Thesis (in revision) (Last updated on 3 June 2013) - Atmospheric ...

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Thesis (in revision) (Last updated on 3 June 2013) - Atmospheric ...

Reduced-Dimensional Retrievals of Precipitation from theTRMM Microwave Imager: Physical Insight and InformationContentBYKe LiM.S., Mechanical Engong>inong>eerong>inong>g, University of New Hampshire (2010)A thesis submittedong>inong> partial fulfillment ofthe requirements for the degree ofMaster of Science(Atmospheric Sciences)at theUNIVERSITY OF WISCONSIN-MADISON2013


This thesis has been examong>inong>ed and approved.ong>Thesisong> director, Grant PettyProfessor of University of Wisconsong>inong>–MadisonTristan L’ecuyerAssistant Professor of University of Wisconsong>inong>–MadisonRalf BennartzProfessor at University of Wisconsong>inong>–MadisonDate


DedicationTo our God, the creator ...The breath of God produces ice,and the broad waters become frozen.He loads the clouds with moisture;he scatters his lightnong>inong>g through them.At his direction they swirl aroundover the face of the whole earthto do whatever he commands them.He brong>inong>gs the clouds to punish people,or to water his earth and show his love.—Job 37:10-13, NIV Biblei


AcknowledgmentsThis thesis is written as completion to the Master of Science ong>inong> Atmospheric and OceanicSciences, at the University of Wisconsong>inong>–Madison.The master program focuses on themicrowave remote sensong>inong>g ong>inong> terms tropical raong>inong> rate retrieval algorithm development. Thesubject of the this thesis, Reduced-Dimensional Retrievals of Precipitation from the TRMMMicrowave Imager: Physical Insight and Information Content, is ong>inong>tended to examong>inong>e:• To retrieve raong>inong> rate, what are the three most significant physical features of a stormobserved by the microwave imagers of satellite TRMM?• How important are each observed features ong>inong> terms of retrievong>inong>g raong>inong> rates?I want to thank my supervisor, Prof.Grant Petty, of beong>inong>g great help durong>inong>g thedevelopment of this thesis. Besides, though my 2.5 year graduate study, Prof. GrantPetty to the best demonstrates to me as a good example of professional scientist with hisong>inong>novative mong>inong>d and hard-workong>inong>g spirit.Moreover, as a friend and mentor, Prof.Grant Petty guided me with patience andhumility, provided me with confirmations and useful critiques, and shared with me aboutlife philosophy usong>inong>g ong>inong>formation theories:“Different from echoong>inong>g, confirmation can only be counted as confirmation whenthere’s a chance of disagreement.”—Dr. Grant PettyI’d like also give thanks to Prof. Larissa Back and Prof. Jonathan Martong>inong> for their ong>inong>structionsand critiques. It was their presence that makes me grow as a young researcher withii


even stronger faith. It was their feedback that ong>inong>spired me to develop further my projectto a deeper level.iii“As iron sharpens iron, so one person sharpens another”—Proverbs 27:17, NIV BibleI’d like to thank Prof. Tristan L’ecuyer for his time and effort on givong>inong>g me feedback ofmy project. I’d like to thank the PhD student Alex Matus, Pei Wang, the MS studentKyle Hosley, Marc Collong>inong>s, William Long>inong>e, Kuniaki Inoue, and Bryan Crouch. It was theirfriendship that fills my office life with a lots of joy!Besides, it is their comments andcritiques ong>inong> the daily conversation that motivated me to make improvement on my projects.Especially, I want to thank my wife, Kristen, for her dedicated commitment with me atups and downs durong>inong>g my graduate study ong>inong> UW–Madison. I deeply appreciate her strongspiritual support, ong>inong> terms of prayers, encouragement, and comfortong>inong>g, which constantlyremong>inong>d of me of my ong>inong>tegrity and faith to be a good man. Besides, I’d like to thank mywife, Kristen, for every song>inong>gle meals she prepared for me ong>inong> the midst of my busy study. I’dlike to thank her also for all the effort that she did to keep me relaxed and cheered after abusy day: watchong>inong>g Madagascar 3, havong>inong>g hot dates at the Walmart or the McDonald, andhavong>inong>g random game nights, etc. Without her support and companionship, I can hardlyreach today’s achievement.I also want to thank my small group from Faith Community Bible Church for theirwisdom and prayers. It was their constant love and prayers that keep me on track of myconviction as a Christian man. It was their warm hearts that made Madison WI a cozyhome for me durong>inong>g my graduate study.Most importantly, I want to give my highest praise and appreciation to our omnipotentGod, creator of the universe. It is by his grace my song>inong> is forgiven, and my debt is paid. Itis his endurong>inong>g love that breaks all the chaong>inong>s of my life and sets me free. It is his powerthat turns uglong>inong>ess ong>inong>to beauty. It is His righteousness and patience through Jesus Christthat ong>inong>fluences me daily to move forward with my new identity.


Table of ContentsDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iiixiv1 Introduction 11.1 Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Experimental basis before 1978 . . . . . . . . . . . . . . . . . 11.1.2 SMMR ong>inong> 1978 . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 SSM/I ong>inong> 1987 . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.4 TRMM ong>inong> 1997 . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.5 GPM ong>inong> 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Retrieval Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.1 Cloud Resolvong>inong>g Models . . . . . . . . . . . . . . . . . . . . . 71.2.2 Screenong>inong>g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.3 Bayesian estimation methods . . . . . . . . . . . . . . . . . . 81.3 Information Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.1 Shannon Entropy . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.2 Relative Entropy . . . . . . . . . . . . . . . . . . . . . . . . . 131.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Passive Microwave Sensors and Raong>inong> Rate Retrieval 172.1 Brightness Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . 18iv


v2.1.1 Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.2 Transmittance . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.3 Radiative Transfer Equations . . . . . . . . . . . . . . . . . . 192.1.4 Earth Surface Types . . . . . . . . . . . . . . . . . . . . . . . 222.1.5 Cloud and Raong>inong> Water Emission and Ice Scatterong>inong>g . . . . . . 242.1.6 Polarization and Raong>inong> Rate . . . . . . . . . . . . . . . . . . . 282.2 Bayesian Algorithm for Raong>inong> Rate Retrieval . . . . . . . . . . . . . . 313 Pseudo-channels 343.1 Over the Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.1.1 Gram-Schmidt Process . . . . . . . . . . . . . . . . . . . . . . 353.1.2 CH1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.1.3 CH2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.1.4 CH3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Probability Distribution Functions . . . . . . . . . . . . . . . . . . . 404 Information Theory 464.1 Shannon Entropy and Relative Entropy . . . . . . . . . . . . . . . . . 464.1.1 Gaussian Distribution . . . . . . . . . . . . . . . . . . . . . . 464.2 Pseudo-channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3 Necessity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.3.1 CH1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.3.2 CH2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.3.3 CH3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.4 Sufficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795 Conclusions 83


List of Tables2.1 Surface type descriptions ong>inong> Figure 2-5 (Petty and Li, 2013a) . . . . . 253.1 Case division for Gram-Schmidt analysis . . . . . . . . . . . . . . . . 384.1 Maximum retrieved raong>inong> rate changes ong>inong> posterior probability distributionsby addong>inong>g CH1 under various surface types. The columns aredenoted as (from left to right): surface categories, surface skong>inong> temperature,total precipitable water, CH1, raong>inong> rate difference (posterior- prior), raong>inong> rate ratio (posterior/prior), probability of posterior raong>inong>rate equal to zero mm/hr, probability of posterior raong>inong> rate greaterthan 1 mm/hr, relative entropy, and Shannon entropy difference. Thethreshold of sample numbers ong>inong> this analysis is 1000 samples. All casesong>inong> 4c have less than the threshold, therefore, the 4c cases cannot beconsidered statistically representative. . . . . . . . . . . . . . . . . . . 67vi


vii4.2 Maximum retrieved raong>inong> rate changes ong>inong> posterior probability distributionsby addong>inong>g CH2 under various surface types. The columns aredenoted as (from left to right): surface categories, surface skong>inong> temperature,total precipitable water, CH1, CH2, raong>inong> rate difference (posterior- prior), raong>inong> rate ratio (posterior/prior), probability of posteriorraong>inong> rate equal to zero mm/hr, probability of posterior raong>inong> rate greaterthan 0.1 mm/hr, relative entropy, and Shannon entropy difference. Thethreshold of sample numbers ong>inong> this analysis is 1000 samples. All casesong>inong> 4c have less than the threshold, therefore, the 4c cases cannot beconsidered statistically representative. . . . . . . . . . . . . . . . . . 734.3 Maximum retrieved raong>inong> rate changes ong>inong> posterior probability distributionsby addong>inong>g CH3 under various surface types. The columns aredenoted as (from left to right): surface categories, surface skong>inong> temperature,total precipitable water, CH1, CH2, CH3, raong>inong> rate difference(posterior - prior), raong>inong> rate ratio (posterior/prior), probability ofposterior raong>inong> rate equal to zero mm/hr, probability of posterior raong>inong>rate greater than 0.01 mm/hr, relative entropy, and Shannon entropydifference. The threshold of sample numbers ong>inong> this analysis is 1000samples. All cases ong>inong> 4c have less than the threshold, therefore, the 4ccases cannot be considered statistically representative. . . . . . . . . 78


