints on that level since we Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 8 7 - perform a minimization of error on the entire level. This comes about when we fix the mean value of the level; if one area has temperature values that are too low, the error is distributed through the rest of the domain making those values too high. Since the errors are not the same from level to level, this leads to noise in the vertical profiles. The vertical smoothing of the winds reduced the error in the smallest scales but did not eliminate the noise at 200 mb wavelengths. If we want to control the noise in the vertical profiles, we need to take some sort of explicit measures to do it. Use of Dirichlet boundary conditions (provided they are not too much in error) may reduce the error somewhat since in fixing the values on the boundary we remove the bias in the estimated gradients. Sasaki and McGinley (1981) give a method for adjustment of superadiabatic layers in soundings by using an inequality constraint, but very few of the layers in these soundings are superadiabatic. Another option would be to place some constraint on the second derivative, but that would have to be carefully applied so as to not eliminate valid changes in the lapse rate. 8.3 Future Applications The plans for a Profiler network have changed since this research was initiated; because of costs, it is unlikely now that a network of ground-based radiometers will be deployed although the wind Profilers will be available very soon for the dynamic retrievals. One possibility for temperature profiling that is now being looked into is the use of a Radio Acoustic Sounding System (RASS) (May, e ta i, 1988,1989, 1990; Strauch, etal., 1989) with the wind Profilers. The concept is based on the fact that acoustic (sound) waves cause variations in the refractive index of air that can be detected by radars such as the Profiler radars. The largest amount Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 8 8 - of backscattered power is attained when the acoustic wave satisfies the Bragg condition, i.e., the wavelength of the acoustic wave is half the radar wavelength (XB = XT / 2). The method currently being considered uses a frequency- modulated, continuous wave (FM-CW) acoustic source. The radar detects a band of frequencies, with the largest signal power returned for the Bragg frequency f B= ca / A,B = 2ca / A.r , where ca is the speed of sound. The speed of sound ca, is related to virtual temperature, by Tv = <Ca / (8.1) 2 0 .0 4 7 )2 Tests with a 404 MHz Profiler radar showed that the RASS temperature measurements were limited by acoustic attenuation to about 2 - 2.5 km AGL depending on meteorological conditions. The accuracy of the temperature measurements is comparable to that of radiosondes (May, et al., 1988,1989). If RASS capability is added to the Profiler network, those temperatures could be used in lower layers, and wind-derived temperatures by dynamic retrieval could be used in higher layers where RASS temperatures are not available. The Profiler winds will probably be able to produce useful height fields by dynamic retrieval. These fields, along with the wind measurements, will be useful in tracking the movement and development of weather systems. The computational burden is not too large, the dynamic retrievals could be performed on local computer workstations, or perhaps even personal computers. (Cram, etal., (1988) performed their retrievals on a PC.) If forecast fields of sufficient quality are available, then use of Dirichlet boundary conditions may improve the retrievals. It is clear that some improvement of the dynamic retrieval method will be necessary to make the retrieved temperature fields useful. Some possible Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -289- improvements have already been mentioned above. Another way that the dynamic retrievals of temperature could be improved is to incorporate them into the data assimilation process of a forecast model. The wind analysis used in the dynamic retrieval could be improved if a forecast first guess field were available and optimum interpolation methods were used for the objective analysis. The dynamically-retrieved height and temperature analyses could be combined with the forecast fields by a variational blending technique such as mentioned by Cram, etal. (1988) with the forecast field providing a constraint for the vertical second derivatives of the field. Another way that the errors of the dynamically retrieved temperature could be reduced is to abandon the idea of trying to produce detailed temperature profiles in the vertical and work on producing good estimates of mean layer virtual temperature. Recall from Eq. (7.3) that the error in the temperature retrievals resulting from the wind measurements is inversely proportional to Ap. As noted previously, the benefit of increasing the thickness of the layer calculation decreases for thicknesses greater than Ap = 200 mb. How could these thicknesses or mean layer virtual temperatures be used? These layer mean temperatures could also be assimilated into numerical models to help improve forecasts. The papers by Gal-Chen, et al. (1986) and Aune, et al. (1987) have shown assimilation of mean layer temperatures into models has a beneficial effect, although their assimilation method used mean layer temperatures derived from satellite observations. In fact, it would be possible to treat the mean layer temperatures in the same fashion as radiance observations with rectangular weighting functions, and one way to assimilate them into numerical models would be to use an approach like outlined for combined retrievals in this paper with forecast fields used as first guess and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -290- Profiler thicknesses replacing the ground-based or satellite radiances. The work on combined radiance-wind retrievals is useful, even though it is unlikely that a ground-based radiometer network will come into existence. The theory, advanced by Gal-Chen (1988) and extended in this paper, of using radiance gradients and any appropriate first-guess field to obtain temperature analyses can also be applied to satellite measurements. The use of satellite measurements did not have very much effect on the retrievals in this paper (beyond making the Smith's method retrievals work) but this is probably because only two channels were used and the channels had very broad weighting functions. The use of these gradient retrieval methods should be explored using satellite radiances. The derivations in Chapter 4 show the method whereby existing satellite retrieval methods could be converted to be used as gradient retrieval methods. The results of this paper show that gradient retrievals are possible and that they can give good results in levels where the radiance measurements provide good information about temperature gradients. The ultimate judgement of the utility of these methods for satellite radiance retrieval can be found only through application to real data cases. