# Estimation of rainfall using dual polarization attenuation and propagation differential phase at microwave and millimetre wave frequencies

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The Pennsylvania State University The Graduate School College o f Engineering ESTIM ATIO N O F RAINFALL USING DUAL PO LAR IZAT IO N ATTENUATION AND PRO PAG ATIO N DIFFERENTIAL PH ASE AT M IC RO W AV E AND M ILLIM ETRE W AVE FREQUENCIES A Thesis in Electrical Engineering by Sean E. A. Daisley © 2002 Sean E. A. Daisley Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2002 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3051637 ___ ® UMI UMI Microform 3051637 Copyright 2002 by ProQuest Information and Learning Company. Ail rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We approve the thesis o f Sean E. A. Daisley. Date o f Signature 41-7 1-Q-2. Kultegin Aydin Professor of Electrical Engineering Thesis Advisor Chair of Committee Charles L. Croskey Professor of Electrical Engineering C. Russell Philbrick Professor q£J^lectrical Engineering Douglas H. Werner Associate Professor of Electrical Engineering iln b 2 Gregory S. Jenkins Assistant Professor of MeteOFOtogy. JohaK(u)s Verlinde Associate Professor of Meteorology William Kenneth Jenkins Professor of Electrical Engineering Head o f the Department of Electrical Engineering Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT Relationships between dual polarization propagation parameters (attenuation and phase shift) and rainfall rate are studied at microwave and millimeter wave frequencies (9.4, 14, 35, and 94 GHz) for the purpose of estimating rainfall rate R using these parameters. The effects of drop size distributions (DSDs), drop oscillations, and canting on these relationships are evaluated. Both gamma model DSDs and disdrometer measurements of DSDs are used. The study focuses on the 35-GHz specific attenuation (Ah and Av) and specific differential attenuation (AA = Ah- Av), as well as the specific differential phase at the various frequencies indicated above. Previous studies have shown that the specific attenuation of microwave signals at 35 GHz is almost linearly related to the rainfall rate. This study confirms that result using a large data set o f disdrometer measured drop size distributions. It is noted that the scatter around the linear relationship is generally low and has a tendency to increase at higher rainfall rates. It is also demonstrated through modeling studies that the effects of drop oscillations and canting on the R-Ah and R-Av relationships are negligibly small. Therefore, Ah and Av (or the average of the two) can be effectively used for estimating rainfall rate. At 35 GHz the differential attenuation AA has an almost linear relationship with rainfall rate for medium to high rainfall rates and less linear for low rainfall rates (below 10 mm/h). The scatter around the linear relationship is comparable to that of Ah or Av except for lower rainfall rates where it is much larger. As expected for differential measurements, AA is significantly affected by variations in drop shape (caused by drop oscillations). Interestingly, drop canting, assuming a Gaussian polar canting angle distribution with zero mean and 10° standard deviation, has a fairly small effect on AA. Therefore, if the effective drop shapes are known, AA can be effectively used for estimating rainfall rates larger than 10 mm/h. A potentially useful application of estimating the effective shape of raindrops is proposed using Ah (or Av) together with AA. This is possible because Ah (or Av) is insensitive to drop shape and AA is sensitive to it. The slope in the Ah-AA (or AV-AA) relationship can be used as an estimator of the effective drop shape. An interesting result from this study is that the relationship between the rainfall rate R and specific differential phase KDP is almost linear at 94 GHz and produces very little scatter. The R -K Dp iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. relationships at 9 and 14 GHz are less linear and have more scatter. It should be noted that the use of a 94 GHz R -K dp relationship will be limited to short paths and low rainfall rates due to the large rain attenuation at this frequency. At 35 GHz (as well as 30 and 40 GHz) K Dp is not useful due to the large scatter in the R -K DP relationship. However, R-Zh and Zt, - Ah relationships at 35 GHz were shown to be almost linearly related and relatively insensitive to variations in DSD. Therefore, rainfall rate could be estimated from the radar reflectivity measurements at 35 GHz with attenuation correction. This would be accomplished using the relationship between Zt, and Ah. A 35 GHz Dual Polarization Propagation Link (DPPL) was used for testing rainfall rate estimation with Ah, Av and AA over short path lengths (from 200 to about 500 m). Rain gauges and a disdrometer were used for comparison. The measurements showed promising results, more so with Ah and Av than with AA which appeared to be noisier than the other two. The effects of drop size distribution variations on the attenuation parameters were observed. produced very good results (within 1 0 Rainfall accumulation estimations based on Ah and Av % of the rain gauges) for long lasting (over an hour) rainfall events. However, further testing and evaluation o f the propagation link was considered necessary before any conclusions could be drawn regarding the microphysical details of rainfall and the utility of AA for measuring rainfall. iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table o f Contents LIST OF FIGURES ...................................................................................................................................viii LIST OF TABLES .....................................................................................................................................xv ACKNOWLEDGEMENTS............................................................................................................................... vi CHAPTER 1 INTRODUCTION.......................................................................................................................1 1.1 Background .......................................................................................................................... 2 1.2 Previous Work- The radar and propagation parameters..........................................................3 1.3 Research Objectives and Plans................................................................................................ 10 CHAPTER 2 2.1 2.1.1 2.1.2 2.1.3 2.2 2.3 2.4 THE MICRO-PHYSICAL STRUCTURE OF RAIN........................................................... 12 Drop Shapes...............................................................................................................................12 Equilibrium Shapes................................................................................................................... 12 Oscillation Shapes..................................................................................................................... 14 Drop Shape Models................................................................................................................... 16 Fall Orientation......................................................................................................................... 17 Drop Temperature..................................................................................................................... 18 Drop-size Distributions (DSD)................................................................................................ 19 CHAPTER 3 SENSITIVITY OF MICROWAVE ATTENUATION AT 35 GHZ TO CHANGES IN THE MICROSTRUCTURE OF RAIN................................................ 22 3.1 Electromagnetic Scattering Theory - Single Scatter............................................................. 22 3.1.1 Canting (Fall Orientation) Simulations...................................................................................23 3.1.2 Polarimetric Parameters Of Interest........................................................................................ 24 3.2 Electromagnetic Scattering Calculations................................................................................25 3.3 Estimation Of Rainfall Rates From Microwave Attenuation At 35 GHz........................... 27 3.3.1 Effects O f Drop Shape On A-R Relationships...................................................................... 33 3.3.2 The Effects Of Drop Canting On A-R Relationships At 35 GHz........................................ 35 3.4 Conclusions.............................................................................................................................. 37 CHAPTER 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 CHAPTER 5 5.1 5.2 5.2.1 ESTIMATING RAINFALL RATE AND ACCUMULATIONS FROM SIMULATED MICROWAVE ATTENUATIONS AT 35GHz USING DISDROMETER DERIVED DSDs...........................................................39 Disdrometer Drop Size Distributions......................................................................................39 Modeling The Relationship Between Rainfall Rate And Attenuation Using Measured DSDs............................................................................................................ 41 Effects O f Changes In The Drop Orientation........................................................................ 47 Effects O f Variations In Drop Shape......................................................................................48 Error Analysis O f The Rainfall Estimation Models..............................................................56 Estimating Rainfall Accumulations........................................................................................61 Identifying The Occurrence Of Raindrop Oscillation.......................................................... 64 Conclusions.............................................................................................................................. 6 6 35 GHz MICROWAVE ATTENUATION MEASUREMENTS IN RAIN.......................6 8 Background...............................................................................................................................6 8 Dual Polarization Propagation Link (DPPL)......................................................................... 6 8 Results O f DPPL’s Stability Tests..........................................................................................70 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.3 5.3.1 5.3.2 5.3.3 5.4 5.5 5.6 5.7 5.7.1 5.7.2 5.7.3 5.7.4 5.8 CHAPTER 6 Other Rainfall-Measuring Instruments Used In The Experiment.........................................72 TE525 Tipping Bucket Rain Gauge........................................................................................ 73 Optical Rain Gauge.................................................................................................................. 74 Joss-Waldvogel Disdrometer (JWD)......................................................................................76 The Propagation Link (DPPL) setup..................................................................................... 78 Procedures For Estimating The Attenuation And Rainfall Rates........................................79 Instrument Noise Reduction Schemes.................................................................................. 82 DPPL Operation For Selected Rainfall Events.................................................................... 83 Event #1 June 30th 2001 ....................................................................................................... 83 Event #2 June 21” 2000....................................................................................................... 103 Event #3 September 28th 1996.............................................................................................106 Event #4 June 16*2001....................................................................................................... 109 Conclusions...........................................................................................................................120 6.1 6.2 6.2.1 6.2.2 6.2.3 6.2.4 6.3 6.4 THE RELATIONSHIP BETWEEN RAINFALL RATE, SPECIFIC DIFFERENTIAL PHASE, ATTENUATION AND THE REFLECTIVITY AT CENTIMETER AND MILLIMETER WAVELENGTHS.........................................122 Background...........................................................................................................................122 Propagation Differential Phase........................................................................................... 123 Specific Differential Phase (KdP) And Rain Rate ( R )......................................................125 Radars And Backscattered Differential Phase Shift (5).....................................................141 The Effects O f Multiple Scattering On Propagation Through Rain................................ 146 Sensitivity Test O f The R-K^p Relationships..................................................................... 147 Dual Frequency Relationships For Use In Rainfall Retrieval Algorithms..................... 154 Conclusions.......................................................................................................................... 167 CHAPTER 7 7. 1 7.2 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK................................. 169 Summary and Conclusions................................................................................................. 169 Suggestions for Future Work.............................................................................................. 171 APPENDIX A Table of axial ratios describing the drop shapes................................................................173 APPENDIX B Dielectric constants of water at selected frequencies........................................................175 APPENDIX C Newton-Raphson (Gauss-Newton) Iterative Technique................................................... 176 APPENDIX D Piece-Wise R-A f its ............................................................................................................181 APPENDIX E Allan Deviation and Its Use in Describing the DPPL Stability....................................... 182 APPENDIX F Joss-Waldvogel Disdrometer bin size categories............................................................. 187 APPENDIX G Drop size distributions of the June 16th 2001 rain event..................................................188 APPENDIX H Table of published R-KDP relationships............................................................................. 193 REFERENCES.................................................................................................................................................196 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures Figure Page 2.1 Beard-Chuang (BCeq), Andsager et al. (ABLav), and Keenan et. al. (Kav) axial ratio (drop shape) models.............................................................................................................................. 17 2.2 Illustration of the parameters used to describe the fall orientation of oblate spheroidal raindrop models.................................................................................................................................... 18 3.1 Variations in the gamma model drop concentrations for ji=-l, 0, and 3, and for different rainfall rates (R = 1, 20, 50, and lOOmm/h).......................................................................................29 3.2 The sensitivity of Ah, Av, AA, and Aavg at 35 GHz to changes in the drop shape parameter |i........................................................................................................................................... 30 3.3 The distribution functions of the H-pol. extinction cross section a eh used to calculate the attenuation at 35 GHz, and the products o f the terminal velocity and equivalent drop volumes v, • D 3 (drop momentum) used in calculating R, for different shape parameters p, and as functions of the equivalent volume diameters. The distribution functions are given as the products % ■N(D)dD where x can be either oehor v, - D 3 ...............................................................................................................................31 3.4 Sensitivity of the A-R relationships at 35 GHz to changes in drop shape parameter p.............................................................................................................................................34 3.5 Plots of microwave attenuation vs. rainfall rate for different distributions of drop canting angles.....................................................................................................................................................36 3.6 Results of the combined effect of changes in DSD, drop shape and fall orientation on the A-R relationship..............................................................................................................................37 4.1 A) Plot of the total drop size distributions of the TROP, SWISS, and MISS data sets along with the gamma model DSDs scaled for comparison o f their shapes. B) Plot of the mean drop size distributions of the TROP, SWISS and MISS DSDs along with the gamma model DSDs.......................................................................................................................................... 40 4.2 Scatter plots showing the A-R relationships for the different distributions. Note that for all three distributions the axes have been adjusted to the similar ranges for each parameter............................................................................................................................................... 41 43 Distribution of median size diameters D0 and equivalent volume diameters D for each of the DSDs. Also shown are the relationships between the D0 and the rainfall rate R .................................................................................................................................................... 43 4.4 Single scatter extinction cross-sections (CTe) and the drop momentum F(D) as functions of the equivalent volume diameter D................................................................................. 45 4.5 The effects of variations in the fall orientation (canting) and drop shape on the AA-R scatter plots..................................................................................................................... 48 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.5 The A-R power law fits for the different DSDs. Results provided by Atlas and Ulbrich (1977) A-U for a MP distribution o f spherical drops have also been included................................................................................................................................ 52 4.7a Scatter plots of the Attenuations estimated from equilibrium shaped drops as functions of the rainfall rates. The error bars give the scatter in the data to within one standard deviation of the fitted relationships given in Table 4.2...............................................54 4.7b Scatter plots of the disdrometer estimated rainfall rates as functions o f the attenuations estimated from equilibrium shaped drops. The error bars give the scatter in the data to within one standard deviation of the fitted relationships given in Table 4.3................................... 55 4.8 Normalized bias in the R-A models relative to the disdrometer estimated rainfall rates................ 58 4.9 The Fractional Standard Errors in the R-A models relative to the disdrometer derived estimates of the rainfall rate..................................................................................................................59 4.10 Accumulations estimated from the R-A models as functions of the disdrometer derived accumulations........................................................................................................................................ 61 4.11 Simulated results of Ah as a function o f AA for different drop shape models. The results show distinct signatures depending on whether or not drop oscillations have been assumed........64 4.12 Model relationships for Ah as a function of AA for different drop shape models............................ 65 5.1 The errors expected in the DPPL estimated rainfall rates as a function of the Integration time......................................................................................................................................72 5.2 TE525 Tipping Bucket Rain Gauge...................................................................................................... 74 5.3 The ORG-75 Optical Precipitation (Rain) G auge............................................................................... 75 5.4 The Joss-Waldvogel Disdrometer (JWD) used for measuring raindrop size-distributions................................................................................................................................... 77 5.5 Pictorial showing the relative locations of the Walker Building (upper left comer) and the smoke stack (lower right comer). The radar was located below the southeastern edge of the blue square, on the roof of the Walker building, and close to the right hand comer of the roof facing the smoke stack. The oval shows the relative location of the JWD, ORG, and TE525 instruments ............................................................................................................ 78 5.6 Time series of the receiver’s temperature (Tair °C ), the atmospheric temperature (T Dvs °C) and the relative humidity (Humid %) during the periods preceding and subsequent to the main rain event on June 30, 2001. Also included are the rainfall measurements, and the standardized return signals....................................................................................................... 85 5.7 Time series of the DPPL receiver output and the co-incident rainfall rate s.................................... 8 6 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.8 Microwave attenuation values estimated from the DPPL measurements (dots) given as functions of the ORG estimated rainfall rates ( R o r g )- Also shown are the attenuations simulated from the disdrometer data similarly given as functions of the rainfall rates estimated from the same data. The time series of the ORG and JWD derived rainfall rates are also compared...........................................................................................................8 8 5.9 The DPPL estimated attenuation values derived after smoothing the DPPL return signal with a non-overlapping 1-min discrete average smoothing filter. Note that the reduced scatter in the experimental results is comparable to that given in the simulated results...................................................................................................................................90 5.10 Time series of the DPPL receiver output after being filtered by a 2-minute moving average low-pass filter......................................................................................................................... 91 5.11 The microwave attenuations estimated from the smoothed DPPL output given in Fig.5.10............................................................................................................................................. 92 5.12 The time series representation of the experimental (DPPL) and simulated (JWD) microwave attenuation values. The ORG measurements have been reduced by a factor o f 3 for easier comparison of the variations in the rainfall rate with the variations in attenuation values................................................................................................................................. 96 5.13 The drop concentration N(D)dD estimated from measurements taken by the JWD. These results represent the measurements taken over 15 consecutive non-overlapping minutes during the course of the shower (i.e. the period spanning the beginning and end of the event).......................................................................................................................................... 97 5.14 Time series of the drop concentration (conc.), the equivalent volume diameter (DQ), the drop counts (# o f drops detected by the JWD), and the rainfall rates estimated from the ORG and JWD measurements taken during theshower.................................................. 98 5.15 Variation in the wind velocities and rainfall rates during the course of the June 30th 2001 rain event.............................................................................................................................................. 99 5.16 The experimental results of AA (divided by 2) showing a general dependence on the rainfall rate consistent with the theoretical results. The Ah-AA plot suggests that the presence of oscillation drop shapes will generate steeperslopes in this relationship.................... 1 0 1 5.17 Plots showing the fitted relationships to the experimental and simulated results for the June 30lh 2001 rain event................................................................................................................... 102 5.18 DPPL receiver output and ORG estimated rainfall measurements taken during the June 2 1512000 rain shower................................................................................................................. 104 5.19 Microwave attenuations estimated from DPPL measurements taken during the June 21“ 2000 rain event. Also shown are the regression curves describing these relationships........................................................................................................................................ 105 5.20 DPPL receiver output and ORG estimated rainfall measurements taken during the September 28th 1996 rain event..........................................................................................................106 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.21 Estimated microwave attenuations and rainfall rates for the September 28th 1996 rain event. Also shown are the results for the regression fits given in Table 5.4 for the respective parameters..........................................................................................................................108 5.22 The time series of the receiver output (V, H) and their ratios (V/H) along with the ORG estimated rainfall rates for the June 16th 2001 rainfall event..........................................................110 5.23 Variations in temperature and relative humidity during the course of the day of June 16lh2001. Also included are the standardized receiver output signals along with the ORG estimated rainfall rates.............................................................................................................. I ll 5.24 The time series of the wind vectors during the course o f the June 16lh 2001 rain shower............ 112 5.25 Microwave attenuation measurements estimated form the DPPL receiver output given in Fig.5.22 (dots) as functions of the ORG rainfall estimates. The asterisks give the theoretical results derived from the JWD measurements. Also shown are the regression fits to the experimental and theoretical results................................................................................. 113 5.26 The time series representations of the experimental and theoretically derived microwave attenuations estimated from measurements taken during the course of the June 16th 2001 rain show er................................................................................................................ 114 5.27 Time series representations of the changing DSD, drop concentrations, D0 and the rainfall rates estimated by both the ORG and JWD for the June 16th 2001 rain event................. 115 5.28 Comparing the rainfall rates estimated by the from the experimental results (RADPPL) and those derived from the disdrometer measurements (RA JW D )...............................................117 5.29 The time series of the rainfall rates estimated from the R-A models given in Table 5.5 ............118 5.30 The cumulative rainfall derived from the link (DPPL) and the disdrometer (JWD) measurements for the June 16lh 2001 rain event...............................................................................119 6.1. a) Shows the variations in Re{SM(D) - S vv(D)}and ± v,D 3 with equivalent volume diameter D. b) Shows the effects of drop oscillations on Re{Sw, ( D ) - S vv(D)} versus D.......................................................................................................... 128 6.2 Simulation results showing the relationship between the KDp and R estimated from the MISS, SWISS, and TROP disdrometer data given in the first, second and third rows respectively. Each column shows the results at the indicated frequencies..........................131 6.3 Scatter plot of the median volume diameter Do for the three DSDs and Kdp at 30, 35 and 40 GHz...................................................................................................................133 x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.4a The results of published R-Kdp relationships along with the results obtained from the MISS, SWISS and TROP data sets for the given frequency bands. Shown are the results reported by Sachidnanda and Zmic' (1987) S-Z, Chanrasekar et al. (1990) Chand., Aydin and Giridhar (1992) A-G, Matrosov et al. (1999) Mat. (where mn and eq refers their mean and equilibrium shapes with L and G referring to the lognormal and gamma model DSDs), Keenan et al. (1997) Keen., and Timothy et al. (1999) Tim., for Pruppaccher-Beard P-B, Chuang-Beard C-B shapes (Tim* give the best fit over the variations of all the parameters)................................................... 136 6.4b The results of the W-band fits obtained from the SWISS, MISS, and TROP data sets for BCeq shapes compared to the results derived from the relationship provided by Aydin and Lure (1990).......................................................................138 6.5 Scatter plots of the rainfall rates as a function of the KDP at 9.4, 14, and 94 GHz. The Effects of variations in drop shape on these relationships are also show n...................................139 6 .6 6.7 6 .8 Modeling the effects of drop shape and fall orientation of the R-KDp relationships. The solid lines give the results in the absence of canting while the broken lines shows the change in the results for drops with a normal distribution of canting angles N (0 = O \std .d ev. = lO°) ...............................................................................................................141 Single scatter backscattered differential phase shift for the BCeq (solid) and Kav (broken) drop shapes..........................................................................................................................................143 Scatter plots of the rainfall rate and the differential phase shift on backscatter at 9.4, 14, and 94 GHz....................................................................................................145 6.9 The errors in the KDp derived rainfall rates (Rkdp) relative to the rainfall rates determined directly from the disdrometer data (R<us). The errors are given as functions o f R^s.................... 148 6.10 Fractional Standard Errors (FSE) in the model derived rainfall rates Rkdp relative to the disdrometer derived rainfall rates (R ^) as functions of the ranges of rainfall rates given in Table 4.4...............................................................................................................................149 6.11 The Normalized Bias (NB) in R kdp relative to R ^ over ranges o f rainfall rates defined in Table 4.4..................................................................................................................150 6.12 The cumulative rainfall derived from the KDp estimated rainfall rates for the different drop shape models, disdrometer data sets, and frequencies of interest. These accumulations are plotted against the accumulations estimated directly from the disdrometer data................... 151 6.13 Scatter plots of the specific attenuation at 35 GHz (a) and 14 GHz (b) as functions of the radar reflectivities at those frequencies, and at different look angles (90° an 73°). The estimated rainfall rates are also given as functions of the reflectivities at 35 GHz (c) and 14 GHz (d) respectively. These results were simulated from the TROP disdrometer data. ... 158 xi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.14 The functional relationship between the specific attenuation at 14 GHz and 35 GHz (a). The scatter plots o f the rainfall versus the frequency differential attenuation and the specific attenuation at 35 GHz and 14 GHz are shown in (b) and (d) respectively. These results were simulated from the TROP disdrometer data............................................................... 159 6.15 Scatter plots of the specific attenuation at 35 GHz (a) and 14 GHz (b) as functions o f the radar reflectivities at those frequencies, and at different look angles (90° an 73°). The estimated rainfall rates are also given as functions of the reflectivities at 35 GHz (c) and 14 GHz (d) respectively. These results were simulated from the IVtISS disdrometer data................................................................................................................................... 160 6.16 The functional relationship between the specific attenuation at 14 GHz and 35 GHz (a). The scatter plots of the rainfall versus the frequency differential attenuation and the specific attenuation at 35 GHz and 14 GHz are shown in (b) and (d) respectively. These results were simulated from the MISS disdrometer data......................................................161 6.17 Scatter plots of the specific attenuation at 35 GHz (a) and 14 GHz (b) as functions o f the radar reflectivities at those frequencies, and at different look angles (90° an 73°). The estimated rainfall rates are also given as functions of the reflectivities at 35 GHz (c) and 14 GHz (d) respectively. These results were simulated from the SWISS disdrometer data................................................................................................................................... 162 6.18 The functional relationship between the specific attenuation at 14 GHz and 35 GHz (a). The scatter plots of the rainfall versus the frequency differential attenuation and the specific attenuation at 35 GHz and 14 GHz are shown in (b) and (d) respectively. These results were simulated from the SW ISS disdrometer data...........................................................................163 6.19 The model fits to the A-Z, and R-Z relationships given in Table 6.5 for the three disdrometer data sets............................................................................................................................165 6 .2 0 Comparing the power law fits o f the Am4 -AH3 s, R-AAH, R-AHj 5 , and R-Ahu given in Table 6.5 for the TROP, SWISS, and MISS data set........................................................................166 E.1 The short-term stability characteristics of the DPPL for single polarization (V, H), differential polarization (WH), and signal product (V ■H ) processing. Also shown are the stability characteristics o f the estimated signal attenuations over a 103.5 m path under ‘clear air’ conditions.........................................................................................................184 E.2 a) The normalized errors in the DPPL signals under “clear-sky” conditions as a function of the integration times. Note that the error decreases monotonically with averaging times between 5-10 minutes. b)The SNR on the DPPL measurements. Note that the SNR is a minimum for those averaging times over which the normalized errors in the signal are at a maximum.........................................................................................................186 xii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables Table Page 3.1 Power law fits to the A-R relationship at 35 GHz for different DSD shape parameter values p................................................................................................................................................... 33 3.2 Fits to the A-R relationships for the assumed oscillation shapes ......................................................35 4.1 Piece-wise power-law fits for the single scatter extinction cross-sections and drop momenta to the equivalent volume spherical diameter...................................................................... 46 4.2 Coefficients and exponents of the power law models of the microwave attenuations as functions of the rainfall rate R. The 95% Cl of the fitted relationships are given as a ± CTa and b ± <rb .................................................................................................................................. 51 4.3 Coefficients and exponents of the power law models of the rainfall rate R as a function of the microwave attenuations.............................................................................................. 53 4.4 The ranges o f rainfall rates over which the models are to be evaluated. Also given are the numbers of 1 -minute spectra with rainfall intensities in these ranges and the mean rainfall intensity over each range of rainfall rates........................................................................................... 57 4.5a Statistical figures o f merits for the accumulations estimated from the power law R-A Model fits to the TROP distributions ...............................................................63 4.5b Statistical figures of merits for the accumulations estimated from the power law R-A Model fits to the SWISS distributions ..............................................................63 4.5c Statistical figures of merits for the accumulations estimated from the power law R-A Model fits to the MISS distributions ................................................................63 4.6 Errors associated with the accumulations estimated from the power law R-A Model fits when drop oscillations have been ignored. The accumulations were estimated from attenuations calculated for the oscillation (average) shapes with the model fits derived for the equilibrium shapes...................................................................................................... 64 4.7 Power law fits to Ah as a function of AA for different drop shapes and DSD ................................65 5.1 Operating characteristics of the 35 GHz Dual Polarization Propagation Link (DPPL) ................69 5.2 Power Law fits to the experimental and theoretical results of the 30th June, 2001 event The 95% Cl of the fitted relationships are given as a ± c a and b ± o b .......................................... 102 5.3 Power Law fits to the experimental and theoretical results of the June 21 2000 event The 95% Cl of the fitted relationships are given as a ± <ra and b ± <rb ..........................................105 5.4 Power Law fits to the experimental and theoretical results of the event of September 28, 1996 The 95% Cl of the fitted relationships are given as a ± o a and b ± <yb ..........................................108 xiii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.5 Power Law fits to the experimental (DPPL) and theoretical (JWD) results for the event of June 16,2001. The 95% Cl of the fitted relationships are given as a ± a a, b i Ob, ct i oa, and p ± <Tp................................................................................................................... 116 .1 Complex dielectric constants of water at 10°C.................................................................................. 123 6.2 Power model fits to the difference Re{5M( D ) - S n (£>)}as a function of D...................................129 6.3 Power model fits describing the R - K Dp relationships at the X-, Ku and W-band frequencies. The fits at S- and C-band have been included for comparison with other previously published results...............................................................................................................134 6.4 The FSE and NB statistics on the cumulative rainfall estimates from the R - K d p models given in Table 6.3............................................................................................... 153 6.5 Power law fits to the relationships Y-X where Y = a- X b, a ± a a and b ± crb give the 95% Cl of the fits a and b respectively, and AAk= AhiS - Ahxi ............................................164 6 xiv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS I wish to dedicate this thesis to my mother, Mrs. Eileen M. Daisley nee George, whose prayers and encouragement were not only instrumental in the development of my own faith in Providence, but were also of critical importance in forging an indomitable sense of duty, responsibility and mission to this and other tasks. I wish to salute the memory of my beloved grandparents Mr. Daniel T. George and Mrs. Gertrude A. George. They offered us much for which to be proud. I am also grateful to my wife. Dr. Nyambura I. Mbugua-Daisley, for her patience, counsel and support during this process. I would also like to thank my father, Mr. Eric Ardon Daisley, and my siblings Ysanne E. A. Daisley, Shelley E. A. Daisley, Dr. Racquel E. A. Daisley-Kydd, and Sheldon E. A. Daisley, for their support. I wish to thank my thesis advisor Dr. Kultegin Aydin, and the other committee members Dr. Charles Croskey, Dr. Gregory S. Jenkins, Dr. Johannes Verlinde, Dr. Douglas Werner and Dr. C. Russel Philbrick, for their support throughout this research. I would like to thank Dr. Christopher Ruf for his earlier assistance with the radar before leaving for the University o f Michigan, and Dr. Philbrick for agreeing to substitute for Dr. Ruf. Thank you Dr. Croskey for not only proof-reading my drafts, but also for offering me invaluable advice on troubleshooting and maintaining the radar. Thanks to Dr. Verlinde for the discussions we had while he was on sabbatical in the Netherlands, and on your return to University Park. I wish to thank Dr. Werner for his encouragement and also for the opportunity to work with him as his teaching assistant and lab coordinator for his courses on Antenna Engineering. Your respect for and confidence in my abilities was a source of encouragement. Special thanks go out to Dr. Jenkins for his guidance and unconditional support during the course of this research, and for honoring me with the privilege o f his friendship. Thank you for your ceaseless efforts to ensure that I was able to procure the disdrometers used in my field campaign. I wish to acknowledge the kind assistance of Dr. Kakar, Dr. Christian Kummerow*, and Dr. Ali Tokay of NASA-Goddard for their assistance in making the disdrometers available for this study (Dr. Christian Kummerow* formerly of NASA-Goddard is currently at the Colorado State University). Special thanks to Dr. Dennis Lamb who, even though not a member of my committee, was always available to provide more than a listening ear by offering the sort of criticism that motivated and facilitated further scientific growth and refinement. I appreciate the assistance given to me by Dr. John Diercks and William J. Syrett in getting the radar on-line and for providing the data from the Weather Station respectively. I also wish to thank Dr. Lynn Carpenter for his friendship and his advice. xv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I wish to thank those who have contributed in other ways for my success here at Penn State. The Department of Electrical Engineering, Dr. Larry Burton, Dr. John Mitchell and Dr. Kenneth Jenkins for providing me with the opportunities to serve as Graduate Teaching Assistant and Lecturer during my tenure here at Penn State. I am also grateful to the staff, especially Robert L. Divany, Marsha Church, Diana Feltenberger, Francine Cauffman, Julie Corol, and Janet Woomer, for their assistance. I wish to thank the administration and staff of the College of Engineering, in particular Dean Mason along with Saundra Johnson and Barbara Bogue of the Minority Engineering Programme, for their many recognitions including the College of Engineering Graduate Fellowships (2000, 2001) and the General Electric Foundation Dissertation Fellowship (2000). I wish to thank the Organization o f American State (OAS) and the government and people of Saint Vincent and the Grenadines for their assistance during the course o f my graduate and undergraduate degrees respectively. There are many others, too numerous to mention, that have contributed in one-way or the other to my success here at Penn State. The guys with whom I have played football (soccer) over the years, my friends in the Caribbean Students Association (CSA), Black Graduate Students Association (BGSA), the African Student Association (ASA), and the Turkish Students Associations among others. However, I wish to recognize the special friendships I have developed with Ambassador Roy Austin and his wife Mrs. Glynis Austin, who have provided me with a home away from home. 1also wish to thank the many fellow students whose friendship I was privileged to share. Among these are my colleagues Dr. Thomas Walsh, Dr. Alkim Akyurtlu, Dr. Diego Janches, Dr. Obika Nwobi, Dr. Richard de Gouville, Dr. Ali Koc, Kolah Segun, Mike Kwanisai, Kofi Wie Adu, Frank Cadwell, Derrick Otieno Okull, and Jasonjot Singh. Lastly, but in no way the least, I wish to thank my uncle Mr. Auton E. George and family, my cousins Kevin George, Mervyn George, the rest of my extended family, other friends, Colin Jones, Sybia Lowman-Marshall, Juliana Tracie Matthias and others, for their love and support, and for keeping me in touch with reality. “...let’s remember that the people we honor for moral courage are more likely to defy conventional wisdom than conform to it.” Salim Muwakkil Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1 INTRODUCTION It has long been known that the latent heat released by rain plays a crucial role in driving low-latitude circulations, while supplying energy to balance the radiative heat losses and fuel the global wind systems (Simpson et al., 1988). The conversion o f energy along the equatorial belt in particular has been shown to be a major factor in the regulation of the global energy budget (Malkus, 1962). Due to the critical role that rain and atmospheric moisture play in the general circulation and climate models, researchers are naturally interested in obtaining more reliable large scale and long-term estimates of average rainfall. It is therefore not surprising that one of the main goals o f the Tropical Rainfall Measurement Mission (TRMM) satellite launched on 27 November 1997, is to study the global energy and water cycles by measuring rainfall and latent heating over the global tropics (Kummerow et al., 1998, 2000). In addition to its vital role in the global hydrological cycle, rainfall also plays a significant role in the lives and welfare o f humans. Hydrologists and civil engineers increasingly need more accurate rainfall measurements in order to design more reliable water catchments, and develop more efficient flood prediction and protection schemes. Local and national governments need accurate information on rainfall measurements for effective disaster preparedness and avoidance planning, to deal with such natural occurrences as droughts and floods. Monitoring o f rainfall productivity and how it is affected by both natural and human activity is of considerable importance across many disciplines. For example, the marginal value of an inch of rain has far reaching consequences to grain yield and prices universally (Eddy, 1979). The burgeoning demand for higher performance communications systems with an ever increasing number of channels and bandwidth capacities have forced communications and RF engineers to use increasingly higher operating frequencies. Apart from the increased bandwidth capacities, there are advantages to be gained from increases in data transmission rates, and using smaller, more portable, and eventually cheaper systems operating at these higher frequencies. Currently, several industry leaders have developed, or are in the process of developing satellite communications systems that operate at these higher frequencies (e.g. K- and Q-band frequencies, see Violette et al., 1988; Rogers et al., 1997; Beaver and Bringi, 1997; Crane and Rogers, 1998). However, there are a number of related problems, such as the increased incidence of signal corruption due to rain and other hydrometeors, at these higher frequencies. 