List of Figures1-1 Data for postboost TMI sensors ong>inong> various frequencies, ong>inong>cludong>inong>g polarization(V is vertically polarized, H is horizontally polarized.), ong>inong>stantaneousfield of view (IFOV), etc. (Kummerow, 2006) . . . . . . 162-1 Passive microwave remote sensong>inong>g mechanism. Green arrows are radiancereflected by the earth. Blue arrows are radiance emitted directlyby the atmosphere. The gray dotted arrow represents radiance emitteddirectly from the earth surface. T A is the averaged atmospherictemperature. T s is the surface temperature. ε is the surface emissivity.t is the transmittance of the atmosphere. . . . . . . . . . . . . . . . 232-2 Land ocean brightness temperature difference reflected ong>inong> the image ofAMSR-E 10 GHz V on the west coast of US. (Courtesy to U.S.Navy/NRL/NASA) 252-3 Map of brightness temperature differences, 10 GHz(V) - 10 GHz(H)for 2002 TMI non-precipitatong>inong>g pixels. . . . . . . . . . . . . . . . . . 262-4 Map of brightness temperature of 85 GHz(H) for 2002 TMI non-precipitatong>inong>gpixels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262-5 Map of the seven surface classes constructed by usong>inong>g TMI brightnesstemperatures for 2002. (Petty and Li, 2013a) . . . . . . . . . . . . . . 26viii


ix2-6 Typical response of different TRMM channels to raong>inong> rate. Red dotsare 10 GHz. Blue dots are 19 GHz. Magenta dots are 37 GHz. Greendots are 85 GHz. The horizontal axis is retrieved raong>inong> rate, not directlymeasured PR raong>inong> rate. The plots here serve to illustratively show thetrends of brightness temperatures. Usong>inong>g retrieved raong>inong> rate would havemore samples that represent trends of TMI brightness temperatures. . 292-7 Scatterong>inong>g plots of polarizations with raong>inong> rate under TMI channel:(upper, left) 10 GHz, (upper, right) 19 GHz, (bottom, left) 37 GHz,(bottom, right) 85 GHz. Horizontal axes are brightness temperaturesof vertically polarized channels, and vertical axes are brightness temperaturesof horizontally polarized channels. . . . . . . . . . . . . . . 302-8 Comparisons of algorithms and PR raong>inong> rate data. (Left) comparisonbetween PR raong>inong>fall rate and raong>inong>fall retrieved ong>inong> 2A12; and (Right)comparison between PR raong>inong>fall rate and raong>inong>fall retrieved with UWalgorithm. (Petty and Li, 2013a) . . . . . . . . . . . . . . . . . . . . 333-1 Scatter plots of three pseudo-channels over the ocean (1B11.20020911.27506.7.HDF):(a) CH1 and CH2, (b) CH3 and CH2, and (c) CH1 and CH3. The bluelong>inong>e refers to the division of different cases for Gram-Schmidt analysis 373-2 Orthogonal modes of 9-microwave-channel distributions with: CH1positive ong>inong> blue dotted long>inong>e, CH2 positive ong>inong> green triangles, CH2 negativeong>inong> black triangles, CH3 positive ong>inong> magenta squares, and CH3negative ong>inong> red diamond. . . . . . . . . . . . . . . . . . . . . . . . . . 43


x3-3 Microwave channels 19GHz V, 85GHz V, and retrieve raong>inong> rate of HurricaneLily ong>inong> October 2002, to the southwest of Florida. (a) 19GHz Vbrightness temperature (K), (b) 85GHz V brightness temperature (K),and (c) retrieved raong>inong> rate ong>inong> mm/hr. . . . . . . . . . . . . . . . . . . 443-4 Pseudo-channels of Hurricane Lily ong>inong> October 2002, to the southwestof Florida. (a) CH1, (b) CH2, (c) CH3, and (d) RGB plot of CH1(red), CH2 (green), and CH3 (blue). . . . . . . . . . . . . . . . . . . 454-1 Information entropy changes between two Gaussian distributions overunevenly divided bong>inong>s. Blue blocks are normal distribution before, andpong>inong>k blocks are normal distribution afterwards. All upper panels areprobability density functions (PDFs), and lower panels are changes ofong>inong>formational entropy of different kong>inong>ds. The green long>inong>es are contong>inong>uousShannon entropy difference (∆CSE). The magenta long>inong>es are discreteShannon entropy difference (∆DSE). The cyan long>inong>es are relative entropy(RE). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564-2 Information entropy changes between two Gaussian distributions overevenly divided bong>inong>s. Blue blocks are normal distribution before, andpong>inong>k blocks are normal distribution afterwards. All upper panels areprobability density functions (PDFs), and lower panels are changes ofong>inong>formational entropy of different kong>inong>ds. The magenta long>inong>es are discreteShannon entropy difference (∆DSE). The cyan long>inong>es are relativeentropy (RE). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57


xi4-3 Probability distribution changes and ong>inong>formation entropy changes withrestrictions of CH1, CH2, and CH3. The land type is 0w. After scalong>inong>g,T skong>inong> = 6, σ water = 7, CH1 = 41, CH2 = 38, CH3 = 23. Color blocksare probability withong>inong> each bong>inong>, usong>inong>g the vertical axis to the left.The black triangle long>inong>es are relative entropy (RE). The black dottedlong>inong>es are change of shannon entropy ∆SE. Both triangle and dottedlong>inong>es follow the vertical coordong>inong>ates to the right. Shown ong>inong> each panelsare: (a) only climate data, (b) climate data with CH1 restriction, (c)climate data with restrictions of CH1 and CH2, and (d) climate datawith restrictions of CH1, CH2, and CH3. . . . . . . . . . . . . . . . . 584-4 Probability distribution changes and ong>inong>formation entropy changes withrestrictions of CH1, CH2, and CH3. The land type is 0w. After scalong>inong>g,T skong>inong> = 3, σ water = 2, CH1 = 23, CH2 = 27, CH3 = 23. Color blocksare probability withong>inong> each bong>inong>, usong>inong>g the vertical axis to the left.The black triangle long>inong>es are relative entropy (RE). The black dottedlong>inong>es are change of shannon entropy ∆SE. Both triangle and dottedlong>inong>es follow the vertical coordong>inong>ates to the right. Shown ong>inong> each panelsare: (a) only climate data, (b) climate data with CH1 restriction, (c)climate data with restrictions of CH1 and CH2, and (d) climate datawith restrictions of CH1, CH2, and CH3. . . . . . . . . . . . . . . . . 59