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. REFERENCES Anthes, R.A., and D. Keyser, 1979: Tests of a fine-mesh model over Europe and United States. Mon. Wea. Rev., 107, 963-984. Anthes, R.A., and T.T. Warner, 1978: Development of hydrodynamic models suitable for air pollution and other mesometeorological studies. Mon. Wea. Rev., 106, 1045-1078. Aune, R.M., L.W. Uccellini, R.A. Petersen, and J.T. Tuccillo, 1987: A VASnumerical model impact study using the Gal-Chen variational approach. Mon. Wea. Rev., 115, 1009-1035. Barnes, S.L., 1964: A technique for maximizing detail in numerical weather map analysis. J. Appl. Meteor., 3, 396-409. _________ , 1973: Mesoscale objective map analysis using weighted timeseries observations. NOAATech. Memo. ERL-NSSL-62, 60 pp. Bleck, R., R. Brummer, and M.A. Shapiro, 1984: Enhancement of remotely sensed temperature fields by wind observations from a VHF radar network. Mon. Wea. Rev., 112, 1795-1803. Charney, J.G., 1962: Integration of the primitive and balance equations. Proc. Int. Symp. Numerical Weather Prediction, Tokyo. Chedin, A., N.A. Scott, C. Wahiche, and P. Moulinier, 1985: The improved initialization inversion method: a high resolution physical method for temperature retrievals from satellites of the TIROS-N series. J. Climate Appl. Meteor., 24, 128-143. Chesters, D., D.A. Keyser, D.E. Larko, and L.W. Uccellini, 1988: An assessment of geosynchronous satellite soundings retrieved with the aid of asynoptic radiosonde profiles. Meteorol. Atmos. Phys., 39,85-96. Cram, J.M., M.L. Kaplan, C. A. Mattocks, and J.W. Zack, 1988: The use of profiler winds to derive mesoscale height and temperature analyses. Preprints, Eighth Conference on Numerical Weather Prediction, Baltimore, MD. -291- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -292Decker, M.T., E.R. Westwater, and F.O. Guiraud, 1978: Experimental evaluation of ground-based microwave radiometric sensing of atmospheric temperature and water vapor profiles. J. Appl. Meteor., 17, 1788-1795. Doyle, J.D., and T.T. Warner, 1988: Verification and mesoscale objective analysis of VAS and rawinsonde data using the March 1982 AVE-VAS special network data. Mon. Wea. Rev., 116, 358-367. Eyre, J.R., 1989: Inversion of cloudy satellite sounding radiances by nonlinear optimal estimation. I: Theory and simulation for TOVS. Quart. J. Roy. Meteorol. Soc., 115, 1001-1026. Fankhauser, J. C., 1974: The derivation of consistent fields of wind and geopotential height from mesoscale rawinsonde data. J. Appl. Meteor., 13, 637-646. Foster, M., 1961: An application of Wiener-Kolmogorov smoothing theory to matrix inversion. J. Soc. Ind. Appl. Math., 9, 387-392. Fritz, S., 1977: Temperature retrievals from satellite radiance measurements an empirical method. J. Appl. Meteor., 16,172-176. Fuelberg, H. E., 1974. Reduction and error analysis of the AVE II pilot experiment data. NASA Contractor Report CR-120496. Marshall Space Flight Center, Alabama, 140 pp. __________ , and P.J. Meyer, 1986: An analysis of mesoscale VAS retrievals using statistical structure functions. J. Climate Appl. Meteor., 25, 59-76. Gage, K.S., and J.L. Green, 1979: Tropopause detection by partial specular reflection with very-high-frequency radar. Science, 203, 1238-1239. _________ ,W.L. Ecklund, and D.A. Carter,1989: Convection waves using a VHF wind-profiling Doppler radar during the PRE-STORM experiment. Preprints, 24th Conference on Radar Meteorology, Tallahassee, FL. Gal-Chen, T., 1986a: Selected comments on the use of the divergence equation to obtain temperature and geopotentials from an observed wind. J. Atmos. Oceanic Technol., 3, 730-733. __________, 1986b: A theory for the retrievals of virtual temperature from winds, radiances, and the equations of fluid dynamics. M iddle Atmospheric Program (MAP), Vol. 20. S. A. Bowhill and B. Edwards, Eds. Published by ICSU Scientific Committee on Solar Terrestrial Physics (SCOSTEP), pp. 5 -1 6 . _________ , 1988: A theory for the retrievals of virtual temperature from remote measurements of horizontal winds and thermal radiation. Mon. Wea. Rev., 116, 1302-1319. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -293__________ , and R.W. Kropfli, 1984: Buoyancy and pressure perturbations derived from dual-Doppler radar observations of the planetary boundary layer: Applications for matching models with observations. J. Atmos. Sci., 41, 3007-3020. __________ , B.D. Schmidt, and L.W. Uccellini, 1986: Simulation experiments for testing the assimilation of geostationary satellite temperature retrievals into a numerical prediction model. Mon. Wea. Rev., 114,12131230. Goody, R.M, and Y. L. Yung, 1989: Atmospheric Radiation - Theoretical Basis. Oxford University Press, 519 pp. Haltiner, G. J., and R. T. Williams, 1980. Numerical Prediction and Dynamic Meteorology. 2nd Edition. John Wiley and Sons, Inc., 477 pp. Hane, C. E., and P. S. Ray, 1985: Pressure and buoyancy fields derived from Doppler radar data in a tornadic thunderstorm. J. Atmos. Sci., 42, 18-35. Hermes, L.G., 1988: Retrieval of horizontal gradients of virtual temperature using winds from the Oklahoma-Kansas PRE-STORM profilers. M.S. thesis, University of Oklahoma, Norman, Oklahoma, 133 pp. __________ , 1991: Comparisons of rawinsonde-deduced kinematic and thermodynamic quantities with those deduced from VHF profiler observations. Mon. Wea. Rev., in press. Hill, C. K., and R, E. Turner, 1983: NASA's AVE/VAS program. Bull. Amer. Meteor. Soc., 64, 796-797. Hogg, D. C., M. T. Decker, F. O. Guiraud, K. B. Earnshaw, D. A. Merritt, K. P. Moran, W. B. Sweezy, R. G. Strauch, E. R. Westwater, and C. G. Little, 1983: An automatic profiler of the temperature, wind and humidity in the troposphere. J. Climate Appl. Meteor., 22, 807-831. Holton, J., 1972: An Introduction to Dynamic Meteorology. Academic Press, New York, 319 pp. IMSL, 1987a: Interpolation and approximation. IMSL Math Library, Vol. 2, Ch. 3. IMSL, Houston, TX. IMSL, 1987b: Integration and differentiation. IMSL Math Library, Vol. 2, Ch. 4. IMSL, Houston, TX. Jedlovec, G. J., 1985: An evaluation and comparison of vertical profile data from the VISSR atmospheric sounder (VAS). J. Atmos. Oceanic Techno!., 2, 559-581. Kaplan, L. D., 1959: Inference of atmospheric structure from remote radiation measurements. J. Opt. Soc. Am., 49,1004-1007. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -294King, J.I.F, 1956: The radiative heat transfer of Planet Earth. Scientific Uses of Earth Satellites, Univ. of Michigan Press, Ann Arbor. Koch, S.E., M. desJardin and P.J. Kocin, 1981: The GEMPAK Barnes objective analysis scheme. NASA Tech. Memo. 83851, NASA/GLAS, Greenbelt, MD 20771, [NTIS-N8221921 ], 56 pp. ________, ________, and_________ , 1983: An interactive Barnes objective map analysis scheme for use with satellite and conventional data. J. Climate Appl. Meteor., 22, 1487-1503. Kuo, Y.