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Attenuation and depolarization due to rain is a major constraint in the design and operation of communications systems, particularly those operating at frequencies above 10 GHz. For example, the rain induced attenuation (in dB) increases approximately as the square of frequency through the K-band frequencies between 10.9-36 GHz (Laster and Stutzman, 1995). Rainfall has a deleterious effect on centimeter and millimeter-wave propagation because the raindrop sizes are comparable to the wavelengths of the signals, thus enhancing their scattering and absorption cross-sections. This is of particular concern in the tropical regions where over 75% of the world’s precipitation events occur (Simpson et al., 1988). This degradation of signal integrity is related to both the microphysical properties and macro-structure o f rainfall and their effects on electromagnetic waves propagating through rain filled regions in the atmosphere. The nature of these interactions, where, how and why they occur, and what can be done to avoid or eliminate these effects are of great concern to researchers o f various disciplines of engineering and meteorology. They are among some of the issues that have given rise to the need for more comprehensive studies on the impact of rainfall on microwave and millimeter wave signal propagation. 1.1 Background With the introduction of the first microwave radars, the diffuse, and often strong, background echoes associated with rain and other hydrometeors have been a source of concern to radar operators. These echoes were later found to be o f meteorological significance, (Marshall et al., 1947; Wexler, 1947; Wexler and Swingle, 1947). The echoes are related to the effective reflectivity factor Z, a parameter associated with the backscattering cross-section per unit volume of scattering particles, and is given by the sixth moment of the drop-size distribution for spherical particles in the Rayleigh scattering regime (Battan, 1973; Doviak and Zmic', 1993; Collier, 1996). In a series of articles Ryde (1941, 1946) and Ryde and Ryde (1944, 1945), the theoretical basis for measuring and calculating echo intensities and attenuation rates associated with different hydrometeors were presented. Among the many significant predictions of these studies were • the backscatter and extinction cross-sections of spherical ice and water particles with diameters as large as D =1.3A (for wavelengths up to A = 10c/n ), • the reflectivity and attenuation by rain and hail as a function of wavelength and precipitation rates, 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • the reflectivity o f snow, • the prediction o f the "bright band" in the melting layers thus, effectively laying the foundation for what was later to be referred to as Radar Meteorology. Nevertheless, the last 2-3 decades have witnessed revolutionary changes in the area of radar meteorology as more technologically sophisticated radars, and radar techniques, have been employed in the study of these and many other atmospheric phenomena. Advances in Doppler radar technology, using modem digital signal processing techniques and display technology, have encouraged the replacement of the previous weather radar network in the United States with a next-generation Doppler system NEXRAD (Doviak and Zmic', 1993). This system provides quantitative and automated real-time information on storms, precipitation, hurricanes, tornadoes, and a host of other important weather phenomena, with higher spatial and temporal resolution than that previously provided (Skolnik, 1990). Another important advance has been the development of poiarimetric radars, i.e. radars that are capable of transmitting and receiving orthogonally polarized signals (Seliga et al., 1984; Hall et al., 1980; Goddard et al. 1982; Jameson, 1983, 1985; Goddard and Cherry, 1984, Saizadeh. 1985; Walsh, 1998). 1.2 Previous Work - The radar and propagation parameters A detailed review of scattering and propagation studies in rain, hail and other ice forms conducted over the past 20 years is provided by Aydin (2000). The measurement of the mean sample powers at the alternately switched signals allows for the calculation of the coherency matrix from which a number of meteorologically significant parameters can be obtained (Doviak and Zmic', 1993). The parameters of interest include the effective reflectivity factor at horizontal and vertical polarization: Zh and Zy in mm6 m'3; the differential reflectivity ZDR = 10 log (Zh/Zv) (Seliga and Bringi 1976); the difference reflectivity ZDP = 1 0 log (|Zh - Zv|) (Golestani et al. 1989); linear depolarization ratio LDRM= 1 0 log (Po/Pm). where Po and PMare the returned powers in the orthogonal and main (i.e., transmitted) polarization channels; the correlation coefficient at zero lag PhV(0 ); the differential phase shift <{>dp and the specific differential phase Kdp (Seliga and Bringi, 1978; Mueller and Jameson, 1985; Sachidandna and Zmic', 1986). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Rainfall rate (R) and water content (M) can be estimated using single observables such as Zt, and Kdp, or combinations of two observables such as Zh and ZDR (or Zh and Zh), and KDp and ZDR (Battan 1973; Seliga and Bringi 1976, 1978; Jameson 1991; Doviak and Zmic 1993; Ryzhkov and Zmic 1995, Aydin, 2000). Problems associated with an R -Z (where Z can be Zh or Z J relationships have been well documented (Atlas and Chmela, 1957; Battan, 1973; Jameson, 1991). Z is proportional to the integral of Da over the drop size distribution (DSD) with a = 6 for Rayleigh scattering (which is the case at S-band), whereas a = 3.67 for rainfall rate, and a = 3 for water content. As a result, the R-Z relationship is strongly dependent on the DSD, which can be quite different from one rainfall event to another with a high degree of variability even during the same shower (Waldvogel, 1974). Polarization diversity techniques have been used to discriminate between ice particles and rain and for detecting hail as these parameters have been shown to be very sensitive to the shape, orientation, and composition of hydrometeors (Aydin et al., 1984; Aydin and Seliga, 1984; Aydin et al., 1986; Bringi et al., 1986; Jameson, 1987; Longtin et al., 1990; Balakrishnan and Zmic, 1990; Aydin and Zhao, 1990; Vivekanandan et al., 1990, 1994). These techniques have also led to improvements in the quantitative measurement of precipitation (Atlas et al., 1984). The differential reflectivity ZDR is sensitive to the mean shape and orientation of the scatterers. For example, raindrops which can be approximated as oblate spheroids with their major axes along the horizontal direction have ZDr > 0 dB due to stronger scattering at horizontal polarization (at side incidence in the Rayleigh scattering regime). If the spheroids are randomly oriented, then ZDR is 0 dB, as is the case for spheres or randomly oriented, irregularly shaped particles. The linear depolarization ratio LDR (LDRh or LDR,.) is also sensitive to shape and orientation. However, LDR is very low in rain due to the highly aligned drops. In addition, oblate spheroids, aligned with their symmetry axes pointing in the vertical direction, cause very little depolarization. Deviations from such alignment increase LDR, so too does the higher dielectric constants created by the melting of ice particles. The correlation coefficient at zero lag P hv ( 0 ) is affected (its magnitude is reduced) by a number of factors including irregular particle shapes, distribution of shapes (such as oblate and/or prolate spheroids with varying axial ratios), and distribution of backscattering differential phase shift 8 (8 is negligibly small for spheroidal particles in the Rayleigh scattering regime, but it oscillates and increases in magnitude for larger size particles in the resonance regime). 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The specific differential phase KDp is more sensitive to aligned hydrometeors (axes along H and V directions) with higher aspect ratios, and dielectric constants. It is not affected by tumbling or spherical shaped particles. These features have been exploited for measuring rainfall in the presence of hail. ZDp and Kdp are similarly affected. As a result. ZDP- Zh and KDP-Zh have been shown to be potentially useful for estimating the rain in a rain-hail mixture (Doviak and Zmic', 1993). Similarly, the combination of Zh and ZDR has been shown to be very useful in differentiating ice and liquid phase hydrometeors and detecting hail (Hall et al., 1984; Bringi et al, 1984; Aydin et al., 1986). Bringi et al. (1986) present X-band measurements in a convective storm in Colorado with a LDR bright band that is not observed in the S-band effective reflectivity factor Zh or the differential reflectivity Z drThey interpreted this as being due to melting graupel. Illingworth and Caylor (1991) show vertical profiles of Zh, Z Dr, LDR, and PhV(0 ) with an S-band radar through stratiform and convective precipitation in England. The stratiform precipitation showed Zh, ZDR, and LDR bright bands, whereas PhV(0) displayed a minimum in the same region. They suggested that this was most likely due to melting snowflakes or aggregates. The convective precipitation exhibited a bright band only in the LDR measurements and was attributed to melting graupel. Doviak and Zmic (1993) and Aydin (2000) have provided lists of the polarimetric observables that can be used for identification and distinguishing between various types of hydrometeors. Seliga and Bringi (1976) suggested that Z Dr together with Zh or Zv would provide improved estimate of rainfall over that provided by Zh or Zv alone. It was shown that Z Dr was independent of the number density N 0 of an assumed exponential drop-size distribution N(D). N(D) = Noe ~ AD Z Dr ( 1 .1 ) has a dynamic range o f about 5 dB, and is sensitive to the median volume diameter D0 at S-band and C-band, where DaA = 3.67 for exponential DSDs. The dependence of Z Dr on D0 can be useful in determining D0, and consequently the other unknown DSD parameter N0, from simultaneous measurements of Zh and Z Dr (Seliga and Bringi, 1976; Sauvageot, 1994; Collier, 1996). Improvements of about 25% over single parameter measurements have been observed (Seliga et al., 1979, 1981, 1986; Bringi et al., 1982, Goddard et al., 1984; Aydin et al., 1987, 1990; Chandrasekarr et al., 1990; Doviak and Zmic', 1993). The accuracy of Z Dr measurements, nevertheless, can be limited by a number of factors: 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • cross-correlation, between Zh and Zv, at zero-lag p hv(0 ), which is dependent on i. raindrop shape-size relationships ii. canting angle distributions iii. raindrop oscillations • the number of pulse pairs averaged (Bringi and Hendry, 1990) • DSD significantly different from exponential or gamma forms. • Presence of non-Rayleigh scatterers at the wavelengths being used (Sauvageot, L994). Comparisons of radar rainfall rate estimates obtained using an R-(Zh, ZDR) relationship have shown good agreement with rain gauge and disdrometer measurements (Bringi et al. 1981; Aydin et al. 1987, 1990). On the other hand, for KDp (at S-band) the exponent a = 4.24 for larger sizes and 5.6 for smaller sizes (Sachidanda and Zmic', 1986; Ryzhkov and Zmic' 1996). Therefore, the R-KDp relationship is not as sensitive to DSD variations for heavy rain involving larger drops and becomes more sensitive in light rain with smaller drops. The two-observable relationships R-(Zh, ZDR) or R-(Zh, Zv), and R to variation in the DSD. -(K dp . Z Dr ) are generally less sensitive Ryzhkov and Zmic' (1995) have compared the one and two-observable relationships using gamma model DSD simulations. They concluded that the R-(KDp, ZDR) relationship performs better than the others. However, in a case study comparing radar and rain gauge measurements, the R-K DP relationship performed just as well for rain rate from 10 to 45 m m h1. Simulations by Chandrasekar et al. (1990) show that R -K DP performs better for R > 70 mm h 1; R-(Zh, Z dr) gives the least error for 20 < R < 70 mm h' 1 and is comparable to R-Zh for R < 20 mm h'1. It is no surprise that K0p out performs Zh and ZDR at higher rainfall rates. This is due to the fact that at such high rates the precipitation is often comprised of a mixture o f rain and hail. Zh is biased to the hailstones, which are larger than the raindrops since it is proportional to the square of the volume of the particles (i.e. D6) in the Rayleigh regime, while KDp, being less sensitive to the larger hailstones, senses mainly the liquid portion of the mixture instead. In addition raindrops have a tendency to be more aligned along their vertical axes and as such tend to be anisotropic in their scattering, whereas hailstones tumble as they fall and are more spherical in shape. Consequently, hailstones are considered isotropic scatters. It is for these reasons that Balakrishnan and Zmic' (1990) proposed using KDp to estimate rain rate in the presence of hail. Aydin et al. (1995) demonstrated this by comparing R obtained using KDP in the presence of hail with a ground-based rain gauge. Other advantages of KDP are, • it is relatively insensitive to beam blockage and ground clutter (Zmic' and Ryzhkov, 1996) 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • it is less sensitive to beam filling • independent of system gain • it is independent o f system calibrations (Doviak and Zmic', 1993). At the same time, KDP is sensitive to drop-shape (Keenan et al. 1997). Ryde (1946), in his calculations o f the attenuation coefficients, predicted that they increased only slightly faster than the first order of the rain rate R at most wavelengths, but was directly proportional to R for wavelengths close to 10 mm. Jameson (1991) reported that the distributions for At5 (attenuation at 25 GHz) and A38 over the drop sizes were well correlated ( = 99.8% ) with those of the rain rates. The implication here being that most of the attenuation is due to drops that are significantly contributing to the rainfall rates. This therefore suggests that both A1 5 and A3 8 may be good estimators o f rain rate. This is consistent with the nearly linear R-A relationship at 35 GHz reported by Wexler and Atlas (1963) and Atlas and Ulbrich (1977). Atlas and Ulbrich (1977), assuming spherically shaped raindrops, determined that at 34.09 GHz and 35 GHz the attenuation and rainfall rates had the same moments of drop-size distributions. They argued that for 8.5mm < A < 9.0mm the integrals for both attenuation (A) and rainfall rate (R) were virtually identical. In addition, they observed that the scaling constants of the integrations, and the moments of drop-size distributions were essentially independent of temperature over the rage 0-18 °C, and showed very little changes as the temperature rose to 40°C. This resulted in the attenuation as a linear function of R, independent of the drop-size distribution and temperature, over this range of wavelengths. To compensate for the general non-spherica! shapes of raindrops, and their varying canting angles, they suggested the use o f other techniques involving differential and average attenuation measurements. Jameson (1991) suggested that the attenuation at 25 GHz was more independent of drop-size distribution and, as a result, was a more accurate predictor of rain rate. However, it must be noted that 25 GHz is very close to the 22.235-GHz water vapor absorption line, and in the tail end o f the 35 GHz transmission window. As a result, the absorption due to water vapor at 25 GHz will exceed that at 35 GHz (Ulaby et al., 1986a). Furthermore, it has also been shown that rain induced specific attenuation at 35 GHz is always greater than that at 25 GHz (Crane, 1996). To compensate for the general non-spherical shape of the raindrops. Atlas and Ulbrich (1977), and subsequently Jameson (1989), suggested using either the average or the differential measurements of the 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. attenuation at alternate polarizations. Jameson (1991) observed that while polarization differential attenuation (AA = AH - Ay) was potentially useful for frequencies around 10-20 GHz, it was dependent on drop shape. Furthermore, he suggested that because of its relatively small magnitude there were no distinct advantages to using AA. (^ He instead proposed the use of the average attenuation = A" * Av ), which was thought to be independent of drop shape in addition to having a larger magnitude. Atlas and Ulbrich (1977) also observed that while AA was linear to R, it was also sensitive to the effects of canting, i.e. decreasing with canting angle. Despite the much lower sensitivity o f the differential attenuation measurements, AA, it was thought that the measurement stability inherent in differential data processing was a mitigating factor in its favor (Ruf et al., 1996). Ruf et al. (1994, 1996) reported that, from a practical point of view the differential measurements were more accurate and desirable. They noted that while Aavg corrected for drop shape irregularities, it was nevertheless highly susceptible to the same instrument instabilities that affect Ah and Av. On the other hand, they observed that differential measurements were largely free of these effects provided the time scale on which they occurred were slower than the time taken to switch between the alternate polarizations. They suggested that the differential measurements might also cause a cancellation of the second order effects of polarization-independent attenuation by oxygen, water vapor and mist, along the link. A number o f other studies on the relationship between microwave attenuation and rainfall rate have been reported (Oguchi, 1973; Oguchi and Hosoya, 1974; Atlas and Ulbrich, 1977; Olsen et al., 1978; Jameson, 1989; Jameson, 1991). Oguchi (1973) reported on improvements on the calculations of the forward scattering field intensities (which are used to calculate the attenuation) of oblate spheroidal rain drops shapes at 34.8 GHz, assuming orthogonally polarized plane waves at normal incidence to the drop axis. These forward scattering field intensities were then used to determine the effective propagation constants for rain filled space. This enabled them to make estimates of the differential attenuation and differential phase shifts between these orthogonally polarized signals. Atlas and Ulbrich (1977) presented the theoretical basis for the supposed linear relationship between rain rate and microwave attenuation at 35 GHz using Marshall-Palmer (MP) DSD of spherical drops. Olsen et al. (1978) reported power law models for attenuation-rainfall rate relationships for different model drop size distributions, over a range of frequencies. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Earlier, experiments had been undertaken to measure the path-integrated rainfall at this frequency (Collis and Lidga, 1961; Goddard, 196S) and had had only modest success. Collis and Ligda (1961) used two fixed targets along the path of his 0.86cm wavelength signal, and compared the relative signal intensities from the targets in rainfall to that in clear air in order to estimate the average rainfall rate over the intervening path. Five-minute mean values calculated from the attenuation measurements were reportedly within a factor of 2 of those measured by rain gauges. Goddard (1965) conducted similar experiments over a 3.4-km path with two comer reflectors at 2.7 and 6.1 km from their 0.86cm radar. The radar computed rainfall gave a maximum deviation of 23% in the accumulations of five rain events relative to that indicated by eight rain gauges located between the two targets. Unfortunately, the range of rainfall intensities studied was restricted to 0.5-4 mm/hr. This raises some concerns about the accuracy of their measurements, as the fluctuations in the target echoes, which can have a detrimental effect on the accuracy o f path-integrated attenuation, are generally more pronounced in light rain where the attenuations are lowest. Atlas and Ulbrich (1977), discusses a number o f experiments for estimating rainfall rate from path averaged attenuation and suggest a number of other methods by which this may be done. Other experiments involving rainfall induced microwave attenuation have been conducted. Sweeney et at. (1992) have reported on the relationship between attenuation at 20 GHz and 30 GHz in the presence of rain. Their distribution of attenuation rates, when plotted against each other, exhibited a hysteresis effect. This can potentially create problems for frequency scaling of attenuation on 20/30 GHz satellite links. They did not report any direct relation between these attenuations and the rainfall rate. More recently Holt et al. (2000) reported on the use of a dual-frequency microwave link operating at 12.8 GHz and 17.6 GHz for measuring path-averaged rainfall over a 23 km path. Their results suggest a relationship between the frequency differential attenuation and the weighted average estimate of rainfall rates from a network o f tipping bucket rain gauges. As in the previous case, no empirical models describing the relationships between either the single frequency attenuations or the frequency differential attenuations and the rainfall rates were presented. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. U Research Objectives and Plans The main objective of this research is to study the electromagnetic scattering characteristics of rain at X(X = 31.9 mm), K„ (X = 21.41 mm), Ka (X = 8.57 mm), and W (X = 3.19 mm) band frequencies for estimating rainfall rate and rainfall accumulation using propagation and radar measurements. My goal is to better understand the effects of drop size, their DSD, shape, and fall orientation on selected propagation and radar parameters. Their effects on the rainfall rates estimated from these parameters will also be assessed. Particular attention will be given to studying the relationship between the attenuation at 35 GHz (X = 8.57 mm) and rainfall rates, due to the availability of a propagation link operating at this frequency. Simulations of both rainfall and attenuation will be undertaken using theoretical drop-size distribution (DSD) models and ground-based disdrometer estimates o f actual DSD in rain. Relationships between the rainfall rates and different attenuation products will be derived and compared as to their accuracy in estimating total rainfall accumulations. The results of a field experiment to measure the rainfall rate from the attenuation of microwave signals in rain using a dual-polarization microwave propagation link (DPPL) operating at 35 GHz will presented. Coincident disdrometer derived drop spectra and optical rain gauge measurements Rorg(mm/h) will be used in conjunction with these measurements, first as a validation check on the measurements, and secondly, as an aid in our study of the underlying physics of the rainfal 1-microwave attenuation process. The experiment to be discussed in this report measures the attenuation over a much shorter path than those earlier presented from the literature, and will use two orthogonal polarizations (V and H). Measurements were also be made over a wider range of rainfall rates. The shorter path avoids many of the problems associated with the inhomogeneous distribution of rainfall fields. Additionally, this study seeks to link the measured attenuation to microphysical structure of the rain by incorporating aspects of rain drop size distributions (i.e. naturally occurring DSD as derived from insitu disdrometer measurements), the drop shapes, and fall orientations, which were not collectively represented in the previous studies. A number studies on the A-R relationship at a number of other K-band frequencies have been reported, Ali et al. (1992), Yeo et al. (1993, 2001), Ajose and Sadiku (1995), Li et al. (1995, 2000), Islam and Tharek (1999), and Holt et al. (2000). Norbury and White (1972), Zavody and Harden (1976) and Harden et al (1978a, 1978b) have reported on rain induced microwave attenuation measurements over propagation links operation around 35 GHz. This author is not aware of any other 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. experiments on rainfall and microwave attenuation at 35 GHz that has since been undertaken. This study concludes with an examination of the scattering characteristics o f rainfall at X-, Ku-, Ka-, and W-band frequencies as observered from the Z, A and KDp radar and propagation parameters. Chapter 2 discusses the microphysical structure of rain and its possible effects on microwave propagation through a rain medium. Chapter 3 outlines the formulation of the problem. Theoretical models of DSD, drop shapes, and fall behavior are defined and implemented in the computation of the attenuation parameters at 35 GHz. In Chapter 4, we employ disdrometer-derived estimates of naturally occurring DSDs to derive A-R/R-A relationships. A comparative analysis will be performed on these relationships. Estimates o f the rainfall accumulations will also be derived. Chapter 5 discusses the field experiments to measure microwave attenuation at 35 GHz using the DPPL. The results of the DPPL, optical rain gauge (ORG), and the Joss-Waldvogel Disdrometer (JWD) will be compared and discussed. Chapter 6 discusses the relationships between R, Kdp, A, and ZDR at Ku-, Ka-, and W-band frequencies. Finally, Chapter 7 presents the conclusions of this study and recommendations for future studies. 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2 THE MICRO-PHYSICAL STRUCTURE OF RAIN A thorough analysis o f the electromagnetic scattering characteristics of rain would require that careful consideration be given to both its physical (storm size and structure) and microphysicai representations (drop shape, size distribution, and fall behavior). The rainfall process is random and highly non-linear as a result of a number of inter-related non-linear processes involved in the drop formation (e.g. nucleation) and growth processes (e.g collection and evaporation, see Pruppacher and Klett, 1997). Unraveling the mystery that is rainfall requires that we understand the relationship between these non-linear processes. In the interests of time, and due to the availability of resources, I have limited this study to exploring the effects of the micro-physical properties of rainfall, such as the shape, size, number, temperature and fall behavior of the raindrops on the propagation of microwave signals, and how these may affect the use of selected microwave radar and propagation parameters in the estimation of rainfall. 2.1 Drop Shapes Photographs of drops in still air and wind tunnels (e.g. Magono, 1954; Pruppacher and Klett, 1997) suggest that raindrops may be well approximated by oblate spheroids. Therefore, since raindrops are generally non-spherical, they would tend to scatter different polarizations unequally. For example, the larger slightly flattened drops whose horizontal dimensions exceed their vertical dimensions will give higher scatter in the horizontal than in the vertical direction. Polarization dependent meteorological parameters would therefore depend strongly upon the shape of the raindrops. 2.1.1 Equilibrium Shapes The shape of a model drop is expressed by its axial ratio, Of = —, where b is the semi-minor axis ratio a and a, is the semi-major axis of the drop. This effectively gives a measure of the deformation or ellipticity o f the drop itself. The shape of a drop is greatly influenced by its size, which is generally 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. considered to be within the range D < S m m , where D is the equivalent volume diameter. There is considerable disagreement on the upper limit of this range, as this varies with rain type and geographical region. However, it is generally accepted that large drops ( D > 4 mm ), if they exist, will be quite rare since they are hydrodynamically unstable, and as such they will tend to break up (Oguchi, 1983; Pruppacher and Klett, 1997). For D < 1 mm drops are generally spherical in shape while those with D > 1 mm tend to be more asymmetric, spreading laterally with flattened bases and rounded tops (Beard and Chuang, 1987). The shape of a drop is influenced by a combination of internal, external and surface tension forces. Drops in a steady or laminar flow field tend to assume an equilibrium shape (Pruppacher and Klett, 1997). Attempts to calculate these equilibrium shapes have been met with varying degrees of success (Sphilhaus, 1948; Imai, 1950; Blanchard, 1950; Magono, 1954; Pruppacher and Beard, 1970; Pruppacher and Pitter, 1971; Green, 1975; Beard and Chuang, 1987, 1989; Beard, 1989). Beard and Chuang (1987) and Chuang and Beard (1990) have derived an expression to calculate the axial ratios of equilibrium shape drops as a function of their sizes. A close match was reportedly found between their shapes and the profiles of drop with diameters up to 5mm as seen from photographs (Beard and Chuang, 1987). Unlike earlier models of raindrops as oblate spheroids, the Beard and Chuang (1987) model provides appropriate large amplitude responses to distortions induced by both aerodynamic and hydrostatic pressure modifications. It has also been found appropriate for examining the effects of electric stress (Pruppacher and Klett, 1997). Noticeably absent is the indentation at the base of the drops so evident in the Pruppacher and Pitter (1971) drop shape model. They suggested that it would require an unrealistically high base pressure to produce such a dimple. In fact, photographic evidence of suspended drops shows them to have positive base curvature, i.e. dimple free (Pruppacher and Klett, 1997). If they do occur, it is believed that the presence of a dimple may be caused by oscillation and not as a consequence of equilibrium forces on the drop. These depressions may also be responsible for the eventual breakup of drops in quiet air (Pruppacher and Klett, 1997). Other axial ratio models have been reported (Green, 1975; Stalyadkin, 1988). Field studies have reported the existence of drops with non-equilibrium shapes and sizes (Jones, 1959; Voltz, 1960; Chandrasekar et al. 1988). There are a number of reasons why drop shapes may differ from their equilibrium shapes: • collision induced oscillation (Beard et al., 1983) 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • air turbulence and wind shear (Jones, 19S9) • eddy vortex shedding (Beard and Tokay, 1991) • drops falling in the wake of larger drops Gunn (1949) and Beard et al. (1991) • drop electrification e.g. drops approaching their charge limit at the edges of a rain shaft (Chuang and Beard, 1989) Green (1975) argues that, due to the vagaries of ambient conditions during a rain event, it is quite possible that average and equilibrium forces for a given drop volume may not be identical. He is also doubtful that a falling drop attains a true equilibrium shape under natural conditions. 2.1.2 Oscillation Shapes Jones (1959) reported that there was a mean drop shape that varied with the mass, and that this shape resulted from oscillations about a mean. He suggested that there are no static shapes, but that, especially at the larger drop sizes, the drops oscillated about preferred shapes. Generally, non-laminar state-flow fields induce drop oscillations, which in turn cause periodic changes in the shapes of the drops (Pruppacher and Klett, 1997). Jameson and Beard (1982) reported that asymmetry in the surface energy, as a function of axial ratios, can result in time-averaged axial ratios that are larger for oscillating drops than for non-oscillating ones. Furthermore, the mean axial ratios o f the oscillating drops tended to shift towards unity in agreement with Jones (1959) observations. This being the case will have an impact on polarimetric techniques. Numerous documented references to drop oscillations can be found in the literature (Blanchard, 1948, 1950; Jones, 1959; Mueller and Sims, 1962; Latham, 1968; Brook and Latham, 1968; Nelson and Govkhale, 1972; Beard, 1982; Goddard and Cherry, 1984). Studies undertaken with single drops in wind tunnels and fall chambers, suggests that drops larger than 1mm in diameter oscillate in response to unsteadiness in their ambient flow. Estimates of these oscillation amplitudes range from ± 10% < A a < ±30% of the equilibrium axial ratios (Nelson and Gokhale, 1972; Beard et al., 1983; Beard, 1984; Chandrasekar et al., 1988) and as much as 8-20% o f their equilibrium radii (Trinh et al. 1982; Annamalai et al., 1984). Generally the frequencies of oscillations tend to decrease, and their amplitudes increase, as the drops increase in size. Studies have shown the existence of discrete spectra o f allowed frequencies and modes with certain prescribed asymmetries (Beard, 1984; Beard et al., 1991; Beard and Kubesh, 1991; Kubesh 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and Beard, 1993; Pruppacher and Klett, 1994). Latham (1968) observed drops undergoing oscillations "approximately o f the prolate-oblate type". Nelson and Gokhale (1972), Musgrove and Brook (1975), and Beard (1982) have provided additional evidence of these oscillations. There is currently no single explanation as to why raindrops oscillate. There are a number of factors that are thought to play a role. Gunn (1949) proposed the idea of wake induced oscillations. In the case of small drops, periodic detachments of eddies in the wake of a drop induces pressure changes at their surfaces resulting in the resonance between the eddy shedding frequency and the natural frequency of the drop (Pruppacher and Klett, 1997). Oscillation of larger drops ( D > 2 m m ), are often difficult to explain since they are found to oscillate in modes for which large mismatch exists between eddy shedding and natural oscillation frequencies. For this reason a number of other possible factors have been proposed: • Sub-harmonic resonance for eddy frequencies of an integer multiple of oscillating frequencies (Beard, 1984; Beard and Kubesh, 1991; Beard et al, 1991, Feng and Beard, 1991; Kubesh and Beard, 1993) • Wind shear or turbulence induced oscillations (Jones, 1959) • Drop collisions which may be accompanied by coalescence or breakup (Warner, 1977; Beard, 1983; Johnson and Beard, 1984) Tokay and Beard (1994) are of the view that wind shear and turbulence have a negligible effect on drop oscillations. Instead, they are of the opinion that changes in drag, caused possibly by changes in drop shapes, may produce positive feed back to oscillations of the fundamental oblate-prolate mode. In addition, they suggested that although collisional forcing may contribute to these drop oscillations, it is not the main factor. Deviations from equilibrium shapes could have a considerable influence on quantitative applications of polarization radar techniques that are highly dependent on the assumed drop shapes (Goddard et al, 1982; Jameson and Beard, 1982). For example, ZDR had been proposed to improve the estimate of rainfall (Seliga and Bringi, 1976), but only if the drop size and axial ratio relationships are known. Beard and Johnson (1984) reported significant differences in ZDR calculated for oscillating and non-oscillating drops. Kubesh and Beard (1993) reported variations in Z dr of about 30% based on model simulations. Previous modeling studies have shown that estimates of ZdR, based on equilibrium shapes, may vary from 10% to 60% when the average shapes of oscillating drops were used instead (Jameson and Beard, 1982; Beard and Johnson, 1984; Seliga et al., 1984; Chandrasekarr et al. 1988, Zmic', 1989; Tokay and Beard, 1993). However, actual radar measurements compared with rain gauge and disdrometer 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. measurements do not show such large deviations in ZDR, which would cause significant errors in the rainfall estimates. 2.13 Drop Shape Models Three different oblate spheroidal drop shape models will be used throughout this study, one equilibrium shape model and two oscillation models. The model for equilibrium shaped drops used in this study is based on that proposed by Chuang and Beard (1990). This model includes the shapes provided by Tokay and Beard (1993) for drops of sizes D < 0 .7 m m , with the equilibrium shapes given in Table 3 of Beard and Kubesh (1991) for 0.7 < D < l . 4 m m , and finally those given by Chuang and Beard (1990) for D > 1.4mm . This model will be referred to as BCcq. The first oscillation (average) drop-shape model is denoted as ABLJV. This model includes the shapes defined by Tokay and Beard (1993) for D <0.7m m, with the average shapes given in Table 3 of Beard and Kubesh (1991) for 0.7 < D < 1.4mm, the relationship given by Andsager et al. (1999) for 1 .5 < D < 4 .0 m m , and the relationship given by Chuang and Beard (1990) for D > 4 .0 m m . This model is applicable for light to heavy rainfall, and for situations where raindrop collisions are only responsible for a small fraction of the oscillations, most of which are produced by vortex shedding (Andsager et al., 1999). The second oscillation (average) drop shape model, denoted as Kav, is given by Keenan et al. (1997) is representative of rainfall where collision-induced raindrop oscillations dominate the steady-state oscillations. Other relationships have been proposed but were observed to give shapes that mostly fell within the range of shapes given above. It was also determined that the above-mentioned shapes included the most important aspects of drop oscillations that have so far been described. All three relationships are shown in Fig. 2.1. These axial ratios have been provided in Appendix A 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 95 09 .S 0.75 07 0.65 06 D(mm) Fig.2.1 Beard-Chuang (BCq,), Andsager et al. (ABLaV)> and Keenan et. al. (Kav) axial ratio (drop shape) models. 2.2 Fall orientation The symmetry axis o f non-spherical drops in the atmosphere may not necessarily be aligned in the vertical direction, but may tilt away from the vertical in response to drop collision, breakup, wind shear and other aerodynamic or internal forces (Oguchi, 1983; Jameson and Beard, 1990; Pruppacher and Klett, 1997). This tilting of the drop axis, "canting", can have an effect on the microwave propagation and scattering as can be observed from the scattering matrix elements, which are functions of the canting angle (see Sec. 3.2). There is some dispute as to the range of canting angles as this seems to vary with the type o f rain, the intensity o f the rain, and the location of the drops above ground. McCormick and Hendry (1974) found that canting angles were narrowly distributed about an average of 0.48°, with a standard deviation of 1.77°. Oguchi (1983) mentions mean canting angles between 7-12°. The canting angles generally increase with drop-size, however, there is no established relationship between drop size and canting angle. The rate of increase decreases when the diameters exceeded 1.Omm, remaining constant for D > 2.0mm. Brussard (1976) proposed a physical model from drop canting, which suggested that at higher altitudes the drops have smaller canting angles. 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. / Fig. 2.2 Illustration of the parameters used to describe the fall orientation of oblate spheroidal raindrop models. For this study the fall behavior of oblate spheroidal drops will be modeled in terms of the polar (0) and azimuthal ($) angles defined using a spherical co-ordinate system centered in the drop (see Fig. 2.2). The polar angle is defined as the angular separation between the zenith direction (assumed to be along the z-axis) and the drops symmetry axis. The azimuthal angle is defined as the angle between the x-axis and the projection of the symmetry axis on the x-y plane. The distributions of 0 and <{>are assumed to be independent of each other and of size D. A uniform distribution. U = 0°,0 ^ = 360°), is assumed for <{>and a Gaussian distribution, N(0 = 0°. std.dev. = 5°, 10°), is used for 0. 2J Drop Temperature We expect the electromagnetic scattering properties o f the drops to change with temperature, as their refractive indicies is temperature dependent. Atlas and Ulbrich (1977) assumed a drop temperature of 10°C, which would be consistent with rain showers in temperate regions. In addition, the effect of 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. temperature on rain attenuation was estimated to be within ±2.5% at T = 1 0 ± 1 0 °C (Sekine and Lind, 1982). Consequently, for this study we shall assume a drop temperature of 10°C, which would correspond to a dielectric constant (14.0729-j24.627) at 35 GHz following Ray (1972). A list of the dielectric constants of water at the other frequencies to be discussed is given in Appendix B. 2.4 Drop-Size Distributions (DSD) Blanchard (1953, 1957) reported variations on maximum drop sizes with rainfall rate for rains over Hawaii at 2800 ft. elevation, which suggested that warm rains seldom include drops larger than 2-3mm in diameter. However, raindrops with diameters in excess of 3 mm have been observed in tropical storms and hurricanes with rain rates exceeding 50 mm/h (Willis, 1984). Attempts to arrive at a reliably accurate description of the raindrop spectra has been frustrated by the inability of current models to simultaneously deal with the many, often competing, physical processes involved in drop formation and growth (see Pruppacher and Klett, 1997). Drop-size distributions are known to vary considerably with geography, rain type and from storm to storm of the same type (Blanchard, 1953, 1957; Atlas and Chmela, 1957; Battan, 1973; and Waldvogel, 1974). The implications of such variations are quite far reaching as many rainfall quantities e.g. rainfall rate, liquid water content, attenuation, reflectivity, specific differential phase, and optical extinction, are all integral quantities of the drop size distribution of the form ( 2 . 1) where each of the above integral parameters M goes as the exponent x of D and the DSD N (D), where D is the equivalent volume spherical diameter of the drop. Laws and Parson (1943) undertook the first notable spectra measurements. They observed considerable variations in size distributions between rain events with similar rain rates. Probably the most widely used spectra description is the Marshall-Palmer (MP) model (Marshall and Palmer, 1948) given as N ( D ) = N ne “AO ( 2 .2 ) 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where A = 4 .1 /f-0'21m m~l and the intercept N0 = 8 x 103 m ' W . The MP model has been observed to give reasonable approximations to DSD if sufficient averaging in time and space is performed. However, it is not always possible to perform such averages. Furthermore, it was observed that N„ was not constant but varied from event to event with changes occurring even in the same event (Blanchard, 1953; Waldvogei, 1974). Waldvogel (1974) suggested that the MP distribution was not sufficiently general to accurately describe most observed spectra, as it was unable to estimate the sudden variations in the drop spectra and reflectivity profiles that are commonly observed. For this reason N0 is often given as a function of the rain rate to account of the so called N0 jump which is thought to indicate the transition from one mesoscale area within the precipitation field to another. The variations in N0 were also found to be independent of those occurring in A. Furthermore, it is also known that the DSD is not exponential as D goes to zero. In other words, there is a lower limit to the size of naturally observed drops. Consequently, the MP distribution over estimates the smaller drop sizes. Ulbrich (1983), and Ulbrich and Atlas (1984), have shown that greater accuracy can be achieved if the integral parameters are derived using a gamma model DSD of the form: N ( D ) = N 0 D fie~AD 0<D < (2.3) where p. is a shape parameter having positive or negative values. In this expression N0 is a function of both p and R having units m'3'^ mm'1, a quantity that is difficult to interpret physically. The advantages of this model are that, • with these three parameters it is possible to describe the variations in DSD observed naturally on small spatio-temporal scales, and • the model also includes the exponential distribution as a special case (i.e. p = 0). Other DSD models have been proposed (Sekine and Lind, 1982; Feingold and Levin, 1986). Sekine and Lind (1982) suggested that at high rain rates the larger drops produced by coalescence would become unstable and break-up initiating a chain reaction process. They suggested that it is because of this coalescence-break up of drops that naturally occurring DSDs deviated from the MP model at higher rain rates. They instead propose the Weibull DSD, which they suggested would better explain the coalescence and drop break-up chain reaction processes occurring in rain. Feingold and Levin (1986) 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. recommended using a lognormal DSD model. This was not based on any mathematical proof or the connection between the break-up/collection process, but instead on the assumption that the number of drops per unit volume per unit size is distributed lognormally (According to the law of proportionate effect, if a variate x is one of many independent influences, each of which generates some effect that is proportional to itself, then x will be lognormally distributed). Jiang et al. (1997) reported better agreement to their experimental results for Weibull DSD over MP and gamma model DSDs. Feingold and Levin (1987) reported that the lognormal DSD performed as well as the gamma model DSD. In the following chapter the gamma model DSD will be used to test the sensitivity of the different attenuation-rainfall relationships to the variations in the physical properties o f raindrops. In Chapter 4 naturally occurring drop spectra, as observed by ground base disdrometers, will be used to verify the results from the theoretical gamma model drop size distribution. 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER3 SENSITIVITY OF MICROWAVE ATTENUATION AT 35 GHz TO CHANGES IN THE MICROSTRUCTURE OF RAIN Before proceeding any further, a brief explanation of the interaction of microwave radiation with a scattering medium (e.g. rain filled medium) will be presented. The definitions o f some of the terms that will be referred to throughout the remainder of this study will also be given. 3.1 Electromagnetic Scattering Theory-Single Scatter The interaction of a monochromatic electromagnetic plane wave with an individual scatterer of arbitrary size and shape can be expressed as: r r - jc a t V £ xm \ h J where X X ' f eX e ' ,kr r X X. (3.1) and E l"c represents the scattered and incident electric fields respectively with their orthogonal components j = /t, v(h-horizontal, v-vertical), r gives the radial distance from the scatterer, 2n k = —— is the propagation constant of the wave, and Sy are the elements o f the scattering amplitude A matrix. Here the first subscript i denotes the scattered/received polarization, while the second subscript j gives the incident/transmitted polarization. In general, the elements of the scattering matrix are functions of the frequency and the direction o f the incident wave. They are also functions o f the physical properties o f the scatterer, e.g. its size, shape, orientation, and dielectric constant. The backscattering cross-sections of the individual scatterer is given as K =4;zjs,/| (m 2) / = It, v,and j = h,v (3.2) 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where Sy is an element of the backscattering amplitude matrix. The extinction cross section is given following Ishimaru (1991), as ac =— , i = h,v (m 2) (3.3) where Sy is an element of the forward scattering amplitude matrix. 