xii4-5 Probability distribution changes and ong>inong>formation entropy changes withrestrictions of CH1, CH2, and CH3. The land type is 0w. After scalong>inong>g,T skong>inong> = 4, σ water = 1, CH1 = 23, CH2 = 23, CH3 = 26. Color blocksare probability withong>inong> each bong>inong>, usong>inong>g the vertical axis to the left.The black triangle long>inong>es are relative entropy (RE). The black dottedlong>inong>es are change of shannon entropy ∆SE. Both triangle and dottedlong>inong>es follow the vertical coordong>inong>ates to the right. Shown ong>inong> each panelsare: (a) only climate data, (b) climate data with CH1 restriction, (c)climate data with restrictions of CH1 and CH2, and (d) climate datawith restrictions of CH1, CH2, and CH3. . . . . . . . . . . . . . . . . 604-6 Raong>inong> rate ong>inong>formation change by addong>inong>g CH1 restriction. (a) averageraong>inong> rate (R) difference, (b) avaerage raong>inong> rate ratio, (c) probabilityof R = 0, (d) probability of R > 1mm hr −1 , (e) relative entropy afteraddong>inong>g CH1, and (f) Shannon entropy difference after addong>inong>g CH1.The land type is 0w. Environmental variables: T skong>inong> = 6, σ water = 6after scalong>inong>g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644-7 Same as above. The land type is 1w. Environmental variables: T skong>inong> =5, σ water = 6 after scalong>inong>g. . . . . . . . . . . . . . . . . . . . . . . . . 664-8 Raong>inong> rate ong>inong>formation change by addong>inong>g CH2 restriction. (a) averageraong>inong> rate (R) difference, (b) avaerage raong>inong> rate ratio, (c) probabilityof R = 0, (d) probability of R > 1mm hr −1 , (e) relative entropy afteraddong>inong>g CH2, and (f) Shannon entropy difference after addong>inong>g CH2.The land type is 0w. Environmental variables: T skong>inong> = 6, σ water = 6after scalong>inong>g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714-9 Same as above. The land type is 1w. Environmental variables: T skong>inong> =5, σ water = 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72


xiii4-10 Raong>inong> rate ong>inong>formation change by addong>inong>g CH3 restriction. (a) averageraong>inong> rate (R) difference, (b) avaerage raong>inong> rate ratio, (c) probabilityof R = 0, (d) probability of R > 1mm hr −1 , (e) relative entropy afteraddong>inong>g CH3, and (f) Shannon entropy difference after addong>inong>g CH3.The land type is 0w. CH1 = 24, Environmental variables: T skong>inong> = 4,σ water = 3 after scalong>inong>g. . . . . . . . . . . . . . . . . . . . . . . . . . . 754-11 Same as above. The land type is 1w. CH1 = 24, Environmentalvariables: T skong>inong> = 6, σ water = 6 after scalong>inong>g. . . . . . . . . . . . . . . . 774-12 Bivariate histograms comparong>inong>g distributions of raong>inong> rates prior to(horizontal axis) and followong>inong>g (vertical axis) the successive addition ofchannels, startong>inong>g with Channel 1 relative to climatology (left), Channel1+2 relative to Channel 1 only (center), Channels 1-3 relative to1+2 (right). The surface type here is ocean (0w). . . . . . . . . . . . 814-13 Same as above. The surface type here is warm vegetated land (1w). 814-14 Same as above. The surface type here is warm land/water mix, coast(2w). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824-15 Same as above. It ong>inong>cludes all surface types (0w, 1w, 1c, 2w, 2c, 3w,3c, 4w, 4c, 5w, 5c, 6w, 6c), where surface 4c has zero selected samplesfor the statistics that are statistically representative. . . . . . . . . . 82


ABSTRACTReduced-Dimensional Retrievals of Precipitation from the TRMMMicrowave Imager: Physical Insight and Information ContentbyKe LiUniversity of Wisconsong>inong>-Madison, , 2013The TRMM Microwave Imager (TMI) is employed to observe and measure tropical raong>inong>fallusong>inong>g ong>inong>formation from nong>inong>e passive microwave channels. These channels are sensitivenot only to raong>inong>fall but also to clouds, water vapor, ice, surface moisture, and snow cover,among other variables. These other variables are a noise source ong>inong> the retrieval of precipitation.Petty (2012) has devised a dimensional reduction algorithm that filters out muchof the background noise while retaong>inong> much of the precipitation ong>inong>formation ong>inong> the form ofthree so-called pseudo-channels, which are constructed as long>inong>ear combong>inong>ations of the origong>inong>alnong>inong>e channels. The purpose of the present work is to ong>inong>vestigate the nature of the physicalong>inong>formation about storm structure contaong>inong>ed ong>inong> each of the pseudo-channels and to examong>inong>etheir combong>inong>ed utility ong>inong> revealong>inong>g subtle differences between precipitatong>inong>g cloud systems ong>inong>different environments.xiv


Chapter 1Introduction1.1 Historical OverviewRaong>inong> rate retrieval from satellite has been movong>inong>g forward maong>inong>ly accordong>inong>g to thetechnology improvement for radiometers.1.1.1 Experimental basis before 1978In the early 1970s, a few passive microwave sensors were used for precipitation estimationresearch. Both Nimbus–5 launched ong>inong> 1972 and Nimbus–6 launched ong>inong> 1976 hadthe Electronic Scannong>inong>g Microwave Radiometer [ESMR, Wilheit (1972)] on board.ESMR on Nimbus–5 platform had only one channel, 19 GHz, horizontally polarized,with a resolution of 25 km. The swath width is 160 km. ESMR on Nimbus–6 platformhad two channels, 37 GHz, horizontally and vertically polarized.One of the earliest quantitative raong>inong> rate estimations was built by Wilheit et al.(1977), who associated the change of brightness temperature of 19 GHz on Nimbus–5over the ocean with surface raong>inong> rate variation. Wilheit et al. (1977) utilized raong>inong>gauge and radar observations to verify the assumption. The raong>inong> rate was calculatedfrom radar observations (Marshall and Palmer, 1948). The ocean was observed to bea cold background ong>inong> brightness temperature of 19 GHz. For moderate raong>inong> rate 1 ∼20mm hr −1 , the brightness temperature warms up. At high raong>inong> rates (> 20mm hr −1 ),1


2brightness temperature of 19 GHz was observed to reach ‘saturation’, due to theenhanced scatterong>inong>g of ice aloft.Another one of quantitative raong>inong> rate estimations was built by Weong>inong>man and Guetter(1977), who associate the change of brightness temperature difference betweenvertical and horizontal polarizations of 37 GHz on Nimbus–6 over the ocean and landwith raong>inong> rate. The water surface was observed to have strong polarization comparedwith land at 37 GHz. When precipitation was added to the water background, brightnesstemperatures was observed to be weakly polarized. A ‘Polarization correction’was defong>inong>ed to elimong>inong>ate contrast between water and land as background noise ong>inong> raong>inong>rate retrieval. Simple form ong>inong> noise filterong>inong>g among surface types was widely usedong>inong> later raong>inong> rate retrieval algorithms (Spencer et al., 1989; Conner and Petty, 1998;Kidd, 1998).1.1.2 SMMR ong>inong> 1978In 1978, two copies of the Scannong>inong>g Multichannel Microwave Radiometer (SMMR)were launched, on the Seasat-A oceanographic satellite and the Nimbus-7 meteorologicalsatellite (Gloersen and Barath, 1977). The first of these failed 3 months afterlaunch; while the Nimbus 7 SMMR contong>inong>ued to function for nong>inong>e years.SMMRwas the most advanced microwave radiometer launched to that time. SMMR has tenchannels at frequencies at 6.6, 10.7, 18.0, 21.0, and 37.0 GHz with horizontal andvertical polarizations each. SMMR was ong>inong>tended to obtaong>inong> ocean and atmosphericparameters (ex. sea surface temperatures, low altitude wong>inong>ds, water vapor and cloudliquid water content). The Nimbus 7 SMMR was decomissioned ong>inong> 1987.Examong>inong>ong>inong>g SMMR data, (Spencer et al., 1983) addressed three most basic andquite ong>inong>fluential concepts on physical-based raong>inong> rate retrieval algorithm for laterliteratures.First, cold brightness temperature at 37 GHz was observed ong>inong> heavy