-H., and R.A. Anthes, 1985: Calculations of geopotential and temperature fields from an array of nearby continuous wind observations. J. Atmos. Oceanic Technol., 2, 22-34. _________, E.G. Donall, and M. A. Shapiro, 1987a: Feasibility of short-range numerical weather prediction using observations from a network of profilers and its implications to mesoscale predictability. Mon. Wea. Rev., 115, 2402-2427. _________ , D. O. Hill, and L. Cheng, 1987b: Retrieving temperature and geopotential field from a network of wind profiler observations. Mon. Wea. Rev. ,115, 3146-3165. Lanczos, C ., 1956: Applied Analysis.. Prentice Hall, Englewood Cliffs, NJ. Lee, T.-H., D. Chesters, and A. Mostek, 1983: The impact of conventional surface data upon VAS regression retrievals in the lower troposphere. J. Climate Appl. Meteor., 22, 1853-1874. Lewis, J., C. Hayden, and J. Derber, 1989: A method for combining radiances and wind shear to define the temperature structure of the atmosphere. Mon. Wea. Rev., 117, 1193-1207. Luenberger, D.G., 1969: Optimization by vector space methods. John Wiley and Sons, Inc., New York, 319 pp. Marquardt, D.W., and R.D. Snee, 1975: Ridge regression in practice. Amer. Stat., 29, 3-20. May, P.T., R. G. Strauch, and K. P. Moran, 1988: The altitude coverage of temperature measurements using RASS with wind profiler radars. Geophys. Res. Lett., 15, 1381 - 1384. ________, K. P. Moran, and R. G. Strauch, 1989: The accuracy of RASS temperature measurements. J. Appl. Meteor., 28, 1329 - 1335. , R. G. Strauch, K. P. Moran, and W. L. Ecklund, 1990: Temperature sounding by RASS with wind profiler radars: A preliminary study. IEEE Trans, Geosci. Remote Sens., 28,19 - 27. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 9 5 - Meeks, M. L., and A. E. Lilley, 1963: The microwave spectrum of oxygen in the earth’s atmosphere. J. Geophys. Res., 68,1683-1703. Menzel, W.P., W. Smith, and L. Herman, 1981: Visible infrared spin-scan radiometer atmospheric sounder radiometric calibration: an inflight evaluation from intercomparisons with HIRS and radiosonde measurements. Appl. Optics, 20, 3641 -3644. Modica, G. M., and T. T. Warner, 1987: The error associated with use of various forms of the divergence equation to diagnose geopotential and temperature. Mon. Wea. Rev., 115, 455-462. Nastrom, G.D., M.R. Peterson, J.L. Green, K.S. Gage, and T.E. VanZandt, 1989: Sources of gravity waves as seen in vertical velocities measured by the Flatland VHF radar. Preprints, 24th Conference on Radar Meteorology, Tallahassee, FL. Neiman, P.J., and M.A. Shapiro, 1989: Retrieving horizontal temperature gradients and advections from single station wind profiler observations. Wea. Forecasting, 4 ,222-233. NOAA, 1982: Daily Weather Maps, Weekly Series. National Oceanic and Atmospheric Administration, Washington, D.C. Parsons, D. B., C. G. Mohr, and T. Gal-Chen, 1987: A severe frontal rainband. Part 111: Derived thermodynamic structure. J. Atmos. Sci., 44, 16151631. Ramamurthy, M. K., and F.H. Carr, 1987: Four dimensional data assimilation in the monsoon region. Part I: Experiments with wind data. Mon. Wea. Rev., 115, 1678-1706. Rodgers, C.D.,1970: Remote sounding of the atmospheric temperature profiles in the presence of cloud. Quart. J. Roy. Meteorol. Soc., 96, 654 _________ , 1976: Retrieval of atmospheric temperature and composition from remote measurements of thermal radiation. Rev. Geophys. Space Phys., 14, 609-624. Rosenkranz, P.M., 1975: Shape of the 5 mm oxygen band in the atmosphere. IEEE Trans. Antennas Propag., 23, 498-506. Roux, F., J. Testud, M. Payen, and B. Pinty, 1984: West African squall line thermodynamics and structure retrieved from dual-Doppler radar observations. J. Atmos. Sci., 41, 3104-3121. Saha, K., and R. Suryanarayana, 1971: Numerical solution of geopotential with different forms of balance relationship in the tropics. J. Meteor. Soc. Japan, 49, 510-515. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 9 6 - Sasaki, Y. K., and J. A. McGinley, 1981: Application of the inequality constraint in adjustment of superadiabatic layers. Mon. Wea. Rev., 109,194 -196. Schroeder, J.A., 1990: A comparison of temperature soundings obtained from sim ultaneous radiometric, radio-acoustic, and rawinsonde measurements. J. Atmos. Oceanic Technol., 7, 495-503. Sienkiewicz, M.E., 1982a: AVEA/ASU: 25mb sounding data. NASA Contractor Report CR-170691. Marshall Space Flight center, Alabama, 306 pp. ____________ ,1982b: AVE/VASIII: 25mb sounding data. NASA Contractor Report CR-170692. Marshall Space Flight Center, Alabama, 308 pp. ,1983a: AVE/VASIV: 25mb sounding data. NASA Contractor Report CR-170739. Marshall Space Flight Center, Alabama, 334 pp. ____________ ,1983b: AVEA/ASV: 25mb sounding data. NASA Contractor Report CR-170740. Marshall Space Flight Center, Alabama, 326 pp. _____________ , 1990: Comments on "A theory for the retrievals of virtual temperature from remote measurements of horizontal winds and thermal radiation". Mon. Wea. Rev., 118, 988-989. ____________ , and T. Gai-Chen, 1988: Use of wind profilers and radiometric information for retrieval of virtual temperature . Preprints, Eighth Conference on Numerical Weather Prediction, Baltimore, MD. _____________, and T. Gal-Chen, 1989: Retrieval of virtual temperature from remote measurements of wind and thermal radiation. Preprints, 12th Conference on Weather Analysis and Forecasting, Monterey, CA. Smith, W. L., 1970: Iterative solution of the radiation transfer equation for the temperature and absorbing gas profile of an atmosphere. Appi. Opt., 9, 1993-1999. __________ , 1983: The retrieval of atmospheric profiles from VAS geostationary radiance observations. J. Atmos. Sci., 40, 2025-2035. __________ , 1986: Vertical resolution of current and planned satellite sounding instruments with respect to mesoscale applications. Memorandum, Cooperative Institute for Meteorological Satellite Studies, 20 March 1986. __________, H.M. Woolf, and H.E. Fleming, 1972: Retrieval of atmospheric temperature profiles from satellite measurements for dynamical forecasting. J. Appl. Meteor., 11, 113-122. Snider, J.B., 1972: Ground based sensing of temperature profiles from angular and multispectral microwave emission measurements. J. Appl. Meteor., 11, 958-967. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -297_________, and E. R. Westwater, 1972: Radiometry. Remote Sensing of the Troposphere. V. E. Derr, Ed., National Oceanic and Atmospheric Administration. USA, Superintendent of Documents, USA Government Printing Office, Washington, D.C., Ch. 15,15-1-15-32 . Strand, O. N., and E. R. Westwater, 1968: Minimum-rms estimation of the numerical solution of a Fredholm integral equation of the first kind. SIAM J. Numer. Anal., 5, 287-295. Strauch, R.G., K.P. Moran, P.T. May, A.J. Bedard, and W.L. Ecklund, 1989: RASS temperature soundings with wind profiler radars. Preprints, 24th Conference on Radar Meteorology, Tallahassee, FL. Teweles, S., Jr., and H.B. Wobus, 1954: Verification of prognostic charts. Bull. Amer. Meteor. Soc., 35, 455-463. Thompson, O.E., M. D. Goldberg, and D.D. Dazlich, 1985: Pattern recognition in the satellite temperature retrieval problem. J. Climate Appl. Meteor., 24, 30-48. ____________, D.D. Dazlich, and Y.