3.1.1 Canting (Fall Orientation) Simulations Laboratory observations of falling drops reveal that they are highly oriented with their symmetry axes in the vertical direction. Following Goldstein (1980), the orientation of an arbitrary particle can be fully described by the three Eulerian angles (6, <j), tp), where 0 and <j>are the polar and azimuthal angles relative to zenith and, tp describes the rotation of the particle about its axis. These angles are usually considered to be independent random variable, such that a scatterer’s orientation at any time is given as a distribution over these angles. Several hydrometeor-canting distributions have been suggested in the literature (Moninger and Bringi, 1984, Vivekanandan et al., 1991; Jameson, 1987). To compensate for the fact that the scatterer’s symmetry axis may not be along the zenith the cross sections are averaged using a canting angle probability density function p(6,<J>,(p) to give the angle averaged cross sections or the ensemble average of the cross sections over the angular distributions 2x2xn K W J J <JM(d,<f>,(p)p(0,<t>,(p)s\n0 d0d<j>d(p (m 2) (3.4) 0 0 0 where the subscript M refers to either the backscattering or extinction cross sections as defined in (3.2) and (3.3). For this probability density function p(6,<|>,<p), the distributions of 0 and <j>are assumed to be independent of each other and of size D. A uniform distribution, U{<j>mn = 0 ° ,^ ^ = 360°), is assumed for <j) and a Gaussian distribution, N(6 = 0°, std.dev. = 5°, 10°) is used for 0. Since the drops are modeled as oblate spheroids (rotationally symmetric) it is not necessary to consider the angle tp. 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A similar procedure can be carried out with the scattering amplitudes to give the angle averaged scattering amplitudes ^S ^ in units o f m for the backscattered and forward-scattered directions respectively (see Tang, 1994 and Walsh, 1998fo r further details). 3.1.2 Polarimetric Parameters Of Interest In the previous chapter it was shown that the rainfall rate is defined by a distribution of raindrops over a range of sizes i.e. the DSD. Similarly, the so-called rainfall products are also given over the distribution of raindrop sizes, and would also include information on the drop shapes, and orientations. To calculate these bulk parameters we integrate the ensemble-averaged cross sections, or Field amplitudes, as the case may be, over the size distributions of the raindrops. The parameters that will be considered in this study are thus defined as follows: The specific attenuation, following Oguchi (1973), is given as. A. = 4.343 x l0 3J ( tx ( D ) ) N(D)dD (dB /km ) (3.5) in terms of the extinction cross sections, where ( a r (D)^ is the angle average extinction cross section of a drop having a diameter D. The specific attenuation can also be calculated from the scattered field intensities as. At = -8.686xlO 3 A ■Im{[(S„( D ))N (D )d o j {dB/km) (3.6) where X is the wavelength in m units. The specific differential attenuation is then defined as AA = Ah - A v (d B /k m ) (3.7) The specific differential phase is defined as. K op = 1 * 9 ^ 1 2 1 ^ Re{(stt( D ) _ s v¥(D)>}iV(D)rfD (deg/k m ) (3.8) 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where the fields 5 „(D ) in (3.5) - (3.8) are all defined in the forward direction and in units of m Oguchi (1976). The backscattered differential phase is defined as the difference in the phase terms of the scattering amplitudes in the backward direction and is given as, <5 = a rg [j5 „ (5 w,)’ /V(D)dD| (deg.) (3.9) where (*) represents the complex conjugation (Doviak and Zmic', 1993). Following Battan (1973) the effective reflectivity factor is given as Z, = T j { a bii(D))N(D)dD (mm6» r 3) (3.10) n \K\ The differential reflectivity is then calculated from the ratio of the reflectivity factors following Seliga and Bringi (1976) as Z DR =10 log (dB) (3.11) 3.2 Electromagnetic Scattering Calculations It is obvious that accurate estimates of the scattering amplitudes are critical to measuring and studying the scattering properties of rain. This is especially important at millimeter wave frequencies, and for non-spherical drop shapes whose size parameters 7ED Re{/i} > 0 .3 , where n is the complex refractive Z index of water at the frequency of interest Kerker (1969). This is because the Rayleigh approximation (van de Hulst, 1981, defines the Rayleigh approximation condition as |-\/^r|^»rmax « 1 where rmi is 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. half the maximum dimension of the drop) would no longer apply to these scatters. Consequently, the more computational formulations must be applied for these large size particles. There are a number of computational techniques that have been developed for calculating the scattering the electromagnetic scattering by hydrometeors. Among the most commonly used are the point- matching formulation (Morrison and Cross, 1974; Oguchi and Hosoya, 1974), the T matrix or extended boundary condition method (Waterman, 1965, 1969; Barber and Yeh, 1975; Bringi and Seliga, 1977, Mishchenko et al., 2000), the unimoment (Mie, 1974; Fang and Lee, 1978; Morgan, 1980), and the Fredholm integral equation formulation (Uzunoglu et al., 1976; Holt et al., 1978; Shepard et al., 1980). O f these the T-matrix method is the most common as it is quite flexible in handling many different shapes within the same programme (Holt, 1982). It is ideally suited for shapes with smoothly changing surfaces, but may not handle shapes with rapidly changing irregular surfaces. Furthermore, it is an exact formulation o f the scattering problem and uses spherical vector wave functions that are well suited for numerical computations. The T-matrix method has been selected for use in this study due to fact that it is a well-recognized computational technique in the research community, and also because it is ideally suited for these moderately large, lossy (i.e. complex refractive indices), non-spherical, and rotationally symmetric drop shapes with relatively smooth surfaces. For a given drop shape model the scattering cross sections, as a function of drop size, were computed for equilibrium shaped (BCeq) drops that were assumed to have only rotational symmetry. Similar calculations were performed for drops that were oblate spheroidal. The errors in the latter model were evaluated relative to the former, more realistic drop shapes. The extinction cross-sections calculated for the oblate spheroidal shapes at 35 GHz were observed to be within 4% of the more realistic drop-shapes. The calculations of the scattering amplitudes of drops with only rotational symmetry required more computationally intensive procedures. Since no significant differences were observed in the accuracy of the calculations obtained for these two drop types, the decision was taken to use this simpler model (oblate spheroidal models having both rotational and equatorial symmetry) for all future calculations. Tests were also under taken to determine whether the scattering cross section of an averaged drop shape for a given drop size, was representative of the average scattering cross section of a distribution of drop shapes at that drop size. Using results provided by Kubesh and Beard (1993) for the measured axial ratios of a 2.5 mm drop, three different distributions of axial ratios (i.e. drop shapes) were generated. 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The first was a normal distribution N (a = 0.923, std.dev. = 0.02), the second a uniform distribution U ( a ^ a = 0 . 8 6 7 , = 0.984), and the third, a harmonic oscillation model adapted from Moninger and Bringi (1984), for a = axial ratio. The scattering cross sections for the respective distributions were averaged and their average values were compared to that expected for a 2.5 mm drop with an axial ratio a = 0.923 = a mg . The results were as follows: • The maximum error in the average axial ratios was less than 1% (the average axial ratio of the harmonic oscillation model gave the highest error relative to the mean axial ratio given by Kubesh and Beard (1993), while average drop shape of the normal distribution was closest to that given by Kubesh and Beard). • The maximum relative errors were observed in the <rbh (1.6%) and creh (1.1%) for the cross sections calculated for the average axial ratio of the harmonic distribution relative to the mean shape given by Kubesh and Beard (1993). 3 J Estimation O f Rainfall Rates From Microwave Attenuation At 35 GHz The rainfall rate is given by the expression /? = — (*D3V'(D)N(D)dD 6J {mini h) (3.12) where vt(D) is the terminal velocity of the drop of equivalent diameter D. Atlas and Ulbrich (1977) suggested that at 35 GHz the integral equation for the microwave attenuation in rain (3.6), has a similar exponent (“moment”) m of the DSD to that of the rainfall rate given by (3.12). A = CAj D mN {D )dD (d B /km ) (3.13) /? = C * J d 367A((D)c/D (nun / h) (3.14) The constants C A, C r. and m were reported to be independent of temperature for 0°C < T < 1 8 °C , and showed no significant changes in their values as the temperature rises to 40°C. In arriving at these conclusions they fitted a power law relationships o f the form. 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. X ( D ) = CD" (3.15) to both the extinction cross-sections and the terminal velocities of raindrops as a function of their equivalent volume diameters D. In (3.15) C and n were estimated as the constant and exponent respectively, that gave the lowest errors between the models their expected values. It must be noted that they had assumed a Marshall-Palmer (MP) DSD of spherical raindrops in their study. We are interested in evaluating the sensitivity of this A-R relationship to changes in the microphysical properties of shape, orientation, size, and DSD of the raindrops. For this reason, we shall assume a gamma model DSD of the form N ( D ) = N aD Me _AO (m '3a n ) (3.16a) where the relationship between the parameters are given by Ulbrich (1983), as N„ = 6 x 1 0 V - " A= 3.67 + n ' — eR 6 (3.16b) (cm~l) (3.16c) 8 = ----4.67 + // (3.16d) e = (3.67 + //)[3 3 .3 1A„r(4.67 + f i ) \ s (3.16e) and r(x) is the complete gamma function. From the frequency distribution plots of gamma model DSD parameters fitted to experimental data, Ulbrich and Atlas (1998) estimated that the central 80% of their spectra fitted a p. lying between -1 and 8, with a modal peak value between 3 and 4. For this study, we shall consider gamma model DSDs over a range of drop diameters 0.1 < D < 6.0mm , for rain rates 1 < R < 150m m / h , and drop-shape parameters // = -1 , 0, a n d 3 . Chandraseka and Bringi (1988), Chandraseka et al. (1990), Ryzkov and Zmic' (1995), Scarchilli et al. (1993) used similar ranges o f drop shape parameter p to predict the performance of Polarimetric parameters. Fig. 3.1 shows a plot of the DSD for the different values of drop-shape parameter at different rainfall rates. 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Gamma Gamma model DSDs R = 1mm/h DSDs R = 20mm/h 10' 10 ' 10 to to Q 102 10 Z 10’ 10° 0 1 3 2 5 4 D(mm) D(mm) Gamma model DSDs R = 100mm/h Gamma model DSDs R = 50mm/h 10' 10' S 10' S 10' z 10 10 10 ' 10‘ D(mm) D(mm) Fig 3.1 Variations in the gamma model drop concentrations for p = -1,0, and 3, and for different rainfall rates (R = 1,20,50, and 100 mm/h). Note that the gamma model DSDs for p = -1 and 0, consistently predicts high concentrations of the small drop sizes. Measured DSDs tend to show much lower concentrations o f these sizes (Wills, 1984; Jones, 1992), suggesting that for these values o f for p the gamma models tend to over estimate the number of smaller drops. Therefore, the gamma models for these values of p. are expected to differ from those of naturally occurring DSDs. Nevertheless, we can still gain from useful insights from the behavior of natural drop spectra. It can be observed that at a given rain rate R an increase in the shape parameter p has the effect o f narrowing the range of drop-sizes while increasing the numbers o f intermediate size drops. The effect o f this “shaping” o f the drops spectra on the calculated attenuation rates can be observed in Fig. 3.2. For this case study we assume the drops will be in equilibrium (e.g. not oscillating) and that they would all be aligned with their symmetry axis along the vertical i.e. they are not canting. It is evident that AA is less sensitive to changes in the DSD than Ah, Av, and AaVgfor R > 25 mm/h. The change in AA for p = 0 relative to p = 3 were higher for R < 10 mm/h where the maximum error was 29 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. approximately 19%. The maximum changes in Ah, Av, and A^g over the same range o f rainfall rates were less than 5%. For R > 25 mm/h the maximum change in AA is less than 13% while the maximum changes in Ab Av, and A„g are approximately 30%, with Ah varying slightly less that o f Av. The relatively small change in AA at these rainfall rates is in part due to the fact that both Ah and Av are in general similarly affected by the changes in DSD. When the difference is taken between these similarly changing quantities that difference is relatively constant and thus insensitive to the changes that are taking place. Over the lower rainfall rates the changes in AA tend to be exaggerated since we are taking the difference between very small quantities. 50 40 40 30 30 20 10 50 100 50 150 100 150 100 150 R(mm/h) 8 40 6 30 E 2 0 0 50 100 150 50 R(mm/h) Fig. 3.2 The sensitivity of Ah, Av, AA, and AaVg at 35 GHz to changes in the drop shape parameter p. Notice that the variation in the specific and average attenuation rates is strongest at the higher rainfall rates. This corresponds to the fact that the greatest variations in the drop spectra occur at the higher rainfall rates. From Fig. 3.1, it can be observed that as the rainfall rate increases the drop spectra broadens as larger drops become more common and there is a general increase in the number o f drops in 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. each size category. It should also be noted that as the shape parameter p increases the distribution becomes narrower, and the number of drops in the intermediate-to-Iarge drop size categories increase. The number o f smaller drops is suppressed. This explains why in plots o f the rainfall rates generated from the gamma models DSDs, the rainfall rates obtained from models with the higher values of p always exceed those obtained for p. = -1, and p = 0, since there is a scarcity o f drops in the critical midto-large size ranges. This can be observed form Fig. 3.3 where the integrands of equations 3.5 and 3.12 are shown as functions o f the drop size. Gamma model DSDs R = 1mm/h 250 t* " - 1 ®.h 2 00 H-0o,B . _ _. n « -1 V O' O 150 ■ 3 v, D: 800 Gamma model DSDs R = 20mm/h 600 S 400 200 50 N ', D(mm) D(mm) Gamma model DSDs R = 50mmm Gamma model DSDs R = 100mm/h 2000 4000 1500 3000 §1000 Q 2000 — V 500 1000 D(mm) D(mm) Fig. 3.3 The distribution functions o f the H-pol. extinction cross section 0 * used to calculate the attenuation at 35 GHz, and the products of the terminal velocity and equivalent drop volumes v, •D 3(drop momentum) used in calculating R, for different shape parameters p, and as functions o f the equivalent volume diameters. The distribution functions are given as the products x • N{D)dD where % can be either OdiOr vt D 3. These plots show the weighted contributions of the different drop sizes to both A and R as a function of their size and number. The variations in A can be estimated from the variation Oat(D ). Similarly, 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. variations in R can be estimated from the variations in v( (D ) • D 3, where vt(D) is the terminal velocity o f the drops given by Lhermitte (1989) as, / rv \ v,(D ) = v0[r il - e -^.8D *“ 4 .8 8 0 i J c m s ~I (3.17a) for -i0.5 v =923 (3.17b) _PZ, and D(cm), with pQand pz being the air density at the ground and the altitude z respectively, at which vt(D) is being expressed. Over the medium-to-high rainfall rates, the range of drop sizes that contribute significantly to R is larger than those contributing to attenuation. In addition the weighted contribution of the smaller drop size categories to the attenuation for the lower values of p, is greater than that of the larger drops. This is almost the opposite o f that which occurs with the rain rate term as the smaller drop sizes contribute almost the same insignificant amount to the rainfall rate for the different values of p, whereas the larger drops are bigger contributors to the rainfall rate at the lower values of p than at p = 3, at which value the DSD becomes a bit more symmetrical. In general the rainfall rate responds faster and more dramatically to changes in DSD than does the attenuation. This would suggest that these relationships may not be as linear as previously thought, but that these parameters (A and R) may have different sensitivities/dependencies on different regions of the DSD or ranges of D. Power law fits to the A-R relationships were derived for the BCeq shapes using a non-linear NewtonRaphson iterative technique (Neter et al., 1996). The results o f these fits for different values of p are given in Table 3.1. 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3.1 'ower law fits to the A-R relationship at 35 p=0 P = -l A=aRb a b b a 0.9075 0.3537 0.8803 0.3555 Ah 0.8859 0.3237 0.8562 0.3266 Av 1.0011 1.0221 0.0368 0.0353 AA 0.8976 0.3383 0.8693 0.3408 Aava GHz for different DSD shape parameter values p -l<//<7 P= 3 a b a b 0.2944 0.9938 0.2465 1.0227 0.2750 0.9758 0.2245 1.0104 0.0242 1.1099 0.0243 1.0980 0.2845 0.2354 0.9855 1.0170 It is evident, from these results, that as p increases AA becomes less linear in R and the other attenuation parameters become more linear with respect to R as their exponent gets closer to 1. A best-fit model for the values of - 1 < n < 7 has been included for comparison with other published results. The relationship given by Atlas and Ulbrich (1977) for the MP distribution of spherical drop shapes was A =0.219/?IW (dB/km) (3.18a) Using the results provided by Oguchi and Hosoya (1974) for horizontally oriented oblate spheroids at a variety of raindrop canting angles and drop temperatures of T = 20°C, Atlas and Ulbrich derived AA = 0.0387/? (dB/km) (3.18b) Note that their AA-R relationship is very close to that obtained for p. = -1, 0 in Table 3.1. Spherical drop shapes would give the same results for Ah and Av. Their A-R relationship seems closer to that of Aavg for p>3. 3.3.1 Effects Of Drop Shape On A-R Relationships To evaluate the sensitivity of the A-R relationship to changes in the drop shape due to oscillations, we shall compare the calculated attenuations for the BCcq shapes, to those o f the ABLav and Kav shapes. Fig. 3.4 shows the results for these three drop shapes, each of which assumes the same gamma model DSD (p = 3) for drops having the same fall orientation. 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 35 10 - 5 - so 100 SO ISO R(mm/h) R(mm/h) 100 1SD 100 150 X 50 100 R(mm/h) 150 SO R(mm/h) Fig. 3.4 Sensitivity o f the A-R relationships at 35 GHz to changes in drop shape parameter |i It is clear that variations in drop shape can lead to significant errors in the estimation o f the rainfall rate from AA while AaVg is the most insensitive to changes in drop shape. On average, the equilibrium shapes give a slightly higher Ah than the oscillation shapes, whereas the average (oscillation) shapes have higher values for Av. This comports well with our expectations, since the oscillations tend to make the drops appear more spherical as can be seen from the plot o f the axial ratios (see Fig. 2.1). Relative to the BC*,, shapes, there is a 25% change in the AA o f the ABLaVshapes for R < 5 mm/h that decrease slightly to about 22% at the maximum rain rate o f 150 mm/h, despite the fact that the axial ratios of both shapes are almost identical for D < 1.0 mm. Evidently these smaller drop sizes do not have a significant impact on AA. Since the drops in this region are almost spherical they would have very similar attenuations at V- and H-poIarizations, the difference between which would be very small. Drops in the range 1 < D < 1.4 mm where the resonance oscillation phenomenon is thought to occur (Beard et al. 1991, Beard and Kubesh, 1991), dominate the contributions from these smaller drops sizes. Not only are the shapes significantly different to those o f the equilibrium shaped drops, but also the fact that the majority o f the drops are in this range suggests a reason why this effect may be amplified. The relative 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. error decreases at higher R as the number of larger drops with shapes approximating those of BC,*,, begin to increase. For the Kav shapes the relative changes in AA increases from about 21-25% for R < 5 mm/h and are less than those of ABI^, over the same range of R. This despite the fact that the Kav shapes, unlike the ABL^ shapes, are significantly different from the BQq shapes for D < 1.0 mm. These errors increase to a maximum 28% at the higher rainfall rates, as the larger, more distorted drops become more significant players in both AA and R. The maximum changes in Ah and Av were less that 1.5 and 3.5% respectively. The spread in Aavg was less than 1%. The power fit models for oscillation dependent A-R models are given in Table 3.2. % > II Table 3.2 Fits to the A-R relationships for the assumed oscillation shapes AE Lav Kav a b a b 0.9934 0.2916 0.2919 0.9927 Ah 0.9782 0.2775 Av 0.2783 0.9802 0.0176 0.0176 1.1233 1.1075 AA 0.2846 0.2848 0.9862 0.9867 Aave Jameson (1989) suggests that the specific attenuation is sensitive to variations in drop shape and DSD. Using a series approximation to the normalized imaginary component of the forward scattering amplitude, he showed that the diameter dependence was determined by a multiplicative factor of D raised to a power. On the other hand, the shape factor is only an additive term and as such variations in this parameter has less of an effect on the forward scattered amplitude. He also showed that the difference (sum) of the vertical and horizontal component of the forward scattered fields were proportional (independent) of the drop shape parameter. This suggests AA (Aavg) will likewise be sensitive (insensitive) to variations in the drop shape parameter. 3.3.2 The Effects O f Drop Canting On A-R Relationships At 35 GHz The sensitivity of microwave attenuation in rainfall to the orientation of the raindrops can be estimated by averaging the scattered fields calculated in section 3.1 over some assumed distribution of canting angles, and then integrating over the DSD. For this study we have assumed equilibrium shaped drops with their symmetry axes normally distributed about the zenith direction (0 = 0° mean position). We 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. considered two normal distributions o f canting angels with about the mean position with standard deviations o f 5° and 10°. The results of this study are shown in Fig. 3.5. 0° 10° T 30 § 20 50 100 50 150 100 150 100 150 R (m m /h) 40 7 6 30 5 e § 4 I% 3 < 2 10 1 0 0 50 100 R (m m /h) 150 o 0 50 R(mm/h) Fig. 3.5 Plots o f microwave attenuation vs. rainfall rate for different distributions o f drop canting angles It can be observed from the above results that AA shows the greatest sensitivity variations in the fall orientation o f the drops. For standard deviations o f 5° and 10° in the canting angles about the z-axis there is a 2.5% and 9% decrease in AA, respectively relative to the non-canting case, over the entire range o f rainfall rates. The effects on Ah, Av, and A^g are negligible. The maximum variation in Av was about 1% and this is 3 times the maximum relative error in Ah with the variation in AaVg being less than that in Ah. As a drop cants away from the vertical, less of its upper hemispheric surface becomes illuminated and its vertical dimension appears larger than that seen before canting, with little or no change in its horizontal dimension. Consequently, the vertically polarized wave would be expected to experience an increase in attenuation while the attenuation on the H-pol. is unchanged, as a result of which AA decreases. Fig. 3.6 shows the combined effects of the changes in the shape parameter o f the DSD, the drop shape, and the fall orientation on the A-R relationship. The results for A„,g are very similar to those o f Ah and have therefore not been included. The spread in AA increased from approximately 20-40% with rainfall 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. rates R < 55 mm/h, while the spread in Ah and Av increased from 3-31% and 5-35% respectively. Therefore the errors in AA due to the combined effects of variations in drop shape, DSD and fall orientation are slightly higher than those observed in the specific attenuation parameter. 40 45 BC BC 40 35 35 O V -0 BC " 0° n - 3 ABL 1 0 V « -1 ABL 10° 0 ABL 10° n - 3 K K K 30 10° ii«-1 10° n« 0 10° n - 3 30 25 20 10 100 150 50 100 R(mm/h) 150 100 50 R(mm4i) 150 Fig. 3.6 Results of the combined effect of changes in DSD, drop shape and fall orientation on the A-R relationship 3.4 Conclusions Ah, Av, AA and A.vg are all sensitive to changes in DSD. AA is the least sensitive to changes in the DSD, especially at higher rainfall rates. The other three parameters, Ah, Av and A,Vg are comparable. Aavg followed by Ah is the least sensitive to changes in drop shape. The maximum spread in A^g due to changes in drop shape, as determined by the ABLav and KaVmodels, relative to the BC,*,, model was less than 1%. AA is very sensitive to changes in drop shape. Relative differences of between 25-28% have been estimated in AA due to changes in drop shapes. 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Variations in the fail orientations were observed to have the smallest impact on the calculated microwave attenuations at 35 GHz. The largest spread (-9% relative to the non-canting case) was observed in AA when the drops were assumed to have their canting angles normally distributed with a standard deviation of 10° about zenith. Aavg (<0.3%) followed by Ah (<0.4%) were the least affected. When the combined effects of changes in DSD, drop shape and fall orientations were considered, the AA parameter showed the largest overall spread. 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER4 ESTIMATING RAINFALL RATE AND ACCUMULATIONS FROM SIMULATED MICROWAVE ATTENUATIONS AT 35GHz USING DISDROMETER DERIVED DSDs In Chapter 3, we began the examination of the relationship between microwave attenuation at 33 GHz and rainfall rate. We discussed the sensitivity of these relationships to changes in the drop size distribution (DSD), the drop shape, and fall orientation. To evaluate the dependency on DSD we used three different moments of the theoretical gamma model DSD. It was previously noted that the gamma model DSD, despite its obvious usefulness in gaining insights into the relationship between different integrated parameter, may in some cases be quite inadequate to describe the high degree o f variability observed in natural raindrop size distributions. In this chapter we will use estimates o f naturally occurring DSDs (as observed by an impact type Joss Waldvogle Disdrometer) to further refine our models and to determine which relationships may be most appropriate for rainfall estimation over different geographical regions. 4.1 Disdrometer Drop Size Distributions The DSDs used in this study were obtained from ground-based disdrometer (RD-69 Disdromet type) measurements at three different geographical locations. The first data set, representing tropical rainfall, was taken during TOGA COARE at the Kapingamarangi atoll in the western equatorial pacific during the period from November 1992-February 1993 (see Tokay and Short, 1996 for further details), and will hereafter be referred to as TROP. A second data set, to be given the designation SWISS, was gathered continuously over the months o f May to November 1988 at Hoenggerberg (530m ASL) in Zurich, Switzerland. The third data set, to be denoted as MISS, was collected from March 1996 to June 1998 at the Goodwin Creek research watershed in Panola County, northern Mississippi (more details on these data sets are provided by Steiner, 1991 and Steiner and Smith, 2000 respectively). All three data sets were given as one-minute averaged drop spectra measurements. For this study we considered only those spectra with a minimum of 10 drops and having an equivalent rainfall rate more than 0.1 mm/h. Consequently the TROP, SWISS, and MISS data sets used contained 8800, 18025, and 31040 1-minute DSDs respectively. Running averages were carried out on the data using 2-, 5-, and 10- 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. minute wide filtering windows in an attempt to remove some o f the measurement noise from the data. This had an insignificant effect on the A-R relationships that were derived (less than 1% and 2% in the Ah-R and AA-R relationships respectively). Figure 4.1A shows the total distributions (the sum o f all the 1-min spectra in each distribution) of drop sizes for each o f these data sets. Also included in this plot are the gamma models (DSDs) used in the previous chapter. The gamma model DSDs have been scaled for better comparison to the disdrometer derived measurements. Figure 4 .IB shows the mean values of these distributions 10 10' — 10s TROP S W ISS — MISS — rn --i — r(i» o — r ti« 3 105 101 32. 10° O 10 •1 10’ 10° 0 2 4 D(mm) 6 10‘ 0 2 4 6 D(mm) Fig. 4.1 A) Plot o f the total drop size distributions of the TROP, SWISS, and MISS data sets, along with the gamma model DSDs scaled for comparison of their shapes. B) Plot o f the mean drop size distributions o f the TROP, SWISS, and MISS DSD along with the gamma model DSDs. Note that the SWISS and MISS comes closest to the MP DSD where p = 0 (i.e. their distributions have general straight line slopes similar to those o f a typical MP DSD). On the other hand the TROP DSD has a form that is more typical of the DSDs with higher values o f the DSD shape parameter p. The TROP 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. distribution has a relatively smaller number of drops at both ends of the measured drop sizes and is narrower that the other two distributions (i.e. more ‘concave down’). 4.2 Modeling The Relationship Between Rainfall Rate And Attenuation Using Measured DSDs The attenuation and rainfall rates were calculated for each distribution. Fig. 4.2 shows the results of these simulations for BC«, shaped drops with their canting angles normally distributed about the vertical direction with standard deviation of 5°. Figure 4.2a-4.2d are the results corresponding to the TROP DSD, while plots 4.2e-4.2h and 4.2i-4.21 are the results for the SWISS and MISS DSDs respectively. 25 TROP _20 4 .v «■ E3 if 2 IS 1150 5 0 .i I 100 50 I 15 S=10 5 0 I 25 _20 100 50 j . 50 0 f J. - k 50 100 5 0 100 0\ f 50 R(mm/h) •100 0 100 50 25 I 15 S 10 .100 / 100 50 X 50 E 20 • » ^15 XJ p 40 < 5 . 0 • • • 100 50 100 E 20 I 15 \ ■ 50 R(mmAi) 0 25 _20 5 y j f ' 25 20 4 MISS I 15 0J E3 <| 2 * ..*• JE 20 ^15 ^WO < 5 0 ^ 1° 4 SW ISS _20 I 15 3 i s 25 / 25 25 _20 y IX . 50 100 0 / R(mnrVh) 50 R(mrWh) 100 Fig. 4.2 Scatter plots showing the A-R relationships for the different distributions. Note that for all three distributions the axes have been adjusted to the similar ranges for each parameter. From these results it appears that there is less spread in the AA-R plots for the SWISS and MISS distributions at the highest rainfall rates (e.g. R > 25 mm/h) than there are in their respective Ah-, Av-, and Aavg-R plots. This is consistent with the results of Fig. 3.2 showing the sensitivity of the A-R relationships to variations in DSD. More variation in the DSD tends to occur at the higher rainfall rates 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. as larger numbers of drops over a wider distribution o f sizes, begin arriving at the surface. For the TROP distribution the scatter in R-AA exceed those of the R-Ah over the entire range of rainfall rates. Note that the scatter in the TROP results exceeds those of the SWISS and MISS distributions (by as much as 30% in some cases) over the lower rainfall rates (R < 10 mm/h). On the other hand, the Ah-, Av-, and Aavg-R plots for the TROP distribution show less spreading (20% less scatter) than the corresponding plots of the other two distributions. One explanation for the enhanced scatter in the TROP AA-R plot may be obtained by taking a closer look at its DSD, which was earlier noted to be somewhat different from the other two DSDs. In Fig. 4.1 it was noted that the TROP DSD was closer to that o f a higher order gamma model DSD, to the extent that a relatively higher proportion o f the drops were within the medium-to-large category (1 < D < 3mm range), with a faster decline in the number of smaller and larger drops. The resulting “concave-down” shape would tend to accentuate the physical characteristics of the drops in this range of sizes. This is supported by the results in Fig. 4.3 that shows the distributions o f the median volume diameter D0 and the equivalent volume drop diameter D for each o f the disdrometer data sets. The median volume diameter is defined as the drop diameter below which half the water content o f that spectrum is contributed by drops below this size. It is therefore of some physical significance as it is indicative of the range of drop-sizes that are the main contributors to both the rainfall rate and the rainfall accumulation. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 a 8 TROP J k . g 0 .2 5 0 1 u. •« K L 2 3 4 5 0 C 1000 S U £ i i I i :/ / I i I. i/ 0 .5 0 b // M/ •§ 0 7 5 0 1 2 3 4 500 I 5 llhn..------ 150 d "5 0 -7 5 £-100 SWISS 1500 *• E ** 1000 *•::* S ’ 50 a 0 .2 5 :j 0 1 2 3 4 Ik 5 MISS 2000 V) D0(mm)) D0(mm)) 1000 D0(mm)) Fig. 4 J Distribution of the median volume diameters D0 and the equivalent volume diameters D for each of the DSDs. Also shown are the relationships between the D0s and the rainfall rate R Figure 4.3a-c corresponds to the TROP DSD, while plots Fig. 4.3d-f, and Fig. 4.3g-i corresponds to SWISS and MISS respectively. From Fig. 4.3a, -d, and -g it is clear that D0 increases with the rainfall rate, although there are many instances where there are large values of D0 at low R. It is clear that there is a lower limit to the value of D0 at a given rainfall rate. A high D0 is generally indicative of the presence of larger drops, which has been confirmed by the examination of a number of individual spectra. The large spread in D0 at the lower rainfall rates is believed to be due to larger drops, with higher terminal velocities, arriving at the surface before the lighter and smaller ones at the beginning of a shower where the rainfall intensity is usually quite low, or the occasional large drop appearing at the surface due to drops collision and coalescence or low evaporation in a low intensity event. Figure 4.3b, -e, and -h show the fractional increase in both D and D0 for all three distributions, while Fig. 4.3c, -f, and -i show the number of spectra in each of the three data sets, with D0 in the size categories indicated. From Fig. 4.3b we see that 56% of the TROP spectra have equivalent volume diameters 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D < 1.0mm while 19% have Da < 1.0mm mm. This suggests that quite a large percentage o f the water content is due to the drops in the range 1-3.0 mm. This may not always be the case, as can be observed from 4.3e where, for the SWISS distribution, 91 % of the drop sizes and 55% of the D„s are less than 1 mm. In the case of the MISS distribution 85% o f the drop sizes and 36% of the D„s were in this size range. The SWISS and MISS distributions were observed to have a larger concentration o f drops D>3.0 mm (0.3% and 0.2% respectively), than the TROP distribution, 0.07%. It is however clear that the majority of the drops in the TROP distribution are concentrated within the medium to large drop size range (1 < D < 3.0m m ), and that the majority of the water content is contributed by these drops, thus providing conclusive proof that this distribution is narrower and concentrated about this critical range of drop sizes than the other two distributions. From Fig. 4.3c it is also clear that the modal D0 of the TROP data is larger than that o f the SWISS and MISS data sets. For additional information on how this may be related to the estimated attenuations we must examine the functional dependency of the extinction cross sections of the drops on their sizes. Figure 4.4 shows the behavior of the extinction cross sections at H-pol. (och), and the differential extinction cross section ( A<rf = a th - crn ) versus D for the different drop shape models being considered (the extinction cross sections at V-pol. and the average extinction cross sections at V-, and H-pol. were not shown as they were observed to vary in a manner similar to that of <reh). Note the sudden changes in the slopes of the extinction cross-sections for D > 2.0 mm. This is particularly noticeable in the case of A<Je. Both crch and ocv begin oscillating for D > 2.0 mm as these drops are in the resonance scattering region at this frequency. The greater scatter observed in the TROP results is due to the fact that such a large percentage of the drops and the median size diameters are within in this resonance region. In addition to this it is clear from Fig. 3.3 that the drops in this range dominate both R and A. Thus this non-linear effect in Aae is enhanced resulting in more scatter in the AA-R results from the TROP spectra over that o f the other distributions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 — F (P ) •a 10 "JO 109 10 :r:?■—• 10 10’ 101 D(mm) 101 10° 101 D(mm) Fig. 4.4 The single scatter extinction cross-sections (Oeh), the differential extinction cross-sections ( Act, = a th - O" ), and the drop momentum F(D) as functions o f the equivalent volume diameter D. Also included in Fig. 4.4 is the expression F(D) = v,(D )D 3 (4.1) the drop momentum, which is related to the rainfall rate in the same manner that o e is related to A (see Eq. 3.13, 3.14, and 3.16). Therefore the relative dependence o f F(D) and <Je over the drop size D can be used as a first order test to the relationship between their integral parameters R and A. From Fig. 4.4 it can be seen that the general slope o f a,* is closer to that o f F(D) for D < 3.0 mm while the gradients diverge for the larger sizes. In the case o f Aoe and F(D) the opposite is true; the general trend in Aoe is closer to that o f F(D) over the larger sizes while being quite different at the smaller sizes. In may be convenient to fit a two-piece power law curve to the extinction cross sections over the non resonant region (D < 2.4 mm) and the resonant region (D > 2.4 mm). The results of these fits are given in Table 4.1. F(D) was fitted over the same regions with exponents b = 3.625 and 3.134 over the ranges 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.1 < D < 2.4 and 2.4 < D < 6.0 respectively. It should be noted that the exponent b of F(D) is given in the literature as varying from 3.6-3.67 for 0.5 < D < 5.0 mm, with the latter being the most commonly used (Sekhon and Srivastava, 1971; Atlas and Ulbrich, 1979; Doviak and Zmic', 1993). Table 4.1 Piece-wise power-law fits for the single scatter extinction cross-sections and drop momenta to the equivalent volume spherical diameter._______________ _______________ ___________________ Ac = 0 ^ - 0 ,, (mJ) er..i = <WmJ) Drops + <r„ )/2 (m2) a D(mm) b a b b Shapes a a b bcm 0.1 < D < 2.4 1 A < D < 6.0 ABLgv 0.1 < D < 2.4 2.4 < O < 6.0 Kav 0.1 < D < 2.4 2.4 < D <6.0 4.71E-7 2.7 IE-6 4.68E-7 2.64E-6 4.69E-7 2.64E-6 3.892 1.922 3.881 1.936 3.874 1.929 4.75E-7 2.51E-6 4.73E-7 2.73E-6 4.70E-7 2.68E-6 3.646 1.778 3.687 1.731 3.706 1.768 2.55E-8 3.63E-7 1.83E-8 2.42E-7 5.322 2.335 5.404 2.564 1.82E-8 2.09E-7 5.311 2.548 4.71E-7 2.59E-6 4.67E-7 2.65E-6 4.69E-7 2.64E-6 3.778 1.860 3.789 1.847 3.794 1.857 Note that for D < 2.4 mm the exponent ‘b’ of aeh is closer to that of F(D) than the corresponding value of Aae. Similarly, the exponent of Acre is closer to that of F(D) than the exponent of och for D > 2.4. Since low to moderate intensity rain is chiefly comprised of small to medium size drops it is therefore expected that Ah-R would have a tighter relationship than AA-R at these lower rates. Conversely, as the moderate to high intensity rain typically has a larger number of bigger drops, it is expected that the AA-R relationship would be superior to that of Ah-R over these higher rain rates. It is for these reasons that one observes more spreading at the lower rainfall rates in the AA-R plots relative to the Ah-R plots while also observing more spreading in the Ah-R plots relative to the AA-R plots at higher rain rates. Similar arguments to those considered for the Ah-R relationship would apply to the Av-R and Aavg-R relationships. Another reason for the increased spreading in the AA-R plots relative to the Ah-R plots at the lower rainfall rates is due to the relative magnitudes of the respective cross sectional areas (<rch and Aae). At D = 1 mm the magnitude of a<.h is 36 times that of Aoe while at D = 0.3 mm it is 3 orders of magnitude higher. Hence there is a loss of sensitivity in Acre over these smaller sizes. As a result, calculations involving ACTe become that much more sensitive to numerical errors particularly at the smaller drop sizes. Since the DSDs tend to be heavily biased by the large number of drops at the smaller drop sizes, then these effects become more significant at the lower rainfall rates that are in turn generally dominated by these drop sizes. The combined effects of the difference in the slopes of F(D) and A<rc(D) along with the 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. loss of sensitivity in Aae over the smaller drop sizes is the enhance scattering or spreading in the AA-R plots relative to the Ah-R plots at the lower rainfall rates. 4J Effects Of Changes In The Drop Orientation The impact of the changes in the drop orientation (fall orientation) on the estimated attenuation-rain rate relationship was studied by considering the drop orientations to be normally distributed about a mean polar canting angle of 0°, and varying the standard deviations of the canting angle distributions from 0° (no canting case), 5° and 10°. The effects o f varying the polar canting angle’s standard deviation from 0° to 10° on the Ah-R relationships was negligible as predicted by Fig. 3.5 (less than 0.5% difference in Ah relative to the non-canting case). The relative errors in Av-R and Aavg-R were similar to those obtained for the Ah-R relationships. The maximum error in the AA-R relationships due to the variations in the polar canting angle's standard deviation from 0° to 10° was on the average less than 9% and represented a constant bias over the entire range of rainfall rates. The effects of variations in the canting angle distribution on AA-R are shown in Fig. 4.5a,-c, and -e for the three distributions. The maximum error in the AA-R relationship due to drops having a 5° standard deviation in their polar canting angle distribution relative to drops that are not assumed to be canting was estimated to be less than 2.5%. For all subsequent calculations, we will assume the polar canting angles are normally distributed about zenith (0°) with a standard deviation of 5°. 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 TROP, 0 20 40 60 60 100 100 4 ......................... ......... .............. fa s w is w ^ E3 * ■ *-,.?■ .* * ! ■ —3 •' | • '• -•* •' 2 *1 d 20 40 60 80 100 40 60 60 100 0 20 40 60 80 100 4 10' ■o £ 1 0 0 MISS 20 R(mm/h) 40 60 R(mm/h) 100 Fig. 4.5 The effects of variations in the fall orientation (canting) and drop shape on the AA-R scatter plots 4.4 Effects Of Variations In Drop Shape The sensitivity of the A-R relationships to variations in the drop shapes (drop oscillations) were estimated by comparing attenuations calculated for the equilibrium axial ratio model BC«, with those of the average axial ratio models denoted as ABLav and KaV. The effects of changes in the drop shape on the Ah-, Av-, and Aavg-R relationships were observed to be negligible (less than 2%). On the other hand, the AA-R relationships were more sensitive to these changes as can be seen from Fig. 4.5b, -d, and -f. There were many instances where the relative change in the AA values computed for the Kav model shapes to those given by the BC*, shapes were in excess of 100%. These occurred at very low rainfall rates (R < 3.0 mm/h), with spectra almost entirely comprised of the very smallest drops. From Fig. 4.4 it is clear that for the smaller drop sizes (D < 1.0 mm), the A<Te of the shapes are significantly higher than that of the BC«, shapes over the same range. Consequently, for spectra dominated by these smaller drops the AA computed for the K,v shapes would be higher than those calculated for the BC«, shapes. It should be noted that there were less o f these instances in the case o f the TROP results as 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. compared to that o f the SWISS and MISS distributions. This is because the TROP DSD, being more congregated than the other two DSDs had fewer of these spectra with such small drop sizes. The change in the Kav AA values relative to those of the BCeq values decreased with increasing rainfall rates. For 5 < R < 25 mm/h the change in the ABLav model estimated AA values relative to those of the BCeq model shapes exceeded those of the Kav shapes (approximately 40% relative to that of the BCeq shapes). There was a significant reduction in these differences at the higher rainfall rates. For R > 25 mm/h the relative differences estimated from the ABLJVand Kav shapes were very similar, =20% for the TROP and MISS distributions and =15% for the SWISS data. The gradual reduction in these differences at the higher rainfall rates is due to these two shape models (BCeq and ABLav) converging for larger drop sizes (see Fig. 2.1). Conversely, the higher errors observed at the lower-medium range rain rates were due to the protrusion in the axial ratio plot of the ABLav shapes in the 1.0 < D < 1.5mm size range. For this model, these drops are considered to be in a state of resonant oscillation (Andsager et al., 1999). Since a large percentage o f drops are present at these sizes this tend to cause an amplification of this effect in the integral parameter. The fact that for the SWISS results these differences were less than those of the TROP and MISS distributions may not be that significant when one considers that a smaller percentage of the SWISS measurements had rainfall rates R > 25 mm/h. If the complete description of the rainfall process (the DSD, drop shapes, fall orientation and terminal velocities) along a path is available then the attenuation along that path will be known. However, because this data is almost never available, it becomes necessary to derive models that can provide the best possible estimates of R given the data available. We have thus far been able to estimate the rainfall rates and attenuations from disdrometer measurements of naturally occurring DSDs. The scatter plots of these results on a log-log scale suggested a linear relationship between the logarithm o f rainfall rate and the logarithm o f attenuations. Therefore power law models were fitted to the data. Table 4.2 shows the fits that describe the A-R relationships for the different distributions. Also included with these fits are the interval estimates of the coefficients (cta and ob). These interval estimates give the values of the coefficients within the 95% Cl (confidence interval e.g. a ± <ra and b ± o b). Note that the interval estimates of the different models (for the different distributions) do not overlap. This indicates that there are statistical differences between the fits (models). The models were fitted using the nonlinear Newton Raphson iterative technique also referred to as the Gauss-Newton Method (Neter et al.. 1996; see Appendix C). This technique uses a direct numerical search procedure for finding the fits that minimizes the perpendicular distance between the expected and 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. estimated values. The standard Linear Least/Minimum Squared Error (LLSE) techniques minimize only the vertical distance between the data point and the fitted line. This results from the assumption, implicit in this technique, that the error to be minimized is contained entirely in the dependent variable (variable on the y-axis), and that the independent variable (x-axis) is error free. There are, however, errors in the disdrometer measurements o f the DSD N(D)dD (Tokay and Short, 1996; Nystuen, 1999, Tokay et al. 2001). Consequently, both A and R will also contain errors (see equations 3.5 and 3.12). By minimizing the perpendicular distance between the data and the fitted line (usually referred to as the error in the fitting process), this technique effectively considers, and simultaneously reduces, the errors in both variables. The advantage of this method, compared to the more conventional LLSE routine, is that the relationships can be inverted analytically to give relationships similar to those obtained directly when the independent and dependent variable have been interchanged (compare the results of Tables 4.2 and Table 4.3). The same is not always true when the standard LLSE technique is used. This is due, in part, to the fact that the errors reduced by the LLSE routine differ depending on which parameter is chosen as the dependent or independent variable. In addition, the linearization procedure (e.g. the logarithm) can significantly 'alter the errors in both the dependant and independent variables. Consequently, the errors being reduced may not be that of the original data. Another problem with the LLSE technique is its tendency to be biased by the number and distribution of the points (data) being fitted. This would result in a good (or better) fit for only those regions where the majority of the data is located. This nonlinear technique on the other hand tended to give a better fit over the entire range of data points. 