3raong>inong>fall events over land. Brightness temperatures on SMMR ong>inong>clude contributionsnot only due to liquid phase atmospheric constituents, but also due to ice phaseatmospheric constituents. Due to this observation, raong>inong> rate retrievals were conceptuallydivided ong>inong>to two parts: emission based algorithm, where warm and nonpolarizedliquid phase atmospheric constituents were seen agaong>inong>st cold and polarized water surfacebackground; and scatterong>inong>g based algorithms, where cold and weakly polarizedice phase atmospheric constituents were seen agaong>inong>st cold and polarized water surfacebackground or warm and weakly polarized land surface. The similar observationassociatong>inong>g surface raong>inong> rate with ong>inong>tensity of brightness temperature depression dueto ice scatterong>inong>g was addressed ong>inong> Wilheit (1986). Third, raong>inong> rate retrieval biasesdue to partially filled fields of views (FOVs) was observed ong>inong> Spencer et al. (1983),which was later named as ‘beamong>inong>g fillong>inong>g effect’ (Kummerow, 1998; Kummerow andPoyner, 2004; Petrenko, 2001; Petty, 1994a,b; Chiu et al., 1990; Ha and North, 1995).These three basic observations regardong>inong>g microwave brightness temperatures and raong>inong>rate domong>inong>ated maong>inong>stream thong>inong>kong>inong>g about the retrieved problem for many years.1.1.3 SSM/I ong>inong> 1987On June 19th, 1987, the first Special Sensor Microwave/Image (SSM/I) was launchedon the F-8 satellite (Block 5D-2 model) ong>inong> the Defense Meteorological Satellite Program(DMSP). The goal of SSM/I was to provide the Navy and Air Force withoperational imagery of weather systems and remote sensong>inong>g data on critical surfaceand atmospheric parameters.The F-8 satellite was set to travel ong>inong> a near polar(98.8 o ) circular (±7m) sun-synchronous orbit. With an altitude of 833km has microwavechannels of 19.35, 22.235, 37.0 and 85.5 GHz (Hollong>inong>ger, 1989). Most of theSSM/I microwave channels are dual polarized with vertical and horizontal directions,except 22 GHz. In SSM/I, the new channel 85 GHz was added which reflects higher


4sensitivity to ice scatterong>inong>g as well as higher resolution.Spencer et al. (1989) poong>inong>ted out that 85 GHz shows small polarization over a opticallythick precipitatong>inong>g hydrometeor, and large polarization over the ocean surface.He defong>inong>ed the so called ‘Polarization corrected temperature’ (PCT) and used it todistong>inong>guish strong ice scatterong>inong>g from emission due to liquid clouds and precipitation.The PCT was poong>inong>ted out to reflect the ong>inong>tensity of scatterong>inong>g by ice without theong>inong>fluence of background noise, and the column optical depth.As new algorithms for SSM/I were developed by numerous ong>inong>vestigators (Wilheitet al., 1991; Liu and Curry, 1992; Prabhakara et al., 1992; Kummerow and Giglio,1994b; Petty, 1994b; Ferraro et al., 1996; Wentz, 1998; Schols et al., 1999; Prigentet al., 2001), a number of validation and ong>inong>tercomparison exercises were undertaken,ong>inong>cludong>inong>g the Precipitation Intercomparison Projects (PIP1–3) (Barrett et al., 10-14 Jul. 1995; Smith et al., 1998; Adler et al., 2001) and the Global PrecipitationClimatology Project (GPCP) Algorithm Intercomparison Projects (AIP1–3). PIP1–3 performed regional estimated raong>inong> fall comparison, over land and ocean. AIP1–3focuses on improve algorithm by mergong>inong>g products.Comparisons were also madebetween PIP1–3 and AIP1–3 by Ebert et al. (1996); Barrett et al. (10-14 Jul. 1995);Smith et al. (1998); Adler et al. (2001); Schols et al. (1999) etc.Petty (1994a,b) poong>inong>ted out that SSM/I over the ocean reflects different degreeof three components: emission from atmospheric liquid water and gaseous absorbers,featured with “warm” and weakly polarized Tbs, scatterong>inong>g due to ice-phase precipitation,featured with “cold” and weakly polarized Tbs, and polarized ocean surfacebackground, featured with “cold” and strongly polarized Tbs.Further, two parameters were composed ong>inong> (Petty, 1994a,b), to quantitativelyreveal the two most important cloud features strongly associated with raong>inong> rates: depolarizationdue to cloud liquid water from highly polarized ocean surface, measured


y normalized polarization difference P , and ice scatterong>inong>g from top of a precipitatong>inong>gcloud, measured by volume scatterong>inong>g ong>inong>dex S. The parameter P is described as:P ≡T V − T HT V,O − T H,O(1.1)5where T Vand T H are the observed vertical and horizontal polarized brightness temperaturesare the same frequency; and T V,O and T H,O are the vertical and horizontalpolarized brightness temperatures of the same frequency at the same scene withoutthe clouds. The parameter S is described as:S = P T V,O + (1 − P )T c − T V or, S = P T H,O + (1 − P )T c − T H (1.2)where, T c is the (unpolarized) limitong>inong>g brightness temperature as a hypothetical nonscatterong>inong>gliquid water layer becomes optically thick, which is suggested to be T c =273 K.1.1.4 TRMM ong>inong> 1997The Tropical Raong>inong>fall Measurong>inong>g Mission (TRMM) was launched on November 27,1997, with precipitation radar (PR) and TRMM microwave imager (TMI). TheTRMM scans the earth averagely 16 times a day at a marchong>inong>g speed of 6.9 km/s,coverong>inong>g 38 o S ∼ 38 o N, the tropical region and some subtropical regions. The TMImeasures the ong>inong>tensity of radiation at five separate frequencies 10.7, 19.4, 21.3, 37,and 85.5 GHz with both vertical and horizontal polarizations, except 21.3 GHz onlyhas vertical polarization. The TMI has a swath width of 880 km on the surface. TheTMI has swath of 104 pixels per scan for channels of 10.7, 19.4, 21.3, and 37 GHz,and 208 pixels for channels of 85 GHz (Simpson et al., 1988; Kummerow et al., 1998,2000). Regardong>inong>g the scannong>inong>g geometry, the TMI has off-nadir ong>inong>cident angle of52.8 o and conical scan of 130 o . The PR has nadir scannong>inong>g with a swath of 215 km.The nong>inong>e TMI channels are set to have different of field of views (FOVs, ong>inong> 1-1).


6TRMM experienced a satellite reposition ong>inong> August 2001, which divides the life ofTRMM ong>inong>to preboost and postboost. The preboost started on December 8th, 1997,and ended on August 7th, 2001. Durong>inong>g the preboost, the swath width of TMI was760 km. The spatial resolution is 4.4km at 85.5 GHz. The postboost started onAugust 24th, 2001 and is still on goong>inong>g. The swath width of TMI durong>inong>g postboostperiod is 878 km. The spatial resolution is 5.1 km at 85.5 GHz. Other variablesregardong>inong>g TMI channels are presented ong>inong> Table 1-1.The ong>inong>stallation of first spaceborne weather radar made the TMI and PR complementaryong>inong> terms of surface raong>inong> rate retrieval (Haddad et al., 1997; Grecu andAnagnostou, 2002; Grecu et al., 2004; Grecu and Olson, 2006). The most recent algorithmshave been confong>inong>ed to Bayesian estimation methods, which will be ong>inong>troducedong>inong> later sections. In particular, the Goddard Profilong>inong>g algorithm (GPROF) (Kummerowet al., 2001, 2006, 2007, 2011) has become the standard operational algorithmfor TMI and will contong>inong>ually be refong>inong>ed for GMI for Global Precipitation Measurement(GPM) mission. The earlier versions of GPROF (Kummerow et al., 2001, 2006)depended on a database constructed from simmulation derived from cloud resolvong>inong>gmodels. However, progress has been made ong>inong> the later version of GPROF Kummerowet al. (2011) that avoided both cloud resolvong>inong>g models and screenong>inong>g.1.1.5 GPM ong>inong> 2014The Global Precipitation Measurement (GPM) mission is designed to carry an advancedradar and radiometer system to measure the global precipitation. The GPMCore Observatory will have a Dual-frequency Precipitation Radar (DPR) and a multichannelGPM Microwave Imager (GMI) (Smith et al., 2007). Equipped with DPR,GPM can provide new ong>inong>formation about particle drop size distribution over moderateprecipitation ong>inong>tensities. The new features of GPM will provide measurement for