-T. Hou, 1986: The ill-posed nature of the satellite temperature retrieval problem and the limits of retrievability. J. Atmos. Oceanic Technol., 3, 643-649. Tikhonov, A. N., 1963: On the solution of incorrectly stated problems and a method of regularization. Dokl. Akad. Nauk. USSR, 15, 501 Twomey, S., 1963: On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature. J. Ass. Comput. Mach., 10,97. _________ , 1977: Introduction to the mathematics of inversion in remote sensing and indirect measurements. Elsevier Scientific Publishing, Amsterdam, 243 pp. Uddstrom, M.J., and D.Q. Wark, 1985: A classification scheme for satellite temperature retrievals. J. Climate Appl. Meteor., 24,16-29. Ulaby, F.T., R.K. Moore, and A. K. Fung, 1981: Microwave Remote Sensing Active and Passive. Vol. 1 Microwave Remote Sensing - Fundamentals and Radiometry. Addison-Wesley, Reading Mass.. van de Kamp, D. W., 1988: Profiler Training Manual #1 - Principles of Wind Profiler Operation. NOAA/ERL, Boulder, 49 pp. Van Zandt, T.E., G.D. Nastrom, J.L. Green, and K.S. Gage, 1989: The spectrum of vertical velocity from Flatland radar observations. Preprints, 24th Conference on Radar Meteorology, Tallahassee, FL. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -298Wark, D. Q., and H. E. Fleming, 1966: Indirect measurements of atmospheric temperature from satellites. 1. Introduction. Mon. Wea. Rev., 94, 351-362. Weber, B.L., and D.B. Wuertz, 1990: Comparison of rawinsonde and wind profiler radar measurements. J. Atmos. Oceanic Technol., 7,157-174. _________ , D.B. Wuertz, R.G. Strauch, D.A. Merritt, K.P. Moran, D.C. Law, D. van de Kamp, R.B. Chadwick, M.H. Ackley, M.F. Barth, N.L. Abshire, P.A. Miller, and T.W. Schlatter, 1990: Preliminary evaluation of the first NOAA demonstration network wind profiler. J. Atmos. Oceanic Technol., 7, 909918. Westwater, E.R., 1965: Ground-based passive probing using the microwave spectrum of oxygen. Radio Sci., J. Res. Natl. Bur. Std., 69D,1201-1211. __________, 1972: Ground-based determination of low altitude temperature profiles by microwaves. Mon. Wea. Rev., 100,15-28. __________ , and N.C. Grody, 1980: Combined surface- and satellitemicrowave temperature profile retrieval. J. Appl. Meteor., 12,1438-1444. _________ , and O.N. Strand, 1968: Statistical information content of radiation measurements used in indirect sensing. J. Atmos. Sci., 25, 750-758. __________, M.T. Decker, A. Zachs, and K.S. Gage, 1983: Ground based remote sensing of temperature profiles by a combination of microwave radiometry and radar. J. Climate Appl. Meteor., 22,126-133. _________ , W.B. Sweezy, L.M. McMillan, and C. Dean, 1984: Determination of atmospheric temperature profiles from a statistical combination of ground-based Profiler and operational NOAA 6/7 satellite retrievals. J. Climate Appl. Meteor., 23,689-703. _________ , J.B. Snider, and A.V. Carlson, 1975: Experimental determination of temperature profiles by ground-based microwave radiometry. J. Appl. Meteor., 14, 524-539. _________ , Z. Wang, N.C. Grody, and L.M. McMillin, 1985: Remote sensing of temperature profiles from a combination of observations from the satellite-based Microwave Sounding Unit and the ground-based Profiler. J. Atmos. Oceanic Technol., 2, 97-109. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX A A.1 Derivation of the Divergence Equation Minimization The functional j(<t>) given by (3.5) would be minimized for a particular choice of + provided the variation of j is zero: „ lim J l ^ > - *<♦> . 0 a-»o a (A.1) a is some small real number and t\ is some arbitrary function, so that <j>+ar) is a function in the "neighborhood" of <|>. Since it can be shown that the operator 6 acts as a differential operator with respect to dependent variables the variation 8 j in (A.1) can quickly be transformed to give the Euler-Lagrange equation for the problem: 8 j = 5JJ( v<j) - G ) 2 dS = JJ 8 [ (V<|> - G) ] 2jJ(V<j> - G) • ( 8 V<j)) d x dy = 2 2 dx dy = jJ(V<J) - G) * V (S(J>) dx dy (A.2) By application of the chain rule, we get 5 j = 2 JJV • [ (V<j> - G ) 8 <|)] d x d y - JJv * (V<|) - G)8 <|> d x d y (A.3) We know 8 j = 0 if each of the two terms in (A.3) above is individually zero. The first term of (A.3) gives the boundary conditions for the problem. By Gauss’ divergence theorem: JJv- [ ( V<j) - 0 ) 8 +] dx d y = | n 1 [ (V<j> - G) 8 <J) ] dT (A.4) A sufficient condition for this term to be zero would be to have 8 + (V+ - g ) • n = 0 at each point along the boundary r . The second term in (A.3) gives the Euler-Lagrange equation for this -299- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -3 0 0 - minimization problem. If 5iJ>= ocn is arbitrary, then at each point in the domain, in order for the integral to be zero we must have V - (V<f> - 6 ) = 0 or V2<j> = V - g . ( A .5 ) A.2 Derivation of the Contribution Function Dv(z) In this case, Sj can be written 8 j = X 2py(Dy(Z)) + 8 X ^ V Dy(z)dzj + X > t v ^ [ z Wv ( z ) ^J2 Wv ( z ) Dy(z) and we may collect terms and write (for y = 2 PyD y ( Z ) + X.\lWv ( + Z dz - 8 dy(z) dzj 8vyj ( A .6 ) l , , n) ) I SD y ( z ) d z X S * V g ’ wv(z) Dy(Z ) dz - SvyJ = 0 (A.7) V and since Sd y( z ) and 8XV are arbitrary functions, then the factors multiplying each must be zero in order for the equation to be equal to zero. Thus: 2|lyD y(z) + X ^ V W V (Z) = (A.8 ) 0 V f W v(z) ^23 Dv ( z ) Multiply (A.8 ) by f f PyW y(z)D y(z) Za w Y( z ) dz dz = 1; f wv (z) zs and integrate over + £ Dy(z) z. J°°W y(z)W v ( z ) »■'Zs Zo dz = 0 v ^ y (A.9) The equation becomes dz = 0 (A.10) V which can be readily solved for Xv, thus Xv = - 2 Py Wy ( z ) Wv ( z ) d z J . (A.11) ( Z) (A. 12) This may be substituted back into (A.8) to get Dy (Z ) = Wy ( z ) Wv ( z ) d z J Wv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -301- A.3 Minimization of (4.49) - Gradient Minimum Variance Retrieval We want to find the field of t which minimizes : oo J oo J J ( V HT ( z * ) —VHT ( z * ) ) • L ( z * , z * ' ) ■ ( VhT ( z * ' ) —V HT ( z * ' ) ) d z * d z * ' L z *S S Z*8 OO oo J wv ( z * ) V hT ( z *) d z * —h v ) • M vy ’ ( JwY( z * ’) V hT ( z *') d z * ’- h y ) v y dS * z* ' (A. 13) = J (T) This equation can be written using tensor notation, with indices i and j indicating direction, and repeated indices implying summation: f f°°dT(z*) J J Jnf r 1S z* dT(z*) 3t (z* ') - 3t (z * ') — 55—'o*-**1 z* . 