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.2 Coefficients and exponents of the power iaw models of the microwave attenuations as functions of the rainfall rate R. The 95% Cl of the fitted relationships are given as a ± P a and b ± gb ABL1V BC„ Rel. TROP Ah-R 0.2436 1.0444 0.0013 0.0016 0.2417 1.0433 0.0013 0.0015 0.2420 1.0423 0.0013 0.0015 A,-R 0.2312 1.0181 0.0011 0.0014 0.2333 1.0220 0.0012 0.0014 0.2324 1.0246 0.0012 0.0014 AA-R 0.0160 1.2477 0.0004 0.0060 0.0114 1.2662 0.0003 0.0065 0.0119 1.2371 0.0002 0.0058 Aavg-R 0.2373 1.0321 0.0021 0.0015 0.2374 1.0331 0.0012 0.0015 0.2372 1.0338 0.0012 0.0015 Ah-R 0.2672 0.9726 0.0013 0.0016 0.2647 0.9722 0.0013 0.0016 0.2650 0.9710 0.0013 0.0016 A„-R 0.2481 0.9458 0.0012 0.0017 0.2516 0.9482 0.0013 0.0017 0.2510 0.9516 0.0012 0.0017 AA-R 0.0222 1.1492 0.0002 0.0029 0.0160 1.1816 0.0002 0.0029 0.0160 1.1532 0.0001 0.0026 0.2576 0.9601 0.0012 0.0016 0.2581 0.9608 0.0013 0.0016 0.2579 0.9618 0.0012 0.0016 Ah-R 0.2731 0.9815 0.0011 0.0010 0.2702 0.9813 0.0011 0.0010 0.2703 0.9804 0.0011 0.0010 A,-R 0.2494 0.9602 0.0011 0.0011 0.2532 0.9627 0.0011 0.0011 0.2530 0.9654 0.00(1 0.0011 AA-R 0.0272 1.1001 0.0001 0.0012 0.0204 1.1155 0.0001 0.0012 0.0197 1.1002 0.0001 0.0011 Aavg-R 0.2611 0.9717 0.0011 0.0010 0.2616 0.9726 0.0011 0.0010 0.2616 0.9734 0.0011 0.0010 SWISS r < MISS OC I Dist. a b o. Ob a b o« CTb a b a* ob The exponent “b” gives the gradient of these relationships on a log-log plot. If the relationship between the parameters were in deed linear then we would expect the exponent “b” to be 1. It is evident from the results provided in Table 4.2 that none of the above relationships between the rainfall rate and the microwave attenuation parameters at 35 GHz are exactly linear. However, it is apparent that, for the most part, these relationships are very close to being linear and that some parameters are more linear than others. On the other hand it is clear that the effects of changes in drop shape on Ah, Av, and Aavg is not significant since the 95% Cl of their exponents ‘b’ all overlap. The AA-R relationships are shown to be the least linear. This is especially true for the TROP data where 1.23 < b < 1.27 depending on the drop shape. For the TROP data the Av parameter has the most linear relationship with respect to R, while the Ah parameter tended to be more linear for the SWISS and MISS data. Note that the exponent b = 1.0444 of the Ah-R TROP model is very close to that obtained by Atlas and Ulbrich (1977) for a MP distribution of spherical drop shapes. For a given distribution, the Aavg models show the least variation in their coefficients for the different drop shape models. This suggests that the Aavg parameter is the least sensitive to variations in drop shape. Figure 4.6 shows the power law fits to the A-R models of the three distributions where the equilibrium drop shapes have been assumed with 5° standard deviation in canting angle distribution. Also included 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. are the relationship Ah = 0.219/?104 given by Atlas and Ulbrich (1977) A-U, for a Marshall-Palmer (MP) distribution o f spherical drops, along with AA = 0.0287/?, which they derived from th e calculations o f Oguchi and Hosaya (1974). Olsen et al. (1978) provided other relationships for different DSDs, and raindrop temperatures. It is interesting to note that for a given rain rate, the TROP relationships for Ah, Av, and A^g consistently have higher attenuation values and consequently the steepest gradients. This is due to the fact that the TROP DSD is concentrated over the range o f sizes that contribute significantly to both A and R. Thus there is a stronger relationship (dependency) between these parameters for this distribution. It is for the opposite reason that the SWISS models have the lowest attenuations. It is clear that for R < 50 mm/h AA is insensitive to the DSD. From Fig. 3.2 of the previous chapter, we observed that for R > 50 mm/h AA became more sensitive to variations in the DSD shape parameter |X. In general the number of larger drops increases with rainfall rates, and the effects o f changes in drop shape are more pronounced in the larger drops as they undergo the largest deformation. It is therefore reasonable that the spread in AA increases with R in Fig. 4.6 as the effects of drop shape become more significant at higher rainfall rates. MISS - ■ SWISS — TROP A-U E I <* 100 40 R(mm/h) 100 R[mm/h) 80 100 00 100 R(mm/h) Fig. 4.6 The A-R power law fits for the different DSDs. Results provided by Atlas and Ulbrich (1977) A-U for a MP distribution of spherical drops are also included. 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Note the behavior of the A-U models. They are very close to the MISS models for R < 50 nun/h; overestimating Ah and underestimating AA for higher rainfall rates. It is questionable whether or not this model is valid at the higher rainfall rates, for the simple reason that the MP DSD may not be able to generate the critical number of drops in the intermediate drop size ranges that are so important to both A and R, at these higher rainfall rates. Similar models were derived for the rainfall rate regressed against the attenuations. The fits to these models are given in Table 4.3. It may be observed that the fits to R-A models are not exactly that which one would obtain from directly inverting the corresponding A-R relationships. This is due to the inability of the regression routine to reduce the errors between the non-linear model and the simulated results, to zero. On the other hand, the fact that these models, on inversion, give results very close to the direct regression models is a testimony to the general robustness of the Newton-Raphson iterative technique over the more general linear least square error (LLSE) procedures (see Appendix C). Table 4 J Coefficients and exponents of the power law models o f the rainfall rate R as a function o f the microwave attenuations. BC* ABU Ka* Dist. Rel. a b a b a b CTa <Jb Ob Oa <Ta Ob TROP R-An 3.8776 0.9545 0.0124 0.0014 3.9132 0.9556 0.0123 0.0014 3.9133 0.9565 0.0122 0.0014 R-A, 4.2396 0.9778 0.0122 0.0014 4.1786 0.9743 0.0121 0.0014 4.1777 0.9720 0.0121 0.0014 R-AA 26.7332 0.8113 0.1009 0.0039 33.1477 0.8018 0.1348 0.0041 35.1714 0.8175 0.1305 R-Aaij 4.0486 0.9654 0.0120 0.0014 4.0404 0.9645 0.0120 0.0014 4.0402 0.9639 00120 SWISS R-Ah 3.8937 1.0164 0.0126 0.0018 3.9329 1.0170 0.0125 0.0018 3.9356 1.0182 0.0126 R-A, 4.3634 1.0439 0.0148 0.0021 4.2865 1.0407 0.0147 0.0021 4.2756 1.0375 0.0144 R-AA 26.4209 0.8508 0.0832 0.0025 31.7483 0.8281 0.1091 0.0025 34.6410 0.8499 0.1122 R-Aavg 4.1122 1.0292 0.0135 0.0019 4.1000 1.0283 0.0134 0.0019 4.0969 1.0274 0.0133 MISS R-Ah 3.7788 1.0132 0.0107 0.0011 3.8198 1.0135 0.0106 0.0011 3.8248 1.0143 0.0107 R-A, 4.2849 1.0336 0.0132 0.0012 4.2037 1.0309 0.0130 0.0012 4.1881 1.0284 0.0126 R-AA 26.2312 0.9092 0.0349 0.0012 32.4551 0.8960 0.0399 0.0012 35.2249 0.9083 0.0380 R-Amg 4.0141 1.0226 0.0118 0.0011 4.0013 1.0216 0.0117 0.0011 3.9973 1.0209 0.0116 0.0037 0.0014 0.0018 0.0021 0.0023 0.0019 0.0011 0.0012 0.0011 0.0011 Figure 4.7a shows the scatter plots for the attenuations estimated for equilibrium shaped drops (BC,*,) versus the estimated rainfall rates. Also included are the models describing these relationships with error bars showing the confidence level of the models relative to the data. The first error bar is position at the R = 5 mm/h mark with subsequent bars position at almost 10 mm/h intervals. The error bars indicate the 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. confidence o f the relationships to within one standard deviation on either side of the fitted curves for R ± 5 mm/h. It is clear that the TROP distribution gave the best fits for the R-Ah, R-Av, and R-A.vg models. The data was generally within two standard deviations o f the fitted curves, for R < 55 mm/h. For R > 55 mm/h there were larger deviations in the data about the curves. The R-AA model gave an improved fit the data at R > 55 mm/h where the data was generally within one standard deviation o f the model. For lower rainfall rates (R < 55 mm/h) there was considerably higher scatter in the data. The SWISS models were not as well fitted to the data. The R-AA model fitted the data poorly for R > 50 mm/h, and for lower rainfall rates (R < 50 mm/h) the spread generally exceeded one standard deviation on both sides o f the curve. There is a significant reduction in the scatter for 30 < R <50 mm/h. The RAh model gave a better fit to the data for R < 10 mm/h. The R-Av and -A avg models gave similar results to that of the R-Ah model. 25 TROP I 20 15 .10 50 100 50 100 4 100 50 100 50 100 50 100 25 SWISS 50 50 20 100 50 100 4 MISS 25 25 20 20 15 .10 100 0 50 R(mm/h) 100 50 R(mrrVh) 100 100 Fig. 4.7a Scatter plots of the Attenuations estimated from equilibrium shaped drops as functions o f the rainfall rates. The error bars give the scatter in the data to within one standard deviation of the fitted relationships given in Table 4.2. 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The MISS distribution gave the best fit o f ail the R-AA models. The data was relatively close to within two standard deviations about the model for the entire range R < 100 mm/h. As in the case o f the SWISS distribution the spread in the MISS data was within two standard deviation o f the R-Ah, -Av, and Aavg models for R < 10 mm/h. For R > 10 mm/h there is an increase in the scatter about the models, however the spread was significantly less than that observed in the SWISS data. Figure 4.7b shows the results obtained when the variable in Fig. 4.7a were interchanged. The error bars give the scatter in the disdrometer derived R about the attenuation estimated rainfall rates. 100 100 100 100 m-Z'■ .-A so so 50 / ! •*2 : 50 Jf^TPOP 10 20 0^F T ........... 0 r ._ 100 100 10 20 oJ 10 20 100 100 50 SWISS 0 10 20 100 100 w / r 50 / 0 10 20 A^dB/km) 0 2 AA(dB/km) 4 0 10 20 A^dB/km) 10 20 fdB/km) ■VH A Fig. 4.7b Scatter plots of the disdrometer estimated rainfall rates as functions o f the attenuations estimated from equilibrium shaped drops. The error bars give the scatter in the data to within one standard deviation o f the fitted relationships given in Table 4.3. The R-Ah and R-AA relationships derived for the MISS distribution were compared with those obtained from the SWISS and TROP DSDs. The difference between R miss- Ah and Rswiss- Ah was less than 5%, while the difference in R miss- Ah and R trop- Ah was less than 10% for Ah < 5dB/km (or R < 20 mm/h) and less than 16% for Ah < 25dB/km (or R < 100 mm/h). The difference between R miss- AA and RswissAA was less than 15% for AA < 1.25dB/km (R < 30 mm/h) and less than 7% for 1.25 < AA < 4dB/km 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (30 < R < 100 mm/h), while the difference in R miss- AA and R trop- AA was less than 28% for AA < 1.25dB/km (R < 30 mm/h) and less than 11% for 1.25 < AA < 4dB/km (30 < R < 100 mm/h). The differences with respect to R-Av and R-Aavg were between 1-2% higher than those of R-Ah, with R-Av given the larger of the two differences. In each case the R trop underestimated the rainfall rates o f the other two models, with RSwiss producing the highest rainfall rates in the case of the R-Ah, R-Av, and RAavg estimates. In general the SWISS and MISS model results were more similar to each other and quite different to those of the TROP distribution. 4.5 Error Analysis O f The Rainfall Estimation Models The errors in the rainfall rates estimated from the above models were assessed relative to those estimated directly from the disdrometer data. This serves as a dual test of the effectiveness of the R-A models in estimating rainfall, while at the same time providing some quantitative measure of the sensitivity of these models to the effects of differences in the shapes of the raindrops. Two figures of merit, namely, 1) the fractional standard error FSE, and 2) the normalized bias NB, are computed to compare the errors in the different rainfall models. The fractional standard error is defined as FSE = 2 = Rn (4.2) i=i where RDt and RAi are the corresponding rainfall rates estimated from the disdrometer measurements and the R-A models respectively, and RD is the mean rainfall rate. Here we are assuming that the disdrometer derived rainfall rates (RD) represents the “true” or expected values, and as such may be conveniently referred to as the ‘data’. Consequently, the FSE provides a measure of the spread in the data about the model-derived estimates o f the rainfall rates similar to the standard deviation, but is normalized to the mean values of the rainfall rates. Thus the FSE provides some measure of the appropriateness of these models. The normalized bias is given as follows. 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. NB compares the mean values of the model-derived rainfall rates to those estimated directly from the disdrometer data. The values of NB therefore represent the bias in RA, such that negative values of NB indicate that RAover estimates the rainfall rate and vice versa. The errors are expressed over different ranges o f rainfall rates as described in Table 4.4. Also included in Table 4.4 are the numbers of individual spectra with rainfall rates in the different ranges, along with the mean rainfall rate in each range, as determined from the disdrometer measurements. Table 4.4 The ranges of rainfall rates over which the models are to be evaluated. Also given are the numbers o f 1-minute spectra with rainfall intensities in these ranges and the mean rainfall intensity over R(mm/h) TROP Ranges SWISS # spect. Rd 0.1<R<1 1<R< 5 5<R<10 10<R<15 15<R<20 2CkR<25 25<R<30 3CkR<35 35<R<40 R>40 0.44 2.53 6.89 12.26 17.37 22.44 27.24 32.41 37.65 50.81 Ro 3494 3305 1123 307 171 no 74 52 51 113 0.43 2.25 6.70 12.12 17.10 22.65 27.47 32.48 37.21 58.69 MISS # spect. 10061 6585 938 215 90 56 32 18 11 19 Rd 0.43 2.45 6.91 11.99 17.19 22.23 27.49 32.29 37.42 67.27 # spect. 12627 12142 3541 1062 494 281 181 147 93 472 Due to the ‘wide’ distribution of rainfall rates with the largest percentage of measurements having R < 10 mm/h it was convenient to use two-piece models for the SWISS and MISS distributions (in the case of the SWISS and MISS distributions > 90% of the 1-minute spectra had R < 10 mm/h). These fits have been included in Appendix D. There was a 50% reduction in the NB and FSE of the estimated rainfall rates for the SWISS and MISS distributions for RD < 20 mm/h when the two-piece fits were used for the R-Ah, R-Av, and R-Aavg models. No significant reductions were observed in the R-AA estimated models. Graphical 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. representations o f these errors are given in Fig. 4.8 and Fig. 4.9. Figure 4.8 shows the normalized bias in the rainfall rates calculated from the relationships given in Appendix D. 0 .2 0.1 0.1 0 .1 0.05 0.05 ITROP 0.05 *QC ---i a. s •0.05 •0.05 • 0.1 •0.1 60 • 0 .2 20 40 60 • 0.1 60 20 • 0 .1 0 .2 S W IS S 0.05 I> %e. 0.05 0.05 S 1 •0.05 - 0.1 -0.05 • 60 20 0.2 20 0.2 40 60 • 0 .1 20 40 60 • 0 .1 40 60 6C M ISS A6L 0.05 0.05 0.05 •0.05 •0.05 % ac $ •0.05 •0 .1 • 0.1 20 40 R (m m /h) 60 • 0 .2 20 40 R(mm/h) 60 • 0.1 20 60 - 0. 1 R( m m / h ) 20 40 R(m m/h) Fig. 4.8 Normalized bias in the R-A models relative to the disdrometer estimated rainfall rates The R-Ah models generally gave smaller biases over the lower rainfall rates than the R-AA, R-Av, and RAavg models. For the TROP data the mean rainfall rates were estimated to within 2% of their expected values. In the case o f the SWISS and MISS data, the rainfall rates were estimated within 6% and 3% respectively. As expected these values were unaffected by the assumed drop shape. The TROP models tended to underestimate the rainfall rates for R < 20 mm/h. The SWISS models were the worst over the mid-to-high end ranges of the rainfall rates. The relatively poor performance of the SWISS models over the higher ranges o f R is partly due to the small number of data points available at these values (see Table 4.4). The mean biases in the R-Ah models were 0.9%, 2.3%, and 1.5% for the TROP, SWISS, and MISS data respectively. The R-AA models were generally able to estimate rainfall to within 10%. The MISS and TROP models performed better than the SWISS models for R > 30 mm/h. The bias in the TROP R-AA estimates S8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. decreased for R> 20 mm/h, while the errors in the SWISS estimates increased for R > 40 mm/h. These estimates were more noticeably affected by changes in the drop shapes. This effect was strongest in the case o f the TROP data where bias increases o f 2-5% between the BC^, and Km, shapes were observed at the lowest rainfall rates. The average bias in the estimated Raa were within 6%, 4%, and 4% for the TROP, SWISS, and MISS data respectively. In general the R-Av and R-A«vg models performed well (less than 4% bias) for the TROP data. There was a tendency for the RAv to over estimate R at the lower rainfall rates. The R ^ g models gave lower biases than RAv especially at the lower rainfall rates ( R < 10 m m / h ). The performances o f the R-Av and R-A,vg models were similar to that of the R-Ah models. o.a 0.15 0.15 0.15 0.05 0.05 0.6 % x e 0.4 UJ ut CO Ik n u. 0.05 0.2 TROP 40 20 20 60 40 60 40 20 60 20 40 60 20 40 60 20 40 60 0.8 1 ____ _____ L ________ _ L 0.15 - - 0.15 0.6 \ / * 11 \ $ ucoi u. \ * 0.4 / 0.05 0.05 0.2 SWIS^ 20 20 40 40 60 0.8 0.15 0.15 0.15 0.05 0.05 0.6 c< UJ (IkO ? « 0.4 UJ 0) tk 0.05 0.2 MISS 20 40 R(mm/h) 60 20 40 R(mm/h) 60 20 40 60 R( m m/ h) R( m m/ h) Fig. 4.9 The Fractional Standard Errors in the R-A models relative to the disdrometer derived estimates of the rainfall rate. The FSE estimates the scatter (spread) in the data (R0) relative to the model-derived estimates (RA). These results indicate that the TROP models were generally better fitted than those o f the other distribution as the R-Ah, R-Av, and R-Aovg models for this distribution consistently gave lower errors than 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. those of the SWISS and MISS models. This is consistent with the TROP DSD being concentrated around the sizes that contribute most to both A and R. The SWISS and MISS have “wider” DSDs with the result that there are a large number of drops whose weighted contributions to the rainfall and attenuation rates are different. The spread in the R-Av estimates of the TROP distribution was slightly less than that of the R-Ah estimates for R < 1 mm/h. For higher rainfall rates the R-Ah, R-Av, and R-Aavg estimates were all comparable (<5% for R > 20 mm/h). For the SWISS and MISS distributions the spread was in excess of 5% with the SWISS distribution having the higher scatter. The largest errors were obtained from the R-AA estimates. O f these the R-AA models generated from the TROP distributions were the highest. This is consistent with the fact that a large percentage of the TROP drops were of the range of sizes for which the differential extinction cross-sections increased in non-linear manner with the equivalent volume diameter. However, for R > 20 mm/h there is a significant reduction in these errors. The errors resulting from ignoring the presence of drop oscillation by assuming that the drops all have equilibrium shapes while they may indeed have been in a state of oscillation were also estimated. These were estimated by applying the relationships derived from the BCeq shapes to the attenuations simulated from the DSDs when the ABLav and Kav shapes were assumed. The bias in R-Ah and R-Av estimates increase by less than 1% over the previous values while those of R-Aavg remained unaffected (i.e. they were very similar to the results of Fig. 4.8) This is in agreement with the notion that R-Aavg is insensitive to changes in drop shape. As expected the Raa estimates were most strongly affected with Raa consistently over estimating R. The bias in Raa are 10-20% higher than those observed in Fig. 4.8. The estimates obtained with the Kav shapes resulted in a =5% higher bias than those of the ABLav shapes with the R-AA models. The Kav shapes also gave higher biases in the R-Ah and R-AA models while the ABLav shapes gave higher biases with the R-Av and R-Aavg models. The TROP models showed the greatest sensitivity to difference in drop shapes, while the SWISS and MISS models were less affected. This is because these distributions had a much larger percentage of the smaller drops (i.e. D < 1.0 mm) that are spherical or have axial ratios (shapes) that are similar for the different models. This is also related to the fact that these smaller drops (D < 1.0 mm) are not as significant to R and A as the larger drops. For R < 5 mm/h there is a 10% decrease in the FSE of the R-AA estimates, and =10% increase in the FSE at the higher rates. This is due to the fact that the number of larger drops which suffer greater deformation, increase with rainfall rate. 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This thus provides further proof that AA is very sensitive to the assumed drop shape. These shapes did not significantly affect the FSE for R-Ah, R-Av, and R-A„g. 4.6 Estimating Rainfall Accumulations The performance o f these models in the estimation o f rain accumulations has been assessed. Individual rain events were identified from the data such that there was a minimum 5-minute separation in the time sequence o f consecutively recorded measurements, and that each event had a total accumulation o f at least 1 mm. In this manner the TROP, SWISS, and MISS data sets produced 76, 103, and 227 events, respectively. The rainfall accumulations indicated as I Ah, £ av, Eaa, and X/vavg, are obtained from the RAh, R-Av, R-AA, and R-AaVg models respectively. The accumulations obtained directly from the respective disdrometer data are referred to as ZD. The scatter plots of the accumulations estimated for the three DSDs are plotted against their expected values I Dfor the respective distributions in Fig. 4.10. 80 70 30 70 - TROP 60 - MISS 25 - SW ISS 60 50 20 50 40 E 15 E 40 c~«e i 30 JC 30 10 20 20 10 10 20 40 £ D(m m ) 60 80 10 20 30 0 10 20 30 40 50 60 70 r D(m m ) Fig. 4.10 Accumulations estimated from the R-A models as functions o f the disdrometer derived accumulations 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The accumulations shown in Fig. 4.10 were estimated assuming BC^, drop shapes. The bias and error statistics on these accumulations are given in Tables 4.5a, b, and c. There were no significant differences in the statistics of the ZAv, and Z ^ g , accumulations resulting from changes in the assumed drop- shapes; hence these results have not been included in Tables 4.5 or Fig. 4.10. Table 4.5 gives the NB and FSE o f the accumulations estimated from the R-A models relative to those estimated directly from the disdrometer data. The results are given for events with total accumulations less than 5 mm and those with accumulations exceeding 5 mm. Also shown, are the mean accumulations over these ranges. The largest errors were obtained for Z^a for events with less than 5 mm accumulation. It is not at present clear what type of rain events were mainly responsible for these errors as these events appear to be a mix of both stratiform and convective showers (based on the duration o f the events as observed from the disdrometer data). The largest errors in Z^a for an assumed drop shape were obtained from the TROP spectra. This is consistent with the earlier results showing the R-AA scatter plots and the R-AA model for this distribution to have the largest scatter. For the TROP data the FSE in Zaa is roughly 3-times as high as that o f Z,ui, while the FSE in R-AA is 4times as high as that of R-Ah. This suggests a reduction in the errors involved in the rainfall estimation process when the models are used to estimate the rainfall accumulations. This would be expected since the estimation of the accumulation involves the integration/averaging of the rainfall rates (and their errors) over time. Similar differences can be observed in the accumulation and the rainfall estimated obtained from the SWISS and MISS distributions. The TROP and SWISS distributions have a similar percentage of their events with accumulations 1 < Zd > 5.0 mm (67% and 68% respectively) and ZD > 5.0 mm (33% and 32% respectively). However, the SWISS distribution had 36% more events than that observed from the TROP data. The MISS events were more evenly distributed with 56% having I < ZD ^ 5.0 mm and the remaining 44% with ZD > 5.0 mm. Nevertheless, there were almost 3-times as many events in the MISS data as were recorded in the TROP distribution. Furthermore, many of the events for the different distributions were observed to be of similar durations. However, the SWISS data had the lowest accumulations while the MISS distribution recorded the highest accumulations as can be seen in Fig. 4.10. It would therefore be difficult to have a meaningful comparison between the errors and biases in the accumulations of these three distributions for each ZA. 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.5a Statistical figures of merits for the accumulations estimated from the power law R-A Model fits to the TROP distributions z * £ AA U TROP Range NB BCu FSE ABU . NB FSE K„ BCu NB FSE NB BC, FSE NB BCu FSE NB FSE I<£p<5mm 0.0171 0.2896 0.0151 0.3027 0.0066 0.2713 0.0134 0.0859 0.0190 0.0680 0.0160 0.0766 Z D > 5mm 0.0136 0.1554 0.0159 0.1656 0.0084 0.1468 -0.0002 0.0350 0.0019 0.0232 0.0007 0.0286 £ 0(m/n) events 2.48 51 19.23 25 Table 4.5b Statistical figures of merits for the accumulations estimated from the power law R-A Model fits to the SWISS distributions £ AA SWISS Range BC, NB FSE ABU. NB FSE NB z* X * BC„ K.v FSE NB BCu FSE 0.0660 0.2062 0.0666 0.2108 0.0559 0.1890 0.0039 NB BCu FSE 0.0592 -0.0041 0.0520 NB FSE 0.0002 0.0564 1<Sd <5 m m £0 > 5mm 0.0101 0.1234 0.0110 0.1245 0.0052 0.1135 -0.0043 0.0511 -0.0020 0.0527 -0.0034 0.0509 events 2.29 70 9.86 33 X AM MISS Range BCu NB FSE ABU. NB FSE 1C NB FSE NB M * Table 4.5c Statistical figures of merits for the accumulations estimated from the power law R-A Model fits to the MISS distributions U BC, BC„, FSE NB Z m„ , BC. FSE NB FSE l<I„<5m m 0.0276 0.2326 0.0193 0.2408 0.0176 0.2160 0.0181 0.0793 0.0173 0.0781 0.0178 0.0773 Z 0 > 5mm 0.0083 0.0654 0.0104 0.0717 0.0064 0.0626 -0.0063 0.0334 -0.0057 0.0396 -0.0061 0.0359 Z g (m m ) events 2.30 128 18.10 99 Again the errors incurred by ignoring changes in drop shape due to drop oscillation when in fact such changes are occurring, were estimated by applying the R-A relationships based on the BCcq model to the attenuations simulated from the DSDs before estimating the rainfall accumulations. No significant changes were observed in I Ah, SAv, and I Aavg. There were, however, some changes in values of ZaaThese results are given in Table 4.6 (the results for XAh, XAv, and ZAavgwere not given due to their relative insensitivity to changes in drop shape). The FSE in I^ a increased by as much as 24% when the effects of oscillation are ignored. This error increased for events with higher accumulations (the FSE in the I^ a of the MISS distribution increased from 6.3% to 29.4% for ZD> 5 mm). In general Zaa underestimated the accumulation by approximately 20% if oscillation effects are ignored. 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Tables 4.6 Errors associated with the accumulations estimated from the power law R-A Model fits when drop oscillations have been ignored. The accumulations were estimated from attenuations calculated for TROP SWISS MISS BC„ -K»v BCm- ABLav BC„ Kfv BCm- ABLav BC„- ABU, BC„ - K r NB NB FSE NB FSE FSE FSE NB FSE NB FSE NB 1mm<1^ < 5mm 0.2199 0.3296 0.2359 0.3251 0.2323 0.3034 0.2332 0.2967 0.1988 0.2873 0.2214 0.2971 0.2152 0.3296 0.2405 0.3511 0.1817 0.2279 0.2039 0.2433 0.1898 0.2392 0.2351 0.2938 I D>5mm Accumulations Range 4.7 Identifying The Presence O f Raindrop Oscillation It has been noted that the parameter AA is very sensitive to the changes in drop shape and hence drop oscillation, while Ab, Av and AaVg are not significantly affected by these changes but are nevertheless sensitive to changes in DSD. It may therefore be possible to use combinations o f AA with Ab, Av and Aavg to determine the presence o f drop oscillations and/or changes in DSD. Figure 4.11 shows the scatter plots for Ab-AA comparing the BC,,,, ABLav, and K«v models o f the MISS data (the other two data sets are not shown, however they exhibit similar behavior to that being presented). 50 45 M ISS 35 30 20 10 Fig. 4.11 Simulated results of Ah as a function of AA for different drop shape models. The results show distinct signatures depending on whether or not drop oscillations have been assumed. 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Despite the high degree o f scatter in the Ah-AA scatter plot and the resulting overlap o f the models at low values o f Ah and AA, it is evident that there are distinct trends in the plot depending on whether or not an oscillation shape has been assumed. The power law fits describing the relationships between these two parameters for the different drop shapes and distributions have been provided in Table 4.7. The coefficients given in Table 4.7 are fits to the model Ah = a A A b (4.4) where of Ah and AA are given in units of (dB/km). T able 4.7 Power law fits to Ah as a function o f AA for different drop shapes and DSD D iitrib. MISS SWISS TROP a 6.7811 6.5091 7.5967 BC„ b o. 0.8887 0.0092 0.8212 0.0180 0.8441 0.0235 at 0.0012 0.0021 0.0033 a 8.2497 7.6484 9.3861 ABU, b a* 0.8734 0.0119 0.7964 0.0255 0.8324 0.0317 Ob 0.0014 0.0023 0.0036 a 8.9105 8.3137 9.9542 K« b a. 0.8865 0.0109 0.8190 0.0254 0.8487 0.0302 Ob 0.0012 0.0021 0.0032 The curves for these relationships for the MISS models are shown in Fig. 4.12. Note that these curves show obvious distinctions that may be used to determine the presence o f oscillations provided enough data is available so that reliable trends can be established for the comparison with these models. 45 4G M ISS 35 30 15 10 AA(dB/lcm) Fig. 4.12 Model relationships for Ah as a function of AA for different drop shape models. 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It is evident from these results that the oscillation shapes have lower values of AA than the equilibrium shapes. This is because the oscillation models have larger axial ratios than the equilibrium shapes as the oscillations tend to increase the sphericity of the drops. The difference in the attenuations of the V- and H- polarizations for a spherical drop will be less than that of a flattened (horizontally) or elongated (vertically) drop shape. Hence the differential attenuation for the oscillating (more spherical) drops will be less than that of an equilibrium shaped drop. 4.8 Conclusions The extinction cross-sections (Teh(D) and Aoe(D) (which are related to the specific and differential attenuations respectively) at 35 GHz and the drop momentum F(D) = vl( D) D3 (which is related to the rainfall rate) were represented as power law functions a D bo f the equivalent volume spherical diameters D. The power ‘b’ for oeh was closer to that o f F(D) for D < 2.4 mm while the power of Aoe(D) was closer to that of F(D) for D > 2.4 mm. These differences in the powers, when modulated by the different DSDs, lead to different sensitivities in the attenuation-rainfall rate relationships at 35 GHz. Raindrop size distributions from three different geographical locations (Tropical rainfall DSD TROP, Sub-Tropical rainfall DSD MISS, continental rainfall DSD SWISS) were used to obtain relationships between the rainfall rate R, the specific attenuation (Ah, Av, and Aavg), and the specific differential attenuation AA at 35 GHz. Estimates of rainfall rate R had less scatter when obtained from Ah, Av, or Aavg, and more scatter when obtained from AA in rain events dominated by smaller drops (< 2.4 mm) such as light widespread rainfall. The opposite was true for rain with larger drops (especially heavy convective events) where the estimates of R from AA had less scatter. The specific (Ah and Av) and average attenuations (Aavg) were shown to be insensitive to changes in drop shape and fall orientations (drop canting). However, as expected, the differential attenuation was shown to be quite sensitive to variations in both drop shape and fall orientations. The performances of these R-A relationships were evaluated in terms of the normalized bias (NB) and fractional standard error (FSE) statistics in Fig. 4.8 and Fig. 4.9. Estimates of the rainfall accumulations (ZA) derived from these relationships were also assessed. In general I aa (accumulation estimated using 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. simulated AA) underestimated the rainfall accumulations to within 7% with less than 29% FSE. On the other hand Eaii lead to less than 2% underestimation with less than 10% FSE. The combination of AA with Ah (Av or Aavg may also be used in place of Ah) is proposed for determining the mean drop shape due to the presence of drop-osciilations and canting. This is a result of the differences in the sensitivities of AA and Ah to variations in the drop shapes. Distinct trends were observed in the AA-Ah scatter plots resulting from the oscillation drop shapes with lower values of AA than those obtained for the equilibrium shapes. These trends can be used to compare and or distinguish between equilibrium and non-equilibrium shaped raindrops. Power law fits to the AA-Ah relationships have been provided. 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTERS 35 GHz MICROWAVE ATTENUATION MEASUREMENTS IN RAIN 5.1 Background In Chapters 3 and 4 we discussed the theoretical basis for studying microwave attenuation at 35 GHz in rainfall. In Chapter 4 simulations for both rainfall rate and microwave attenuation from disdrometer derived DSDs were presented. In this Chapter we will discuss the results of an experiment to measure these relationships as we continue to study the role o f variation in DSD on microwave attenuation measurements at 35 GHz. Comparisons will be made between the experimental and theoretical results. In this experiment the two-way path attenuations of orthogonal linear polarizations of a 35 GHz microwave signal are measured using a monostatic radar and a fixed reflector target of known cross sectional area. This radar-reflector arrangement will be referred to as the Dual Polarization Propagation Link (DPPL). The DPPL was designed to provide path integrated rainfall measurements. The original intent was to estimate rainfall rate from the differential attenuation measurements of alternate linearly polarized signals at 35 GHz, reflected from a comer reflector over a predetermined path. 5.2 Dual Polarization Propagation Link (DPPL) The DPPL was designed and built by faculty and students at the Pennsylvania State University. A detailed account o f the hardware and its operational characteristics can be found in Mathur (1994) and Ruf et al. (1996). The DPPL is essentially a portable monostatic pulsed radar that reflects alternate linearly polarized signals from a comer reflector. The co-located transmitter and receiver hardware is assembled on an aluminum chassis. The transmitter is mounted in such a manner as to afford a limited amount of adjustment in both the azimuth and elevation directions. The instrument is housed in a 100cm x 50-cm x 75-cm aluminum casing with temperature control electronics included. Table 5.1 provides a list of the radar’s operating characteristics. 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 5.1 Operating characteristics of the 35 GHz Dual Polarization Propagation Link (DPPL) Operating Frequency (GHz) Transmitted Power (dBm) Antenna Type (Tx & Rx) Tx & Rx Antenna Gain (dBi) Antenna HPBW (deg.) Polarization Polarization select settling time (ps) Polarization switching speed (Hz) Tx Pulse width (ns) Rx Noise Figure (dB) Rx pre-detection bandwidth (MHz) 35 21 30cm (12") Gaussian lens loaded homs 39 2.3 at V- and H-Pol. linear V and H, Selectable 80 2200 80-300, Selectable 21 10-150, Selectable A crystal controlled phase-locked Gunn Oscillator is used to generate the 35 GHz source signal at 23 dBm. This source is pulsed by a PIN-diode SPST switch with a 40 ns on-to-off and a 10-ns off-to-on settling time. These settling times effectively limit the pulse width o f the transmitted signal to a minimum pulse width of approximately 80 ns (Ruf et al., 1996). Polarization switching is provided by a Faraday rotation solenoid. The relatively large inductance of the solenoid, and the resulting large settling times limit the pulse repetition frequency (PRF) to approximately 2200 Hz. Similar polarization switching is performed at the receiver. The receiving antenna is similar to the transmitting antenna. Since the antennas are co-located, it was necessary to isolate them from each other. This was achieved by using aluminum plates inside the radar to separate them, and externally another aluminum plate is place between the two lenses. The receiver includes a 35 dB low-noise amplifier operating at 35 GHz, followed by a single stage 250 MHz IF down converter. This down conversion is achieved with a sixth-harmonic mixer pumped by a synthesized 5.79 GHz dielectric resonant oscillator (DRO). The DRO is temperature stabilized to ±2 °C to prevent frequency drifting. The temperature of the receiver is monitored with the aid of an ambient temperature probe inside the receiver housing, in addition to placing a thermistor in the DRO enclosure. Additional gain, followed by power detection, is provided by an IF subsystem. Additional electronic hardware is housed in a 51”x 22” rack enclosure. The rack contains a Stanford Research Systems (SRS) Gated Integrator and a Boxcar Averager with computer interfaces, a 486-class personal computer, and a DC power supply sub-system. The SRS, along with the Boxcar Averager, provide analog data sample-and-hold, A/D conversion, and digital data buffering. The 486 DX computer 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. provides overall control and data storage while the power supply provides DC power to the microwave devices of the radar. The transmitter/receiver unit along with the accompanying electronics and computer equipment is housed in a custom built tent. Two trihedral comer reflectors have been made available for use with the link. A 6" (15 cm) comer reflector (RCS = 29.4 m2 at 35 GHz), for use over short ranges on the order of a few hundred meters, and a 18”(46 cm) reflector (RCS = 2444 m: at 35 GHz), for longer distances. The larger reflector is fitted with custom-built mount while the smaller 6” reflector is usually used with a regular tripod. Both arrangements allow for adjustments in azimuth and elevation. Each arrangement is provided with a protective covering so as to preserve the integrity of their radar cross-sectional areas. The comer reflectors are generally deployed at least one beam-width above the transmitter/receiver’s horizontal plane to avoid ground clutter and multi-path interference. The instruments ability to detect rainfall has already been demonstrated (Ruf et al. 1996). 5.2.1 Results Of DPPL’s Stability Tests The DPPL operates with a pulse repetition frequency (PRF) of 2200 Hz and a pulse width of 140 ns. The 2200 Hz PRF produces 1100 interleaved vertical and horizontal samples every second. The 1-sec averages of the individual horizontal and vertical received signals are recorded every 5 sec. The accuracy and stability of the DPPL’s performance over various averaging periods can be determined from its a - x plot, in which the error a is given as a function of the averaging time x (see Appendix E). The stability of the rainfall rate estimates (corresponding to the Allan deviations shown in Fig. E. 1 of Appendix E) can be determined by scaling the values according to the R-A relationships derived from data retrieved during a storm that passed over the link during the recent campaign. These results are shown in Fig. 5.1 where RAv, R ^ , R^a, and RAavg, represents the rain rate estimated from the Av, Ah, AA, and Aavg respectively. These may be seen as the noise limit on the ability of the DPPL to resolve the temporal spectral variability of the rain-rate process or the hardware-dependent absolute accuracy of the link’s retrieval of rain rate (see Ruf et al., 1996). It is clear that for time scales of less than 20seconds the average attenuations (Aavg) gives the most accurate estimates of the rainfall rate. The highest absolute accuracy possible from the instrument should give rainfall rates within 1 mm/h for attenuation measurements on the V-pol signal (Av) integrated over 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8-10 minutes. For the same integration period the highest absolute accuracy possible from AA should be within 6 mm/h. It is evident that the differential processing, for this instrument, would give the least accurate rainfall estimates. This is consistent with the results o f Fig. E.2 o f Appendix E, showing the differential signals to have the largest normalized errors, and the lowest SNR for all the averaging times shown. Figure E.l shows the Allan deviations in the differential signal to be on the order of those observed in the H-pol signal. It is apparent that the differential processing multiplies rather than reduces the errors in the alternate signal polarizations. This may be due to the fact that the errors in the V- and H-polarized signals are somewhat different, especially over integration periods of less than approximately 10 minutes (see Fig. E.l). For longer averaging periods ( 1 0 < r < 5 0 m in .) the errors in the V- and H-polarized signals vary in a similar manner although the errors in H-pol are 2 to 3-times as high as that in V over these time scales. Reducing the errors in the differential estimates would require the reduction in the noise on the alternate polarizations (i.e. more stable or coherent signals), or carrying out the differential processing after averaging over longer time scales (10 < r < 50 m in.). The former would require a more stable hardware. The latter would be more difficult as a practical alternative since most rain events tend to occur on time scales of a few minutes (5-20 minutes) which are significantly shorter that those being suggested here (i.e. up to 50 minutes). These results therefore suggest that the lowest accuracy would be achieved using differential processing. 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. DPPL Stability Test on June 2001 data 102 10 10' 10-’ 10l 10’ 102 Integration Time t(min) Fig. 5.1 The errors expected in the DPPL estimated rainfall rates as a function o f the integration time Goldhirsh (1975) suggests that the number of independent samples that are used to estimate the ratios of the return powers must be fairly large to wield reasonably accurate estimates. This sensitivity to the sampling size results from the fact that taking the ratios of the powers has the effect of amplifying the error variance. 5.3 Other Rainfall-Measuring Instruments Used In The Experiment The DPPL is usually deployed with a tipping bucket and an optical rain gauge (ORG). For this experiment we were able to secure the loan of a RD-69 Distromet disdrometer (JWD) form the NASATRMM programme. The tipping bucket, ORG and JWD provide an estimate of the rainfall at its particular location, and as such their estimates are referred to as point measurements. Hence their area (region) o f accuracy or validity will vary depending on the structure o f the storm system under observation, the type o f rain, the topography, and other environmental conditions. In order to properly interpret and make meaningful 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. comparisons between the measurements obtained from these instruments it is imperative that we have some idea of the general physics of their individual operations. 5J.1 TE525 Tipping Bucket Rain Gauge The TES25 is a standard tipping bucket rain gauge. It includes a removable outer funnel with a 9.66” diameter orifice, a tipping bucket assembly, and an outer housing assembly. The funnel is fitted with a mesh screen to prevent debris from clogging its vent, and to reduce the effects o f splashing. The funnel location is such that it empties directly into one of two small plastic collector buckets. The buckets are attached together about a pivot. For every 0.1 mm of rain that collects in a bucket, the weight of the water causes it to tip over, emptying itself as the second bucket moves into place under the funnel. The process is repeated for as long as the rain continues. The tipping of the bucket activates a reed switch that produces a contact closure on each tip, which is then recorded. Adding up these counts over a period of time gives a measure of the rainfall over that period. The TE525 measures rainfall at rates up to 2” per hour (50.8 mm/h) with an accuracy of 1% according to the manufacturer’s specification. This design helps to lower the effects of evaporation on the measurement of the rainfall. However, its accuracy, especially in measuring very light and very heavy rainfall rates, is limited. In general tipping buckets have poor time resolution, even at moderate rain rates (2-5 mm/h), and is known to vary with rainfall rates (Zawadzki, 1987). The data has a 1-minute resolution, which indicates a minimum detectable rain rate of 6 mm/h for a metric gauge such as the TE525, with a 0.1 mm resolution. Tipping buckets are also limited in their ability to accurately measure high intensity rainfall rates. In their experiments Adami and Da Deppo (1985) reported larger errors in the tipping bucket measurements as the rainfall rate increased. They attributed this error to the dump interval (the time it take for the bucket assembly to swing to the alternate position after a bucket is filled to its tipping point). Since this dump interval is fairly constant over the range of rainfall rates, a greater percentage of the bucket volume is lost as rainfall rates increases. Rainfall rates below about 51 mm/h are thought to produce negligible loss when compared to the systematic errors that are though to affect these precipitation measurements (Sevruk, 1985). The tipping bucket is nevertheless considered a reliable instrument for estimating total rainfall 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. accumulation (Tokay and Short, 1996). Figure S.4 shows a picture o f a TE525 similar to the one used with the DPPL. Fig. S.2 TE525 Tipping Bucket Rain Gauge Before the comparisons were made with the DPPL, the TE525 was calibrated to within 0.01-cm3 using a syringe with a precision-glide needle. Following the manufacturer’s guidelines the TE525 was calibrated to tip for each 4.73-cm3 o f water collected in its bucket. 