7research about human activity ong>inong>fluencong>inong>g precipitation. Similar to TMI, GMI willhave horizontal and vertical polarizations cross the 10.7 ∼ 89 GHz spectrum, andpossibly 3–5 additional high frequency channels positioned at 166 GHz and 183 GHz.Regardong>inong>g scan geometry, GMI will have swath width of 885 km, off-nadir ong>inong>cidentangle of 52.8 o , and conical scan range of 140 o . The DPR will have nadir scannong>inong>gwith swath width of 215 km for Ku-band radar and 245 km for Ka-band radar. TheDPR will have spacial resolution of 5 km. The core of GPM is scheduled for launchong>inong> early 2014.1.2 Retrieval Methodology1.2.1 Cloud Resolvong>inong>g ModelsFor the past twenty years, convective-scale models have greatly improved the study ofdynamics and microphysics of mesoscale convective systems. Modern cloud resolvong>inong>gmodels are non-hydrostatic and ong>inong>clude an explicit representation of microphysicalprocess(Kummerow et al., 2001, 2007, 2011). In several studies, (Panegrossi et al.,1998; Adler et al., 2001; Aonashi et al., 1996), algorithms based on cloud resolvong>inong>gmodels have been validated. It was poong>inong>ted out that the precision of raong>inong> rate retrievallargely depends on the quality of the cloud resolvong>inong>g models and associated radiativetransfer assumptions.1.2.2 Screenong>inong>gScreenong>inong>g is a modification methodology to improve the consistancy between raong>inong>fallestimates from different sensors. Screenong>inong>g was firstly addressed ong>inong> more completeddetails by Ferraro et al. (1998), for raong>inong>/no raong>inong> idenfication. In microwave precipitationretrieval over land, screenong>inong>g serves to make a decision whether the observed


8signiture reflects the raong>inong> condition or specific land type that shares the similar observedsigniture (Sudradjat et al., 2011). The Grody-Ferraro screenong>inong>g methodology(Ferraro et al., 1998; Grody, 1991) is frequently applied, and was employed withong>inong>GPROF (Kummerow et al., 2001).Used ong>inong> some prior precipitation retrieval algorithms, screenong>inong>g is set to conceptuallycategorize non-raong>inong> scenarios from raong>inong> scenarios, or categorize stratiform precipitationfrom convective precipitation (Kida et al., 2009; Basist et al., 1998). Besides,screenong>inong>g method is also applied to conceptually categorize land types, such as snowcover, desert, etc Grody (1991); Sudradjat et al. (2011).The quality of raong>inong> rateretrieval algorithm with screenong>inong>g largely depends on the correct screenong>inong>g, whichneeds further ong>inong>vestigation or offers large variations upon different observations.1.2.3 Bayesian estimation methodsModern algorithms to retrieve raong>inong>fall often rely on Bayes’ theorem (Bayes and Price,1763). Bayes’ algorithm, which states that:P (A|B) ∝ P (A) · P (B|A) (1.3)where A is the retrieved variable, e.g.raong>inong> rate; B is the observational variable,e.g. brightness temperatures from TMI; P (A) is the prior probability distributionfunction (PDF) of retrieved variable A; P (B|A) is the PDF of the observationalvariable B conditioned on a specific value of A; and P (A|B) is the posterior PDF ofA conditioned on a specific observational variable B.Based on the Bayes’ theorem, the prior PDF ong>inong> Equation 1.3 is replaced by largedata base rather than a contong>inong>uous function ong>inong> remote sensong>inong>g retrievals (Chiu andPetty, 2006).The large data base for prior PDF collect candidate solutions withassociated observed or modeled multichannel radiances.This variation of Baye’s


9theorem that ong>inong>volves the large data base for prior PDF was called a Bayesian MonteCarlo (BMC) method (L’Ecuyer and Stephens, 2002).Among the recent work done for TRMM and GPM raong>inong> rate retrieval algorithmdevelopment, Bayesian estimation methods were majorly agreed upon (Evans et al.,1995; Kummerow et al., 1996; Olson et al., 1996; Haddad et al., 1997; Marzano et al.,1999; Bauer et al., 2001; Kummerow et al., 2001; Tassa et al., 2003; Di Micheleet al., 2005; Grecu and Olson, 2006; Olson et al., 2006; Chiu and Petty, 2006; ?;Seo et al., 2008; Petty, 2013). In Bayesian methods for microwave imager raong>inong> rateretrieval, a‘lookup’ table is created from the traong>inong>ong>inong>g data. The ‘lookup’ table long>inong>ksthe detectable variables to the retrieved variable. To retrieve raong>inong> rate, the detectedvariables are matched accordong>inong>g to the ‘lookup’ table, and the raong>inong> rate is retrieved.Bayesian methods were employed ong>inong> one of the maong>inong>stream raong>inong> rate retrieval algorithms,Goddard Profilong>inong>g Algorithm (GPROF) (Kummerow and Giglio, 1994b,a;Kummerow et al., 1996; Huffman and Coauthors, 1997; Kummerow and Coauthors,2001; Wilheit et al., 2003; Shong>inong> and Kummerow, 2003; Masunaga and Kummerow,2005; Kummerow et al., 2006; Kummerow, 1993; Kummerow et al., 2009). GPROFwas maong>inong>ly developed by Kummerow et al. (2001); Olson et al. (1996, 2006) overthe ocean, and Ferraro et al. (1998); Ferraro and Li (2002) over land. The GPROFwas firstly mentioned by Kummerow et al. (1996), and has undergone significantimprovements. GPROF aims to retrieve ong>inong>stantaneous raong>inong>fall and the vertical structureof the raong>inong>fall. GPROF uses radiative transfer model and cloud resolvong>inong>g models(CRMs). GPROF 2004 was reported with representiveness errors (Kummerow et al.,2006) on representong>inong>g light smal raong>inong> system, song>inong>ce deep convective system is moreong>inong>terestong>inong>g ong>inong> CRMs. Besides, GPROF 2004 uses empirical screenong>inong>g routong>inong>es developedfor various sensors ong>inong> its raong>inong>/no raong>inong> discrimong>inong>ation, and was reported byKummerow et al. (2006) with ong>inong>correct ratio of stratiform, convective, and shallow


10raong>inong>fall. Later on ong>inong> GPROF 2010 constructed an a-priori database from observedTRMM radar and radiometer measurement. In GPROF 2010, CRM conservativelyserved to constraong>inong> parameters such as cloud water and ice that are not detecteddirectly by PR. Compared with GPROF 2004, GPROF 2010 made the followong>inong>gprogress over the ocean, which became the TRMM V7 product:• No more raong>inong> screens;• No more convective/stratiform separation; and• Pixels are classified only by background sea surface temperature (SST) andtotal precipitable water (TPW).GPROF 2010, compared with 2004, made the ocean part of the algorithm moreconsistant, although the land part is not changed much. However, some troublesomesurfaces and coastlong>inong>es remaong>inong> problematic for light precipitation retrieval (Grody,1991; McCollum and Ferraro, 2005; Sudradjat et al., 2011). It is crucial to fong>inong>dthe way to optimize the noise-to-signal ratio, where the noise refers to any physicalvariation unrelated to raong>inong> rate and signal refers to variation due to precipitation.UW algorithm was developed by Petty and Li (2013a,b) to retrieve the tropical raong>inong>rate, by reducong>inong>g the dimensionality of the nong>inong>e microwave channels to three pseudoong>inong>dependentchannels a.k.a. pseudo-channels, and applyong>inong>g Bayesian algorithm overthe three pseudo-channels (Petty and Li, 2013a,b; Petty, 2013). Dimensionality reductionfrom nong>inong>e microwave physical channels to three pseudo-channels greatly improvedthe efficiency of Bayesian estimation retrievong>inong>g process (Petty, 2013). The dimensionalityreduction would reduce the non-precipitatong>inong>g-related noise from TMI, maong>inong>taong>inong>sample density, and significantly reduce the computational memory requirement.As validation, improvement was shown when the UW algorithm was comparedto 2A12 version 7 algorithm (Petty and Li, 2013a). In Figure 2-8 left, 2A12 v7 was


11poong>inong>ted out to have a slight overall positive bias of about 4% over ocean, while UWhas no overall bias Petty and Li (2013a). Besides, the substantial smaller scatter ong>inong>the UW was also shown ong>inong> Figure 2-8 (Petty and Li, 2013a). In UW, the traong>inong>ong>inong>gdata was made statistically identical to the retrievong>inong>g data, completely ong>inong>dependentfrom models or screenong>inong>g (Petty and Li, 2013a).By Petty and Li (2013b), UW algorithm was compared with other prior publishedraong>inong> rate retrievong>inong>g algorithms for TRMM. In the followong>inong>g are sum-ups of advantagesof UW algorithm compared with other algorithms.• The UW algorithm doesn’t make conceptual distong>inong>ction between land types(such as ocean and land). Rather, all surface distong>inong>ctions are embodied ong>inong> theparticular transformation coefficients and a priori databases associated witheach of seven surface classes.• No ‘screenong>inong>g’ (raong>inong>/no-raong>inong> classification or stratiform/convective precipitationclassification) is applied. Rather, the probobility of non-zero raong>inong> is reflectedthrough the posterior raong>inong> rate distribution generated by Bayesian algorithm.• The result of raong>inong> rate retrieval is not just expected raong>inong> rates. Posterior distributionsof raong>inong> rate are also provided.• In the UW algorithm, TMI-PR matchup data were employed as the a-prioridata base, and no cloud resolvong>inong>g models or simulated radiations are ong>inong>volved.• The UW algorithm pre-averages the a priori database ong>inong>to a 5D lookup table,which consists of two environmental variables (surface skong>inong> temperature and totalprecipitable water content) and three reduced-dimensional pseudo-channelsderived from nong>inong>e microwave TMI physical channels (Petty, 2013).


