# Bayesian retrieval of complete posterior PDFs of rain rate from satellite passive microwave observations

код для вставкиСкачатьGraduate School Form 9 (Revised 6/03) PURDUE UNIVERSITY GRADUATE SCHOOL Thesis Acceptance This is to certify that the thesis prepared Jui-Y uan Chiu By E n titled BAYESIAN RETRIEVAL OF COMPLETE POSTERIOR PDFS OF RAIN RATE FROM SATELLITE P A SSIV E MICROWAVE OBSERVATIONS Complies with University regulations and meets the standards of the Graduate School for originality and quality For the degree o f D o c to r o f P h ilo so p h y Signed by the final examining committee: Approved by: \ t 4-&^'a JA ai'-AaA I^j _____________________ A i/J U i Head o f the Graduate Program This thesis (_ / is not to be regarded as confidential. <m 0 3 Date — Major Professor Format Approved by: or Chair, Final Exam ining Committee /} D epartm ent T hesis Form at A dvisor Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. BAYESIAN RETRIEVAL O F C O M PLETE PO ST E R IO R PD FS OF RAIN RATE FROM SATELLITE PASSIVE MICROWAVE OBSERVATIONS A Thesis Subm itted to th e Faculty of Purdue University by Jui-Y uan C. Chiu In P artial Fulfillment of the Requirem ents for th e Degree of D octor of Philosophy August 2003 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3113785 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignm ent can adversely affect reproduction. In the unlikely event that the author did not send a com plete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3113785 Copyright 2004 by ProQuest Information and Learning Company. All 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. ACKNOW LEDGM ENTS I would like to express my gratitude to Drs. Grant W. Petty, Harshvardhan, Lawrence W. Braile, and Sonia Lasher-Trapp for serving on my Com m ittee. Their helpful advice and insightful comm ents have been beneficial to my research. I p ar ticularly wish to th an k my adviser, Dr. G rant Petty. His expertise, encouragement, support, and counsel added considerably to my graduate experience. I doubt th a t I will ever be able to express my appreciation fully, bu t I owe him th e m ost over whelming debt of gratitude. A very special thanks goes out to Dr. W illiam S. Olson, who has been always a patient and kind m entor to me. I m ust also acknowledge K athy Kincade for her assistance in adm inistration. I am also grateful to my office m ate Benjamin T. Johnson for his advice in com puter and editing, and especially for his friendship. Finally, b u t not least, I would also like to thank my family for providing me unconditional support and love through my entire life. W ithout their encouragem ent, I would not have finished this dissertation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. iii TABLE OF CO N TEN TS Page LIST OF T A B L E S ............................................................................................................ vi LIST OF F I G U R E S ........................................................................................................ vii A BSTRACT 1 2 IN TRO D U CTIO N ................................................................................................... 1 1 .1 Im portance of precipitation m e a s u re m e n ts ............................................. 1 1.2 Microwave rem ote s e n s i n g ............................................................................ 2 1.3 Overview of Bayesian algorithm s .............................................................. 5 1.4 Objectives and p r o c e d u r e s ............................................................................ 8 DATA D E S C R IP T IO N ............................................................................................. 11 2.1 11 TM I d a t a .......................................................................................................... 2.1.1 Ideal response to surface rain r a t e .................................................. 11 2 1 .2 A ttenuation in d e x ................................................................................. 13 2.1.3 D istributions of attenuation index of T M I .................................. 16 2.2 P R rain r a t e ....................................................................................................... 19 2.3 PR -T M I m atch-up procedure ..................................................................... 21 ............................................................................................. 25 3.1 Prior d is tr ib u tio n ............................................................................................. 25 3.2 Conditional likelihood ................................................................................... 26 3.3 E stim ators of th e posterior d is trib u tio n .................................................... 29 P R O O F - O F - C O N C E P T .................................................................................................. 31 . 3 4 ......................................................................................................................... xiii ALGORITHM BASIS 4.1 Rain rate estim ates from NWS W SR - 8 8 D n e tw o rk ................................ 31 4.2 Polarization c a lc u la tio n s ............................................................................... 32 4.3 Comparison w ith PR -T M I m atch-up d a ta 36 .............................................. 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Further reproduction prohibited without permission. iv 5 4.4 Bayesian algorithm from sim ulations ........................................................ 39 4.5 Retrieval based on s im u la tio n s ...................................................................... 40 SENSITIVITY T E S T ................................................................................................ 48 5.1 T he prior and conditional d is tr ib u tio n s ..................................................... 49 5.2 Designed e x p erim en ts....................................................................................... 51 5.3 Results for theoretical t e s t ............................................................................ 57 5.3.1 Intrinsic uncertainty of th e a lg o r ith m ............................................. 57 5.3.2 Sensitivity to th e prior k n o w le d g e ................................................... 64 5.3.3 Sensitivity to th e conditional d is trib u tio n ...................................... 69 5.4 6 7 Sum m ary ........................................................................................................... 85 .......................................... 86 6.1 G P R O F .............................................................................................................. 86 6.2 P E T T Y TM I algorithm ................................................................................ 89 6.3 P E T T Y HIST4 a l g o r i t h m ............................................................................. 91 6.4 Linear model .................................................................................................... 91 REAL-W ORLD A PPLIC A TIO N AND V A LID A T IO N .................................... 93 7.1 Bayesian a lg o r ith m .......................................................................................... 93 7.2 D atasets for v a lid a tio n .................................................................................... 96 7.2.1 Selected global rain c a s e s ..................................................................... 96 7.2.2 H eavy/w idespread rain e v e n t s ........................................................... 96 7.2.3 PR -T M I global m atch-up d ataset of 1998/04 ................................. 99 Validation m e tric s .............................................................................................. 99 BENCHMARK A LGORITHM D ESCR IPTIO N S 7.3 7.3.1 B i a s .............................................................................................................100 7.3.2 Root-m ean-squared d iff e r e n c e ..............................................................100 7.3.3 Correlation coefficient ........................................................................... 100 7.3.4 Heidke Skill S c o re ..................................................................................... 101 7.3.5 Theoretical HSS d is trib u tio n ................................................................. 103 7.4 R e s u lts ..................................................................................................................... 108 7.4.1 Typhoon c a s e ............................................................................................ 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 B auer’s 12 oceanic c a s e s ........................................................................ 114 7.4.3 Heavy/widespread rain e v e n t s ...............................................................119 7.4.4 April 1998 ............................................................................................... 123 7.4.5 S u m m a r y .................................................................................................. 127 UNCERTAINTY ANALYSIS........................................................................................ 134 8.1 Assessment of the posited conditional lik e lih o o d ...................................134 8.2 U ncertainty of the posterior m ean due to the prior distribution . . . . 8.3 Uncertainty of the posterior variance due to the prior distribution . . 143 8.4 R e s u lts ................................................................................................................ 145 8.5 9 7.4.2 141 8.4.1 IV m o d e l .................................................................................................. 145 8.4.2 2V m o d e l .................................................................................................. 148 Sum m ary ..............................................................................................................151 CALIBRATION OF MODEL SIM U L A T IO N ...........................................................152 9.1 C alibration of radar-sim ulated rain r a t e ................................................... 152 9.2 Verification of radiative transfer c a lc u la tio n s ......................................... 153 9.3 Effects of freezing level h e i g h t .................................................................... 154 9.4 Im plication ...........................................................................................................156 10 CONCLUSIONS AND FU T U R E W O R K ................................................................ 164 VITA ...................................................................................................................................... 174 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vi LIST O F TABLES Table Page 1.1 C haracteristics of T R M M /P recipitation R ad ar.......................................... 4 4.1 Param eters a and b in the approxim ation of liquid w ater extinction coefficient............................................................................................................... 33 5.1 Coefficients of oM, b^ and cM............................................................................. 50 5.2 Inform ation of designed experim ents in sensitivity tests, including the experim ent ID, the training dataset, and th e specifications of th e prior and conditional likelihoods applied to th e Bayesian algorithm ............... 52 Inform ation of date, orbit number, and th e nadir locations for rain events in the validation d a ta se t....................................................................... 97 7.1 7.2 S tandard 2 x 2 contingency table for evaluation of the skill of a binary classification procedure......................................................................................... 1 0 2 7.3 Sum m ary of bias, root-m ean-square difference (RMS), and correlation coefficients against validation datasets of orbit 336 and B auer’s cases for each algorithm. The unit of bias and RMS is m m /h r........................... 109 7.4 Sum m ary of bias, root-m ean-square difference (RMS), and correla tion coefficients against validation datasets of heavy/w idespread rain events and 1998/04 dataset for each algorithm . The unit of bias and RMS is m m /h r........................................................................................................ 1 1 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIG U RES Figure Page 2.1 Idealized brightness tem peratures dependence on surface rain rate (From Petty, 2001).............................................................................................. 12 2.2 Overall m ultichannel distributions of P index represented by th e num ber of pixels. D ata were selected from 110 TM I orbits during 1999 and 2000. Contours are logarithm ically spaced; actual value is 10x where x is th e contour label, x are p lotted for values of [0.5, 1, 2, 3, 4, 5] at a fixed P i 0 interval................................................................................ 17 2.3 Same as figure 2.2, bu t at a fixed P 3 7 interval.............................................. 18 2.4 Probability distributions of th e near-nadir P R 15-km interpolated rain rate are depicted by lighter-colored dots. D ata were calculated from the 2A25 P R near-surface rain ra te estim ates in January, April, July and O ctober of 1998, and a cut-off value of 0.04 m m /h r was applied. The resulting sample size is around 2.4 million. T he fitted lognorm al distribution is also presented here by a dashed curve............................... 23 3.1 Com parisons of the prior rain ra te probability distributions used in this study............................................................................................................... 27 4.1 C ontours of the num ber of pixels based on TM I d a ta (the first and th ird columns) and radar-radiative sim ulations (the second and fourth columns). Contours are logarithm ically spaced; actual value is 10x where x is th e contour label, x are p lo tted for values of [0.5, 1, 2, 3, 4, 5].......................................................................................................................... 38 4.2 1km NWS W SR - 8 8 D network com posite reflectivies for 10:40am, Au gust 13, 2002, w ith suspicious ra d a r retu rn s removed............................... 41 4.3 Plots of sim ulated 15 x 15 km rain ra te based on the W SR- 8 8 D com posite reflectivity for 10:40am, A ugust 13, 2002, and a Z-R relation ship proposed by M arshall and Palm er (1948) 42 4.4 Sim ulated P w for the d a ta of 10:40am, A ugust 13, 2002......................... 43 4.5 Sim ulated Pig for the d ata of 10:40am, A ugust 13, ......................... 44 4.6 Sim ulated P 3 7 for the d ata of 10:40am, A ugust 13, 2002......................... 45 2 0 0 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. viii 4.7 Plots of retrieved rain rate for th e d a ta of 10:40am, A ugust 13, 2 0 0 2 , based on the new Bayesian algorithm ............................................................ 46 T he probability distribution of rain rates and the joint and m arginal pdfs of the P vector based on analytical solutions of th e sophisticated model. The joint pdfs are plotted for [0.05, 1, 5, 7, 10, 12]..................... 53 5.2 Same as figure 5.1, b u t for the random ly generated D 2 d a ta se t 54 5.3 M ultichannel relationships by plotting th e num ber of pixels at a given P 10 (left column) or given P 3 7 (right column) for D2 training dataset. Contours are logarithm ically spaced; actual value is 1 0 x' where x is th e contour label, x are plotted for values of [0.5, 1, 2, 3, 4, 5]............. 56 2-D contours of th e stan d ard deviations of th e posterior probability distribution in the P 3 7 vs. P i 9 space at a fixed P 10 value for experiment R0. Contours are plotted in an interval of 5 m m /h r. Regions outside th e zero value of th e contour indicate an impossible P vector due to a zero m arginal density...................................................................................... 58 Histograms of retrieved rain ra te at different rain ra te ranges for R0 experim ent. Titles contain inform ation about range of tru e values, sample size, and th e estim ator. Numbers in parentheses represent the m ean and standard deviation of the histogram . Percentages present the fraction of retrieved rain rates located in the correct range............. 60 5.6 Same as figure 5.5, b u t using MLE estim ations......................................... 62 5.7 Examples of derived posterior rain rate distributions at some given P vectors in experim ent R0. The observation vector (P 1 0 , P 1 9 , P 3 7 ) is presented by the three num bers in parentheses........................................... 63 Plots of the prior probability distributions used in th e sensitivity tests. To b etter illustrate th e differences, ranges of [0, 10] and [10, 100] m m /h r are respectively shown in (a) and (b ).............................................. 65 Same as figure 5.5, but for experim ent R 1................ 66 5.10 Same as figure 5.6, but for experim ent R 1................ 68 5.11 Same as figure 5.5, but for experim ent R 4................ 70 5.12 Joint and m arginal probability distributions of th e physical model for e x p e rim e n t SO (u p p e r tw o row s) a n d S5 (b o tto m tw o ro w s). C o n to u rs are plotted for [0.05, 0.5, 1.0, 2.5, 5.0, 7.5, 10, 12.5, 15] in conditional pdfs.......................................................................................................................... 71 5.13 Same as figure 5.5, b u t for experim ent SO.................................................... 72 5.1 5.4 5.5 5.8 5.9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ix 5.14 Same as figure 5.13, b u t using m axim um likelihood estim ates as single pixel retrieved rain ra te s.................................................................................... 73 5.15 Similar to figure 5.5, b u t for MEAN (left panel) and MLE (right panel) estim ates of Experim ent S5 at ranges of heavy tru e surface rainfall (greater th a n 30 m m /h r)................................................................................... 74 5.16 C onditional and m arginal pdfs of experim ents S 6 (first row), S7 (sec ond row), and S 8 (the bottom ). Note th a t different channels are shown here for each experim ent to highlight th e change due to the difference in the covariance m atrix .................................................................................... 76 5.17 Similar to figure 5.5, bu t for the MEAN (left panels) and M LE (right panels) estim ates of experiment S 6 a t heavy precipitation rate ranges (greater th a n 50 m m /h r)................................................................................... 77 5.18 Same as figure 5.5, b u t for experim ent S 8 ................................................... 79 5.19 Same as figure 5.18, b u t for MLE estim ates............................................... 81 5.20 Joint and m arginal probability distributions of th e physical model for experim ent R5. T he joint pdfs are p lo tted for [0.05, 1, 5, 7, 10, 12]. . 82 5.21 Same as figure 5.5, b u t for experim ent R 6 .................................................. 84 6.1 6.2 7.1 Normalized histogram s of surface rainfall of cloud profiles included in the old (solid curve) and new (dotted curve) G PR O F database. The to ta l num ber of cloud profiles are 11069 and 3097 for th e old and new database, respectively......................................................................................... 88 S catter plots of num ber of cloud profiles on th e rain rate vs. P i 9 dom ain a t a given interval of P 10 and P 3 7 in the old (left panel) and new (right panel) G PR O F d atab ase............................................................... 90 Locations of heavy/w idespread rain events selected from TM I orbits during January-D ecem ber 1998....................................................................... 98 7.2 Plots of th e sample mean of th e retrieval vs. the posterior mean estim ated from th e synthetic dataset a t a given P if the corresponding sample size is (a) greater th an 100; (b) greater th a n 30; (c) greater th a n 10; and (d) less th an or equal to 10. Note change of range in each case....................................................................................................................104 7.3 Plots of 2-D Heidke skill scores (HSS); th e m axim um of HSS w ith re spect to tru e rain ra te threshold; and th e best algorithm threshold vs. tru e rain ra te threshold based on th e synthetic d a ta of the Bayesian 3V model, (a)-(c) are illustrated for M EAN estim ates, while (d)-(f) are for M L E ..............................................................................................................106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. X 7.4 M aps of P R interpolated rain ra te w ith 15-km resolution and retrieved rain ra te of G PR O F (old and new databases), P E T T Y TM I, P E T T Y HIST4, and linear model algorithm s for T R M M /T M I orbit 336. . . . I l l 7.5 Same as figure 7.4, b u t for Bayesian 3V-MEAN, 3V-MLE, 2V-MEAN, 2V-MLE algorithm s............................................................................................... 112 7.6 S catter plot of retrieved rain rate vs. P R rain rate for all algorithm s, based on th e validation d a ta of the TM I orbit 336.................................... 113 7.7 2-D distribution of Heidke skill scores (HSS) for th e 12 selected cases from th e B auer et al. (2001). The value noted in th e bottom -right corner of each plot indicates th e highest HSS of the algorithm ................. 115 7.8 Plots of th e best algorithm rain ra te threshold w ith respect to th e threshold of P R rain rate for th e B auer’s cases. T he algorithm used in th e retrieval is shown in th e title of each p lo t........................................... 116 7.9 P lots of th e m axim um Heidke skill score vs. the threshold of P R rain ra te based on th e dataset of 12 B auer’s cases.................................................117 7.10 2-D distribution of Heidke skill scores (HSS) for the heavy/w idespread rain events. The value noted in the bottom -right corner of each plot indicates th e highest HSS of the algorithm ..................................................... 120 7.11 Plots of th e best algorithm rain ra te threshold w ith respect to th e threshold of P R rain rate for th e cases w ith heavy and w idespread precipitation............................................................................................................. 1 2 1 7.12 Plots of th e m axim um Heidke skill score vs. the threshold of P R rain ra te based on the dataset of heavy/w idespread rain events........................ 1 2 2 7.13 2-D distribution of Heidke skill scores (HSS) in random ly selected 118 files in A pril 1998. The value noted in th e bottom -right corner of each plot indicates th e highest HSS of th e algorithm .............................................124 7.14 P lots of th e best algorithm rain ra te threshold w ith respect to th e threshold of P R rain rate for th e cases of April, 1998............................... 125 7.15 Plots of th e m axim um Heidke skill score vs. the threshold of P R rain ra te based on th e dataset of April, 1998....................................................... 126 7.16 P lots of th e m axim um Heidke skill score vs. th e P R rain rate threshold in th e ra n g e o f [0, 10] m m / h r b a s e d o n B a u e r ’s c a se s......................................128 7.17 Same as figure 7.16, bu t for th e heavy and widespread precipitation cases............................................................................................................................128 7.18 Same as figure 7.16, bu t for th e d ataset of April, 1998............................. 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xi 7.19 Plots of th e best algorithm threshold vs. P R rain ra te threshold in th e rain ra te range of [0, 10] m m /h r for th e B auer’s cases..........................131 7.20 Same as figure 7.19, b u t for heavy/w idespread precipitation cases. 7.21 as figure 7.19, b u t for th e d ataset of April, 1998........................ 133 8.1 Same . .132 Plots of joint probability density functions / ( P 37, i?) based on th e I V model (upper) and near-nadir PR -T M I d ata (bottom ). C ontours are logarithm ically spaced; actual value is 10X where x is th e contour label, x are plotted for values of [-5, -4, -3, -2, -1, 0, 1] and [-4, -3, -2, -1, 0, 1] for th e I V model and observations, resp ectiv ely ...........................135 8.2 Same as figure 8.1, b u t for th e joint p d f of (P i 9 ,R) based on th e 2 V model (u p p er)..........................................................................................................137 8.3 Same as figure 8 . 1 , b u t for th e joint p d f of (P i 0 ,R) calculated from the 3 V m odel........................................................................................................... 138 8.4 Plots of conditional pdfs / ( P i o | P i 9 , P 3 7 , P ) based on th e PR -T M I d a ta (dashed curve) and th e posited Bayesian model (dotted curve), a t a given set ( P , P 19,P 37) of (a) (1.5, 0.70, 0.40); (b) (4.5, 0.35, 0.25); (c) (6.5, 0.45, 0.20); (d) (11.5, 0.3, 0.1); (e) (15.5, 0.15, 0.05); and (f) (19.5, 0.1, 0.05). N noted in figures represents th e sample size of observed d a ta ........................................................................................................... 140 8.5 Plots of th e posterior m ean (m m /hr) vs. P 3 7 in the I V model under the specification th a t the prior distribution is logiV( —2.8, 2.0). T he uncertainty of the prior is sim ulated by th e symmetric contam ination function w ith a factor of 0.25. Solid and dotted curves indicate the suprem um and infimum, re s p e c tiv e ly ..............................................................146 8 .6 Range of posterior stan d ard deviation for the given posterior m ean in th e I V model when P 3 7 is (a) 0.6, (b) 0.4, and (c) 0.2. T he square and asterisk symbols present th e m inim um and m axim um values, re spectively. Thus, the area inside th e curves indicates th e region where th e stan d ard deviation might be located..........................................................147 8.7 Range of posterior m ean in th e 2 V model for (a) the lower bound, (b) upper bound, and (c) th e m aginitude of variation in units of (m m /h r). T he contam inated function and factor used here are th e same as those applied in th e I V m odel....................................................................................... 149 8 .8 Same as figure 8 .6 , bu t for 2 V model and given (P i9, P 37) = (a) (0.7, 0.4) (b) (0.6, 0.4), (c) (0.5, 0.2), (d) (0.4, 0.2), (e) (0.2, 0.1), (f) (0.2, 0.05), (g) (0.05, 0.05) and (h) (0.06, 0.07).................................................... 150 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xii 9.1 Cum ulative distribution functions of P R observed (solid curve) and NWS W SR- 8 8 D netw ork-sim ulated (dotted curve) rain rate. P R curve is based on the near-nadir P R interpolated rainfall intensity during Jan., Apr., Jul., and Oct. of 1998. W SR - 8 8 D-simulated curve is calcu lated a) using a fixed Z-R relationship w ithout adjustm ents tow ard P R m easurem ents, and b) w ith adjustm ents from a lookup table. Sample sizes are shown in th e parentheses..................................................................... 157 9.2 2-D contours of num ber of pixels associated w ith rain rate less th an 1 m m /h r at a given Pio interval. Left and right panels are based on W SR- 8 8 D simulations w ith a freezing level at 3 and 5 km, respectively. The middle panel is derived from PR -T M I d ata in 1998. C ontours are plotted for values of [1, 10 ,50, 100 , 500, 103, 104, 105]..................................158 9.3 Same as figure 9.2, b u t for rain ra te of [1, 5] m m /h r and different intervals of P \ q.........................................................................................................159 9.4 Same as figure 9.2, b u t for rain ra te of [5, 15] m m /h r and different intervals of Pio.........................................................................................................160 9.5 Same as figure 9.2, but for rain rate greater than 15 m m /hr and different intervals of lower P w ...........................................................................161 9.6 Plots of fraction of d a ta vs. bright band height based on P R 2A25 algorithm retrieval of 55 orbits during August 1998 and July 1999. Season inform ation is shown in titles. D JF represents th e w inter sea son, while MAM expresses th e spring. The vertical resolution of 2A25 product is 0.25 km ..................................................................................................162 9.7 Plots of fraction of d a ta vs. altitu d e of freezing level based on TM I m easurem ents in January and July, 1998. (a) and (b) are derived from the equation (9.1), while (c) and (d) are com puted from W ilh eit’s approach. Note th a t the W ilheit’s m ethod constrains th e freezing level to th e climatological altitu d e (around 4.9 km) at a given 302 K of sea surface tem p eratu re ....................................................................................163 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A BSTRA CT Chiu, Jui-Y uan C.. Ph.D ., P urdue University, A ugust, 2003. Bayesian Retrieval of Complete Posterior PD F s of Rain R ate From Satellite Passive Microwave Observa tions. M ajor Professor: G rant W. Petty. This dissertation presents a new Bayesian rain rate retrieval algorithm for th e TRM M Microwave Im ager (TM I), along w ith associated error analysis of synthetic sensitivity tests and real-world applications. The Bayesian approach offers a rig orous way of optim ally combining the actual m ultichannel observations w ith prior knowledge. It has been applied in m any studies to retrieve instantaneous rain rates from microwave radiances. However, this is believed to be the first self-contained algorithm whose ou tp u t is not ju st a single rain rate, bu t rath er continuous posterior probability distributions of th e rain rate. The success of th e Bayesian algorithm depends on th e accuracy of b o th the conditional probability density function (pdf) of microwave observations and the prior pdf of rain rate, as well as on th e interpretation of th e posterior probability distribution of rain intensities. T he current study presents explicit functions to reasonably approxim ate th e physical relationships between rain rates and microwave radiance based on b o th model simulations and observations from TM I and TRM M Precipitation R adar (PR). T he prior distribution is lognormal based on P R rainfall measurements. Two common statistical estim ators are tested for converting th e posterior pdf of th e retrieved rain ra te to a single rain rate estim ate for a pixel. To advance th e understanding of theoretical benefits of th e Bayesian approach, sensitivity tests are conducted using two synthetic datasets for which th e ’’tru e ” physical model and th e prior distribution are known. Sensitivity results have dem on strated th a t even when th e prior and conditional likelihoods have been applied per Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xiv fectly, an apparent bias in retrieval at high surface rain rate occurs. In addition, the tests suggest th a t the choice of th e estim ators and the prior inform ation are b o th crucial to th e retrieval. In this study, the new Bayesian algorithm has also been applied to real TM I d a ta over the ocean. Its estim ates are validated against independent d atasets and the perform ance of the new Bayesian algorithm is compared w ith th a t of other bench m ark algorithm s. The results are satisfactory in th a t our algorithm has com parable perform ance to other algorithm s while having th e additional advantage of providing posterior rain ra te probability distribution. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1. IN T R O D U C T IO N 1.1 Im p ortan ce o f p recip itation m easurem ents M easurem ents of precipitation are essential for our understanding of ocean and land-atm osphere interaction, and for th e q uantitative understanding of the global hy drological cycle. Ocean and land exchange freshwater and heat w ith the atm osphere across the interface via precipitation and evaporation. The terrestrial exchange pro cess affects the soil m oisture and runoff, and is thus an im portant issue in w ater balance and w ater m anagem ent. In th e m arine environm ent, precipitation is also a crucial factor th a t determ ines the salt balance in the upper layers of th e ocean, and thus influences th e oceanic circu lation on seasonal and inter-annual tim e scales. Ropelewski and H alpert (1996), Dettinger et al. (1998), and Paegle and Mo (2002) dem onstrated strong evidence relating interannual to decadal variability of regional precipitation to El N ino-Southern Os cillation (ENSO). O ther studies from Fowler and Kilsby ( 2 0 0 2 ), and Lucero and Rodriguez (2002) also suggested th a t precipitation at some areas is highly correlated with th e N orth A tlantic Oscillation (NAO). A num ber of studies have dem onstrated promising results th a t assim ilations of precipitation inform ation lead to significant improvements in forecasts and simula tions (Treadon, 1996; K rishnam urti et al., 2001). Hou et al. (2001) also suggested th a t the assim ilations of surface rainfall improve not only th e hydrological cycle in global analyses and short-range forecasts, b u t also key climate param eters such as clouds and radiation. In addition, inform ation about rainfall helps to validate th e current forecast/sim ulation ability of w eather prediction models and regional/global climate models (Kleinn et a l, 2002; Wei et al., 2002). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 In short, the need for inform ation about spatial and tem poral distributions of precipitation occurs in m any areas of earth science, and it points out th e im portance of a regular basis of system atic and accurate rainfall measurem ents. In th e next section, a num ber of techniques for m easuring precipitation will be discussed in detail. 1.2 M icrowave rem ote sensing The standard technique for m easuring rainfall has been to use surface rain gauges at discrete points. It measures precipitation in th e most direct way, b u t suffers from th e representativeness problem because rain gauges only observe a point quantity. In contrast to the low spatial resolution of rain gauge observations, surface w eather radars provide high tem poral and spatial resolution precipitation m easurem ents, bu t their coverage is still limited. In particular, th e oceans rem ain largely unobserved by surface radar. Unlike surface-based m ethods, satellite rem ote sensing offers rela tively frequent global coverage, and is an excellent tool for m easuring precipitation, especially in the m arine environment. Precipitation m easurem ents using spaceborne visible or infrared sensors are made by inferring rainfall from the characteristics of cloud. They do not directly sense the rainfall. However, due to th e ability of microwave radiation to p en etrate clouds, satellite passive microwave observations have been of interest for estim ating precip itation over the globe in the past decades. A num ber of passive microwave sensors have flown on spaceborne platform s, including the earliest Electronically Scanning Microwave Radiom eter (ESMR) aboard Nimbus 5 and Nimbus 6 , th e Scanning M ul tichannel Microwave Radiom eter (SMMR) on Seasat and Nimbus, th e Special Sensor M icrowave/Imager (SSM /I) onboard th e Defense Meteorological Satellite Program (DMSP) F - 8 to F-15 satellites, and th e recent Tropical Rainfall M easuring Mission (TRM M ) launched w ith b o th passive and active microwave sensors in 1997. TRM M operates in a circular orbit w ith a 35° inclination angle at an original altitude of 350 km. The altitu d e was boosted to 402 km in A ugust 2001 in order Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 to extend the mission. The prim ary instrum ents on TRM M include th e TRM M Microwave Imager (TMI), th e Precipitation R adar (PR ) and th e Visible and Infrared R adiom eter System (VIRS). T he em phasis of this study is on th e TM I and P R measurements. The TM I is a m ultichannel passive microwave radiom eter based on th e design of the highly successful SSM /I. The TM I antenna scans conically, and the resulting incident angle and swath w idth are about 52.8° and 758.5 km. TM I provides mi crowave measurem ents at 10.65, 19.35, 37.00, and 85.50 GHz vertical and horizontal polarization, and at 21.3 GHz w ith vertical polarization. T he effective fields of view (EFOV) in down-track and cross-track directions for th e above channels (ascending frequencies) are 44 x 27 km, 30.4 x 19.9 km, 27.2 x 18.3 km, 16.0 x 12.6 km, and 7.2 x 6.0 km (Bauer and B ennartz, 1998). Detailed descriptions of other TM I characteristics can be found in Kummerow et al. (1998). 