5.3.2 Optical Rain Gauges (ORG) A Scientific Technology Inc. Optical Precipitation Sensor (Optical Rain Gauge-ORG) model ORG-705 was also used with the DPPL. The ORG-705 has overall dimensions of 38" x 19" x 7", and weighs 9.51bs. The distance between the optical transmitter and the detector is 30.1”. The manufacturer’s specifications give the operating temperature to be —50 ° C < T < 50° C . It is estimated to make measurements with accuracies of 1%, and 4% for rainfall rates over the ranges o f 10-100 mm/hr, and 1500 mm/h respectively. Figure 5.5 shows a picture o f an ORG-705 similar to the one being used with the DPPL. Unlike the tipping bucket the ORG measures rainfall rate rather than rain accumulation. This ORG was designed to measure rainfall rate directly from scintillation o f its optical Near Infrared (NIR) beam ( k = 0.85 pm). As raindrops fall through the beam o f light they induce optical scintillations in the detected 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. light intensity. The variation in the light intensity o f a given drop shadowing the source/detector is a function o f the drop size, fall speed, and coherence of the source. By choosing the proper geometry, variations in the light intensity caused by naturally occurring raindrops can be made proportional to rainfall rate (Wang et al. 1978). The statistical average o f the measured scintillation signals is then used to estimate the “instantaneous rain rate”. Fig. 5.3 The ORG-75 Optical Precipitation (Rain) Gauge In theory, optical gauges are highly correlated to the “moment” o f drop size distribution associated with rainfall. However, tests performed with different ORGs suggested that this was not necessarily the case, and that different ORGs may be more correlated to different drop size distributions (NASA Conf. Pub, 1994). Use o f the drop shadow to infer drop size may suggest that the shapes o f the DSDs may affect these measurements. Nystuen (1999) suggests that changes in the performance o f the ORG are associated with variation in the N0 parameter of the gamma model DSD. In general, the ORG is considered to have fast time resolution and high accuracy in both light drizzle and heavy showers, and it is not sensitive to evaporation or splash errors as in the case of tipping buckets. Preliminary results suggest that for rain rates between 1-100 mm/hr, the calibrations of most ORGs are 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. stable and linear (Nystuen, 1999). 533 Joss-Waldvogel Disdrometer (JWD) The JWD was designed to estimate the drop size distribution within rain (Joss and Waldvogel 1967, 1969). The JWD assumes that drops fall at their terminal velocity (Gunn and Knizer, 1949). The instrument transforms the momentum o f individual drops falling on the sensor, into electrical pulses whose amplitudes are functions of the respective equivalent drop diameters. In the presence of vertical air motions this assumption o f terminal fall speed may result in erroneous measurements of the DSD. A conventional pulse height analysis is then performed to yield the size distribution o f the raindrops. The actual signal amplitude to drop size relationship depends on the characteristics of the sensor head. The sensitivity of the sensor is somewhat dependent on the point of impact of a drop on the sensor, so that pulse amplitudes o f drops of equal diameter will form a distribution around the average amplitude. The accuracy of the drop size measurement can be within ±5% of the actual diameter depending on the make and model being used. The instrument is fitted with a processor unit that supplies power to the sensor and filter spurious signals (mainly due to acoustic noise) before processing the drop-induced signals. The processor distinguishes 127 classes of drop diameter. In an attempt to reduce the amount of data for storage while retaining statistically meaningful samples, the 127 drop channels of the sensor output are combined into 20 drop size classes distributed more or less exponentially over a range of mid-sizes from 0.35-5.135 mm (i.e. the bin sizes are not evenly distributed). The calibration of each unit determines the exact channel boundaries (Sheppard, 1990; McFarquhar and List, 1993). The mid-size values provided by the manufacturer for use with this instrument have been included in Appendix F. Figure 5.4 shows picture of a JWD. 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 5.4 The Joss-Waidvogel Disdrometer (JWD) used for measuring raindrop size-distributions The instrument has a sampling cross-sectional area o f 50-cm2 and the measurements are usually integrated over 1-min intervals. The resolution volume o f the JWD is determined by the fall velocity of the drops such that for drops falling at 7 m/s (i.e D=2.2 mm) the observing volume of the JWD will be approximately 2 m3 for a 1-min sample. It may therefore be useful in the analysis of the variations of some integral parameters (e.g. R) and useful to the discussion o f radar measurements. The JWD can be very sensitive to environmental noise so great care must be taken in selecting the location where they are deployed. The unit is fitted with electronic noise controllers; however, when the ambient noise gets loud the smaller drops are not counted. This can potentially lead to a significant under representation o f the smaller drop sizes. In addition, the noise floor is automatically raised during high intensity rain events. This may also prevent the small drops from being detected. This feature was built into the instrument to increase the likelihood of detecting the larger drops (Nystuen, 1999). Another reason why the JWD may not be sensitive to the smaller drop sizes relates to the finite time (“dead-time”) needed for the instrument to recover from a drop strike and be ready for a next drop. The manufacturers have provided a correction matrix to compensate for this effect. However, a number of researchers have declined to use it as it has a tendency to modify the DSD in such a manner as to cause increases in some of the integrated DSD parameters such as rain rate (for more discussion on this issue see Sauvageot and Lacaux, 1995; Tokay and Short 1996, and Tokay et al., 2001). Sauvageot and Lacaux (1995), presents experimental arguments suggesting that the relatively small number o f small drops observed with the JWD is essentially real and that the instrumental cause is marginal. 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Another possible source of error in the JWD is that the accumulation of water on the sensor head may change its characteristics. This is of particular concern for drops directly striking an area where liquid had previously collected. The JWD is also susceptible to possible errors that may be due to secondary drop impacts resulting from drops breaking up, and striking the sensor with the splatter registering as additional strikes. Despite all these concerns the JWD is still considered the standard for DSD measurements at the ground (Tokay et al., 2001). 5.4 The Propagation Link (DPPL) setup The radar unit was deployed about 70 ft. above the ground on the roof of the Walker Building at the University Park Campus of the Pennsylvania State University in the May of 2000. The 6-inch comer reflector was installed on a tripod 103.5 m away on the catwalk of a brick chimney. The catwalk was approximately 10m above the level of the radar. The radar’s elevation angle was measured at 4.84°. These measurements were made with the aid of a SET5AS Theodolite (Electronic Total Station). Fig. 5.5 Pictorial showing the relative locations of the Walker Building (upper left comer) and the smoke stack (lower right comer). The radar was located below the southeastern edge of the blue square, on the roof of the Walker building, and close to the right hand comer of the roof facing the smoke stack. The oval shows the relative location of the JWD, ORG, and TE525 instruments 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The tripod allowed for adjustments to the comer reflector in both elevation and azimuth. The comer reflector and tripod were held firmly in place to prevent them from being moved about by the strong winds. An aluminum covering was placed over the comer reflector as an additional protection against wind and especially to prevent the reflector from being wetted by rain, as this would likely have a significant effect on the scattering cross-section of the reflector. The aluminum covering was used as a compromise against having to use a plastic or some other dielectric covering which would most likely suffer even more dramatic changes in their electrical properties when wet. This aluminum cover was assumed to have very little effect on the broad-beam characteristics of the comer reflector, and was never in direct contact with the reflector. The line of site path between the radar and the comer reflector was unobstructed for several range bins along the axis of the antenna, and over several beam widths across. The link was oriented along the southeasterly direction. Coincident rainfall accumulation and rain rate measurements were made in rain with the aid of the TE525 and ORG respectively, and the JWD, when available, provided estimates of the DSD and rainfall rate. Both the TE525 and the ORG data products are fully integrated into the DPPL system. The measurements for these instruments are recorded along with the averaged return signal power, every 5 sec. All three instruments were located on the radar end of the propagation link at a distance of about 3m to the side of the tent containing the radar and 2m above the level of the radar, on top of the shaft of an emergency stairway. These instruments were arranged within 1m of each other, sufficiently far away from each other so as not to cause any obstruction to any of the instruments, and as far as possible from the edge to avoid the wind turbulence generated at the edge of the shaft. A Davis Weather Station (DWS) operated by The Pennsylvania State University Campus Weather Service, is also located on the roof of the Walker Building. Five-minute averaged measurements of temperature, humidity, pressure, rainfall accumulations, wind speeds and wind directions are available from this DWS. 5.5 Procedures For Estimating The Attenuation And Rainfall Rates The DPPL’s receiver is fitted with a tunnel diode square-law detector, therefore the output of the detector is directly proportional to the power of the return signal. We assume that the return signal at the receiving antenna has a magnitude V0 and is of the form Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. v = V oe-r -« r -A )z ( 5 . 1) = Va'e -2i’z (V) where k, and kj are the real and imaginary parts of the complex propagation constant, z(km) is the distance between the radar and the comer reflector, and the attenuation of the signal over the propagation path is given by the exponential term e ' 2k,:. In the absence of rain this attenuation is mainly due to 0 2 and water vapor in the atmosphere. The output of the square law detector P x V2 can therefore be considered to be of the form P = P e ~ xkz (5.2) <W) In the absence of rain we would consider the output of the detector to be the “clear-air” return signal power When it rains along the path of the link the signal suffers additional attenuation due to the raindrops. Consequently, the output of the detector in the presence of rain can be expressed as a function of the “clear-air” return power in the form — 2zA(dB/ km) p r = p c 10 (W ) ■» (5.3) where the rain-induced attenuation A is given in dB/km. Hence, the attenuation can be obtained from the signal as A(dB / km) = — log 0-7 pr (5.4) From (5.1)-(5.3) we can relate the return signals at the V-, and H-polarization to the return powers Pvand Ph respectively. Therefore the specific attenuation at the V-, and H-polarizations are given, following (5.4) as Aj (dB / km) = log (5.5) p : 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where j represents v or h accordingly. From the above expressions it follows that the differential attenuation is given by AA(dB/km) = Ah(dB / km) - Av(dB /k m ) Pc r v r h ' i f Pr = 10 - |ogl PC■P ' V " hJ 1 0 . ( p r / p; = — log p : / p: (5.6) and the average attenuation as. ( d B I b n ) = ± W b * > + A ,ld B I _b n ) (5.7) = — log P r •P r 4z The voltage output of the ORG varies from -0.35 to 4.13Vdc as the rainrate varies from 0.1 to 3000 mm/h and can be converted to rainrate following the manufacturer's transfer function R ( m m l h ) = 10 (5.8) The rain rate can be estimated directly from the JWD measurements as D , /f, n 3.6xl0-3^, R jwd ( m m / h ) = - - 6 „3 -----2 , D. dt -Area " (5.9) where ni is the number o f drops in the i* bin, Dj is the mid-size of the i* bin in mm, dt is the sampling time (I minute) in seconds, and Area (m2) is the cross-sectional sampling area of the JWD. The summation is carried out over the 20 bin size categories. The DSD estimator N(D)dD is calculated form the JWD measurements as 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. N(D')dD, = A r ea - d t- V '( D ') (5.10) ( m ~ 3) where Hi is the drop count in the i* bin corresponding to drops within the range of sizes D, ± d D t , and v,(Di) is terminal velocity (m s '1) o f the drops given in equation 3.17 (Lhermitte, 1990). The total drop concentration (conc.) is given as the summation of N(Dj)dDi over all the sizes. 5.6 Instrument Noise Reduction Schemes The most persistent problem experienced during this campaign was the inability to stabilize the DPPL’s system gain. The gain may vary between l-3dB within a 24-hr period. This variation was strongly correlated to the system’s ambient temperature as measured by the thermistor probes in the receiver housing, and the relative humidity. This temperature in turn fluctuated from day to day with the weather. The receiver is built on a plate with some limited temperature control provided by means o f heater strips strategically placed near to the more sensitive components, and controlled by a MINCO CT-137 proportional controller. An air conditioner unit and a number o f ventilation fans were added to the unit in an effort exert greater control over the temperature inside the radar, which have on occasions exceeded 40°C. These have only had a limited effect in reducing the gain fluctuations. It is still unclear which component or components are most directly responsible for these gain fluctuations, or whether there may be some other external factors (e.g. climatological) influencing the fluctuations. This problem is compounded by the fact that the alternate polarizations of the detected signal appear to have different dependencies on the temperature. Attempts were made to rectify this problem by modeling this dependency of the returned signal to the temperature under clear weather conditions. These modeled dependencies were then subtracted from the measurements. This resulted in the reduction of the magnitude o f the compensated signal, which nevertheless appeared more stable for the V- and H-polarizations. However, it was common to get a number of spikes in the data when the ratios were taken (see e.q. 5.7). This was especially common in the measurements taken during rain. Additional steps were taken to minimize the effect of the random electronic noise in the system by removing an estimate of the “receiver noise” from the return signal. The receiver noise was estimated from measurements taken out to 7.5 km away from the radar. As no significant scattering source 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. interrupted the signal’s path out to 7.5 km, this effectively provided a measure of the receiver’s noise floor at both polarizations. This noise floor estimate is then subtracted from the return signal in an attempt to minimize the effects of the system drifts on the estimated attenuation values. 5.7 DPPL Operation for Selected Rainfall Events One of the original goals of this experiment was to acquire a large data set of attenuation and rainfall measurements for verifying and adjusting the R-A relationships obtained in Chapter 4. My inability to stabilize the system’s gain for extended periods at a time has forced me to restrict my study to the individual storms on an event-by-event basis. There were two storms that passed over the link while all four instruments (the DPPL, JWD, ORG, and the TE525) were in operation. The first was on June 16, 2001 and the second on June 30, 2001. The JWD was returned to the folks at NASA-TRMM soon there after and was not available for any of the subsequent rain events. 5.7.1 Event #1 June 30lb 2001 It would be convenient to first examine the June 30* 2001 event. This was a single-cell convective system that passed over the link from 6:15-6:30pm. The wind gust during the first half of the shower was estimated to be at about 22 mph, decreasing to 11 mph by the end of the shower. The winds were almost constantly in the WNW direction for the duration of the shower. The maximum rain rates measured by the ORG and JWD were 55 mm/h and 64 mm/h respectively. The total rainfall accumulation (SR) measured by the rain gauges and the disdrometer were as follows, for the ORG I Rorg= 2.68 mm, JWD ZRjwd = 3.50 mm, and for the TE525 £ Rtip = 2.8 mm. The total accumulation measured by the Davis Weather Station (DWS) for this event was SRDws = 2.79 mm. It is therefore necessary to determine the “best estimate” of the rainfall measurement. Tipping bucket rain gauges are generally considered reliable at measuring total rainfall accumulations. However, their accuracy in measuring the accumulations for events with high rainfall rates is limited (Nystuen, 1999). In this case the rainfall rates were moderately high and the accumulation measured by the TE525 was quite similar to that of the ORG. It was therefore convenient to use the mean accumulation as the 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. “ground truth”. The rainfall rates measured by each instrument were then adjusted in such a manner that the total rainfall accumulation for each instrument was equal to the mean value o f 2.94 mm (i.e. the mean accumulation was distributed over the event). Figure 5.6 shows the standardized returned signals of the DPPL (reduced by a factor o f 5) along with the temperature and humidity measurements taken during the day. The temperature Tair DPPL (°C) gives the ambient temperature in the DPPL's receiver housing compartment, T Dvs gives the atmospheric temperature (°C), and Humid gives the relative humidity (%), both of which were measured by the Davis Weather Station. The return signals have been reduced by a factor of 5 so that variations in the signal magnitudes can be more effectively compared with the changes in the rainfall, temperature, and relative humidity measurements. Notice that the variations in the V-pol return is positively correlated to the changes in the relative humidity while the H-pol returns seems to be affected by the combined variations in both the relative humidity and the temperature. Large fluctuations in the H-pol signal power coincide with variations in the receiver temperature exceeding 25°C (77°F) and with sudden changes in relative humidity. Variations in the atmospheric temperature, humidity and total pressure are known to cause fluctuations in the absorption and refraction of electromagnetic waves (Hill and Clifford, 1980). The imaginary part of the atmosphere’s frequency-dependent complex refractivity is a function of the temperature, pressure, and humidity (ITU-R, 2001). 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 90 80 70 60 50 40 30 20 10 10 15 20 25 Time o f Day (hr.) Fig. 5.6 Time series o f the receiver’s temperature (Tair °C ), the atmospheric temperature (T Dvs °C) and the relative humidity (Humid %) during the periods preceding and subsequent to the main rain event on June 30 ,2 0 0 1. Also included are the rainfall measurements and the standardized return signals. Figure 5.7 shows the time series o f the DPPL’s l-sec averaged return signal and the rain rate estimated by the ORG. These results show the variation in the return power during the 45-minutes preceding the shower and over the l-hr period following the end o f the shower. The variations in the “clear-air” return signals are due to the combined effects of the DPPL’s hardware noise and the variations in the ambient temperature and humidity. Consequently the variation in the return signal power has a strong quotidian component. Apart from this cyclical feature, it is also common for the temperature to fall with the approach o f rain systems. The temperature decline recorded over the 90-min period during which the main event occurred was approximately 9.5-lO°C. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 400p , ■ , , ■r ....... — Time (hr.) "i— i i - ................................ 60 45 I: 30 > - • a7.5 15 ee 18 ! 18.5 Time (hr.) "i i 19 19 % 60 45 « . 30 % . 15 ce 1 i__________ £Z 18 18.5 Time (hr.) --------- i-------------19 19 % Fig. 5.7Time series o f the DPPL receiver output and the co-incident rainfall rates Note that the decrease in the signal power in the shower is well correlated with the rain rate. It is also evident that their ratio V/H is not as well correlated with the rainfall rate. This is due to the fact that the noise in the alternate signals is not well synchronized over these time scales (see Fig. 5 .1 and Fig. 5.2). This lack of correlation between the noises in these two signals is a potential source of error in estimating the differential attenuation (AA). The utility o f this parameter was based on the assumption that taking the ratio of these signals (V/H) would reduce the sensitivity to the hardware noise fluctuations, since the receiver drifts were expected to occur over time scales longer than the polarization switching speed o f 2200 Hz (see Ruf et al. 1994, 1996). This would be true if, in addition to the receiver drifts occurring slower than the PRF, the variance in both V and H were well correlated with each other. Taking the ratios of such signals would then lead to a cancellation o f those noise terms. Since AA is calculated directly from this ratio it was therefore expected that AA would be a very stable parameter. Note the sharp dip in the H-pol return signal that occurred just prior to the start o f the shower. This coincided with a much smaller positive spike in the V-pol signal. The effect of taking the ratio V/H o f 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. these two signals is the large spike seen in the plot of V/H at that the same instant. This is one o f those instances where the ratio effectively causes an increase in the noise rather than a reduction as previously assumed. We have observed the effects of the variation in the system gain on the individual signals and their ratios. The fact that the attenuations are calculated relative to some reference signal level is expected to be a mitigating factor in reducing some of the noise caused by these gain fluctuations. Care would therefore be necessary in choosing the reference levels to ensure that they are representative of the ‘signal reference’ or base levels during the rain event. Figure 5.8 shows the results of the 1-sec attenuations calculated from the signals in Fig. 5.7. This shows the microwave attenuations plotted against the ORG estimated rainfall rates. Also shown are the simulated attenuations plotted against the rainfall rates Rjwd; both of which were calculated from the JWD measurements. The simulated attenuations, denoted by the red asterisks, were calculated assuming the BCeq equilibrium drop shape model that was introduced in Chapter 3. 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30th June, 2001 20 20 40 40 R (mmrti) R (mm/h) • • • p r * 20 18.3 ; • « • * . 18.35 18.4 18.45 Time of Day (hr.) • • • D P PL JWVOBC., 60 40 R (mm/h) R (mm/h) 18.25 • ............ • •• • * |l • • , ^ o tite K C • 18.5 10 AA(dBAm) Fig. S.8 Microwave attenuation values estimated from the DPPL measurements (dots) plotted against the ORG estimated rainfall rates (R org)- Also shown are the attenuations simulated from the disdrometer data plotted against the rainfall rates estimated from the same data. The time series estimates of the ORG and JWD derived rainfall rates are also compared. These simulations were performed for the link’s 5° elevation angle. The raindrops were assumed to have a normal distribution o f canting angles about the about their vertical symmetry axes (zenith 0°) with a standard distribution o f 5°. There is considerable scatter in the attenuations estimated from the DPPL signals. The AA parameter shows the highest scatter. Much o f this scatter is related to the noise in the measurements as previously discussed. Some of the scatter is believed to be related to microphysical changes occurring in the rain during the course of the shower. The negative values generated by noise in the estimated attenuation baseline were eliminated. It is clear that the DPPL derived attenuations are generally scattered about the simulated results. This is a promising sign as it suggests that the experimental results have a general trend consistent with the theoretical predictions (the AA results not withstanding). It is also evident that there is some scatter in the simulated results. This scatter is significantly less than that observed in the DPPL data. 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Because the basic physics behind the operations of both the DPPL and the JWD are so different it is unlikely that this scatter is related to some common design feature of the instruments (i.e. it is unlikely that the scatter is all instrument related). The lower scatter in the simulated results is partly due to the difference in the resolution times of the instruments (i.e. the DPPL gives 1-sec average measurements whereas the JWD give 1-minute average data). If it is assumed that some of this scatter is due to physical changes occurring in the rain, then all things being equal, the longer averaging time of the JWD should lead to a reduction in this rain induced scatter, thus causing the simulated results to have less scatter than the DPPL results. In the simulation carried out in the previous chapter, we used DSDs that were assumed to be constant over each 1-minute period. Implicit in this assumption is notion that the DSD is spatially homogeneous. However, rainfall is often considered to occur in random patches of different sizes (Jameson and Kostinski, 1999, 2001). As a result of these random “patches”, it is expected that there would be regions of high drop concentrations interspersed with regions o f lower concentration. In this experiment we have deployed the link over a very short path (=100m). This initially allowed us to make a number of assumptions: • That the link is sufficiently short as to be completely covered by the typical rain cell or patch (i.e. it is unlikely that it would rain at on one end of the link without it raining on the other). • It is similarly assumed that the rain would be homogeneous over such a short distance (the use of two or more disdrometers at different locations along the link would be a suitable test of this assumption) • The statistics of the spatial and temporal variations in the DSD along the link are similar, such that they may be made to approach the same value by choosing an appropriate integral. Using a 1-min non-overlapping filter, we can average the DPPL and ORG data so that their time resolutions are similar to that of the JWD. We can thus test this notion of the homogeneity of the rain over the link. This can also help us to address the issue of whether or not the differences in averaging times of these instruments is in part responsible for the differences in the scatter of the resulting parameters. It may also serve as a test as to whether or not the spread is due to some hardware defect in the DPPL or as a result of some physical process occurring in the rain. In the event of an instrument defect, such averaging is unlikely to cause any significant decrease in the scatter. 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The results of this test are shown in Fig. 5.9. Note that there is a significant reduction in the spread o f the experimental results and that they are closer to that observed in the simulated results. 20 20 40 R (mmAi) 40 R (mm4>) 20 60 ♦ ♦ ^ 40 “ * * 20 1 18.3 * _____ * * * ^ 5 * ** 18.25 ¥ 15 110 ,» » * tjk ♦ 18.35 18.4 18.45 Time of Day (hr.) T 18.5 ■ 0< 6 AA(dB/km) Fig. 5.9 The DPPL estimated attenuation values derived after smoothing the DPPL return signal with a non-overlapping l-min discrete average smoothing filter. Note that the reduced scatter in the experimental results is comparable to that given in the simulated results It has already been discussed that these instruments have different noise sources, with the DPPL instrument measurements being noisier of the two. Filtering the DPPL signals can eliminate some o f the system noise from the data and reduce the scatter in the calculated attenuations. One of the most popular filters used for reducing random noise is moving (or running) average low-pass filter. The moving average is an optimal smoothing filter for time domain coded signals to the extent that it gives the largest reduction in random noise for a given step response (Smith, 1999). However, in order to resolve events in a signal the filter’s step response should be o f a shorter duration than the spacing of the events. The amount of smoothing is controlled by the width of the filter kernel/window, and the noise reduction is proportional to the square root o f the number o f points in the filter kernel. The larger filter kernels (wider the window) result in greater noise reduction, leading to a smoother output signal. However, the 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. larger the filter kernel the slower the filters step response. Therefore, while the moving average may be optimal for reducing the amplitude o f the random noise, for large filter kernels it also reduces the sharpness o f the signal, and may also cause a reduction in its amplitude. In other words it has a similar effect on both the signal and the noise as the width o f the filter increases. This excessive smoothing is due to the fact that the rise time o f the filter’s step response increases with the filter kernel. It is important to maintain as much of the sharpness of the signal edge in order to observe many o f the subtle changes in the signal that may be related to equally subtle changes in the rainfall. Therefore, care must be taken in selecting the filter kernel in such a manner as not to cause so much smoothing that the important details o f the signal variation are lost along with the noise. From the results of Fig. E.2 it was observed that a significant improvement in the normalized error and the SNR of the signals could be obtained if the data was smoothed with a 2-min average filter. The marginal improvement in the data for longer integration periods is shown to be minimal. Hence, the use o f longer averaging periods is not justified and may only lead to the excessive smoothing o f the data which we seek to avoid. 30th June, 2001 3001— 250 - 60 50 6.150 ^ 40 30 x 100 - 20 50 - 10 5-200 - 18.5 Time (hr.) 400 T 60 300 6.200 - 100 - 40 > 4 80 18.5 Time (hr.) 80 - 60 40 20 18.5 Time (hr.) Fig. 5.10 Time series of the DPPL receiver output after being filtered by a 2-minute moving average low-pass filter. 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 5.10 shows the signals after being smoothed by a 2-min moving average filter. This allows for an appropriate amount o f smoothing with an adequate degree o f signal sharpness and without significant loss in signal amplitude that would be typical of a longer moving average filter. The attenuations calculated from the resulting filtered signals are given in Fig. 5.11. These results are plotted against the ORG estimated rainfall rates, and are indicated by the “dots”, while the simulated results are shown with asterisks. 15 10 % 0 20 40 60 20 15 15 10 10 20 40 60 40 60 40 60 R (mmfli) 20 R (mm/h) 15 60 10 40 16.3 18.35 16.4 18.45 Time of Day (hr.) . 16.5 Fig. 5.11 The microwave attenuations estimated from the smoothed DPPL output given in Fig. 5.10 The obvious effect o f filtering the DPPL signals is a significant reduction in the spread of the estimated attenuations. It is also evident that there is a systematic variation in the attenuation signatures with R, and that they are not exactly linear in R. Figure 5.11 shows the ORG rainfall rates and the DPPL estimated attenuations related to the first half of the storm using black dots, and the second half of the storm using red dots. The corresponding results for the simulations are shown with blue and green asterisks respectively (the asterisks are joined by lines to illustrate the relative sequence in which they appear over the course o f the shower). 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It is apparent from these results that the attenuation signature of the shower before 18.35-hr is different from that of the second half, to the extent that the attenuations calculated form the DPPL data follows two generally distinct trails depending on whether they correspond to the period before or after the peak rain rate. This consequently leads to situations of similar or identical rainfall rates resulting in different attenuation measurements. This apparent hysteresis effect is believed related to the physical processes occurring during the course of the rainfall, which is being captured (observed) by the DPPL measurements. This would be possible in situations whereby the DSD of the rain may be changing in such a manner to create some inter-play between its different parameters e.g. the size o f the drops and the number of drops. Note the results for the AA parameter do not show quite the same difference in the values before and after the rain peak, as is evident in the other parameters. Likewise there is no spreading evident in the simulated results of AA. The question arise as to whether this is a manifestation of some physical processes occurring in the rain, or whether it may be some artifact of the system, or even the link set-up. It is clear from the results of Fig. 5.9 and Fig. 5.10 that by filtering the data we were successful in removing much of the high frequency noise from the signals. Furthermore, a careful check of the results in Fig. 5.11 indicates that the source is unlikely to be random as the results show distinct trends that are well correlated with the process being measured (i.e. the rainfall rate). Is it possible that variations in the system’s temperature can result in these changes? This possibility has been explored. This is unlikely, as the temperature variations which results in the large gain fluctuations are of period much longer than the few minutes being discussed here. Fluctuations in the system gain of between l-3dB may typically occur at different times of the day, with the maximum system gain occurring in the late night to early morning hours when the temperature is at its minimum, and with the lowest system gain typically occurring typically at mid-afternoon when the temperature is at its maximum. The typical temperature variation over this period is usually between 10-20 °C. On the other hand, the temperature was observed to have fallen by 1.5 °C during the 15-min period in which the rain fell. This is unlikely to have had a significant effect on the system gain over that period. Similar patterns were observed in other rain showers that occurred at different times of the day, and for different durations. In none these events were there any evidence of the attenuation signature being significantly affected by the temperature. Another possible source of error may be the variance due to the scatter return from the chimney. The 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. radar cross section of the chimney would change due to the enhanced electrical properties o f the wet bricks relative to when they are dry. This would generally lead to an increase in the clutter produced by the chimney, particularly if it was well correlated with the rain (e.g. by preferentially oriented wetting, Ruf et. al, 1996). The width o f the antenna beam (2.3° HPBW) at the reflector is estimated to be about 4.15m with an area of about 13.5 m2. The comer reflector and its aluminum covering are estimated to occupy about 0.51 m2 of the beam, which when considered in addition to the relatively large radar cross section of the comer reflector, ensures that an appreciable signal level is generally present above the noise floor. This having been said, this chimney effect has proven to be a difficult noise source to properly estimate. This is in part due to its curved structure of the chimney, which tends to experience different “wetting” depending on the direction of the wind. It is therefore not uncommon to notice the side of the chimney facing the link completely dry even after a heavy downpour. Additional measures were taken to reduce the effect of this clutter on the interpretation of the results by considering only those events for which there was high correlation between the DPPL signal levels and the ORG measurements indicating the beginning and ending of an event. If the ORG signal returned to its normal level indicating a cessation of the rain activity, while the DPPL signals had remained low and not returned to there pre-rainfall levels, this would suggest some residual effects of possibly the reflector having been wet or moved, some system or hardware failure, or some change in the environmental conditions (e.g. the temperature, pressure and humidity) that may affect the signal propagation along the path. Additional precaution is therefore taken to reduce these effects by judiciously choosing the reference power levels used in calculating the attenuations given by equations (5.4)-(5.7). This was achieved by comparing the signals before and after the shower. If there were significant differences in these signal levels then a mean reference level was obtained by averaging these signals to give a reference level more “representative” of the baseline during the rain. Holt et al. (2000) has reported on a set of procedures they have employed to address these problems of excess attenuation along their link, along with the identification and correction of the attenuation baseline. Norbury and White (1972) used a more elegant technique to compensate for excess path attenuations along their microwave link and the periodic fluctuations in their system gains. This was achieved by alternately sampling portions of the transmitted and received signals, coupling these samples to a differential amplifier, the output of which is then used to make automatic adjustments to a calibrated attenuator placed in the input wave-guide of the receiver. This method of attenuation measurement should be more insensitive to variations in tuning and fluctuations in system gains. 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. From the above results (Fig. 5 .11) it is evident that the experimental results show a generally strong agreement with the simulations after taking the above-mentioned precautions. This therefore suggests that the differences observed in the attenuation values in Fig. 5.9 are partly the result of microphysical changes occurring in the rain over the course of the shower. The question to be answered here appears to be: • What characteristics of the rainfall have undergone dramatic changes before and after the rain rate peak? • What microphysical properties are capable of causing this separation in the attenuation signature before and after the rain rate peak? • Why are the DPPL attenuation values after the rainfall peak higher than those prior to the peak, whereas the opposite is true for the simulated results? We may gain a better understanding of the variations in both the attenuation and rainfall measurements over the course of this rain event by examining the time series of these measurements. A clearer understanding of the variations in the experimental results over the smaller time scales is possible by passing the data through a short ( 15-sec.) smoothing filter. Figure 5.12 shows the time series of the simulated (JWD BCcq) and the experimentally derived attenuations along with the ORG estimated rainfall rates (the rainfall rates have been reduced by a factor of three for easier comparison with the attenuation results). These results were obtained after applying a 15-sec smoothing filter to the DPPL data. 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Time Series of estimated attenuations 20 - DPPL Ajj — JWDBC A,, _ DPPL \ JWD BC.q ^ —- ^ R O 73 15 10 ---- i 18.3 18.4 18.35 18.45 18.5 Time (hr.) Fig. 5.12 The time series representation o f the experimental (DPPL) and simulated (JWD) microwave attenuation values. The ORG rainfall measurements have been reduced by a factor of 3 for easier comparison o f the variations in the rainfall rate with the variations in attenuation values. Some of the evidence indicating microphysical changes occurring during the event can be observed from the DSD measurements taken during the shower. The DSD is the most obvious, and also the most prominent physical characteristic of the rain that may be able to offer information on what is contributing to these observations. The DSD can offer clues, not only on the drop concentration and drop sizes, but can also give some indication as to the possible drop shapes associated with the event. Figure 5.13 shows the variation o f the drop concentration N(D)dD in units o f m'3, during the 15-minutes of the shower. Note that only the values o f N(D)dD > 1 m" ^ are shown. Each frame (individual plot) represents the DSD averaged over non-overlapping 1-min time intervals arranged chronologically. The numbers in the top right-hand comer of each graph indicates the number of minutes into the event when the corresponding measurements were taken, e.g. 1 and 15 refers to the first and fifteenth minutes respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 10' liiiliiiiiilj l i i i i l i i i i i i l : : 3 : : c : r : 3 : : E 10 Uzztztzlzz 3 10 * i - “ i— 10 0 l i i i i i f i i i n r - -I — 1 1___ L4 5 10“ lla lllllillll 10 ' ' 1 2 10 3 w 0 fe- - k 10’ ' i 0 1 10 ' 3 4 5 l = i = =!|=i = 3==1 22 3 l 4 4 5 5 k - H- - n 5 ^ - - k - H ---1 i 1 10 0 1 2 3 4 5 0 1 2 3 4 5 10 ' l i i l l'l l l l s l l 10 10 ! l § i y 11i |= | y i n i~ 2 lllllli 10 I 0 L 1 2 3 4 5 10 ZZ3ZZ I---- ||l i |y i |l l iftH jiiiliiii w 11 ! i | J fill h i l l ““»EEeE5EE •I- - f —H--- f 1<,2fe i i i i b l i i i l l 10 0 102 - - ^ - - 1- i t 1 2 3 4 5 iiiii'iiiiiilf 10 0 10* 1 2 3 4 5 llillillils is 10 0 102 1 2 3 4 5 51 =11 =='= = 0 1 2 3 4 5 H illjlllllll E S I I I I I 10’ Il i t ! i;i 111111 10 nX iT i 10' r " “i i D((nm) i__ 10 " llitliilillll T l. i" D(mm) 10' 1§1! 10’ 10° 10° D(mm) D(mm) D(mm) Fig. 5.13 The drop concentration N(D)dD estimated from measurements taken by the JWD. These results represent the measurements taken over 15 consecutive non-overlapping minutes during the course o f the shower (i.e. the period spanning the beginning and end o f the event). Changes in the shapes o f the DSDs over time can be observed from these results. It is also evident that there are some general trends (features) to these variations leading up to, and subsequent to, the maximum rainfall rate. This peak rainfall rate measurement corresponds to the DSD given in frame #7 above. Figure 5.14 shows the total numbers of drops counted, the total drop concentration (conc.), which represents the area under the curves in Fig. 5.13, and the D0 evaluated for each 1-minute DSD measurement Note that the DSD at the beginning o f the shower has an almost exponential shape that slowly evolved to a more concave-down shape as the time progressed. This concave-down shape, as we saw in Chapter 3, suggest a higher moment o f the DSD. 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 4 10 ' 3 10 ’ Q°2 V. V.v.' 10 18.25 18.3 1 18.35 18.4 nm«(hr.) 18.45 18.35 18.45 18.4 1%.25 18.5 18.3 18.35 18.4 Time(hr.) 18.45 18. 100 o 10 18.25 18.3 18.35 18.4 Time(hr.) 18.45 18.35 18.4 Time<hr.) 18.45 Fig. 5.14 Time series of the drop concentration (conc.), the median volume diameter (D0), the drop counts (# of drops detected by the JWD), and the rainfall rates estimated from the ORG and JWD measurements taken during the shower. The first 2-minutes o f the shower witnessed the arrival of a number o f large drops o f sizes D > 3 mm. This led to the D0 o f the first 1-minute drop spectra being quite high (4.6 mm), however due to the very low drop count the associated rainfall rate and attenuations were very low. Over the next 3-minutes the D0 rapidly decreases, the drop concentration increases initially and then declines. Such changes in these two parameters occurring in opposite directions appear to cancel each other resulting in little or no changes in both R and A. However, when they both decrease this result in both R and A decreasing. This is because we have both a decrease in the number and sizes of the raindrops. Between the 5(i>-7 a>minutes both the drop concentration and D0 begin to increase with N(D)dD becoming more concave down and wider. It is over this period that both A and R increases to their maximum values. Over the next minute the DSD gets a little wider and D0 increases over that o f the previous minute as a number of larger drops start arrive at the surface. However, the drop concentration decreases as the total number of drops arriving at the surface declines. This causes a decreasing in both the rainfall rate and the signal attenuations that continue for the remainder of the shower. This suggests that while the larger drops may individually have higher water content, their low frequency o f appearance renders them less significant 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to the rain rate than those o f the more populous mid-size ranges. The Davis Weather Station (DWS) measurements show a sharp fall off in both the wind speed and wind gusts over the second half o f shower. These measurements represent horizontal wind velocities. Figure 5. IS shows the variations in the wind velocities and the rainfall rates over the course o f the shower. The wind speeds (Wind) and gusts have been reduced by a factor of 20 and the wind directions by 10 (deg.) to facilitate the comparison o f the variations in these parameters to the changes in the rainfall rates and ultimately to the attenuation signatures. The wind directions are given as 0° coming from North and 90° coming from East. 60 50 R ^ o m n /h ) " VM nd(cmA)/20 “ G u * l(c m /t)/2 0 PK ddflV IO ) 40 30 20 10 18 18.1 18.2 18.3 18.4 18.5 18.6 Time of Day (hr.) 18.7 18.8 18.9 19 Fig. 5.15 Variation in the wind velocities and rainfall rates during the course of the June 30th 2001 rain event Zavody and Harden (1976) have also reported on the effects of changing DSD on single polarization (Vpolarization) dual frequency (36 GHz and 110 GHz) attenuation measurements made over a short link (220 m). When these attenuations were plotted against each other (without reference to the rainfall rate) two distinct trails of data points corresponding to the leading and trailing edges of the thundery shower was observed. They concluded that this was the result of changes in the DSD during the course o f the 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. event (they observed the median drop diameter change steadily from 4 mm to 1.5 mm within a 6-min period). Sweeney et al. (1992) reported similar effects in the attenuation measurements taken on a 20/30 GHz satellite link. This DSD hysteresis effect can reduce the effectiveness o f simple frequency scaling techniques that are used to adjust the uplink transmitter power. This is because these techniques estimate the uplink attenuations from measured downlink attenuations, which may be erroneous if the DSD is changing along the path. The experimental values of AA are observed to be almost a factor of 2 times larger that the simulated results. It is not immediately clear why this is the case. However, the values of Ah, Av, and Aavg are consistent with those of the simulated results. The experimental results of AA indicate a higher degree of scattering than would be expected from the simulated results. We consider much of this scatter to be related to the measurement instabilities as previously discussed. The asynchronous fluctuations in V and H generate additional variance when their ratios are taken. Figure 5.16 shows the results o f AA after adjusting its magnitude for this difference. Note that the general slope o f the experimental results is consistent to given by the simulated results. In the previous chapter simultaneous measurements of Ah and AA were proposed for use in determining the presence of oscillation as AA was observed to be sensitive to drop shape while Ah was not. Figure 5.16 also shows the results for both parameters plotted against each other along with the simulated values for the equilibrium and oscillation shapes. The attenuations corresponding to the rain rates before the rain peak appear to line-up in the direction o f the equilibrium shape drops. The attenuations corresponding to the latter part of the shower tend to follow the results for the oscillation shapes. 