• Durong>inong>g the retrieval usong>inong>g UW algorithm, there is no weightong>inong>g for candidatesolutions.12Three pseudo-channels are derived usong>inong>g a two-stage prong>inong>ciple component analysis(Petty, 2013), representong>inong>g the three multichannel signatures most related to surfaceraong>inong> rate. Each of the three successive pseudo-channels ong>inong> the Bayesian estimationalgorithm serves to alter the posterior PDF of raong>inong> rate.It is rare that an algorithmproduces an explicit posterior PDF of the retrieved variables. Therefore wehave unique opportunity to rigorously evaluate the ong>inong>formation content of successivepseudo-channels.Assumong>inong>g other thong>inong>gs we may examong>inong>e whether three pseudochannelsare both necessary and reasonably sufficient to maximize the ong>inong>formationfrom the 9 TMI channels.1.3 Information TheoryAs addressed by Kullback (1997), ‘speakong>inong>g broadly, whenever we make statisticalobservations, or design and conduct statistical experiment, we seek ong>inong>formation.’ Informationtheory is a mathematical subject that ong>inong>volves the rigorous quantificationof ong>inong>formation. The fundamental ong>inong>formation theory was pioneered ong>inong> the 1950s,maong>inong>ly by Fisher (1956), Shannon (1956), and Viener (1956).The concept of Information entropy was first presented by Shannon (1948) toquantify the expected value of the ong>inong>formation contaong>inong>ed ong>inong> a message. Therefore,the ong>inong>formation entropy is hereafter called Shannon entropy.1.3.1 Shannon EntropyA detailed mathematical description of Shannon entropy is addressed ong>inong> (Xu, 2006),both ong>inong> discrete forms and contong>inong>uous forms. The discrete Shannon entropy (DSE)


13was written as:DSE ≡ − ∑ p i ln p i (1.4)where p irefers to the probability of occurrence of the ith possible outcome and∑ pi = 1. The contong>inong>uous Shannon entropy (CSE) is written as:∫CSE ≡ −p(x) ln p(x)dx (1.5)where p(x) refers to a contong>inong>uous PDF with variable x.1.3.2 Relative EntropyAssumong>inong>g p and q are two PDFs, relative entropy is a non-symmetric measure of thedifference between probability distributions p and q. Relative entropy is a measurewith direction.Relative entropy from q to p, denoted as RE(p||q), is a measureof the ong>inong>formation lost when q is used to approximate p. Relative entropy from pto q, denoted as RE(q||p), is a measure of the ong>inong>formation lost when p is used toapproximate q. RE(p||q) ≠ RE(q||p) (Kullbak and Leibler, 1951). Relative entropyhas many synonyms ong>inong> literatures, mean ong>inong>formation for discrimong>inong>ation, Kullback-Leiber divergence, crowd entropy, ong>inong>formation gaong>inong>, ong>inong>formation number, ong>inong>formationdivergence, KL distance, expected weight of evidence, discrimong>inong>ation (Verdu, 2010),etc.Wald (1945) for the first time brought ong>inong> the conception of ‘relative entropy’ tosolve the sequential problem of testong>inong>g. The literature described the relative entropyas the expected number of observations necessary for reachong>inong>g a decision. Jeffreys(1945) further ong>inong>novated the relative entropy with a symmetric expression. Despitethe symmetry, the formula for relative entropy ong>inong> (Jeffreys, 1945) turned out not quiteuseful ong>inong> recent literature (Verdu, 2010). The ong>inong>novated formula by Jeffreys (1945)


14nowadays is addressed as separate kong>inong>d of measure of relative entropy, called Jeffery’sdivergence (Verdu, 2010).Kullbak and Leibler (1951) provided the first commonly accepted defong>inong>ition ofrelative entropy, which is the reason that relative entropy is sometimes called KLdivergence. By Kullbak and Leibler (1951), relative entropy was called mean ong>inong>formationfor discrimong>inong>ation. The motivation to ong>inong>troduce the measure was to generalizethe defong>inong>ition of ong>inong>formation by Shannon (1956)and Viener (1956).A detailed mathematical description of relative entropy is addressed by Xu (2006),both ong>inong> discrete forms and contong>inong>uous forms. The discrete Relative entropy (DRE)was written as:DRE ≡ R(p, q) = ∑ p i ln(p i /q i ), (1.6)while the contong>inong>uous form is∫CRE(p, q) ≡p(x) ln[p(x)/q(x)]dx, (1.7)where p i and q i ong>inong> Equation 1.6 are probabilities for a discrete distribution, andp(x) and q(x) ong>inong> Equation 1.7 are PDFs for contong>inong>uous distribution.In Bayesianstatistics, RE(p, q) is defong>inong>ed as a measure of the ong>inong>formation gaong>inong> ong>inong> movong>inong>g from aprior distribution q to a posterior distribution p (Chaloner and Verdong>inong>elli, 1995).Information theory has been widely used ong>inong> research ong>inong> meteorology radar signalanalysis (Xu, 2006), electrical signal analysis, and remote sensong>inong>g area. In particular,withong>inong> remote sensong>inong>g area, Shannon entropy was employed by L’Ecuyer et al. (2005)and Cooper et al. (2005) to explore optimal MODIS channels for cloud propertyretrievals withong>inong> the optimal estimation framework. In the present study, we evaluatethe ong>inong>formation content of the three successive pseudo-channels used ong>inong> retrievals ofraong>inong> rate from TMI, with both relative and Shannon entropies are employed.


1.4 Objectives15This paper has two major objectives:• Investigate the physical meanong>inong>gs of the three successive pseudo-channels derivedwithong>inong> the UW algorithm to retrieve surface raong>inong> rate over the ocean;and• Quantify the ong>inong>formation content of the three successive pseudo-channels ong>inong>terms of alterong>inong>g the posterior surface raong>inong> rate distributions for each of severalsurface classes.To achieve the objectives, the first two chapters are dedicated to the basic ong>inong>troductionson brightness temperatures and three successive pseudo-channels. The thirdchapter focuses on seekong>inong>g the physical meanong>inong>gs of the three successive pseudochannels,and describong>inong>g the procedure of each pseudo-channels alterong>inong>g the raong>inong> rateprobability distribution functions to retrieve raong>inong> rate. The fourth chapter focuseson quantifyong>inong>g the ong>inong>formation content of three pseudo-channels from both necessaryand sufficient perspectives. The last chapter is the conclusion.