43 t ( + ( JIw w *^(zz ** ) - f a -*' '■ d z * — ) m]J ( J w V ( z * ‘) a — d z * ' —hT) dS dXj 3 Z . = J(T) (A. 14) The minimization is derived in the usual way by calculating 8 j, then determining what conditions are necessary in order for 8 j to equal zero. Thus: 8 j = = 0 f c d f° 3 t ( z * ‘) 3 t(z * ') . d z s z* U *s Jwv ( z * ) ( ^ 8t ( z * ) ) . m£[ ( JwY(z*')— ] dz*'-h^) .dz1 where we have invoked a symmetry argument (i.e. L ji (z* •, z * ) 1 ; m^ = dS, (A. 15) L i j ( z * / z * , ) = ) to enable some of the terms to be consolidated. Then, when we apply the product rule, we get: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -3 0 2 - j R r<z*>I Lij ( Z * , 2 * ' ) ( J 5T(z*>ad JL z*s (Z * / z i * 1 9t (z * 1) 9t (z * 1) ) dz* 3xi 0Xi ,0 T ( z * 1 ) dT(z * ' ) ) ( . d z* ) d z * ' I dz* + U *s J a il z% (Jwy(z*’)^ | * — dz*'-hp l .dz1 z *s J5t (z * ) ^ - . WV ( Z * ) M ^ ( JwY ( z * ' ) dS = 0 . (A.16) ' } dz * - h ] We then apply Leibnitz’ rule to the first and third terms, to bring the derivative outside the vertical integral: 0 T ( z * 1) J fc l/ J . )dz’ 0Xj f T(Z* ’ °° r ^ d T ( z * 1) dz* + op 8 t(z * )w v (z*)m^ ( J w r ( z *') — ' ] d z ) . dz * * + 9z*o f 9 t(z * ') 3t ( z * ' ) - — ^ ------) d z * 1 + 0Xi 5 t ( z * s ) j Lij ( z * s , z * 1) ( - Jd x ~ dz* 8 t ( z * s ) Wv ( z * s ) ( JwY( z * ' ) 9 T ^ * ' } d z * ' - h ] ) - Z*B J ST(z*>al;j J 8 T ( z * ) ^ - |wv(z*)M g (Jw Y (z*,) 3T^ * ' ) -) dzi d z * ' —h^) .dz* . dz * dS = 0 . (A.17) The first four terms of (A.17) provide the boundary conditions for the problem. We use Gauss’ divergence theorem on the first and second terms to obtain the lateral boundary conditions: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -3 0 3 - m oo #r J oo J 8t ( z i n • h ( z *, z*') • (VHT ( z * ,) - V Hif ( z * 1) ) dz* oo <-> Wv ( z *) n *M vy " - J«T( z * ’) VhT ( z*') d z *'-h Y dz* ) d r = 0 , (A.18) v y J where n is a unit vector normal to the boundary conditions, t( z * ) on r is held fixed (8 t ( z * ) = . r. For Dirichlet boundary o); for Neumann boundary conditions, the temperature field must satisfy oo J n • L* ( z * , z * ' ) • ( VhT ( z * 1) -V fjT ( z * 1) ) d z * ' + oo £ 2 Wv (z *) n •Mvy ’ JwY(z*') VHT (z * ')d z * '-h Y (A. 19) v y at all points on r. The third and fourth terms are zero if we specify the temperature at the surface ( 5 t ( z * 3) = o). Otherwise, the temperature field would have to satisfy dz*s dxi ■\ 3t ( z * ■)dz* ' Wv (z*s) mJJ ( JwY(z*') — 0 " - " - d z * + hY) . = 0 (A.20) which is true if there is no surface pressure gradient (generally not the case) or if the temperature profile is chosen such that the bracketed terms equal zero. This temperature profile would not satisfy the necessary condtions for the interior points of domain (as specified in the Euler-Lagrange equations, see below) given an arbitrary function Lij ( z * s , z * •). Thus it is preferable to use the condition 8t ( z *3) = o. The Euler-Lagrange equations come from the fifth and sixth terms of (A.17). Given an arbitrary 5t (z * ); if the sum of these two terms equals zero, the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -3 0 4 - temperature field must satisfy o d 0Xi ' I Li-j ( z * , z * ' ) 9 t (: 3xi -dz*' oo 3xi ’ J' +Wv ( z * ) M : r 13 3t ( z * - ) dz*' 3xj vv 5 J * L ij ( z * , z * • WY ( Z * v (A.21) d z * 1 + Wv ( z * ) M [ hT 13 3 OXj at each interior point. If we express this in vector form, we have Eq. (4.50): oo oo J L ( z * , 2 *') •V HT ( z * ' ) d z * ' + X S Wv ( z * ) M vy- JwY( z * ' ) V HT ( z * ' ) d z * ' o< J L ( z * , z * ' ) • V hT ( z * ' ) dz*' + ^ ^ W v(z*) M Vy • h Y . (A.22) v y A.4 Derivation of Gal-Chen's (1988) Method Gal-Chen's method is a simplified version of the minimum variance gradient retreival derived in section A.3. To obtain the equations for this method from the equations derived above, the weighting dyad function replaced by and 5 ( z 8ij8(z*, z* * , z * •) p0 ( z * ) Lij (z * , z * •) is , where 8 i j is the Kronecker delta function 1) is the Dirac delta function. Additionally, is replaced by 8ij pv. We substitute these functions into (A. 17): J air J 8t ( z*') J . S T ( z * ) w v ( z * ) S i j Svrpv 0 Z* + ( Jwr(z * c 8 T ( z * s) 3z * 8t ( z * s )Wv ( z * s ) ' 3t ( z * ' ) |8ij8(z*,Z*')po(z*) V + 8t ( z * ‘ ) 5 T ( Z *> j 8 i 3 5 ( z * , z * ,) P o ( z * ) ( ^ ^ ; 9x3 ) ) dz*' . d z ’ 5x-j ■ d z *'-h 3 t ( z * ■) aXj y' .d z ’ ^ j dz*' r 3t (( 7 z *') *') 8 ij S ^ p v ( l w Y ( z * ’) — ^ — d z * ' - h Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -305- J 3 3xi| f 3 T ( z * ') J 8 i j 5 ( z * , z * l) P o ( z * ) ( - ^ — 00 ■ f 3 L 3 ^ 3T(z*) - d z * 1 hT) , d z * dS = ^ 3xi Jaz* Z*8 oe f d + w v ( z * ) 5 ij S^rpy ( j wY( Z*') c „ fdTtz*) st ( z -)P o<=*) J 5^7 j S > p / z * •\ r° ■ 3 t (z * ' ) I - — 3 ^ ------) d z * | d z * 00 f x^ J < 8 t ( z * ) Wv ( z * ) (Jv 8 3t (z* 1 ( IWv ( z * 1) 3z*< + + _ f „ 3 t(z* ) 3t(z*) 6T «z*«>Po'z*> i - ^ T 1 — t(z * ,) Pv W ^ z * , ) .dz1 dz*'-h^ 5^— 1 ( fwv ( z * ' ) ^ | p - d z * ' - h ” ) Z*8 f°c - 3 3 t ( Z *) 3t(z*) | 6t<z*>5^ r (z*> {~ s ^ ~ - OO Q0 J bK z* 1 a* > } dz* ^ ) ^ - . PvV^fz*) ( Jwv ( z * 1) ~ dz*'-h^) ldzv dS = 0. (A.23) Again, the first four terms define the boundary conditions. We justify the use of 8 t ( z * s) = 0 in the third and fourth terms by the same arguments as given in the previous section. The lateral Neumann boundary conditions derived from the first and second terms are: n ’ [ p 0 ( z * ) ( V hT ( z * ) - V hT ( z * ) ) + 00 2 = 0 pvWv ( z * ) . Jwv ( z * ’) V HT ( z * ' ) d z * ' - h v (A.24) l?*s The Euler-Lagrange equations arise from setting the sum of the fifth and sixth terms of (A.23) equal to zero: 00 -JL , _ ^ 3t(z *)) 3xi (Po(z ] 3xi 3xi' 3t ( z*'i .Wv ( z * ) wv ( z * 1) ■gx dz*'! J P v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -3 0 6 - 3t ( z *) = 3 x : p o (z* ) (A.25) Iz*)h 0Xi These conditions must be satisfied at all points in the interior of the domain. Since pv is assumed not to be a function of ( x , y , z) it may be taken out of the integral. The derivative in the second term is brought into the integral through application of Leibnitz’ rule, which produces the “extra term” based on VHzs* which was discussed in Sienkiewicz (1990). Eq. (A.25) then becomes: . . . 9 t ( z *') s i * p* J a y wv(z* lwv <z* 'r ~axj ' d z *' 3 t ( z *) P » <Z*> 0 X , | + Pvia|-(w',(z * )h ') + dz * 0Xi (A.26) 0Xi When this equation is written in vector form we obtain Eq. (4.52) V H - p o V HT + ZvPvf ,V H - [ Wv ( z *) Wv ( z * ' ) V hT ( z *' ) ] d z * ' = V H - p 0 V HT ZS + Zy pv[VH■Wv ( z *) hv + VHZS* • Wv ( z*) Wv ( z 3* ) VhT ( z s* ) ] . (A.27) A .5 Derivation of an Alternate Form of Gal-Chen’s Method (Sienkiewicz and Gal-Chen, 1988, 1989) Eq. (4.62) can be rewritten in tensor form: [z 0X i J(T) 0 T (z * ) V 0X i J dz1 oo JL + P> 3xi J*WV( z *) T ( z *) dz *—T bv (A.28) dS / J and to minimize this we set the first variation 5 j ( t ) = 0 : oo 2J 5 f s J 7 - ^ 7(5t(z*)) p0 (z * ) f0T( z * ) t d x i 0T(z*) ' 0Xi J dz* *S Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -307fd Y a I r” , Pv h j ^ - j5T<z*)W v < z * ) d z * ^ 1) d z * ' - T BV . Jwv ( z * ') T ( z * 7 ds. (A.29) A This can be rewritten by use of the product rule: 8j ( t ) = o ■ * a ( dxi j ftt s 5t ( z *) Pq(z * ) ^9t ( z * ) 3t(z*) dxi dx\ dz* + _z* r r 3 \\ f f J S T { z * ) W v ( z * ) d z * P v ^ ^ • J w v ( z * ' ) T ( z * ' ) d z * ' —T bv // l?