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 2.5 IPPL 14 12 1.5 10 0.5 -0.5 20 40 60 1 0 1 2 3 R (mm/h) Fig. 5.16 The experimental results o f AA (divided by 2) showing a general dependence on the rainfall rate consistent with the theoretical results. The Ah-AA plot suggests that the presence of oscillation drop shapes will generate steeper slopes in this relationship. Figure 5.17 shows the experimental and simulated results along with their fitted relationships. The solid lines give the results for the fits to the simulated results while the broken lines represents the experimental fits. The power law fits describing these relationships are given in Table 5.2. Note that the fits to the theoretical results are within the 95% Cl of the coefficients obtained for the experimental results for the Ah, Av and Aavgparameters. In the case of Av the relationships are almost identical. 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 15 • * DPPL JW D DPPL* JW D * 10 % 20 40 60 20 40 60 20 40 60 R (mm/h) 20 15 15 10 S . 10 20 40 60 Fie. 5.17 Plots showing the fitted relationships to the experimental and simulated results for the June 30 2001 rain event. In general the theoretical results were more linear than those obtained experimentally. The experimental results for AaVg-R and the simulated results for AA-R show the highest degree o f linearity. Table 5.2 Power Law fits to the experimental and theoretical results o f the 30th June, 2001 event. The 95% Cl of the fitted relationships are given as a ± q , and b ± Oj, Model A =a R t AA-R A„-R Av-R A*v*-R a Fits to the experimental results-DPPL b o. <h 0.0185 0.2012 0.3033 0.2416 1.3809 1.0996 0.9108 1.0140 0.0250 0.2294 0.3167 0.2633 0.3598 0.3091 0.2875 0.2973 Fits to the theoretical results -JWD b O. Ob 0.0392 0.9789 0.0074 0.0524 0.0654 0.0838 0.3557 0.9299 0.0777 0.9197 0.0850 0.3040 0.0709 0.9253 0.0842 0.3298 a 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.7.2 Event #2 June 21st 2000 Another single cell storm had passed over the DPPL on the 2151June 2000. The link was deployed over the same path as that presented earlier. The shower associated with this system lasted for about 25minutes, with a maximum rain rate of 174 mm/h estimated by the ORG. The total accumulation estimated from the ORG measurements was 11.45 mm while the TE525 measured 10.4 mm. The JWD was not available to corroborate either o f these accumulations. It is assumed that the TE525 would have underestimated the actual accumulation as these tipping bucket rain gauges are noted for underestimating rainfall with high rain rates. This is generally thought to be due to water loss between “tips” which becomes more of a problem at the higher rainfall rates (Adami and Da Deppo, 1985; Nystuen, 1999). The wind gust and wind speeds were fairly constant throughout the event with mean values of 10 m/s (22.3 mph) and 5 m/s (11.2 mph) respectively. The wind directions varied continuously during this event between the West and Northwesterly directions. Figure 5.18 shows the data taken over the course of this shower after it was smoothed with a 2-min moving average filter. About 3.5-minutes of data was lost at the tail end of the storm. 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21#t Jun«. 2000 60 180 40 — 120 € 60 /U L 17.2 17.6 17.6 18.2 18.6 18.6 Tlm« (hr.) 30 160 22.5 S“ — 90 > £ 7.5 - ./Vjru 17.2 17.4 17.6 17.8 18.2 18.4 18.6 18.6 180 —(135 _ r > 0.5 — 17.2 17.6 17.8 18.2 16.4 18.6 18.8 Tlmt (hr.) Fig. 5.18 DPPL receiver output and ORG estimated rainfall measurements taken during the June 21st 2000 rain shower. The results o f the attenuations estimated from these measurements are shown in Fig. 5.19 as functions of the ORG rain rate measurements. In the absence of the DSD measurements for this event it is difficult to comment on the physical changes occurring in the rain and how they may have influenced the measurements. Despite differences in the magnitudes o f the rainfall rates and the corresponding attenuation rates to those o f the June 30th 2001 event, there are a few similarities. As with the previous event this storm has a structure that is dominated by a well-defined convective core. Again there are two distinct trails o f attenuation values corresponding to the regions o f the shower before and after the rainfall peak. Similarly the attenuation values obtained after the rain peak exceeded those preceding the rain peak. For the first 6-minutes o f the shower the response in the H-pol returns to the rapidly changing rainfall rates lagged behind that o f the V-pol return. This resulted in V/H, and consequently the estimated AA, being unusually low over these high rainfall rates. 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 ** June. 2000 10 SO 100 60 150 50 100 150 200 150 200 60 150 200 50 100 R (mm/h) R(mm/h) 200 50 200 R(mm/h) 100 O' SO 17.5 17.6 17.7 Time of Day 17.8 17.9 Fig. 5.19 Microwave attenuations estimated from DPPL measurements taken during the June 21“ 2000 rain event. Also shown are the regression curves describing these relationships. The other parameters do not appear to be similarly affected, but instead display trends consistent with our expectations. Figure S. 19 show the results o f the power law relationships o f these attenuation values regressed against the rainfall rates. The power law fits to these relationships are given in Table 5.3. Table 5.3 Power Law fits to the experimental and theoretical results of the June 21 2000 event. The ‘the fitted re ationships are given as a ± o a and b ± Ob 'its to the experimental results Model a b oa Ob A =a R b 1.3894 0.0089 0.0047 0.3943 AA-R 0.5439 0.8530 0.1258 0.0496 Ah-R 0.7224 0.7784 Av-R 0.1146 0.0343 0.6285 0.1164 0.8153 0.0398 AaVii"R Note that the exponent ‘b ’ of the AA-R relationship for this event is similar to that of the previous event. However, the fits to the other parameters show less similarity to those of the previous shower. 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.7.3 Event #3 September 28th 1996 Another single cell event was observed by the DPPL on the 28th September 1996. On this occasion the DPPL was deployed over a 255m path between the roofs o f the Walker and Kern buildings on the Penn State University Park campus. The 46-cm comer reflector was located on the roof o f the Kern building with the receiver/transmitter on the roof o f the Walker building. Scattered showers were observed for most o f the day begin around 5 :10am and ending at 10:25pm. One convective shower occurred at about 3 :10pm. This was part o f a larger system that lasted from 2 :15-3:45pm. 28th September 1996 4 0 0 1— 350 300 5250 £ 200X 150 - 100 — 18 0 -7 0 / - 60 -5 0 -4 0 - 50 ? 4 .2 —>rv>|. 1 4 .4 1 4 .6 1 4 .8 15 1 5 .2 1 5 .4 1 5 .6 Time of Day (hr.) 400 r 350 300 - 80 70 60 50 40 5*250 - £.200 10 15.1 - > 150 - 100 - 50 - 4.2 1 4 .4 1 4 .6 1 4 .8 2 15 Time of Day (hr.) 1 5 .2 1---------- 1 5 .4 1 5 .6 T 80 1 .7 5 1.5 1 .2 5 i 60 50 40 1 0 .7 5 0 .5 0 .2 5 4.2 20 1 4 .4 1 4 .6 1 4 .8 1 5 .2 1 5 .4 1 5 .6 Time of Day (hr.) Fig. 5.20 DPPL receiver output and ORG estimated rainfall measurements taken during the September 28 1996 rain event. The total rainfall accumulation measured by the TE525 was 30.2 mm while that measured by the ORG was 33.32 mm over the 24-hr period. The 5-min averaged measurements from the Weather Station were not available for this event. The data corresponding to the 90-min period during which the convective portion o f the shower occurred is shown in Fig. 5.20. Note that the ratio o f the return power was almost 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. unity for most of the shower outside of the main down pour. The system gain, as observed from the signal levels outside of main rain shower, exhibited much of the usual fluctuations throughout the day. It is however clear that the alternate polarizations of the return signals were well synchronized with each other and were of almost the same magnitude over these periods. The rainfall was generally very light and more akin to a drizzle. Generally the small spherical drops that would tend to attenuate both polarizations equally would dominate this type of rain. In the convective core of the shower there would be a tendency to have larger drops with non-spherical shapes. The oblate spheroidal shapes of the larger drops would tend to attenuate the H-pol signal more than they would the V-pol signal. As a result the ratio V/H would tend to increase as shown in Fig. S.20. The presence of the guardrail on the catwalk of the smoke stack with the regularly spaced structures (bars) may have contributed to some of the differences in the orthogonal returns of the previous events (i.e. Event #1 and Event #2). No such structures were present on the roof of the Kern Building while the measurements in Fig. 5.20 were being taken. This may help to explain why the V-pol and H-pol returns in these measurements are so similar. The attenuations calculated from these results are shown in Fig. 5.21 along with the rain rates estimated by the ORG. Note that unlike in the previous showers, higher attenuation values were obtained from the region of the storm preceding the maximum rain rate, than for those corresponding to the rain rates following the maximum rainfall rate. Without the benefit of disdrometer measurements showing the DSD profile for the storm it is difficult to determine conclusively the reasons for this difference. 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28th Septem ber 1996 30 20 X 0.5 20 60 80 30 30 20 20 20 60 80 20 60 80 0.5 1.5 >10 20 60 80 80 60 oT20 15 15.2 15.4 Time (hr.) 15.6 Fig. 5.21 Estimated microwave attenuations and rainfall rates for the September 28th 1996 rain event. Also shown are the results for the regression fits given in Table 5.4 for the respective parameters. In this case it may be assumed that the larger drops appeared at the first phase (first 3-minutes) o f the shower resulting in the higher attenuation values. The rates of attenuation on the V-pol and H-pol signals were very similar, and resulted in values o f the AA being lower than in the previous events with similar rainfall rates. The results for the power law fits to this data are also shown. The coefficients and exponents for these relationships are given in Table 5.4. Table 5.4 Power Law fits to the experimental and theoretical results of the event of September 28, 1996. The 95% Cl of the fitted relationships are given as a ± g , and b ± qj, Model Fits to the Experimental F.esults a b Ob Ob A =a R b 0.0415 0.0870 0.6813 0.1246 AA-R 0.8034 0.2626 0.1025 Ah-R 0.6583 0.5805 0.8136 0.2348 0.1038 Av-R 0.2485 0.1030 0.6193 0.8083 A™-R 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. With the exception o f the AA-R relationship, these fits show more similarity to those of Event #2 than to the results o f Event #1. 5.7.4 Event #4 Ju n e 16th 2001 The first significant rain event of the Summer 2001 campaign that was observed by all four instruments occurred on June 16, 2001. This storm lasted from 3:55-6:30pm with the heaviest showers occurring within the first hour of the event. The total rain accumulations estimated for this event by the different instruments were as follows; the ORG S0rg = 60.55 mm, TE525 £te 5 2 s = 49.6 mm, JWD EJWd = 56.89 mm, and DWS Zdws = 57.91 mm. For this event rainfall rates in excess o f 100 mm/h were recorded by the ORG (Rmax = 112.2 mm/h) and JWD ( R ^ = 107.2 mm/h), and with the TE525 measuring rainfall rates as high as 138.4 mm/h! This is quite suspicious for an instrument that is known to underestimate the rainfall accumulation for events with high rainfall rates. Furthermore, the total accumulation measured by the TE525 underestimated the mean accumulation by approximately 13% indicating significant water loss by this instrument. It is therefore clear that the TE525 measurement is unreliable. An average accumulation of ZJvg = 58.45 mm was estimated from the measurements of the other three instruments. The rainfall rates of the individual instruments were then adjusted to give the same mean accumulation when integrated over the entire event (i.e. the mean accumulation was distributed over the entire event). Figure 5.22 shows the filtered returned signals along with their ratios, and the ORG estimated rainfall rates for this event. Both the V-, and H-pol return signals are generally well correlated with the rain rate. However, their ratios V/H show less correlation to the changes in the rainfall rate. At the end of the shower the reference level of the H-pol return signal was approximately 1.8-dBm below its mean value before the storm, while the V-pol signal just about returned to its former clear air reference level. This can also be observed from the ratio V/H. It is not quite clear what may be responsible for this as this situation was observed to have persisted for the remainder of the evening and throughout the night. This was also observed in the data for the event of June 30u> 2001. However, during that event the signals returned to their pre-shower levels about 30-minutes after the shower ended. 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16th June. 2001 120 100 — 60 1 J 40 20 * 15 15.5 16 16.5 17 700 600 17.5 Time (hr.) 18 18.5 19 19.5 120 100 60 o 40 § 20 « 200 100 15 ~ 80 % 5-500 1.400 >300 16.5 15.5 17 4 17.5 Time (hr.) 18 19 19.5 T 20 120 90 € 60 — 30 15.5 16 16.5 17 17.5 Time (hr.) 18 16.5 19 19.5 20 Fig. 5.22 The time series o f the receiver output (V, H) and their ratios (V/H) along with the ORG estimated rainfall rates for the June 16th 2001 rainfall event. The signal fluctuations appear to be affected by the variations in the temperature and relative humidity. Changes in temperature and humidity may affect the electrical properties (permittivity) of the atmosphere. However, this is unlikely to have differing effects on the orthogonal returns. Figures 5.23 shows the relative humidity and temperature in the vicinity o f the link before and after this event. Half an hour before the shower began the relative humidity was about 72% while the temperature hovered around 26 °C. Within the first 5-minutes of the shower’s commencement there was a 1°C increase in the temperature (from 26.5 to 27.5 °C) before the temperature began falling (the temperature at the end of the shower was approximately 19°C). Over the same period the relative humidity increased from 72% to 97%, eventually saturating at 100% before the shower ended. The relative humidity remained at 100% for the next three hours. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 90 80 70 60 vonvys H(mvys R^flnm/h) 50 T D vW *C ) T „ D P P t.("C ) H um ky% ) 40 30 20 10 15 10 20 25 Time (hr.) Fig. 5.23 Variations in temperature and relative humidity during the course o f the day of June 16th2001. Also included are the standardized receiver output signals along with the ORG estimated rainfall rates. Figure 5.24 gives the time series o f the wind velocities over the course of the shower. At the beginning of the shower, the wind direction changed from ENE (East-North-Easterly) to WNW, with wind speeds and wind gust as high as 13 mph and 20 mph respectively. For the next 1.5 hr., the wind blew in the ENE direction with wind speeds o f almost 2-mph. The rainfall accumulation estimated for the June 16th event is almost 20-times that of the June 30th event which lasted for only 15 minutes. I ll Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Re^dnnVh) W1ncKcm/s)/10 Gust(cnVs)/10 Dlrfdeg./S) 100 80 60 40 20 5.5 16 16.5 17.5 17 Time of Day (hr.) 18 19 18.5 Fig. 5.24 The time series of the wind vectors during the course o f the June 16th 2001 rain shower. Figure 5.18 shows that the structure of this storm is different to those previously discussed. The multiple peaks in the rainfall profile suggest a multi-cell convection system. The resulting attenuation values given in Fig. 5.25 show a more complex profile to that o f the June 30th event that was associated with a single-cell event. The higher degree of scatter in these results is believed to the result of the greater variation in the physical characteristics of the rainfall in this event. 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I.S 16 40 60 80 R(mmAi) 100 120 120 40 60 80 R (mm/h) 100 120 18.5 5 10 AA(dBAon) 40 60 80 R(mmAi) 100 120 40 60 80 R(mm/h) 100 16.5 17 17.5 Time of Day 20 Fig. 5.25 Microwave attenuation measurements estimated form the DPPL receiver output given in Fig. 5.22 (dots) as functions o f the ORG rainfall estimates. The asterisks give the theoretical results derived from the JWD measurements. Also shown are the regression fits to the experimental and theoretical results. The time series of the attenuation values estimated from both the DPPL and JWD data are presented in Fig. 5.26 along with the estimated rainfall rates. As expected the simulated attenuations were well correlated with the JWD estimated rainfall rates. Note the difference in the response o f the DPPL estimated Ah, Av, Aavg, and AA parameters to the various rainfall peaks. All four parameters detected the first rain peak. The response o f the AA to this rain peak exceeds its response for other higher rainfall rates. This was due to the H-pol signal response lagging that the V-pol response as the rainfall rate increases. The relatively large variations in V and the corresponding smaller variations in H result in a large increase in V/H. In Fig. 5.26 the third and fourth rainfall peaks are not easily identifiable from the estimated AA values. Although the fourth rainfall peak is almost o f the same magnitude as the second rain peak in R org and R jwd . the corresponding Ah, Av, and Aavg values coinciding with the fourth rain peak are less than that 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. corresponding to the second rainfall peak. It is possible that Ah, Av, and A*vg maybe more accurately representing the average change changes in the rainfall rates over the 2-way path of the link and not just at one specific location as in the case of the ORG and JWD. 30 30 25 (A — 25 20 | 15 £ 15 Av JW O 20 10 16 16.5 17 17.5 16 16 16.5 17 17.5 16 6 15 org 5 AA D P P L 10 ".W 20 4 3 2 1 5.5 16 16.5 17 Time(hr.) 17.5 18 O1— 15.5 16 16.5 17 17.5 18 Fig. 5.26 The time series representations o f the experimental and theoretically derived microwave attenuations estimated from measurements taken during the course of the June 16th 2001 rain shower. The first 84-minutes of the DSD have been analyzed. Plots showing these distributions have been included in Appendix G. Figure 5.27 shows the time series representations of the variations in the median size diameters (D0) and the drop concentrations with the rainfall rates. It is possible to identify four distinct peaks in the rainfall rate; both from the plot of the rain rate measurements and the drop spectra data. These peaks occurred at approximately 12-minute intervals beginning with the first peak that occurred 3-4 minutes into the shower. Unlike the previous events that began with a slow drizzle, this shower began suddenly with a heavy downpour. The number of drops measured by the JWD increasing from about 38 in the first minute to over 500 drops per minute by the time the first peak appeared 3-minutes into the shower. Initially the drops striking the JWD were almost uniformly distributed over the range o f detectable sizes with drop 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sizes exceeding 5 nun being observed. It is for this reason that the large values for D0 were observed over this period. 100 50 0 2 10 ' 15 16 17 TIme(hr.) 18 5.5 16 16.5 g 18. 17 17.5 Time(hr.) 600 7 ?5.5 16 16.5 17 17.5 Time(hr.) 18 18.1 f5.5 16 16.5 17 17.5 Time<hr.) 18 Fig. 5.27 Time series representations o f the changing DSD, drop concentrations, D0, and the rainfall rates estimated by both the ORG and JWD for the June 16th 2001 rain event. A number o f changes in the DSD were observed at each rain peak. These would invariably result from some combination o f the DSD becoming narrower (wider) to include (exclude) the larger drop sizes, along with the total drop counts and drop concentrations being modified. At the first rain peak the DSD had an almost exponential shape. Thereafter the spectra became narrower and more “concave down”. The drop concentration increased for D < 2.0 mm, while the number o f larger drops decreased. This decrease in the larger drops can be observed from examinations o f the individual spectra and the corresponding values o f D0. A more detailed discussion o f these changes can be found in Appendix G. There is a somewhat unusual increase in the D0 between 17.5-18.5 hrs. These appear 30-90 minutes after the main convective activity had passed over the link. The rainfall over this period was rather light (< 6 mm/h) and could best be described as stratiform. Such showers are generally characterized as being comprise chiefly o f smaller drops. However, these large D0 values suggest the presences of much larger 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. drops. This may provide further proof that these showers may have in fact resulted from a larger multi cell convection system. The typical convection storm is often considered to be comprised o f “cells” each of which may be considered to be a couplet composed of an individual updraft and an associated downdraft. Large convective systems such as the one currently being discussed, are thought to be capable o f inducing large anvils over areas considerably larger than that o f the up and down draft regions. In such cases, stratiform rain from the anvil may appear at the surface long after the main rain shaft has passed, giving rise to long “tails” in the rain rate measurements as can be observed in the above plots. The large values of D0 observed in such rain may result from larger drops falling out o f the anvil long after the main shower has passed. The power law fits to the results of the experimental and simulated rainfall and microwave attenuations were also determined. The results o f these fits are shown in Fig. 5.25. The fits to these relationships are given in Table 5.5. Note that despite the fact that their exponents are quite different in Table 5.5, the slopes of the R-Ah and R-Aavg line fits for the experimental and theoretical results appear quite similar. Table 5.5 Power Law fits to the experimental (DPPL) and theoretical (JWD) results for the event of June 6, 2001. The 95% Cl of the fitted relationships are given as a ± q a, b ± ob, q ± a a, and ft ± g BModel a AA-R Ah-R Av-R Aavc"R R-AA R-Ah R-Av R— A jve 0.7821 0.7618 0.4576 0.6105 a 16.0411 2.5775 6.2023 3.8867 Experimental b 0.5177 0.7559 0.7743 0.7627 0.1599 0.0603 0.0464 0.0480 P 0.6353 1.0954 0.9211 1.0220 2.0317 0.2544 0.4017 0.2941 Oa <*> 0.0505 0.0193 0.0246 0.0191 <Tb 0.0648 0.0340 0.0258 0.0278 a 0.0332 0.3445 0.3169 0.3304 a 25.1826 3.3066 3.7641 3.5184 Theoretical b O. 0.0029 1.0582 0.0363 0.9255 0.9016 0.0361 0.0361 0.9145 CTb 0.0213 0.0256 0.0277 0.0265 P 0.9349 1.0622 1.0865 1.0733 aB 0.0203 0.0338 0.0395 0.0363 Oa 0.5307 0.3099 0.3847 0.3429 The relationships for the experimental results are quite different to those obtained for the previous events (Event #1-3). The fits to the theoretical results, especially the Ah-R relationships, show more similarity to the fits to the theoretical results o f Event #1. The differences in these relationships are indicative of the difference in the rainfall types. Figure 5.28 shows the rainfall rates estimated from the experimental and theoretical attenuation values. The solid lines give the experimental results and the theoretical results are shown by the dots. Different colors are used to distinguish between the R-A relationships. Note that R-Av generally tended to 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. overestimated the rainfall except for the highest rainfall rates. The R-Awg model performed well for R < 60 mm/h relative to the theoretical model. Here the differences in the results o f the experimental and theoretical R-Ah models are more apparent, and appears to represent a general bias in the values. 120 100 80 60 40 20 10 15 Atterajation(dB4on) 20 25 30 Fig. 5.28 Comparing the rainfall rates estimated by the from the experimental results (RA DPPL) and those derived from the disdrometer measurements (RAJWD) The time series o f the rainfall rates from the simulated and experimental attenuations results are shown in Figure 5.29. It is clear that the rainfall rates derived from the differential attenuation measurements are not as well correlated with the ORG estimated rainfall rates as those estimated by the single polarization or average attenuation parameters. The DPPL derived rainfall rates underestimate the third and fourth rain peaks. This is consistent with the lower than expected attenuations that corresponded to these ORG rainfall rates as was observed in Fig. 5.26. 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 120 ora 100 100 80 80 60 60 40 40 20 20 16 16.5 17 Time(hr.) 17.5 15.5 18 100 120 80 100 16 16.5 17 Tlme<hr.) 18 R . . JW D 80 60 17.5 60 S ' 40 “ 20 40 20 16 16.5 17 Time(hr.) 17.5 18 5.5 16 16.5 17 nm e(hr.) 17.5 18 Fig. 5.29 The time series of the rainfall rates estimated from the R-A models given in Table 5.5 The rainfall accumulations estimated from the results given in Fig. 5.29 are shown in Fig. 5.30. Here the estimated rainfall accumulations are given as functions o f time (i.e. time series o f the rainfall accumulations). 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 70 60 SO 40 R DPPL 30 JW D 20 10 5.5 16 16.5 1 7 .5 18 1 8 .5 Fig. 5.30 The cumulative rainfall derived from the link (DPPL) and the disdrometer (JWD) measurements for the June 16th 2001 rain event. Note that in Fig. 5.28 the rainfall estimated from the experimental differential attenuations (AA) underestimated the maximum rainfall rate by approximately 15%, however, Fig. 5.30 indicate that the cumulative rainfall estimated from these experimental values over estimate the event accumulation by approximately 26%. This is indicative o f the error (scatter) in the measurements o f AA. The relative error in the cumulative rainfall estimated from the experimental results of Ah-, Av, and Aavg were all less than 7% with the Ah giving an accumulation within about 1% of the ORG estimated accumulation. For Event #1 (30th June 2001) the accumulations estimated from the A^g and Av measurements were within 10% of the ORG estimated results. The accumulation estimated from Ah was within 20% o f the expected value while the error in the AA estimated results were around 80%. Note that Event #4 lasted for almost 2-1/2 hrs, while Event #1 had a duration o f about 15-minutes. It is expected that the average error in these results would be reduced over a longer period. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.8 Conclusions The stability o f the DPPL has been assessed from <r-x plots where a represents the Allan deviation in the DPPL measurements integrated over some period x. From the o-x plots it is clear that the differential processing (V/H) produced the noisiest attenuation measurements, while the single polarization processing o f the V-pol signal resulted in the most stable attenuation measurements. As a result of this the differential processing was observed to give the least accurate estimates of the rainfall rates. The increased noise in the differential signal is due to the fact that the variances in the orthogonal signals are not identical nor are they well correlated. Therefore taking the ratio o f these two incoherent signals lead to an increase in the variance. Also, the fluctuations in the DPPL's signal gain were strongly correlated to variations in the temperature and relative humidity. Experimental results of the attenuation of the orthogonal polarizations o f a 35 GHz microwave signal propagating through different rainfall events have been presented. These experimental results have been compared to theoretical results for selected rainfall events. The experimental results showed general trends that were consistent with the results of the simulations. The largest discrepancies were observed in the measurements of AA, and the relationship between these measurements of AA and the rainfall rates. Evidence of real-time variations in the raindrop size distributions DSD were observed in the DPPL estimated attenuation measurements. Similar effects were not as strongly manifested in the results of the simulated attenuations. Nevertheless, supporting evidence for the changing DSD was obtained from an examination o f disdrometer measurements taken during two of the rain events discussed. Systematic variations in the DSD were observed in these two events. The period over which these changes in DSD occurred varied with the rainfall type. The variations in the DSD caused variations in the microwave attenuation-rainfall rate signature which differed depending on the type of rainfall. For complex multi-cell storms different signatures were observed in different phases of the storm. The rainfall peaks (maximums) and troughs (minimas) appear to represent regions of transition in the variations in DSDs. The rainfall-attenuation signature for each rain event was generally different. Consequently different relationships were obtained for each event. The rainfall rates and the cumulative rainfall estimated from 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. these relationships were compared for the June 16<h 2001 rainfall event. The results indicate that the cumulative rainfall estimated from the Ah, Av, and Aavg parameters were generally within 7% of the expected accumulation with the R-Ah estimate being within 1%. On the other hand the accumulations estimated fro the R-AA models overestimated the expected rainfall accumulations by approximately 26%. For Event #1 (30Ih June 2001) the accumulations estimated from the Aavg and Av measurements were within 10% o f the ORG estimated results. The accumulation estimated from Ah was within 20% of the expected value while the error in the AA estimated results were around 80%. Note that Event #4 lasted for almost 2-1/2 hrs, while Event #1 lasted for about 15-minutes. The error in the rainfall accumulation estimates reduces over a longer period. The propagation link shows promise despite obvious problems in the measurements. 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6 THE RELATIONSHIP BETWEEN RAINFALL RATE, SPECIFIC DIFFERENTIAL PHASE, ATTENUATION AND THE REFLECTIVITY AT CENTIMETER AND MILLIMETER WAVELENGTHS It has always been assumed that the higher rates of atmospheric attenuation observed at higher frequencies will prevent the operation o f any earth based free-space system operating at these frequencies (Skolnik, 1990). However, the demand from the communications engineering community for high bandwidth capacities, along with recent technological improvements making radar operations at centimeter and millimeter wavelengths more practicable and hence available. Consequently, researchers have recently turned their attention to studying wave propagation in the atmosphere at these shorter wavelengths. 6.1 Background There is a growing interest in the interaction o f higher frequency signals with rain. The introduction of a number or research instruments operating at higher frequencies have provided the opportunity for such studies. For example, the recent proposals for the use of Ku-band frequencies for rainfall measuring schemes in urban areas and mountain valleys (Holt et al. 2000), along with the proposed introduction of satellite systems using Ku and Ka-band frequencies to satisfy the demand for broad band communication capabilities (Rogers et al. 1997, Arbesser-Rastburg and Paraboni, 1997 Karasawa and Maekawa, 1997, Cane and Rogers 1998 etc.). Also, the increasing demands for precipitation radars aboard airplanes and satellite platforms (e.g. TRMM, Simpson et al. 1988, 1996; BEST, Im and Li, 1998) would require the use of smaller antennas operating at higher frequencies. For the past two years rain rate studies have been conducted at The Pennsylvania State University (PSU) using a 35 GHz Dual Polarization Propagation Link-DPPL (see Ruf et al. 1996, Aydin and Daisley, 1999, 2000, 2002). In recent years there has been an increase in the number of radars operated at 94 GHz (Lhermitte 1990, Clothiaux et al. 1995, Gloageum and Lavergnat 1996a, 1996b, Kollias et al. 1999) for cloud physics and short path propagation studies. O f particular interest to many in the remote 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sensing community is the increased sensitivity to the small scale structures involved in the growth and development of hydrometeors that can be obtained at these smaller wavelengths (Lhermitte, 1987). It is for these reasons that I wish to study the relationships between selected radar and propagation parameters and rainfall rate (R) at X, Ku, Ka, Q and W-band frequencies. These relationships will be examined using the disdrometer measured DSDs along with the experimental drop shape models discussed in Chapter 3. Special mention will be made of the relationship between the rainfall rate and the specific differential phase shift (KDP). In particular, the very linear R-KDp relationship at W-band (94 GHz) and its possible use for rainfall estimation over short propagation paths will be discussed. We shall also consider the relationships between attenuation (A), reflectivity (Z), and R at Ku-and Ka- band frequencies. Comparisons will be made to previously published results, where possible, as we examine the effects of drop oscillations, drop canting, and DSD, along with the signal frequency, on these relationships. For this study we shall assume the temperature of the raindrops to be 10 °C. The complex dielectric constants for water at this temperature, and for the different frequencies of interest were calculated following Ray (1972) and are given in Table 6.1. Wavelength (mm) 3.189 7.495 8.565 9.993 21.410 31.995 6.2 Frequency (GHz) 94 40 35 30 14 9.4 £' 6.71186 12.21690 14.07290 16.74950 39.66280 55.14100 f Complex dielectric constants of water at 10°C ,<■*> n Table 6.1 £" 10.15310 22.12600 24.62700 27.62410 38.98790 37.93160 Propagation Differential Phase The propagation constant (k) for propagation in the forward direction was first derived by van de Hulst (1957) and was reviewed by Oguchi (1975) for a rain-filled medium. Due to differences in the refractive indices of air and the raindrops in the region through which a wave may be propagating, the phase o f the forward scattered field generally lags behind that of the un-scattered free-space component. The 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. addition of this forward scattered wave to the transmitted wave modifies the rate at which the phase of the total wave changes with increasing distance. In addition, the phase on orthogonal components o f the signal may change to varying extents depending on the scatters in the medium. As the phase change is a relative quantity, it is only meaningful when compared to some initial or other reference state. The relative difference in the phase shifts between two orthogonally polarized signals has been proposed for estimating rainfall rate (Seliga and Bringi, 1978; Jameson, 1985). The estimate of the differential propagation phase O dp (i.e. the differential phase shift on propagation) contains the back-scattered differential phase S = 6„ - SM (i.e. the differential phase shift on back-scattering), where the scattering amplitudes in the backward direction are S n =|5„.|e-'lVv and = |5 M|eJ‘>“ . Doviak and Zmic' (1993) defines the back scattered propagation phase (using e*luMtime convention) as ^ D p ( O = 0 h h - 0 n + ^ ( r o) = 2J ; Re{[M r)-M r)]< /r} + < * ( 0 ( * where km(for m = h, v) is the propagation constant at V and H, which are actually additions to the free ly space propagation constant, k0 = — , as a result of scattering from particles along the signal path. The A propagation constant is given as **.. = *. + 4 S ^ J D ) N{D) dD (6.2) for X the wavelength of the signal, Shh.w(D) the scattering amplitudes in the forward direction in units of meters, and N(D)dD is the number of drops per cubic meter in the size range D ± . The factor of 2 in (6.1) indicates that this is the 2-way definition. The backscattering differential phase shifts can be estimated from the zero-lag correlation coefficient of the co-polar return signals. Doviak and Zmic' (1993) defines this co-polar correlation coefficient (P hv) as 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( s « - S ’m ) (6.3) where Sm,,, (for m = h, v) is the scattering amplitude in the backward direction for the i* scatterer in the resolution volume, and (*) denotes the complex conjugate. The expectation is taken over all the scattering particles in the volume. The specific differential phase KDp is defined as the difference between propagation constants for horizontally and vertically polarized waves (Oguchi, 1983; Doviak and Zmic', 1993). For RayleighGans scatters 5 is very small and KDp can be estimated directly as the range derivative of <t>DP. Following Doviak and Zmic' (1993), the propagation-differential phase shift along a path of Rayleigh scattering particles may be expressed as ■jRe{S*(D)-S„(D)}/V(D)<«) 0 deg/ b n (6.4) where Re{} represents the real part of the term in parentheses. For scatterers that have dimensions comparable to the wavelength of the propagating signal, 5 is no longer insignificant and may even dominate d>Dp at some higher frequencies (Timothy et al., 1999). In such cases K0p can no longer be obtained directly from the measured 0 Dp unless 8 is compensated for, or removed. Techniques for reducing the effect of 5 on <t>DP at different frequencies have been proposed (Balakrishnan and Zmic, 1990; Tan et al., 1991; Hubbert et al., 1993). 6.2.1 Specific Differential Phase (KDp) And Rainfall Rate ( R ) Recent studies (Zmic' and Ryzhkov 1996, Ryzhkov and Zmic', 1998) have shown that the specific differential propagation phase shift (K dp) between vertical (V) and horizontal (H) polarized signals provides improvement in rainfall rate prediction over the more conventional Z-R relationships. Advantages of R-KDp relationships include: • Its insensitivity to the presence of isotropic scatterers like hail 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • It is not dependent on system gain or calibration • Almost immune to beam blockage and is less sensitive to beam filling and ground clutter cancellers • Most importantly is its lower sensitivity to the drop-size distribution (DSD). Seliga and Bringi (1978) first proposed the specific differential phase as an estimator for rainfall rate. This was subsequently followed by a number o f studies (Goddard and Cherry, 1984; Jameson, 1985; Sachidanada and Zmic', 1986, 1987; Doviak and Zm ic' 1993; Aydin et al. 1995; Zmic' and Ryzhkov 1996; Ryzhkov and Zmic', 1998 to name a few), which were all limited to S-band frequencies. More recently May et al. (1997), has compared R-KDp and Z-R based estimates using 5-cm length (C-band) polarimetric radar data from the MCTEX (Maritime Continent Thunderstorm Experiment) field program. It has been suggest that R-KDP relationships can be used for making rainfall measurements at X-band (Matrosov et al., 1999), K„-band (Iguchi et al., 1999; Timothy et al. 1999), and W-band frequencies (Aydin and Daisley, 2000). The enhanced phase signal obtained for higher frequency signals propagating through rain relative to that obtained at the lower frequencies has been one o f the main incentives for promoting the interest in R -K dp estimation at these frequencies. Matrosovo et al. (1999) have focused on R-KDp relationship to both X- and Ka-band frequencies. They concluded that for rainfall rates less than 15 mm/h the backscatter differential phase shift contribution 8 at X-band was usually very small and could be neglected. This was not the case at the Ka-band frequency where the 8 values were about one order of magnitude higher than those at the X-band frequency, while the propagation differential phase shift was only about two times larger than that at Xband. Their results also suggested that KDp was more sensitive to the drop shapes than to the DSDs. In addition to being more portable and less expensive, the X-band system, when compared to the C-band platforms, was shown to have the advantage of having reduced random statistical errors on the KDp measurements for a given rain rate, which resulted in more accurate measurements at lower rain rates. This was in part due to the fact the propagation differential phase (d>DP) is approximately proportional to the frequency of the signal. It is this enhanced phase shift at the higher frequencies relative to that obtained at the lower frequencies that has promoted KDp measurements at these higher frequencies. However, there is an associated cost of higher differential backscattered phase shift component 8. 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Nevertheless, the R-KDp estimates were shown to be superior to the Z-R based estimates even after correcting for attenuation and beam blockage of the radar echoes. In addition, the relative magnitudes of the random statistical errors are lower at the higher frequencies allowing for more accurate measurements to be made at the lower rainfall rates (May et al., 1999). It is this increased sensitivity that has encouraged some investigators to explore the use of this parameter at higher frequencies, to estimate rainfall (Matrosovo et al., 1999; Timothy et al., 1999). As mentioned in the previous chapters, the rainfall rate is calculated as R = ^ D \ l (D)N(D)dD m m/h (6.5) 0 where the terminal velocities v, (D) are given as v,(D)= 9.25 •[! - e ,-00680’' 0488D' ] m /s (6.6) following Lhermite (1988). Comparing (6.4) and (6.5) it is clear that the sensitivity of KDp to changes in drop shape and equivalent volume diameter can be assessed from the variations in the difference in the forward scattered fields Re{SM(D) - S n (D)}. At the same time the responsiveness of KDp to Rcan be estimated by comparing these variations to variations in D 3v ,(D ) with the equivalent volume diameter as shown in Fig. 6.1. In Fig. 6. la the variations in Re{SM(D) - S n (D)j over drop size for different frequencies are given for equilibrium shape drops (BCeq). In Fig. 6.1b the effect of changes in drop shapes on Re{SM( D ) - S „ ( D ) j relative to that obtained for BCeq shape can be observed when Kav oscillation shapes are assumed. The broken lines give the Kav results. 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. x10 x 10' 1.5 1.5 36G H z i0 .5 0.5 co co O' -0.5 -1.5 -1.5 D (mm) D (mm) Fig. 6.1. a) Shows the variations in Refo*( D ) - S n (D)}and ± v tD 3 with equivalent volume diameter D. b) Shows the effects of drop oscillations on Re{Shh( D ) - S vv(D)} versus D. In general Rej5w (D) - 5„(D)} increases with the angle o f incidence from vertical (0 = 0°) to horizontal (0 = 90°) incidence. Fig. 6.1 shows the results at horizontal incidence. It is evident from these results that Re{SM(Z ))-5 „(D )} is always positive over the range o f equivalent volume diameters in question for the frequencies 9.4 and 14 GHz. This ensures that their R-KDp relationships will be positive. In the case of the 14 GHz results, the variation is not monotonic, but instead has a local maximum at D ~ 5.0mm . This deviation from monotonicity occurs at sizes that are rarely encountered in rain, and is therefore expected to have negligible effect on KDpat this frequency. More importantly are the gradients o f the curves, relative to that o f D i vl( D ) , in determining how sensitive measurements of KDP will be to R at these frequencies. The variations are generally exponential with drop-size (with the exception o f the 14 GHz curve for D > 4.5 mm) and may be suitably described by power-fit models: F(D) = a D b (6.7) 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where F(D) = Re{SM(D) - S„,(D)}. Table 6.2 shows the results of these fits. Table 6.2 Power model fits to the difference Re{Sw ( D ) - S m(D)}as a function of D Frequency (GHz) 9.4 14 30 30 35 35 40 40 94 94 Range F{D) = a D b a 4.93E-9 1.23E-8 2.71E-9 -3.90E-9 2.73E-8 -1.61E-8 -2.84E-8 -2.84E-8 4.02E-9 -4.49E-8 D(mm) 0.1 <D <6.0 0.1 < D < 4.5 0.1 < D < 2.4 3.0 < D <6.0 0.1< D<2.1 2.5 < D <6.0 0.1 <Z><1.8 2.2 < D <6.0 0.1 < D <0.8 0.9 < D <6.0 b 4.52 4.19 3.53 4.41 3.32 3.62 3.55 3.29 2.66 2.79 The model corresponding to the 14 GHz curve is only valid for D < 4.5 m m . A very similar response was observed at the C-band frequency of 5.6GHz (not shown). Fitting Re{S^(D) - S n.(D)} at C-band to equivalent volume diameter for D < 5 .1 mm gave an exponent of 4.44. Note that the corresponding fit for D i vl (D) over the equivalent volume diameters 0.1 < D < 6.0 mm goes to the power 3.67. From the above results it would appear that the variation in Re{5M(D) -£„.(£>)} with D at 40 GHz is almost parallel to that of D 3v ,(D )as the values of their exponents are indeed quite close. However, a closer examination of the results of Fig. 6.1 indicates that the curves corresponding to 94-, 40-, 35-, and 30 GHz have positive values for D < 0.85, 2.12, 2.45, an d 2.92 mm respectively. The corresponding local maximums were at D = 0.8, 1.8, 2.1, and 2.4 mm respectively. This parallels the behavior o f the 14 GHz curve for D > 4.5 m m , however at these higher frequencies the curves have both positive and negative values. Note that the Re{S^(D) - S^(D)} curves peak at equivalent diameters that are approximately equal to ^ . These maximum values occur at sizes that are at the transition to the resonance scattering regions at the respective frequencies. 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The 94 GHz curve is positive over the range of small drop sizes (D < 0.85 mm) that are unlikely to contribute significantly to either the rain rate R or KDp except possibly at very low rain rate values. For D > 1.0 mm the slope o f the Re{SM(D )-S„,(D )} curve is very close to that of - D 3vf(£>). Fitting Re{SM(D) - S„(D)} to - D 3v,(D) in this region produces an exponent b = 1.05 at 94 GHz. Therefore, the rainfall estimated from the R-KDp relationship at 94 GHz should have little dependence on DSD. On the other hand the slopes of the curves at 30, 35 and 40 GHz change from positive to negative over the range of drop sizes that are very significant to both R and KDP (i.e. D = 2.92, 2.25, and 2.12 mm respectively). As a result of this, the integrands of R and KDp are significantly different over these critical ranges of drop sizes. Consequently, the resulting positive and negative values of Re{SM( D ) - 5 ^ (0 ) } will be weighted comparably by the DSD for moderate-to-high rainfall rates, with the likelihood of having both positive and negative KDp values at similar rainfall rates. The scatter plots of the rainfall rate as a function of KDp at all six frequencies, and for all three disdrometer DSDs are shown in Fig. 6.2. For these simulations equilibrium shaped drops (the BQ,, shapes) with their fall orientations normally distributed in the polar elevation direction N(ju =0°,cr = 5o), and uniformly distributed in the azimuthal direction £/(0, = 0°, <pz =360°) has been assumed. 