16Figure 1-1: Data for postboost TMI sensors ong>inong> various frequencies, ong>inong>cludong>inong>g polarization(V is vertically polarized, H is horizontally polarized.), ong>inong>stantaneous field ofview (IFOV), etc. (Kummerow, 2006)


Chapter 2Passive Microwave Sensors andRaong>inong> Rate RetrievalThe nong>inong>e passive microwave channels on TMI ong>inong>clude dual polarization channels at10 GHz, 19 GHz, 37 GHz, and 85 GHz, and a vertically polarized 23 GHz channel(Figure 1-1), which have varyong>inong>g responses to atmospheric constituents, such aswater vapor, cloud and raong>inong> water, and ice particles aloft.Besides, the nong>inong>e passivemicrowave radiants ong>inong> TRMM have various degree of attenuation/scatterong>inong>g overvarious global surface features, such as water surface, desert, coastal regions, agriculturalland, etc. The TRMM microwave imager (TMI) is responsible for passivelymeasurong>inong>g brightness temperatures of atmospheric constituents and the earth surfacebackground. The associated precipitation radar (PR) is responsible for measurong>inong>g thereflectivity due to raong>inong> drops. PR estimation of raong>inong> rate from reflectivity is based onMarshall-Palmer (Marshall and Palmer, 1948) drop-size distribution.TMI covers a swath width of 833km, over three times as wide as the PR swathwith a width of 247 km. Limited by swath width, PR often cannot fully provide thehorizontal precipitation structure of mesoscale precipitatong>inong>g events. The precipitationdata from PR overlaps with brightness temperatures by TMI over a limited swath atthe center of TMI scans. Over two thirds of the TMI swath doesn’t have direct accessto the ground raong>inong>fall rate. Therefore, a precipitation algorithm is developed to long>inong>k17


18the TMI brightness temperatures that relates clouds and hydrometeors to surfaceraong>inong> rate.The reduced-dimensional Bayesian raong>inong> rate retrieve algorithm serves to predictwhat the PR would retrieve at those locations that are covered only by TMI withoutdirect PR raong>inong> fall measurement. Besides, the raong>inong> rate retrieval can be extrapolatedto satellites without PR but only microwave sensor.In the algorithm, nong>inong>e TMIchannels are reduced to three ong>inong>dependent channels, which largely improves the computationalefficiency and the robustness of the retrieval. Comparisons are shown toquantitatively demonstrate the improvement on retrieval quality.2.1 Brightness Temperatures2.1.1 EmissionBy Planck’s function, with a fixed wavelength, the ong>inong>tensity of radiation of this blackbodyshould be unique upon a given blackbody’s temperature, and vice versa (Petty,2006). Most real surfaces that emit less radiation at a given wavelength and temperaturethan the blackbody does accordong>inong>g to Planck’s function. The monochromaticemissivity ε λ of a surface is defong>inong>ed as the ratio between the actual emitted radiationand the emitted radiation accordong>inong>g to Planck’s function ignorong>inong>g other sources, fora fixed surface temperature T and wavelength.ε λ ≡I λB λ (T )where for a chosen wavelength λ, I λ is the surface radiation, and the B λ (T ) is theplanck’s function of a blackbody temperature T , which gives ong>inong>tensity of blackbodyradiation. Conversely, one can convert ong>inong>tensity of monochromatic radiation from anysurface to an equivalent blackbody temperature (a.k.a. Brightness temperature), usong>inong>gthe ong>inong>verse of Planck’s function. Accordong>inong>g to the Rayleigh-Jeans Approximation,


19Planck’s function for microwave frequencies is a long>inong>ear operator:T B ≡ B −1λ(I λ) = B −1 (ε · B λ(T )) = ε · Tλ2.1.2 TransmittanceAssumong>inong>g the absence of scatterong>inong>g or emission, for a given wave length λ, the radiationattenuation process through a optical path from poong>inong>t s 1 to s 2 can be calculatedasI λ (s 2 ) = t(s 1 , s 2 ) · I λ (s 1 )where I λ (s) refers to the ong>inong>tensity of radiation at poong>inong>t s, t(s 1 , s 2 ) is the transmittanceof the constituent between s 1 and s 2 , that causes the radiation attenuation at thechosen wave length. The transmittance t can be calculated as:t(s 1 , s 2 ) ≡ exp(−τ(s 1 , s 2 ))where τ(s 1 , s 2 ) is the optical thickness of the constituents from poong>inong>t s 1 to poong>inong>t s 2(Petty, 2006).2.1.3 Radiative Transfer EquationsWhen scatterong>inong>g of a constituents take place, the complete radiative transfer equationong>inong> differential form (Petty, 2006) is written as:dI( ˆΩ)dτ= I( ˆΩ) − (1 − ˜ω)B − ˜ω ∫p(4πˆΩ ′ , ˆΩ)I( ˆΩ ′ )dω ′ (2.1)4πwhere ˜ω ≡ β s /β e is the song>inong>gle scatterong>inong>g albedo, β s is the scatterong>inong>g coefficient and β eextong>inong>ction coefficient, B is the Planks’ function, dτ is an ong>inong>crement of optical depth,I( ˆΩ) is the ong>inong>tensity of radiation from a chosen direction represented by a unit vectorˆΩ, p( ˆΩ ′ , ˆΩ) is the scatterong>inong>g phase function for an arbitrary combong>inong>ation of ong>inong>comong>inong>gˆΩ ′ and scattered directions ˆΩ. On the right hand side of Equation 2.1, the first term


gives the attenuation, the second term gives the emission, and the last term gives thescatterong>inong>g source. The optical depth dτ can be calculated as:20dτ = −k e ds = −(k e /µ)dzwhere k e is the extong>inong>ction coefficient. s is the geometric distance along an optical pathof a constituent, which can be calculated with µ, the cosong>inong>e value of zenith angle, andvertical coordong>inong>ates ong>inong> cloud remote sensong>inong>g.Equation 2.1 describes that the radiance along a particular long>inong>e of sight (withoptical depth of τ) either ong>inong>crease or decrease, dependong>inong>g on whether ong>inong>itial ong>inong>tensityof radiation I( ˆΩ) is greater or less than emission (1 − ˜ω)B plus scatterong>inong>g˜ω ∫4π 4π p( ˆΩ ′ , ˆΩ)I( ˆΩ ′ ). With assumption of non-scatterong>inong>g, a radiance passong>inong>g throughthe atmospheric constituents would depend on the transmittance t and emission fromconstituents B(T ), as the first two terms ong>inong> Equation 2.1 with ˜ω = 0. Incorporatedwith Rayleigh-Jeans approximation, non-scatterong>inong>g version of Equation 2.1 is writtenas:dT B = k e (z)(T − T B )dz/µ (2.2)After Integratong>inong>g Equation 2.2 from surface to the top of atmosphere, one can calculatethe upward-directed atmospheric component of the brightness temperature T ↑ B:T ↑ B = 1 µ∫ ∞0T (z)k e (z)t(z, ∞)dz (2.3)After ong>inong>tegratong>inong>g Equation 2.2 from top of atmosphere to surface, one can calculatethe downward-directed atmospheric component of the brightness temperature T ↓ B:T ↓ B = t(0, ∞)TB toa + 1 ∫ ∞T (z)k e (z)t(0, z)dz (2.4)µ 0Both T ↑ B and T ↓ B are atmospheric brightness temperatures towards the top of atmosphereand towards the surface of earth. The brightness temperature T B observed


ong>inong> the space by the satellite have direct relationship with T ↑ B and T ↓ B, and can becalculated as:21T B = T ↑ B + t(0, ∞)[ε s T s + (1 − ε s )TB] ↓ (2.5)When a radiance is received by the satellite microwave sensors, the signal consistsof contributions from three parts about the earth (Figure 2-1): surface emission,atmospheric emission/scatterong>inong>g, and surface reflection.The first part of radiation received by TMI comes from direct emission from theearth surface (gray dotted arrows ong>inong> Figure 2-1). the emission depends on two majorfactors, the earth surface emission ε s T s and atmospheric transmittance t A = t(0, ∞)ong>inong> Equation 2.5.The second part of radiation receive by TMI (blue solid arrows poong>inong>tong>inong>g upwardong>inong> Figure 2-1) comes from upward emission of the atmospheric constituents TB, ↑ afunction of atmospheric temperature distribution T (z), extong>inong>ction coefficient k e (z),and transmittance t(z, ∞) accordong>inong>g to Equation 2.3.The third part of radiation received by TMI is the radiance that origong>inong>ally areemission from atmospheric constituents to earth surface, get reflected by the earthsurface, transmit through the atmospheric constituents for the second time, thenreach sensor (blue downward arrows and green arrows ong>inong> Figure 2-1). The third partof radiation that ong>inong>volves earth reflection depends on earth surface reflectivity (1−ε s ).With scatterong>inong>g at presence, accordong>inong>g to Equation 2.1, the emission B(T ) isreduced due to a ong>inong>creased ˜ω. Therefore ong>inong> Equation 2.5, an ong>inong>creasong>inong>g ong>inong> scatterong>inong>gdue to ice would reduce the upward atmosphere emission T ↑ B and reflected emissionTB.↓ The ice particles can take ong>inong> various forms, such as ice particles aloft ong>inong> anvil ordeep convective storms, and ice formed atmospheric hydrometeors (Petty, 1994a,b;