1 Precipitation R adar is an active microwave sensor th a t scans in a cross-track strategy from nadir to 17°. T he sw ath w idth is about 215 km and th e num ber of independent samples for each scan is 49. T he minimum detectable threshold of P R reflectivity is about 17 dBZ (around 0.5-0.7 m m /h r) in th e absence of attenuation. The horizontal resolution is ab o u t 4.3 km a t nadir, while the vertical resolution is 0.25 km. O ther detailed inform ation is sum m arized in table 1.1. SSM /I and TM I m easurem ents have been extensively used to develop a num ber of retrieval algorithms for rain intensities. In th e developm ent of retrieval algo rithm s, th e key element is to find th e relationship between th e rain intensity and th e observed brightness tem peratures (or some transform ed variables from microwave m easurem ents). The observed variables have to show a great ability to reflect the precipitation signal and have a lower degree of sensitivity to noise from instrum ents, surface, and the atm ospheric variations. In addition, the construction of the physical relationship can be either empirically derived via ground m easurem ents from rain 1N ote, however, that paper gave an incorrect characterization of the TM I’s effective field-of-view (EFOV). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 Table 1.1 Characteristics of T R M M /P recipitation R adar. PR Frequency 13.8 GHz (K u band) Beam w idth 0.71° Peak transm it power 500W Pulse duration 1 .6 Pulse repetition frequency 2776 Hz Minimum detectable signal ~ 1 7 dBZ A ntenna height above m ean sea level 350 km Speed of one scan 34° in 0.6 s Scan range (cross track) ±17° Swath w idth 215 km Horizontal resolution at nadir 4.3 km Vertical resolution a t nadir 0.25 km /is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 gauges or radar, or based on simulations from conceptual or explicit cloud models and radiative transfer models, w ith or w ithout empirical calibration. 1.3 O verview o f Bayesian algorithm s An algorithm th a t applies Bayes theorem is called a Bayesian algorithm . The Bayes theorem states th a t given th e d a ta P th e distribution of the param eters is proportional to th e conditional likelihood tim es th e prior distribution ir(9|P) <x / ( P |9 ) jt(«). (1.1) The first term on th e right side specifies th e probability distribution of th e response variables (or observation vector) P given th e values of covariates and param eters 0, which is a conditional probability density function (pdf). Since th e distribution expresses statistical and physical inform ation ab o u t the relationship between P and 6, we will refer to this p art as th e physical m odel in the study. T he second term on the right hand side summarizes our prior knowledge of the param eter before the d ata are seen. The prior distribution of th e param eters is denoted as 7t(9). The prior distribution may only represent our beliefs, or it can be estim ated from exper iments. T he interaction of the physical m odel and th e prior probability distribution determ ines th e so-called posterior distribution 7r( 0 |P ), which tells us th e new pdf of 9 in light of th e observations P . Normally, th e effect of P is to reduce th e spread of tt(0 |P ) relative to 7r(0); the degree of reduction is a m easure of th e inform ation content of P . The Bayesian approach has been applied in many studies to retrieve in stan ta neous rain rates and vertical hydrom eteor profiles from microwave radiance m easure ments. Evans et al. (1995) developed a Bayesian algorithm based on num erical cloud models and forward radiative transfer modeling. The conditional pdfs in their algo rithm were constructed simply by assum ing th a t th e observation vector (brightness tem peratures in their study) is norm ally d istrib u ted around th e sim ulated bright ness tem peratures w ith a fixed variance. In th eir assum ptions, these channels in the observation vector were independent of each other. As to the prior distribution, a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 m ultivariate lognormal form based on cloud model sim ulations was used to approx im ate th e distributions of hydrom eteor profiles. Since th e focus of their study was placed on finding the most likely atm ospheric state, they com puted th e maximum likelihood estim ate of th e posterior p d f by optim izing a cost function. However, as they mentioned, some relationships in the parameter vector did not quite follow a lognormal behavior. In addition, due to th e high dim ensionality b o th in the obser vations vector and the param eters in their m ethod, it is h ard to trace the retrieval errors and to understand th e sensitivity of th e algorithm to those specifications. The G oddard Profiling algorithm (G PR O F) is a simplified Bayesian scheme, pro posed by Olson et al. (1996) and Kummerow et al. (1996). Similar to the previous work, G PR O F is also based on cloud models and forward radiative transfer calcula tions. The assum ptions regarding th e observation vector are th e same as in Evans et al. (1995): the conditional probability distributions are norm ally distributed, and the observational and modeling errors at various channels are uncorrelated. In ad dition, the retrieved variables in G PR O F also include th e surface rain rate and ver tical hydrom eteor profiles. However, unlike th e Evans study, th e prior distribution was not specified as a lognorm al distribution th a t contained covariance information, and th e posterior probability density function was not directly com puted. Instead, they introduced a database from cloud-radiative model sim ulations to serve as th e prior distribution, representing th e occurrence frequency of cloud profiles. Then, th e single-pixel retrieval was estim ated from an approxim ated m ean E of the posterior distribution using the expression: E’(x) ^ Xjexp{~°-5[y° ~ y*(xj)]T( ° + s )~1[y0 - y s(xi)]} 3 ^ A where x and y represent th e cloud profiles (covariates) and th e observation vector; j is the index of candidate profiles in th e database; th e subscript of o and s indicate th e observed and sim ulated quantity, respectively; O and S describe the diagonal error m atrix caused from m easurem ent noises and model simulations, while in fact th e simulation errors were assum ed zero; finally, A is th e norm alization factor stated Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 as A = ^ e x p { - 0 . 5 [ y o - y s (xi )]T( 0 + S )-1 ^ j - y^x,-)]}. (1.3) Based on th e estim ator of equation(1.2), the retrieved rain rates and hydrom eteor profiles in G PR O F are determ ined by how close th e observations are to th e candi dates of the database (the term of yQ—ys), and how often the selected cloud profiles occur in th e database (j). A nother TRM M standard retrieval algorithm (2B31) combines P R observations, TM I m easurem ents at 10.65 GHz, and an adapted radar retrieval algorithm to es tim ate the profiles of rain clouds via th e Bayesian approach. Details are contained in H addad et al (1997). The m ain response variables they used are th e rad ar re flectivities, one-way integrated atten u atio n s for each radar beam , microwave radi ance m easurem ents from TM I, and th e derived microwave brightness tem peratures from the rad ar model, while th e covariates encompass the rain profiles and th e drop size distribution (DSD) param eters. T he prior knowledge about SDS param eters is obtained from the analysis of th e D arw in and TOGA-COARE d ata. The prior distribution of the rain profiles is based on model calculations and represented in a form of m ultiplications of Dirac 8 and G aussian functions via th e dependence of radar beams. A similar Bayesian algorithm for G PR O F was developed by Bauer et al. (2001). Their retrieval technique was also based on combined cloud-radiative transfer sim ulations. In addition, the same functions as equations (1.2) and (1.3) were used to com pute th e retrieved near-surface precipitation. However, there are three ma jor modifications in their work. First, they criticized the representativeness of the database in G PR O F, and provided a m erged database th a t was constructed from cloud model simulations and 130 orbits of TM I data. Second, th e dim ensionality of th e observation variables were reduced by replacing brightness tem p eratu re informa tion w ith two or three empirical orthogonal functions (EOFs). T hird, the modeling errors in forward transfer calculations were no longer assumed to be zero, b u t were Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 assigned 2, 4, 6 , and 10 K for 10.65, 19.35, 37.00, and 85.50 GHz, respectively. However, these channels are still assum ed uncorrelated. In summary, the above Bayesian retrieval m ethods all rely on th e ability of inde pendent cloud-radiative databases to represent th e most possible solutions th a t have to be close to the real observations. The retrieved variables for those algorithm s are surface rain rate and vertical hydrom eteor profiles. In th e construction of the conditional and the prior distributions, simplified assum ptions are made for both elements. In addition, to reduce th e com putational requirem ent, th e single-pixel retrieval is desired instead of com puting th e complete posterior probability density function. The effects of imperfection of th e conditional likelihood and the prior inform ation on the retrieval have not been fully investigated. Due to th e complexity in both cloud and radiative transfer models, and the n atu ral uncertainties in th e atm ospheric variables, it is difficult to evaluate how well th e Bayesian approach itself works in the retrieval applications, and how sensitive th e Bayesian algorithm is to those assum ptions and uncertainties. It is essential to know the behavior of the Bayesian estim ator in the rain rate retrieval before adapting the Bayes rules to real-world applications. 1.4 O b jectives and procedures The success of the Bayesian algorithm depends on th e accuracy of both the conditional probability density function (pdf) of microwave observations and the prior p d f of rain rate, and the in terpretation of th e posterior probability distribution of rain intensities. In this study, we present a Bayesian algorithm for rain rate retrieval over the ocean which has two unique characteristics: 1) Unlike other Bayesian m ethods, ours is based on explicit functional models of th e conditional likelihood f ( P \ 9 ) and th e prior distribution n(9) fit to actual d ata derived from both observations and simulations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 2) Unlike other methods, th e result of this m ethod is not a single ’’b e st” rain rate b u t rath er the complete posterior probability distribution. The objectives of this study are: 1) To present explicit, closed-form functions to reasonably approxim ate th e phys ical and statistical relationships between rain rates and microwave m easurem ents, based on b o th observations and model simulations. 2) To provide a system atic sensitivity te st to show th e theoretical benefits of the Bayesian estim ators in th e retrieval, and to examine w hether th e Bayesian algorithm can be im plem ented into real-world applications w ithout losing its advantages when im perfect inform ation is used. 3) To implement our Bayesian algorithm s to real-world applications and inter compare w ith other benchm ark algorithms. 4) To assure the adequacy of the specification of the conditional likelihoods th a t describe the statistical and physical relationships between microwave radiance and surface rainfall, and to examine th e robustness of th e retrieval algorithm w ith respect to th e uncertainty of th e prior precipitation distribution. The results will provide guidance for the quantitative uncertainty of the retrieved rain rate. This thesis is organized as follows. C hapter 2 introduces th e microwave observed variables used in th e study, and gives the detailed descriptions of their characteristics and m ultichannel relationships from th e TM I m easurem ents, of surface rainfall infor m ation based on P R m easurem ents, and of th e PR -T M I m atch-up procedures. The m athem atical and physical basis to specify probability distributions in our Bayesian algorithm are sum m arized in C hapter 3. A fter th e presentation of th e theoretical framework, C hapter 4 offers model simulations based on reflectivity d a ta of th e Na tional W eather Service’s (NWS) W eather Surveillance D oppler R adar (W SR- 8 8 D) network and a simplified one-dimensional plane-parallel radiative transfer model to prove th e applicability of th e Bayesian approach. Following th e proof-of-concept, a thorough sensitivity test using two random ly generated d atasets are described in C hapter 5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 To p u t th e perform ance of th e new Bayesian algorithm into context, a num ber of benchm ark algorithm s are introduced in C hapter 6 . C hapter 7 develops our new Bayesian algorithm from the PR -T M I m atch-up dataset and employs the new algo rithm to real-world applications. This chapter also uses several validation m etrics to evaluate th e perform ance of all algorithm s applied to identical validation datasets. Then, when the com parable perform ance of th e new algorithm is found, a q u an tita tive analysis is conducted in C hapter 8 to com pute th e uncertainty of th e retrieval. The purpose of C hapter 9 is to provide a basis to calibrate the NWS W SR- 8 8 D netw ork-estim ated surface rainfall and to validate th e radiative transfer calculations based on PR -T M I m atch-up data. It dem onstrates th a t once a more reliable dataset of surface rainfall m easurem ents becomes available, th e Bayesian algorithm could be modified from the model simulations. Finally, conclusions and future work will be presented in C hapter 10. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 2. DATA D E SC R IP T IO N The emphasis of this chapter is to introduce two prim ary microwave observation datasets used in the study and their general characteristics. T he first d ataset is microwave radiance m easurem ents from TM I. Due to the intrinsic ambiguities in the physical relationship between microwave brightness tem peratures and rain rates, we use a transform ed variable, attenuation index P, to construct our Bayesian models. The other dataset is reflectivity m easurem ents from P recipitation R adar (PR). PR offers detailed inform ation about vertical and horizontal structures of rain cloud, b u t we will only focus on the near-surface rainfall rate in the study. Finally, the coincident P R and TM I observations are combined together to provide a m atch-up dataset for b o th th e development and validation of our Bayesian models. 2.1 T M I data The purpose of th e section is to describe th e ideal response of microwave bright ness tem peratures to surface rain rate; to briefly introduce th e atten u atio n index; to summarize the advantages of its use; and to explore the m ultichannel distributions of P values from TM I observations. 2.1.1 Ideal response to surface rain rate P etty (2001) dem onstrated th e dependence of th e idealized brightness tem per atures (T b ) on surface rain ra te (R) over th e ocean, when horizontal homogeneity of rain cloud is assumed (Figure 2.1). The brightness tem peratures a t all channels in cre ase as rain ra te becomes larger, due to th e emission of cloud and rain water. However, beyond some satu ratio n point, the brightness tem p eratu res decrease w ith increasing surface rain intensity as scattering effects become more significant. Higher microwave frequency tends to satu rate at lower surface rain rate. Furtherm ore, the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 300 19 GHz 250 •vT & 37 GHz d. 200 I H U5 V •u. a ■ ■da 150 85.5 GHz' 03 100 100 R ain R ate (m m /hr) Figure 2.1. Idealized brightness temperatures dependence on surface rain rate (Petty, 2001). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 ideal behavior of th e microwave signature in response to rain has revealed a strong polarization difference in brightness tem peratures over the open ocean, and this po larization difference varies w ith frequencies. W hen surface rain rate increases, the polarization difference decreases and theoretically, the vertically and horizontally polarized brightness tem peratures will reach to th e same value for th e cases of heavy precipitation. The curve in figure 2.1 suggests an ambiguous relationship between microwave signature and surface rainfall rate, in th a t two very different rain rates can be asso ciated w ith a single TB ■ To reduce the ambiguity, polarization differences a t various channels have been used to eliminate the signal from ocean surface and to isolate th e effects of the rain cloud. In the next section, a variable representing a normalized polarization difference will be introduced. 2.1.2 A tten u a tio n index D efin ition The attenuation index at a given frequency is defined as (Petty, 1994a) p = TT v ~ l H , J- V.O — 1 H ,0 (2.1) where T y and T# are the vertically and horizontally polarized brightness tem per atures; Tyfl and TH q are th e clear-sky background brightness tem peratures. The background brightness tem peratures are m ainly affected by w ater vapor, surface wind speed, and sea surface tem perature, and th u s th e attenuation index reflects th e inform ation about th e column liquid water. According to th e definition, theoreti cally, P index will be in a range of [0, 1], while 1 represents a pixel where th e cloud is absent, and 0 depicts a very opaque atm ospheric condition. The advantages to using the attenuation index in th e retrieval algorithm are: 1) P value decreases m onotonically w ith increasing rainfall intensity. 2) Since P m ainly reflects the signal coming from the rain cloud, th e atten u atio n index is not sensitive to the background variabilities. 3) For th e special case of a horizontally homogeneous rain layer, P and tran sm ittan ce t obey an approxim ate power-law relationship, P ~ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 t a . Therefore, in this limiting case, P index yields a direct indication of th e rain cloud transm ittance. For convenience, we denote th e vertically polarized brightness tem peratures at 10.65, 19.35, 21.3, 37.0, and 85.5 GHz as Tlov, T\gy, T 2 1 V 1 Tyrvi &ftd T ^ y , and replace V w ith H for horizontal polarization. Similar rules are applied for cloud-free brightness tem perature, but O is added to present the background quantities. For example, T i 0 y,o expresses the vertically polarized clear-sky brightness tem perature at 10.65 GHz. E stim a tio n o f T y o and T u ,o In an a tte m p t to obtain the clear-sky background brightness tem peratures, col um n w ater vapor, surface wind speed, and sea surface tem p eratu re have to be esti m ated first. The estim ation of column w ater vapor is developed from a Radiosonde observations (RAOBS)-TM I m atch-up process, similar to Alishouse et al. (1990) and P e tty (1994b). The RAOBS were extracted from the N ational Centers for Envi ronm ental Prediction (NCEP) A utom ated D ata Processing (ADP) U pper Air Obser vation subsets in January and July, 1999. Since our purpose is to find th e m atch-up d a ta w ith TM I measurements over th e ocean, only stations located w ithin [40°S, 40°N] and associated land area percentage less th a n 19% of th e 19 GHz footprint were considered in the subsequent m atch-up procedure. T he m atch-up criteria is th a t th e distance between RAOBS stations and TM I pixels have to be w ithin 100 km ra dius, and th e difference of their observation tim e is not greater th a n 4 hours. The final sam ple size after m atch-up procedure is 39,360. Then, based on th e regression analysis, our w ater vapor algorithm is given by th e following: V = 128.57+ 33.941n(290-T i9y ) - 7 2 .1 3 1 n (2 9 0 -T 2iy) + 10.481n(290-T 37//), ( 2 .2 ) where th e w ater vapor content (V in k g /m 2) is estim ated by th e vertically polarized brightness tem peratures (K) at 19.35 and 21.3 channels, and horizontally polarized brightness tem perature at 37 GHz. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 Estim ations of surface wind speeds are based on th e m atch-up betw een the TM I m easurem ents and the buoy wind d a ta from NOAA M arine Environm ental Buoy database, collected by National D ata Buoy Center (NDBC). Sim ilar to th e estim ation of w ater vapor, criteria in location and land area percentage were applied to screen th e buoy stations. TM I d a ta from January and July 1999 were processed to find the m atch-up pixels. The pixel was selected if it satisfied th a t th e distance was w ithin 100 km radius of the buoy location and th e observation tim e was w ithin 2 hours of buoy measurements. In addition, TM I pixels w ith possible rain were excluded. Due to various observed heights for th e surface wind speed a t buoy stations, all wind speeds were converted to th e stan d ard height of 10 m based on a logarithmical vertical wind profile. Through th e m atch-up procedure, a d ataset w ith a sample size of 68,674 was obtained and utilized for regression analysis to derive th e following relationship: U = 130.908 + 0.170Tiov T 0.1287jo.fr —0.0347\gy— 0 .1 157W - 0.079T21V - 1 -1 2 1 T37V + 0.543T37H , (2.3) where U is the surface wind speed (m /s); T indicates th e brightness tem peratures (K) from TM I observations; and th e subscripts represent th e channels and polarized directions used in the regression. T he regression equations for estim ations of background brightness tem peratures were derived from 489 TM I orbits in July 1999 w ith a final sample size of 419,272 pixels. The cloud-free 7g is approxim ated by w ater vapor, surface wind speed and sea surface tem perature (7s), as th e following equations shown: Tiw ,o T Wh ,o = 154.1 + 0.076V + 0.241/ + 0.477s = 73.8 + 0.14 V + 0.901/ + 0.247s ln(300 - T IW>0) = 4.89 - 0.0072V - 0.001717 - 0.00257s In(300 - T iqh o ) = 5.39 - 0.0078V - 0.006317 - 0.000527s ln(300 - Tzrv>0) = 4.65 - 0.0058V + 0.0005517 - 0.000697s Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 ln(300 - T37Hi0) = 5.22 - 0.0065V - 0.0080P + 0.00031TS, (2.4) where the column w ater vapor content V (kg/m 2) and surface wind speed U (m /s) are estim ated by equations ( 2 .2 ) and (2.3), and the sea surface tem p eratu re Ts (°C) is obtained from a 0.5 x 0.5° climatological database (obtained from N A SA /G SFC). Then, p u ttin g the estim ated background brightness tem peratures into equation 2.1, estim ated P values for the T M I pixel can be obtained. For convenience, we hence forth denote the P value a t 10.65, 19.35, and 37 GHz channels as Pio, Pig, and P 3 7 , respectively, and P as th e vector (Pio, P 1 9 , P 3 7 ). 2.1.3 D istrib u tion s o f a tten u ation index o f TM I To b e tte r understand th e overall distribution of P , 110 TM I orbits during 1999 and 2000 were selected to investigate th e m ultichannel relationship of the P values a t 10.65, 19.35, and 37.00 GHz. Since th e overall 3-D d istribution is of interest, no rain screen task is applied in th e following process. TM I m easurem ents were checked every 10 pixels and only those classified as ’ocean’ pixel were picked. As a result, th e final sample size is 4.6 x 106 pixels. Figure 2.2 and 2.3 depicts th e 3-D structure of P values by plotting 2-D contours of the num ber of pixels at a given P 10 and P37, respectively. Theoretically, in a homogeneous case of rain cloud, a unique non-linear relationship in P is expected. However, in reality, due to various beam-fillings effects, P scatters in 3-D space and its distribution forms a cloud on the 2-D contours (as shown in th e above figures). There are some features noted in th e 3-D m ultichannel relationships from TM I mea surements. First of all, th e P value w ith th e m axim um num ber of pixels is near (1.0, . , 1 .0 ), indicating th a t th e estim ations of background brightness tem peratures are 1 0 reasonable. Secondly, the shape of the 2-D contours is asym m etric and elliptical w ith v a ry in g o rie n ta tio n s . It d e m o n s tra te s t h a t th e P v a lu e s a t th e s e th re e c h a n nels are definitely correlated. An assum ption th a t the response vector elements are uncorrelated is not appropriate. In addition, for a given smaller P i 0 value which is often associated w ith a more significant rain rate, th e ellipse shape tends to be like a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 P 1 0 = 0 .3 P 1 0 = 0 .7 TMI TMI 1.2 1.0 0.8 0.8 E 0.6 CL 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 P19 P10 = 0.4 0.0 0.2 0.4 0.6 0 .8 1.0 1.2 P19 1.0 1.2 TMI P 1 0 = 0 .8 TMI 1.2 1.0 0.8 E 0.8 0 .6 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 P19 P 1 0 = 0 .5 1.0 1.2 0.0 0.2 0.4 0.6 0 .8 P19 P 1 0 = 0 .9 TMI 1.0 1.2 TMI .2 .0 0.8 0.8 a 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.2 0 .4 0.6 0.8 P19 P 1 0 = 0.6 1.0 1.2 0.0 0.2 0.4 0.6 0 .8 P19 P 1 0 = 1.0 TMI 1.0 1.2 TMI .2 .0 0.8 0 .6 0.8 a 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0 .8 1.0 1.2 P19 0.0 0.2 0.4 0.6 0 .8 P19 1.0 1.2 Figure 2.2. Overall m ultichannel distributions of P index represented by th e num ber of pixels. D ata were selected from 110 TM I orbits during 1999 and 2000. Contours are logarithm ically spaced; actual value is 10x where x is the contour label, x are plotted for values of [0.5, 1, 2, 3, 4, 5] a t a fixed Pio interval. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 P 3 7 = 0 .0 O) cl TMI P 3 7 = 0 .4 0.8 0.8 0 .6 cl 0.4 0.2 0.2 0.0 0.0 P 3 7 = 0.1 cn 1.0 1.2 TMI 0.8 cl 0 .6 0.4 0.4 0.2 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 P10 0.0 0.0 0.2 0.4 0.6 0.8 P10 P 3 7 = 0.8 TMI 0.8 1.0 1.2 TMI 0.8 cl 0 .6 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 P10 P 3 7 = 0 .3 cl P 3 7 = 0.6 0.8 a 0.6 O) 0.0 0.2 0.4 0.6 0 .8 P10 TMI 0.6 P 3 7 = 0.2 cn 0 .6 0.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 P10 a TMI 0.0 0.2 0.4 0.6 0.8 1.0 1.2 P10 TMI P 3 7 = 1.0 0.8 0.8 0 .6 a. 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0 .8 1.0 1.2 P10 0.0 0.2 0.4 0.6 0.8 P10 TMI 1.0 1.2 Figure 2.3. Same as figure 2.2, bu t at a fixed P 37 interval. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 banana, showing the existence of non-linear relationships in th e P values at m ultiple channels. The TM I observations show th a t m ost P w values are greater th a n 0.4, pointing out the scarcity of the extrem ely low Pio values. T he scarcity corresponds to the fact th a t the polarization difference at 10.65 GHz is rarely lower th a n 30 K based on TM I m easurem ents. A low polarization difference at 10.65 GHz requires widespread intense precipitation throughout the field of view a t 10.65 GHz, which rarely occurs in th e real world, even though typhoon and strong convective cases have been included in the 110 TM I files. At higher frequencies of 19.35 and 37.0 GHz, th e lowest polarization difference is found to be a few Kelvin instead of zero, even under th e condition of very thick rain cloud. Therefore, the num erator of equation 2.1 is seldom equal to zero as well as P values a t 19.35 and 37.0 GHz. Liu and Simmer (1996) suggested th a t the low polarization differences may be explained by spherical particles alone. In addition, coupling the instrum ental noises, th e errors from th e estim ated w ater vapor and surface wind speed, w ith the uncertainties in th e regression equations, P might be slightly greater th a n 1, b u t smaller th a n 1.1 in m ost cases. 2.2 P R rain rate Due to th e operating frequency of 13.8 GHz, P R suffers from strong attenuation in heavier rain. Therefore, P R reflectivity observations have to be corrected before they are used to estim ate rain rate. TRM M stan d ard product 2A25 from th e rainprofiling algorithm of the Precipitation R adar contains estim ations of th e vertical rainfall rate profile, the atten u atio n corrected rad ar reflectivity, and m any other interm ediate param eters, such as attenuation param eters and coefficients used in the Z-R relationship. Among them , th e variable of near surface rain intensity is utilized in b o th the development and validation of th e TM I Bayesian rain rate retrieval algorithm . For convenience, th e near-surface rain ra te from 2A25 is referred as P R retrieved (surface) rain intensity hereafter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 The details of the P R retrieval algorithm for 2A25 were given in Iguchi et al. (2000). T he key points are summarized here. First, due to strong atten u atio n ef fects, th e observed P R reflectivity is corrected using a hybrid of th e surface reference m ethod and th e Hitschfeld-Bordan m ethod w ithout considering th e beam filling ef fect of th e rain cloud. The param eters (a and j3) representing th e relationship of attenuation (k) and corrected reflectivity (Z e) 1 k = a Z @, are adjusted to find a m atch in the path-integrated attenuation (PIA) between the two m ethods. Then, after ob taining th e prelim inary corrected reflectivity, a non-uniform ity index of th e rain is estim ated from the variability of th e PIA. The atten u atio n derived from th e surface reference technique will be modified again based on th e non-uniformity, and a corre sponding tru e reflectivity could be com puted. Finally, the estim ation of rain ra te is based on a power-law relationship w ith attenuation-corrected reflectivity, specified by R = aZg. T he param eters (a and b) are determ ined by rain type, th e height of freezing level, and the height of the storm top. The ranges for a and b are constrained from 0.005 to 0.2, and from 0.5 to 1.0, respectively. Since there is no long-term and w ide-area dense rain gauge network or other reliable d a ta to provide the true rainfall intensity in th e m arine environm ent, it is im portant to keep in mind how the P R retrieved rain ra te behaves in comparison w ith other surface measurem ents before a m eaningful comparison between the PR (assumed tru e value) and the retrieved rain rates from TM I can be made. Bolen and Chandrasekar (2000) made simultaneous comparisons w ith P R and S-band polarim etric rad ar (S-POL) and concluded th a t there was a good agreem ent between the attenuation-corrected reflectivities at altitudes less th a n 2 km to w ithin 1 dBZ with ground rad ar observations. In addition, due to th e low sensitivity of P R to light rain rates, a 3.8% underestim ation is indicated in th e mean areal rainfall. Furtherm ore, some cross-validations were made com paring P R rain rate estim ations w ith surface m easurem ents over India, China, and A ustralia. Those results dem onstrated th a t P R accum ulated rainfall was around 20% lower th a n th a t obtained from th e surface m easurem ents (according to personal com m unications w ith Dr. Iguchi). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 The rad ar a t th e Kwajalein TRM M validation site was supposed to provide an invaluable oceanic dataset for our validation purpose. However, Schumacher and Houze (2000) dem onstrated th a t when the ’ground tr u th ’ ra d a r was adjusted up ward by 2 dBZ, then the rainfall estim ate of P R at an altitu d e of 3 km would be consistent w ith th a t derived from th e Kwajalein radar. Houze et al. (2001) also pointed out th a t necessary calibration corrections have to be m ade in th e Kwajalein radar m easurem ents. Oki et a l (1998) attem p ted to validate th e spaceborne P R d a ta using ground-based radars, and their results ended up suggesting th a t P R could be used as a space calibrator due to high reliability of P R data. In summary, P R rainfall rate estim ates might not be able to represent th e absolute tru e rain intensity due to the uncertainty in the attenuation correction and Z-R relationship. However, the P R is b e tte r calibrated th a n a typical ground based rad ar (personal communications w ith Dr. Y uter), and has an extraordinary advantage of th e sim ultaneous observa tions w ith TM I. Therefore, th e P R provides a unique way to validate TM I retrieval algorithm s, which contains long term , various environments and precipitation sys tems. In chapter 9, we will show th a t th e Bayesian algorithm can be adjusted when even more reliable m easurem ents of rainfall intensity become available. Note th a t the 2A25 outputs include an estim ate of th e error in th e near-surface rain rate, indicating th e uncertainties of param eters used and of th e m easurem ent errors inherent to th e instrum ent or th e observations. W hen attem p tin g to use the error estim ate as a quality control of P R rainfall intensity, it flags out a large fraction of pixels associated w ith significant precipitation, in which more interest is placed. Therefore, we simply use estim ates in th e study for now, b u t th e m axim um rain intensity is not allowed to exceed 150 m m /h r. 2.3 P R -T M I m atch-up procedure The 2A25 P R retrieved rain rates has a 4-km resolution on th e precipitation map. For the purposes of developing or validating a TM I retrieval algorithm representing rain intensity at a 15-km resolution, th e P R estim ated rain rates were averaged by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 Gaussian weighting functions and interpolated to the location of TM I pixels. Since the P R swath is much narrower th a n th e TMI, the interpolation was only conducted for the TM I pixels th a t were located ± 20 TM I high-resolution pixels from th e nadir (around 160 km wide). In addition, the lightest rain intensity th a t P R is able to detect is around 0.5—0.7 m m /h r, and hence th e smallest area-averaged P R rain rate at a 15 km resolution should be ab o u t 0.035—0.049 m m /h r. Therefore, we assigned a minimum threshold of 0.04 mm / h r for the interpolated P R rainfall intensity, i.e., any pixel w ith a lower interpolated rain rate th a n the threshold was excluded in the study. For convenience, the rain ra te estim ated through this stage is referred to as the P R (surface) rain rate hereafter. Note th a t the P R reflectivity m easurem ents might suffer from th e sidelobe con tam ination when th e m ain beam of the P R is off-nadir, and the resulting errors would propagate to the rain ra te estim ations and interpolations. To reduce th e un certainty, only near-nadir pixels were included in the developm ent of th e Bayesian retrieval algorithm . In the study, th e near-nadir pixel is defined as th e one w ith a TM I pixel index between 102 and 106 (i.e., 104 (nadir) ± 2 ) . As to the validation process, while the qualitatively horizontal structures of precipitation system are of interest, th e off-nadir inform ation will be shown as well. Otherwise, only near-nadir d a ta will be used to calculate verification measures and to quantify th e retrieval ability for all algorithms. Figure 2.4 depicts the probability distribution of the near-nadir P R 15-km inter polated rain rates. D ata periods contain January, April, July, and O ctober of 1998. It is clear th a t the distribution of rain rate is very close to th e lognorm al function denoted as follows: - eXP logN(r\fj., cr) = < 0 , r = 0 5 ^ (In r - g f , where logN expresses the lognorm al function and rain rate r is in m m /h r. /i and a are th e param eters of the lognorm al distribution in unit of m m /h r, b o th determ ining Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 R a i n R a t e D i s t r i b u t i o n — P R ( 1 9 9 8 , n e a r —n a d i r ) 0.20 15 0.10 0 .0 5 0.00 0.001 0 .0 1 0 0 .100 1.000 10.000 100.000 Rain R a t e ( m m / h r ) Figure 2.4. Probability distributions of th e near-nadir P R 15-km interpo lated rain ra te are depicted by lighter-colored dots. D ata were calculated from th e 2A25 P R near-surface rain rate estim ates in January, April, July and O ctober of 1998, and a cut-off value of 0.04 m m /h r was applied. T he resulting sam ple size is around 2.4 million. The fitted lognormal distri bution is also presented here by a dashed curve. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 th e mode and the variance of th e rain rate probability distribution. W hile trying to fit th e distribution w ith a lognorm al function, th e pair of param eters, (/j, a), was found to be around (-2.7, 2.2). p and a change w ith varying d atasets from -2.9 to -2.6, and between 2 and 2.3, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 3. A LG O R ITH M B A SIS As we m entioned in chapter 1, th e Bayesian posterior density function is de term ined by a prior rain rate distribution th a t represents our knowledge or belief, and a conditional likelihood th a t statistically describes th e physical relationships between rain ra te and microwave signal. Therefore, there are three key elements in our algorithm : th e prior distribution, th e conditional likelihood, and th e esti m ator interpreting th e posterior distribution. In this chapter, th e generic forms to model the prior and conditional distributions are introduced, and two commonly used estim ators for th e posterior pdf to represent th e single-pixel retrieval are also discussed. Note th a t th e generic forms for conditional pdfs in our algorithm are used to fit b oth model sim ulations (chapter 4) and actual observations (chapter 7) in the development of our Bayesian algorithm. 