*s |(8 T(zM) ^ - f Pv J S T ( z * ) W v ( z * ) d z * oo r a a oo fwv ( z * ' ) T ( z * ') d z * ' - T bv ► d S . (A.30) dX id X i Z*s r. J- We use Leibnitz’ rule to move the derivative in the first term outside of the vertical integral: 2 Jfefi 8t ( z *> Pq (z *) — ( z *) dxi dxi 0T(z*s ) 3z 0 0T (z * ) 8 dz* + 3t ( z * s ) Po(zs*) — t ( z s*) | 3 x i \\ a J 8t ( z * ) Wv (z* ) d z * Pv0^ - I fwv ( z * 1) T ( z * ' ) d z * ‘- T bv 0Xi )J dxi , dz* - 3xi 7* L S OO (a a Pv J s T( z * ) wv ( z * ) d z * d x id x i f oo \" jW v ( z * 1) T ( * ) z 1 d z * ' —T V bv ► ds. (A.31) /- The first and third terms determine the lateral boundary conditions. We rewrite these in vector form: __ ____ oo 2 Jvh . J {5T(Z*> Po( z *) ( V hT ( z *) - VHT ( z * , ) } dz* + Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -3 0 8 - oo oo X pv J*5t(z*)Wv(z*) Vh . Jwv ( z * ') T ( z * ' ) d z * Tbv . dz 11 dS = 0 (A.32) By Gauss’ divergence theorem, we have: oo J j> n - _________ {S t ( z *) po(z*) ( OO v hT ( z *) - V hT ( z *) ) } dz1 oo E P v J 5 T ( Z * ) W V ( Z* ) V H . Jwv ( z * 1 ) T ( z * ' ) d z * T bv . d zv d r = 0 (A.33) This equation is satisfied if 8t ( z * ) =0 (Dirichlet conditions) or if o Po ( z * ) VhT ( z * ) • n +XP vWy (z* )VH . Po ( z *) VhT ( z * ) Wy(z*' ) T ( z * ' ) d z* 1' •n = - n +SPvWv ( z * ) V H Tbv • n (Neumann conditions) at every point on the boundary (A.34) r. The second term of (A.31) shows the terrain influence; as mentioned before this term is zero if there is no gradient of surface pressure. This method gives us two ways in which we can deal with the terrain influence question; we can specify the surface values of temperature as in the Gal-Chen (1988) method, but it is also possible to satisfy this term by specifying that the surface temperature gradient is equal to the first-guess gradient. The fourth and fifth terms of (A.31) give us the Euler-Lagrange equations for this problem; given that S t is arbitrary, those terms equal zero only i f : ?k + PvWv <z*) a 8t ( z * ) \ <z* dx; j 3 x i rI 1' 01* " 'l - 3x a lr a lr j W - m z - j d z * f + (A.35) at every point in the interior of the domain. When this is written in vector form, we obtain Eq. (4.63): Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -309V H • P oVhT 2y pv Wv ( z *) V H2 f Wv ( z * ' )T (z * ' ) dz* ' = J Zs* V H • Po V hT+ L v p v Wv ( z * ) Vh2 Tbv (A.36) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX B INTERCOMPARISON OF SOUNDINGS WITH SIMILAR BRIGHTNESS TEMPERATURES A comparison similar to the one presented by Thompson, et al. (1986) was performed for this study using simulated ground-based radiances from a 5 year set of Oklahoma City spring (Feb-May) soundings. The comparison showed 111 soundings (from a set of over a million intercomparisons) had radiance values for all four ground-based channels within a 0.2 K noise level of each other. When two MSU radiance channels (with noise level 1.0 K) were also considered the number of pairs of soundings which matched dropped to 35. These figures, however, cannot be used in direct comparison with Thompson et al. as this sounding data set was much more homogeneous. The rms differences between 25 mb and the surface for the 35 OKC soundings with matching ground and satellite radiances ranged from 0.64 K for the closest pair of soundings (which, incidentally, were taken within a few hours of each other) and 3.38 K for the most dissimilar pair, with the average rms difference from 25mb - surface being 2.32 K. The 111 OKC pairs that matched ground-based radiances (which is 0 .01 % of the total number of pairs) had rms differences ranging from 0.64 K to 5.21 K, with an average value of 2.79 K. (Note, however, that the average rms value in these cases were calculated using sounding levels evenly spaced in pressure so the differences in lower levels would be weighted more heavily than if height or log pressure coordinates were used.) -310- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -311- 200 200 E 400 j= 400 2 2 600 Ground-! 600 600 Grouhd +Saiallife, CS0.2K- 600 1000 1000 0 Figure B.1. S 10 Difference (K) 15 20 0 4 8 12 Difference (K) 16 20 Maximum differences and RMS differences (K) between temperatures at the same pressure level for OKC sounding "dissimilar pairs" with ground based brightness temperature differences < 0.2 K. Figure B.1 shows the maximum absolute difference and rms difference between pairs of OKC soundings at each level between 950 mb and 50 mb for the “dissimilar pairs” matching only the four ground-based channels and for the pairs that also matched two satellite channels. It is apparent from both figures that the rms difference between the “dissimilar pairs” is small at the surface but becomes 2 K or greater above 700 mb, and close to 4 K in the upper troposphere and stratosphere. The maximum differences were at 200 mb, near the tropopause level. Figure B.2 shows one "dissimilar pair" of profiles; it is apparent how the positive deviations in one layer can cancel out negative deviations in a higher layer. This shows the limitation of the ground-based channels in detecting differences in sounding profiles in the middle and upper troposphere, even given the fairly low noise level of 0.2 K. When the noise levels for the ground-based radiances are increased to a more realistic 0.5 K, the number of “dissimilar pairs" increases to 2256 (still only about 0.2% of the possible pairs), while those detected using satellite and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -312- i oo \ y -90 200 -80 -7 0 300 -60 500 -50 600 700 800 900 1000 -30 Figure B.2. -2 0 20 30 Temperature soundings at OKC from 0000 UTC, 28 Feb 1981 (solid line) and 0000 UTC, 24 Apr 1981 (dashed line). These soundings produce brightness temperatures that differ by less than 0.2 K. ground-based radiances increases to 645. The average rms differences between 25 mb and the surface for soundings with matching ground-based radiances increased to 3.05 K (with range from 0.61 K to 7.18 K) while those also matching satellite radiances had an average rms difference in the profile of 2.57 K (with range from 0.64 K to 5.37 K). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -313- 200 200 E 400 2 600 600 Ground-baded.cs 0.5K 800 Ground+saiollite.es 0.5K eoo 1000 1000 0 Figure B.3. 