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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
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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
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RETRIEVAL OF VIRTUAL TEMPERATURE FROM VERTICAL WIND PROFILES
AND GROUND-BASED MICROWAVE RADIOMETER DATA
A DISSERTATION
APPROVED FOR THE SCHOOL OF METEOROLOGY
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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-
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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
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-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
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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
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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
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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
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-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
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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
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-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
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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
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-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
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-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
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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
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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-
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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-
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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-
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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
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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
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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
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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-
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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.
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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
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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
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-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
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-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
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-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
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-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)
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-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 .
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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-
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-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
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-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
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-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
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(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
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-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)
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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 -
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-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
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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
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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
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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
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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.
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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,
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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
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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)
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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
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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 ‘ ,
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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
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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.
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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.
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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.
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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
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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
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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
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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,
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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
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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
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-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)
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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
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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
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-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
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-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)
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-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.
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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).
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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
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-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*
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= 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
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-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.
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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
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-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
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-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
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-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
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-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:
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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)
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-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:
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-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.
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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.
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-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
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-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
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-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-
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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.
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-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
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-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.
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-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.
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-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
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-154-
i>) vAd 3 '
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20
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RMS error (m)
15
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RMS error (m)
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RMS error (m)
20
25
Twoscafoappx. •
1000
30
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15
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25
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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
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-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
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d) VASp
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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
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f
400
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s
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600
600
800
1000
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800
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RMS error (m)
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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
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40
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Sj score
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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
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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).
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-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
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-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
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-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
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-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).
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-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.
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-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).
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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).
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-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.
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-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).
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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).
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-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.
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-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
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-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.
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-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
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-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
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-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
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-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
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- 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
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-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.
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-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
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-204' ' Mis1-
- t-T -r-r
l) VAS
b)
1 1 1 1
i
-r-.-r-T - |
i- r - r - T -
—
1..........
o i
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- i- r - 7
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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
£
>
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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
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-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.
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-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.
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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
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-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
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•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
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-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
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-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
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-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
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-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.
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-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
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............
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600
: ............... P
:
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r
................. !..................
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a ’Sij T
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Sm iths mqthbtl^_
).5K rms e m x To ' — ■
\
j
c
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............... j 1 ’ 1 1 | 1 ' " ’" H
t
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: ^
1
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• ........ | .............. ..
' ' ' ''
j
'
■ " > “‘j r 0
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------- 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
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-
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
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-218-
|
.
I
I
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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
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1
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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-
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temperature as first guess. Solid line: Standard deviation of
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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 -
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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 -
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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
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(a) True’ field
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(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-
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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
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(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
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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 -
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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.
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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
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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
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600
800
800
1000
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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
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-240-
1'* '
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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
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-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
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-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
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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.-
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1
RMS error (K)
3
2
Gai-Oien's method.
S
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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
.......
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--< > •
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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
**■
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^ *
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—
— 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.
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-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
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Wind T first guess
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Wind T fir'it guess
’ MTriinium ih T o n ^ ion
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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.
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-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
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-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 ....
....
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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
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[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
'
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4
0.2
0.2
0.4
0.4
.
.
0.6
I Gradient retrieval
I Wind T fins guess
.
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0.6
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’ — Smith's method
— -q — Minimum information
r.r.^.r...Gat£thdn'sjsflttiod_
0.8
0
20
40
60
60
100
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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
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Wind T first guees
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Wind T f|rst guees
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...........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
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0 5KRMSerrTB j
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score
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Pressure (mb)
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d) VAS 5
—I— I— I— r
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0.5 KR^SerrTg
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— 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.:%
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1000
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60
80
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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.
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-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
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-2 7 2 -
t--- 1—i-- 1—i-- 1-- 1-- 1-- r
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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
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-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
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-2 7 4 -
a) V A S b
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1000
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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.
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-2 7 5 -
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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.
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-2 7 6 -
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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.
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-2 7 7 -
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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.
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-2 7 8 -
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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.
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-2 7 9 -
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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.
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-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.
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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-
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-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
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-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
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-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
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-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
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-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
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-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
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-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
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-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
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-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.
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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
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-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
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-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* +
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-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-
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-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
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-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-
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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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