3.1 P rior d istrib u tion The probability distribution of the surface rain rates has been estim ated by many researchers using various forms for which the m ean and stan d ard deviation m ight not vary a lot, b u t tail behavior differs appreciably. A lognormal function is widely used to approxim ate th e rain rate distribution (Houze and Cheng, 1977; Kedem and Chiu, 1987), and th e PR -TM I m atch-up d a ta (shown in th e previous chapter) also dem onstrated th e adequacy of th e approxim ation. Therefore, in our algorithm , the prior rain ra te likelihood is modeled by a lognormal density function, which is d e n o te d as log N( [i, cr) a n d d e sc rib e d b y e q u a tio n (2.5). Since (p, a) varies w ith different rain ra te observation datasets, a num ber of sets of (/a, cr) will be tested in the study. First, Kedem et al. (1990) docum ented the estim ations for th e param eters using a m inim um chi-square estim ation based on Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 the G A RP (Global Atmospheric Research Program ) A tlantic Tropical Experim ent (GATE) observations. In their fitting processes, the GATE d a ta were tru n c a ted at 1 m m /h r and represented a 4 x 4 km averaged instantaneous rain rates. A lthough retrieved rain rates from microwave m easurem ents are more often based on 15 x 15 km area, their study is still valuable for the investigation of uncertainty of th e prior rainfall distribution. Two sets of th e param eters from their study were adapted: (1.140, 1.047) and (1.043, 1.205), which were estim ated from different tim e periods and only had slight differences in //, and a. In th e previous chapter, PR -T M I m atch-up dataset represented th e rain rate measurem ents on th e basis of 15 x 15 km area. In addition to a given threshold of 0.04 m m /h r, th e m atch-up d ataset suggested a set of (-2.7, 2.2) for describing the prior rain rate distribution. It dem onstrates th e applied cutoff value has dram atic influence on th e fitted values of param eters, and thus the resulting characteristics of tails will be different significantly. In addition to th e set of (-2.7, 2.2), th e param eters of (-2.8, 2.0) will also be used due to th e variations of the /j and a. Furtherm ore, for a purpose of theoretical tests, two sets of (0, 1) and (0, 2) were also employed in the study. To b e tte r understand the difference in th e modes, means, variances, and the tails between those prior specifications, figure 3.1 depicts the lognorm al distributions with th e above various param eter com binations. The differences between th em will be discussed in more detail in chapter 5. 3.2 C on d ition al likelihood The conditional likelihood in our Bayesian algorithm is a m ultivariate probability distribution since rain ra te and observation vector are all involved. In this kind of highly dim ensional condition, we would like to specify th e underlying density functions in term s of some param etric families. A param etric form provides a simple function to characterize the density function, and th e advantage of th e use is the ability to obtain an analytical solution. T he param etric forms have to be used w ith caution to ensure th a t th e equations reflect the behavior of d a ta as close as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 1.2 1.0 ^ !l logN( 0 . 0 0 , 1 . 0 0 0 ) 0.8 . logN( 1 . 1 4 ,1 .0 4 7 ) logN( 0 . 0 0 , 2 . 0 0 0 ) logN( 1 .0 4 ,1 .2 0 5 ) c 0 .6 <D Q J\ _______ l o g N ( - 2 . 8 0 , 2 .0 0 0 ) 0.4 0.2 0.0 0 2 4 6 Rain r a t e ( m m / h r ) 8 10 0.010 (b) logN( 0 . 0 0 , 2 . 0 0 0 ) 0 .0 0 8 logN( 0 . 0 0 ,1 .0 0 0 ) logN( 1 .1 4 ,1 .0 4 7 ) 0 .0 0 6 *ccn <D logN( 1 .0 4 ,1 .2 0 5 ) 0 .0 0 4 l o g N ( - 2 . 8 0 , 2 .0 0 0 ) ■\ \ \ .............. logN( —2 . 7 0 ,2 .2 0 0 ) 0.002 20 40 60 80 Rain r a t e ( m m / h r ) 100 Figure 3.1. Comparisons of th e prior rain rate probability distributions used in this study. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 possible. A non-param etric approach w ith less rigid assum ptions is an alternative, which allows d a ta to speak for themselves. However, it is beyond the scope of the paper. A univariate case is used first to illustrate th e approach of specifying th e condi tional pdfs. Let us take th e P 3 7 variable and model its relation to rain rate w ith a closed-form function. Since m ost values of P index are w ithin around a range of [0, . ] (see chapter 2 ), the conditional p df can be simply constructed by 1 1 / ( P 3 7 |P ) oc P 37(a - P 37) exp 1 :(-P 2ct32 *37 ~~ h 3 Y ,P 3 7 G [0,a], (3.1) where / ( P 3 7 |P ) is the likelihood of P 3 7 a t a given rain rate R; a is a constant. /i 3 and <r3 are functions of rain ra te and determ ined by the fitting to th e d ata. The distribution basically follows a norm al distribution, bu t it is bounded in [0 , a] and modified by the term of P 37(n — P 37). In addition, th e likelihood is exactly equal to zero a t the two extreme boundaries. T he distribution is skewed to th e right if most pixels achieve saturation associated w ith larger rain rates, while the distribution skews to th e left under a condition th a t th e atm osphere is almost tran sp aren t to this frequency. The distribution tends to be more sym m etric if th e mode is around 0.5 to 0.6. Unlike the norm al distribution, the param eters of /i 3 and <r3 here are not exactly equal to th e m ean and th e stan d ard deviation of th e distribution, bu t they are nevertheless related to th e location of th e mean and the variance of th e pdf. Furtherm ore, based on th e actual observations from TM I, the a is assigned as 1.1 in the study. W hen the observation vector contains (Pio, P i 9 , P 3 7 ) , the distribution of P at a given rain rate can be approached hierarchically: / ( P |P ) = / ( P 3 7 |P ) / ( P 1 9 |P 37, P ) / ( P i o | P i 9, P 3 7 , R) (3.2) where / ( P i 9 |P 37, R ) represents th e distribution of P 1 9 when P 3 7 and R are fixed; and th e / ( P 1 0 IP 1 9 , P 3 7 , R) describes th e p d f of P index a t 10.65 GHz while th e other three variables are known. Similarly to th e univariate case, those conditional pdfs Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 can be approxim ated as /(P 3 7 I-R ) oc P37(a - P 37) exp / ( P 1 9 I-P3 7 , R ) oc P 19(a - P i9) exp f (P 1 0 1P1 9 j P 3 7 >R) oc P 10( a - P i o ) e x p where /i 2 and <72 1 ( _ ,;,p2 2 <7‘ '2' 2 :(Pio —A41 )2 (3-3) 2 cT !2 are functions of rain ra te and P 3 7 ; and //1 and <7 i are functions of rain rate, P 37 and P 1 9 . Again, these param eters are all determ ined by fitting the d ata b oth from model sim ulations and observations. Finally, the conditional likelihood of m ultivariate observation vector at a given rate could be obtained combining th e equation set (3.3) w ith th e equation (3.2). Note th a t the last term set of (P 1 9 , P 3 7 , / ( P 1 0 I P 1 9 , P 3 7 , R) requires inform ation about P 10 a t a given P ). Since three variables are fixed while fitting this conditional pdf, the sample size has to be large to provide enough d a ta points for the fitting. 3.3 E stim ators o f th e p osterior distrib ution Based on the Bayes theorem (equation 1.1), th e posterior distribution can be derived as rrfpin ( 1 > fS41 Jf(P\R)w{R)dR’ ( ■) where th e denom inator is used to normalize th e posterior distribution, and the inte gration w ith respect to rain rate (R ) in th e four-dimensional space will be perform ed numerically. Theoretically, th e upper lim it of rain rate in the lognormal distribution goes to infinity, then the integration should be conducted to the same upper limit. Practically, a reasonable upper bound R max has to be given to define the integration interval during the num erical calculations. We set R max to be 80—100 m m /h r in our study, depending on th e train in g dataset. In addition, th e interval of th e rain ra te for integration is increases to 0 .2 0 .0 1 a t a very light rain situation (less th a n 0 .2 m m /h r), and m m /h r a t th e range of [0 .2 , P max] m m /hr. P index ranges from 0 to 1.1 for all channels, and the resolution of 0.01 is used in the calculation and norm alization of th e conditional pdfs. Once th e posterior Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 distribution is ready, two most common estim ators were taken: th e m ean and the maximum likelihood estim ate of th e posterior probability distribution, denoted as MEAN and MLE, respectively. Then, th e corresponding Bayesian estim ates are stored in a 3-dimensional lookup table w ith a resolution of 0.2 for each P value for the single-pixel retrieval. In th e next chapter, th e algorithm basis will be applied to model sim ulations to dem onstrate the retrieval ability. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 4. P R O O F-O F-C O N C E PT The purpose of the chapter is to prove th e concept of the Bayesian approach from model simulations. This sim ulated training dataset was generated from NWS W SR88D network observations and a simplified one-dimensional plane-parallel radiative transfer model. To statistically characterize the physical relationships between rain rate and microwave signature shown in th e simulations, the conditional likelihood is derived using explicit, closed-form functions to fit the data. The generic form has been described in the previous chapter, and the associated coefficients fitted to the model simulations will be given in th e chapter. Finally, the algorithm is applied to one of the WSR-88D network d atasets to dem onstrate the retrieval ability of th e Bayesian approach. 4.1 R ain rate estim ates from N W S W SR -88D network Beam-filling errors arising from inhomogeneities of rain clouds have been a sig nificant problem in rain rate retrieval from passive microwave m easurem ents. The coupling of non-uniform ity of precipitation w ith th e nonlinear relationship between microwave radiance and rain ra te produces a footprint-averaged microwave signature th a t is not uniquely related to th e footprint-averaged rain rate. T he errors can be reduced if beam-filling effects are considered in the algorithm in either an implicit or explicit way. To b e tte r account for th e heterogeneous beam filling effects in our algorithm , we in c lu d e d re a listic a lly v a ry in g p r e c ip ita tio n s p a tia l s tru c tu re s in th e tra in in g d a ta using th e NWS W SR-88D network product of 1 km US N ational Base Reflectivity. T he reflectivity d a ta were com posited from the 154 sites in th e US w ith an hourly tem poral resolution and a spatial resolution of 1 km. Observations in precipitation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 and clear air modes were b o th collected in the product. Since characteristics of precipitating systems are of interest, m easurem ents from th e clear air mode were excluded in the study. However, some d a ta still showed th a t there were apparent clear air returns (most of them less th a n 20 dBZ) even in the precipitation mode, especially during the nighttim e, which might mislead us to the wrong precipitation p attern s and then translated to incorrect probability distributions. To avoid th e errors from those noises, the reflectivity values were cut off at 20 dBZ before performing the subsequent rain intensity estim ations and forward microwave radiance calculations. The surface rain rate for each rad ar pixel was modeled by th e most commonly used Z-R relationship (M arshall and Palm er, 1948) Z = 200.R1'6, (4.1) where Z is th e observed rad ar reflectivity factor (dBZ) and R is the rain rate in m m /hr. Since reflectivities higher th a n about 55 dBZ (corresponding to a rain rate of around 100 m m /hr) usually indicate the presence of hail, th e estim ates of rain rates might be questionable in th a t case. Therefore, an arb itrary upper limit of the radar pixel rain rate was set as 150 m m /h r, and estim ated intensities were averaged over 15 x 15 km domains to serve as th e ’’tru e ” instantaneous rain rates. 22 radar reflectivity files were random ly picked during July and August 2002. Each file includes 4736 x 3000 observations all over th e U nited States. After fil tering out th e suspicious returns, rainy pixels comprised 2.5% of d a ta for each file. Furtherm ore, the strength of rad ar retu rn for precipitation mode was recorded in particular reflectivity levels w ith an interval of 5 dBZ, resulting in an obviously dis crete property in the histogram of th e rain rates. To com pensate for this deficiency, reflectivity values were system atically adjusted by a shift of ± 2 and ± 1 dBZ, so a to ta l of 110 files were processed in th e study. 4.2 Polarization calculations The mass extinction coefficient of suspended cloud w ater Ke i was com puted from Liebe et al. (1991) assum ing a tem perature of 0°C, and th e corresponding Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 Table 4.1 Param eters a and b in th e approxim ation of liquid w ater extinction coefficient. ^e,l a b 10.65 0.0244 0.002956 1.18759 19.35 0.0785 0.01585 1.09403 37.00 0.261 0.06896 1.01876 85.50 0.932 0.2799 0.84693 Channel (GHz) values are listed in Table 4.1 w ith unit of m 2/kg. The relationship between the volume extinction coefficients of rain water kej. and rain rates (R) was estim ated from Mie theory, assuming spherical rain drops w ith a liquid w ater tem perature of 10°C, and using the M arshall-Palm er drop size distribution and microwave complex index of refraction formula of Liebe et a l (1991). A power-law form was found to approxim ate the relationship well (P etty, 1994b), A;e>r = a R b. (4.2) Values of the coefficients a and b for each channel are also shown in Table 4.1. The optical depth r r was m odeled as Tr ~ Z f k etr(R), (4.3) representing the contribution of rain w ater only. Since there was no information about th e values of the freezing height Z f in th e rad ar reflectivity product, a value of 3 km was assumed for Z f here. In real-world applications, th e freezing height is a free param eter in th e Bayesian retrieval algorithm , so a series of values of freezing heights will be used to calculate th e microwave radiance, and th en the corresponding relationships of P values and rain rates can be derived. More discussions are given in C hapter 9. The optical following. depth caused from th e suspended cloud w ater was modeled as th e The suspended cloud w ater content (L in k g /m 2) was sim ulated as a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 random noise generated from a lognormal distribution lo g N (//£ , 0 l) . The h l , ol were given by ^ = aL = 0 .1 5 + 0.03# 0.25 + 0.045#, (4.4) where R is th e surface rain ra te in m m /hr. Once th e cloud w ater content is specified, th e optical depth a ttrib u te d to th e suspended cloud w ater (-p) can be given as Tt = KejL. (4.5) A simplified 1-D plane-parallel model was used to com pute the brightness tem peratures. T he radiative transfer equation is w ritten as T b = (1 —t)Tj\ + etTs + (1 — t ) ( l —s)tTA, (4-6) where the sim ulated brightness tem perature TB is determ ined by th e transm ittance t, the specular emissivity of th e surface (sea w ater in th e study) e, th e surface tem perature Tg, and th e air tem perature # 4 . T he tran sm ittan ce t is defined as t = e x p |- s b ]' (4-7) where 0 is th e incident angle and r is the optical depth. T he surface tem perature was approxim ated from th e extrapolation of the tem p eratu re profile, assum ing a tem perature at the freezing level of 0°C and a lapse ra te of 6.5 K /km . T 4 was estim ated by the air tem peratu re at the mid point between th e surface and the freezing level. Since the emissivity of the ocean is polarized, b o th the vertically and horizontally polarized radiances were obtained. As mentioned in chapter 2 , th e attenuation index is used in th e study to serve as the observation vector instead of brightness tem peratures . In order to com pute P index, values of background brightness tem peratures are also needed. The cloudfree brightness tem peratures (TVp and TH,o) are approxim ated as follows. Based on equation (4.6), we can derive th e polarization difference A TB as A T b = Tv — T h = (ev —£fj)[t(Ts — T a ) + t 2TA}. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.8) 35 Define th e t a, ti, and tr as th e tran sm ittan ce attrib u te d to th e atm osphere, the suspended cloud water, and the rain w ater, respectively, and t\ as th e to ta l tran s m ittance (the product t a, ti, and tr). Then, th e atten u atio n index can be given as A Tb T v - TH A T b ,o T v ,o ~ T h ,o h { T s - T a ) + t \T A t a{Ts ~ T a ,o ) + t l T A,o ’ where TAjo represents the air tem perature when th e suspended cloud w ater and rain w ater are absent. We assume TA ~ TA}o in th e simulations. In addition to th a t the order of th e first term for both the num erator and denom inator is much smaller than the second term , the attenuation index can be approxim ated as ^ ^ = 2 ) 2 =«?£ (4.10) It is found th a t t a will be cancel out in th e calculations of P, and th u s th e absolute value of t a does not have a great effect on P. Note th a t P etty and K atsaros (1992) also suggested a power-law relationship betw een P value and th e transm ittance, (4.11) where a « 2 ,and the transm ittance t in equation (4.11) presents th e contribution from th e suspended cloud w ater (ti) and rain w ater (tr). Since th e atten u atio n index considers th e ratio of the polarization differences, and th e contribution of th e atm o spheric transm ittance will be canceled out, th e cloud-free brightness tem peratures can be easily approxim ated employing a tran sm ittan ce of unity (zero optical depth) in equation (4.6), in which case TB = sTsIn th e radiative transfer equation (4.6), th e tran sm ittan ce a ttrib u te d to th e sus pended cloud w ater should be included to calculate the brightness tem peratures. In our calculations, we com puted th e contribution of rain w ater first by replacing t w ith Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 t r to yield microwave brightness tem peratures and prelim inary values of P index at different channels. Then, using th e relationship described by equations (4.10) and (4.11), the contribution from the suspended cloud w ater was added and th e modified P value (P*) for each pixel can be estim ated from P* = t}tl = t\P. (4.12) Finally, when th e P values were ready, a zero-mean G aussian random noise was added to P value w ith standard deviations of 0.01, . 0 0 2 , and 0.02 for 10.65, 19.35, and 37.00 GHz channel, respectively. At this point, a sim ulated d ataset is obtained, containing inform ation about rain rate, P w , Pw , P 3 7 for each d a ta point. It has to be noted th a t since the rad ar com posite reflectivity d a ta only provide horizontal rain rate estim ates, some assum ptions have to be m ade in our model about other param eters, such as surface and air tem peratures th a t describe the atm ospheric environment. One m ight argue th a t th e sim ulated microwave signature would be very sensitive to those param eterizations. This concern points out th e rationale of the usage of P index in th e study, which is able to distinguish th e m ajor signal of th e rain cloud itself from the background variability, and thus P is not sensitive to the assum ptions we made. In addition, the 3-D stru ctu re of rain cloud is ignored in the radiative transfer calculations, and it is a potential source of errors for the com puted attenuation index. Since there is no easy way to deal w ith th e issue, we will use this 1-D model to provide simulations for this study. 4.3 C om parison w ith P R -T M I m atch-up d ata Before using the model simulations to construct th e Bayesian algorithm , two more processes have to be conducted for th e sim ulated d ataset. One is th a t a rain ra te cutoff has to be applied to th e dataset, and th e other is to ensure the repre sentativeness of the sim ulated microwave signatures. The m ajor reason for applying a cutoff rain ra te is provided as the following. T he prior rain rate distribution will always have th e largest probability a t zero rain ra te in th e real world. Theoretically, th e posterior rain ra te distribution for non-rainy pixels should obey a d elta function Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 with an impulse at zero rain rate. Practically, if th e zero rain rates are not excluded in the constructions of the prior and conditional pdfs, and th e algorithm is applied for every single pixel, th e com puted posterior rain ra te p d f for non-rainy pixels will have finite probabilities a t non-zero rain rates, and it leads to widespread non-zero estim ations everywhere. Therefore, we will separate the rain screening as another issue and only include rainy d a ta points in th e development of our Bayesian algo rithm . Then, apply our retrieval algorithm only for the rainy pixels. In addition, th e cutoff value is set as small as possible to make the lower lim it of th e retrieval range smaller. Evans et al. (1995) used a cutoff value of 1 m m /h r for their model simulations, b u t found out th a t th e cutoff led to an overestim ation in the areas of light rain rate. Therefore, in this study a cutoff of 0.04 m m /h r was applied to keep more inform ation and to obtain b e tte r estim ation for the regimes of light rain rate. As a result, the final sample size was 4.7 million after th e cutoff. To evaluate the representativeness of th e radar-radiative simulations, th e m ulti channel relationships of the sim ulated microwave signatures are com pared w ith those of the TM I measurements. Figure 4.1 depicts th e 2-D contours of num ber of pix els at a given Pio or P 3 7 based on actual TM I observations (shown in th e first and th ird column) and from radar-radiative sim ulations (the second and fourth colum n). The contours are shown in a logarithm ic scale. Note th a t the dataset of the TM I m easurem ents in figure 4.1 is same as th e one shown in figure 2.2 and 2.3. Plots are drawn here again for the convenience of comparisons. T he 3-D stru ctu re of P values from rad ar simulations dem onstrates a great sim ilarity to th e actual TM I ob servations in b oth locations and orientation in th e 2-D slices. T he distribution of the radar sim ulations is generally broader th a n th e TM I data. This discrepancy might be a ttrib u te d to the random noise added to th e P index. In addition, note th a t th e sample sizes from TM I and rad ar sim ulations are on th e same order. However, TM I d a ta include non-raining pixels, while rad ar d ataset was employed a rain rate cutoff. T he difference might also contribute to th e broadness th a t th e sim ulated contours have. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 P 1 0 = 0 .3 P 1 0 = 0 .3 0 .0 0 .2 0 .4 0 .6 0 .8 1.0 1.2 P 1 0 = 0 .4 P 3 7 = 0 .0 RADAR 0 .0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 P 1 0 = 0 .4 TMI RADAR P 3 7 = 0.2 1.2 1.2 TMI TMI P 3 7 = 0 .0 0.0 0.2 0.4 0.6 0.8 P 3 7 = 0.2 1.0 1.0 1.0 1.0 0.8 0.8 0.8 £ 0.2 0.6 a 0.4 0.4 S 7 .............. 0.2 0.0 4) 0 .0 0.2 0.4 0.6 0 .6 1.0 1.2 P 1 0 = 0 .5 TMI P 1 0 = 0 .5 RADAR RADAR 1.2 1.2 0.8 2 0.6 RADAR 0.6 r # Jffnix CL 0.6 / j ) 0.4 0.4 0.2 0.2 0.0 0 .0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 P 3 7 = 0 .4 & P 3 7 = 0 .4 RADAR P 3 7 = 0 .6 RADAR 0.0 0.2 0.4 0.6 0.8 1.0 1.2 P 1 0 = 0 .6 TMI P 1 0 = 0 .6 RADAR P 3 7 = 0 .6 TMI 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 .0 0.2 0 .4 0 .6 0 .8 1.0 1.2 P1 0 = 0.8 TMI 0 .0 0.2 0.4 0.6 0.8 1.0 P 1 0 = 0 .8 0.0 0.2 0.4 0.6 0.8 1.0 1 RADAR P 3 7 = 0.8 1.2 1.0 SI 'tlvl <n 0 8 TMI 0 .0 0.2 0.4 0.6 0.8 1.0 1 P 3 7 = 0 .8 RADAR 1.2 # 1.0 0.8 0.4 0.4 0.2 0.2 # 0 .0 0.2 0.4 0.6 0 .8 1.0 1.2 P 1 0 = 1 .0 TMI P10 = 1 .0 RADAR P 3 7 = 1.0 TMI P 3 7 = 1.0 RADAR Figure 4.1. Contours of the num ber of pixels based on TM I d a ta (the first and th ird columns) and radar-radiative simulations (the second and fourth columns). Contours are logarithm ically spaced; actual value is 10x where x is the contour label, x are p lo tted for values of [0.5, 1, 2, 3, 4, 5]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 In this chapter, radar-radiative sim ulations are only used to prove th e concept of the Bayesian approach, so the reflectivity-estim ated surface rainfall has not been calibrated by any other precipitation m easurem ents yet. Therefore th e exam inations of physical relationship between rain ra te and atten u atio n index are not conducted here. However, despite the uncertainty in th e rainfall estim ations from radar re flectivity, the simple forward radiative model simulations have shown a surprising consistency w ith the TM I observations in 3-D atten u atio n index distributions. An other aspect of application of model sim ulations and further analysis of relationship between rain ra te and microwave signatures will be provided in chapter 9. 4.4 B ayesian algorithm from sim ulations The generic forms representing th e statistical and physical relationship between surface rainfall and microwave observations have been presented by equations (3.2) and (3.3). Using those analytical forms to fit to th e radar-radiative simulations, the corresponding param eters describing each individual conditional probability distri bution are obtained as follows: a = 1.1 Hz = 1.6exp(—0.24#) —0.62 0-3 = 10.3 • 0.07L88/T (1 .8 8 ).^(i- 8 8 -i) exp(_o.07.R) + 0.06 /U = bo + b\P%j + 62 b0 = 0.25 exp(—0.024#) bx = 0.76 - 0.01# + 0.0005#2 b2 = 0.4 + 0.0035# S2 = ( 0 . 5 - 1 . 0 8 ^ 7 + 3.11P3 - 0 . 3 \ / # + 0 . 0 5 # - 0 .0 0 3 # 15) -e x p (—10#37) 02 = 0.13 —0.084#37 — 0.004#37 Hi = c0 exp [ - w ^ ( # i9 - cxf ] + c3 + <Si ZC2 Co = -0 .3 5 - 0.3#37+ 1.5 exp [ - 0 . 5 ( # 3 7 - 0 . 3 ) 2] + 0 . 0 1 # - 0 . 0 8 ^ Ci = - 0.000004#3 1. 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 c2 = 1.08 + 0.0037? —0.27 exp(P37) —0.00487? exp(P37) c3 = 0.99 + 0.03P37 - 0.002P + 0 . 0 3 3 ^ - 0.037P37\ / P - c0 S1 = 0.06 - 0.023P19 + 0.03P37 - 0.02 exp(P37) - 0.0002P oi = 0 .1 3 - 0 . 0 9 / p ^ + 0.015, (4.13) where the unit of rain rates (7?) is m m /h r, and P index is dimensionless. In those equations, th e complexity gradually increases when more variables are considered. Combining this set of fitted param eters w ith an assum ed prior surface rainfall probability distribution, the posterior probability density function can be obtained. In the next section, we will dem onstrate an example to assess th e applicability of concept of th e Bayesian approaches. 4.5 R etrieval based on sim ulations To evaluate the applicability of the Bayesian algorithm, th e physical model de rived from th e sim ulated dataset is applied to one of the NW S W SR - 8 8 D network datasets to reveal the retrieval ability. It has to be made clear th a t even though th e surface rainfall inform ation from over-land radars is used in th e simulations, our Bayesian algorithm is an ocean algorithm th a t is being evaluated. Therefore, in the following figures, only radar composite reflectivity is plotted over th e land w ith coast lines. O ther sim ulation-related variables are assumed over th e ocean w ithout plot ting th e US continent. In addition, the aim of the section is to provide an overall idea about the qualitative perform ance of sim ulated retrievals, and th u s th e quantitative comparisons are not given here. D etailed discussions about th e perform ance of the Bayesian algorithm derived from th e actual P R and TM I observations will be given in C hapter 7. Figure 4.2 shows the 1 km NWS W SR- 8 8 D network com posite reflectivities on A ugust 13, 2 0 0 2 , after suspicious rad ar returns have been removed (see section 4.1). T he estim ated rain rate w ithout adjusting reflectivity is m apped in figure 4.3, and th e corresponding m odel-calculated attenuation index a t 10.65, 19.35, and 37.00 GHz are depicted in figures 4.4, 4.5 and 4.6, respectively. These plots show some Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (dBZ) 41 n0r_ 020 813 _1040 Figure 4.2. 1km NWS W SR- 8 8 D network composite reflectivies for 10:40am, A ugust 13, 2002, w ith suspicious rad ar retu rn s removed. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Simulated Rain R a te ( m m / h r ) 42 Figure 4.3. Plots of sim ulated 15 x 15 km rain rate based on th e W SR8 8 D composite reflectivity for 10:40am, A ugust 13, 2002, and a Z-R re lationship proposed by M arshall and Palm er (1948). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.4. Sim ulated P i 0 for th e d a ta of 10:40am, A ugust 13, 2002. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. :n , *• '•• ' A* ^ • ** . \X^L* . -?( w ^ fr j *^ ..;i V> ■*. * * *• "j ^5r?o * P * • • ‘ ■»- "t • A / i •*, * V ■' : j?£W ^ X ? a*/ V i '. v .*' *•»' Figure 4.5. Sim ulated P i 9 for th e d a ta of 10:40am, A ugust 13, 2002. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.6. Simulated P 3 7 for th e d a ta of 10:40am, A ugust 13, 2002. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Retrieved Rain R a te ( m m / h r ) 46 Figure 4.7. Plots of retrieved rain rate for the d ata of 10:40am, August 13, 2002, based on the new Bayesian algorithm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 strong precipitation systems associated w ith very low P\g and P 3 7 values in the Midwest and th e east coast, and other scattered convection structures in th e west and south. Figure 4.7 reveals th e retrieved rain rate using the prior rainfall distribution of log 77(1.14, 1.047) and th e m ean estim ates. The retrieval dem onstrates very consistent horizontal precipitation distributions to those of th e reflectivity-estim ated rainfall intensity. Since th e retrieval was only processed for th e pixels in which either one of P 10, P i 9 , P 3 7 values is less th a n . , some very light precipitation areas (~ 0 8 0.01 m m /h r) w ith high P values were not retrieved and are not shown in th e 4.7. Up to this point, the Bayesian rain ra te retrieval algorithm using explicit, closedform functions fitting to th e d a ta has shown high retrieval ability. M ost im portantly, our algorithm is not only able to give the single-pixel retrieval, b u t also provides a continuous posterior probability distribution. Note, however, th a t th e retrieval yielded from this given prior rain ra te distribution and th e specified conditional likelihood associates a bias in some degree. The errors m ight be introduced from 1) imperfect prior o r/a n d 2) im proper conditional likelihood functions o r/a n d 3) the m ean estim ator. Instead of reducing th e errors tuning th e prior and conditional probability distribution or using another estim ator to interpret th e posterior pdf, a more fundam ental and im portant issue for the development of Bayesian algorithm s is to understand how well th e Bayesian approach works when perfect inform ation is applied; and w hether the Bayesian algorithm still preserves its retrieval ability when im perfect functions are used. To clarify this issue, a detailed theoretical sensitivity analysis will be introduced in th e following chapter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 5. SE N SIT IV IT Y TEST The success of a Bayesian retrieval m ethod depends on w hat assum ptions are made in th e conditional likelihood, how th e prior precipitation distribution is speci fied, and how the posterior density function is interpreted. In order to system atically analyze th e effects of th e aforementioned uncertainties on th e retrieval, theoretical sensitivity tests were conducted using two synthetic datasets where th e ” tru e ” condi tional and th e prior distribution were idealized. One synthetic d ataset was generated by a simple conditional pdf, which was characterized by a covariance m atrix and thus only linear relations were included. The other dataset was generated by th e modelderived physical model th a t is m entioned in th e previous chapter to statistically describe th e physical relationships between rain ra te and atten u atio n index. The la tte r d ataset considers b o th linear and non-linear interactions of th e observation variables. One thing worthy of m ention is th a t a theoretical simulation te st was perform ed as well in th e study of Evans et al. (1995). The m ain purpose of th eir test was to de term ine which covariates had th e greatest influence on th e accuracy of th e retrieval. Nevertheless, their results still suffered from the imperfection of th e cloud model and radiative transfer model, and the assum ptions in b o th conditional and prior distri butions. It is hard to trace the sources of errors due to the high complexity in b o th models and dimensions. Since th e synthetic d a ta in the study were generated from the condition where all of the probability density functions were exactly known, our tests are able to evaluate th e contribution of each com ponent (the conditional, prior and posterior probability distributions) w ithout th e influences of th e uncertainty in th e assum ptions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 The aim of the theoretical exam inations here is to provide answers to th e fol lowing questions: 1) W hat is th e performance of th e Bayesian-approached retrieval algorithm in different rain rate ranges, especially under extrem ely heavy precipitation condition? 2) How do th e various interpretations (e.g., using different estim ators) to th e posterior distribution affect the retrieved rain rates? 3) Is th e posterior density function sensitive to b o th th e conditional and prior probability distributions? 4) W hich p art, conditional distribution, prior pdf, or th e interpretation of th e posterior function, will have more significant im pacts on the retrieval? 5) W hen th e uncer tainty of prior distributions and the imperfectness of the physical model exist, which is the case in m ost retrieval applications, is th e retrieval still meaningful? 5.1 T h e prior and conditional distributions Since one of the m ain purposes of the sensitivity tests is to b e tte r understand the behavior of retrieval a t ranges of higher rain intensity, logN(0, 2) was served as th e prior distribution to generate the random variates in an a tte m p t to prevent a scarcity of high rain rates in the training dataset. Two physical models will be used for the element of conditional likelihood. In addition to th e complete conditional pdfs derived from th e m odel sim ulations (in chapter 4), another way to specify the conditional likelihood is shown here as well. As m entioned in chapter 1, th e m ultichannel relationships in other Bayesian algorithm s were expressed by a linear covariance m atrix. To investigate th e influence of the covariance m atrix on the retrieval, a similar specification for conditional p df was built as well in th e study. Like th e univariate situation, th e conditional probability density function representing the physical relationships between P and R can be w ritten as f ( P \ R ) oc P i o ( a - P 1 0 ) P 1 9 ( o - P 19 ) P 3 7 ( o - P 37)-e x p i ( p - p ) c r 1( p - p ) T , (5.1) where th e param eter /j, and th e covariance m atrix C are similar to th e (p, a 2) of the univariate case. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 Table 5.1 Coefficients of a^, bfl and cfl. Channel(GHz) K 10.65 0.75 0.03 0.30 19.35 1.35 0.05 -0.30 37.00 1.55 0.10 -0.50 fi is defined as At = (1 1 1 , 1*2 , 1*3 )', W = an,iexp [-b n,iR ] + cliii, (5.2) where th e subscripts 1, 2, and 3 describe th e quantities at 10.65, 19.35, and 37.00 GHz, respectively. Values for th e param eters of a^, bfil and cfl are ta b u la ted in Table 5.1, while i expresses th e corresponding channel. T he /q a t a given channel exponentially decreases w ith respect to rain rates, bu t the decreasing ra te and the achieved minim um vary w ith microwave frequencies. P 37 tends to decrease more rapidly w ith R th a n P 19 and P i0 do, and th e lower minimum lim it can force th e peak of P 37 to be very close to zero if rain rates become large. On th e contrary, P 10 are seldom lower th a n 0.4 based on th e actual TM I observations. In addition, u C 21 C31 C \2 C 22 C 32 C l 3 CO C C 33 The covariance m atrix defines the variance and correlations of P index a t th e three channels. T he diagonal elements indicate th e variations associated w ith P i0, and P 3 7 P 1 9 , themselves, while th e non-diagonal elements express th e covariances of the three variables. It is noted th a t th e arb itrary covariance m atrix in th e stu d y has to be positive definite in order to keep th e exponential term in a stable mode, which is expected physically and produces reasonable probabilities for th e conditional distri bution. Several sets of values for th e covariance m atrix are given and tested later. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 Finally, since the synthetic d a ta include rain rates up to 100 m m /h r, th e retrieved range of R in th e sensitivity test is set to be [0, 100] m m /h r as well for th e purpose of comparisons. 5.2 D esign ed exp erim ents Various experim ents w ith different com binations of th e training d ataset (i.e., re trieval target), and the conditional and prior distributions used in th e retrieval calcu lations, are designed for the sensitivity te st (summarized in table 5.2). Experim ent ID is nam ed by starting w ith a letter S or R, expressing the source of the training dataset based on th e simple model (D1 dataset) and radar-radiative sim ulations of the complicated model (D2 d ataset), respectively. Note th a t D1 and D2 b o th were generated along w ith the a same prior distribution logN{0,2). The values of th e covariance m atrix elements given in D 1 d ataset are th e same as the experim ent SO. D1 was random ly generated by a rejection m ethod (von Neumann, 1963), and th e to tal sample size was 1 million. For D2 dataset, the random variates were generated from those conditional distributions of th e com plicated model (as equations (3.2), (3.3) and (4.13) shown). First, generate rain rates based on the prior distribution, then generate P 3 7 from / ( P 3 7 |P ) by using a rejection m ethod. Next, similarly, generate P i 9 and P 10 from th e corresponding conditional density function, th e n a full vector (P , P37, P 1 9 , P 1 0 ) can be returned. The sample size was 5 million in D2 dataset. Before conducting th e sensitivity tests, it is essential to ensure th a t th e training datasets indeed follow the designed probability density functions; th e random vari able generating process does not introduce any other significant errors into th e data; and the physical model is able to represent th e behavior of actual observations to some degree. Figures 5.1 and 5.2 illustrate th e distributions of the surface rain rate, multichannel relationships, and m arginal distributions of P index a t three channels, which are calculated from theoretical functions of the sophisticated m odel and from the corresponding D2 training d ataset, respectively. Plots have revealed th a t the analytical solutions and random ly-generated d a ta m atch w ith each other very well. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 Table 5.2 Inform ation of designed experim ents in sensitivity tests, including the experim ent ID, the training d ataset, and th e specifications of the prior and conditional likelihoods applied to th e Bayesian algorithm. Exp ID D ata 7r(R) Cu C 22 C 33 C\2 C l3 C 23 SO D1 logN( 0 , 2 ) 0 .0 1 0.04 0.06 0.015 0 .0 2 0 0.045 SI D1 logN( 0,1) 0 .0 1 0.04 0.06 0.015 0 .0 2 0 0.045 S2 D1 l o g N ( l . U , 1.047) 0 .0 1 0.04 0.06 0.015 0 .0 2 0 0.045 S3 D1 l o g N ( l M 3 , 1.205) 0 .0 1 0.04 0.06 0.015 0 .0 2 0 0.045 S4 D1 U n if o r m [ 0,100] 0 .0 1 0.04 0.06 0.015 0 .0 2 0 0.045 S5 D1 logN (0, 2) 0 .0 1 0.04 0.06 0.000 0.000 0.000 S6 D1 logN{0, 2) 0 .0 2 0.04 0.06 0.015 0 .0 2 0 0.045 S7 D1 logN{ 0,2) 0 .0 1 0.0 8 0.06 0.015 0 .0 2 0 0.045 S8 D1 lo g N ( 0 , 2) 0 .0 1 0.04 0.12 0.015 0 .0 2 0 0.045 R0 D2 logN( 0,2) Com plicated model from equation (4.13) R1 D2 logN( 0,1) Com plicated model from equation (4.13) R2 D2 logN(1.14:, 1.047) Com plicated model from equation (4.13) R3 D2 logN ( I M S , 1.205) Com plicated model from equation (4.13) R4 D2 U n if o r m [ 0 ,100] Com plicated model from equation (4.13) R5 D2 l o g N (0 , 2 ) Param eterization from equation (5.4) R6 D2 logN( 0,2) 0.01 0.04 0.06 0.015 0.020 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.045 53 (a ) Rain ro te (c ) P19 vs. P37 (b ) P10 vs. P19 2.0 1.0 r- 0.6 c V o 0.4 0.4 0.2 0.2 0.5 0.0 0 10 5 20 15 0.0 0.0 0.0 0.2 0.4 0.6 0 .8 0 .0 1.0 0.2 0.4 0.6 0.8 1.0 R oin R o t e ( m m / h r ) 6 (d ) M arginal p d f (P 1 0 ) ( f) M arginal p d f (P 3 7 ) (e ) M arginal p d f (P 1 9 ) 2.0 4 5 3 D e n sity 4 >, 3 a> o 2 1 a> o 1 n E. . . i . 0.0 2 0.2 . 0.4 0.6 0.8 0 .5 n F ^ - r - T '^ . . . 1.0 0.0 0.2 0.4 0.0 0.6 0.8 1.0 0.0 0.2 0.4 0 .6 0.8 Figure 5.1. T he probability distribution of rain rates and the joint and m arginal pdfs of th e P vector based on analytical solutions of the sophis ticated model. The joint pdfs are plotted for [0.05, 1, 5, 7, 10, 12]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.0 54 ( c ) P19 vs. P37 (b ) P10 vs. P19 (o ) Rain rate 2.0 0.8 0.8 E o a> 0.6 X 0.4 0 .4 0.2 0.2 0.5 0.0 0 10 5 20 15 0.0 0.0 0.0 0.2 0.4 0 .6 0.8 1.0 0 .0 0.2 0.4 0.6 0.8 1.0 R oin R o te ( r r > m /h r ) 6 (e ) M arginal p df (P 1 9) (d ) M arginal pdf (P 10) 2.0 4 ( f) M arginal pdf (P 3 7 ) 1 i ' ' ■- r ' " i ■ ■ ■ i « ' ■ 5 3 4 2 2 1 0 0.0 1 0.5 oE^rrrT. 0.2 0.4 0.6 0.8 1.0 0 .0 0.2 ...... 0 .4 0.6 0.8 0.0 1.0 0.0 0.2 0.4 0.6 0.8 P10 Figure 5.2. dataset. Same as figure 5.1, b u t for the random ly generated D2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.0 55 This, in turn, implies a sufficient sample size and a proper generating m ethod for the training dataset. The m ultichannel relationships of P values in th e D2 d ataset are conveyed by figure 5.3. Com pared to th e figure 4.1, the sim ilarity to each other reveals a proper specification of the physical model. Recall th a t th e prior distribution of D2 d ataset tends to generate more points w ith higher rain rates, so the contours of th e D2 dataset are broader in 2-D space, and a little more d a ta are seen in th e lower P\ q values th an the original radar-radiative simulations. However, th e general m ultichannel relationships are very consistent. Control runs (SO and RO) represent the experim ents th a t th e Bayesian retrieval m ethod uses the exactly same conditional and prior pdfs w ith th e training d ata. An analysis of these two control experim ents provides insight into th e inherent uncer tainty of the retrieval, since each pdf is perfect and no assum ption is m ade in the algorithm. The sensitivity of th e algorithm to th e specifications of th e prior d istri butions is evaluated by experim ents SI to S4, and R1 to R4. S5 to S8 are designed to investigate the dependency of the retrieval on the assum ptions of th e correlations of the response vector in th e case of the simple physical model. S5 uses a diago nal m atrix, expressing th a t th e three observed variables are uncorrelated w ith each other. Note th a t S5 is sim ilar to w hat the previous work assumed. S6, S7, and S8 express the situations where one of the diagonal elements is twice th a t of th e control run while the others rem ain th e same as th e control run. To investigate th e sensitivity of th e Bayesian algorithm to varying param eterizations in the physical model, a different approach of conditioning / ( P 2|P 3, R) of the complicated model is tested in experim ent R5. T he change is only m ade via the param eterization of p 2 as follows: Hi = exp(60 + hi In R) + 52 b0 = -0 .4 5 1 + 0.365P37 - 0.14 exp [ - 0 .5 (- 37 ~ o°7 ° 8Q) ] + 0.07 b\ —0.44 exp(—14P37) —0.04 — 0.04 exp [ — 0.5( 37^ ^ — ) ] p = n QO ^ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 P 1 0 = 0 .3 0.0 0 .2 0.4 0.6 P 1 0 = 0 .4 P37 = 0 .0 RADAR 0.8 1.0 0.0 1.2 1.2 1.0 1.0 N a 0.6 cl 0.2 0.4 ....................................... 0.6 0.8 1.0 1.2 P 1 0 = 0 .5 0.2 0.4 0.6 P 1 0 = 0 .6 0 .2 0.4 0.6 P 1 0 = 0 .8 0.0 0.8 1.0 1.2 RADAR 0.6 0.2 0.2 0 .0 0.6 0.4 0.4 0.0 0.4 P37 = 0 .2 RADAR 1.2 0.0 0.2 RADAR 0.2 0.4 0.6 P10 = 1 .0 0.0 0.0 1.0 1.2 0.0 1.0 1.2 0.0 1.0 0.2 0.4 0.6 0.2 0.4 0.6 P 3 7 = 0 .8 RADAR 0.6 0.6 P 3 7 = 0 .6 RADAR 0.8 0.4 P 3 7 = 0 .4 RADAR 0.8 0.2 1.2 0.0 0.2 0.4 0.6 P 3 7 = 1.0 RADAR 0.8 1.0 1.2 RADAR 0.8 1.0 1.2 RADAR 0.8 1.0 1.2 RADAR 0.8 1.0 1.2 RADAR 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 .0 0.2 0.4 0.6 0.8 1.0 1.2 Figure 5.3. M ultichannel relationships by plotting the num ber of pixels at a given P w (left column) or given P 3 7 (right column) for D2 training dataset. Contours are logarithm ically spaced; actual value is 1 0 r where x is the contour label, x are plotted for values of [0.5, 1, 2, 3, 4, 5]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 62 = R — 4 1.08 • exp(—3 OP3 7 ) exp [ - 0.5(— -— ) ] O (5.4) Results from the test here will give an indication about th e degree of how accurate th e physical model has to be and how much uncertainty th e algorithm is able to tolerate. Experim ent R6 is designed to apply th e P -R relationships of th e simple model into the algorithm , while a complicated d ataset D2 is the targ et to be retrieved. It helps to evaluate the adequacy of simple assum ptions of the conditional p d f in the retrieval algorithm , when in fact th e d ataset has much larger complexities. 5.3 R esu lts for theoretical test 5.3.1 Intrinsic uncertainty o f th e algorithm An im portant property of the Bayesian analysis is th a t a probability statem ent can be m ade about the realization of th e retrieval based on th e derived posterior distribution. The objective of the section is to assess the intrinsic uncertainty of th e retrieval and the difference in inference m ade by th e MEAN and MLE, when th e ideal physical model and exact prior distribution are known in th e Bayesian algorithm . The inherent uncertainty of th e Bayesian retrieval algorithm is illustrated in figure 5.4 by plotting the standard deviation (m m /h r) of the posterior p d f a t a given P 1 0 value for th e experim ent RO. Since observed P 1 0 values are rarely lower th an 0.4, only results where P 10 is greater th a n 0.4 are shown in the figure. In th e case of larger P 10 values, say 0.8 to 1.08, th e contour shows an overall small uncertainty in th e retrieved rain intensity, except in some regions where P 3 7 is reaching saturation. For instance, at a given Pw value of 0.9, th e com binations from [0.0, 0.6] of P 19 and [0.0, 0.1] of P 3 7 produce around 10 m m /h r deviations. Note th a t this kind of P vector is often seen in th e real-world satellite observations, and th e Bayesian retrieval will be associated w ith a considerable uncertainty even when th e perfect prior/conditional pdfs has been applied. Figure 5.4 also reveals th a t the m axim um deviation for each slice increases w ith decreasing P i0 values. Generally speaking, th e large variances occur in two regions Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 P10 = 0 .7 0 P10 = 1.08 1.0 0.8 0.8 iv 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 P19 0.8 1.0 -10 0 .0 0.2 0.4 0.6 P19 0.8 1.0 0.8 1.0 0.8 1.0 0.8 1.0 P10 = 0 .6 0 P10 = 1.00 0.8 0.8 ^ 0.6 1-0 Q. 0.4 0.4 0.2 0.2 0.0 -10 0.0 0.2 0.0 0.4 0 .6 P19 0.8 1.0 0.0 0.2 0.4 0.6 P19 P10 = 0 .5 0 P10 = 0 .9 0 1.0 0.8 0.8 0.4 0.4 0.2 0.2 0.0 -10 0 .0 0.2 0.0 0.4 0.6 P19 0.8 1.0 0 .0 0.2 0.6 P19 P10 = 0 .4 0 P 10 = 0 .8 0 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.4 -10 0 .0 0 .2 0.0 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 Figure 5.4. 2-D contours of the stan d ard deviations of th e posterior probability distribution in the F 3 7 vs. Pig space at a fixed Pig value for experim ent RO. Contours are plotted in an interval of 5 m m /h r. Regions outside the zero value of th e contour indicate an impossible P vector due to a zero m arginal density. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 of th e 2-D contour space. One is associated w ith a low Pio, a satu rated P 3 7 , bu t a m oderate Pig value. T he other region happens in th e com bination of a low P i0, high Pig, b u t a m ediate P37. Those response vectors express precipitating systems associated w ith particular horizontal structures, which are not commonly seen in observations. Under the condition, a relatively large uncertainty m ight have existed in the physical model, and then it m igrates into th e posterior distribution. Therefore, when this kind of vector is observed, caution has to be exercised in th e corresponding retrieval, and further additional inform ation or a more precise physical model might be required. Figure 5.5 depicts th e histogram of the retrieved rain rates for RO experiment. The title for each subplot is made of three components. R R indicates th e specific range of rain rates, where th e retrieved rain intensity is draw n if th e corresponding rain rate of the training d a ta point is in this range. It is followed by a num ber showing how m any d a ta points are picked to compose th e histogram . MEAN or MLE describes which estim ate is used to interpret the posterior distribution, the mean or th e m axim um likelihood value. T he pair of th e num bers on th e upperright corner in each subplot indicates th e m ean and th e stan d ard deviation of the histogram , while th e percentage shown right below it tells us th e proportion th a t the retrieved rain rates are in th e same range w ith th e actual rainfall intensities w ith respect to th e whole histogram . Note th a t for th e tru e range R R lower th a n 30 m m /hr, th e histogram is plotted in a logarithm ic scale. The results from RO experim ent using MEAN estim ates suggest th a t th e Bayesian algorithm in the case is able to retrieve th e light rain ra te very well, although the general m ean value of the histogram s m ight be slightly larger th a n th e range of the training data. For the m oderate intensity (7—15 m m /h r), th e retrieval m ethod captures around 46% d a ta points, and th e m ean value is about right. T he algorithm tends to underestim ate the rain rates when the actual intensity in training d a ta increases to a heavier range (15 to 75 m m /h r). Meanwhile, th e associated stan d ard deviation of th e histogram starts to increase as well. Even so, th e retrievals still Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 R R = [ 0 .1 , 0 .2 ] 430957 - MEAN RR = [ 1.0, 0.8 5 RO ( 0 .2 4 , 0 .1 5 ) 670027 - R R = [1 5 .0 , 3 0 .0 ] 0.30 f MEAN 222467 - MEAN (1 4 .0 1 , 8 .6 9 ) ( 2 .6 0 , 1 .9 3 ) 32.4% 4 8 .1 % . 4 H is to g ra m 2 .0 ] R0 28.6% 0.6 3 E 2 X o S’ 0.1 5 0.2 1 0 0.0 1.0 10.0 0.1 100.0 R R = [ 0 .2 , 0 .4 ] 1.0 10.0 0.1 562461 - = [ 2.0 , MEAN 4 .0 ] 597394 - R R = [3 0 .0 , 5 0 .0 ] 0.20 R0 MEAN 3.0 ro: ( 5 .5 7 , 3 . ( 0 .3 7 , 0 .2 5 ) 2.5 1.0 10.0 98415 - MEAN ( 2 4 .2 7 ,1 3 .2 3 ) 21.2% : 2 7 .9 7 39.6% ' 100.0 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) 0.1 5 2.0 E o E □ cn o g10.10 X x 0 .05 0.5 0.0 0.1 1.0 10.0 100.0 0.1 0 .6 ] 100.0 1 0.0 0.00 0 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) R R = [ 0 .4 , 1 .0 374641 - 2.0[--------------------- R = [ 4.0 , MEAN 7 .0 ] 40 MEAN 395442 R R = [5 0 .0 , 7 5 .0 ] 0.1 4 • R0 21.6% 60 100 80 R e tr ie v e d RR ( m m / h r ) 0.4 ( 0 .6 2 , 0 .4 3 ) 20 49896 - MEAN (45.8,6,17.58) 35.0% 34.1 0.3 E o g*0.2 ^ 0.08 x 0.1 0.0 0.1 1.0 10.0 100.0 0.1 R R = [ 0 .6 , 1 .0 ] 1.0 1 0 .0 498245 - R R = [ 7.0, 1 5 .0 ] MEAN RO R0 391911 - MEAN ( 1 0 .0 9 , 6 .8 7 ) • 30.0% 20 40 60 80 100 R e tr ie v e d RR ( m m / h r ) 0.30 1.4 0 10 0 .0 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) - 46 .4 % - R R = [7 5 .0 ,1 0 0 .0 ] 0.10 R0 24139 - MEAN ( 6 9 .6 3 ,1 4 .2 0 ) 42 .0% 0.08 J 0.20 1.0 E E o o o> 0.8 E 0.06 g* 0.15 ■± 0.6 x 0.04 0.10 0.4 0.02 0.05 0.2 0.00 0.0 0.1 1.0 10.0 R e tr ie v e d RR ( m m / h r ) 1.0 1 0.0 R e tr ie v e d RR ( m m / h r ) 0.00 0 20 40 60 80 R e tr ie v e d RR ( m m / h r ) Figure 5.5. Histograms of retrieved rain ra te a t different rain ra te ranges for R0 experiment. T itles contain inform ation about range of tru e values, sample size, and the estim ator. Numbers in parentheses represent the m ean and standard deviation of th e histogram . Percentages present the fraction of retrieved rain rates located in th e correct range. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 61 encompass around 30% d a ta of the tru e range. For th e extremely large rain rate, there is around 42% of d a ta in the correct rain range, and th e mode of th e histogram falls w ithin the tru e range as well. In th e control experim ent, the Bayesian algorithm shows the ability of retrieving the rain intensity over all ranges, even for th e case w ith extrem ely heavy precipitation (greater th a n 75 m m /hr). The histogram of MLE for experim ent RO (as figure 5.6 shown) discloses a po tential problem when th e inference about th e posterior distribution is m ade by the m aximum likelihood estim ate. Results from th e light rain regime (from 0.1 to 4 m m /hr) dem onstrate th a t the perform ance of th e algorithm is quite good. However, the underestim ation for interm ediate intensities (4 to 30 m m /h r) is significant. More over, there are two distinct regimes in th e histogram when the tru e range R R goes up greater th a n 30 m m /hr. T he one associated w ith a lower rainfall ra te dom inates in the range of [30, 75] m m /h r, and it m ight result in retrieved rain rates at least 20 m m /h r smaller th a n the tru e rain intensity. T he other peak becomes dom inant w ith th e increasing tru e rain rates, and reflects th e consistency of th e retrieved intensity with the tru e rain rates, giving around 40% d a ta points in th e correct range. Under th e extremely heavy precipitation, th e mode w ith th e lower rain intensity becomes much smaller, and the algorithm produces m ost of th e retrieval a t an appropriate intensity range. It is im portant to understand th e physical m eaning of the two various regimes in the M LE histogram and their associated im pact on th e overall perform ance of th e Bayesian m ethod. Figure 5.7 shows posterior distributions for some specific P vectors in experim ent R0. Some posterior distributions have a single m aximum over the entire rain ra te range, b u t some have two local maximums. T he single maximum depicts a condition th a t th e P vector expresses very clear inform ation about w hat rain rate will be m ost likely happened when b o th physical relationships and prior knowledge are taken into account. T he double local maximums imply th a t two com petitive rain intensities exist. T he peak on the left may indicate a scene of widespread stratiform precipitation associated w ith a smaller rain rate, while the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 R R = [ 0.1 , 4 RO 0 .2 ] 430957 - R R = [1 5.0 , 3 0 .0 ] MLE ( 0.09, 0.11) RO ( 1.20, 0.61) RO 56.1% 16.8% o cr> o 0.4 I 1 0.5 0 0.2 0.0 0.0 1.0 1.0 1 0.0 R e tr ie v e d RR ( m m / h r ) 0 .4 ] : 2.5% E 0.6 E R R = [ 0 .2 , 4 RO MLE 10.0 100.0 1.0 MLE RR = [ 2.0 , ( 0.19, 0.19) 597394 - RR = [3 0 .0 , 5 0 .0 ] MLE 14.5% 98415 MLE (14.92,18.95) RO 10.7% 49.2% 0.8 3 ?2 4 .0 ] 10 .0 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) 562461 ( 6.17, 9.24) 0.8 3 0.1 222467 - 2.0 o x 0-4 0.2 0 0.05 o.oo a 0.0 0.1 1 .0 1 0 .0 1.0 1 0 0 .0 R e tr ie v e d RR ( m m / h r ) RR = [ 0.4 , 4 RO 0 .6 ] 20 10.0 374641 - MLE R R = [ 4 .0 , ( 0.36, 0.26) 7 .0 ] RO 27.6% 395442 R R = [5 0 .0 , 7 5 .0 ] MLE 60 80 49896 - 12.6% . 0.6 MLE (41.5[4,28.94) 2.99, 3.52) 0.8 3 40 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) 43.4% § 0.15 o' 2 £ 0.4 1 0.2 0 0.0 0.1 1.0 1 0 .0 1 0 0 .0 0.1 R e tr ie v e d RR ( m m / h r ) R R = [ 0.6 , 2.0 RO 1 .0 ] 1 .0 1 0 .0 20 1 0 0 .0 498245 - MLE R R = [ 7.0 , 1 5 .0 ] ( 0.62, 0.35): RO . %: 42 8 391911 - MLE R R = [7 5 .0 ,1 0 0 .0 ] 0.14 3.8!4, 5.50) E o coh 60 80 241 3 9 (7 4 .7 5 .2 1 .4 4 ) RO 0.12 0.10 : 3.5% . 0.8 40 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) 40.5% : E %>0.08 o i I 0.4 x 0.06 0.04 0.5 0.2 0.0 0.0 0.1 1.0 10.0 R e tr ie v e d RR ( m m / h r ) 0.1 1 .0 1 0 .0 0.02 0.00 0 R e tr ie v e d RR ( m m / h r ) 20 40 60 80 100 R e tr ie v e d RR ( m m / h r ) Figure 5.6. Same as figure 5.5, b u t using MLE estim ations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 0.12 (0.8 0 , 0.50, 0.10) (0 .8 0 , 0.20, 0.05) (0 .6 0 , 0.02, 0.01) • f 0.08 c o0) S ' a. 0.06 0.04 0.02 0.00 0 20 40 60 Roin Rate ( m m / h r ) 80 100 0.035 (0 .6 8 , 0.28, 0.01) 0.030 (0 .6 0 , 0.28, 0.0 1 ) 0.025 a 0.020 -9 0.015 0.010 0.005 0.000 0 20 40 60 80 100 R ain R a te ( m m / h r ) Figure 5.7. Examples of derived posterior rain rate distributions at some given P vectors in experim ent R 0 . The observation vector (P 10, P i 9 . P 3 7 ) is presented by the three num bers in parentheses. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 second peak suggests th a t a strong convection precipitating system may occur in th a t 15 x 15 km area. This plot suggests th a t th e M LE estim ate will increase from 5 to 60 m m /h r if th e Pio decreases from 0.64 to 0.60 and Pw and P 3 7 rem ain the same. However, this kind of decrease in P value is possibly attrib u te d to th e noise from the instrum ent, atm ospheric condition, and the calculations of brightness tem peratures, not necessarily meaning a dram atic change in th e rain intensity of th e scene. The fact th a t there are two very different regimes in th e M LE histogram under high rain rate cases brings up a drawback for th e algorithm. It implies th a t a slight change in the one of P values may make th e retrieved rain ra te ju m p from one regime to the other, and introduce a discontinuity in th e retrieval product. Moreover, the characteristics of two distinct regimes are also the reason th a t th e peaks in the histogram of the MEAN estim ates are significantly underestim ated. T he values are around between 15 and 43 m m /h r for the d a ta range of [30, 50] and [50, 75] m m /h r, respectively, since th e MEAN is trying to find th e averaged point of these two regimes. 5.3.2 S en sitiv ity to th e prior know ledge The purpose of the section is to evaluate th e sensitivity of th e Bayesian algorithm to the prior surface rainfall distribution. Analysis of experim ents S I to S4 and R1 to R4 helps to dem onstrate how th e retrieved rain rate changes to th e various specifi cations of the prior distribution when th e form ulation of physical model rem ains the same. Since results of experim ents SI to S4 (simple physical model) indicate similar inform ation to experim ents R1 to R4 (physical model derived from sim ulations), this section only gives detailed outcome of R1 to R4. ’S’ series experim ents are needed here since it m anifests th a t the property of being sensitive to th e prior inform ation is not localized in a specific physical model, and thus th e following statem ents are representative. The prior probability density functions utilized in th e sensitivity test are all plotted again in th e figure 5.8 to highlight their individual characteristics and for th e convenience of illustration. Com pared to th e control prior density function, the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 P rio r PDF logN(0.00,2.000) logN(0.00,1.000) 0.8 logN( 1.14,1.047) >. c 0.6 <D Q logN( 1.04,1.205) 0 .4 0.2 0.0 0 2 4 6 Rain r a t e ( m m / h r ) 8 10 P rio r PDF 0.012 logN(0.00,2.000) 0.010 logN(0.00,1.000) togN(1.14,1.047) 0 .0 0 8 logN(1.04,1.205) >, c 0 .0 0 6 (D O 0 .0 0 4 0.002 0.000 20 40 60 80 Rain r a t e ( m m / h r ) 100 Figure 5.8. Plots of th e prior probability distributions used in th e sen sitivity tests. To b e tte r illustrate the differences, ranges of [0, 10] and [10, 100] m m /h r are respectively shown in (a) and (b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 RR = [ 1.0, 2 .0 ] 670027 - R R = [1 5.0 , 3 0 .0 ] MEAN 222467 - MEAN 0.30 0.8 ( 5.97, 2.70) ( 1.78, 0 .9 0 ) ( 0.38, 0.12) 0.25 48.4% 0 .0% . ' 0.9% • 0.6 0 .20 0 CP O 1 0.4 E o X x x E o E ? 0.15 0 .1 0 0.2 0.05 0.0 0.1 1.0 10.0 100.0 0.00 0.1 R e tr ie v e d RR ( m m / h r ) R R = [ 0 .2 , 0 .4 ] 1.0 1 0 .0 1 0 0 .0 0.1 562 4 6 1 - RR = [ 2 .0 , MEAN 4 .0 ] 597394 - MEAN RR = [3 0 .0 , 5 0 .0 ] 0.4 r 3.0 1 0 .0 1 0 0 .0 98415 - MEAN 0 .2 0 ( 3.13, 1.34) 0.48, 0.18) 2.5 1 .0 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) 8 .4 8 , 4.0 8 ) 56.8% 38.9% - 0.3% 0.15 6o 2.0 cr> o cn I X 0.05 0.5 0.0 0.00 0.1 1.0 10.0 100.0 0.1 R e tr ie v e d RR ( m m / h r ) R R = [ 0 .4 , 0 .6 ] 374641 1.0 10.0 100.0 0 R e tr ie v e d RR ( m m / h r ) - MEAN R R - [ 4.0 , 7 .0 ] 395442 - ( 0.66, 0.27)! MEAN 60 80 100 R R = [5 0 .0 , 7 5 .0 ] 49896 - MEAN 80 100 0.14 4.02, 1.66) 41.1% 3 1 . 1% : 40 R e tr ie v e d RR ( m m / h r ) 0.4 f 2 .0 20 0 .1 2 0 .1 0 E E o CP o o x X 0.08 0.06 0.04 0.5 0.02 0 .0 0.00 0.1 1 .0 1 0 .0 1 0 0 .0 0.1 R e tr ie v e d RR ( m m / h r ) R R =[ 0 .6 , 1 .0 ] 1.0 1 0 .0 1 0 0 .0 498245 - MEAN R R =[ 7.0, 1 5.0 ] 391911 40 60 - MEAN R R = [7 5 .0 ,1 0 0 .0 ] 24139 - MEAN 0 .1 0 (;4.68, 2.10) 0.25 41.3% . E o 20 R e tr ie v e d RR ( m m / h r ) 0.30 ( 0.96, 0.44)' o 0 R e tr ie v e d RR ( m m / h r ) : : 1 1 .2 % * 0.08 0.20 £ 0.06 E o 0.8 S’ 0.15 x 0.6 0.04 x 0.10 0.4 0.02 0.05 0.2 0.0 0.00 0.1 1 .0 1 0.0 R e tr ie v e d RR ( m m / h r ) 0.00 0.1 1 .0 1 0 .0 R e tr ie v e d RR ( m m / h r ) 0 20 40 60 80 R e tr ie v e d RR ( m m / h r ) Figure 5.9. Same as figure 5.5, bu t for experim ent R l. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 67 prior distribution in R1 experim ent, lo g N (0,1), has a larger proportion of probability over th e lower rain rate (0.4—4 m m /h r) and relatively smaller probabilities at very light rain rates and beyond 5 m m /h r. The prior probability is very close to zero at a value of 30 m m /hr. The property of the prior pdf leads th e algorithm to miss all points for the true d a ta range of [0.1, 0.2] m m /h r and to produce a peak at around 0.35 m m /h r in the histogram , when the MEAN is th e Bayesian estim ator (as Figure 5.9 shown). However, th e algorithm is still able to retrieve reasonably for th e tru e intensities between 0.2 and 7 m m /h r. The characteristic th a t there is a smaller probability a t higher rain rates obviously limits th e ability of th e algorithm in retrieving heavy precipitation. Note th e cases th a t th e R R is above 15 m m /hr, th e averaged rain intensities of those histogram s are around 6, 8.5, 14, 33 m m /h r for th e range of [15, 30], [30, 50], [50, 75], and [75, 100] m m /h r, respectively, indicating a significantly low bias in the retrieval. In addition, th e histogram s are highly skewed to the right, encompassing a much lower proportion of true d a ta points th a n th a t of experim ent R0. Similar results are obtained in th e MLE estim ates for th e R1 experim ent, and the change of th e histogram over th e higher rain rates is emphasized here (as figure 5.10 shown). Like experim ent R0, the MLE estim ates also reveal two distinct regimes in the histogram , b u t it is only shown in th e case of extrem ely heavy precipitation. The signal of th e one associated w ith a lower rain rate is much stronger in R1 th an R0. It m eans th a t the algorithm has few chances to generate from interm ediate to extrem ely heavy rain intensities, as indicated by th e zero proportion of d a ta points in th e range of [30, 50] and [50, 75] m m /h r. T he prior density function param eterized from observed rain rates is tested in th e R2 experim ent. Com pared to R0, th e prior probability density function of R2 has a less ’spiky’ distribution w ith a less heavy tail (see figure 5.8 (a)). In other words, th e probability density function is relatively small a t very light rain rates (between 0.1 to 1 m m /h r) and a t large rain intensity (above 25 m m /h r), while being larger for m oderate precipitation (from 1 to 25 m m /hr). Therefore, similar to R l, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 R R =[ 0 .1 , 0 .2 ] 430957 - RR = [ 1.0, 2.0 MLE 4 2 .0 ] 670027 - MLE R R = [1 5 .0 , 3 0 .0 ] 1.0 59.9% 34.8% MLE ( 3.02, .1.07) .17, 0.50) 0.24, 0.11) 222467 - • 0 .0 % 0.8 3 e 0.6 E o O' o 0.4 X 0.5 1 0 0.2 0.0 0.0 0.1 1.0 0.1 10.0 R R = [ 0 .2 , 0 .4 ] 1.0 0.1 10.0 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) MLE 562461 RR = [ 2.0 , 4 .0 ] 597394 MLE 4 ( 1.91, 0.69) ( 0 .3 2 , 0 .1 5 ) 33.2% 10.0 100.0 RR = [3 0 .0 , 5 0 .0 ] 98415 - MLE 0 .3 0 1---------------------------------------------( 3 .9 1 , 1 .3 4 ) 46.7% . 0.8 3 1.0 R e tr ie v e d RR ( m m / h r ) X 0.4 1 0.2 0 0.1 1.0 10.0 0.0 100.0 0.1 R R = [ 0 .4 , 0 .6 ] 1.0 10.0 20 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) 374 6 4 1 - MLE RR = [ 4 .0 , 7 .0 ] 40 60 80 100 R e tr ie v e d RR ( m m / h r ) 395442 R R = [5 0 .0 , 7 5 .0 ] MLE 4 49896 - MLE 0.25 2.34, 0.78) 0.46, 0.21) 39.7% ( 4.97, 6.60) : 0 .0% 0.20 3 6 0.1 5 o' 2 0 .1 0 1 0.05 0 0.1 1.0 10.0 1.0 R e tr ie v e d RR ( m m / h r ) 0] 10.0 0 100.0 R e tr ie v e d RR ( m m / h r ) 498245 - MLE R R =[ 7.0 , 1 5.0 ] 391 9 1 1 20 40 60 80 100 R e tr ie v e d RR ( m m / h r ) - MLE R R = [7 5 .0 ,1 0 0 .0 ] 24139 - MLE 2.0 ( 0.67, 0.30) (14.79,28.93; 2.5>B, 0.91) 49.2% • 0.1% . 0.8 1.5 0.12 0.10 E o 0.6 0.08 O' o f x 0.4 •■f 0.06 0.04 0.5 0 .2 0.02 0.0 0.00 0.0 0.1 1.0 10.0 R e tr ie v e d RR ( m m / h r ) 1.0 10.0 R e tr ie v e d RR ( m m / h r ) 20 40 60 80 R e tr ie v e d RR ( m m / h r ) Figure 5.10. Same as figure 5.6, bu t for experim ent R l. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 th e MEAN estim ates show th a t the algorithm overestim ates intensity at the true range of [0.1, 0.4], and considerably underestim ates heavy rain intensities (figures not shown). However, the skewness of th e histogram under heavy precipitation is not as strong as R l, thus some good retrievals are yielded for th e high R R of rain ranges. T he perform ance in the interm ediate rain rates is not affected much. The MLE estim ates of experim ent R2 reveal a similarity to R l, having th e feature of two distinct regimes, b u t the probability of the peak w ith a lower rain ra te is smaller th a n R l and larger th a n RO (figure not shown). E xperim ent R3 yields very similar retrieval to the experim ent R2 in either MEAN or MLE histogram s, showing th a t the effects of th e small changes in th e param eter ization of th e prior distribution due to n atu ral variations on th e Bayesian algorithm are negligible. A non-inform ative prior distribution is used for the experim ent R4, which assigns equal weight (0.01 for the experim ent) to all values over the param eter space. Based on the Bayes theorem , one can know th a t in this experim ent th e posterior probability density is proportional to the likelihood represented in the d a ta only, so th e retrieval is dom inated by the behavior of th e physical model. Note th a t th e use of th e uniform prior p d f leads to equivalent results in th e classical approach. Figure 5.11 reveals th a t uniform prior knowledge yields much higher retrieved rain rates in light and m oderate rain situations. Com pared to th e retrieval of RO experim ent, th e entire histogram of R4 retrieval shifts to th e larger rain range by 7 to 15 m m /h r when the tru e rain rates are greater th an 2 m m /h r. For th e true range of heavier rain rates, experim ent R4 captures more d a ta points in th e correct range. Since th e experim ent only contains th e d a ta information, results here suggest th a t th e P vector is sufficient to reflect th e signal of intensity under th e heavy precipitating systems. 5.3.3 S en sitiv ity to th e conditional distribu tion The m otivation of the section is to advance the understanding of th e sensitivity of th e Bayesian retrieval to the specification of the conditional likelihood th a t provides statistical inform ation about th e physical relationship between surface rainfall and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 RR = [ 0 .1 , 0 .2 ] 430957 - R R = [ 1.0, MEAN 2 .0 ] 670027 - R R = [1 5 .0 , 3 0 .0 ] MEAN 222467 - MEAN 0.8 5 F4 R4 ( 0.54, 0.69) ( 6.46, 4.48) 57.7% 10.3% 13.6% . 4 (2 5 .i:5 ,i:i.1 5 ) 0 .6 E o O' g* 0.4 2 x 0.2 1 0 1.0 10.0 0 .0 100.0 0.1 R R = [ 0 .2 , 0 .4 ] 1.0 10.0 100.0 0.1 562461 - R R = [ 2 .0 , MEAN 4 .0 ] 597394 - RR = [3 0 .0 , 5 0 .0 ] MEAN 0.4 3.0 R4 R4 (12.81, 7.69) MEAN (38.49,13.80) . %: 48 9 0.1 5 0.3 2.0 E o E o ? 100.0 98415 - 5.3% : 26.2% ‘ H is to g r a m 10.0 0.20 R4 ( 0.88, 1.06) 2.5 1.0 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) g 4 0.1 0 0.2 x x 0.05 0.1 0.5 0.00 0.0 0.0 0.1 1.0 0.1 10.0 RR = [ 0 .4 , 0 .6 ] 1.0 10.0 100.0 0 20 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) 374641 - MEAN R R = [ 4 .0 , 7 .0 ] 395442 - MEAN R R = [5 0 .0 , 7 5 .0 ] 0.4 2.0 [-------------------------------------------- R4 ( 1.56, 1.65) (17.52, 9.61): 0 .14 5.8% : 0 .12 13.0% 40 60 100 80 R e tr ie v e d RR ( m m / h r ) 49896 - MEAN R4 0.3 0 .10 E H istogram o 0 .08 S’ 0.2 i 1 0.06 0.04 0.1 0.02 0 .0 0 0.0 0.1 1.0 10.0 100.0 0.1 R R = [ 0 .6 , 1 .0 ] 498245 - 1.0 10.0 100.0 0 20 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) MEAN R R = [ 7.0 , 1 5 .0 ] 0.30 r 391911 40 60 100 80 R e tr ie v e d RR ( m m / h r ) - R R = [7 5 .0 ,1 0 0 .0 ] MEAN 24139 - MEAN 0 .1 0 R4 R4 30.4% (77.3:0, 9.79) 63.9% 0.08 . Histogram 1.0 E 0.06 0.8 x 0.6 0.04 0.4 0.02 0.2 0.0 0.0 0 0.1 1.0 10.0 R e tr ie v e d RR ( m m / h r ) 0.1 1.0 10.0 R e tr ie v e d RR ( m m / h r ) 0 20 40 60 80 R e tr ie v e d RR ( m m / h r ) Figure 5.11. Same as figure 5.5, b u t for experiment R4. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 71 P10 vs. P19 1.0 P10 vs. P37 P19 vs. P37 SO SO SO 0.8 0.8 0 .6 r. 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.8 os 0 . 6 ^ 0.0 0.0 0.0 0.0 0.2 0.4 0 .6 P10 0.8 1.0 0.0 0.2 0.6 P19 0 .8 0.0 1.0 0.2 4 0.6 P10 0.8 1.0 2.0 so SO 0.4 M arginal p df (P 3 7 ) M arginal pdf (P 1 9) M arginal pdf (P 1 0) 6 5 0.4 SO 3 4 3 2 2 0.5 1 1 Of . . . i . . . ■ __ . . . . . . . 0 .0 0.2 0 .4 0.6 P10 0.8 n F — r-r-rT , . 1.0 0 .0 0.2 0 .6 P19 0.0 0.8 0 .0 1.0 •o* 0.4 0.2 0.0 0 .2 0.4 0.6 P10 0.8 0.2 1.0 0.8 0.4 0.4 0.2 0.2 0 .4 0.6 0.8 0 .0 1.0 0.2 0 .4 0.6 P10 0.8 M arginol pdf (P 3 7 ) M arginal p df (P 1 9 ) 4 2.0 S5 S5 S5 0.8 3 4 3 10 c a> o 2 2 0.5 1 1 nf 0.0 -— 0.2 0 .4 .... 0.6 0.8 i . . . . . . . . . 1.0 0.0 1.0 0.0 0.2 Marginal pdf (P 1 0 ) 6 0.6 P37 P10 vs. P37 0.8 0.0 0.0 0.4 S5 S5 0.8 5 . ■_ . P19 vs. P37 P10 VS. P19 S5 0 .0 0.4 0.2 0 .4 0.6 0.8 0.0 . 1.0 0.0 0.2 0.4 0.6 P37 0.8 Figure 5.12. Joint and m arginal probability distributions of th e physical model for experim ent SO (upper two rows) and S5 (bottom two rows). Contours are plotted for [0.05, 0.5, 1.0, 2.5, 5.0, 7.5, 10, 12.5, 15] in conditional pdfs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.0 72 R R = [ 0 .1 , 2.0r 0 .2 ] 86024 - R R = [ 1.0, MEAN 2 .0 ] so 0.94, 2.23) MEAN 133861 i.or--------------------( 1.95, : R R = [ 15.0, 3 0 .0 ] 0 .1 4 2.92): H is to g ra m E 0.6 SO MEAN (13.31, 5.93} 0.12 0.10 29.0% . 44198 - il: 24.5% : E o a* 0 . 0 8 0 1 0 .0 6 0 .0 4 0.02 0.00 0.1 1.0 10.0 0.1 100.0 R R = [ 0 .2 , 0 .4 ] 1.0 10.0 0.1 100.0 1.0 R R = [ 2 .0 , 0.5 E 1 1 2 8 0 7 -M E A N 2.0 [------------------------------------------------------- 4 .0 ] 119 7 2 8 - = [3 0 .0 , 5 0 .0 ] MEAN MEAN 34.8% 28.0% 2 1 .8 % 19853 - (29.19,16.09} ( 3.70, 3.65) ( 1.02, 2.33) 10.0 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) H istogram 0 .0 4 0.1 1.0 10.0 100.0 0.1 R R = [ 0 .4 , 0 .6 ] 1.0 10.0 100.0 0 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d R R ( m m / h r ) 74899 - MEAN R R = [ 4 .0 , 7 .0 ] 78879 - RR = [5 0 .0 , 7 5 .0 ] MEAN 0 .4 2.0 SO 0 .0 6 SO .15, 2.41) 20 40 60 80 R e tr ie v e d RR ( m m / h r ) 1 MEAN (56.211,17.40} 6.82, 3.98) 59.4% 28.9% 27.2% 9881 - 100 0 .3 E H istogram o ? 0.2 x 0 .5 0.0 0.0 0.1 1.0 10.0 100.0 0.1 R R = [ 0 .6 , 1 .0 ] 99679 - MEAN R R = [ 7.0 , 1 5 .0 ] 0.20 SO 2 .0 SO 1.0 10.0 100.0 20 40 60 80 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) ( 1.33, 2.49) 29.1% 78788 - RR = [7 5 .0 ,1 0 0 .0 ] MEAN 4780 - 100 MEAN 0 .0 6 SO 9.87, 3.42): • (71.30,12.60} 4.6.4% i 0 .0 5 7 5 .3 % : 0 .1 5 0 .0 4 o E ? 0 .0 3 x x 0.02 0.01 0.00 0 Histogrom E o g 0.10 0.5 0 .0 5 0.00 0.0 0.1 1.0 10.0 R e tr ie v e d RR ( m m / h r ) 100.0 0.1 1.0 10.0 R e tr ie v e d RR ( m m / h r ) 20 40 60 80 R e tr ie v e d RR ( m m / h r ) Figure 5.13. Same as figure 5.5, b u t for experiment SO. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 73 R R =[ 1.0, 2.0 2 .0 ] R R * [1 5 .0 , 3 0 .0 ] MLE 133861 44198 - MLE 2 .0 SO ( 8.71, 7.76) ( 0.35, 1.38): SO ( 0.05, 0.68 1 2 .6 % 0.0 % 0 .0% H is to g r a m G o X 0.5 0 .5 0.0 0.0 0.1 1.0 1.0 10.0 R e tr ie v e d RR ( m m / h r ) RR = [ 0 .2 , 0 .4 ] 0.1 10.0 1.0 R R =[ 2.0 , 1 12 8 0 7 - M L E 4 .0 ] RR = [30.Q , 5 0 .0 ] 119728 - M L E 19853 - MLE (15.80,19.54) 1.42, 2.49) ( 0.06, 0.6 10.0 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) 26.6% Histogram 2 2 .8 % 0.1 1.0 0.1 10.0 1.0 10.0 100.0 20 40 60 80 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) R R = [ 4 .0 , 7 .0 ] 7 88 7 9 - R R = [5 0 .0 , 7 5 .0 ] MLE 9881 - MLE 2.0 2 .0 SO SO 0.09, 0.76)! (38.85,30.71) 3.96, 3.39) 0.0% : 41.5% 37.7% Histogram E o cn o « X 0 .5 0 .5 0.0 0.0 0.1 1.0 ).1 10.0 R R =[ 0.6 , 1 .0 ] 99679 1.0 10.0 20 40 60 80 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) MLE = [ 7.0 , 1 5 .0 ] 78788 R R = [7 5 .0 ,1 0 0 .0 ] MLE 2 .0 4780 - 100 MLE 0 .1 2 SO 5, 3.66) ( 0.13, 0.9 SO 0.10 40.9% 0.07 (63.85,30.05) ■31.9% : Histogrom 1 .5 E o 1.0 0 .0 8 g ' 0 .0 6 i 0 .0 4 0.5 0.02 0.0 0.1 1.0 10.0 R e tr ie v e d RR ( m m / h r ) 100.0 0.00 0.1 1.0 10.0 R e tr ie v e d RR ( m m / h r ) 0 20 40 60 80 R e tr ie v e d RR ( m m / h r ) Figure 5.14. Same as figure 5.13, b u t using maximum likelihood estim ates as single-pixel retrieved rain rates. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 74 RR = [3 0 .0 , 5 0 .0 ] 19853 - R R = [3 0 .0 , 5 0 .0 ] MEAN 19853 - MLE 0 .0 6 ( 1 4 .7 4 , 4 . ( 1 6 .9 7 , 5 .7 2 ) 0 .0 5 0 .0 4 E S ' 0 .0 3 i 0.02 0.01 0.00 0 100 20 40 60 80 R e tr ie v e d RR ( m m / h r ) R R = [5 0 .0 , 7 5 .0 ] 9881 - 20 40 60 80 R e tr ie v e d RR ( m m / h r ) R R = [5 0 .0 , 7 5 .0 ] MEAN 0.20 0 .0 6 S5 (25.05, 8.11) 0 .0 5 • 0 .4 % S5 100 9881 - MLE (21.1D, 6.44) ■ 0 .0 % ■ 0 .1 5 0 .0 4 E o E 0 ^ w 1 g1 o.io 0 .0 3 w X 0.02 0 .0 5 0.01 0.00 0 20 R R = [7 5 .0 ,1 0 0 .0 ] 4780 - 0.00 0 100 80 40 60 R e tr ie v e d RR ( m m / h r ) S5 0.12 0.10 (31.55, 9.81 0 .0 5 • 60 80 40 R e tr ie v e d RR ( m m / h r ) RR = [7 5 .0 ,1 0 0 .0 ] MEAN 0 .0 6 0 . 0% 0 .0 4 100 20 S5 4780 ( 2 5 .9 4 , • MLE 7 .7 1 ) 0 .0% 0 .0 8 E E o o S ’ 0 .0 3 S ' 0 .0 6 x I 0.02 0 .0 4 0.01 0.00 0 20 40 60 80 R e tr ie v e d RR ( m m / h r ) 100 0.02 0.00 0 20 40 60 80 R e tr ie v e d RR ( m m / h r ) 100 Figure 5.15. Similar to figure 5.5, b u t for MEAN (left panel) and MLE (right panel) estim ates of Experim ent S5 at ranges of heavy tru e surface rainfall (greater th a n 30 m m /h r). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 microwave signature. R esults from simple model (S5 to S8) th a t specifies physical model via a covariance m atrix will be given first, followed by those from R5 and R6 whose training d ataset is generated from a more complete set of conditional distributions (equation 4.13). To b etter illustrate th e analysis, th e m ultichannel relationships of th e microwave observation vector described by th e covariance m atrix of SO experim ent is shown in the first two rows of figure 5.12, and th e corresponding retrieved MEAN and M LE rain rate are depicted in figures 5.13 and 5.14. Note when using M LE as th e retrieved rain rate, SO experim ent reveals th a t th e algorithm is not able to yield satisfactory retrieval for the very light rain ra te (less th a n 2 m m /h r). It is due to th e com bination of the simplified covariance m atrix and prior distribution, which tends to favor the very small surface rainfall rate. In experim ent S5, th e covariance m atrix th a t approxim ates the physical relation ship is a diagonal m atrix, implying no correlation between response vector elements. The joint pdfs (as shown in th e b ottom two rows of the figure 5.12) of S5 exper iment reflect the assum ption of uncorrelation in th e elliptic contours. There are slight changes in th e m arginal pdfs of all three channels in experim ent S5. The tongue shape directed tow ard th e low P values is due to saturation, resulting from the param eterization of /q in equation (5.2). For th e MEAN estim ates, substantial differences occur between experim ent S5 and SO as th e true rain intensity is greater th an 30 m m /h r (as figures 5.13 and 5.15 shown). The algorithm significantly under estim ates the tru e heavy rain rates, and only few percentages of retrievals are in th e correct range. Under extrem ely heavy precipitating systems, th e m axim um retrieved rain intensity is only up to 65 m m /h r or so. Similar behaviors are seen in th e M LE retrieved surface rainfall (figure 5.14 and th e right panel of figure 5.15) as well. The effect of assum ptions in diagonal elements of covariances of th e response vector on the Bayesian retrieval is shown by the results of experim ents from S6 to S8. As expected in th e experim ent S6, the double in the CYi element of covariance m atrix produces a broader spectrum for th e m arginal probability distribution of P w Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 Morginol p df (P 1 0 ) P 10 vs. P37 P10 vs. P19 | i i i | -i—i—r ■f . * . f r 6 S6 5 S6 0.8 4 3 0 .4 0 .4 0.2 0.2 1 0.0 0.0 nE 0 .0 0 .2 0 .4 P10 0 .6 P10 0 .8 0 .0 1 .0 2 0 .2 P19 VS. 0 .4 0 .6 P10 0 .8 1.0 0 .0 . . . . . 