5 10 Difference (K) 15 20 0 5 10 Difference (K) 15 20 Maximum differences and RMS differences (K) between temperatures at the same pressure level for OKC sounding "dissimilar pairs" with ground based brightness temperature differences <, 0.5 K. Figure B.3 shows the maximum absolute difference and rms difference between the pairs for this noise level for the ground-based and satellite channels respectively. A comparison with Figure B.1 shows that the rms differences between pairs increased only a small amount (generally less than 0.5 K) and the maximum absolute differences increase generally by about 4 K, except near the tropopause level. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX C VERTICAL PROFILES OF TEMPERATURE AT STATION LOCATIONS This appendix contains skew-T diagrams showing vertical profiles of temperature or virtual temperature for selected times at Oklahoma City and Stephenville. Figures C.1, C.2, and C.9 - C.21 are profiles of virtual temperature interpolated from retrieved analyses at gridpoints using bilinear interpolation. Figures C.3 - C.8 are from retrievals performed at the station locations. In each figure, the thick solid line represents the ‘true’ sounding profile; either the observed temperature profile at a station or a profile interpolated from the ‘true’ temperature analysis. The thin solid line is the retrieved sounding; the caption tells which method has been used. The dashed line is the mean or first-guess field used in the retrieval. The sounding on the left-hand side of the figure is from Oklahoma City, 1200 UTC, 7 March 1982. The sounding on the right-hand side is either from Oklahoma City or Stephenville, at 1200 UTC on 27 March 1982. The retrievals at station locations have the Oklahoma City sounding instead of the Stephenville sounding because the raob from Stephenville at that observing time terminated early. -314- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 100 -9 0 -9 0 200 -00 200 -7 0 300 -6 0 -6 0 400 500 500 600 600 700 700 BOO 800 000 900 -50 -40 J000 1000 in f Oklahoma City -1 2 0 0 UTC, 7 March 1982 Figure C.1 -3) Stephenville - 1200 UTC, 27 March 1982 Vertical profiles of virtual temperature interpolated from gridded data analyzed in pressure coordinates for dynamic retrieval using the two-scale approximation, Lanczos' derivative, and Neumann boundary conditions. Dashed line: Mean temperature of gridded field. Thin solid line: Retrieved sounding. Thick solid line: 'True' gridded field. 100 -9 0 200 -7 0 200 -8 0 -70 900 -6 0 300 -316- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 -60 500 500 '50 GOO 600 700 700 800 -U0 800 600 600 1000 -31 Oklahoma City - 1200 UTC, 7 March 1982 Figure C.2 Stephenville - 1200 UTC, 27 March 1982 Vertical profiles of virtual temperature interpolated from gridded data analyzed in sigma coordinates for dynamic retrieval using the two-scale approximation, Lanczos’ derivative, and Neumann boundary conditions. Dashed line: Mean temperature of gridded field. Thin solid line: Retrieved sounding. Thick solid line: ‘True’ gridded field. -9 0 200 -80 -7 0 300 -6 0 <400 SOO -5 0 600 700 000 1000 ~2Q Oklahoma City - 1200 UTC, 7 March 1982 Figure C.3 TO Oklahoma City - 1200 UTC, 27 March 1982 Vertical profiles of temperature retrieved from conditioned regression coefficients using only groundbased radiometric measurements. Dashed line: Mean temperature of 5 year data (first guess).. Thin solid line: Retrieved sounding. Thick solid line: Observed temperature profile. i -317- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Oklahoma City - 1200 UTC, 7 March 1982 Figure C.4 Oklahoma City - 1200 UTC, 27 March 1982 Vertical profiles of temperature at Oklahoma City retrieved from conditioned regression coefficients using ground-based and satellite radiometric measurements. Dashed line: Mean temperature of 5 year data (first guess). Thin solid line: Retrieved sounding. Thick solid line: Observed temperature profile. 100 -90 -80 200 200 800 300 <<00 100 500 500 600 600 -70 -80 -50 X ' t ; 700 700 800 600 90 0 900 1000 1000 Oklahoma City - 1200 UTC, 7 March 1982 Figure C.5 Oklahoma City - 1200 UTC, 27 March 1982 Vertical profiles of temperature at Oklahoma City retrieved by Smith’s method using only groundbased radiometric measurements. Dashed line: Mean temperature of 5 year data (first guess). Thin solid line: Retrieved sounding. Thick solid line: Observed temperature profile. -319- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 -320- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Oklahoma City - 1200 UTC, 7 March 1982 Figure C.6 Oklahoma City - 1200 UTC, 27 March 1982 Vertical profiles of temperature at Oklahoma City retrieved by Smith’s method using ground-based and satellite radiometric measurements. Dashed line: Mean temperature of 5 year data (first guess). Thin solid line: Retrieved sounding. Thick solid line: Observed temperature profile. 100 200 200 -7 0 -7 0 300 300 -6 0 qoo -6 0 V 500 -5 0 500 -5 0 600 600 700 700 800 800 900 900 1000 1000 Oklahoma City -1 2 0 0 UTC, 7 March 1982 Figure C.7 Oklahoma City - 1200 UTC, 27 March 1982 Vertical profiles of temperature at Oklahoma City retrieved by the minimum information method using only ground-based radiometric measurements. Dashed line: Mean temperature of 5 year data (first guess). Thin solid line: Retrieved sounding. Thick solid line: Observed temperature profile. -321- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -SO 100 -90 100 -9Q .80 200 -70 900 -60 300 -60 soo 500 600 600 700 700 -50 -MO SOO 900 900 1000 1000 Oklahoma City -1 2 0 0 UTC, 7 March 1982 Oklahoma City - 1200 UTC, 27 March 1982 Figure C.8 Vertical profiles of temperature at Oklahoma City retrieved by the minimum information method using ground-based and satellite radiometric measurements. Dashed line: Mean temperature of 5 year data (first guess). Thin solid line: Retrieved sounding. Thick solid line: Observed temperature profile. -322- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. too -9 0 200 -8 0 300 MOO -323- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 500 -5 0 600 700 -MO 800 900 1000 Oklahoma City -1 2 0 0 UTC, 7 March 1982 Figure C.9 Stephenville - 1200 UTC, 27 March 1982 Vertical profiles of virtual temperature interpolated from gridded data analyzed in pressure coordinates for retrieval by Smith’s method from ground-based and satellite radiometric measurements and mean field first guess. Dashed line: Mean temperature of gridded field. Thin solid line: Retrieved sounding. Thick solid line: ’True’ gridded field. -9 0 -9 0 200 200 -80 -70 -7 0 300 300 400 500 SOO -5 0 -5 0 600 600 100 Oklahoma City - 1200 UTC, 7 March 1982 800 800 900 900 iooo 1000 Stephenville - 1200 UTC, 27 March 1982 Figure C.10 Vertical profiles of virtual temperature interpolated from gridded data analyzed in pressure coordinates for retrieval by the minimum information method from grouna-based and satellite radiometric measurements and mean field first guess. Dashed line: Mean temperature of gridded field. Thin solid line: Retrieved sounding. Thick solid line: ‘True’ gridded field. -3 2 4 