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 9 .4 G H z HOO 14G H z 30 G H z 35G H z 40G H z H00 94G H z 10 -10 10 -15 -10 -5 10 -10 10 -15 -10 -5 0 100 f 0 SW ISS 5 10 100 0 5 10 Kop(deg./km) t 0 10 20 -10 14GHZ f 0 10 -10 \ 0 I 0 10 20 -10 0 10 -10 O 10 -10 0 10 -15 -10 -5 0 KDp(deg./km) Kop(deg./Km) KQp(deg../km) Ko p (deg./km) KQp(deg./km) Fig. 6.2 Simulation results showing the relationship between the KDP and R estimated from the MISS, SWISS, and TROP disdrometer data given in the first, second and third rows respectively. Each column shows the results at the indicated frequencies. As expected the Kdp at 9.4 GHz and 14 GHz are always positive. The scatter in the results is due to R and KDp having different sensitivities over the drop sizes at these frequencies (see Fig. 6.1). However, the results at these two frequencies are very similar. The only significant difference being that the KDp values at 14 GHz are higher than those at 9.4 GHz. This is due to the fact that 4>Dp scales approximately as -y , such that the smaller wavelength gives higher Kdp values. Note the high degree of scattering in the R-KDp results at 30-, 35-, and 40 GHz. This is due to the R e { S * (0 )-S w(D)} curves having local maximas within the range of critical drop sizes 1 .0 < D<3.0mm where D 3v,(D) is monotonic. Hence, the slopes of these curves are significantly different to that of D 3v,(D) over these sizes. In addition, the Re{5M(D) -S„,(D )} curves become negative at sizes where the DSDs have high frequencies o f drop counts. Because of this, Re{SM( D ) - S n (D)} becomes heavily weighted by these sizes such that the resulting KDp have both 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. positive and negative values. It is therefore unlikely that any useful R-KDP relationships can be derived at these frequencies (except possibly at 30 GHz for low rainfall intensities in temperate climates where smaller drop sizes are more likely). Figure 6.3 shows the median volume diameter D0 plotted against the corresponding KDP values at these frequencies. On closer examination of these plots it can be observed that (with the exception of the 30 GHz TROP results) the KDP values become negative at D0 = X/4. It should be of interest to note that for the 30 GHz results the TROP distribution exhibits the least scatter with only four data points having negative KDP values. This is because the drops in this distribution are concentrated over the range 0.5 < D < 2 .0m m . At this frequency the Re{SM(D) - S n (D)} curve is positive for D < 2.92 mm with its local maximum is at 2.4 mm. The dynamic range of the R-KDP relationship varies with frequency. The largest range of KDP values were observed at 14 and 94 GHz. The TROP distribution gave a dynamic range of approximately 10 deg/km at both frequencies. However, for the SWISS and MISS distributions the range of KDP were above 15 deg/km, with their estimated values at 14 GHz exceeding those at 9.4 GHz by 24% and 36% respectively. 132 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30G H z 40 G H z 35G H z MISS ► *v -10 KjjpCdeg./km) KOP(deg./km) KOP(deg./lon) Fig. 6 3 Scatter plot of the median volume diameter D0 for the three DSDs and K dp at 30, 33 and 40 GHz The results presented in Fig. 6.2 suggest that an almost linear relationship exists between the rainfall rate and the KDP at 94 GHz. This is due to both Re{SM(D )-S „ (D )} and D3vt(D) having similar sensitivities over the range of drop-sizes 1.0< D<3.0mm that contribute significantly to both KDp and R respectively (i.e. they have similar dependencies on the drop size diameter over this range). The compactness of these plots indicates that at this frequency KDp is insensitive to the smaller drops (D < 0.9 mm) where Re{Sw(D) -S „ (D )} is positive and not monotonic, and is instead dominated by the larger drops. Power model fits of the form R=aK DP (mm/h) (6.8) were derived for the R-KDp relationships at 9.4, 14, and 94 GHz. These relationships are given in Table 6.3. Table 6.3 also includes the fits to this relationship at 2.7GHz (S band) and 5.6 GHz (C-band) for 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. comparison with previously published results (see Appendix H). Results have also been included for the oscillation (average) shapes. Table 6.3 Power model fits describing the R-KDp relationships at the X-, Ku and W-band frequencies. Frequencies 2.7 GHz 5.6 GHz 9.4 GHz 14 GHz 94 GHz DSD Shape b a b b a b a a a (95%CI) (95%CI) (95%CI) (95%CI] (95%CI) (95%CI) (95%CI (95%CI) (95%CI) b (95%CI) MISS BCa, 39.9077 0.8252 20.7444 0.8086 13.8214 0.8537 10.0925 0.8477 8.7255 (0.0746) (0.0017) (0.0575) (0.0019) (0.0356) (0.0015) (0.0312) (0.0015) (0.0103) 0.9605 (0.0006) ABL,, 46.6204 0.7991 24.5937 0.7823 16.8622 0.8295 12.3794 0.8221 (0.1012) (0.0020) (0.0714) (0.0021) (0.0470) (0.0071) (0.0415) (0.0017) 11.4063 (0.0143) 0.9359 (0.0007) Kav 51.2189 0.8230 26.5508 0.8059 17.9345 0.8498 13.0241 0.8442 (0.0998) (0.0018) (0.0668) (0.0019) (0.0425) (0.0015) (0.0373) (0.0015) 12.0042 (0.0122) 0.9565 (0.0006) BCo, 34.2161 0.7318 19.0769 0.7188 13.6737 0.7594 10.2770 0.7781 (0.1293) (0.0021) (0.0652) (0.0022) (0.0395) (0.0020) (0.0318) (0.0020) 9.2005 (0.0193) 0.9198 (0.0016) 38.0214 0.7009 21.6339 0.6888 15.9684 0.7271 12.1960 0.7459 11.8607 (0.1689) (0.0022) (0.0842) (0.0023) (0.0512) (0.0021) (0.0401) (0.0022) (0.0249) 0.8797 (0.0017) K1V 42.1856 0.7355 23.4023 0.7226 16.8895 0.7586 12.7509 0.7807 (0.1730) (0.0020) (0.0831) (0.0022) (0.0477) (0.0019) (0.0359) (0.0019) 12.4974 (0.0264) 0.9050 (0.0018) BC„, 49.1116 0.8229 26.1997 0.8111 16.4788 0.8238 12.0690 0.8389 (0.2034) (0.0034) (0.0964) (0.0036) (0.0662) (0.0034) (0.0534) (0.0032) 8.8689 (0.0210) 0.9344 (0.0016) ABL,, 60.2506 0.8046 32.4476 0.7923 20.7293 0.8077 15.3332 0.8197 (0.3167) (0.0037) (0.1356) (0.0039) (0.0822) (0.0037) (0.0663) (0.0035) 11.9050 (0.0300) 0.9118 (0.0020) Kav 65.2329 0.8355 34.3459 0.8227 21.5691 0.8359 15.7282 0.8500 (0.3048) (0.0033) (0.1255) (0.0035) (0.0718) (0.0032) (0.0560) (0.0030) 11.8706 (0.0199) 0.9502 (0.0014) SWIS S ABLJV TROP These fits indicate that the R-KDp relationship generally becomes more linear at the higher frequencies with the relationships at 94 GHz being the most linear. At each frequency the SWISS models were the least linear. This is due to the fact that this data set has a relatively large number of drops outside the range of sizes that contribute significantly to both R and KDP, but at sizes where the sensitivities of R and KDp are very different. In general the exponent “b’ of the MISS and TROP models are very similar. The SWISS models tend to have lower values of ‘b \ This follows from the previous discussion of the SWISS data having a wider range of DSD. Although the MISS distribution (DSD) has a similar shape to that of the SWISS, it nevertheless has a higher proportion of drops over this range of sizes, as in the case of the TROP data set. Therefore, the relationships obtained from the TROP and MISS data sets tend to be more linear in R. 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For a given distribution and frequency the exponent ‘b’ is relatively insensitive to variations in the drop shape model, however the constant ‘a’ varies significantly from one drop shape model to another. This suggests that while the KDp may be insensitive to variations in DSD (i.e. ‘b’ is relatively insensitive to changes in DSD), it is nevertheless dependent on the drop shapes. This shape dependency can lead to biasing the estimates of R from the R-KDP relationships. A number of R-KDp relationships have been proposed, the majority of which have been derived for Sband frequencies: Sachidananda and Zmic' (1987) introduce the power law relationship R = 31AK°0ip66 for Marshal 1-Palmer DSD of spheroids. Chandrasekar et al. (1990) derived the expression R = 40.5 K°Dl5 from a nonlinear regression to R and KDp values derived for gamma model DSDs with Green's (1976) axial ratios. For these simulations the parameters o f the gamma DSD (N0, D0, and n) were randomly varied over their respective ranges for a maximum drop diameter D = 8.0 mm. Aydin and Giridhar (1992) give two-piece relationships for S- and C-band frequencies. These relationships have been included in Appendix H. Keenan et al. (1997) reported the relationship R = 34.5K£“3 for C-band (5-cm) radars, which is almost identical to that given in Table 6.3 for the TROP Kav model at 5.6 GHz (i.e. the coefficients are within the 95% Cl of those given in Table 6.3). This model was derived using T-matrix simulations with MCTEX (Maritime Continent Thunderstorm Experiment) DSD data, and assuming Kav shapes. Matrosov et al. (1999) also presents a number of relationships for different drop shapes assuming gamma and lognormal distributions for R < 15 mm/h and at S-, C-, X-, and Ka-band frequencies. Timothy et al. (1999) has also reported a number of relationships for different gamma model DSDs at different temperatures and for different drop shapes, for Ku-band (13.8GHz) measurements. Aydin and Lure (1990) reported a number of piece-wise relationships between KDp and R for side incidence observations at 94 GHz and 140 GHz. These relationships were given for M-P, Joss Thunderstorm (J-T), and Joss Drizzei (J-D) DSDs of oblate spheroidal raindrops. Apart from the relationships reported by Aydin and Giridhar (1992) and Keenan et al. (1997) the results given in Table 6.3 are the only known relationships that have been obtained from either radar or disdrometer derived measurements. 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. S-band Relationships C-band Relationships -T T - VO M a tt q .O ML m nO mn. L Kaan M 8S — >• SWISS m m TROP 8-Z C hand *0 M at t q O — Ma t a q. L —■ M a tm n .O M a tm n L MI88 SWISS ■ • TROP 2 3 K0P(deg./km) 150 100 X-band Relationships i.«q.O .•q.L I. run. L — MSS — SWISS — TROP 4 2 4 6 KDP(deg./km) Ky-band Relationships 150 —— Tim* — TlmP-B — Tim C-B MISS 100 — SWISS — TROP K0P(deg./km) Fig. 6.4a The results o f published R-KDP relationships along with the results obtained from the MISS, SWISS and TROP data sets for the given frequency bands. Shown are the results reported by Sachidnanda and Zmic' (1987) S-Z, Chanrasekar et al. (1990) Chand., Aydin and Giridhar (1992) A-G, Matrosov et al. (1999) Mat. (where mn and eq refers their mean and equilibrium shapes with L and G referring to the lognormal and gamma model DSDs), Keenan et al. (1997) Keen., and Timothy et al. (1999) Tim., for Pruppaccher-Beard P-B, Chuang-Beard C-B shapes (Tim* give the best fit over the variations o f all the parameters) Figures 6.4a and 6.4b show the results for the relationships given in Appendix H. Also shown are the results for the relations given in Table 6.3 for the BC*, shapes. The S-, C-, X-, and Ku-band relationships are given in Fig. 6.4a. Four different relationships were shown from Matrosov et al. (1999); Mat. Eq. G for gamma (G) DSDs of equilibrium (eq) shaped drops, Mat. Eq. L for a lognormal (L) DSD of equilibrium shaped drops, and Mat. mn. G and Mat. mn. L for the gamma and lognormal DSDs o f their ‘mean’ shape drops. Three relationships were obtained from Timothy et al. (1999): Tim* give the results corresponding to their best-fit model to various gamma DSDs and shapes, Tim. P-B shows their best-fit model to Pruppacher-Beard (P-B) shapes, and Tim. C-B shows their best-fit results to their 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chuang and Beard (C-B) shapes. Note that the C-B shape used by Timothy et al. (1999) differ from the BCo, shapes being considered in this study. C-B shapes are not defined for D < 1 mm, however the BC^, model used in this study is given for 0.1 < D < 6.0 m m . For each of these frequency bands the TROP relationships gave higher rainfall rates than those of the MISS and SWISS relationships respectively, for a given KDp value. Note that for the S-band results the Mat. mn. G model closely approximate the SWISS BCeq results. Similarly, the A-G, S-Z, Chand., Mat. eq. G, and Mat. eq. L approximates that obtained for the MISS data over different ranges o f Kdp- The TROP results are not close to any of these previously published results. The A-G C-band relationship give similar results to the SWISS results for R < 30 mm/h, and approximates the MISS results for R > 70 mm/h. The X-band relationships, Mat mn L and Mat mn G given by Matrosov et al. (1999) overestimate the rainfall rates given by the SWISS, MISS and TROP BCo, models. However, the Mat eq L, and in particular the Mat eq G models gave results that were very similar to the MISS model. In the K„-band the TROP model is similar to the Tim* model for R < 50 mm/h and KDP < 6.0deq/km, but is closer to the Tim P-B model for higher rainfall rates (R > 70 mm/h) and KDp values (KDp > 8 deg/km). 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. W-band RelationsNps 150 ArL M-P M ISS SW ISS TROP 100 QE SO -18 -16 -14 -12 Fig. 6.4b The results of the W-band fits obtained from the SWISS, MISS, and TROP data sets for BC*q shapes compared to the results derived from the relationship provided by Aydin and Lure (l 990). At 94 GHz the linear relationship reported by Aydin and Lure (l 990) for a M-P distribution of Green (1975) shapes A-L M-P approximates the SWISS and TROP models for R < 25 mm/h and R > 50 mm/h. Note how closely the results for the SWISS and TROP relationships approximate each other over the entire range o f R and Kdp values. These two data sets have quite different DSDs over a similar range o f rainfall rates. This is strong evidence to the insensitivity o f KDp to variations in DSD, and its linear dependence on R at this frequency. The MISS relationship is slightly different to these two BCo, models (TROP and SWISS). This is because the MISS results were fitted over a wider range of range of rainfall rates involving much higher rainfall intensities. Scatter plots o f R as a function of KDp are shown in Fig. 6.5 for all three distributions at 9.4-, 14-, and 94-GHz, and for the equilibrium and oscillation (average) shapes first mentioned in Chapter 3. The model fits to these results are also shown. Note how poorly the SWISS results are fitted at the higher rainfall rates. This is due to the small number of measurements at these rainfall rates. 138 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. At the lower frequencies the scatter due to the non-linearity between the R and Kdp integrands, for a given drop shape exceeds the scatter in Kdp caused by differences in drop shapes (i.e. shape effects). The effects o f drop shape appear to increase with rainfall rate. This is due to the fact that larger drops are generally more common at higher rain rates, and these drops undergo greater deformation than the smaller drops (i.e. it is the larger drops that tend to oscillate). 9.4 GHz 100 100 14 GHz 20 100 100 94 GHz 100 -15 -10 -5 -10 -5 100 50 ?15 100 100 100 50 O 5 10 Koptdeg-Am) 0 10 KopWeg.Am) 20 -15 -10 -5 KQp(deg.ykm) Fig. 6.5 Scatter plots of the rainfall rates as a function of the K dp at 9.4, 14, and 94 GHz. The effects of the variation in drop shape on these relationships are also shown. From Fig. 6.5 it is evident that the oscillation shapes give lower K dp values than the equilibrium shaped drops. Because oscillating drops tend to be more spherical than the equilibrium shaped drops, the difference in their forward scattered fields Re{5w(D) - 5 w(D)}, and consequently their K dp values will be lower than those o f the equilibrium shapes. The relative error in the KDp values due to drop shape can be assessed by comparing the KDp values of the K«v shapes to those o f the BC«, shapes for the same rainfall rates. Relative errors exceeding 100% 139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. were obtained for some of the KDp values at lower rainfall rates (R < 4.0 mmh'1). The errors were less than 30% for higher rainfall rates. This was observed at all three frequencies and for all three data sets. Far fewer instances o f errors exceeding 100% were observed in the KDP values calculated at 94 GHz than were the case at 9.4 GHz. This is consistent with the higher scatter in the 9.4 GHz values of KDP shown in Fig. 6.2 and Fig. 6.4. The errors in the 14 GHz estimates were comparable to those at 9.4 GHz. In addition, there were far fewer instances of errors exceeding 100% in the KDP values calculated from the TROP distribution than were the case for both the MISS and SWISS distributions. This is again due to the fact that a larger percentage of the drops in the TROP distribution are in the range of sizes over which the BCeq and Kav shapes (curves) are closest to each other (i.e. the curve of their axial ratios (Fig. 3.1) are closes for D < 2.0 mm). An assessment of the effect of canting on the estimated KDP values can be obtained by looking at the relative difference in the values calculated for BC^ shaped drops with 10° standard deviation canting about their symmetry axes, and comparing them to the values obtained when no canting was assumed. For all three distributions the relative errors in the values calculated at 9.4 GHz and 14 GHz were always less than 9%. Smaller errors were obtained at the lower rainfall rates (R < 4.0 mm/h). This is because these low intensity rainfall rates tend to be dominated by small spherically shaped drops whose orientations are not associated with any discemable canting effects. For the case o f the 94 GHz simulations, errors in excess o f 100% were obtained at the lower rainfall rates (R < 4 mm/h) for the SWISS and MISS data sets. However, for the higher rainfall rates the errors were comparable to those obtained at the lower frequencies (< 9%). It was previously noted that the lower rainfall measurements are dominated by spectra that consist almost entirely of the smallest drop sizes. These are drops are generally within the range of sizes for which the variation in RejSy, (D) - S m(£>)} is most nonlinear (D < 1.0 mm). They are also located within the region where the curve has both positive and negative values. Therefore, distributions o f these drop sizes will tend to give similar rainfall rates with vastly differing KDP values, thus resulting in the large errors observed. Figure 6.6 shows the combined effects of oscillation and canting on the R-KDP relationships at the specified frequencies. The solid lines represent the best fit models to the R-KDP results obtained for drops that fall with their symmetry axes along the vertical direction (i.e. not canted). The broken lines represent the models for drops with fall orientations described by a Gaussian distribution with a mean canting angel of 0° with a standard deviation of 10° (i.e. 10° tilting from the vertical direction). These 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. results indicate that there is a systematic decrease in Kdp with both canting and oscillation. The different colors, as shown in the legend, indicate the different drop shapes. 9.4 GHz 100 14 GHz so 100 100 50 50 94 GHz i f MISS 10 0 r 10 0 -15 20 100 100 100 so 50 50 A -10 i f SW ISS 10 100 E 0 r 10 0 20 100 SO S' 0 TROP 0 5 KDp(deg./km ) 10 10 -10 KDp(deg./km ) -5 KDp(deg./km ) Fig. 6.6 Modeling the effects o f drop shape and fall orientation o f the R-KDp relationships. The solid lines give the results in the absence o f canting while the broken lines shows the change in the results for drops with a normal distribution o f canting angles N (0 = 0°, std.dev. = 10°) The error in the KDp values o f the Kav shapes with 10° standard deviation canting relative to the BCo, shape values with no canting was less than 40% for rainfall rates above 4.0 mm/h. 6.2.2 Radars And Backscattered Differential Phase Shift (6) Thus far we have limited our discussion to the differential phase shift along a propagation path. However, as it was earlier mentioned, the total differential phase shift (4 > d p) that is usually measured by radars, contains both the propagation differential phase and the differential phase shift upon backscatter 8 as shown in (6.1). 141 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In order for KDP to be an accurate estimate of the slope of the propagation differential phase O dp then 8 must be very small or close to zero. At the lower frequencies (S- and C-band frequencies) where the drops are predominantly Rayleigh scatters even at high rainfall rates, 8 is very small and can often be ignored. In addition <t>DP is a range cumulative (or path integrated) quantity (Doviak and Zmic', 1993) independent of the relative position of the scatters while 5 depends on the physical properties of scattering medium e.g. particle size etc. (Timothy et al., 1999). Therefore the KDPcan be easily retrieved at these frequencies as the slope of (change in) d>DPover a number o f successive range gates. At higher the frequencies (above 10 GHz) for which most of the drops are in the Mie resonance regime, this 8 term becomes more significant and may even dominate the propagation differential phase component. Larger drops can contribute significantly to 6 (Holt, 1992) even for modest rainfall rates (Matrosov, 1999). Techniques for reducing/eliminating these effects have been proposed (Balakrishnan and Zmic', 1990; Tan et al., 1991; Hubbert et al., 1993; Timothy et al., 1999). Balakrishnan and Zmic' (1990) used regression techniques to estimate KDP from Odp- Tan et al. (1991) corrected Odp for 5 by using empirical power-law relationships between Sand KDP. Hubbert et al. (1993a) proposed an iterative filtering technique (Finite Impulse Response-FIR adaptive filter) that attenuates spatial fluctuations in 4>dp caused by 5. Timothy et al. (1999) used a non-recursive digital filter (Savitzky-Golay filter see Press et al. 1992) to perform spatial smoothing (averaging) of the <t>DP profile. The backscattered differential phase is obtained from the difference between the arguments of the complex scattering amplitudes at the vertical and horizontal polarizations S = arg (5 „) - arg(5w) rc c . , = arg [5 „ S J (deg ) (6-9) where (*) signifies the complex conjugate. Figure 6.7 shows 8 for individual drops as a function of their equivalent diameter D(mm). The solid lines show the results for BCeq shapes while the broken lines give the results for the Kav shapes. The line colors indicate the frequency at which these values were calculated. 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 20 9.4 GHz 14 GHz 94 GHz 15 10 ■ ' _ Jw---— It CO -10 D (mm) Fig. 6.7 Single scatter backscattered differential phase shift for the BC*, (solid) and Kav (broken) drop shapes. It is clear that the non-equilibrium shapes (Kav) tend to have lower values of 8 (except possibly at 94 GHz where the non-equilibrium shape give higher 8 values at some o f the larger drop-sizes). The more spherical oscillation shape (larger axial ratios) tends to induce smaller changes in the phase of the alternate polarized signals. These results also indicate that 8 is negligible at X-band for drops D < 3.0 mm and at Ku-band for D < 2.0 mm, but becomes more significant at large drop sizes. At X-band the rate of increase in 8 is greatest for drops 3.0 < D < 4.0 mm, while at Ku-band the rate of increases is greatest for drops o f sizes D > 4.0 mm. The resonance scattering oscillation effects in the 8 profile at 94 GHz increase asymmetrically about zero with drop size. The integrated backscattered differential phase shift over a distribution o f raindrops can be calculated following Tan et al. (19 9 1) as ' = a r g | j s j(D)Sl,(D)N(D)dD deg. (6.10) 143 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 6.8 shows the scatter plot of the rainfall rates as a function of 8 for the three DSDs. It is clear that 8 is quite large and may affect the estimation of KDp at these frequencies, especially over short paths. This is especially the case for the low to moderately high rainfall rates and tends to be most severe at the X- and Ku-band frequencies. For R < 50 mm/h the spread in the values of 5 is significant. The high degree o f scatter indicates that 8 is not directly related to R. The spread in 8 values decreases with increasing R, however, 8 also increases. The spread is less at the higher frequencies (i.e. 14 GHz and 94 GHz) and there appears to be a lower limit to 8 for a given rainfall rate at 94 GHz. The largest backscattered phase shifts were obtained from the 14 GHz simulation results. The lower scatter in the 94 GHz results is in agreement with the results of Fig. 6.7 with the oscillating single scatter phase shifts curve at 94 GHz having both positive and negative values about 0°. These alternately positive and negative values causes a reduction in the DSD weighted phase shifts. Consequently, the backscattered differential phase shift at W-band is less than would be expected relative to that obtained at X-band. It is because these oscillations are asymmetric about 0° and increase with drop-size that there is only a reduction in the ensemble averaged backscattered phase shifts instead o f a complete cancellation (i.e. 8 = 0°). 144 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. s(deg) s(deg) S(deg) Fig. 6.8 Scatter plots of the rainfall rate and the differential phase shift on backscatter at 9.4, 14, and 94 GHz. The TROP data gave the lowest values for 5, and also had the lowest scatter at each frequency. This is consistent with the fact that the TROP DSD being narrower than the other DSDs, with a larger percentage of its drops in the in the mid-size range. The largest scatter occurs at the lower rainfall rates where most of the distributions will tend to be dominated my smaller drop sizes. The spread in the TROP results were approximately 50% of that observed in the SWISS and MISS results. Such high values of 5 at these frequencies may create problems in estimating rainfall from KDp at these higher frequencies (X-, K„- and W-band). Fortunately, there are techniques to reduce the effects of 8 on the determination of KDp. as previously mentioned (Balakrishnan and Zmic', 1990; Tan et al., 1991; Hubbert et al., 1993; Timothy et al., 1999). It is possible to avoid the problems associated with 8 affecting the estimation of KDP at these higher frequencies by using a propagation link similar to the DPPL (Chapter 5). In such a system where the 145 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. receiver and transmitter are co-located and the return signal is from a passive comer reflector of known cross-sectional area, the phases of the transmitted and received signals can be readily monitored with an appropriate phase detector circuit. The phase of the return signal will contain a system offset phase contribution (<j>0), a known engineering quantity that can be easily subtracted, along with the forward scattered phase shift term. The KDp may then be obtained by comparing the phases of the transmitted and the returned signals. The advantage of this arrangement is that by range gating the signal out to the location of the comer reflector, the phase on the returned signal represents the sum o f the forward scattered (2-way propagation) phase and the transmitted phase, thus effectively avoiding the backscattered phase terms. In addition, the fact that these measurements are being taken in the forward direction (i.e. not as radar measurements being backscattered from the rain) they will be less affected by the multiple scattering effects that plague radar measurements at these higher frequencies 6.23 The Effects O f Multiple Scattering On Propagation Through Rain Traditional algorithms used to estimate rainfall from measured radar parameters are based on the conventional radar equation, which only takes into account the single scattering contributions from raindrops (Oguchi, 1980; Oguchi et al., 1994). At centimeter and millimeter frequencies, and especially at high rainfall rates, the effects of multiple scattering cannot be ignored. Oguchi (1980) has shown that for frequencies above 10 GHz, the incoherent wave components generated by the effects of multiple scattering, become noticeable because the scattering cross-section of raindrops are no longer much smaller than the absorption cross-sections. Oguchi and Ito (1990) and Oguchi et al. (1994) have reported on the effects of multiple scattering on forward and backscattered wave components through rain at a number of different frequencies (16 GHz, 34.8 GHz and 140 GHz). Ma and Ishamaru (1991) have also studied the effects of multiple scattering of arbitrarily polarized, obliquely incident infrared waves (k = 10pm) in random media. In both cases their results indicated that in the forward-scattered direction, the received power of the incoherent components generated by multiple scattering effects were weak compared to that of the transmitted coherent components. Consequently, the incoherent components will not degrade the quality of the propagating signal. However, in the backscattered direction the incoherent components were significantly larger. 146 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Oguchi and Ito (1990) indicated that the intensity o f the incoherent components were almost the same as the single scattering contributions estimated from the radar equation. It is therefore clear that radar measurements (backscattering properties) at these frequencies would be significantly affected by multiple scattering. Hence, forward propagation measurements of differential phase shift will have a distinct advantage at these higher frequencies. 6.2.4 Sensitivity Test O f The R-Kpp Relationships Comparisons of the model derived rainfall rates with those obtained directly from the disdrometer measurements have been made to test the appropriateness of the relationships given in Table 6.3. The relative error between the individual estimates decreased with both the rainfall rate and the frequency of the propagating signal. The largest errors were obtained from the SWISS models. This is related to the fact that the largest errors occurred at the lower rainfall estimates (i.e. R < 10 mm/h). The SWISS distribution having the largest percentage of smaller drops also had the largest proportion of rainfall in this range (92% < 5 mm/h and 97.6 % < 10 mm/h see Table 4.4). The SWISS data had the least number of measurements with rainfall rates in the range R > 30 mm/h, which is probably one of the reasons that relationships poorly fit the data in this region. This result in the SWISS models having relatively higher errors at the higher rainfall rates. Scatter plots of these errors as a function of the rainfall rates are shown in Figure 6.9. Note that the relative error never exceeds 100% for the 94 GHz models. The maximum errors occur at R < 5 mm/h. 147 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9.4 GHz 14 GHz 94 GHz MISS 100 100 K . SW ISS 100 200 100 Rdls(mm/h) Rdls(mm/h) Rdls(mm/h) Fig. 6.9 The errors in the K Dp derived rainfall rates (R kdp) relative to the rainfall rates determined directly from the disdrometer data (Rdls). The errors are given as functions of R ^. The accuracy of the relationships given in Table 6.3 for estimating rainfall rates was also assessed using the FSE and NB (see equations 4.2 and 4.3 respectively) statistical Figures of merits. Graphical representations of these results are presented in Figures 6.10 and 6.11 respectively. These results represent the performance of the models ( R kdp) over the indicated ranges of rainfall rates (see Table 4.4). 148 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9.4 GHz 14 GHz 94 GHz 0.5 0.5 0.5 6 0.4 0.4 0.4 0.3 0.3 0.2 0.2 MBS K 0.3 0.1 0.1 40 40 0.6 0.6 0.5 0.5 60 0.4 ft 0.4 SW BS £ 0.3 0.3 £ 0.2 0.2 0.1 0.1 0.6 0.6 0.5 I 0.5 h°* 1 Imop V 20 60 cc 0.3 - ; 1 " 40 0.6 0.5 — 0.4 BC.c ABL, 0.4 0.3 0.3 0.2 0.2 0.1 60 40 R(mm/h) R(mm/h) R(mm/h) Fig. 6.10 Fractional Standard Errors (FSE) in the model derived rainfall rates Rkdp relative to the disdrometer derived rainfall rates (R^,) as functions o f the ranges of rainfall rates given in Table 4.4. The FSE gives the relative scatter in Rkdp over the range o f rainfall rates described in Table 4.4. This error steadily declines from a maximum o f 30-60% at the lowest rainfall rates (R < 1 mm/h) depending on the assumed drop shape, the DSD and the frequency at which the KDp is calculated. At the lower frequencies (9.4 GHz and 14 GHz) the minimum error at the highest rainfall rates will rarely ever be lower than 10%. On the other hand R-KDp relationships at 94 GHz will typically be accurate to within 15% for rainfall rates around 10 mm/h and have accuracies to within about 5% at higher rainfall rates. The spread in the curves representing the errors for the different drop shapes indicate that 3-15% o f this error may be due to the assumed drop shape (this depends on the rainfall rate, the DSD and the frequency). The models derived from the ABLav shaped drops gave the largest errors, while those of the Km, shaped drops had the 149 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. lowest errors. The higher errors in the ABLav models are the result o f the relatively large number of drops in the 1.0 < D < l.S mm size range where these drops are believed to be in resonance (i.e. the protrusion on the ABL*, curve in the range 1.0 < D < 1.5 mm see Fig. 2.1) and are also dominating the R and Kdp integrands. Compared to the results o f the R-A relationships at 35 GHz given in Fig. 4.7a, these relationships tend to be more sensitive to the drop shapes, while at the same time being less sensitive to the DSD (i.e. unlike the results o f Fig. 4.7a the curves in Fig. 6.10 for the different DSDs at a given frequency are quite similar). The R - K dp models at 94 GHz tend to have less error than the R-A models over the higher rainfall intensities (R > 10 mm/h). 9.4 GHz 0.3 14 GHz b I 8 0.3 0.3 MISS 0.2 94 GHz 0.2 0.2 0.1 0.1 - 0.1 - 0.1 • - 0.2 - 0.2 ■ 0 40 20 0 60 40 20 0 60 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.1 .1 0.1 -0.1 0.2 - SWISS 0.1 - 0.1 - • 0.2 - 0 20 40 20 40 60 0 0 40 20 40 60 60 0.2 0.4 0.4 0.3 0.2 0 - 0.1 - 0.1 - 0.2 - 0.2 0 j1_— ------------------- 0.3 TROP 0.2 J 1 0.1 8 0.2 0.4 0.2 8 0.1 20 40 R(mm/h) 60 9C„ ABL^ ---------- K_ ' 0 _ . . . -------- L. 40 20 R(mnWh) 60 R(mm/h) Fig. 6.11 The Normalized Bias (NB) in Rkdp relative to R^, over ranges o f rainfall rates defined in Table 4.4.. 150 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 6.11 indicates that the mean rainfall rate derived from these models are generally within 10% o f their expected values for R > 1 mm/h. The mean rainfall rates derived from the 94 GHz models are within 5% of their expected values for R > 5 mm/h. The event accumulations estimated from the models are shown in Figure 6.12. These accumulations have been plotted against those estimated directly from the disdrometer measurements. The bias and error statistics associated with these plots are listed in Table 6.4. 9.4 GHz 94 GHz 14 GHz MIS§ 40 40 40 80 20 80 80 60 60 SW $S 40 20 20 80 40 60 80 0 40 60 20 80 1 1 \ 60 TROf» 40 V • 20 20 I_ 40 (mm) 60 80 20 ABL, 40 I d (mm) Fig. 6.12 The cumulative rainfall derived from the KDp estimated rainfall rates for the different drop shape models, disdrometer data sets, and frequencies o f interest. These accumulations are plotted against the accumulations estimated directly from the disdrometer data. Note that for the MISS and TROP distributions the points are generally more tightly positioned about the L Roa = Z rKdf line in the 94 GHz plots than in those at 9.4 and 14 GHz, especially for the higher accumulation events. This can also be observed from the results given in Table 6.4. The FSE, which 151 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. measure the variance in the estimates about their expected accumulations, decreases as the frequency increase. The largest improvement (reduction) in the FSE was observed in the TROP results (approx. 10% reduction in the FSE with increase in frequency from 9.4 GHz to 94 GHz). The FSE in the TROP accumulations are about 3-5% lower than those of the MISS accumulations. The errors in the estimates of the accumulations of the SWISS distribution behave in a manner opposite to that of the other two distributions. This is due to the SWISS distribution having a very large number of low intensity rainfall measurements, which are in turn dominated by small drop sizes. For example, approximately 92% of the SWISS rainfall measurements have rainfall rates R < 5 mm/h. The overwhelming majority of the drops in the SWISS distributions are of the range of sizes for which the Re{S**(0) - 5 re(D)}curve at 94 GHz is not monotonic and where it is positive (i.e. D < 0.85 mm). Therefore, the relationship between the rainfall rates and KDp dominated by these drops will be non linear. This is compounded by the fact that the R-KDP models do not perform well at the lower rainfall rates as they do at the higher rainfall rates. 152 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 6.4 The FSE and NB statistics on the cumulative rainfall estimates from the R-KDp models given in Table 6.3 9.4 GHz 14 GHz 94 GHz Z Disd{mm) DSD Shape NB% FSE% NB% FSE% NB% FSE% BCal 0.2427 0.0666 MISS 0.0493 0.0986 0.1723 0.2295 1 < Z < 5 mm ABLav 0.0479 0.2596 0.0305 0.2474 0.0972 0.1893 1 < Z < 5 mm 0.0238 0.2150 0.0072 0.0890 0.1557 0.2026 1 < £ < 5mm Kav SWISS b c m 0.0562 0.1829 0.0650 0.1139 0.1947 0.1776 1 < Z < 5mm 0.0372 0.1912 ABU 0.0486 0.2053 0.1842 0.1172 1< Z < 5mm 0.0082 0.1630 0.0215 0.2168 0.1510 0.1232 1 < Z < 5mm Kav bc m -0.0362 0.2409 TROP -0.343 0.0268 0.1289 0.2243 1 < Z < 5 mm 0.2617 0.1530 -0.0496 0.2490 0.0202 1 < Z < 5mm ABLa, -0.0487 -0.0539 0.2140 0.0290 0.1039 -0.0528 0.1996 1 < Z < 5mm Kav Z > 5mm BCjq 0.0308 0.0993 MISS 0.0252 0.0954 0.0205 0.0442 Z > 5mm 0.1109 0.0257 0.0241 0.0568 0.1078 ABLav 0.0315 Z > 5mm 0.0225 0.0931 0.0172 0.0239 0.0486 0.0902 Kav Z > 5 mm -0.0013 0.1247 SWISS BCcq 0.0064 0.0475 0.1335 0.1232 Z > 5mm 0.1249 0.0451 0.1424 -0.0019 0.1231 ABLav -0.0106 Z > 5mm -0.0219 0.1038 -0.0122 0.0589 0.1480 0.0992 K„ -0.0074 Z > 5mm BQ, 0.1281 TROP -0.0068 0.0037 0.0431 0.1136 0.1411 Z > 5mm -0.0084 0.0597 0.0036 1.0283 ABLav -0.0081 -0.0137 0.1171 Z > 5mm -0.0131 0.0048 0.0332 0.1035 Kav The NB and FSE in the RK values calculated from the SWISS distribution generally decreases with R and with frequency from the 9.4 GHz models to the 14 GHz models. The 94 GHz R-KDP models give a 50% improvement in the FSE and NB for measurements in the range 5<R<40 mm/h. However, the FSE increase by approximately 10-15% and the NB by a factor of about 2 over those of the 14 GHz models for R < 5 mm/h. Since the SWISS distribution is dominated by these measurements the errors in the accumulation estimated from the SWISS distribution tends to be higher than that of the other data sets. The errors in the 94 GHz results are even higher than those obtained for the 9.4 GHz models. There were also increases in the FSE and NB of the RK values derived from the MISS data for the Ror lower intensity measurements. However, there were relatively less of these measurements in this data set, therefore the errors in the accumulation estimates of the MISS data set were not dominated by the errors in the R-KDP models over the lower rainfall rates. For the TROP data set the FSE and NB in the RKor values decreases with frequency from 9.4 GHz, 14 GHz, to 94 GHz, as the distribution is 153 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. dominated by drop-sizes in the range where Re{5M(D) - S n (D)} is monotonic and where R and KDp are linearly related. The R-Kdp models at 9.4 GHz and 14 GHz are as accurate as the R-AA models at 35 GHz (see Tables 4.5). The 94 GHz R-KDp models are less accurate than the 35 GHz R-Ah models over low accumulation events, however they may out perform the R-A models for events having 63 > 5mm. Dual Frequency Relationships For Use In Rainfall Retrieval Algorithms The Tropical Rainfall Measuring Mission (TRMM) satellite was launched on 27 November 1997 as part of an on going effort to measure the quantity and distribution of precipitation in the global Tropical and Sub-Tropical Regions (Simpson et al., 1988, 1996; Kummerow et al., 1998, 2000). Among the sensors deployed with on this satellite is the first precipitation radar (PR) to measure rain from space. The TRMM PR is a 128-element active phased array system operating at 13.8 GHz from which the threedimensional structure of rainfall distribution over both land and ocean is estimated (Kummerow et al., 1998; Meneghini et al., 2000). The horizontally polarized beam is electronically scanned cross track ±17° with respect to nadir. The mission is expected to conclude by March 2004. The next phase of this project is expected to involve a second satellite, similar to TRMM, carrying among other things, a dual-frequency (DF) rain radar operating at 13.8 GHz and 35 GHz frequencies. The use of two radar frequencies will enable the determination of the first moment of the DSD (i.e., the mode), and thus the quality of the rainfall rates is expected to exceed that o f the standard ground-based weather radar (Kummerow et al., 2000). Following Meneghini and Kozu (1990) the mean return power at each radar frequency (wavelength) will be given by the single frequency (SF) radar equation . r P U , , r) = c a , ) | K j 2 Zf a ,,r ) - - j- e x p ( - 0 .4 6 j A 0 1, , s ) d s ) (6.11) where r is the radar range, C is the radar constant, Z* is the equivalent reflectivity factor 154 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Z'= A* ? s . , r f t M D)N{D)dD X I I' Jo (6.12) /W* —I A refers to the specific attenuation at the wavelength in question and K is given by K = — ------, where m~ + 2 m is the complex index of refraction, and Kw indicates that the refractive index of water is to be used. The two unknown quantities in (6.11) are Zc(Xi,r) and A(Aj,r). The rainfall rate may be estimated from the single frequency measurements if it can be assumed that the rainfall is uniform over the range. In this case the magnitude of the specific attenuation A(dB/km) can be estimated as the slope of the plot logio[P(^i.r)] versus r. From A, the rainfall rate can then be obtained by means of a power relationship R(A) of the form R = a-Ab. In general, the characteristics of the rainfall will change as a function of r. When the fluctuations in the reflectivity factor become comparable to that of the attenuation over the same range, then the uniformity assumption is no longer valid (Meneghini and Kozu, 1990). Non-uniform distribution of rainfall along the antenna beam may bias the retrieved rain rate. Such biases have been found by a number of authors using both simulated and ground-based radar data (Nakamura, 1991; Amayenc et al., 1993; Testud et al., 1996). Amayenc et al. (1996) found biases due to non-uniform beam filling (NUBF) using data from a nadir-looking aircraft radar for a rainstorm off the U.S. Atlantic coast. In such instances it would be convenient to reduce the number of unknown parameters by relating A to 7^ in the formA = or • z / . This allows for Eqn. 6.11 to be expressed as a function of one unknown, 7^. Following Meneghini and Kozu (1990) empirically derived R-Z relationships of the form/f = a- Z* can then be used to estimate the rainfall rate at range r by -b RU i, r) = a - Z r (A,, r)[p{A,, r)] * (6.13) where r p(A, ,/•) = ! - 0.446 ■/3 j a Z% o (6.14) is an estimate of the exponential in (6.11), and the measured, or apparent, reflectivity factor (corresponding to the theoretical Ze in 6.12) is given as 155 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In the above expressions a, b, a, and (3 are obtained for an assumed DSD, temperature, and wavelength using Mie theory, and the estimated rainfall rate is a function o f the radar constant. In the case of a dual-frequency (wavelength) radar with X, < a 2, an estimate of the frequency differential attenuation in the range interval rk to rj is given by Eccles (1979) as (6.16a) for _ P(A1,r >)/*(A2,rt ) 12 (6.16b) P(A2,rj )P(Al,rk) _ Z r(Al ,rJ)Ze(A1,rk ) z gl2 Ze (A2, r, )Ze (6.16c) , rk) If the radar reflectivity factors are independent of wavelength between rk and rj, or if the reflectivity factor at each wavelength is uniform in this range, then = I and the differential attenuation will be given directly by the measured quantity P |2. Provided that the return power can be expressed as in Eqn. 6.11, and assuming that the antennas are well matched, (6.16) indicates that the dual-frequency estimates will be independent o f the radar constants. Following Goldhirsh and Katz (1974), a two-parameter DSD can then be estimated from these dualfrequency measurements. If we define (6.17) 156 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and assuming that the drop size distribution, N(D), is independent of range in the interval [rk, rj], then Log]\__ Log T2 A, D j f f atU l ,D ) N ( D ) dD _ _0_________________ (6.18) D f a exl(Az ,D )N (D )d D o where CText(Ai, D) is the extinction cross section. For a gamma model DSD N uD Me~‘w , for some assumed p, the N0 cancel in (6.18) and the resulting expression yields and an estimate o f A from the measured quantities. The parameter N0 may follow from the use of either T| or T: as: N„ = -5 Log----------------- ^ ------------------------------ . (6.19) 0.434(r, - rk ) f a ca (A,, D) DMe' ™ dD o With the DSD thus determined the rainfall rate may then be estimated following Ulbrich (1983), as R = 33.3 l j Dy67 N u DMe d D . (6.20) The above equations (6.7-6.16) give an outline of the basic philosophy behind the backscattering rainfall retrieval algorithms. Meneghini and Kozu (1990) and Marzoug and Amayenc (1994) reviewed other classes of single- and dual-frequency attenuation based algorithms. These algorithms rely on the use of Z-A and R-A power-law relationships, first to provide estimates of attenuation-related parameters, before finding the corresponding rainfall profile using the relevant R-A relationships. In the backscattered algorithms the estimate depends on the variability in the DSD through the highly sensitive Z-R relationships, and on the radar calibration through the Z estimate. However, these algorithms are usually complimentary since the former (later) performs best at low (high) rain rates in the absence (presence) of significant path-integrated attenuation. Though demanding, the dual frequency algorithms are more efficient and reliable since they allow global correction for all scaling error terms (Marzoug and Amayenc 1994). 157 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. From the preceding it is clear that there is a need for model relationships between A, Z, and R at both radar-operating frequencies, and for different DSDs and rain types. These models may serve the dual purpose o f reducing the number o f unknown parameters in a given algorithm, and provide a first order approximation for the estimated rainfall rate. Scatter plots o f the specific attenuation (AH) at horizontal polarization, and rainfall rates as functions of the reflectivity (ZH) are shown in Fig. 6.13 for the TROP distribution. These parameters were calculated for the nadir (90°) and 17° off-nadir (73°) pointing radar beams. Equilibrium shaped (BC«,) drops with fall orientations normally distributed about their zenith with a 5° standard deviation were assumed. Note that the differences in elevation angles have no significant effect on the calculated parameters (Z and A). It is clear from the relatively lower scatter about the fitted curves, that these parameters are more linearly related at 35 GHz than at 14 GHz. The same is true for the relationships between these parameters (Z and A) and the rainfall rate R at these frequencies. The effects o f canting and changes in drop shape were observed to cause less spreading than that inherent in the parameter. 