0 .2 1.0 3 0.8 O' 0 .8 4 S7 0.8 0 .6 M arginal p df (P 1 9 ) P19 vs. P37 S7 0 .4 O' 2 Q. 0 .4 0 .4 0.2 0.2 1 0.0 0.0 0 .0 0 .2 0 .4 0 .6 P10 0 .8 1.0 0 .0 0 .2 P19 vs. P37 P37 1.0 0 .4 0 .6 P19 0 .8 1.0 0 .0 0 .2 0.8 0.8 0.6 r-. 0 . 6 0 .4 0 .4 0.2 0.2 0 .6 0 .8 1.0 Marginal pdf (P 3 7 ) P 10 vs. P37 2.0 S8 S8 0 .4 S8 vc Q 0 .5 0.0 0 .0 0.0 0.0 0 .2 0 .4 0 .6 0 .8 1 .0 0 .0 0 .2 0 .4 0 .6 0 .8 1.0 0 .0 0 .2 0 .4 0.6 0.8 Figure 5.16. Conditional and m arginal pdfs of experim ents S6 (first row), S7 (second row), and S8 (the bottom ). Note th a t different channels are shown here for each experim ent to highlight th e change due to th e difference in th e covariance m atrix. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.0 77 RR = [5 0 .0 , 7 5 .0 ] 0.06 ............................... ..... ■ S6 0.05 i RR = [5 0 .0 , 7 5 .0 ] 0.20 9881 - MEAN i i • 1 1 (50.1D, 18.13) 1 i ■ 57.0% 9881 - MLE (25.83,27.B9) 26.9% * 0.15 0.04 £ o S’ 0.03 w E g“ 0.10 X X 0.02 0.05 0.01 0.00 0.00 20 40 60 80 R e tr ie v e d RR ( m m / h r ) R R = [7 5 .0 ,1 0 0 .0 ] 4780 - 20 40 60 80 R e tr ie v e d RR ( m m / h r ) 100 MEAN RR = [7 5 .0 ,1 0 0 .0 ] 0.12 0 .0 6 S6 0 .0 5 0.10 0 .0 4 0 .0 8 E S6 4780 - 100 MLE ( 3 7 . 1fc,32.85) •11.5% ' E o o S’ 0 . 0 3 g ’ 0 .0 6 i x 0.02 0.01 0 0 .0 4 0.02 20 40 60 80 R e tr ie v e d RR ( m m / h r ) 100 0.00 0 20 40 60 80 R e tr ie v e d RR ( m m / h r ) 100 Figure 5.17. Similar to figure 5.5, b u t for th e MEAN (left panels) and MLE (right panels) estim ates of experim ent S6 a t heavy precipitation rate ranges (greater th a n 50 m m /h r). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 and a less dogmatic joint pdf of P 10 and P 19, as figure 5.16 shows in th e first row. The m arginal properties of P i 9 and P 3 7 rem ained th e same (figures not shown). The effect of th e increase in the uncertainty of P 10 value is particularly dem onstrated in th e retrieval for the heavy rain rates, shown in figure 5.17. T he mean of the histogram of the retrieved rain intensity in S 6 is around 61 m m /h r a t th e RR range of [75, 100] m m /h r and only 11.4% of pixels are in the correct range, while control ru n shows a mean of 71 m m /h r and contains 46% d a ta consistent w ith th e training data. Similarly, the histogram for M LE estim ates shifts tow ards smaller rain rates as well for th e case of heavy precipitation in experiment S 6 , implying a considerable retrieval bias for the large rain rates. T he reason for the low bias is th a t P 10 is a very im portant factor in the rain ra te retrieval under th e condition associated w ith heavy rain, while P 19 and P 3 7 might reach satu ratio n at th a t point. In other words, P 10 has a broader and higher dynamic range for th e rain rate retrieval. Once th e uncertainty in P 10 increases, the inform ation from th e channel becomes more vague, and in turn, a smaller rain rate is suggested by th e algorithm since th e prior distribution favors th e relatively smaller rainfall intensities. Com paring the joint and m arginal pdfs of S7 w ith the control ru n (SO) at three channels, one can find similar broadness in th e marginal p df of P 19 in experim ent S7 (see figure 5.12 and the second row of figure 5.16). Surprisingly, in experim ent S 7, the double in the uncertainty of the P 19 does not affect th e retrieval significantly in either MEAN or MLE estim ations (figures not shown). The result m ight be a ttrib u ted to fact th a t P 19 values represent inform ation of interm ediate rain intensities compared to the other two channels. W hile th e correlations of the variables (i.e., non-diagonal elements in the covariance m atrix) are well-defined and unchanged, th e combination of P values still provide enough precise m ultichannel relationships in this retrieval algorithm , so there are not significant changes shown in th e corresponding retrieval. For experim ent S8 , the corresponding probability distributions a t three channels are also illustrated in figure 5.16. N ote th a t the m arginal d istribution for P 3 7 is more like a bell shape th an th e control run, showing th a t th e ’b u m p ’ at around 0.15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 RR = [ 0.1 , 0 .2 ] 86024 - MEAN R R =[ 1.0, 2 .0 2 .0 ] S8 ( 1.66, 1.82) S8 0 .0% 133861 - RR = [1 5.0 , 3 0 .0 ] MEAN ( 2.18, 2.19); 0 .1 4 45.3% . 0 .1 2 0.8 S8 44198 - MEAN ( 8.65, 4.50): 0 .1 0 F E H is to g ra E 0.6 0 .0 8 o V) x 0 .4 0 .0 6 0 .0 4 0 .5 0.2 0 .0 2 0.0 0.0 0.1 1.0 10.0 100.0 0.1 0 .4 ] 10.0 0.00 100.0 0.1 1.0 112 8 0 7 R R = [ 2 .0 , MEAN 2.0 0 .5 58 0 .0 % 4 .0 ] S8 119728 - M E A N 3 = [3 0 .0 , 5 0 .0 ] 100.0 19853 - MEAN (19.67,10.36) ( 2.82, 2.50) 13.6% 29.5% . 0 .4 10.0 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) R R = [ 0 .2 , 1.0 Histogrom E 0 .3 o O' o x 0.2 0 .5 0.1 0.0 0.0 0.1 1.0 10.0 0.1 100.0 RR = [ 0.4 , 0 .6 ] 1.0 10.0 100.0 20 40 60 80 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) 74899 - R R = [ 4.0 , MEAN 2.0 7 .0 ] 78879 - =[5 0 .0 , 7 5 .0 ] MEAN 0 .0 6 0 .4 F S8 3.84, 2.79) S8 0.9% •21.0% : 0 .0 5 22 .2% MEAN (38.07,14.65) 0 .0 4 Histogram E o g 4 0 .0 3 X 0.02 0 .5 0.01 0.0 0.00 0.1 1.0 10.0 0.1 100.0 R e tr ie v e d RR ( m m / h r ) R R = [ 0.6 , 1 .0 ] 10.0 100.0 0 99679 - MEAN R R = [ 7.0 , 1 5.0 ] 0.20 ( 1.90, 1.99) 78788 - 40 60 60 R e tr ie v e d RR ( m m / h r ) RR = [7 5 .0 ,1 0 0 .0 ] MEAN O.Ub S8 29.6% 4780 - MEAN (52.(31,14.51) 0 .0 5 • 24.8% : 1 .5 S8 5.27, 2.98)! 100 20 R e tr ie v e d RR ( m m / h r ) 2.0 58 1.0 : 3.3% ’ 0 .1 5 0 .0 4 E E Histogram o ? o g 0.10 0 .0 3 I x 0 .0 2 0 .5 0 .0 5 0.0 0.00 0.01 0.1 1.0 10.0 R e tr ie v e d RR ( m m / h r ) 100.0 0.00 0.1 1.0 10.0 R e tr ie v e d RR ( m m / h r ) 0 20 40 60 80 R e tr ie v e d RR ( m m / h r ) Figure 5.18. Same as figure 5.5, b u t for experim ent S8. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 80 of P 3 7 in SO experiment due to high rain intensities has been sm oothed out w ith increasing uncertainty in P 3 7 values. Unlike the results of th e experim ents S 6 and S7, the increase in the C 3 3 element of th e covariance m atrix has notable im pacts on th e retrieval over all ranges of the rain rates (see figure 5.18). For th e light rain rates (less th a n 1.0 m m /hr), using MEAN of th e posterior distribution in S 8 overestim ate th e rain intensity by around 0.7 m m /h r in general. Com pared w ith SO, the additional positive bias decreases when the tru e rain rates are in the range of [1 , 2 ] m m /h r. T he algorithm tends to underestim ate th e intensity when th e tru e rain rate range increases above 2 m m /hr. T he d a ta proportion in the true range is reduced to almost half, being only 3% for the case of extrem ely heavy precipitation, while SO has around 45% d a ta in the correct range. The m ode of th e histogram generally moves to the smaller intensity by about 20 m m /h r due to th e double uncertainty in P 3 7 . The MLE estim ates from S8 show a greater sensitivity to th e uncertainty of P 3 7 th a n th e MEAN. Figure 5.19 indicates th a t th e doubling in th e variations of P 37 results in a failure of the algorithm under th e situations of m oderate rain rates (2 to 15 m m /h r), while SO experim ent contains at least 20% d a ta points in the correct range. The algorithm of S 8 produces m uch lower rain rates com pared to the true training data. For the heavy precipitation case, th e d ata proportion in the correct range is only up to 9%, which is one-fifth of th a t SO can retrieve successfully. W ith increasing uncertainty of P 37, th e variable provides less precise information, implying th a t the probability is more com patible a t larger P 3 7 value w ith respect to the value where the peak locates, if considering a condition at a given rain rate. In addition to th a t P 3 7 m ainly provides inform ation about light rain, and th e prior distribution always favors th e smaller rain rates, th e algorithm obtains much smaller rain rates over all ranges. Specifications of the physical model in b o th RO and R5 are derived from the same radar-radiative simulation d a ta using different approaches, and they represent very similar relationships between P values and surface rain rates. T he minor difference is th a t the mode of the m arginal p d f of P 19 in R5 is located a t a slightly larger Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 RR = [1 5 .0 , 3 0 .0 ] 2.0 2.0 S8 H is to g ra m I X 0 .5 0 .5 0.0 1 0.0 0 .0 1 0 0 .0 1.0 R R = [ 0.2 , 0 .4 ] 1 0 .0 1.0 112 8 0 7 - MLE R R = [ 2.0 , 4 .0 ] 100.0 119728 - M L E R R = [3 0 .0 , 5 0 .0 ] 19853 - MLE 0 .4 1 S8 ( 0 .0 3 , 0 .3 9 ) 10.0 R e tr ie v e d RR ( m m / h r ) 2.0 S8 0.1 1 0 0 .0 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) 2.0 3 .0 % o O' o 1.0 • E o o> 0.0 MLE ( 0 . 9 :1 , 3 . 7 2 ) 0 .0 % E 0.1 S8 ( 0 .0 4 , 0 .5 4 ) 0 .0 % 0 .5 44198 - 2 .0 S8 0 .0 3 , 0 .4 2 ) ( 0 .0 4 , 0 .5 6 ) 0 .0% 8 .3 4 ,1 2 .4 8 ) 0 .0 % E Histogram o X 0 .5 0 .5 0.0 0.0 0.1 1.0 1 0.0 0.1 1 0 0 .0 R e tr ie v e d RR ( m m / h r ) R R = [ 0 .4 , 0 .6 ] 1.0 10.0 100.0 74899 - MLE R R = [ 4 .0 , 2 .0 7 .0 ] 78879 - MLE S8 0 .0 3 , 0 .3 8 ) RR = [5 0 .0 , 7 5 .0 ] 9881 - 100 MLE 0.20 2 .0 S8 20 40 60 80 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) S8 0 .0 7 , 0 .8 0 ) 0 .0 % ( 2 5 .9 5 ,2 0 .7 8 ) : : 9 .3 % 0 .0% : 0 .1 5 Histogram 1 .5 S' 0.10 x 0 .5 0 .0 5 0 .5 0.0 0.00 0.0 0.1 1.0 10.0 ).1 R e tr ie v e d RR ( m m / h r ) 1.0 10.0 100.0 0 R e tr ie v e d RR ( m m / h r ) R R = [ 7.0 , 1 5 .0 ] 2.0 78788 - MLE 20 40 60 80 R e tr ie v e d RR ( m m / h r ) R R = [7 5 .0 ,1 0 0 .0 ] 4 78 0 100 MLE 2.0 S8 S8 ( 0 .0 3 , 0 .4 8 ) 0 .1 2 , 0 .0 % : ( 4 2 .8 0 , 2 4 .2 7 V 1 .1 6 ) 0 .7 % E Histogram o S' 1.0 X 0 .5 0 .5 0 .0 0.0 0.1 1.0 1 0 .0 R e tr ie v e d RR ( m m / h r ) 10 0 .0 1 .0 1 0.0 R e tr ie v e d RR ( m m / h r ) 20 40 60 80 R e tr ie v e d RR ( m m / h r ) Figure 5.19. Same as figure 5.18, b u t for M LE estim ates. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 P19 vs. P37 P10 vs. P19 Rain ra te : th e o re tica l 2.0 R5 R5 R5 0.8 0.8 a> 0 .6 c QJ ^ 0.6 a. 0 0 .4 0 .4 0.2 0.2 0 .5 0.0 0.0 0.0 0 5 10 20 15 R oin R o t e ( m m / h r ) 0 .0 0 .2 0 .6 P10 0 .8 0 .0 1 .0 0 .2 2 .0 4 R5 R5 0 .4 0 .6 0 .8 1.0 M arginal p df (P 3 7 ) M arginal p df (P 1 9 ) M arginal pdf (P 1 0 ) 6 5 0 .4 R5 3 D e n s ity 4 3 14) Q 2 ‘I av 1.0 2 1 0 .5 1 n E. 0 .0 ■< 0 .2 0 .4 0 .6 ...... t . 0 .8 1 .0 OF 0 .0 . - ‘r T 'T ■ 0 .2 0.0 0 .4 0 .6 0 .8 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 Figure 5.20. Joint and m arginal probability distributions of th e physical model for experim ent R5. The joint pdfs are plotted for [0.05, 1, 5, 7, 10 , 12 ]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.0 83 value th a n th a t of the control run (as figure 5.20 shown). Since results from MEAN and M LE estim ates are similar, only outcomes for MEAN will be m entioned here (figures not shown). The experiment reveals th a t the different param eterizations of the conditional likelihoods in R5 still produce very consistent retrievals, except a slight increase in the intensity over th e very light rain rate. It is partially due to the aforem entioned shift of th e peak in the m arginal p df of P 19. However, the difference is negligible. R esults of experim ent R5 suggest th a t th e retrieval is not so sensitive to th e different ways of specifying the conditional distributions as long as th e relationships represented by the physical model are not far from reality. The adequacy of using a simplified conditional probability distribution to rep resent th e more com plicated behavior of d a ta is evaluated by experim ent R6. A num ber of notable changes in the retrieval histogram are sum m arized in figure 5.21. In general, th e linear physical relationships yield much larger rain rates, and the variations of the retrieval histogram are almost twice th a t of th e control experim ent RO. For very light rain rates (less th a n 0.2 m m /h r), R6 only produces 0.1% d ata points in th e correct range, while th e control ru n produce alm ost half of th e d a ta in the correct range. T he positive bias becomes even more obvious when the true range is in m oderate intensities. Under the condition th a t th e tru e rain rates are only 2 to 15 m m /h r, some retrieval from R6 are even greater th a n 50 m m /h r. The large variations in the retrieved intensity at a certain true rain ra te range is seen in the case of heavy precipitation as well. Retrieval from MLE estim ations behaves similarly to th e MEAN (figures not shown). It is not appropriate to draw a conclusion th a t a simplified physical model is definitely not suitable for the use of a Bayesian algorithm, since th ere are considerable differences in th e physical relationships between th e training d ataset and th e simple model for th e experim ent R6. The retrieval bias might decrease if th e simple physical model is improved to be more close to th e training dataset. However, in fact, the param eters used to build th e simple model are estim ated from th e sam e radarradiative sim ulations. It implies th a t some behaviors of the d a ta can not be explained Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 R R =[ 1.0, 2 .0 ] MEAN 670027 R R = [1 5 .0 , 5 0 .0 ] o.8 r 0 .3 0 • •" r ( 3.99, 3.94) 0.51, 0.29) 0.1% 222467 - "• • .......... • .................... R6 0 .2 5 1 8 .9 % MEAN 1 ( 2 3 .2 4 ,1 :9 .5 3 ) 2 2 .3 % : 0 .2 0 E E o o I t ° - 15 i X 0 .1 0 0 .0 5 0 .0 0 0.1 1 .0 0.1 1 0 .0 0 .4 ] 562461 1 0 .0 1.0 1 0 0 .0 - RR = [ 2 .0 , MEAN 4 .0 ] 597394 - MEAN R6 R R = [3 0 .0 , 5 0 .0 ] ( 8.52, 7.23) H is to g ra m E o a S’ 0.2 g 0.10 x x 0.1 0 .0 5 0.0 0.00 0.1 100.0 0 .6 ] 1.0 10.0 100.0 0 20 40 80 60 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) R R = [ 0 .4 , 374 6 4 1 - : 0 .1 5 E 10.0 MEAN 27.3% 0 .3 1.0 98415 - (39.97,22.71) R6 11.9% 2 8 .5 % 0.1 1 0 0 .0 0.20 0 .4 ( 0.67, 0.50) 1 0 .0 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) R R = [ 0 .2 , 1.0 RR = [ 4.0 , MEAN 2.0 7 .0 ] 395442 - MEAN RR = [5 0 .0 , 7 5 .0 ] 49896 - 100 MEAN 0 .4 R6 0.99, 0.90) R6 0 .1 4 (11.99,10.39) • 24.8% 11.8%: R6 0.12 0 .3 H istogram 0.10 E E o o ? 1.0 ct> 0 .2 x x 0.5 0.1 0.0 0.0 0 .0 8 0 .0 6 0 .0 4 0.02 1.0 0.1 0.00 0.1 10 .0 R e tr ie v e d RR ( m m / h r ) R R = [ 0 .6 , 1 .0 ] 498245 1.0 10.0 100.0 MEAN R R = [ 7.0, 1 5 .0 ] 391911 - MEAN 0 .3 0 R6 0 20 40 60 80 R e tr ie v e d RR ( m m / h r ) R e tr ie v e d RR ( m m / h r ) ( 1.68, 1.72)' 22.8% . R R = [7 5 .0 ,1 0 0 .0 ] o .io r R6 : (81.95,13.67) : 50.2% ' Histogram 0 .2 0 E E 0 .0 6 o 0.8 Jr °-15 w 0.6 •-f 0 . 0 4 X 0 .1 0 0 .4 0 .0 5 0.2 0.0 0.00 0.1 1.0 10 .0 R e tr ie v e d RR ( m m / h r ) 100.0 1.0 MEAN ' (15.00,14.04) 0 .2 5 24139 - 100 1 0 .0 R e tr ie v e d RR ( m m / h r ) 20 40 60 80 R e tr ie v e d RR ( m m / h r ) Figure 5.21. Same as figure 5.5, bu t for experim ent R6. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 by a simplified physical model if it is approached by assum ptions of fixed variances or a simple covariance m atrix, even though th e m atrix is calculated from th e same data. As shown in experiment R6, if th e physical model cannot show th e multichannel relationships to a certain degree, th ere might be significant errors and bias in the retrieval product. 5.4 Sum m ary Results from the above sensitivity tests have revealed th e retrieval ability of the Bayesian algorithm over various rain rate ranges when th e prior and conditional likelihoods are bo th idealized, although th e single-pixel retrieved rain rate might be associated w ith a bias due to th e inherent uncertainty in th e physical relationship and th e interpretation using MEAN and MLE. The resulting posterior probability might have m ultiple peaks, which could result in a significant difference in the re trieved surface rainfall when th e observation vector only has a slight change. This deficiency might be improved via introducing more observation variables to offer more additional inform ation to reduce the am biguity of relation of th e microwave observations to rain rate. The MEAN tends to provide small values of to ta l bias and b e tte r retrieval in the m oderate rain rate range, while th e MLE has b etter retrieval ability in the range of very light rain rate (less th a n 2 m m /h r) and extreme heavy intensity (greater th an 50 m m /h r). In addition, the specifications of both th e prior and th e conditional pdfs are crucial to the posterior distribution. However, once th e prior distribution and th e physical model represent adequate inform ation about th e tru e d ata, the Bayesian algorithm is not so sensitive to either th e very small variation in th e param eters of th e prior distribution or the way to approxim ate th e physical relationship between microwave signature and rain rate. Therefore, when th e n a tu ra l variations of the prior rain ra te distribution and th e imperfectness of th e physical model exist, the Bayesian retrieval is still meaningful, b u t some degree of bias m ight exist. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 6. B E N C H M A R K A LG O R ITH M D E SC R IP T IO N S A num ber of current and new retrieval algorithm s are utilized and developed in the study to provide inter-comparisons for th e purpose of validations. D etailed descriptions for each algorithm will be given in th e chapter. 6.1 G PR O F The G oddard Profiling algorithm (G PR O F), as m entioned in chapter 1, is a sim plified Bayesian retrieval algorithm. A database is introduced in the algorithm to represent the presum ed probability distribution of cloud profiles; the possible cloud profile whose microwave signature is close to th e satellite observations are selected from the database; and then an averaged cloud and precipitation structure for each pixel is obtained, based on the relative occurrences of each candidate profile in the database. Details about the formulations and calculations are referred to equa tions (1.2) and (1.3). Since th e database plays a m ajor role in G PR O F (Kummerow et a l, 1996, 2000; L ’Ecuyer and Stephens, 2002), it is worthwhile to give some de scriptions here about th e old and new database of G PR O F. We will com pare our retrieved rain ra te w ith G PR O F using b o th databases in chapter 7. The old database of G PR O F for the m arine environm ent included three differ ent precipitation systems and was mainly based upon numerical model sim ulations of the G oddard Cumulus Ensemble Model (G CEM ), th e University of W isconsin Nonhydrostatic M odeling System (UW-NMS), and 1-D Eddington radiative transfer model. Cases contained a tropical squall line in T O G A /C O A R E sim ulated on a 128 x 128 km dom ain w ith 1-km resolution; a subsequent squall line sim ulation on 384 x 384 km w ith 3-km resolution; a tropical cyclone on 3.3-km resolution w ith 205 x 205 grid points; and a thunderstorm complex observed during th e C ooperative Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 Huntsville Meteorological Experim ent (COHMEX) on 50 x 50-km dom ain w ith 1-km resolution (Olson et a l, 1999). Note th a t the GCEM and UW-NMS have different assum ptions of microphysics scheme, and various m ethods to deal w ith dynamics. The new database of G PR O F for th e oceanic pixels was obtained from th e simula tions of the Pennsylvania S tate University-National C enter for A tmospheric Research mesoscale model version 5 (MM5). It included th e following simulations: Hurricane Bonnie in 1991 (158 x 158 km inner dom ain w ith 1.33-km resolution); a tropical squall line system th a t occurred on 12 Septem ber, 1974 during GATE (140 x 140 km domain w ith 2-km resolution); and another tropical squall line on 22 February, 1993 in th e T O G A /C O A R E (140 x 140 grid points w ith 2-km resolution). Since the simulations were ru n from the same mesoscale model, all were based on th e G oddard explicit m oisture scheme. To highlight th e difference between the old and new G PR O F databases, th e rain rate distribution and its relation to the microwave signature are dem onstrated here. In G PR O F, th e response vector y representing microwave signature includes the attenuation index a t 10.65, 19.35, and 37.00 GHz ( P i o ,P i 9 , P 3 7 ), and th e scattering index at 85.5 GHz (Ass). The scattering index is proposed by P etty (1994a) and defined as S = P T Vfi + (1 - P )T C - Tv , (6.1) where the scattering index S is determ ined by th e atten u atio n index P , vertically po larized clear-sky/observed brightness tem p eratu re T ^ o /T y , and the lim iting bright ness tem perature of a non-scattering and optically thick cloud layer T q - Here, T c is assumed to be 273 K. High values of S represent significant scattering signals from ice particles. Figure 6.1 illustrates the normalized histogram of the rain ra te in th e old and new database. These two databases have similar probabilities while th e surface rain intensity is less th a n 3 m m /h r. However, it is obvious th a t the new d atab ase has more cloud profiles w ith m oderate rain rate (3—10 m m /h r) and fewer profiles w ith heavy precipitation (greater th an 20 m m /h r) th a n th e old database. T he largest Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 H istogram o f r a i n r a t e in t h e G P R O F d a t a b a s e 0 .0 5 Old d a t a b a s e 0 .0 4 New d a t a b a s e 0 .0 3 x 0.02 0.01 0.00 0 20 40 60 80 100 120 140 Rain r a t e ( m m / h r ) Figure 6.1. Normalized histogram s of surface rainfall of cloud profiles in cluded in th e old (solid curve) and new (dotted curve) G PR O F database. The to ta l num ber of cloud profiles are 11069 and 3097 for th e old and new database, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 rainfall rate in th e new database only reaches about 87 m m /h r, while 2.5% of the rain rate in the old database exceeds 88 m m /h r and some go up to around 140 m m /hr. There is no easy way to visualize a high-dimensional relationship by plotting. Since the plot of rain rate vs. S index is more scattered, it is h ard to identify the change between old and new database, and how sensitive th e retrieval is to the scattering index. Therefore, we only attem p t to dem onstrate th e significant difference in the 4-D relationship between rain rate and P values at 10.65, 19.35, and 37 GHZ channels. Figure 6.2 depicts th e 2-D contours of th e num ber of cloud profiles on the rain rate vs. P \ 9 domain, a t a given interval of P w and P 37. The distribution shows some key features of these two databases. First, the two databases have a similar pattern: the range of th e P i 9 value shifts to the left (decreasing P 1 9 ) while P 10 and P 3 7 b o th decrease. However, th e distribution of th e old database shows a larger coverage and more scatter th a n th e new database. In addition, it is noted th a t the cloud profiles clutter a t different locations, indicating th a t th e occurrence frequency of cloud files in 4-D space differs between the old and new database. For example, a t a given interval of 0.07—0.15 and 0.55—0.65 for P 3 7 and P 1 0 , respectively, the new database has more cloud profiles associated with 0 — 10 m m /h r of rain rate and 0.2—0.45 of P 1 9 , b u t the old database yields more profiles w ith heavier rain rate (10—20 m m /h r) if the P 1 9 is between 0.2 and 0.45 as well. 6.2 P E T T Y T M I algorithm The m ain concept of the P etty T M I retrieval algorithm is to find optim al rain rate estim ates, reducing the discrepancy between observed and forward-calculated microwave signatures. In th e algorithm , th e attenuation index is used to adjust th e rain rate field, and the scattering index is utilized to provide initial rain rate information. The rain mask in th e retrieval algorithm is determ ined by three criteria when either one of them is satisfied. First, rain is possible at a given pixel if significant scattering signature is present (i.e., S 3 5 > 10/7). In addition, the pixel is assigned as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 New D a t a b a s e Old D a t a b a s e 50 50 - " 1 40 P 3 7 = [0 .0 7 ,0 .1 5 ] P 3 7 = [0 .0 7 ,0 .15] _ P 1 0 = [0 .6 5 ,0 .7 5 ] P 1 0 = [0 .6 5 ,0 .7 5 ] 40 ■ 30 r 20 20 q: 10 r 0 0.0 _ L ^ r--- r | ■■ ■ i i . | i . < , i i i | > 50 P 3 7 = [0 .0 7 ,0 .1 5 ] P 1 0 = [0 .5 5 ,0 .6 5 ] ^ a) £ 20 L £. 20 c 'o _c 'o ^ 10 30 QL : s i r - . . 10 r ; 0E 0.0 . 0.2 0.4 0.6 P19 0.8 1.0 P 1 0 = [0 .5 5 ,0 .6 5 ] h . T l_ New D a t a b a s e Old D a t a b a s e 100 P37 = [0 .0 0 ,0 .0 3 ] P 3 7 = [0 .0 0 ,0 .0 3 ] P 1 0 = [0 .4 5 ,0 .5 5 ] P 1 0 = [0 .4 5 ,0 .5 5 ] 80 SI | I 60 a> a) o oc 20 0.0 60 40 20 0.2 0.4 0.6 P19 0.8 1.0 0.0 0.2 J \ 0 ............... ................................................................................................ . 0.4 0.6 0.8 1.0 0.0 0.2 P19 1. 0 100 80 o. ' • ■ i ...’ i 1 1 1 i 1 ■ ■ i 1 1 ' i > . P 3 7 = [0 .0 7 ,0 .1 5 ] 40 5 (m m /h i OJ o 4 0 _C I 0.6 P19 New D a t a b a s e Old D a t a b a s e » - r 50 0.4 0.2 0.4 0.6 P19 0.8 1.0 Figure 6.2. Scatter plots of num ber of cloud profiles on the rain rate vs. Pig dom ain a t a given interval of P w and P 37 in the old (left panel) and new (right panel) G PR O F database. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 containing possible rain while th e absolute polarization difference a t 85 GHz is less than 10.5 K. Furtherm ore, if th e atten u atio n index (Pss) is smaller th a n th e value expected for the case associated w ith 0.5 k g /m 2 cloud w ater plus th e tolerance, then the pixel is classified as a rainy pixel. Once the rain mask procedure is done, an initial first-guess rain rate field is assigned for each rainy pixel by R 0 ( m m /h r ) = S ^ / 7 . 0. (6-2) Based on the initial rain ra te field, corresponding attenuation indices at 10.65, 19.35 and 37 GHz can be com puted from forward calculations. Then, th e calculated attenuation indices are com pared w ith the observed values, and then th e current rain rate field is adjusted to achieve a b e tte r agreement between the observed and com puted microwave signature. This procedure will be processed iteratively until the differences at different channels are all w ithin their own assigned tolerance. In the current im plem entation, th e tolerances are given as 0.2 for 85 GHz and 0.05 for other TM I channels. More related details are found in P etty (1994b). 6.3 P E T T Y H IST 4 algorithm A new algorithm based on th e histogram s of PR -TM I m atch-up d a ta was de veloped in th e study. The histogram analysis of the m atch-up d a ta was conducted using values of P i0, Pig, P 3 7 , and S 3 7 w ith intervals of 0.05 and 1 for P and S indices, respectively. Based on the histogram distribution in discrete intervals, the corresponding m ean and stan d ard deviation were able to be com puted, and the num ber of pixels could be counted as well. These statistics were stored in a 4-D lookup table. W hile applying th e algorithm , th e retrieved rain ra te and associated devia tion could be found according to the location of th e m ultichannel observations in th e lookup table. 6.4 Linear m odel A fitting linear model was developed from PR -T M I m atch-up d a ta in Jan u ary and July, 1998. The idea of th e algorithm is to use linear regression analysis to describe the physical relationship between rain ra te and microwave signature. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Regression 92 variables include th e P indices at 10.65, 19.35, and 37 GHz, and S values a t 37 and 85 GHz. Due to th e non-linearity between rain rate and P values, and the satu ratio n at low P index, th e P values were transform ed first by Box-Cox analysis to obtain a more linear relationship. As to the scattering index, its dependency on rain rate was not as strong as th a t shown in th e atten u atio n index. Thus, no transform ation was carried out for S index in th e linear model. The coefficients in th e linear model were calculated from the R language statistical package (mgcv), which was developed by th e departm ent of statistics, University of Wisconsin at Madison. A to ta l of 1.2 million d a ta samples were used in th e regression analysis, and the final value of multiple R-squared is 0.634. Variable transform ations and corresponding regression equations in th e linear model are expressed as the following: P Xl — 1 n o = PI9 pA2 _ 1 = ----- 1A2 = —0.3 P'm = R= ----- , Ax = - 0 .8 Ra7! - 1 ----- ,A3 = -0 .7 0.201 + 4. l O P ^ - 4.831P*9 + 0.182P3*7 + 0.110S37 + 0.017S85 + 0 . 8 1 0 ^ * 9 + 0 .9 6 0 /^^ 3 7 - 0.234P*9P3*7 + O .IW P ^ P ^ P * ,, (6.3) where R is th e retrieved rain rate in m m /h r, and th e asterisk symbol represents the transform ed variables. In the linear regression model, all pixels were included to be com puted w ithout rain screening. Therefore, some constraints have to be added on th e calculated rain rate field in order to exclude unrealistic precipitation intensity and structure. We assume the pixel is not rainy and a zero of final retrieved rain rate would be assigned if the calculated rain intensity is less th a n 0.5 m m /h r. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 7. REAL-W O RLD A PPL IC A T IO N A N D VALIDATIO N In the previous chapter, we have introduced a num ber of algorithm s. In this chapter, those algorithm s will be used to estim ate th e surface rain rate from TM I observations, and the estim ates will be validated against independent datasets. T he purpose is to put the perform ance of the new Bayesian algorithm into context. A satisfactory outcome would be th a t its performance is found to be com parable to th a t of other algorithm s while also having the added advantage of posterior rain rate probability density functions. 7.1 B ayesian algorithm As we m entioned in chapter 1, th e Bayesian posterior density function is the product of the conditional likelihood (or called physical model in th e study) th a t statistically describes th e physical relationships between rain ra te and microwave signal, and a prior rain ra te distribution th a t represents our knowledge or belief. In chapters 3 and 4, in order to provide a theoretical basis, th e conditional likelihood was derived from model sim ulations, and the prior rain rate distribution was based on the intense observations from field experiments. In this section, in an a tte m p t to construct an algorithm w ith calibrated precipitation m easurem ents, th e conditional and prior pdfs were obtained from th e fit of the near-nadir PR -T M I m atch-up d a ta of January and July, 1998. The prior pdf was estim ated from th e fitting of the P R interpolated rain rate. Since there were natu ral variations in th e distribution of rain ra te when d a ta from different m onths were employed, we used (-2.8, 2.0) for the param eters (//, a) of th e prior distribution in our Bayesian algorithm. In chapter 8, th e im pact of th e uncertainty in the prior distribution on the retrieval will be discussed and quantified. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 The conditional probability distribution is denoted as f ( P \ R ) . Here, we would like to construct the IV , 2V, and 3V models, where IV indicates th a t only informa tion about P 3 7 is used in th e retrieval algorithm; 2V represents th a t P 19 and P 3 7 are included to establish th e physical model; and finally, the physical relationship in th e 3V model is represented by P 1 0 , and P 1 9 , P 3 7 . W hen more th a n one observed variable is used, th e conditional likelihood is built in a hierarchical way. The conditional pdf in the IV , 2V, and 3V model are given as: (7.1) IV m o d el : / ( P |P ) = / ( P 3 7 |P ) 2 V m odel : / ( P |P ) - 3V m odel : f ( P \ R ) = / ( P 1 0 , P / ( P 1 9 , P 3 7 | P ) 1 9 , P 3 7 | P ) = / ( P 3 7 | P ) / ( P i 9 |P 3 7 , = / ( P 3 7 |P ) / ( P i 9 |P 3 7 , P ) / ( P R) i o | P 1 9 , ( P 37, 7 .2 ) P ) (7.3) The generic forms for / ( P 3 7 I P ) , / ( P i 9 | P 37 , R), and / ( P i o | P i 9 , P 3 7 , equation (3.3), and th e corresponding param eterization for P ) are defined by n 3, ciy, cr2, and cr3 are summarized as follows: P + 0.06 2-1 + 0.05 AP = 1.02 P — 3 .3 4 \2' 0.5 + 0.06 <73 = 0.33 exp T39 ) . 1.07 e x p (-0 .6 1 P ) - 0 . 0 7 A^3a R - 1.4 \ 2 0.1 — 0 . 1 exp - 0 . 5 0.8 1.5 fa = 0.93 exp V2 = 0.5 bo + bi P 3 7 + S2 bo = 0.25 exp(—0.024R) bl = 0.77 - 0.028P + 0.0009P2 - 0.000009P3 b2 = 0.4 + 0 .0 0 3 5 P ^2 = $21 = ^21 + $22 + <^23 (-0.0005 - 1.6202^ 7 + 4.4076P37 - 0.0323/ R + 0.001P + 0.0006P15) • (exp[—(6 .2 5 P 3 7 )2] + 0.02exp[—100(P 3 7 - 0.9)2]) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 S2 2 = _ o .0 4 ( e x p [ - 0 .5 ( ^ y ^ ) 2] + 1) • exp[-0.5(-— -■p 2 )2] <523 = +0.05[1 - exp(—0.5P)] e x p [-0 .5 (P37Q 3° '8 )2] cr2 = 0.233 e x p (-0 .2 6 9 5 P 37) - 0.1487 Pl Co = C0 exp [ - —!-r (P 1 9 - c i)2] + c3 + 5i ZC2 - 0 . 0 9 - 0.23P37 + 0.89 exp [ - 0.5(P37 - 0.3)2] - 0.023P + 0.14V P Ci 1.1 = = c2 = 1.06 — 0.049P —0.33 exp(P37) + 0 .0 4 6 P ex p (P 37) c3 = 1.04 - 0.022P37 - 0.0066P + 0.047V P - 0.056P37> /P - c0 = ^ii + Sn = -0 .1 3 - 0 .0 0 0 5 P i 9 - 0.35P37 + 0.2 exp(P37) - 0.001P S12 = —0.1 exp[—0.5(——------:— )2] • (1 — 0 .5 ex p (—0.2P)) Si = 0.1 • [1 — 0 .7exp(—0.1P)] 812 + 6i3 P ___ A AIX 813 = 0.03exp[-0.5( - 1-^ 15° '4 )2] • e x p [-0 .5 (P37Q 2^ '° 5 )2] • (1 - 0 .5 e x p (-0 .2 R)) < 7 1 ( 0 . 1 4 - 0 . 1 2 / ^ + 0.015) • (0.9e x p (-O .O lP )+ 0.1), = (7.4) where the unit of rain rates (P ) is m m /h r, and P index is dimensionless. Note th a t th e dependence on P is included in these param eters. Recall th a t P37 saturates quickly w ith increasing rainfall intensity, and thus th e dynamic range of retrieval in th e I V model is quite small. Therefore, the I V model is not very useful in real-world applications, especially for heavy rain events. How ever, the I V model could be used to examine if th e specification of th e / ( P 37, P ) is appropriate. In addition, it helps to dem onstrate how th e assum ptions of physical model and prior distribution affect the retrieval since only one element is involved in the observation vector. Having the posterior probability distribution, th ere are two commonly used esti m ates to represent th e retrieved rain rate: one is th e m ean value (MEAN), and the other is the m axim um likelihood estim ate (MLE). T he Bayesian 2 V model using the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 posterior m ean as the estim ate is denoted as the Bayesian 2E-M EAN algorithm , and the 2V-M LE presents th a t th e retrieved rain rate is estim ated from th e maximum likelihood value of the posterior pdf. Similar rules for the names are applied to the Bayesian ZV algorithm s as well. 7.2 D a ta sets for validation A num ber of cases were processed to evaluate overall perform ances for all algo rithm s described in the previous chapter. The cases include various precipitating systems, such as typhoon/hurricane, frontal rain bands of extra-tropical cyclones, heavy and widespread rain events, and some scattered strong convection cells. De tailed descriptions for each dataset are as follows. 7.2.1 S elected global rain cases A typhoon case with TM I orbit num ber 336 and twelve other cases over the ocean from Bauer et al. (2001) are ad ap ted to help th e validation of th e retrieved rain rates. T he ’tru e ’ rain rate in th e validation d ataset is obtained from the coincident P R 15 x 15 km averaged rain rate. T he key inform ation about cases is summarized in Table 7.1. T he rationale of selections of these cases is to conduct prelim inary exam inations to see if algorithm s work reasonably in each particular w eather system. 