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 100 -9 0 -8 0 200 300 -325- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 -6 0 500 -5 0 600 700 800 900 1000 -31 Oklahoma City - 1200 UTC, 7 March 1982 Stephenville - 1200 UTC, 27 March 1982 Figure C.11 Vertical profiles of virtual temperature interpolated from gridded data analyzed in pressure coordinates for fields derived from retrievals of temperature gradients by Smith’s method from gradients of ground-based and satellite radiometric measurements and mean field first guess. Dashed line: Mean temperature of gridded field. Thin solid line: Retrieved sounding. Thick solid line: True’ gridded field. -90 -9 0 200 -00 -70 200 ■BO ;7Q V 300 300 400 400 soo SOO 600 600 700 700 -so 600 Oklahoma City -1 2 0 0 UTC, 7 March 1982 900 900 1000 1000 Stephenville - 1200 UTC, 27 March 1982 Figure C.12 Vertical profiles of virtual temperature interpolated from gridded data analyzed in pressure coordinates for fields derived from retrievals of temperature gradients by the minimum information method from gradients of ground-based and satellite radiometric measurementsand mean field first guess. Dashed line: Mean temperature of gridded field. Thin solid line: Retrieved sounding. Thick solid line: ‘True’ gridded field. -326- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 100 -327- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A Oklahoma City - 1200 UTC, 7 March 1982 Stephenville - 1200 UTC, 27 March 1982 Figure C.13 Vertical profiles of virtual temperature interpolated from gridded data analyzed in pressure coordinates for retrivals by Gal-Chen's method from gradients of ground-based and satellite radiometric measurements and mean field first guess. Dashed line: Mean temperature of gridded field. Thin solid line: Retrieved sounding. Thick solid line: ‘True’ gridded field. 100 -9 0 -9 0 200 -00 200 •80 -70 -7 0 > V / 300 v9 •60 V >400. 500 500 600 600 700 700 800 000 900 900 tooo 1000 -3 Oklahoma City -1 2 0 0 UTC, 7 March 1982 Stephenville - 1200 UTC, 27 March 1982 Figure C.14 Vertical profiles of virtual temperature interpolated from gridded data analyzed in pressure coordinates for retrieval by Smith’s method from ground-based and satellite radiometric measurements and wind-derived first guess field. Dashed line: Wind-derived first guess. Thin solid line: Retrieved sounding. Thick solid line: ’True’ gridded field. -328- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 -9 0 200 •80 300 SOO 600 700 800 900 1000 ^7 i Oklahoma City - 1200 UTC, 7 March 1982 Stephenville - 1200 UTC, 27 March 1982 Figure C.15 Vertical profiles of virtual temperature interpolated from gridded data analyzed in pressure coordinates for retrieval by the minimum information method from ground-based and satellite radiometric measurements and wind-derived first guess field. Dashed line: Wind-derived first guess. Thin solid line: Retrieved sounding. Thick solid line: 'True' gridded field. -3 2 9 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 •90 -9 0 200 200 300 300 -7 0 SOO SOO “ SO -50 600 600 700 700 800 800 90 0 900 1000 1000 itr Oklahoma City -1 2 0 0 UTC, 7 March 1982 Stephenville - 1200 UTC, 27 March 1982 Figure C.16 Vertical profiles of virtual temperature interpolated from gridded data analyzed in pressure coordinates for fields derived from retrievals of temperature gradients by Smith’s method from gradients of ground-based and satellite radiometric measurements and wind-derived first guess field. Dashed line: Wind-derived first guess. Thin solid line: Retrieved sounding. Thick solid line: ‘True’ gridded field. -3 3 0 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 100 200 300 500 -50 600 800 900 1000 Oklahoma City -1 2 0 0 UTC, 7 March 1982 Stephenville - 1200 UTC, 27 March 1982 Figure C.17 Vertical profiles of virtual temperature interpolated from gridded data analyzed in pressure coordinates for fields derived from retrievals of temperature gradients by the minimum information method from gradients of ground-based and satellite radiometric measurements and wind-derived first guess field. Dashed line: Wind-derived first guess field. Thin solid line: Retrieved sounding. Thick solid line: ‘True’ gridded field. -331- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. !QO -9 0 -90 -80 -7 0 200 -80 200 -7 0 300 300 -6 0 -6 0 500 500 -50 -SO 600 600 700 -*10 700 -MO 800 800 900 900 1000 -31 Oklahoma City -1 2 0 0 UTC, 7 March 1982 1000 -3 Stephenville - 1200 UTC, 27 March 1982 Figure C.18 Vertical profiles of virtual temperature interpolated from gridded data analyzed in pressure coordinates for retrivals by Gal-Chen’s method from gradients of ground-based and satellite radiometric measurements and wind-derived first guess field. Dashed line: Wind-derived first guess field. Thin solid line: Retrieved sounding. Thick solid line: ‘True’ gridded field. -332- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. too 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 100 -9 0 -9 0 -00 200 200 -00 -7 0 300 -60 500 SOO -50 -5 0 600 600 700 700 BOD 000 j9 Oklahoma City - 1200 UTC, 7 March 1982 900 1000 1000 Stephenville - 1200 UTC, 27 March 1982 Figure C.19 Vertical profiles of virtual temperature interpolated from gridded data analyzed in sigma coordinates, for fields derived from retrievals of temperature gradients by Smith’s method using gradients of ground-based and satellite radiometric measurements and wind-derived first guess field. Dashed line: Wind-derived first guess (interpolated sfc - 0.825 a). Thin solid line: Retrieved sounding. Thick solid line: ‘True’ gridded field. 100 •-90 200 -8 D -7 0 200 -7 0 300 -6 0 M00 300 -6 0 500 -5 0 SOO -50 600 600 700 800 700 -MO 800 900 900 1000 -30 "20 Oklahoma City - 1200 UTC, 7 March 1982 1000 20 10 Stephenville - 1200 UTC, 27 March 1982 Figure C.20 Vertical profiles of virtual temperature interpolated from gridded data analyzed in sigma coordinates, for fields derived from retrievals of temperature gradients by the minimum information method using gradients of ground-based and satellite radiometric measurements and wind-derived first guess field. Dashed line: Wind-derived first guess field (interpolated sfc - 0.825 o). Thin solid line: Retrieved sounding. Thick solid line: ‘True’ gridded field. -334- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. too -335- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Oklahoma City - 1200 UTC, 7 March 1982 Stephenville - 1200 UTC, 27 March 1982 Figure C.21 Vertical profiles of virtual temperature interpolated from gridded data analyzed in sigma coordinates, for retrievals by Gal-Chen's method using gradients of ground-based and satellite radiometric measurements and wind-derived first guess field. Dashed line: Wind-derived first guess field (interpolated sfc - 0.825 a). Thin solid line: Retrieved sounding. Thick solid line: ‘True1gridded field.

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