25 20 £4 - 15 §3 [10 ZH36 <mm6 m°> x 104 ZH14 (mm6 m*3) x 105 ZH14 (mm6 m'3) x 10s 120 80 100 60 80 40 60 40 20 20 0 0 2 4 6 ZH3S (mm6 m '3) 8 x 104 Fig. 6.13 Scatter plots of the specific attenuation at 35 GHz (a) and 14 GHz (b) as functions o f the radar reflectivities at those frequencies, and at different look angles (90° an 73°). The estimated rainfall rates are also given as functions o f the reflectivities at 35 GHz (c) and 14 GHz (d) respectively. These results were simulated from the TROP disdrometer data. 158 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The dual frequency technique for rainfall estimation was conceived to improve the correction o f the scaling error terms used in the estimation process by exploiting the assumed relationship between the attenuation at both frequencies (Testud et al, 1996). Figure 6.14a shows the results o f the specific attenuation at 14 GHz as a function o f the specific attenuation at 35 GHz. At the higher attenuation values (higher rainfall rates) the relationships of these parameters are less linear. However, the differences in the attenuation values (Fig. 6 .14b) appear to increase linearly with the rainfall rate. This is primarily due to the fact that they are dominated by the attenuation at 35 GHz (which has already been shown to be linearly related to the rainfall rate). 80 5 60 I* 40 Q£ 20 1 0 15 10 AH38 (dBAm) 20 15 5 10 AAH(dBAm )=AH38- A n 14 25 80 80 60 60 40 20 40 90° cr tr. 20 73° 20 15 10 (dBAm) A na, (dB/km) 20 F« 25 Fig. 6.14 The functional relationship between the specific attenuation at 14 GHz and 35 GHz (a).The scatter plots of the rainfall versus the frequency differential attenuation and the specific attenuation at 35 GHz and 14 GHz are shown in (b) and (d) respectively. These results were simulated from the TROP disdrometer data. Similar results are shown for the MISS data set in Fig. 6.15 and Fig. 6.16. The SWISS results are shown in Fig. 6.17 and Fig. 6.18 159 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 12 40 10 30 10 1.5 0.5 ZH35 <mm6 m‘3) 200 200 150 150 100 100 50 50 5 10 ZH14 (mm6 m~3) x 10 5 10 ZH1« <mm6 m' 3> 15 x 10s K 0.5 1.5 ZH35 <mm# m' 3) X 10s Fig. 6.15 Scatter plots o f the specific attenuation at 35 GHz (a) and 14 GHz (b) as functions o f the radar reflectivities at those frequencies, and at different look angles (90° an 73°). The estimated rainfall rates are also given as functions o f the reflectivities at 35 GHz (c) and 14 GHz (d) respectively. These results were simulated from the MISS disdrometer data. 160 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 200 12 10 150 100 50 10 30 (dBAon) 40 10 20 AA^dB/km) = 50 200 200 150 150 100 30 - A „14 40 100 oc ae 50 90° 73° 50 Fit 10 40 50 10 15 Aj^14 (dB/km) Fig. 6.16 The functional relationship between the specific attenuation at 14 GHz and 35 GHz (a).The scatter plots o f the rainfall versus the frequency differential attenuation and the specific attenuation at 35 GHz and 14 GHz are shown in (b) and (d) respectively. These results were simulated from the MISS disdrometer data. 161 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a 8 20 6 15 § 10 s4 2 O 0 100 100 80 80 60 60 fie 40 20 2 4 6 8 10 :H 14 <m m * ~ ' 3 x 10s ZH14 (mm6 m"3) x 105 v t* 20 10 Fig. 6.17 Scatter plots o f the specific attenuation at 35 GHz (a) and 14 GHz (b) as functions o f the radar reflectivities at those frequencies, and at different look angles (90° an 73°). The estimated rainfall rates are also given as functions of the reflectivities at 35 GHz (c) and 14 GHz (d) respectively. These results were simulated from the SWISS disdrometer data. 162 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 8 80 6 4 2 0 20 10 15 5 10 AAH(dB4on) = AH38-A H14 20 Ah 36 (dB/km) 100 100 80 80 60 60 & 40 15 90< 73‘ 20 20 15 20 AH14 (dB/km) Fig. 6.18 The functional relationship between the specific attenuation at 14 GHz and 35 GHz (a).The scatter plots of the rainfall versus the frequency differential attenuation and the specific attenuation at 35 GHz and 14 GHz are shown in (b) and (d) respectively. These results were simulated from the SWISS disdrometer data. Power-law fits to the relationships between these parameters have been derived for the three data sets are shown in Table 6.5. It is clear from these results that some of these relationships are more sensitive than others to variations in the DSD. It would therefore be necessary to create a database of these models for the different rain types and geographical locations to be used with the rain profiling algorithms. 163 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. Table 6.5 Power law fits to the relationships Y-X where Y = a - X b, a ± a 1 and b ±CTb give the 95% Cl of the fits a and b respectively, and AAk = Ah3S - AA14 D is t Fits TROP SW ISS M ISS Ahl4~Zhl4 AhJ5*ZhJ5 R-ZU4 R- Z u j R-Abu R* Ah35 AhU* Ah35 R-AAh a 6.46E-4 1.16E-3 0.0419 0.0063 24.632 3.8776 0.1086 4.3950 b 0.7267 0.8781 0.6269 0.8349 0.8401 0.9545 1.1324 0.9641 3.05E-5 3.15E-5 0.0024 0.0002 0.0766 0.0124 0.0012 0.0146 Ob 0.0044 0.0027 0.0056 0.0036 0.0032 0.0014 0.0048 0.0016 a 6.62E-4 8.74E-4 0.0637 0.0034 20.04 3.8937 0.1149 4.5444 b 0.6683 0.9054 0.5207 0.9065 0.7939 1.0164 1.3012 1.0396 o, I.73E-5 1.58E-5 0.0093 0.0001 0.0499 0.0126 0.0013 0.0207 ob 0.0022 0.0019 0.0028 0.0033 0.0018 0.0018 0.0056 0.0030 a 3.19E-4 6.25E-4 0.0243 0.0023 21.095 3.7788 0.1430 4.4273 b 0.7500 0.9385 0.6295 0.9422 0.8666 1.0132 1.1353 1.0313 o, 1.10E-5 5.46E-6 0.0009 0.0000 0.0383 0.0107 0.0011 0.0171 ob 0.0028 0.0008 0.0031 0.0016 0.0013 0.0011 0.0028 0.0016 The model relationships provided in Table 6.5 are compared against each other in Fig. 6.19 and Fig. 6.20. Figure 6.19 compares the specific and the frequency differential attenuation models obtained from the TROP, SWISS, and MISS data sets. Note that the models that involve the specific attenuations and reflectivity at 35 GHz (A^s, and ZH3 s) show less sensitivity to the variation in DSD. 164 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 12 40 10 30 TROP SWISS MISS 10 0.5 5 10 ZH14 (mmGm’3) 1.5 x 10s 200 200 150 150 100 100 50 50 0.5 1.5 ZH33 <mm®m' 3> 5 x 10s 10 15 x 10s 15 ZH14 (mm®m*3) Fig. 6.19 The model fits to the A-Z, and R-Z relationships given in Table 6.5 for the three disdrometer data sets. Figure 6.19a shows the Aros-Zios results for all three data sets closely approximated each other. This is very promising as it suggests that the reflectivity measurements at 35 GHz, which are relatively easier to make than the attenuation measurements, can be directly related to the specific attenuation at that frequency. It is clear from the tight fit in the R-AUs, and to a slightly lesser extent R-Z^s, that AU5 estimates thus obtained can be used to estimate the rainfall rates. Both the Ahi4 and Zhi4 results show a greater sensitivity to variation in DSD type. 165 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 200 12 10 150 100 SO 10 20 30 (dB/km) 40 30 10 20 AA^dB/km) - An30 - AH14 50 200 200 150 150 100 100 50 50 40 M SS (K 10 40 50 15 10 AH14 (dB/km) Fig. 6.20 Comparing the power law fits of the AHu-Ah3 j, R-AAh, R-Ahjj, and R-AHh given in Table 6.5 for the TROP, SWISS, and MISS data set. Figure 6.20 compares the power law-models AHi4 -Ah3 s, R-AAh, R-Ahss, and R-AHi4 given in Table 6.5 for the TROP, SWISS, and MISS data sets. From the spread in the curves o f Fig. 6.20c and Fig. 6.20d it is clear that the R-AHi4 relationships are more sensitive to changes in rain drop distribution type. This can also be observed from the relationships given in Table 6.5 which shows the exponents “b” of the RAhjs, being closer to l and hence more linear than the R-AHu, relationship. It is this inherent linearity that causes the R-Akjs models to be less sensitive to changes in DSD. The results in Fig. 6.20b show that the frequency differential attenuation does show some sensitivity to variations in the raindrop size distribution or rainfall type. This may provide an additional source of error to the dual-frequency techniques. 166 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.4 Conclusions The relationships between rainfall rate and selected radar and propagation parameters have been assessed at X-, K„-, Ka-, Q-, and W-band frequencies. These relationships were examined with the aid of the three disdrometer derived drop size distribution data sets first introduced in Chapter 4. The relationship between the rainfall rate and the specific differential phase was analyzed by comparing the variations in the difference between the real part of the forward-scattered fields Re{Sw (D ) - S m(D)} to variations in the drop momentum Di vl(D) with equivalent volume diameters D. It was shown that the R- KDp relationships were more linear at 9.4, 14, and 94 GHz frequencies for which the gradient of the logarithm of Rc{SUl( D ) - S n (D)} was close to the gradient of the logarithm o f D \ t(D) . This implies that both functions have comparable powers of D in a power law representation. The R-Kdp relationship at 94 GHz was linear (e.g. R = 8.72 • K°0'*) with little scatter. The 9.4 and 14 GHz relationships had more scatter, but the 30, 35, and 40 GHz relationships appeared to be of little value due to the large scatter and sign changes in KDP. As a result, 94 GHz KDP was proposed for measuring rainfall rates over short propagation paths. Because the rain attenuation at this frequency is a limiting factor (e.g. up to 10 dB/km for R = 10 mm/h) the path length must be short. At these frequencies the backscattered differential phase is high and may limit the estimation of KDP using radar. An alternative procedure for estimating K0P at these higher frequencies is using a propagation link similar to the DPPL. The advantages of such a system are expected to be • elimination of the backscattered differential phase term, • reduction of the multiple scattering effects since the measurements will be taken in the forward direction (i.e. not as radar measurements being backscattered from the rain). The relationships between rainfall rate R, radar reflectivity Zt,, and specific attenuation Ah were also presented. The relationship between the specific attenuations at 14 and 35 GHz, along with the relationship between the frequency differential attenuation (the difference between the specific attenuations at 35 GHz and 14 GHz) and the rainfall rates were also examined. These relationships have been proposed for use in both Single-Frequency (SF) and Dual-Frequency (DF) rainfall retrieval algorithms such as those to be implemented in the TRMM Follow-on Mission. 167 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Variations in the radar-pointing angle about nadir (0° ± 17 °) were observed to have little effect on the calculated parameters. From the relatively low scatter about the fitted curves, it is clear that the Z*, and Ah parameters are more linearly related at 35 GHz than at 14 GHz. The Ah-Zt, relationship at 35 GHz was observed to be quite insensitive to variations in DSD (Fig. 6.19a). It was suggested that improved estimates of rainfall rate may be obtained from radar reflectivity measurements at 35 GHz with the aid of these Ah-Zh relationships. However, one has to be mindful that radar measurements at these frequencies become more susceptible to multiple scattering and attenuation effects. The frequency differential attenuation AAk = Ah}S DSD. was shown to be more sensitive to changes in This sensitivity to DSD may present a source o f error in the dual-frequency rain retrieval algorithm. 168 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 7 SUMMARY, CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK 7.1 Summary and Conclusions Results of studies on the polarimetric scattering characteristics of rainfall at X-, Ku, Ka, and W-band frequencies, and their possible use for estimating rainfall have been presented. This study sought to better understand the effects o f changes in drop size distributions (DSDs), drop shape, and the fall orientation of raindrops on selected radar and propagation parameters. One equilibrium and two average drop shape models were used to assess the effects of raindrop shape on the scattering characteristics o f rainfall. The average shapes were representative of oscillation effects. The fall orientations were simulated with the use of a Gaussian canting angle distribution. The scattering computations for the different drop sizes and shapes at different frequencies were generated using the T-matrix computational technique. Raindrop size distributions from three different geographical locations (Tropical rainfall DSD TROP, Sub-Tropical rainfall DSD MISS, continental rainfall DSD SWISS) were used to obtain relationships between the rainfall rate R, the specific attenuation (Ah, Av, and Aavg), and the specific differential attenuation AA at 35 GHz. Estimates of rainfall rate R had less scatter when obtained from Ah, Av, or AaVg, and more scatter when obtained from AA in rain events dominated by smaller drops (< 2.4 mm) such as light widespread rainfall. The opposite was true for rain with larger drops (especially heavy convective events) where the estimates of R from AA had less scatter. The specific (Ah and Av) and average attenuations (Aavg) were shown to be insensitive to changes in drop shape and fall orientations (drop canting). However, as expected, the differential attenuation was shown to be quite sensitive to variations in both drop shape and fall orientations. The combination of AA with Ah (Av or Aavg may also be used in place of Ah) is proposed for estimating the mean drop shape due to the presence of drop-oscillations (and canting). This is a result of the differences in the sensitivities o f AA and Ah to variations in the drop shapes. Distinct trends were observed in the AA-Ah scatter plots resulting from the oscillation drop shapes with lower values of AA than those obtained for the equilibrium shapes. These trends can be used to compare and distinguish between equilibrium and non equilibrium shaped raindrops. Power law fits to the AA-Ah relationships have been provided. 169 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The performances of these R-A relationships were evaluated in terms of the normalized bias (NB) and fractional standard error (FSE) statistics. Estimates of the rainfall accumulations (£ A) derived from these relationships were also assessed. In general Eaa (accumulation estimated using simulated AA) underestimated the rainfall accumulations to within 7% with less than 29% FSE. On the other hand lead to less than 2% underestimation with less than 10% FSE. Therefore, Ah and Av (or the average of the two) can be effectively used for estimating rainfall rate and accumulation; and assuming that the effective drop shapes are known, AA can also be effectively used for estimating rainfall rates larger than 10 mm/h. The relationships between rainfall rate and selected radar and propagation parameters have been assessed at X, Ku, Ka, Q, and W-band frequencies. These were examined with the aid o f the disdrometer derived drop size distributions. The R-KDP relationship at 94 GHz was almost linear (e.g. R = 8.72 • K D ° * ) with little scatter. The 9.4 and 14 GHz relationships had more scatter, but the 30, 35, and 40 GHz relationships appeared to be o f little value due to the large scatter and sign changes in K dp- A s a result, 94 GHz KDP was proposed for measuring rainfall rates over short propagation paths. Because the rain attenuation at this frequency is a limiting factor (e.g. up to 10 dB/km for R = 10 mm/h) the path length would have to be short. The relationships between rainfall rate R, radar reflectivity Zh and specific attenuation Ah, at 14 and 35 GHz, and the frequency differential attenuation (the difference between the specific attenuations at 35 GHz and 14 GHz) were also presented. These relationships have been proposed for use in both SingleFrequency (SF) and Dual-Frequency (DF) rainfall retrieval algorithms such as those to be implemented in the TRMM Follow-on Mission. Variations in the radar-pointing angle about nadir (0° ± 17 °) were observed to have little effect on the calculated parameters. From the relatively low scatter about the fitted curves, it is clear that the Zh and Ah parameters are more linearly related at 35 GHz than at 14 GHz. The Ah-Zh relationship at 35 GHz was observed to be quite insensitive to variations in DSD. It was suggested that improved estimates of rainfall rate might be obtained from radar reflectivity measurements at 35 GHz with the aid of these Ah-Zh relationships. However, one has to be mindful that radar measurements at these frequencies become more susceptible to multiple scattering and attenuation effects. 170 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A 35 GHz dual polarization propagation link was used for testing rainfall rate estimation with A h, A v and AA over short path lengths (from 200 to about 500 m). Rain gauges and a disdrometer were used for comparison. The measurements showed promising results, more so with Ah and A v than with AA, which appeared to be noisier than the other two. The effects of drop size distribution variations on the attenuation parameters were observed. Rainfall accumulation estimations based on Ah and Av produced very good results (within 10% of the rain gauges) for long lasting (over an hour) rainfall events. However, further testing and evaluation of the propagation link was considered necessary before any conclusions could be drawn regarding the microphysical details o f rainfall and the utility o f AA for measuring rainfall. 7.2 Suggestions for Future Work Future studies should include efforts to distinguish between the different rainfall types from the disdrometer data sets. Having grouped these events according to their rain types, relationship between R and A, Z, and KDp should be derived for each type at the frequencies employed in this study. The attenuation and rainfall rate measurements presented in these studies have contributed additional insights into the characteristics of microwave attenuation by rainfall. However, additional efforts should be made to stabilize the DPPL hardware electronics to guard against the temperature induced gain fluctuations. Steps should be taken to more accurately estimate the effects of the variations in temperature, humidity and atmospheric pressure on the attenuation of the transmitted signal in the absence of rain. If these effects can be accurately estimated then more effective methods of compensating for these changes can be used to improve the quality of the DPPL data and provide more accurate estimates of the rain induce attenuation. It is clear that taking the ratio of the return powers has the effect of amplifying the error variance in V/H resulting in of AA being a noisier parameter than A h, A v, or Aavg- It would therefore be of interest to determine what effect would variations in the sample rate and the integration time of the return power have on this error variance. Subsequent field campaigns involving the DPPL should be conducted over open terrain with the link isolated from tall structures such as buildings, and trees. Among the issues of concern in the recently concluded experiments were 171 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • To what extent has the neighboring trees and buildings affected the air currents/flow along the link? • To what extent did the chimney’s “rain shadow” affect the measurements obtained from the link? • To what extent did the chimney’s scatter returns affect the DPPL’s measurements? Steps should be taken to limit or eliminate these questions/problems in the future. Further studies are also required on the effects of wind shear and vertical air motion on the DSD arriving at the surface or over the link. This could involve the use o f a FM/CW 94 GHz radar capable of making measurements from the ground up, so that the DSD closer to the ground can be observed. For the dual-frequency rainfall estimation schemes, further studies are needed on the effects of temperature variations on the relationships between Ah, Zh, and R at the Ku and Ka band frequencies discussed. 172 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX A Table Of Axial Ratios Describing The Drop Shapes The following is the table of axial ratios a = b / a given in Figure 2.1. Equivalent Volume Diameter D(mm) 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 Axial Ratios bcm 1.0000 0.9999 0.9996 0.9988 0.9977 0.9961 0.9939 0.9912 0.9879 0.9841 0.9794 0.9748 0.9686 0.9629 0.9569 0.9506 0.9440 0.9373 0.9304 0.9233 0.9161 0.9088 0.9014 0.8939 0.8863 0.8786 0.8709 0.8631 0.8553 0.8474 0.8396 0.8318 0.8239 0.8161 0.8083 0.8006 0.7928 ABLav 1.0000 0.9999 0.9996 0.9988 0.9977 0.9961 0.9939 0.9904 0.9880 0.9847 0.9839 0.9861 0.9856 0.9727 0.9672 0.9626 0.9578 0.9527 0.9475 0.9420 0.9363 0.9305 0.9244 0.9181 0.9116 0.9049 0.8980 0.8909 0.8836 0.8761 0.8684 0.8604 0.8523 0.8440 0.8354 0.8267 0.8177 a = b!a Kav 0.9936 0.9939 0.9938 0.9934 0.9926 0.9915 0.9901 0.9883 0.9863 0.9839 0.9813 0.9784 0.9752 0.9718 0.9681 0.9642 0.9600 0.9556 0.9510 0.9462 0.9412 0.9360 0.9306 0.9250 0.9193 0.9135 0.9074 0.9013 0.8950 0.8886 0.8821 0.8755 0.8688 0.8620 0.8551 0.8482 0.8412 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40 5.50 5.60 5.70 5.80 5.90 6.00 0.7852 0.7776 0.7700 0.7625 0.7551 0.7478 0.7406 0.7335 0.7264 0.7195 0.7127 0.7059 0.6993 0.6928 0.6865 0.6802 0.6740 0.6680 0.6621 0.6563 0.6507 0.6451 0.6397 0.8085 0.7992 0.7896 0.7798 0.7698 0.7596 0.7492 0.7419 0.7346 0.7274 0.7202 0.7131 0.7061 0.6991 0.6923 0.6855 0.6787 0.6721 0.6655 0.6591 0.6527 0.6464 0.6401 0.8342 0.8271 0.8200 0.8128 0.8057 0.7986 0.7914 0.7843 0.7772 0.7701 0.7631 0.7561 0.7491 0.7423 0.7355 0.7288 0.7221 0.7156 0.7092 0.7029 0.6968 0.6907 0.6849 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX B Dielectric Constants O f Water At Selected Frequencies The following table lists the complex dielectric constants for water at 10°C calculated at the given frequencies following Ray (1972). Tablel Complex dielectric constants of water at 10°C Wavelength Frequency er = e '-je" X(mm) 3.189 7.495 8.565 9.993 21.410 31.995 F(GHz) 94 40 35 30 14 9.4 £' 6.71186 12.21690 14.07290 16.74950 39.66280 55.14100 £" 10.15310 22.12600 24.62700 27.62410 38.98790 37.93160 175 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX C Newton-Raphson (Gauss-Newton) Iterative Technique The Newton-Raphson (Gauss-Newton) Method uses a Taylor series expansion to approximate the nonlinear regression model with linear terms and then employs ordinary least squares technique to estimate the parameters (Neter et al., 1996). We wish to estimate the parameters of the regression function f(X,y) that models the relationship between the series of related data pairs or training sets ( Y j,X j) in the form Cl where f(Xj,y) is the mean response o f the i* case of the predictor variable X according to the nonlinear 'Va response function f(X,y). The regression parameters y = are to be derived such as to minimize the y p-i error 8; in the response (observation) data Y,. We begin the process with an initial (starting) guess of the S„ values of the regression parameters y denoted as g = where the superscript denotes the iteration Sm order (number). The starting values gj0’ may be obtained from previous or related studies, theoretical expectations, or a preliminary search of parameter values that lead to a comparatively low criterion value Q where C2 The next step involves the Taylor Series approximation of the mean responses f(Xj,y) around these starting values gj0’. Retaining only the first two terms of this Taylor series expansion (i.e. the first order Taylor polynomial) gives 176 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. *=0 d f ( X t,y) dyt (n - 8 7 ) C3 r= i'0’ To simplify the above expressions we shall make the following substitutions f r = f ( X r 8 W) C3.1 P 7 = r t - g ? C3.2 r tO) df(X„y) C3.3 J ik ~ r=K -i (0) \k *=0 such that the approximation to the nonlinear regression model o f (C l) may be written as y . - f r + t i T P t * * , C4 k= 0 Rearranging (C4) we note the differences (residuals or deviations of the observations around the nonlinear regression function with parameters replaced by the starting estimates) as Yt(0> = Yt - f ' 0> to obtain the following linear regression model approximation C5 Note that in (C3.2) the regression coefficients /3'°’ represents the change (difference) between the true regression parameter and the initial estimates of the parameters that will be used later to adjust the estimates of the regression parameters. The above expression may be rewritten, using matrix notation, as follows Y<0, 0 .0 ,0 , +e C6 177 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where >w , (0) yr(0> _ C6.1 (0 ) Yn J n nxl r(0) r ( 0) ^ 1.0 ^ l.p-l r(0) ■'fl.O r<u> /<0) ‘'n.p-l J ' 0) = C6.2 nxp 1 0 1 V £ l0) = • and £= • *5. §1 _£ n . 1 pxl C6.3 nxl Note that J<0) in C6.2 represents the Jacobian matrix. The regression parameters P<0>may then be estimated via the ordinary least squares procedure as, O) bi0)= ( j {0),j ' o' y j ' o)‘YM > C7 (Herein lies one o f the weaknesses o f the Newton-Raphson method; the necessity to invert the Jacobian matrix at each step. However, there are a number o f procedures that may be used to avoid inverting these matrices explicitly. See Burden and Faires, 1988fo r further details). The estimate b<0) is then used to update the previous (initial) regression parameters by means o f C3.2 as follows, g i"= * r + c 178 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. or in matrix notation form, as g '" = g m +bt0) C8 At this point we examine whether the revised regression coefficients represents adjustments in the proper direction by comparing the least squared errors before and after the iteration ,=l for j = 0, 1 ....... C9 i=l If the Newton-Raphson method is working properly then the SSE<J) should be smaller than SSE01’. If that is the case then the above procedures may be repeated with g(l> replacing the initial parameter estimates to produce a new set of revised parameters gl 2) and a new least square criterion measure SSE<2). This iterative process may be continued until the differences between successive coefficient estimates g ^ 11ga) and/or the difference between successive least square criterion measures SSE(j+'1 - SSE^' become negligible or reaches some lower limit. The least squares and maximum likelihood estimators o f nonlinear regression models are not normally distributed, and as such are not unbiased, nor do they have minimum variances. Consequently, inferences about these regression parameters are usually based on the Large-Sample Theory. This theory states that the least squares and maximum likelihood estimators for nonlinear regression models with normal error terms are approximately normally distributed and almost unbiased, when the sample size is large. It should be noted that this theory applies even when the error terms are not normally distributed (Neter et al., 1996). The approximate variance-covariance matrix of the regression coefficients is given as s 1(g) = M S E - ( J ' j y ' CIO where MSE is the estimated mean squared error. The interval estimate for the regression parameters are therefore given as 179 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a where /j^l- — ; n - p denotes the a 1—— -100 percentile of the t-distribution with n-p degrees of freedom. 180 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENXIX D Piece-Wise R-A fits The following represents the coefficients ‘a’ and exponents ‘b’ to the piece-wise power law relationships of the form R = a-A b for the three MISS and SWISS data sets. These relationships were used to estimate the rainfall rates and rainfall accumulations presented in Chapter 4, from the simulated attenuation values. The 95% Cl for these fitted values are given as a ± cra and b ± o b. The TROP relationships were fitted over the entire range o f rainfall rates and attenuation values and are only provide here for the purpose of making comparisons. Distribution MISS Relations R-AA R-A h SWISS R-AA R-A„ TROP R-AA R-Ah Shape a BCm AA < 0.23 dB/km 18.3303 AA > 0.23 dB/km 26.0042 ABL.V AA < 0.17 dB/kn 21.1593 AA > 0.17 dB/km 32.2551 Kav AA < 0.16 dB/km 23.7642 AA > 0.16 dB/km 35.0333 BCm A h < 2.05 dB/km 3.9588 A h > 2.05 dB/km 3.7055 A B U A h < 2.03 dB/km 3.9953 A h > 2.03 dB/km 3.7470 Kav A h < 2 .0 2 dB/km 3.9985 A h > 2.02 dB/km 3.7524 BCra AA < 0.16 dB/km 17.6130 AA > 0.16 dB/km 26.5858 A B U AA < 0.11 dB/kr 20.3712 A A > 0 .1 1 dB/km 32.1309 Kav AA < 0 .1 1 dB/km 23.0405 AA > 0.11 dB/km 35.1273 BCm A „ < 2.27 dB/km 4.0071 A h > 2.27 dB/km 3.5606 A B U A h < 2.26 dB/km 4.0430 A„ > 2.26 dB/km 3.6007 Kav An < 2.26 dB/km 4.0456 A h > 2.26 dB/km 3.6024 BC„ 26.7332 33.1477 ABU Kav 35.1714 BU 3.8776 ABU 3.9132 Kav 1 3.9133 da 0.1438 0.0677 0.1890 0.0761 0.2018 0.0717 0.0040 0.0264 0.0040 0.0263 0.0039 0.0266 0.1879 0.2151 0.2417 0.2861 0.2584 0.2975 0.0051 0.0602 0.0050 0.0599 0.0050 0.0605 0.1009 0.1348 0.1305 0.0124 0.0123 0.0122 b 0.6848 0.9192 0.6674 0.9056 0.7029 0.9182 0.9285 1.0204 0.9301 1.0206 0.9311 1.0213 0.6608 0.8830 0.6477 0.8514 0.6852 0.8771 0.9196 1.0634 0.9215 1.0637 0.9225 1.0650 0.8113 0.8018 0.8175 0.9545 0.9556 0.9565 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ob 0.0039 0.0023 0.0039 0.0024 0.0037 0.0022 0.0016 0.0026 0.0015 0.0026 0.0015 0.0026 0.0045 0.0096 0.0045 0.0095 0.0042 0.0088 0.0019 0.0089 0.0018 0.0088 0.0018 0.0089 0.0039 0.0041 0.0037 0.0014 0.0014 0.0014 APPENDIX E Allan Deviation and Its Use in Describing the Dppl Stability The modified Allan variance (MAVAR), or the modified two-sample variance, has been suggested as an appropriate statistic for these plots (Allan, 1987; Allan et al., 1997). The MAVAR is an internationally accepted standard that has been used extensively to measure and characterize the stability performance of clocks and oscillators (Greenhall, 1997), and has recently found increasing applications other areas e.g. radiometry (Croskey, private communications) and radio astronomy (Rau et al, 1984; Schieder et al., 198S; Gursel, 1996). Gursel (1996) employed the Allan variance and Allan deviation statistics in a - T plots to assess the performance of different interferometers. Allan variance measurements have been employed by a number of other groups to characterize radio-astronomical instruments (Rau et al, 1984; Schieder et al., 1985). Allan variance methods were also considered instrumental in the development of the first space-borne Acusto-optical spectrometer (AOS) on board NASA’s Sub millimeter Wave Astronomy Satellite-SWAS (Schieder et al., 1996; Frerick et al., 1999). Allan (1987) used results of Allan deviations (a) in instrument data as a function of the averaging time (x) to compare the stability of a number of then state-of-the-art precision oscillators. The MAVAR equation is given as (E.l) where the variance is on the variable y. Here each value of y represents the average o f the data over a non-overlapping interval x, so that the ys are taken as an adjacent series with no delay between successive mean measurements. A denotes the first finite difference of the measures of y; i.e. Ay = y„+1 - y n. Each of the first finite differences is then squared. The brackets o signifies that the expectation of these differences which, for a finite data set, is taken as the average of the differences squared. Therefore the Allan variance represents half of the mean square of the differences of successive averages. This more accurately captures the short-term stability in the data (deviation between adjacent averaging periods), which is often more relevant than the deviation with respect to a global mean. Dividing by the factor 2 causes the MAVAR to be equal to the classical variance if the ys are random and uncorrelated i.e. white noise (Allan et al., 1997). The advantages of the MAVAR over the classical variance are: 182 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • It converges for all the common types of noise whereas the conventional variance may only converge for certain types of noise. • From the observation of the dependence of the MAVAR on averaging time t, the type and level of the noise can be inferred, while the classical variance does not easily distinguish between noise types. For example the standard variance does not distinguish between short-term variations and long-term drifts (Ruf et al., 1996). • The confidence of the estimated variance increases for longer data length. This approach differs from the stability tests carried out by Ruf et al. (1996) on the DPPL. Their assessment of the short-term stability of the instrument was made by estimating the standard deviation of samples of the data after First smoothing the data with running averages of variable lengths. It is believed that the smoothing performed on the data may serve to obscure some of the errors inherent in the data. Furthermore, it was observed that the errors obtained from this procedure did not converge, even for integration periods as long as 12hrs. This is inconsistent with the expectation that the errors should converge over long periods of averaging times. Allan (1966, 1987) indicates that the classical standard deviation may not converge for some noise sources and can therefore mischaracterize the instrument stability. It is therefore assumed that this type of sampling does not accurately reflect the actual errors in the measurements. The stability test was performed nn l-s averages of the individual horizontal and vertical received powers recorded over a 77-hr period in the absence of rain. The equivalent attenuations were calculated for each data point. Figure E.l shows the results of the stability test performed on this data. 183 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. DPPL StabiNty Test on June 2001 data 10 f* 0 1 1 s V(dB) H(dB) V/H(dB) _ _ _ _ V dB /km ) AA<dB^cm) _ A ^ (d B * m ) 10‘ Integration Time t( t(min) m in ) Fig. E.I The short term stability characteristics o f the DPPL for single polarization (V, H), differential polarization (V/H), and signal product ( V ■H ) processing. Also shown are the stability characteristics of the estimated signal attenuations over a 103.5 m path under ‘clear air’ conditions. The Allan deviations of the received powers and the corresponding attenuations are plotted against the averaging times. The Allan deviation is defined as the square root o f the MAVAR, in the same way that the standard deviation is related to the classical variance. Note that the attenuations Av, Ah, AA, and Avg, are estimated from V, H, V/H, and V ■H respectively (see Section 5.5 for details), where V, and H are the V- and H-polarized return signals, and V/H and V ■H represent their ratio and product respectively. These results indicate the rms level of the drifts in the DPPL hardware over various averaging periods, and the extent to which the signals and the corresponding attenuation parameters are affected by these system drifts. It is evident from these results that the noise in the attenuation parameters are generally higher than those of the signals. This is primarily due to the fact that the attenuations are given as the spatial averages and normalized by the path length, whereas the return powers are not. Consequently, the errors in Av, Ah, AA are a factor o f 4.83 while Avg is a factor o f 2.42, higher than the corresponding errors in V, H, V/H, and V •H respectively. In each case the Allan deviations decreased monotonically 184 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. with averaging times between 5-10 minutes (depending on the parameter), and increased for longer averaging periods. The error is expected to decrease with averaging time as long as the measurement noise is stationary and its mean is constant. Hence these results suggest that the absolute levels of the instrumental and or clutter biases are well correlated over periods o f less than 5-minutes. For averaging times over 3-hrs the statistic becomes somewhat noisy, as fewer independent samples are available with this much averaging. However, the Allan deviations are clearly much larger on these time scales, indicating the rms level of the long-term drifts in the hardware. Note that the errors in V and H are different (i.e. the slopes of their respective curves are different at the various averaging times), with H being noisier than V. It is also clear that for averaging times in excess of 0.7 minutes the errors in V/H are less than that in H. The errors in V ■H are higher than V, H and V/H for all averaging times, since it includes the errors in both H and V. However, the Aavg has the lowest errors of all the attenuation parameters for averaging times of less than 0.7-mins. For all other time averages the errors in Aavg exceeds only those of Av, and the errors in the other attenuation parameters mirror those in the corresponding signal as previously discussed. Note that these “errors” are actually deviations that have not been normalized. Therefore it is quite reasonable to have larger deviations in H or V ■H than in V/H whose magnitude is quite small. The normalized error is defined as the ratio of the Allan deviation of the signal to the mean value of the signal for a given integration time. This provides additional information on the stability of the system. This statistic interprets the variation of the signals relative to its mean value. Another figure of merit is the Signal-to-Noise Ratio (SNR). The SNR is inversely proportional to the normalized error. The normalized error and the SNR of signal were calculated and are given in Fig. E.2 as a function of the integration time. Note that the normalized errors on the alternate signal polarizations and their products are less than 2% for integration periods of up to 100-mins. The errors in the ratio V/H are much higher (> 30-times higher than the errors in H), however this error may be reduced to less than 20% for integration periods of 2-10 minutes. Note that the peak in the SNR coincides with the minimum normalized errors. 185 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. DPPL Stability Test on June 2001 data V(dB) H(dB) V/H(dB) VH(dB) a) Integration Time t(min) b) Integration Time t(min) Fig. E.2 a) The normalized errors in the DPPL signals under “clear-sky” conditions as a function of the integration times. Note that the error decreases monotonically with averaging times between 5-10 minutes, b) The SNR on the DPPL measurements. Note that the SNR is a minimum for those averaging times over which the normalized errors in the signal are at a maximum. 186 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX F Joss-Waldvogel Disdrometer Bin Size Categories The mid-size values of the 20 drop size categories used for the JWD. mid_sizes=[0.351 0.452 0.547 0.648 0.760 0.901 1.098 1.309 1.488 1.648 1.880 2.214 2.542 2.830 3.154 3.494 3.862 4.284 4.788 5.135] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX G Drop Size Distributions of the June 16*112001 Rain Event The following figures show the DSD estimated from JWD measurements taken during the course of the June 16th 2001 rain event. Represented in these plots are those bin sizes for which at least 1 drop per m-3 was detected. Each frame (individual plot) represents the DSD averaged over a non-overlapping 1-min time interval. The frames are arranged chronologically. The numbers at the top right hand comer of each graph (frame) indicates the number o f minutes into the event when the corresponding measurements were taken, e.g. I and 4 refers to the first and fourth minutes respectively. The DSD for the first 84-mins of the shower have been shown. l 10‘ Q 10 10' 5 10' 10 10 ’ 10 10' 10' 5 10' B. 10 S 10 10 ’ 10' 5 10 10' 10' 5 10' 12 11 S 10 10 10’ 10' 10' 10 D(mm) D(mm) 5 10' D(mm) D(mm) The first null i.e. rain trough, coincided with a local minimum in the value of D0 (=1.6mm), however the drop concentration was at its maximum (>900 m'3) and coincides with frame #8. The process soon reversed itself and the rainfall rates began increasing. The second rain peak occurred as the D0 peaked just over 2.5mm (frame #16). The evolution of the DSD curves over this period differed to that leading 188 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. up to the previous rainfall peak and for a short period after. In this case the width of the spectra increased as increasingly larger drops were observed, but unlike the previous situation, these changes were also accompanied by an increase in the modal drop size from just less than 1mm to approximately 2mm. The results also show that this coincided with a decrease in the drop concentration. Not withstanding this decline in drop concentration, because of the significant increase in the number of drops both A and R increased. Hence the second rainfall and attenuation peaks coinciding with frames #15 and #16. is 10 9 10' 10 10 10 10‘ 10 10 10‘ 20 Q 10 10 10 ' 1o 10‘ 5 10 5 10' 2A 10l D(mm) 10 10 10' 10' 10' 10 O(mm) D(mm) D(mm) For the next 12 minutes leading up to the third rainfall peak, very little change was observed in the drop concentrations of the individual spectra. However, the modal drop sizes of the spectra slowly decreased to about 1mm, and the DSD were observed to be less narrow. This third rainfall peak corresponds to frame #27. The third rain peak coincided with a slight increase in the drop concentration and an equally small decrease in D0. It is also evident for the lateral spreading in the DSD curves for this period, that an increasing number of larger drops were arriving at the surface. Therefore this increase in the larger drops was compensated for by an even larger increase in the smaller drops, resulting in decrease in the median size diameters D0. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. There are a few interesting features related to this third rain peak. First the JWD measurement for this rain peak under estimated that of the ORG. This may be related to the ORG’s reported sensitivity to N0 the intercept of an exponential DSD that is believe to describe the ratio of small drops to larger drops (Nystuen, 1999). This may explain why this peak is higher for RoRG than it is for R jwd- Secondly, it is clear that the measured values of Ah, Av, and Aavg corresponding to this rain peak are significantly lower than their values for the previous rain peak, despite the fact that the rainfall rates were quite similar for the ORG estimated R. 10‘ S 2 3 1 0 10 10 10' 10' 10' 10' 22 Q 10 10 10’ 10 10' 10' 10' 10' 36 Q 10 10 10' 5 D(mm) 10' 5 D(mm) 10 10 10' 10' D(mm) D(mm) The shape of the DSDs become more concave-down as the rainfall rate goes through a local minimum (i.e. frames 32-34), but then becomes wider once again with a slight decrease in the drop concentration (from =650 m'3 to =550 m’3) as the rainfall rates begin to increase. The changes in the DSD leading up to the fourth rainfall peak (frame #38) is somewhat similar to the changes that resulted in the third rainfall peak. However, in this case there was a larger increase in DQthan that observed at the second rain peak (D0 increased for about 2.2mm to about 3.0mm). The experimental attenuation values corresponding to this rain peak were similar to those obtained for the previous peak. It therefore appears as thought the measured attenuations and the rainfall rates have different sensitivities to different drop sizes. 190 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. AQ 10 Q 10 10 10 10' 10' 10' Q 10' 10 10 10 10' 10' 10' 10 10 10' 10' □ 10' 10' 10' D(mm) D(mm) D(mm) D(mm) 10 S 10 10 10 10 1 0' 10' 10' 10' 56 10' 10 10 10 10' 10' 10' 50 Q 10’ 10 10’ 10 ’ 10' 10' 10' o 10° 0 1 4 D(mm) 5 D(mm) D(mm) D(mm) 191 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. IQ- 10' 10' 10 10 10 10 10 10' .68 10' 10 10' 10 10 10' 10' 2. Q 10 10 ' 22. 10 10 10 10‘ Q 10‘ 5 10' D(mm) 5 D(mm) D(mm) 10" D(mm) ^ s . 10 Q 10 26 10' 10 10 10 10' 10' 10 60 9 10 ' 10 10 10 10' 10' 10' 10' 64 t □ 1° 10' D(mm) 10 10 10’ 10' 10' 10' D(mm) D(mm) D(mm) I92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX H Table Of Published R-KDp Relationships. The following are published relationships of the form R = a K bDP. They have been provided for comparison with those given in Table 6.3 Authors S-band X-band C-band a b S-Z 37.10 Chand. a b a b Shape DSD 0.866 Equil M-P 40.50 0.85 Green Gamma A-G 36.15; 0.84; 18.45^ 0.82k Green Disdrometer A-G 33.77j 0.97j 16.03, 0.95, Green Disdrometer 34.60 0.83 Keen. Disdrometer Keen. Mat. 41.5 0.85 21.6 0.84 14.0 0.85 Equil. Gamma Mat. 58.1 0.80 30.9 0.80 20.5 0.80 Mean Gamma Mat. 40.0 0.89 20.2 0.89 12.7 0.89 Equil. Lognormal Mat. 59.1 0.86 30.0 0.85 19.0 0.85 Mean Lognormal Authors S-Z Sachidananda and Zmic'. 1987 Chand. Chandrasekar et al. 1990 A-G; Aydin and Giridhar 1992 for 0.01° < KDp< 1.5° km'1 A-Gj Aydin and Giridhar 1992 for 1.50° < Kdp < 7.0° km' A-Gk Aydin and Giridhar 1992 for 0.01° < Kdp < 3 0 km'1 A-G, Aydin and Giridhar 1992 for 3° < KDP < 7.0° km'1 Keen. Keenan et al. 1997 Tim. Timothy et al. 1999 Mat. Matrosov et al. 1999 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The relationship given by Timothy et at. (1999) for drops at T = 10°C at the K„-band frequency o f 13.8 GHz are as follows: K 0P =0.06158- ff" 3443 (deg/km) Tim* gives K 0P =0.076 • /? '108 (deg/km) Tim. P-B ATD/, =0.048 • /? '17 (deg/km) Tim. C-B the best fit over the variation of all parameters The relationships given by Aydin and Lure (1991) using Marshall Palmer (M-P) and Joss Thunderstorm (J-T) distributions o f oblate spheroids for side looking radars at the W-band frequency of 94 GHz are Ko, = -0.148/?+ 0.113 for 1< R < 40 m m /h M-P K Dp = -0.132/?-0.559 for 40 < /? < 100 minih M-P = -0.101/?-0.075 for 1< R < 4 0 m m /h J-T K dp = -0.070/?-1.396 for 40 < R < 100 m m /h J-T Shapes Keen, a/b = 0.993 + 0.082 D - 1.874 D2 + 1.469 D3 Pruppacher and Beard (1970) Model (experimental) b/a = 1.0 Keenan et al. (1997) P-B if D < 0.5 mm, = 1.03 -0.062 D for 0.5 < D < 3.54 mm = 1.254 - 0.124 D for 3.54 D < 4 mm, and = 1.05 - 0.073 D otherwise Chuang and Beard (1990) Model (theoretical) b/a = 1.0 Pruppacher and Beard (1970) C-B if D < I mm = 1.01668 - 0.098055 D -0.252686 D2 + 0.375061 D3- 0.168692 D4 for 0.3 < D < 9mm Equil. a/b = 1.03 - 0.62 D Chuang and Beard (1990) for D > 0.05 cmMatrosov et al. (1999) 194 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Mean a 1_“ —= ---- — b l+I 6 D2 for v - D ------4 where D is in cm Matrosov et al. (1999) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. REFERENCES Adami, A. and L. Da Deppo, 1985: On the systematic errors of tipping bucket-recording rain gauges. Proc. International Workshop on the correction of Precipitation Measurements. Ajose, S. O., Matthew N. O. Sadiku, and U. Goni, 1995: Computation of Attenuation, Phase Rotation, and Cross-Polarization of Radio Waves Due to Rainfall in Tropical Regions IEEE Trans. Antennas Propag., Vol. 43, No. 1, pp. 1-5. Ali, Adel A., and Mohammed A. Alhaider, 1992: Millimeter Wave Propagation in Arid Land-A Field Study in Riyadh IEEE Trans. Antennas Propag. Vol. 40, No. 5, pp. 492-499. Allan D. W„ 1966; Statistics of atomic frequency standards. 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Atmos, and Oceanic Tech. 7, 792-795. 208 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. VITA Name: Sean E. A. Daisley Education: • Master of Science in Electrical Engineering The Pennsylvania State University (PSU)- Penn State (1997) • Bachelors of Science in Electrical Engineering (Hon.) The University of Puerto Rico (RUM)-Mayaguez (1994) Honors and Awards • 3rd Place - 17th Annual Graduate Exhibition 2002 • College of Engineering Graduate Fellowship 2000, 2001 • General Electric Foundation Dissertation Fellowship 2000 • Graduate Teaching Assistant 1995. 1998-2002 • Organization of American States Fellowship 1996-97 • National Collegiate Minority Leadership Award 1991-92 • All American Scholar 1991-92 • National Deans List 1990-94 • Caribbean Basin Initiative (CBI) Scholar 1989-94 Professional Affiliations • Student Member Institute of Electrical and Electronic Engineers - IEEE • Student Member National Society of Professional Engineers - NSPE Extra Curricula Activities • Member Caribbean Students Association - CSA • Member Black Graduate Students Association - BGSA • Treasurer International Football (Soccer) Club - IFC • Associate Member African Students Association - ASA Reproduced with permission of the copyright owner. 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