7.2.2 H ea v y /w id esp rea d rain even ts P etty (2001) pointed out th a t th e effects of beam-filling and 3-D geometric cloud structures are im portant sources of am biguity in the physical interpretation of mi crowave signature for the surface rainfall estim ations. To reduce th e im pact from those problems, precipitating system s associated w ith widespread heavy rainfall were of interest in his paper, which were likely free of 3-D structures and having uniform precipitation throughout the field of view. We will use the same criteria for th e TM I observations to isolate the heavy and w idespread rain events, in order to understand how each algorithm performs in th e p articu lar precipitation system. The special rain events were sam pled from over-ocean pixels of TM I m easure m ents during the period of January-D ecem ber 1998. Pixels were selected if the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 Table 7.1 Inform ation of date, orbit num ber, and the nadir locations for rain events in the validation dataset. Description Date O rbit Lat. Long. 19971217 00336 15 136 Pacific Ocean 19971216 00295 12 144 Cyclone Paka, Pacific Ocean 19980210 01171 -15 60 19980216 01273 -5 -145 19980826 04283 33 -75 19980829 04328 -23 -142 19990121 06620 3 81 Deep convection, Indian Ocean 20000802 15438 6 -16 Squall line, W est African coast 20000825 15795 18 70 20000828 15835 13 -134 20000830 15876 34 -72 Frontal system , N orth American east coast 20000911 16059 35 -33 E xtra-tropical depression, east A tlantic 20000917 16151 3 70 Cyclone Anacelle, Indian Ocean Deep convection, Pacific ITCZ Hurricane Bonnie, west A tlantic Frontal system, east Pacific Monsoon, west Indian coast Convection, central Pacific Monsoon scattered convection, Indian Ocean Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 L o c a tio n s fo r h e a v y /w id e s p re a d ra in e ve n ts in 1 9 9 8 45 •• %5. -9 0 Figure 7.1. Locations of heavy/w idespread rain events selected from TM I orbits during January-D ecem ber 1998. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 criterion T2i v - ? W < 5 K had been m et for b o th th e pixel itself and th e surround ing pixels in a 5 x 5 pixel array, as proposed by P e tty (2001). This ensures th a t the rainfall in th e event is b oth heavy and widespread. Furtherm ore, in order to avoid contam ination from the sidelobe in th e P R reflectivity a t off-nadir locations, only rain events th a t occur close to the nadir of P R scans were included. As a result, a to tal of 88 rain events were found, and th eir locations are shown in figure 7.1. Each rain event comprises around 360 pixels for th e validation. 7.2.3 P R -T M I global m atch-up d ataset o f 1 9 9 8 /0 4 118 orbit files in April 1998 were random ly selected to validate th e overall per formance of all algorithms. Note th a t our Bayesian retrieval algorithm was derived from the d a ta of January and July in 1998, and therefore, th e T M I-P R m atch-up d a ta of A pril is an independent d ataset for th e validation. In addition, since the dataset comprises enough large samples, in order to filter out the uncertainty in the off-nadir P R interpolated rain rate, only near-nadir pixels were considered in the comparison. T he validation d ataset contains a to ta l sample size of 142,100 pixels. 7.3 V alidation m etrics There are a num ber of verification m easures for researchers to quantify statisti cal differences between calculated or retrieved estim ates (or a forecast) and the true value. Among m any others, th e bias, th e root-m ean-square error, and th e correla tion coefficient are the three mostly commonly used measures, which are employed to continuous underlying variables. T he Heidke Skill Score (HSS) is often used in ’categorical forecasts’ when only yes/no type forecasts are considered, and this ap plies prim arily to discrete variables. This section gives the definition and the usage for each verification measure. Note th a t all validation m etrics are highly context sensitive, depending on the characteristics of the validation d ataset as well as the perform ance of the algorithm. Therefore, validation statistics cannot easily be in terpreted in isolation b u t rath er should be used as a basis for com paring different algorithm s applied to the identical validation d a ta set. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 7.3.1 B ias The bias is defined as B IA S,,. = - x.,), (7.5) -‘V1=1 where th e x j and x Q denote th e forecasts or retrieval ( /) and observations (o), re spectively, and i indicates each d a ta point. The to tal sample size is presented by N . Bias is therefore a measure of th e m ean difference between th e retrievals and the validations. Note th a t the validation dataset itself may contain biases of unknown m agnitude. 7.3.2 R oot-m ean -sq u ared difference The root-m ean-squared difference (RMS) is defined as R M S ,,. = (7.6) RMS is th e positive square root of th e average of th e squared differences. It is a measure of dispersion of the retrieved values from the validation data. As a second order error measure, RMS is more strongly influenced by extrem e differences. In addition, it is sensitive bo th to bias and linear correlation. Since th e absolute cali bration of the validation d a ta cannot be guaranteed, RMS difference m ust be treated w ith caution. 7.3.3 C orrelation coefficient T he correlation coefficient Rf,0 between / and o is com puted by R = £ £ i ( g /< -*o) (7 7) y / Z t i (xfi ~ x f )2y / Z h fa* - x o f where x 7 and x Q are the sample m ean of estim ations and observations, respectively. C orrelation coefficients represent th e strength of th e linear relationship between ob servations and retrievals/forecasts. This m easure is sensitive to th e finest details of the errors between the tru e value and th e estim ates (B arnston, 1992). However, since it only describes the linearity between observations and retrievals, retrieved Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 values associated w ith a uniform or linear bias would still achieve a high correlation w ithout showing the discrepancy. This characteristic brings interesting points for the validation processes of the retrieval algorithms. It is th a t th e correlation coeffi cient is still a good measure to verify retrieval when no absolutely calibrated d a ta are available. High correlations indicate th a t th e tru e and estim ated value behave in the same direction. However, caution m ust be exercised when evaluating retrieval algorithm s based only on th e correlation coefficient. It is typical th a t each retrieval algorithm has its own dynam ic range. W hen a large portion of the actual values are beyond the dynamic range, th e correlation can underestim ate the retrieval skill substantially because of th e satu ratio n at higher rain rates. In addition, due to th e quadratic relationship between correlations and errors, th e correlation coefficient is affected significantly by th e outliers. In the presence of outliers, the correlation coefficients might m isrepresent th e tru e retrieval ability as well. In summary, the correlation coefficient measures how strong th e linear relation ship is between th e tru e and estim ated value. We recommend making scatter plots whenever feasible, because it will tell us if there is an obvious bias and if there is any possible m isinterpretation in th e correlation coefficient. 7.3.4 H eidke Skill Score W hile forecasts are expected to have rough accuracy or th e verification is focused on measuring th e skill forecast of discrete num ber of events (so-called categorical forecasts), the Heidke Skill Score (HSS) is a commonly used verification measure. The skill score is com puted from a contingency table (as shown in table 7.2 for 2 categories) th a t sum m arizes th e occurrences of events for observations and forecasts (or retrievals). The contingency table is constructed at a given threshold th a t dis tinguishes ’yes’ and ’no’ events and defines the categories. HSS not only m easures the proportion of correct forecasts, including b o th correct hits and rejections, b u t also takes into account th e expected skill obtained by chance in th e absence of any forecasting skill (B arnston, 1992; G audet and C otton, 1998; Stephenson, 2000). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 Table 7.2 S tandard 2 x 2 contingency table for evaluation of th e skill of a binary classification procedure. Observed Yes Observed No Predicted Yes A (hits) C (false alarms) Predicted No B (misses) D (correct rejections) According to th e definition of th e proportion of correct forecasts, th e proportion correct (P C ) can be w ritten as PC = A + D = A + D A +B +C +D N . ’ ^ ' where N is the to ta l num ber of events. T he definition of HSS is sim ilar to P C , bu t HSS su b tracts the correct forecasts due purely to random chance, E. Therefore, it is defined as H S S = (A+n D} e E , (7-9) where E can be derived as follows. Based on random guess, th e expected num ber for hits is given by A! = N • P(o) ■P ( f ) where P ( o ) and P ( f ) c™ > express th e probability th a t the event willoccur based on observations and forecast, respectively. Similarly, the expected num ber for correct rejections D' can be derived from jy where P(o) and P ( f ) = n ■p (o ) ■p (J) = n .A ± £ 1 .(E ± A N N ’ (7 1 I) [ ’ describes th e probability th a t events willno t occur based on observations and forecast, respectively. Then, th e expected num ber of correct Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 forecasts by random chance E is equal to A' + D'. Finally, combining w ith equations (7.9), (7.10), and (7.11), the H S S can be derived as th e following equation: 2(AD —BC) B 2 + C 2 + 2AD + (B + C)(A + D) ' ' Unlike the bias, RMS difference, and correlation coefficient, th e Heidke Skill Score is based on binary classification and applied m ainly to discrete underlying variables. In the present context, HSS can be applied to determ inations of rain vs. no rain by both th e retrieved algorithm and th e validation. However, th e definition of rain and no rain is not unique. Employed intensity thresholds, calibration errors and minim um detectable rain ra te of b o th validation and retrieved d a ta will all result in different definitions of rain vs. no rain. T he validation m etric can be applied unambiguously only under the condition th a t b o th th e validation d a ta and retrieval have th e same definition of rainy events. If th e condition is not satisfied, as is often th e case in real-world applications, a single scalar HSS m ight m isrepresent th e retrieval ability of algorithms. Therefore, instead of using a fixed rain ra te threshold, Conner and P etty (1998) tre a ted th e intensity threshold as a continuous variable. W hen th e variation of the threshold is applied, th e HSS can be considered as a function of separate rain rate thresholds for b o th th e observed and retrieved d ata. T hen by reference to 2-D plots of this function, it is possible to evaluate retrieval skills of algorithm s at delineating observed rain ra te as a continuous function of th e threshold of observed rain intensity, to reflect the bias of th e retrieval from the position of th e m axim um Heidke Skill Score axis, and to further calibrate th e algorithm s. 7.3.5 T h eoretical H SS d istribution Before employing HSS into the validations of real-world cases, it would be use ful to understand th e two-dimensional HSS distributions for th e Bayesian retrieval when the perfect conditional and prior likelihoods are used. To address this issue, a synthetic dataset was produced from the Bayesian ZV model using random num ber generation. In order to assure th a t th e generation process of th e synthetic d ataset is appropriate and th e num erical solutions indeed follow th e Bayes rule, th e posterior Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 7 1.0 6 0.8 5 0.6 4 3 0 .4 2 0.2 0.0 0 0 .0 0 .2 0 .4 0 .6 0 .8 D a ta s e t ( m m / h r ) 1.0 0 1 2 3 4 5 6 D a ta s e t ( m m / h r ) 7 100 25 80 E E 60 o> <D <D cr 40 20 0 5 10 15 20 D a ta s e t ( m m / h r ) 25 0 20 40 60 80 D a ta s e t ( m m / h r ) 100 Figure 7.2. Plots of th e sample m ean of th e retrieval vs. th e posterior m ean estim ated from th e synthetic d ataset at a given P if th e corre sponding sample size is (a) greater th a n 100; (b) greater th a n 30; (c) greater th a n 10; and (d) less th a n or equal to 10. Note change of range in each case. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 mean of the synthetic dataset at a given P is com pared w ith th e sample mean of the corresponding retrieval. Figure 7.2 depicts the scatter plots of th e sample m ean of th e retrieval vs. the posterior m ean of the dataset points a t various observation vectors P. For any given P, a bin size of 0.01 was chosen for P 10, P 1 9 , and P 37, and th en the sample size would be th e counts of d a ta points located w ithin the 3-D grid box. Different P associates w ith various sample sizes, so th e d a ta points are grouped and plotted by different sample size. It is found th a t th e posterior m ean of th e dataset has a good agreem ent w ith the m ean of the retrieval when th e sample size is greater th an 100, as shown in figure 7.2(a). The shift from th e diagonal line at d ataset rain rate of 0 .2 m m /h r and retrieved rain ra te of 0 .1 m m /h r is a ttrib u te d to the fact th a t th e algorithm only applies to th e pixel where either one of th e atten u atio n indices a t 10.65, 19.35 and 37.00 GHz is less th a n 0.8. Then, some of pixels associated w ith non-zero tru e rain ra te (dataset) have zero retrieved surface rainfall, and thus results in an underestim ation in this range of rain rate. W hen th e sample size decreases, th e means from the synthetic d ataset and retrieval reveal some scattering, which is expected and reasonable from a dataset of random variates. If th e sample size is less th a n or equal to 10 (as shown in figure 7.2(d)) and m ost of th e points might represent only one pixel at given P, the relation dem onstrates very significant scattering phenomenon. In short, th e good m atch-up for th e observation vectors w ith larger sample sizes and th e substantial scattering for those w ith smaller sample sizes axe expected in the synthetic d ataset and its corresponding retrieved rain rate. Therefore, we do have the confidence th a t th e generation process of random variates is proper, and th e algorithm follows th e Bayesian principles. Figure 7.3(a) and (d) depict th e theoretical 2-dimensional HSS distributions for th e Bayesian 3M-MEAN and 3M-MLE algorithm s, respectively. Results show th a t there are two areas associated w ith high skill scores. One is located a t rain rates less th a n 20 m m /h r w ith larger HSS, and th e other occurs w ith a skill score of around 0.4 a t larger rain rate. Moreover, th e coverage of th e 2-D contours is an indication of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 BAYESIAN 3 V -M L E BAYESIAN 3V -M E A N 100 ^ 100 -C^ E E 2 o 80 £ E 80 60 o 60 _C 40 O' SI</) (L> _c in d) sz 40 E E o ji 20 o SP < 20 _Oi < 60 80 100 20 40 0 True rain r a t e th r e s h o ld ( m m / h r ) 60 80 100 0 20 4 0 True rain r a t e th r e s h o ld ( m m / h r ) BAYESIAN 3V —MEAN BAYESIAN 3 V -M L E 0.8 0.8 C/1 x 0.6 o o g 2 g 0.4 0 .4 2 0.2 0.2 0.0 0.0 40 60 80 100 True rain r a t e th r e sh o ld ( m m / h r ) 60 80 1 00 0 20 4 0 True rain r a t e th r e s h o ld ( m m / h r ) BAYESIAN 3V -M E A N BAYESIAN 3 V -M L E 0 20 _c (n x:oin E E o o o cn (U m cn 0) 20 40 60 80 100 True rain r a t e th r e s h o ld ( m m / h r ) CD 20 40 60 80 1 00 True rain r a t e th r e s h o ld ( m m / h r ) Figure 7.3. Plots of 2-D Heidke skill scores (HSS); th e m axim um of HSS w ith respect to tru e rain rate threshold; and th e best algorithm threshold vs. tru e rain ra te threshold based on th e synthetic d a ta of th e Bayesian ZV model, (a)-(c) are illustrated for MEAN estim ates, while (d)-(f) are for MLE. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 relative location between the tru e and retrieved rain rate. For example, at a given PR -T M I threshold of 40 m m /h r, th e contours fall w ithin around 10—60 m m /h r using the posterior m ean as the estim ator, indicating th a t the retrieved rain ra te varies in a wide range even in th e synthetic test. Since th e 0.1 contour could extend to very large tru e rainfall rate, it shows th e retrieval ability of the Bayesian 3M-MEAN and 3M-MLE algorithm s over the heavy precipitation systems if th e specifications of the prior pdf and th e physical model are correct. At a given tru e rain ra te threshold, a m axim um HSS over th e algorithm rain rate dom ain could be found. The curve of m axim um HSS vs. tru e threshold represents the discrim ination ability of the algorithm for a certain rain rate threshold. Figure 7.3(b) and (e) dem onstrate th a t the score is very high in th e situation of very light rain rate in b oth MEAN and M LE algorithms. The m axim um HSS decreases w ith increasing tru e rain intensity, reaches a lowest point of 0.35 a t 20—40 m m /h r, th en increases w ith a small degree while th e tru e rain rate is between 60—80 m m /h r. T he plots indicate th a t th e 31/-MEAN and 3R-M LE algorithm s are able to differentiate th e rain rate in areas w ith very light or heavy precipitation, b u t might have difficulty to discrim inate the rain intensity of 20—40 m m /h r even though perfect specifications were used. The algorithm rain ra te threshold associated w ith the m axim um HSS a t a fixed observed threshold is defined as th e best algorithm threshold. Thus, th e best algo rithm is able to provide th e inform ation about bias. The plot of th e best algorithm threshold vs. th e tru e rain rate threshold (as shown in figure 7.3(c)) dem onstrates th a t th e Bayesian 3P-M EA N algorithm yields very good estim ates for th e tru e rain rate threshold of 0—20 m m /h r, b u t underestim ates higher rain rate. A notable bias occurs a t higher tru e rain rate in th e retrieval even though perfect inform ation has been applied. It indicates th a t the inform ation from the dataset is lim ited due to th e ambiguity of th e physical relation to th e rain ra te and the satu ratio n of th e variables. In addition to th a t th e prior gives more weights in th e light rain rate, th e Bayesian 3M-MEAN algorithm produces a posterior p d f skewed to th e left and th u s leads to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 an apparent underestim ation for the pixels w ith higher rain rate. Furtherm ore, fig ure 7.3(f) reveals th a t the perform ance of the Bayesian 3V-M LE algorithm in th e regimes w ith 5—20 m m /h r rain ra te is not as good as th a t of th e 3Y-M EAN model, and th e retrieval using MLE has a substantial negative bias when th e tru e rain rate is between 20—50 m m /hr. However, it is evident th a t th e 3P-M L E algorithm has more skills to capture the very heavy rain rate when the conditional likelihood and th e prior distribution are perfectly modeled. 7.4 R esu lts 7.4.1 T yp h oon case In order to obtain a direct sense of how different the retrieval from each algo rithm behaves, P R interpolated and algorithm -retrieved rain ra te for all pixels are m apped for th e typhoon case (as figure 7.4 shows). Those rain m aps exhibit th a t all algorithm s are able to retrieve the eye, the two separate rain bands, and the overall cyclonic structure of the typhoon. Exceptionally, the Bayesian 2Y-M LE algorithm failed in th e retrieval of th e eye. In addition, m ost algorithm s have similar m agni tude of rain ra te to the P R interpolated data. However, th e G PR O F w ith th e old database displays higher rain intensity generally, while th e Bayesian algorithm s w ith MLE b o th in th e 2 V and 3 P models dem onstrate noticeable underestim ations about th e rainfall rate. The scatter plot of th e retrieval vs. P R rain ra te for orbit 336 is presented in figure 7.6. Apparently, th e old G PR O F overestimates th e intensity when the P R rain rate is greater th a n 10 m m /h r. Furtherm ore, if th e M LE is used in th e Bayesian algorithm , th e dynam ic retrieval range is only around 10 m m /h r, which is much smaller th a n th a t derived from th e posterior mean. These results could be confirmed by th e set of columns in table 7.3, showing a positive bias in th e old G PR O F and a negative bias in the Bayesian-MLE type algorithm. Table 7.3 also indicates th a t the new G PR O F, P E T T Y TM I, P E T T Y HIST4, linear model, and th e Bayesian-MEAN type algorithm s have similar bias, RMS, and correlation coefficients. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 Table 7.3 Sum m ary of bias, root-m ean-square difference (RMS), and correlation coefficients against validation datasets of orbit 336 and B auer’s cases for each algorithm . The unit of bias and RMS is m m /hr. O rbit 336 Bias RMS Corr G PR O F-O LD DATABASE 1.84 5.55 0.87 G PR O F-N E W DATABASE -0.22 2.68 0.88 P E T T Y TM I ALGORITHM -0.15 2.98 0.84 P E T T Y HIST4 ALGORITHM -0.51 2.72 0.90 LIN EA R M O D E L -PR /T M I DATA -0.29 2.71 0.88 0.00 2.89 0.85 -1.04 3.75 0.85 0.22 2.85 0.86 -1.06 3.90 0.82 B auer’s Bias RMS Corr G PR O F-O LD DATABASE 0.23 1.94 0.78 G PR O F-N E W DATABASE -0.22 1.64 0.76 P E T T Y TM I ALGORITHM -0.12 1.70 0.74 P E T T Y HIST4 ALGORITHM -0.11 1.45 0.80 LINEAR M O D E L -PR /T M I DATA 0.00 1.58 0.78 BAYESIAN 3Y-MEAN 0.08 1.84 0.71 -0.29 1.80 0.74 0.11 1.75 0.75 -0.29 1.78 0.75 BAYESIAN 3Y-MEAN BAYESIAN 3E-M LE BAYESIAN 2E-M EAN BAYESIAN 2E-M LE BAYESIAN 3E-M LE BAYESIAN 2P-M EA N BAYESIAN 2Y-MLE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 Table 7.4 Sum m ary of bias, root-m ean-square difference (RMS), and correlation coefficients against validation datasets of heavy/w idespread rain events and 1998/04 dataset for each algorithm. The unit of bias and RMS is m m /hr. H eavy/w idespread rain events Bias RMS Corr G PRO F-O LD DATABASE 1.85 5.59 0.74 G PR O F-N EW DATABASE -0.15 4.08 0.75 P E T T Y TM I A LGORITHM 0.04 4.08 0.75 P E T T Y HIST4 A LGORITHM 0.54 3.68 0.80 LINEAR M O D E L -P R /T M I DATA 0.84 4.34 0.74 BAYESIAN 3E-M EA N 1.30 5.03 0.68 -0.19 4.49 0.69 1.74 5.30 0.71 BAYESIAN 2R-M LE -0.22 4.54 0.69 1998/04 random events Bias RMS Corr G PRO F-O LD DATABASE 0.24 2.02 0.73 G PR O F-N EW DATABASE -0.21 1.18 0.78 P E T T Y TM I A LG O RITH M -0.18 1.18 0.78 PE T T Y HIST4 A LGORITHM -0.14 1.06 0.82 LINEAR M O D E L -P R /T M I DATA -0.08 1.09 0.81 0.03 1.26 0.78 -0.01 1.21 0.75 0.04 1.26 0.78 -0.01 1.22 0.75 BAYESIAN 3Y-M LE BAYESIAN 2Y-M EAN BAYESIAN 3E-M EA N BAYESIAN 3R-M LE BAYESIAN 2Y-M EAN BAYESIAN 2Y-M LE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill PS2A25 Rain R ate (1 5 k m ) GPROF - GPROF - PETTY TMI ALGORITHM NEW DATABASE PETTY HIST4 ALGORITHM OLD DATABASE LINEAR MODEL - TMI/PR DATA 16.c ■5.4*.. J 1 Figure 7.4. M aps of P R interpolated rain rate w ith 15-km resolution and retrieved rain ra te of G PR O F (old and new databases), P E T T Y TM I, P E T T Y H IS T 4 , a n d lin e a r m o d el a lg o rith m s for T R M M /T M I o rb it 336. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 7.5. Same as figure 7.4, b u t for Bayesian 3V-MEAN, 3V-MLE, 2V-MEAN, 2V-MLE algorithm s. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 113 GPROF - £E BAYESIAN 3V -M LE PETTY HIST4 ALGORITHM OLD DATABASE 60 40 20 60 60 50 50 40 40 30 30 2 20 2 20 UJ cc 0 10 20 30 40 50 P R RAIN RATE ( m m / h r ) GPROF - 0 60 10 20 30 40 50 P R RAIN RATE ( m m / h r ) LINEAR MODEL - NEW DATABASE 0 60 10 20 30 40 50 P R RAIN RATE ( m m / h r ) 60 BAYESIAN 2V-MEAN TM l/P R DATA 60 60 40 40 -C e 40 2 30 30 2 S 20 aor 20 2 20 r / UJ 10 ce 0 10 20 30 40 50 P R RAIN RATE ( m m / h r ) PETTY o 60 10 20 30 40 50 P R RAIN RATE ( m m / h r ) 0 60 60 BAYESIAN 2V -M LE BAYESIAN 3V-MEAN TMI ALGORITHM 60 10 20 30 40 50 P R RAIN RATE ( m m / h r ) 60 60 40 40 50 E E, 40 2z 3 0 2 a 20 30 2 20 Cu O' 0 10 20 30 40 50 P R RAIN RATE ( m m / h r ) 60 0 10 20 30 40 50 P R RAIN RATE ( m m / h r ) 60 0 10 20 30 40 50 P R RAIN RATE ( m m / h r ) Figure 7.6. S catter plot of retrieved rain ra te vs. P R rain ra te for all algorithm s, based on th e validation d a ta of th e TM I orbit 336. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 114 7.4.2 B a u er’s 12 oceanic cases For the twelve oceanic cases, th e P E T T Y HIST4 has th e best perform ance in term s of th e least root-m ean-squared error and th e highest correlation coefficient (as shown in the table 7.3). The linear model has zero bias and th e second largest corre lation coefficient in this case. The retrieval from the Bayesian 317-MEAN algorithm associates w ith a bias of only 0.08 m m /h r, bu t have a larger root-m ean-squared er ror and a smaller correlation coefficient, implying th e lack of retrieval ability in the heavy rain rate. Figure 7.7 shows th e contours of 2-dimensional HSS of each algorithm for the oceanic cases, and indicates th a t high skill scores occur in th e range of 0—10 m m /h r for all algorithm s. T he coverage of 0.1 contour suggests th a t th e new G PR O F, PE T T Y HIST4, the linear mode, and th e Bayesian 2V model have b e tte r skills to retrieve higher rain ra te th a n th e P E T T Y TM I and the Bayesian 3V algorithm do in the twelve cases. Moreover, th e relationship between th e best algorithm rain rate and the P R threshold is depicted in figure 7.8. The linear relation in the plots dem onstrates th a t m ost algorithm s, especially P E T T Y TM I and the BayesianMEAN models, offer good agreem ents w ith PR d a ta when th e tru e rain ra te is less th an 20 m m /hr. Comparisons of th e HSS distributions between th e Bayesian 317-MEAN and th e 217-MEAN algorithm s dem onstrate th a t th e 3R model does offer additional infor m ation in th e retrieval procedure. For example, as figure 7.7 shows, the retrieval from th e 317-MEAN algorithm varies between [2, 20] m m /h r at a given P R rain rate threshold of 10 m m /h r, while th e 217-MEAN algorithm yields rain rate up to 26 m m /hr. However, th e additional variable of P 10 also constrains th e retrieved rain rate significantly for th e cases associated w ith heavy precipitation. T he degradation at higher rain rate in th e 317 algorithm m ight originate from th e poor fit to th e d ata in the specification of conditional likelihood. Recall in th e 2-D HSS distributions of the synthetic d a ta (as shown in fig ure 7.3(a)), the 0.4 and 0.2 contours of th e 317-MEAN algorithm should be able Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 GPROF - Q 0 20 EG 20 Q1 1 2 (m m /h r) ALGORITHM THRESHOLD \ E £ 30 X (X O o < 0.761 10 20 30 P R THRESHOLD ( m m / h r ) GPROF - 0 .7 4 9 0 .7 7 5 0 40 20 10 30 P R THRESH OLD ( m m / h r ) 0 40 10 20 30 P R THRESHOLD ( m m / h r ) 40 BAYESIAN 2V-MEAN LINEAR MODEL - TM l/PR DATA NEW DATABASE 40 40 40 (m m /h r) 40 40 0 30 E 30 o d 20 20 EG 20 0: x ALGORITHM THRESHOLD BAYESIAN 3V -M LE PETTY HIST4 ALGORITHM OLD DATABASE 40 0.778: 0 .7 3 6 0 10 20 30 P R THRESHOLD ( m m / h r ) 0 40 PETTY TMI ALGORITHM 10 20 30 P R THRESHOLD ( m m / h r ) 2 I CK O o < 0.741 o 40 40 40 40 (m m /h r) 20 BAYESIAN 2V -M LE BAYESIAN 3V-MEAN 40 10 30 P R TH RESHOLD ( m m / h r ) \E .c E THRESHOLD o 0 'Or 20 w a. 20 1 ALGORITHM 2 X £ 10 <x o o < 0 .7 3 6 0 10 20 30 P R THRESHOLD ( m m / h r ) 40 0.751 0 10 20 30 P R THRESHOLD ( m m / h r ) 0 .7 4 7 40 0 10 20 30 P R THRESHOLD ( m m / h r ) Figure 7.7. 2-D distribution of Heidke skill scores (HSS) for the 12 se lected cases from the Bauer et al. (2001). The value noted in th e bottom right corner of each plot indicates th e highest HSS of th e algorithm . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 116 GPROF - OLD DATABASE PETTY HIST4 ALGORITHM BAYESIAN 3V -M LE 80 E E 60 a 60 o x to UJ cc 40 X 40 2 X E o 20 o < 20 BEST ALGORITHM THRESHOLD ( m m /h r) 80 20 40 60 P R THRESHOLD ( m m / h r ) GPROF - 80 NEW DATABASE o 0 1 </> 40 ( m m /h r) E, 60 a_ i X to a: x ALGORITHM 8 20 < < to UJ CD (D 40 60 P R THRESHOLD ( m m / h r ) 20 40 60 P R THRESHOLD ( m m / h r ) 80 PETTY TMI ALGORITHM 80 80 40 60 P R THRESHOLD ( m m / h r ) BAYESIAN 3V-MEAN 80 BAYESIAN 2V -M LE 80 N £E 6 0 E £ 60 Cl a o x CO 40 20 80 _c 60 40 O' o 20 20 60 o 40 BEST THRESHOLD \E E to UJ ( m m /h r ) 80 \E 60 80 BAYESIAN 2V-MEAN 80 or o THRESHOLD 40 60 P R THRESHOLD ( m m / h r ) LINEAR MODEL - TM I/P R DATA 80 o x to UJ K 40 X 40 2 I E o 20 o < BEST ALGORITHM 20 20 40 60 P R THRESHOLD ( m m / h r ) O q 20 a: 20 < to UJ to CD 20 40 60 P R THRESHOLD ( m m / h r ) 80 20 40 60 P R THRESHOLD ( m m / h r ) 80 20 40 60 P R THRESHOLD ( m m / h r ) Figure 7.8. Plots of th e best algorithm rain ra te threshold w ith respect to the threshold of P R rain rate for th e B auer’s cases. The algorithm used in the retrieval is shown in the title of each plot. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 117 PLOT OF HSS_MAX VS. PR THRESHOLD in in GPROF - NEW DATABASE PETTY TMI ALGORITHM PETTY HIST4 ALGORITHM LINEAR MODEL - TMI/PR DATA' BAYESIAN 3V-MEAN BAYESIAN 2V-MEAN 0-6 X < 0 .4 0.2 0.0 0 10 20 30 40 PR THRESHOLD ( m m / h r ) PLOT OF HSS_MAX VS. PR THRESHOLD BAYESIAN BAYESIAN BAYESIAN BAYESIAN 3V-MEAN 3V-MLE 2V-MEAN 2V-MLE 0.2 0.0 0 10 20 30 40 PR THRESHOLD ( m m / h r ) Figure 7.9. Plots of the m axim um Heidke skill score vs. th e threshold of P R rain ra te based on th e dataset of 12 B auer’s cases. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 to go up to th e P R rain ra te threshold of 20 and 95 m m /h r, respectively. Similar HSS distributions to the synthetic d a ta could be obtained if th e physical relationships and th e prior rain rate pdfs in the real-world applications are perfectly represented by the Bayesian model. However, it is found th a t th e 0.4 contour for th e valida tion cases only approaches to around 10 m m /hr, when th e 0.2 contour is reaching P R threshold of 30 m m /hr. The discrepancy between the theoretical te st and realworld applications might be be a ttrib u te d to the poor fit at high rainfall rate, and it might also im ply th a t some physical relationships in real world applications are not modeled in th e algorithm. To identify the discrim ination ability of each algorithm w ith respect to th e true rain rate, plots of the m axim um HSS vs. in figure 7.9. P R rain rate threshold are presented The distribution of HSS shows th a t each algorithm generally has great discrim ination ability for th e rain rate of 0 to 10 m m /h r, b u t has different perform ance w ith respect to surface rainfall. For example, th e retrieval ability of th e new G PR O F and P E T T Y TM I algorithm reaches th e weakest point a t rain rate of ~ 10 m m /h r, then the HSS goes up w ith increasing rain intensity. The P E T T Y HIST4, linear model, and th e Bayesian 2Y-MEAN have sm aller HSS in th e range of 12—18 m m /h r, while th e maximum skill of th e Bayesian 3 V model continues to decrease w ith increasing rain rate. It is evident th a t th e 2R-M EAN algorithm perform s superior to the 3 V model in term s of discrim ination ability. Interestingly, th e discrim ination ability of the 3F-M LE algorithm is the highest among th e Bayesian algorithm s (as figure 7.9(b) shows). Note th a t th e curve of th e m axim um HSS based on the synthetic d ataset changes from 0.8 to around 0.35 when th e P R rain rate threshold is betw een 0 and 40 m m /h r (referred to figure 7.3(b)). For this validation dataset, th e m axim um HSS from the 3Y-M EAN algorithm has a similar tren d to the synthetic dataset. However, the curve in th e real-world application reaches 0.35 when the P R threshold is around 12 m m /h r, and has th e lowest point of 0.15 score at 40 m m /h r. O n th e contrary, for th e 3Y-M LE algorithm , th e tren d of th e maximum HSS in th e validation dataset is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 very different from th a t seen in th e synthetic test. It is not quite clear yet w hat is responsible for the discrepancy. 7.4.3 H ea v y /w id esp rea d rain events The values of bias listed in tab le 7.4 for the heavy/w idespread rain events tell th a t the P E T T Y HIST4 algorithm has the least bias, when th e G PR O F w ith the old database and th e Bayesian-MEAN type have significant positive bias in the retrieved rainfall rate. P E T T Y HIST4 has th e greatest performance regarding th e errors and correlation coefficients. Validations for th e heavy/w idespread cases by Heidke skill scores are shown in figures 7.10, 7.11 and 7.12. Figure 7.10 shows th e 0.1 contours of 2-dimensional HSS for all algorithm s are substantially flatter in the P R rain ra te threshold of 0—5 m m /h r, except the P E T T Y TM I and new G PR O F algorithm . W hen P R threshold is around 5 m m /h r, th e re trieved rain rate from P E T T Y HIST4, linear model, and th e Bayesian-type algo rithm s ranges from 0 to around 20 m m /h r, bu t the variations in th e P E T T Y TM I and new G PR O F are about 0—15 m m /h r. This result points out a very unique feature of the P E T T Y TM I algorithm . In G PR O F, the beam-filling effects are con sidered by th e inclusion of th e areal fraction of convective rainfall in th e observation vector. T he P E T T Y HIST4, linear model, and the Bayesian algorithm do not add the inhomogeneity of precipitation structures into th e retrieval process. Those algo rithm s tre a t each pixel individually, and the connections between th e pixel and its surrounding environm ent are not considered. In this way of retrieval, th e P E T T Y HIST4, linear-model, and th e Bayesian algorithm s generate a more significant varia tion in th e retrieval while different spatial structures of precipitation are of interest. In contrast to other algorithm s, th e P E T T Y TM I algorithm considers th e beam filling factor implicitly. In th e iterations of adjustm ent between th e observed and calculated attenuation index for all three channels at the same tim e, th e precipita tion variation in different footprints has been included in P E T T Y TM I algorithm in order to find an optim al estim ate. As a result, the P E T T Y TM I algorithm has a more consistent perform ance in any kind of validation datasets. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 GPROF - OLD DATABASE (m m /h r) N E E I 30 d QL 20 o .O-' 20 o 0 1 to ui )— 2 X cc E o 0 .8 5 0 10 20 30 o GPROF - 0 40 (m m /h r) _<l 10 20 30 P R THRESHOLD ( m m / h r ) 0 .8 5 6 0 40 10 20 30 P R THRESHOLD ( m m / h r ) 40 BAYESIAN 2V-MEAN LINEAR MODEL - TM I/PR DATA NEW DATABASE 40 40 40 0.00 \ 30 I 30 E E a o o o 20 uj CE 20 [2 20 q: i X 2 X ALGORITHM THRESHOLD o o 0.872 < PR THRESHOLD ( m m / h r ) E o 0 .8 4 9 0 10 20 30 P R THRESHOLD ( m m / h r ) o 0 .8 7 2 < 0 40 10 20 30 P R THRESHOLD ( m m / h r ) 0 40 30 20 E E o o 0 O ' X 20 ui 20 q: 1 ALGORITHM 2 2 X I E o 0.842: 20 10 30 P R THRESHOLD ( m m / h r ) 40 40 BAYESIAN 2V -M LE Io 30 W UJ 20 10 30 P R THRESHOLD ( m m / h r ) 40 40 0 0.868 < BAYESIAN 3V-MEAN PETTY TMI ALGORITHM 40 (m m /h r) 0.00 CE I X ALGORITHM THRESHOLD 40 JZ 30 0 THRESHOLD BAYESIAN 3V -M LE PETTY HIST4 ALGORITHM 40 40 £E o 0 .8 7 3 < 0 20 10 30 P R THRESHOLD ( m m / h r ) 40 O O 0.850: < 0 10 20 30 P R THRESHOLD ( m m / h r ) Figure 7.10. 2-D distribution of Heidke skill scores (HSS) for the heavy/w idespread rain events. T he value noted in the bottom -right cor ner of each plot indicates th e highest HSS of th e algorithm . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 121 GPROF - OLD DATABASE r \ E E o BAYESIAN 3V -M LE PETTY HIST4 ALGORITHM 80 80 80 E E 60 a E E 60 o 60 o x co X x a: 40 x 2 X tr x 2 2 £K o q 20 q < < X a o q UI 40 20 < 40 X 20 (/> UJ UJ CO m 20 40 60 PR THRESHOLD ( m m / h r ) GPROF - CD 40 20 60 P R THRESH OLD ( m m / h r ) 80 NEW DATABASE LINEAR MODEL - 80 20 40 60 P R THRESHOLD ( m m / h r ) 80 80 BAYESIAN 2V-MEAN TM I/PR DATA 80 jz \ E E o _! E E 60 Q O 60 N- ' 6 0 a X in UJ I 40 x 20 oE o o 20 CK X 40 40 2 X X E o o 20 < < < to m UI m CD 20 40 60 PR THRESHOLD ( m m / h r ) 20 40 60 PR THRESHOLD ( m m / h r ) 80 PETTY TMI ALGORITHM 20 40 60 P R THRESHOLD ( m m / h r ) 80 BAYESIAN 3V-MEAN 80 80 BAYESIAN 2V -M LE 80 80 x N E E E E E E, 60 60 o 60 o a 01 to tr i 40 40 40 2 2 x X cc a o q O 20 q O' o q 20 co UJ CD 20 40 60 P R THRESHOLD ( m m / h r ) 80 20 < < UJ CD 20 40 60 P R THRESHOLD ( m m / h r ) 80 20 40 60 P R THRESHOLD ( m m / h r ) Figure 7.11. Plots of th e best algorithm rain rate threshold w ith respect to the threshold of P R rain ra te for th e cases w ith heavy and widespread precipitation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 122 PLOT OF HSS_MAX VS. PR THRESHOLD GPROF - NEW DATABASE PETTY TMI ALGORITHM PETTY HIST4 ALGORITHM LINEAR MODEL - TMI/PR DATA' BAYESIAN 3V-MEAN BAYESIAN 2V-MEAN to 0 .6 to x X < 2 0 .4 0.2 0.0 0 10 20 30 40 PR THRESHOLD ( m m / h r ) PLOT OF HSS_MAX VS. PR THRESHOLD BAYESIAN BAYESIAN BAYESIAN BAYESIAN 0.8 3V-MEAN 3V-MLE 2V-MEAN 2V-MLE co 0.6 cxr X < 2 0 .4 0.2 0.0 0 10 20 30 40 PR THRESHOLD ( m m / h r ) Figure 7.12. Plots of th e m axim um Heidke skill score vs. th e threshold of P R rain ra te based on th e dataset of heavy/w idespread rain events. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 123 Similarly to B auer’s oceanic cases, figure 7.11 dem onstrates th a t m ost algorithm s provide reasonable retrieved rain ra te for th e P R rain rate less th a n 20 m m /h r. Among the algorithm s, the P E T T Y HIST4 algorithm shows extraordinary skills in the heavy precipitation. In addition, th e results from the G PR O F w ith th e old database suggest th a t there is th e m ost significant bias in th e comparison. In the figure 7.11, the Bayesian 2 V and 3 V model are associated w ith a noticeable positive bias in the range of m oderate rain rate. Explanations ab o u t th e source of th e bias are given here. Recall th a t there are only 88 events satisfying the criteria of heavy and widespread precipitation in th e entire year of 1998. T he fact indicates how sporadic this kind of precipitating system is in real-world applications, and as well in the training dataset. Therefore, the current Bayesian model has difficulty to represent the physical relationship of th e special cases, especially in th e 3 V model. Due to the sampled events th a t present areas w ith widespread rain, th e precipitation systems discussed here are less beam-filling th a n the average. Thus, even if the tru e rain rate is only 5—10 m m /h r, th e values of P index a t 19.35 and 37.00 GHz are almost saturated, and the P \ q could be around 0.35—0.45. According to such a low Pio value plus satu rated P i9 and P 37, the Bayesian 3V model yields an estim ate w ith higher rain rate th a n the tru e intensity, and then results in a positive bias in th e range of m oderate rain rate. More discussions w ith detailed pdfs are described in section 7.1. The plot of the m axim um HSS vs. P R rain rate threshold are represented in figure 7.12. The distribution of HSS shows th a t the P E T T Y TM I and P E T T Y HIST4 algorithm s have overall high m axim um skill scores in all ranges, while th e linear model and Bayesian m odel degrades substantially. In addition, unlike th e previous twelve cases, the discrim ination abilities in all of our Bayesian algorithm s do not differ so significantly. 7.4.4 A pril 1998 In th e random ly selected cases in April 1998, the P E T T Y HIST4 and the linear model still have th e highest correlation coefficients and least error com pared to other Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 124 GPROF - OLD DATABASE PETTY HIST4 ALGORITHM x: 30 E 30 o o £01 20 2 2 x i X X ti o 0 .69 7 10 30 20 PR THRESHOLD ( m m / h r ) GPROF - (m m /h r) 0 E 2 20 a o 0 .737 0 40 NEW DATABASE LINEAR MODEL - E £ 10 20 30 P R THRESHOLD ( m m / h r ) 40 BAYESIAN 2V-MEAN 40 30 ‘0.0t) Q 0 £os 20 £ 20 01 I x r— ALGORITHM 2 I 5 o 0 .6 9 8 10 20 30 P R THRESHOLD ( m m / h r ) 0 .7 3 5 < 0 40 PETTY TMI ALGORITHM 10 20 30 PR THRESHOLD ( m m / h r ) o 0 40 E E Q ' 0 £01 20 0.00 0.00 a o £ 20 01 <o x 10 o cc o 10 20 30 P R THRESHOLD ( m m / h r ) 40 0.00 X O' 0.698 40 40 E E 30 10 20 30 P R THRESHOLD ( m m / h r ) BAYESIAN 2V -M LE 40 20 0 .716 < BAYESIAN 3V-MEAN 40 (m m /h r) o TM I/PR DATA o 20 0 .714 < 40 o o THRESHOLD 20 10 30 PR THRESHOLD ( m m / h r ) o 40 30 0.00 5O 10 10 < 40 THRESHOLD \ E E d 20 0 ALGORITHM BAYESIAN 3V -M LE 40 40 ALGORITHM THRESHOLD (m m /h r) 40 o o o 0 .724 < o 20 10 30 P R THRESHOLD ( m m / h r ) 40 0 .706 < o 10 20 30 P R THRESHOLD ( m m / h r ) Figure 7.13. 2-D distribution of Heidke skill scores (HSS) in random ly selected 118 files in A pril 1998. The value noted in the bottom -right corner of each plot indicates the highest HSS of th e algorithm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 125 GPROF - OLD DATABASE (m m /h r) THRESHOLD 60 I/) cn 1 40 2x 20 < 00 UJ 03 GPROF - (m m /h r) 60 X 40 20 40 60 P R THRESH OLD ( m m / h r ) 80 LINEAR MODEL - NEW DATABASE 80 THRESHOLD 80 0 BEST ALGORITHM o 20 q 20 20 40 60 P R THRESHOLD ( m m / h r ) 80 80 BAYESIAN 2V-MEAN TM I/PR DATA 80 60 o x CO O' 40 X 40 x 40 20 q 20 2 X 5o 20 o < to UJ m 40 60 P R THRESHOLD ( m m / h r ) 20 40 60 P R THRESH OLD ( m m / h r ) 80 20 80 40 60 P R THRESHOLD ( m m / h r ) BAYESIAN 3V-M EAN PETTY TMI ALGORITHM 80 \E E _c .c 60 o 40 80 BAYESIAN 2V -M LE 80 80 (m m /h r) 40 E E o 60 60 20 THRESHOLD X 80 BEST ALGORITHM 80 E E 60 60 40 P R THRESHOLD ( m m / h r ) E E w 60 o 60 01 tn CK 40 X 40 2 X 20 < to BEST ALGORITHM BAYESIAN 3V -M LE PETTY HIST4 ALGORITHM 80 O' 20 o q UJ CD 03 20 40 60 P R THRESHOLD ( m m / h r ) 80 20 < 20 40 60 P R THRESHOLD ( m m / h r ) 80 20 40 60 P R THRESHOLD ( m m / h r ) Figure 7.14. Plots of the best algorithm rain ra te threshold w ith respect to the threshold of P R rain rate for the cases of April, 1998. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 126 PLOT OF HSS_MAX VS. PR THRESHOLD GPROF - NEW DATABASE PETTY TMI ALGORITHM PETTY HIST4 ALGORITHM LINEAR MODEL - TMI/PR DATA' BAYESIAN 3V-MEAN BAYESIAN 2V-MEAN < /) 0.6 co X X < 2 0 .4 0.2 0.0 0 10 20 30 40 PR THRESHOLD ( m m / h r ) PLOT OF HSS_MAX VS. PR THRESHOLD BAYESIAN BAYESIAN BAYESIAN BAYESIAN 3V-MEAN 3V-MLE 2V-MEAN 2V-MLE c/i 0 .6 tn X 0 .4 0.2 0.0 0 10 20 30 40 PR THRESHOLD ( m m / h r ) Figure 7.15. Plots of the m axim um Heidke skill score vs. th e threshold of P R rain ra te based on the d ataset of A pril, 1998. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 127 algorithm s. A lthough the correlation coefficients from th e Bayesian models are not as high as the two aforementioned algorithm s, th e Bayesian 3R-M EAN retrieval shows a smaller bias in the validation. The 2-D HSS distribution (figure 7.13) and th e plot of th e best algorithm vs. P R rain rate threshold (figure 7.14) reveal th e good skill of th e Bayesian-MEAN algorithm s for the case of rain ra te less th a n 20 m m /h r. T he Bayesian 3P-M EA N algorithm seems to have trouble when th e rain rate is around 20—30 m m /hr. Inter estingly, all algorithm s failed to retrieved th e pixels associated w ith rain intensity of ~ 60 m m /h r in the dataset. Figure 7.15 depicts the change of th e m axim um HSS w ith respect to the P R rain rate threshold. In the 1998/04 dataset, th e discrim ination ability for all algorithm s reduces when the tru e rain threshold increases. However, th e P E T T Y HIST4 al gorithm still has b etter ability to differentiate rain ra te higher th a n 20 m m /hr. In addition, all of the Bayesian algorithm s have similar m axim um HSS curves for the random ly picked dataset. 7.4.5 Sum m ary The above validation results have dem onstrated th a t our Bayesian algorithm is able to yield retrieved surface rainfall com parable to other benchm ark algorithm s in term s of the bias, root-m ean-squared difference, correlation coefficient, and the Heidke Skill Score for the various precipitation systems, while our Bayesian algo rithm has the additional advantage of posterior rain rate probability distributions. T he m axim um Heidke Skill Score (HSS) a t a given P R rain ra te threshold for an algorithm is an im portant indicator of th e potential ability to differentiate true rain rate. However, the proportion of th e num ber of hits to th e to ta l d a ta points in the contingency table for all algorithm s drops dram atically (to less th a n 1%) in th e P R rain ra te threshold higher th a n 10 m m /h r, and thus th e m axim um HSS might not be so meaningful when the tru e rain rate is above 10 m m /h r. Therefore, the sum m ary of this chapter will place more em phasis on th e retrieval ability between 0 to 10 m m /h r for the Bayesian 3P-M EA N , 3P-M L E and other benchm ark algorithms. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 128 PLOT OF HSS_MAX VS. PR THRESHOLD f01 0C X X < GPROF - NEW DATABASE PETTY TMI ALGORITHM PETTY HIST4 ALGORITHM LINEAR MODEL - TMI/PR DATA BAYESIAN 3V-MEAN BAYESIAN 3V-MLE 10 4 6 PR THRESHOLD ( m m / h r ) Figure 7.16. Plots of th e m axim um Heidke skill score vs. th e P R rain ra te threshold in the range of [0, 10] m m /h r based on B auer’s cases. PLOT OF HSS_MAX VS. PR THRESHOLD co 0-6 in x X < 2 GPROF - NEW DATABASE PETTY TMI ALGORITHM PETTY HIST4 ALGORITHM LINEAR MODEL - TMI/PR DATA BAYESIAN 3V-MEAN BAYESIAN 3V-MLE 0 .4 0.2 0.0 0 2 4 6 PR THRESHOLD ( m m / h r ) 8 10 Figure 7.17. Same as figure 7.16, b u t for th e heavy and widespread precipitation cases. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 129 Figures 7.16, 7.17, and 7.18 depict th e m axim um HSS vs. P R rain rate threshold plotted only for [0, 10 m m /hr] based on th e aforementioned three validation datasets. Overall, regardless of the bias, th e Bayesian ZV type algorithm s dem onstrate b e t te r maximum retrieval skills th a n th e G PR O F-N E W DATABASE and th e P E T T Y TM I algorithm, b u t their skills are not as high as those of P E T T Y HIST4 and the linear fitting model. The Bayesian 3Y-M EAN and 3Y-M LE algorithm s have similar maximum HSS distributions, b u t th e Bayesian 3Y-M EAN algorithm tends to have slightly higher scores th a n th e 3Y-M LE algorithm a t th e rain rate range of 2—6 m m /h r, especially for th e dataset of 1998/04. W hen the detailed inform ation about bias is concerned, plots of the best algo rithm threshold vs. P R rain rate threshold are used to point out the associated bias for each validation dataset (as shown in figures 7.19, 7.20, and 7.21). Results reveal th a t the retrieved rain rate from P E T T Y HIST4 has a very consistent linear delineation w ith the P R rain rate, even though there is an apparent system atic bias for the validation dataset of heavy/w idespread precipitation systems. The best algo rithm threshold of the Bayesian ZV algorithm dem onstrates a good linear relation to th e PR rain rate threshold as well for th e B auer’s cases and th e d ataset of 1998/04, b u t has significant non-linear delineation and bias in th e cases of widespread and heavy rain rate. On the contrary, th e Bayesian 3Y-M LE shows th e linearity at th e rain range of 1—6 m m /h r in th e p articu lar heavy/w idespread precipitation cases. Surprisingly, the best algorithm threshold of the G PR O F-N EW DATABASE algo rithm shows perfect linear dependence on th e P R threshold for the heavy/w idespread precipitation systems, while all other algorithm s suffer from a notable positive bias. Results in this chapter have m anifested th e sensitivity of th e validation m etrics to the context, the validation dataset, and th e perform ance of the algorithm itself. Comparisons between our Bayesian algorithm s and other retrieval algorithm s only provide a general basis for us to b e tte r understand th e strength and weakness th a t each algorithm might have, and to have th e direction how the algorithm could be used and further improved. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 130 PLOT OF HSS_MAX VS. PR THRESHOLD 1.0 0.8 V) 1 X < 2 GPROF - NEW DATABASE5*PETTY TMI ALGORITHM PETTY HIST4 ALGORITHM LINEAR MODEL - TMI/PR DATA BAYESIAN 3V-MEAN BAYESIAN 3V-MLE 0 .4 0.2 0.0 0 2 4 6 PR THRESHOLD ( m m / h r ) 8 10 Figure 7.18. Same as figure 7.16, b u t for th e d ataset of April, 1998. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 131 GPROF - 2 NEW DATABASE 4 6 8 P R THRESHOLD ( m m / h r ) LINEAR MODEL - TM I/P R DATA 2 BAYESIAN 3V PETTY TMI ALGORITHM 2 4 6 8 4 6 P R THRESHOLD ( m m / h r 2 4 MEAN 6 PR THRESHOLD ( m m / h r ) P R THRESHOLD ( m m / h r ) PETTY HIST4 ALGORITHM BAYESIAN 3V -M LE 2 4 6 8 P R THRESHOLD ( m m / h r ) 2 4 6 8 8 PR THRESHOLD ( m m / h r ) Figure 7.19. Plots of th e best algorithm threshold vs. P R rain rate threshold in the rain ra te range of [0, 10] m m /h r for th e B auer’s cases. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 132 GPROF - 2 NEW DATABASE LINEAR MODEL - TM I/P R DATA 4 6 8 P R THRESHOLD ( m m / h r ) P R THRESHOLD ( m m / h r ) PETTY TMI ALGORITHM BAYESIAN 3V—MEAN 2 4 6 8 P R THRESHOLD ( m m / h r ) 2 PETTY HIST4 ALGORITHM 2 4 6 P R THRESHOLD ( m m / h r ) 4 6 8 P R THRESHOLD ( m m / h r ) BAYESIAN 3V -M LE 2 4 6 8 P R THRESHOLD ( m m / h r ) Figure 7.20. Same as figure 7.19, bu t for heavy/w idespread precipitation cases. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 GPROF - 2 LINEAR MODEL - NEW DATABASE 4 6 8 P R THRESHOLD ( m m / h r ) 2 PETTY TMI ALGORITHM — 1— 1— 1— 1— 1— .— 1— 1— 1— 1— 1— 1— 1— i— 1— 1— I— 1— 1—> M *' 1 10 r f r 4 6 8 P R THRESHOLD ( m m / h r ) BAYESIAN 3V—MEAN 1 . 1 l— ,--- .----r t ", . r l | 10 . -m , . £ S . E X 8 o m TM I/PR DATA * M o * m m 6 O' I ' *' * • 4 2 in UJ CD M ' • X 5 o o < • .............................................. 2 . . . 1 . 4 6 8 PR THRESHOLD ( m m / h r ) . . * m' ■ 2 PETTY HIST4 ALGORITHM 2 4 6 8 P R THRESHOLD ( m m / h r ) , 4 6 8 P R THRESHOLD ( m m / h r ) 10 BAYESIAN 3V -M LE 2 4 P R THRESHOLD ( m m / h r ) Figure 7.21. Same as figure 7.19, b u t for th e d ataset of April, 1998. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 134 8. U N C E R T A IN T Y ANA LY SIS Based on Bayes theorem , th e posterior probability is proportional to th e product of the conditional likelihood and th e prior probability distribution, which are both derived from the PR -T M I m atch-up d a ta in our Bayesian 2 V and ZV models. Given this, th e most natural questions to ask might be: How do we know if the physical model th a t we posited to statistically describe th e relationship between rain rate and observation vector provides a good fit to th e d ata? How does the assum ption of th e prior distribution affect th e posterior probability? These two questions concern th e uncertainty of the posterior pdf, and they are directly related to the model assessment and the robustness test of the Bayesian algorithm , respectively. In the section, th e analysis for th e two questions will be presented. 8.1 A ssessm en t o f th e p osited conditional likelihood There are qualitative and quantitative ways to assess th e fitness of th e condi tional likelihood in the model. Q uantitative means provide a hypothesis test and corresponding discrepancy variables to suggest if th e posited model should be ac cepted or rejected under th e observed dataset. However, it is still controversial since th e test results strongly depend on w hat kind of statistical variable is used, and different variables might lead to completely opposite conclusions. In addition, if the statistical value does not support rejection of th e model, it does not guarantee th a t th e model provides a high goodness-of-fit. In contrast to q u an titativ e m ethods, qual itative approaches (e.g., visual graphs) help to bring a more intuitive image of how th e observed/sim ulated d a ta behave. Although the graphical way has a drawback in th a t th e difficulty increases in m ultivariate cases, it offers a m ore direct comparison and avoids m isinterpretation assessments based on a single num ber. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 135 JOINT PDF ( 1 V MODEL) 14 12 10 8 6 4 2 OL 0.0 0.2 0.4 0.6 0.8 1.0 P37 JOINT PDF (OBSERVATIONS) 14 12 10 8 6 4 2 0 0 .0 0 .2 0 .4 0 .6 P37 0.8 1.0 Figure 8.1. Plots of joint probability density functions / ( F 3 7 , R) based on th e I V model (upper) and near-nadir PR -T M I d a ta (bottom ). C ontours are logarithm ically spaced; actual value is 1 0 x' where x is th e contour label, x are plotted for values of [-5, -4, -3, -2, -1, 0, 1] and [-4, -3, -2, -1, 0, 1] for th e I V model and observations, respectively . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 136 In th e Bayesian I V model, since only two variables are involved, th e model is assessed by the comparison between observed and sim ulated joint distribution. Figure 8.1 depicts th e joint distribution of f ( P 37, R ) and th e p d f derived from th e near-nadir PR -T M I m atch-up data. Contours of th e density function are logarith mically spaced, so the value of -2 presents the probability of 0.01. Note th a t since th e sample size of the observations is around 1.2 million, the contour value less th an -3 indicates th a t the grid box has a num ber of pixels less th a n 4. In th a t case, the contours of -4 and -5 are not very meaningful for th e comparison, bu t rath er give an indication of an overall coverage of th e data. The results in figure 8.1 dem onstrate th a t th e conditional probability density function in th e I V model visually fits the observations very well. However, there are some minor differences between those two distributions. First, the contour w ith probability of 0.01 in the PR -TM I d a ta goes up to 10 m m /hr, b u t only to 8 m m /h r in th e model. In addition, the nearly saturated P37 is less frequent in the PR -T M I d a ta th a n th a t in th e simulations. Fur therm ore, some artificial effects are shown in th e -3 contour a t larger P37 values in th e IV model. The goodness-of-fit of the 2V model is evaluated from th e joint pdf, f ( P i 9,R ) , which is the m arginal pdf of f(Pig,P37, R) and could be com puted from /(P i9 , R ) = f f ( P 37\R)f(Pi9\P37, R M R ) • d ( p 37). JP37 (8.1) Similarly, th e 3V model is checked by the f ( P w ,R ) , yielded from the following equation: f(P w,R )= [ [ f ( P 37\ R ) f ( P 19\P37, R ) f ( P w \P19P37)n( R) ■d(P37)d(Pl9). (8.2) JPig J P 3 7 The joint probability distribution derived from th e 2 V model (as shown in figure 8.2) reveals th a t the functions describing th e physical relationship between P 19, P37l and R provide a good fit to th e observed data. The coverage and shape of the contour w ith probability of 0.001 in the 2V model are b o th very consistent w ith those of the PR -T M I data. However, th e contour of probability w ith 0.01 shows a discrepancy when Pig is between 0.25 and 0.35. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. JOINT PDF ( 2 V MODEL) 40 30 E E <u 20 □ c 'o cr 0.0 0.2 0.4 1.0 0.8 0.6 P19 JOINT PDF (OBSERVATIONS) 40 30 E E a) 20 o ~ 0 .0 0.2 0.4 0 .6 P19 -T 1 I I - ___ 0 .8 0 1.0 Figure 8.2. Same as figure 8.1, but for th e joint p df of (P 19,R) based th e 2 V model (upper). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 138 JOINT PDF ( 3 V MODEL) 40 30 20 10 oL 0.0 0.2 0 .4 0.6 0.8 1.0 P10 JOINT PDF (OBSERVATIONS) 40 30 20 ;Vi- 10 oL 0.0 0.2 0 .4 0.6 0.8 1.0 P10 Figure 8.3. Same as figure 8.1, b u t for the joint pdf of (Pio,R) calculated from th e 3 V model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 139 The distribution of / ( P 10|P ) is illustrated in figure 8.3, and th e results suggest th a t th e construction of th e 3 V model is appropriate as well to yield a good agreem ent w ith th e probability distribution of th e PR -T M I data. It is notew orthy th a t the coverages of th e contours w ith value of -4 and -5 are not quite th e same as th e observations. At a given rain ra te of 40 m m /h r, the observations are distributed in a P w range of [0.3, 0.8], while th e -5 contour in model intercepts th e rain ra te of 40 m m /h r at Pw = [0.35, 0.9]. Note th a t th e physical model overestim ates occurrences at P values greater th an 1.0 in figures 8.1, 8.2 and 8.3. T he discrepancy is introduced from th e shape and skewness of the param etric function fitted to the d a ta do not m atch those of the observed probability distribution at very high P values. However, since large P values often indicate no rain or very light surface rainfall, th e discrepancy here does not have a great im pact on th e perform ance of our Bayesian algorithm . One might argue th a t th e m arginal probability distribution used to assess the 3 V model is an integrated quantity, and it might not be able to provide the details of the 4-D m ultivariate relationships. To clarify this issue, we com pare th e observed histogram and the m odel-calculated probability distribution of Pw at a given interval of P 19, P37, and R. Figure 8.4 depicts th e histogram s from observations (dashed curve) and model-derived conditional distributions (dotted curve) for six grid boxes only in this 4-D variable space. It dem onstrates th a t the fitted conditional likelihood, /(P io |P i 9 , -P3 7 , R), is very close to th e observed distribution if the sample size is large enough. Figure 8.4(e) shows a shift of approxim ately 0.1 on th e P 10 value, and th e associated variance of th e fitted distribution tends to be larger th a n th a t of the observed d a ta under heavy precipitation (as shown in b o th (e) and (f)). It is plausible to param eterize th e fitted functions w ith a reduced variance and thus to make the /( P io |P l 9 , P 3 7 , R) more dogmatic. However, due to th e larger field of view, observations a t th e 10.65 GHz channel have a more serious beam-filling problem th an other channels. Thus, it might not be beneficial for the retrieval to p u t more weight on th e conditional likelihood of P i0, and test results (not shown) proves th e point. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 140 12 (b) R = P19= P37= N = R = 1.50 P 1 9 = 0 .7 0 P 3 7 = 0 .4 0 N = 2401 10 >. 4 .5 0 0 .3 5 0 .2 5 8 Jl •” ( I 'it _o 5o 4 I it CL 0 L 0 .0 0 .2 10 0 .4 0 .6 P10 0 .8 0.0 1.0 8 6 .5 0 0 .4 5 0 .2 0 158 6 _>N 6 15 0r> 4 O CL 4 0 .6 P10 0 .8 1 i i - 2 2 0 .4 ' ' 1 1 ................... ' ( d ) R = 1 1 .5 0 p i 9 = 0 .3 0 /' P 3 7 = 0 .1 0 11 N = 90 1 '''/ 4. ■ 8 R = P19= P37= N = 0.2 .1 1.0 ■ - - 'l! 1 i \i 'i 0 0.8 0 0 .0 1.0 P10 I 1. :(e) . 11 1. 11 /'R = \ P19 = 1 P37= / 'n = ' :i \ ' ■'1 '■ l _ : : 1 • 1 • / - 15 .5 0 0 .1 5 ' 0 .0 5 43 0 .2 . . J . ............ V v . 1 .>. 0 .4 0 .6 0 .8 1.0 P10 6 R = 19 .5 0 P 1 9 = 0 .1 0 P 3 7 = 0 .0 5 N = 22 5 4 3 2 11 -t u - 0 0 .0 0 .2 0 .4 0 .6 P10 0 .8 1.0 0.6 0 .8 1.0 P10 Figure 8.4. Plots of conditional pdfs /( P io |P i 9 , -P3 7 , R) based on th e PR TM I d a ta (dashed curve) and th e posited Bayesian model (dotted curve), a t a given set (P , P19, P37) of (a) (1.5, 0.70, 0.40); (b) (4.5, 0.35, 0.25); (c) (6.5, 0.45, 0.20); (d) (11.5, 0.3, 0.1); (e) (15.5, 0.15, 0.05); and (f) (19.5, 0.1, 0.05). N noted in figures represents the sample size of observed data. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 141 If the sample size is too small from a statistical point of view, there m ight be some problems in the fit using the param etric functions. For example, in th e figure 8.4(b), th e sample size is only eight, and there are two very different regimes shown in th e distribution. It is hard to determ ine if these two peaks are statistically different due to the small sample size. However, it might give some im plications here. T he peak a t a lower P w value m ight represent a scene associated w ith less beam-filling (e.g., the widespread precipitation system). Under this kind of rain cloud, the P values at three channels are around 0 .2 -0 .4 at a given rain rate of 4.5 m m /h r. T he other peak on th e right side describes a case more like a convective system, which is associated w ith a strong beam-filling effect and produces a much higher P w value. Since the sample size is so small, it is difficult to fit th e distribution and to further evaluate the adequacy of th e posited pdf. W hile th e model suggests fitting th e distribution w ith th e do tted line, the lack of goodness-of-fit on th e left peak will result in an overestim ate of rain rate in widespread rain events, as shown in chapter 7. On th e other hand, th e shift of the right peak reduces th e retrieved rain intensity. Considering figure 8.4(b) and (e) together, th e m ism atch in the fitting to th e observed d a ta results in th e lack of th e ability to retrieve higher P R rain ra te in th e Bayesian 3U algorithm . 8.2 U n certain ty o f th e p osterior m ean due to th e prior distrib u tion The sensitivity of our Bayesian rain rate retrieval algorithm to th e prior is evalu ated to ensure th e robustness of the algorithm and to quantify th e uncertainty of th e posterior mean. There are several ways to estim ate th e uncertainty caused from th e specification of the prior, b u t the analysis here is adapted from th e work of Berger (1990). First, a class of priors was selected to model prior uncertainty. This class has to include as m any reasonable priors as possible, b u t meanwhile it should not contain unreasonable priors. For com putational convenience and adequacy, we em ployed an e-contamination class of priors in th e study. Detailed descriptions and related m athem atical proofs about the class are given in Berger (1990). T he class of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 142 priors is expressed as r« = {IT = (1 - e K , + eq: q € L}, (8,3) where ttq is th e base prior distribution used in our algorithm; e represents the am ount of uncertainty in 7To and ranges between 0 and 1; q is th e contam inated function; and L describes th e class of th e q. The choice of th e L has a great effect on th e estim ation of th e uncertainty of the posterior mean. Sivaganesan and Berger (1989) suggested th a t there were four possible sets to define L. In our case, since th e distribution of rain ra te R is close to lognormal, th e probability of In R would generally follow a norm al distribution, which is symmetric and unim odal. Due to this characteristic of the base prior distri bution, contam inations associated w ith a set of sym m etric unim odal distributions are reasonable and appropriate to model th e uncertainty of th e prior. Furtherm ore, the contam ination was added w ith respect to In i? first, then th e probability d istribution was transform ed back to th e rain rate domain. Once we have chosen the symmetric unim odal subset for contam ination, q could be represented as a m ixture of uniforms: roo \ (8.4) where I is the indicator function; G is an a rb itrary distribution on (0, oo); p is the param eter describing th e mode of th e norm al distribution over Ini?; and po represents the param eter used in th e base prior, 7r0. Note th a t the param eters (p, a) of the lognormal distribution for rain rate are exactly th e same as the param eters in the norm al distribution for In R. From th e equations (8.3) and (8.4), the posterior mean can be w ritten as q{tt) Sr t ■f ( P \ r ) n ( r ) d r (1 - e) J f ( P \ r ) n 0{r)dr + e f£° H 2(z)Gdz Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (8.5) 143 where g(7r) and m(ir) are the posterior mean and m arginal probability of P associated w ith the prior ir. In addition, 1 f e+ ,1 (8.6) where 9- = expf^o - z) ; 6+ = exp(p<> + 2 ). (8.7) By defining f ( z) = (1 - e) / + ' Hi ( z ) g(z) = ( l - e ) J f ( P \ ( ) M m + t H 2(z), (8.8) equation (8.5) can be rew riten as ^ ff(z)Gdz e M = lW ) G T z 7 (8.9) ' Finally, for a quantity formed as equation (8.9), th e variations of the q u an tity could be obtained by finding the m inimum and m axim um values of f ( z ) / g ( z ) (proof can be found in Sivaganesan and Berger (1989)). 8.3 U n certain ty o f th e posterior variance due to th e prior d istrib u tion It is im portant to not only evaluate th e uncertainty of the posterior mean, bu t also provide an accuracy measure by considering th e change of the posterior vari ance. Sivaganesan and Berger (1989) proposed an elegant m athem atical framework to estim ate th e sensitivity of th e posterior variance to the prior distribution. Let ( f and V w represent the posterior m ean and posterior variance w ith respect to the prior 7r. For g0 £ [h^Tr L f ( r ), sup^ gn(r)}, if we let r 0 = {tt = (1 - e)7r0 + eq: q E Q and g*{r) = p0}, (8.10) where r 0 expresses th e class of the prior, and g0 represents the fixed posterior mean, then we would like to find infT V n (r), sup^ V n(r) to examine the range of th e posterior Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 144 variance at a given posterior mean. T he problem could be solved by a theorem (proof was dem onstrated in Sivaganesan and Berger (1989)) th a t sup V n(r) = 7r€ro inf V ^(r) = 7rer0 sup V n (r) 7rero2 inf V*(r), 7rer02 (8.11) where r 0 2 = {tt — (1 — e)7To + eq G Tq : q € (?2 }> (8.12) and Q 2 = {q = aU(/Jlo - z ,H o + z ) + ( l - a ) U ( f i 0—z*,/j,q+z*) : 0 < a < 1 and z , z * > 0}, (8.13) where U represents the uniform function. One of th e three quantities a , z , z* has to be determ ined by the constraint th a t th e posterior m ean for n is g0, and thus the m axim izations over the r 02 are effectively a two-dimensional problem. Here, we have chosen to constrain the coefficient a , and a could be obtained by Q0 ~~ (! - e) I r t ' f ( P \ r ) n 0(r)dr + e a ± //+ f ( P \ r ) d r + e(l - a ) ^ f r f ( P \ r ) d r f. o* > (! —e) I r f ( P |r)7T0(r)<fr + / #_+ f ( P \ r ) ± d r + e(1 - a ) ^ /„.* /(P |r ) ±<fr (8.14) where 9*_ = exp(//0 - z*) ; 9*+ = exp(/j0 + z*). Once a is determ ined, the posterior variance V* over r 02 (8.15) can be calculated from V?m ( z , z ' ) = (1-e) JR(r-So)2f(P\r)iro(r)dr+ea-^(r-g0)2f(P\r)-^dr+e(l-a)-fe} {r~eo)2f(P\r)^dr --------------------------------------------- ------------------------------ 55— --------------------(8.16) (l-«) f R f(P\r)iro(r)dr+ea-± f ( P \ r ) - ^ d r + e ( l ~ a ) ^ r f g,+ f ( P \ r ) ± d r Then, the ranges of the posterior variance could be found by searching th e m axim um and minimum of the variance VpQ2 over th e 2-dimensional (z vs. z*) domain. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 145 8.4 R esu lts Based on the framework in th e two previous sections, th e sensitivity of th e poste rior m ean and variance to the prior can be evaluated for th e Bayesian models. Since th e Bayesian 3 V model has deficiency in th e fit at higher rain rate, the uncertainty analysis for the 3 V model might not be so meaningful. Therefore, only results from the I V and 2V models are shown in the section. In addition, to dem onstrate the variations of posterior pdf in th e same units as rain rate, th e sensitivity of posterior variance is depicted using th e posterior standard deviation. 8.4.1 IV m odel Figure 8.5 depicts th e sensitivity of th e posterior m ean to the prior distribution in the I V model when a sym m etric and unim odal contam ination function is applied to model the uncertainty of th e prior. The am ount of uncertainty in the prior is assumed to be 25% in th e study. Note th a t since th e tails of the prior drastically affect the posterior distribution, th e uncertainty analysis here mainly concentrates on adding a larger tail to the assum ed prior and on estim ating how much the posterior m ean will increase due to th e higher rain rate of th e prior. The curves of lower (dotted) and upper (solid) bounds in figure 8.5 reveal th a t the variation of the posterior mean becomes more significant when th e value of P37 is less th an 0.2. If th e P 3 7 is close to saturation, th e uncertainty of th e prior may result in a 3 m m /h r difference (increasing from 8.5 to 11.5 m m /h r) in th e retrieval, which is about 35% of change. The increase of th e posterior m ean comes from th e heavier tail of th e contam inated prior, b u t beyond some point, P37 is satu rated and is no longer able to distinguish rain rate greater th a n 12 m m /h r from others. T he result indicates th a t other variables are necessary to increase the dynam ic range of th e retrieval. The ranges of the posterior stan d ard deviations for given values of th e posterior m ean are shown in figure 8 .6 a t P37 value varying from 0.6, 0.4, to 0.2. First, when th e prior is not contam inated, th e posterior stan d ard deviation increases rapidly from 0.5 to 3.5 m m /h r w ith decreasing P37 value. The result is expected since a wide range of rain ra te could occur in th e case w ith low P37. In addition, when P37 is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 146 1V m o d e l 14 Choice = L - s u 12 epsilon = 0 .2 5 muR = - 2 . 8 10 8 6 4 2 0 0.0 0.2 0.4 0.6 P37 0.8 1.0 Figure 8.5. Plots of the posterior m ean (m m /hr) vs. P 37 in the I V model under the specification th a t th e prior distribution is log N ( —2.8, 2.0). The uncertainty of the prior is sim ulated by th e symmetric contam ination function w ith a factor of 0.25. Solid and dotted curves indicate th e suprem um and infimum, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 147 P 3 7 = 0 .6 0 9 0 .9 0.6 0.3 0.60 0.80 0.7 0 0.75 0.65 POSTERIOR.MEAN ( m m / h r ) 0.85 P37 = 0.40 1.4 ;(b) 1.2 h- 1.0 LlJ Q 0.8 0.6 a. 1.30 1.60 1.70 1.50 1.40 POSTERIOR.MEAN ( m m / h r ) 1.80 P37 = 0.20 _c 8 o i— < /> o 2L CL 3.0 4.0 4.5 3.5 POSTERIOR.MEAN ( m m / h r ) 5.0 Figure 8.6. Range of posterior stan d ard deviation for th e given posterior m ean in the I V model when P 37 is (a) 0.6, (b) 0.4, and (c) 0.2. The square and asterisk symbols present th e minimum and m axim um values, respectively. Thus, the area inside th e curves indicates th e region where th e standard deviation m ight be located. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 148 0.6, the posterior mean varies from 0.64 to 0.82 m m /h r, and th e posterior standard deviation changes between 0.5 and 0.8. If th e P 3 7 decreases to 0.2, the standard deviation changes significantly from 3.5 to 6 m m /h r due to th e uncertainty of the prior. T he result implies th a t cautions have to be given while th e pixel for retrieval is associated w ith low P37. 8.4.2 2V m odel The minimum, m aximum and range of th e posterior m ean in th e 2 V model over th e P 19 and P 3 7 domains is shown in figure 8.7. Note th a t only th e pairs of (P19, P 37) observed in the PR -T M I m atch-up d a ta are plotted in this figure. Points, for ex ample, having (P19, P 37) = ( 0 . 2 , 0.8), are unlikely to occur in real-world applications, and are not discussed here. The distribution of th e lower bound indicates th a t the dynam ic range of the retrieval extends to rain ra te greater th a n 15 m m /h r, and the posterior m ean increases w ith decreasing P19 and P 37. In th e case th a t P 3 7 = 0 .2 , th e I V model only yields 3—5 m m /h r (as figure 8.5 shows), b u t a range between 3—15 m m /h r can be retrieved in th e 2 V model, depending on th e value of P19. In addition, the range between th e m inim um and th e m axim um of th e posterior mean (as shown in figure 8.7(c)) reveals th a t except some points, m ost variations of the posterior m ean are less th a n Figure 8 .8 6 m m /h r, indicating th a t the 2 V model is quite robust. depicts the range of the posterior stan d ard deviation in th e 2 V model while th e contam ination factor of 0.25 is added to the prior pdf. Results indicate th a t th e percentage of change in stan d ard deviations is around 25% — 33%, and goes up to around 45% when th e P values a t 19.35 and 37.00 GHz b o th are near saturated. Similar to the I V model, th e 2 V model also shows the reduced accuracy for the pixies associated w ith low P 19 and P 3 7 values. Meanwhile, th e posterior standard deviations for these pixels are th e m ost sensitive to th e uncertainty of the prior distribution as well. Unfortunately, low P values usually indicate th e presence of substantial precipitation, which is of particular interest. Therefore, to reduce the posterior standard deviation and the sensitivity to the prior, more variables might be needed to add more inform ation to make th e posterior p d f more informative. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 149 POSTERIOR.MEAN.MIN IN L-SU 1.0 RAIN RATE (M M /H R ) 0.8 ps. 0.6 Q_ 0.4 0 .2 0.0 0.0 0.2 0.4 0.6 P19 0.8 1.0 POSTERIOR.MEAN.MAX IN L-SU 1.0 RAIN RATE (M M /H R ) 0.8 r-s. 0.6 to Q_ 0.4 0 .2 0.0 0 VARIATIONS OF POSTERIOR.MEAN IN L-SU 1.0 RAIN RATE (M M /H R ) 0.8 ^ to a. 0.6 0.4 0.2 0.0 0 Figure 8.7. Range of posterior m ean in th e 2 V model for (a) th e lower bound, (b) upper bound, and (c) th e m aginitude of variation in units of (m m /h r). The contam inated function and factor used here are th e same as those applied in the I V model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 P 1 9 = 0 .6 0 ,P 3 7 = 0 .4 0 P19 = 0 .7 0 ,P 3 7 = 0 .4 0 2.0 2.0 \£E JZ e E E, o g z z < > < o O > ag o O' S £ 0 .5 0 .5 o Q. 0.0 2.0 0.0 1 .5 0 1 .6 0 1 .7 0 1 .8 0 1 .9 0 POSTERIOR.MEAN ( m m / h r ) 2 .00 2.2 2 .4 2.8 2.6 POSTERIOR.MEAN ( m m / h r ) 3 .0 3 .2 P19 = 0 .4 0 ,P 3 7 = 0 .2 0 P19 = 0 .5 0 ,P 3 7 = 0 .2 0 10 6 E E 4 z o •i > iij 3 o O' o 2 CE 1 O F ......................... 3 .2 3 .4 . . ....I 1 4 3 .6 3 .8 4 .0 4 .2 POSTERIOR.MEAN ( m m / h r ) 4 .4 9 7 8 POSTERIOR.MEAN ( m m / h r ) 6 4 .6 P 1 9 = 0 .2 0 ,P 3 7 = 0 .0 5 P 1 9 = 0 .2 0 ,P 3 7 = 0 .1 0 14 14 12 12 10 10 8 8 6 6 12 13 10 7 .0 14 16 15 POSTERIOR.MEAN ( m m / h r ) 7 .5 8 .5 9 .0 8.0 POSTERIOR.MEAN ( m m / h r ) 9 .5 10.0 23 24 P 1 9 = 0 .0 6 ,P 3 7 = 0 .0 7 P 1 9 = 0 .0 5 ,P 3 7 = 0 .0 5 18 16 O O 14 14 10 15 16 17 18 19 POSTERIOR.MEAN ( m m / h r ) 20 21 18 19 22 20 21 POSTERIOR.MEAN ( m m / h r ) Figure 8.8. Same as figure 8.6, b u t for 2 V model and given (Fig, F 37)= (a) (0.7, 0.4) (b) (0.6, 0.4), (c) (0.5, 0.2), (d) (0.4, 0.2), (e) (0.2, 0.1), (f) (0.2, 0.05), (g) (0.05, 0.05) and (h) (0.06, 0.07). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 151 8.5 Sum m ary Analysis of the model assessment in this chapter indicates th a t th e specified conditional likelihoods are able to provide a good fit to th e actual observations and present the statistical and physical relationships between surface rainfall and microwave variables. Furtherm ore, results of the robustness test suggest th a t our Bayesian algorithm is robust even under th e existence of uncertainty in th e prior rain ra te distribution. Note th a t a 25% contam ination factor in th e prior distribution is added in the entire uncertainty analysis, and thus results here m ight present a larger extent of variations in th e posterior mean and variances th a n th e reality. W hen the contam ination factor can be estim ated in a more quantitative way, th e analysis conducted in th e chapter can easily be applied again to com pute th e uncertainty. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 152 9. C A L IB R A T IO N OF M ODEL SIM ULATIO N In chapter 3, we constructed a Bayesian retrieval algorithm from NWS W SR - 8 8 D network 1-km composite reflectivity m easurem ents and a simplified 1-D radiative transfer model. A fixed Z-R relationship ( Z = 200i?L6) was used in th e estim ations of rain rate, which was not calibrated w ith any other rain rate m easurem ents. W hen a reliable d ataset of observed rainfall intensity is available, how can th e model-derived Bayesian retrieval algorithm be modified to yield calibrated rain ra te estim ates? On th e other hand, since the PR -T M I m atch-up d ataset provides sim ultaneous observa tions of rain ra te and microwave signature, it offers a basis to validate th e radiative transfer calculations in our simplified 1 -D plane-parallel model. Again, we assume th a t P R interpolated rain rate presents tru th , and both th e calibration and valida tion of th e retrieval algorithm are based on the P R measurements. Once a b etter d ataset becomes available, the same principles and procedures can be easily applied to the new training dataset. 9.1 C alibration o f radar-sim ulated rain rate In th e study, we suggest calibrating th e estim ates derived from NWS WSR- 8 8 D network reflectivity adjusting th e cum ulative density function (cdf) of model- sim ulated rain rate to th a t of PR -T M I m atch-up data. Figure 9.1(a) presents the cdfs of near-nadir interpolated rain ra te of P R in 1998, and the W SR - 8 8 D networksim ulated rain rate derived from a fixed Z-R function and an assum ed freezing level of 3 km. It is evident th a t the sim ulated rainfall intensity from NW S W SR - 8 8 D m easurem ents has a larger portion of d a ta associated w ith higher rain rate. For example, a t a given rain rate of 2 m m /h r, th e cumulative density has reached to 0.9 for the P R d ataset, meaning th a t only 10% of P R rain rate is greater th a n 2 m m /h r. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 153 However, 20% of the W SR - 8 8 D sim ulations are associated w ith rain rate higher th a n 2 m m /hr. It implies th a t the fixed Z-R relationship m ight not be always suitable for every pixel. In an atte m p t to force th e sim ulated rain rate d istribution to be close to th e PR TM I dataset, a lookup table based upon the cdf of th e observations was generated to describe the conversion between the NWS W SR- 8 8 D network reflectivity-derived rain intensity and calibrated rainfall rate. Then, th e sim ulated rain intensity was adjusted at each grid point to produce a cumulative probability distribution closer to th e cdf of P R rain rate. As a result, th e cdf of modified rainfall ra te from W SR- 8 8 D simulations is able to converge to th e PR curve after two adjustm ent iterations (as shown in figure 9.1(b)). 9.2 V erification o f rad iative transfer calculations Once the adjusted W SR - 8 8 D network-simulated rain ra te is ready, th e assum p tions of radiative properties and radiative transfer calculations in our simplified plane-parallel model can be verified by the m ultivariate relationship between rain rate and microwave signature shown in Pw , Pig, and P37. T he 4-D distribution of th e occurrence frequency is dem onstrated by plots of 2-D contours while the other two variables are selected w ithin certain intervals. Figure 9.2 depicts the contours of th e num ber of d a ta points where th e corresponding rain ra te is less th an 1 m m /h r. T he left panel is the distribution from the W SR- 8 8 D network sim ulations a t a given freezing height of 3 km, and th e middle panel is th e result draw n from the PR -T M I m atch-up data. The orientations of the contour distributions from th e observations and simulations are consistent. However, it is obvious th a t th e contours of PR -T M I m atch-up d a ta has a long and narrow shape and the coverage of th e contours extends further to lower .P3 7 values. T he model simulations indicate a broader distribution in the direction of P 1 9 , b u t narrower along the P 3 7 axis. W hile d a ta points having rain rate between 1 and 5 m m /h r are selected, similar discrepancies shown in th e previous contour plots are also found here in th e com par R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 154 ison between th e observed and sim ulated d a ta (as shown in figure 9.3). Note th a t the contour of PR -T M I m atch-up dataset has a tongue-like shape tow ard very low Pw (less th a n 0.2) values, b u t the sim ulated P\g values are seldom smaller th a n 0.2. In addition, if the Pio is in the range between [0.55, 0.65], some sim ulated points w ith higher P37 values are not shown in the PR -T M I data. The existence of these points might be due to th e Gaussian random noise we added in th e model. For the case of larger rain rate (5 -1 5 m m /h r), figure 9.4 suggests th a t these two datasets have a similar m ultichannel relationship, although th e contours of th e simulations still cover a larger area at th e P37 vs. P 19 domain. Furtherm ore, it is notable th a t the location of the m aximum occurrence from th e observations is a t P37 values of 0 .1 w ith a varying Pw, bu t th e sim ulations have more points associated w ith almost zero in P37 values. Under very heavy precipitation (rain ra te greater th an 15 m m /h r as shown in figure 9.5), the sim ulated d a ta produce more points w ith much higher P37 values, and those points are not generated from th e observations. The possible source of errors is given in the next section. 9.3 E ffects o f freezing level height The discrepancy in th e 4-D distribution between th e PR -T M I m atch-up d a ta and the W SR- 8 8 D sim ulations might be caused from th e assum ption of th e freezing height. To examine th e adequacy of th e assigned value of th e freezing height, the bright band inform ation is extracted from th e 55 orbits of th e P R 2A25 product to clarify the issue. Again, in order to avoid th e uncertainty from off-nadir contam ina tion, only near-nadir pixels are included here. Figure 9.6 delineates th e histogram of the height of P R bright bands in various seasons. T he distributions reveal th a t th e bright bands are generally located at altitudes of 4—6 km in tropical environm ents. If we assume th a t th e freezing level is about 0.25 km higher th a n th e peak of th e bright band, then th e freezing heights estim ated from P R reflectivity are around 4.25 — 6.25 km. Note th a t th e histogram has a strong peak in altitude of 6 km for all seasons, and it is not clear yet if an upper lim it has been applied to th e altitu d e of R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 155 the bright bands in th e P R 2A25 algorithm . Since there is no independent d ataset to help validate th e locations of th e bright band, it is not possible here to quantify th e errors of th e estim ations. However, radiosonde observations rarely reveal freezing level heights greater th a n 5 km. To provide further inter-com parison w ith estim ated P R bright bands, freezing heights based on TM I m easurem ents are also calculated using the m ethod W ilheit et al. (1991), and th e following regression equation we proposed: Z f = exp 3.469 - 7.6041 InV (9.1) where the freezing height Z f is in units of km, and V is the column w ater vapor (kg/m 2). This regression model is derived from th e TM I and RAOBS m atch-up d ata, same as those used to develop th e w ater vapor algorithm . The corresponding correla tion and root-m ean-square error for the estim ations are 0.86 and 0.4 km, respectively. In W ilheit’s m ethod, th e freezing level heights are determ ined by brightness tem per atures at 19.35 and 22 GHz channels via a lookup table, which is based on model simulations using a simplified atm ospheric condition. The histogram s of freezing height for oceanic and rainy pixels in Jan u ary and July of 1998 based on TM I observations are shown in Figure 9.7. O ur regression model and W ilheit’s m ethod b o th yield m ost freezing levels at around 5 km, even though the second m ode of th e distribution is located at different altitudes. Based on the radiosonde and TM I observations, an upper limit of 5 km for th e freezing level height seems more reasonable. Therefore, we modify the param eter of th e freezing level height from 3 km to 5 km in our model, re-generate simulations, and re-com pare with the PR -T M I m atch-up datasets. The re-generated sim ulations w ith th e freezing level at 5 km are shown in the right panels of figures 9.2, 9.3, 9.4, and 9.5. These results dem onstrate th a t th e assum ption of higher freezing level yields very consistent 2-D contours w ith those of the observations, in term s of contour extension, orientation, and the location of th e maximum occurrence frequency. It suggests th a t th e assum ptions in our simplified R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited without perm ission. 156 radiative transfer model is appropriate. However, generally, the sim ulations still have broader coverage in the direction of Pi 9 , and P 3 7 is still satu rated a t lower value th a n the PR -T M I d a ta w ith increasing rain rate. 9.4 Im plication Simulations based on the NWS W SR - 8 8 D reflectivity and the simplified 1 -D plane-parallel radiative transfer model have shown a surprising good agreem ent w ith the PR -TM I m atch-up d a ta in the m ultivariate physical relationship. In addition, model simulations suggest th a t th e free param eter of heights of freezing levels has a significant im pact on the 4-D (rain ra te and P vector) probability distribution. To obtain b e tte r retrieval, th e Bayesian rain ra te algorithm has to be developed according to various freezing levels, and th en th e free param eter can be applied to select more appropriate Bayesian models to com pute the retrieval. More detailed analysis has to be conducted for th e validation of heights of freezing levels. It is still not quite understood w hat th e uncertainty of the estim ates of freezing heights is, and how much error in freezing heights is acceptable in th e retrieval application. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 157 CDF of Rain Rate PR Da ta ( 2 3 7 6 3 0 8 ) S im u l a te d Da ta ( 4 7 8 4 5 1 9 ) 0.6 U- O o 0 .4 0.2 0.0 0 2 4 6 8 10 12 Rain R a t e ( m m / h r ) CDF of Rain Rate 1.0 0.8 PR D at a ( 2 3 7 6 3 0 8 ) S im u l a te d D ata ( 3 6 5 4 9 9 7 ) 0.6 0 .4 0.2 0.0 0 2 4 6 8 10 12 Ra i n R a t e ( m m / h r ) Figure 9.1. Cum ulative distribution functions of P R observed (solid curve) and NWS W SR-88D network-sim ulated (dotted curve) rain rate. P R curve is based on th e near-nadir P R interpolated rainfall intensity during Jan., Apr., Jul., and Oct. of 1998. W SR-88D-simulated curve is calculated a) using a fixed Z-R relationship w ithout adjustm ents tow ard P R m easurem ents, and b) w ith adjustm ents from a lookup table. Sample sizes are shown in th e parentheses. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 158 RR ' = [ 1.0 O, 1] 1 P 1 0 = [ 0 .9 5 ,1 .0 5 ] J 1.0 RR = [ O, 1] ' p i o = [ o . 9 5 , i .0 5 ] fffy k 1.0 ff/m m m rv 0.6 l \U 0.4 0.2 0.2 on 0.0 0.2 0.4 0.6 P19 0 .8 0.0 0.0 1.0 M i K> CL ............................. 0.6 P19 0.8 1.0 P1 0 = [0 .8 5 ,0 .9 5 ] P37 0.2 0 .4 0.6 P19 0.8 . 1.0 NEXRAD s i m u a l t i o n s (5.0KM) RR = [ 0, 1] P 1 0 = [0 .8 5 ,0 .9 5 ] P 1 0 = [0 .8 5 ,0 .9 5 ] 0.8 0.8 0.6 rv 0 .6 0 .6 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0 .6 0.8 0.0 1.0 ■ C O Q. & 0.4 0.2 nn 0.0 0 .0 0.2 0.4 0.6 P19 i 0.8 1.0 0 .0 j / 0.4 i■k9 r 0.2 0.4 0 .6 P19 1.0 1.0 : ■ : $ 0.2 ......... 0.0 0.8 1.0 0 .0 PR ( 1 9 9 8 , n e a r - n a d i r ) NEXRAD s i m u a l t i o n s (3 .0KM) RR = [ 0, 1] P 1 0 —[0 .6 5 ,0 .7 5 ]. / 0 .8 J n l f( u . r* ro. 0.6 a. 0.4 0.6 NEXRAD s i m u a l t i o n s (5.0KM) 0.6 0.2 0 .4 1.0 RR = [ 0, 1] P 1 0 = [ 0 .7 5 , 0 .8 5 ] y ^ |‘ 0.8 1.0 RR =[ 0, 1] P 1 0 = [0 .7 5 ,0 .8 5 ] 0.8 Fit 0.6 0.2 P R (1 9 9 8 ,n ear-n ad ir) NEXRAD s i m u a l t i o n s (3.0KM) 1.0 . RR = [ 0, 1] P 1 0 = [0 .7 5 ,0 .8 5 l> fA 0.8 P37 ...... 0.0 0 .0 PR( 19 9 8 ,n e a r- n a d ir ) NEXRAD s i m u a i t i o n s (3 .0KM) ■ 0.2 w 0 .4 m m M W 0.4 i f f 0.2 i rv 0.6 M m CO Q. 0.4 '(fflfrS§ 0.8 m il M 0.6 RR ° ’ 11 P 1 0 = [ 0 .9 5 ,1 .0 5 ] . 0.8 0.8 P37 NEXRAD s i m u a l t i o n s (5.0KM) PR ( 1 9 9 8 , n e a r - n a d i r ) NEXRAD s i m u o l t l o n s (3.0KM) 0.2 0 .4 0.6 P19 0 .8 1.0 NEXRAD s i m u a l t i o n s (5 .0KM) RR = [ 0, 1] P 1 0 = [0 .6 5 ,0 .7 5 ] 1.0 P 1 0 = [0 .6 5 ,0 .7 5 ] 0.8 0.8 P37 r*. 0.6 0.4 0 .2 0.2 0.0 0.2 0.4 0.6 0 .8 1.0 0.0 0.0 0.0 0.2 0 .4 0.6 P19 0.8 1.0 0 .0 0.2 0.4 0.6 0 .8 Figure 9.2. 2-D contours of num ber of pixels associated w ith rain ra te less th a n 1 m m /h r a t a given P \ q interval. Left and right panels are based on WSR-88D sim ulations w ith a freezing level at 3 and 5 km, respectively. The middle panel is derived from PR -T M I d a ta in 1998. C ontours are plotted for values of [1, 10 , 50, 100 , 500, 103, 104, 105]. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 1.0 159 RR '=[ 1, 5] P 1 0 = [0 .8 5 ,0 .9 5 ] 1 .0 ' NEXRAD s i m u a l t i o n s (5.0KM) PR( 1 9 9 8 , n e a r - na d i r ) NEXRAD s i m u a l t i o n s (3.0KM) RR = [ 1, 5] P 1 0 = [0 .8 5 ,0 .9 5 ] RR = [ i . 5] P 1 0 = [0 .8 5 ,0 .9 5 ] ’ . ^ P37 0.8 0.6 r» 0 .6 0.4 0.2 0.2 0.0 0 .0 0 .2 0.4 0 .6 P19 0.8 o.o 0.0 1.0 P 1 0 = [0 .7 5 ,0 .8 5 ] P37 P 1 0 = [0 .7 5 ,0 .8 5 ] 0.8 0 .8 0.6 (v. 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .0 0.2 0.4 0.6 P19 0.8 1.0 0 .0 0 .0 PR ( 1 9 9 8 , n e a r - n a d i r ) NEXRAD s i m u a l t i o n s (3.0KM) 0.8 1.0 0.2 0.4 0.6 0.8 1.0 NEXRAD s i m u a l t i o n s (5.0KM) RR = [ 1, 5] P10 —[0.6 5 ,0 .7 5 ] P 1 0 = [0 .65,0 .7 5 ] 0 .6 P19 P 1 0 = [0 .7 5 ,0 .8 5 ] . ■: t. 0.8 0.0 0.4 NEXRAD s i m u o l t i o n s (5.0KM) P R (1 9 9 8 ,n e a r-n a d ir) NEXRAD s i m u a l t i o n s (3.0KM) 0.2 1.0 P 1 0 = [0 .6 5 ,0 .7 5 ] P37 0 .8 0.4 0.2 0.0 0.0 . RR '=[ 1, 5] P 1 0 = [0 .5 5 ,0 .6 5 ] ' 0.4 0.6 0.8 ^ 1.0 P 1 0 = [0 .5 5 ,0 .6 5 ] RR = [ 1, 5] P 1 0 = [0 .5 5 ,0 .6 5 ] 0.8 0.8 P37 0.6 0.4 0.2 0.2 o.o' 0.0 0 .0 0.2 0.4 0.6 0.8 1.0 NEXRAD s i m u o l t i o n s (5.0KM) PR( 1 9 9 8 , n e a r - n a d i r ) NEXRAD s i m u a l t i o n s (3.0KM) 0.2 1.0 Figure 9.3. Same as figure 9.2, b u t for rain rate of [1, 5] m m /h r and different intervals of P \ q. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 160 NEXRAD s i m u a l t i o n s (5.0KM) PR (1998,neor-nadir) NEXRAD s i m u a l t i o n s (3 .0K M) RR =[' 5, 15] P1 0 = [0 .7 5 ,0 .8 5 ] 1.0 P 1 0 = [0 .7 5 ,0 .8 5 ] P 1 0 = [0 .7 5 ,0 .8 5 ] P37 0.8 0.8 0.8 0.6 ^ 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0.2 0.4 0.6 0.8 0.2 0.0 0.0 1.0 0.2 0.6 P19 0.8 1.0 o.o 0.0 0.2 0.4 0 .6 P19 0.8 1.0 NEXRAD s i m u a l t i o n s (5.0KM) RR '=[ 5, 15] P 1 0 = [0 .6 5 ,0 .7 5 ] P 1 0 = [0 .6 5 ,0 .7 5 ] P 1 0 = [0 .6 5 ,0 .7 5 ] 0.8 0.8 P37 0.4 P R (1998,n ear-n ad ir) NEXRAD s i m u a l t i o n s (3 .0K M) 0.6 r- 0 .6 ► o CL 0.4 0.4 0.2 0.2 o.o 0.0 0.2 0.4 0 .6 0.8 1.0 0.0 0.0 0.2 0.4 0.6 P19 0.8 1.0 NEXRAD s i m u a l t i o n s (5.0KM) P R (1 9 9 8,near-nadir) NEXRAD s i m u a l t i o n s (3 .0K M) .RR '=[ 5, 15] P 1 0 = [0 .5 5 ,0 .6 5 ] 1.0 P 1 0 = [0 .5 5 ,0 .6 5 ] P 1 0 = [0 .5 5 ,0 .6 5 ] 0.8 0.8 P37 r*. 0.6 0 .2 0.0 . 0.0 0.2 0.4 0.6 P19 0.8 1.0 0.4 0.4 0 .2 0 .2 o.o 0.0 0.2 0.4 0 .6 P19 0.8 1.0 1.0 1.0 0.4 0.4 n2 0.2 0.2 0 .0 0.2 0.4 0.6 P19 0.8 1.0 0.0 0.0 0.6 0 .8 1.0 . RR = [ 5, 15] P 1 0 = [0 .4 5 ,0 .5 5 ] : rv. 0.6 Q. 0.4 on 0 .4 0.8 P37 o *01 0.6 0.2 NEXRAD s i m u a l t i o n s (5.0KM) RR =[ 5, 15] P 1 0 = [0 .4 5 ,0 .5 5 ] 0.8 0.8 0.0 0 .0 PR(19 9 8 ,n e a r- n a d ir ) NEXRAD s i m u a l t i o n s (3 .0KM) . RR =[ 5, 15] P 1 0 = [0 .4 5 ,0 .5 5 ] P37 ^ 0 .2 0.4 0.6 P19 0.8 1.0 0.0 0.0 • »v 0.2 0.4 0.6 P19 0.8 Figure 9.4. Same as figure 9.2, b u t for rain rate of [5, 15] m m /h r and different intervals of P iq. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 1.0 161 ’ ' : 0.8 0.8 P37 .RR =[ 15,100] P 1 0 = [0 .6 5 ,0 .7 5 ] .RR = [ 15,100] P 1 0 = [0 .6 5 ,0 .7 5 ] RR = [ 15,100] P 1 0 = [0 .6 5 ,0 .7 5 ] 0 .6 0.4 0.4 0.2 0 .2 o.o 0.0 0.2 0.4 0 .6 0.8 1.0 o.o 0 .0 0.2 0.4 0.6 P19 0.8 0.0 1.0 0.2 0.4 0.6 P19 0 .8 1.0 NEXRAD s i m u a l t i o n s (5.0KM) P R (1 9 9 8 ,n ear-n ad ir) NEXRAD s i m u a l t i o n s (3 .0KM) P37 NEXRAD s i m u o l t i o n s (5.0KM) P R (1 9 9 8 ,near-nodir) NEXRAD s i m u o l t i o n s (3.0KM) . RR = [ 15,100] P 1 0 = [0 .5 5 ,0 .6 5 ] 1 n .RR = [ 15,100] P 1 0 = [0 .5 5 ,0 .6 5 ] 1.0 0.8 0.8 0.8 0.6 r- 0.6 Q. r- 0.6 Q. 0.4 0.4 1.0 . r r =[ 1 5 , 1 0 0 ] P 1 0 = [0 .5 5 ,0 .6 5 ] : 0.4 , no . 0 .0 0.2 0.4 .................... .................... 0.6 0 .8 1.0 P19 NEXRAD s i m u a l t i o n s ........................................................ 0.0 0 .0 (3.0KM) P37 0.2 0 .4 0.6 P19 0.8 0.0 0.0 1.0 0.6 0.4 0.4 0.4 0 .2 0 .2 0.2 0.6 P19 0.8 1.0 0.0 0.0 1.0 0.2 0.4 0.6 P19 0 .8 1.0 RR = [ 15,100] P 1 0 = [0 .3 5 ,0 .4 5 ] 0.2 0.2 0.4 0.6 P19 0.8 1.0 0 .6 0.0 0 .0 0.4 0.6 P19 0 .8 1.0 NEXRAD s i m u a l t i o n s (5.0KM) 1.0 0 .4 0.4 0.2 0.2 0.0 0 .0 0.2 . RR = [ 15,100] P 1 0 = [0 .3 5 ,0 .4 5 ] r-'. 0.6 fO P37 o 0.6 1.0 0.8 0.8 0.8 0.0 0.0 rv PR( 1 9 9 8 , n e a r —n a d i r ) NEXRAD s i m u a l t i o n s (3.0KM) .RR = [ 15,100] P 1 0 = [0 .3 5 ,0 .4 5 ] 0.8 0.8 0.6 0.4 0.6 P19 RR - [ 15,100] P 1 0 = [0 .4 5 ,0 .5 5 ] 0.8 0.2 0.4 NEXRAD s i m u a l t i o n s (5. 0K M) PR ( 1 9 9 8 , n e a r - n a d i r ) 0.8 0.0 0.0 0.2 RR = [ 15,100] P 1 0 = [0 .4 5 ,0 .5 5 ] RR = [ 15,100] P 1 0 = [0 .4 5 ,0 .5 5 ] P37 0.2 0.2 0.2 0.2 ....................................................... 0.4 0 .6 0.8 1.0 P19 ^ 0.0 0.0 0.2 ....................................................... 0.4 0 .6 0.8 1.0 P19 Figure 9.5. Same as figure 9.2, b u t for rain ra te greater th an 15 m m /h r and different intervals of lower P i0. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. : 162 PR ( n e a r - n a d i r , M A M ) PR ( n e a r - n a d i r , D J F ) 0.20 0 .2 0 0 .1 0 0.10 0 .0 5 it 0 .0 5 o o c o 0 .00 0.00 0 2 4 6 8 0 10 8 10 0 .2 0 O oa 6 0 .2 0 0 .1 0 c o 4 PR ( n e a r - n a d i r , S O N ) PR ( n e a r - n a d i r , J J A ) 0 .2 5 D 2 Bri ght b a n d h e i g h t ( k m ) Bright b a n d h e i g h t ( k m ) 0.10 t Li - 0 .0 5 0 .0 5 0 .0 0 0.00 0 2 4 6 8 Bright b a n d he i g h t ( k m ) 10 0 2 4 6 8 10 Br ight b a n d h e i g h t ( k m ) Figure 9.6. Plots of fraction of d a ta vs. bright band height based on P R 2A25 algorithm retrieval of 55 orbits during August 1998 and July 1999. Season inform ation is shown in titles. D JF represents th e w inter season, while MAM expresses th e spring. The vertical resolution of 2A25 product is 0.25 km. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 163 T M I- 19980' TMI —1 9 9 8 0 ' 0.20 0.4 GPROF-Wilheit D 0 .3 c 0.10 0.2 TO ) O o o u: 0 .0 5 Li - O.O 0 0.00 0 2 4 6 8 10 2 4 6 8 10 Freezing height (km ) Freezing height (km ) TMI—1 9 9 8 0 7 T M I - 19 9 8 0 7 0.4 0 .3 0 GPROF-Wilheit o -§ 0 .2 5 o 0.20 o o c 0.2 co oo o < oj L i- L i- 0 .0 5 0.00 0.0 0 2 4 6 8 Freezing height (km ) 10 0 2 4 6 8 10 Freezing height (km ) Figure 9.7. Plots of fraction of d a ta vs. altitude of freezing level based on TM I measurem ents in Jan u ary and July, 1998. (a) and (b) are derived from th e equation (9.1), while (c) and (d) are com puted from W ilheit’s approach. Note th a t th e W ilheit’s m ethod constrains th e freezing level to th e climatological altitu d e (around 4.9 km) at a given 302 K of sea surface tem perature. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 164 10. C O N C LU SIO N S A N D F U T U R E W ORK The emphasis of th e dissertation has been presenting a new Bayesian rain rate retrieval algorithm which has two unique characteristics. One is th a t the conditional likelihoods representing statistical and physical relationships between surface rainfall and microwave signatures, and th e prior rain ra te probability distribution are derived from both observations and model simulations. T he other is th a t th e o u tp u t of the algorithm is not ju st a single-pixel retrieved rain rate, bu t rath er a continuous posterior probability distribution of surface rainfall. Unlike other Bayesian algorithm s which are all based on simulations of cloudradiative models and simplified specifications in conditional likelihoods, this study offers generic forms which are comprised of explicit, closed-form functions and are fit ted to both model sim ulations (based on NWS W SR-88D network reflectivity and a simplified 1-D plane-parallel radiative transfer model) and real m easurem ents (from PR-TM I m atch-up dataset). These generic forms make it possible to analytically obtain a continuous posterior probability d istribution of surface rainfall w ith lim ited com putational requirem ents. Various estim ators to the posterior pdf can be used later to serve as the single-pixel retrieved value, depending on th e desired character istics of the retrieval. Simulations based on W SR-88D reflectivity d a ta and a simplified one-dimensional plane-parallel radiative transfer model are surprisingly close to actual observations in term s of m ultichannel relationships in atten u atio n index. The sim ulated d ataset helps to prove the prelim inary retrieval ability of th e Bayesian approach using th e generic forms. W hile th e prior and conditional probability distributions show a great effect on the Bayesian algorithm developed from the model simulations, a sensitivity R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 165 test is needed to clarify the theoretical advantage of the Bayesian approach and the retrieval ability if imperfect inform ation is applied to th e algorithm. Two synthetic random ly-generated d atasets are used in th e theoretical sensitivity tests. One of th e datasets has a physical model specified via a covariance m atrix (same as other Bayesian algorithm s), and th e other dataset is generated from the conditional pdfs based upon the radar-radiative model simulations. Results of the sensitivity test suggest th a t we should be aware of th e inherent retrieval uncertainty associated w ith some observation vectors and th a t significant bias occurs at higher rain rate, even when the exact prior and conditional likelihoods were employed in the Bayesian algorithm. The deficiency is a ttrib u te d partially to th e retrieval ambiguity in some scenes because of beam-filling effects, and partially to th e lack of further inform ation from d a ta for the cases of heavy rain rate, due to the satu ratio n of microwave signatures. The theoretical test also dem onstrates th a t the retrieved surface rainfall is sen sitive to the assum ptions in the prior rain ra te distribution. The effect of the prior distribution on the retrieval is different a t various rain ra te ranges, m ainly deter mined by the characteristics of th e prior pdf. W hen two prior pdfs have very similar distributions, the Bayesian algorithm is no t sensitive to th e slight change of th e pa ram eters used to express the distribution. Furtherm ore, th e test results show th a t use of a simple covariance m atrix to describe th e conditional pdfs might not be able to provide enough accurate inform ation ab o u t th e physical m ultichannel relationships, and thus can lead to substantial errors and bias in th e retrieval. A dataset from the m atch-up of P R and TM I m easurem ents is also used to find another set of coefficients for those generic forms, and to develop Bayesian 3R-M EAN, 3E-M LE, 2E-M EAN and 2E-M LE algorithm s, where MEAN and MLE indicate th a t the expectation value and m axim um likelihood estim ate of th e poste rior pdf are of interest, respectively; and ZV and 2 V present th e num ber of variables used in th e retrieval algorithm. T he retrieved surface rainfall is com pared w ith th a t of other benchm ark algorithm s and validated against independent datasets, includ R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 166 ing random cases and particular types of precipitation systems. Comparisons with other algorithm s suggest th a t our Bayesian algorithm s perform com parably to oth ers b u t have th e additional advantage of continuous posterior rain rate probability distribution. Validations have dem onstrated th a t th e retrieval from th e Bayesian 3V-M EAN algorithm has a low bias except for cases of heavy/w idespread precip itation systems, which are not random ly sampled. We found th a t th e heavy and widespread precipitation systems are relatively rare in th e entire year of 1998, and th a t their characteristics m ight not be included well in b o th th e training dataset and th e Bayesian model. The Bayesian-type algorithm shows good skill a t retrieving th e rain rate at 0 ~ 20 m m /h r, b u t has difficulty obtaining heavy rain rate in all cases. The lack of ability to yield the high rain rate increases th e corresponding root-m ean-square error and reduces th e correlation coefficients. T he degradation of th e Bayesian algorithms at the higher end of the tru e rain rate results from th e inherent lim ited inform ation offered by th e microwave variables. Since th e microwave signature is satu rated in the scene of heavy precipitation, and th e prior rain rate distribution always gives larger probability for the light rain rate, th e Bayesian algorithm s ten d to underestim ate higher rain intensity. The M LE-type algorithm s have a smaller dynam ic retrieval range th a n th a t of th e M EAN-type algorithms, b u t have b e tte r skills for th e retrieval of very light rain intensity. If the purpose of the analysis is th e overall (or global) rain rate w ith the least bias, one might take th e expectation value of th e posterior probability distribution as the retrieved surface rainfall for a single pixel. If th e precision in th e areas of light rain rate is emphasized, one m ight use th e m axim um likelihood estim ates for the retrieval. A rigorous uncertainty analysis in th e study dem onstrates how the m ean and variance of th e posterior density function change owing to th e uncertainty of the prior rain ra te distribution, and indicates th a t our Bayesian algorithm is relatively robust. In th e study, the additional variable of P 10 in the 3V -typc algorithm s pro R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 167 vides further inform ation in some scenes associated w ith heavy precipitation, bu t the algorithm suffers from th e uncertainty introduced by th e larger field of view at 10.65 GHz. Results from the model assessment also point out an im perfect fit to the PR -T M I m atch-up d a ta in 4-D space due to th e small sample size. These two factors combine to make th e benefits of th e inclusion of P 10 questionable. In addition, the distinguishable regimes shown in the 4-D conditional probability distribution suggest th a t other variables m ight help to isolate out different signals. Finally, th e four-dimensional rain rate and m ultichannel physical relationships are examined by a comparison between PR -T M I m atch-up d ataset and th e radarradiative model simulations. The analysis helps to calibrate th e W SR-88D-derived surface rainfall and to verify the adequacy of the model assum ptions about mi crowave properties and radiative transfer calculations. R esults reveal a very good agreement between observations and simulations, b u t the 4-D relationships show a great sensitivity to th e heights of freezing levels. T he la tte r statem ent points out th e need for inform ation about freezing levels in b o th actual m easurem ents and the development of our Bayesian algorithm. The verification conducted in th e study has two crucial im plications in th e development of our Bayesian rain ra te retrieval algo rithm . One is th a t the verification provides a basis th a t th e estim ates of rain rate can be calibrated and th e Bayesian algorithm can be modified when more reliable surface rainfall m easurem ents are available. T he other is th a t th e assum ptions about microwave property and radiative transfer calculations in th e radar-radiative model could be used for further analysis if actual observations are no longer available. The altitude of the freezing level is not included as either a variable or a free param eter in the current version of our Bayesian algorithm . Since the model ver ification dem onstrates th e im portance of th e inform ation ab o u t th e freezing level, reliable estim ates of th e freezing level heights are highly needed. The ideal approach to address this issue for th e future is to quantify th e accuracy of th e retrieved freezing level heights estim ated from the P R reflectivity by com paring w ith a large d ataset of RAOBS, and to investigate th e effects of th e inclusion of freezing level heights on R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 168 the retrieved surface rainfall. Furtherm ore, based on model sim ulations, th e freezing level heights can be added as a free param eter. Then, depending on th e freezing level of the pixel, various posterior p df and single-pixel retrievals will be given. The corresponding perform ance can be further evaluated against th e current Bayesian algorithm and other benchm ark algorithms. While the attenuation index at 10.65 GHz is expected to provide more detailed inform ation about th e heavy surface rainfall for the retrieval algorithm , b u t is as sociated w ith more uncertainty due to the larger field of view, th e scatter index at 85.50 GHz m ight be a favorable alternative. The PR -T M I m atch-up d ataset is ready to offer the scattering inform ation, b u t th e radar-radiative transfer model is not able to present the scattering signal due to th e simplicity of the model. Therefore, future work should evaluate th e contribution of th e scattering index a t 85.50 GHz on th e PR -T M I m atch-up d ataset first. Then th e radiative transfer model should include more details on scattering effects. More d a ta associated w ith heavy rain ra te should be included in th e future as well to yield a b e tte r goodness-of-fit of the conditional likelihood /(-P 1 0 I -P 1 9 , P 3 7 , R)- The need will be more obvious if th e scattering index is added to th e Bayesian algorithm . Fortunately, one of the advantages of our Bayesian algorithm is th a t conditional pdfs are of interest, and therefore we could focus on th e pixels where th e heavy rainfall rates occur to improve th e fit w ithout processing redundant d a ta points. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. LIST OF REFERENCES R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 169 LIST OF R EFEREN CES Alishouse, J. C., S. Snyder, and R. R. Ferraro, 1990: D eterm ination of oceanic to tal precipitable w ater from th e SSM /I. IE E E Trans. Geosci. Rem ote Sensing, 26, 811-816. B arnston, A. G., 1992: Correspondence among th e correlation, RM SE, and Heidke forecast verification measures; refinement of the Heidke score. Wea. Foecast., 7(4), 699-709. Bauer, P., P. Amayenc, C. D. Kummerow, and E. A. Smith, 2001: Over-ocean rainfall retrieval from multisensor d a ta of th e Tropical Rainfall M easuring Mission. P art II: A lgorithm im plem entation. J. Atm os. Ocean. Tech., 18(11), 1838-1855. Bauer, P. and R. B ennartz, 1998: Tropical Rainfall M easuring Mission microwave imaging capabilities for th e observation of rain clouds. Radio Sci., 33(2), 335-349. Berger, J. O., 1990: R obust Bayesian analysis: Sensitivity to th e prior. J. Stat. Planning and Inference, 25, 303-328. Bolen, S. M. and V. Chandrasekar, 2000: Q uantitative cross validation of spacebased and ground-based rad ar observations. J. Appl. Meteor., 39(12), 2071-2079. Conner, M. D. and G. W. Petty, 1998: Validation and intercom parison of ssm /i rain-rate retrieval m ethods over th e continental united states. J. Appl. Meteor., 37(7), 679-700. D ettinger, M. D., D. R. Cayan, H. F. Diaz, and D. M. Meko, 1998: N orth-south pre cipitation p attern s in w estern N orth America on interannual-to-decadal timescales. J. aim., 11(12), 3095-3111. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 170 Evans, K. F., J. Turk, T . Wong, and G. L. Stephens, 1995: Bayesian approach to microwave precipitation profile retrieval. J. Appl. Meteor., 34(1), 260-279. Fowler, H. J. and C. G. Kilsby, 2002: Precipitation and the north atlantic oscillation: A study of clim atic variability in northern england. Int. J. Climatol., 22(7), 843866 . G audet, B. and W . R. C otton, 1998: Statistical characteristics of a real-tim e precip itation forecasting model. Wea. Forecast., 13(4), 966-982. H addad, Z. S., E. A. Smith, C. D. Kummerow, T. Iguchi, M. R. Farrar, S. L. Durden, M. Alves, and W. S. Olson, 1997: The TRM M ’Day-1’ radar-radiom eter combined rain-profiling algorithm . J. Meteor. Soc. Japan, 75, 799-809. Hou, A. Y., S. Q. Zhang, A. M. da Silva, W. S. Olson, C. D. Kummerow, and J. Simpson, 2001: Im proving global analysis and short-range forecast using rainfall and m oisture observations derived from TRM M and SSM /I passive microwave sensors. B ulletin o f the Am erican Meteorological Soc., 82(4), 659-679. Houze, R. A., S. Brodzik, C. Schumacher, and S. Yuter, 2001: University of W ashington K w ajalein G round V alidation R ain Maps: Septem ber 19, 2001 Re lease. http://w w w .atm os.w ashington.edu/ co u rtn ey/kw a j/ NewProductDocumentationOl 0919. p d f . Houze, R. A. and C.-P. Cheng, 1977: R adar characteristics of tropical convection observed during GATE: m ean properties and tren d s over the sum mer season. Mon. Wea. Rev., 105(8), 964-980. Iguchi, T., T. Kozu, R. Meneghini, J. Awaka, and K. Okamoto, 2000: Rain-Profiling A lg o rith m for th e T R M M P r e c ip ita tio n R ad ar. J. A p p l. M e te o r ., 39(12), 2038 2052. Kedem, B. and L. S. Chiu, 1987: Are rain ra te processes self-similar? sources Research, 23(10), 1816-1818. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. W ater R e 171 Kedem, B., L. S. Chiu, and G. R. N orth, 1990: Estim ation of m ean rain rate: application to satellite observations. J. Geophysic. Research, 95, 1965-1972. Kleinn, J., C. Frei, J. G urtz, P. L. Vidale, and C. Schar, 2002: Coupled climaterunoff simulations: a process study of current and a warmer clim ate in th e Rhine basin. A M S Sym posium on Global Change and Climate Variations, 13, 13-17. K rishnam urti, T. N., S. Surendran, D. W. Shin, R. J. Correa-Torres, T. S. V. V. Kum ar, E. Williford, C. Kummerow, R. F. Adler, J. Simpson, R. Kakar, W. S. Olson, and F. J. Turk, 2001: Real-tim e m ultianalysis-m ultim odel superensem ble forecasts of precipitation using TRM M and SSM /I products. Mon. Wea. Rev., 129(12), 2861-2883. Kummerow, C., W. Barnes, T. Kozu, J. Shiue, and J. Simpson, 1998: T he tropi cal rainfall m easuring mission (TRM M ) sensor package. J. A tm os, and Oceanic Technol., 15(3), 809-817. Kummerow, C., W. S. Olson, and L. Giglio, 1996: A simplified scheme for obtaining precipitation and vertical hydrom eteor profiles from passive microwave sensors. IE E E T. Geosci. Rem ote, 34(5), 1213-1232. Kummerow, C., J. Simpson, O. Thiele, W . Barnes, A. T. C. Chang, E. Stocker, R. F. Adler, A. Hou, R. Kakar, F. Wentz, P. Ashcroft, T. Kozu, Y. Hong, K. Okamoto, and T. Iguchi, 2000: T he statu s of the Tropical Rainfall M easuring Mission (TRM M ) after two years in orbit. J. Appl. Meteor., 39(12), 1965-1982. L’Ecuyer, T. S. and G. L. Stephens, 2002: An uncertainty model for Bayesian Monte Carlo retrieval algorithms: A pplication to the TRM M observing system. Quart. J. R o y . M e te o r. Soc., 1 2 8 (5 8 3 ) , 1 7 1 3 -1 7 3 7 . Liebe, H. J., G. A. Hufford, and T. M anabe, 1991: A model for th e complex perm it tivity of w ater a t frequencies below 1 THz. Int. J. Infrared and M illim eter Waves, 12, 659-675. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 172 Liu, Q. and C. Simmer, 1996: Polarization and intensity in microwave radiative transfer. Contributions Atm os. P hys., 69(4), 535-545. Lucero, 0 . A. and N. C. Rodriguez, 2002: Spatial organization in Europe of decadal and interdecadal fluctuations in annual rainfall. Int. J. Climatol., 22(7), 805-820. M arshall, J. S. and W. M. Palm er, 1948: The distribution of raindrops w ith size. J. M eteor., 5, 165-166. Oki, R., K. Furukawa, S. Shimizu, Y. Suzuki, S. Satoh, H. Hanado, K. Okamoto, and K. Nakamura, 1998: Prelim inary results of TRM M . P a rt I: A com parison of P R w ith ground observations. Marine Technol. Soc. J., 32(4), 13-23. Olson, W. S., C. D. Kummerow, G. M. Heymsfield, and L. Giglio, 1996: A m ethod for combined passive-active microwave retrievals of cloud and precipitation profiles. J. Appl. M eteor., 35(10), 1763-1789. Olson, W. S., C. D. Kummerow, Y. Hong, and W .-K. Tao, 1999: A tmospheric latent heating distributions in the tropics derived from satellite passive microwave radiom eter m easurem ents. J. Appl. Meteor., 38(6), 633-664. Paegle, J. N. and K. C. Mo, 2002: Linkages between sum m er rainfall variability over South America and sea surface tem perature anomalies. J. Clim., 15(12), 1389-1407. Petty, G. W., 1994a: Physical retrievals of over-ocean rain rate from m ultichannel microwave imagery. P a rt I: Theoretical characteristics of normalized polarization and scattering indices. Meteorol. Atm os. Phys., 54(1-4), 79-100. Petty, G. W ., 1994b: Physical retrievals of over-ocean rain ra te from m ultichannel microwave imagery. P a rt II: A lgorithm im plem entation. Meteorol. Atmos. Phys., 54(1-4), 101-121. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 173 Petty, G. W., 2001: Physical and Microwave R adiative Properties of P recipitating Clouds. P a rt I: Principal component analysis of observed m ultichannel microwave radiances in tropical stratiform rainfall. J. Appl. M eteor., 40(12), 2105-2114. Petty, G. W. and K. B. K atsaros, 1992: The response of the Special Sensor Mi crowave/Imager to th e m arine environment. P a rt I: An analytic model for th e atmospheric com ponent of observed brightness tem peratures. J. Atm os. Ocean. Tech., 9, 746-761. Ropelewski, C. F. and M. S. Halpert, 1996: Quantifying Southern Oscillation- precipitation relationships. J. Clim., 9(5), 1043-1059. Schumacher, C. and R. A. Houze, 2000: Com parison of R adar D ata from the TR M M Satellite and Kw ajalein Oceanic V alidation Site. J. Appl. Meteor., 39(12), 21512164. Sivaganesan, S. and J. O. Berger, 1989: Ranges of posterior measures for priors w ith unimodal contam inations. Ann. Statist., 17(2), 868-889. Stephenson, D. B., 2000: Use of the “odds ratio ” for diagnosing forecast skill. Wea. Forecast., 15(2), 221-232. Treadon, R. E., 1996: Physical initialization in th e NMC Global D ata A ssim ilation System. Meteorol. A tm os. Phys., 60(1), 57-86. Wei, H., W. J. Gutowski, C. J. Vorosmarty, and B. M. Fekete, 2002: C alibration and validation of a regional clim ate model for Pan-A rctic hydrologic sim ulation. J. Clim., 15(2), 3222-3236. W ilheit, T. T., A. T. C. Chang, and L. S. Chiu, 1991: Retrieval of m onthly rainfall indices from microwave radiom etric m easurem ents using probability distribution functions. J. A tm os. Oceanic Technol., 8(1), 118-136. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. VITA R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 174 VITA Jui-Yuan C hristine Chiu was born in Ping-Dong, Kaohsiung, Taiwan. She re ceived her Bachelor and M aster of Science degrees in Atmospheric Physics from th e N ational C entral University of Taiwan in 1992 and 1994, respectively. She passed the national government exam ination in 1993 and had worked at the Environm ental Protection A dm inistration of Taiwan for three years since 1994. R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.

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