22Bennartz and Petty, 2001). The ice particles ong>inong> the clouds are strongly scatterong>inong>g,directly resultong>inong>g ong>inong> a reduced bulk emissivity of the cloud top (Bennartz and Petty,2001).There are other factors contributong>inong>g to the brightness temperatures (Petty, 1994a)such as: the water vapor, reflected image of raong>inong> cell by the surface, FOV averagong>inong>gprocess, etc.2.1.4 Earth Surface TypesAs the second term t(0, ∞)ε s T s ong>inong> radiative transfer equation (Equation 2.5), earthsurface types provide background ong>inong> TMI images for raong>inong> rate retrieval. Correctlyassessong>inong>g the surface type background would great aid the separation of atmosphericconstituents (such as clouds, raong>inong>, snow, ice aloft etc) from the earth surface. It isknown that ong>inong> microwave sensors, surface background have a large variation ong>inong> brightnesstemperatures. As an acknowledged challenge of microwave raong>inong> rate retrievals,some surface types also have reduced brightness temperatures from TMI as clouds, orprecipitatong>inong>g events would, such as coastal regions, desert regions, and accumulatedsnow on the ground, etc. Therefore, it is extremely important to recognize the surfacetypes from microwave sensor perspectives.Observed by microwave remote sensors, earth surface is ong>inong> general categorized ong>inong>toland, water, and sea ice (but sea ice is not a factor ong>inong> the view of TMI). Land usuallyprovides a warmer background ong>inong> microwave brightness temperatures, than water(ocean, lake, etc) does (Figure 2-2). Ocean ong>inong> TMI has large emissivity polarizationdifference compared with land (Figure 2-3), and provides a colder background thanland (Figure 2-2).Withong>inong> land categories, various surface types are reflected ong>inong> emissivities and surfacetemperatures under TRMM microwave channels. Over the land ong>inong> TMI, veg-


23!"##$%&'($)*+,"%&'-&.+/&'0&1#$12'T b = ε · T s +(1− t − r) · T A +(1− ε)(1 − t − r) · T AFigure 2-1: Passive microwave remote sensong>inong>g mechanism. Green arrows are radiancewhere, T b is brightness temperature, T A is the averaged atmospheric temperature,reflectedT s is thebysurface the earth.temperature,Blue arrows areε isradiancethe surfaceemittedemissivity, directly by thet isatmosphere.the transmittanceof the atmosphere, r is the reflectivity of atmosphere due to ice scatterong>inong>gThe gray dotted arrow represents radiance emitted directly from the earth surface.T A is the averaged atmospheric temperature. T s is the surface temperature. ε is the3'surface emissivity. t is the transmittance of the atmosphere.


24etation area is observed to have large emissivities, but low emissivity polarizationdifferences, compared with desert areas. Over at Tibetan plateau and some mountaong>inong>areas, cold brightness temperature is shown on song>inong>gle channel map (Figure 2-4).Some surface types can not be easily shown through polarization difference or song>inong>glechannel, but can be distong>inong>guished through combong>inong>ations of nong>inong>e TMI channels.In order to quantitatively categorize surface types, a more detailed algorithm wasdeveloped to categorize the surface types. This surface type division algorithm has aresolutions 1 ◦ ×1 ◦ , utilizong>inong>g clusterong>inong>g technique applied to the means and covariancesof TMI brightness temperatures for non-precipitatong>inong>g pixels (Petty and Li, 2013a).As a result from the surface type algorithm, the surface is categorized ong>inong>to 6 classesas a background with warm and cold surface temperatures on each division Figure2-5 by Petty and Li (2013a). The classes are named ong>inong> Table 2.1. For each surfaceclass except ocean (Class 0), there are cold and warm background difference denotedas ‘w’ and ‘c’ respectively.In short, the surface backgrounds are reflected differently among nong>inong>e TMI channels,and are categorized ong>inong>to six surface types. The categorized surface types serveas various background for the raong>inong> rate retrieval algorithm.2.1.5 Cloud and Raong>inong> Water Emission and Ice Scatterong>inong>gDurong>inong>g a precipitatong>inong>g event, vertical structure of the event consists of the near surfacehydrometeor (snow, raong>inong>fall, etc), the cloud and raong>inong> water, and the ice aloft. Cloudand raong>inong> water emission is considered to have important correlation with raong>inong> rate.In microwave remote sensong>inong>g, the cloud water emission contributes to the receivedbrightness temperatures (Equation 2.5). Therefore, measurong>inong>g cloud and raong>inong> wateremission by microwave channels ong>inong> TMI would provide important ong>inong>formation for raong>inong>rate retrieval.


25ClassDescription0 Ocean1 Vegetated land2 Land/water mix (coast)3 Desert4 Raong>inong> forest5 Tibetan Plateau and similar6 Himalayan range and similarTable 2.1: Surface type descriptions ong>inong> Figure 2-5 (Petty and Li, 2013a)Figure 2-2: Land ocean brightness temperature difference reflected ong>inong> the image ofAMSR-E 10 GHz V on the west coast of US. (Courtesy to U.S.Navy/NRL/NASA)


26Figure 2-3: Map of brightness temperature differences, 10 GHz(V) - 10 GHz(H) for2002 TMI non-precipitatong>inong>g pixels.Figure 2-4:Map of brightness temperature of 85 GHz(H) for 2002 TMI nonprecipitatong>inong>gpixels.Fig. 1. Map of the seven surface classes constructed empirically from a climatology of TMIbrightness temperatures. See Table 1 for descriptive labels.Figure 2-5: Map of the seven surface classes constructed by usong>inong>g TMI brightnesstemperatures for 2002. (Petty and Li, 2013a)


27Besides, ice can be at presence, when there is severe convective storm, hurricanes,or anvil clouds. The ice aloft would be subject to scatterong>inong>g of radiance comong>inong>g fromthe lower part of atmosphere. Therefore ice scatterong>inong>g is counted to be responsible tobrightness temperature depression (Petty, 1994a, 2001). Therefore it is very importantto ong>inong>vestigate the nong>inong>e microwave channel behaviors towards cloud and raong>inong> wateremission and ice scatterong>inong>g. It is acknowledged that 10 GHz is less absorbed by cloudand raong>inong> water, more transparent to the precipitatong>inong>g clouds, and less sensitive toice scatterong>inong>g than higher frequency channels. As frequency ong>inong>creases to 85 GHz, theTMI is more absorbed by cloud and raong>inong> water, reflectong>inong>g more opaque atmospherefor precipitatong>inong>g events, and more sensitive to ice scatterong>inong>g (Petty, 2001).An orbit of TMI and PR is chosen to reveal the relationship between PR measuredraong>inong> rate and TMI channels (Figure 2-6). The ocean is used as a background. (Therest of this chapter, ocean is used as a background unless specially mentioned.) Asis shown ong>inong> Figure 2-6 left, the brightness temperature ong>inong> 10 GHz and 19 GHz getwarmer, when the raong>inong> rate is ong>inong>creasong>inong>g. Typically, it is known that raong>inong> rate hasa positive correlation with cloud and raong>inong> water. A high precipitatong>inong>g event usuallyis associated with thick cloud and raong>inong> water, which ong>inong>crease the emission from thecloud. It is known that ocean tends to provide a cold background (160K ∼ 200Kfor 10 GHz (V) and 19 GHz(V)), any cloud and raong>inong> water would obscure the oceanbackground and warm up the brightness temperatures.Under 37 GHz and 85 GHz, the ong>inong>crease of raong>inong> rate is subject to an eventual decreaseof brightness temperatures. The decrease ong>inong> brightness temperature ong>inong> 37 GHzand 85 GHz is believed to reflect the brightness temperature depression by ice scatterong>inong>g(Petty, 2001). When a raong>inong> rate ong>inong>creases to be ≥ 5mm hr −1 , the ice may bevery possible at presence which causes to decreases the brightness temperatures.In short, it is confirmed ong>inong> Figure 2-6 that lower frequency channels (10 GHz and


19 GHz) are very sensitive to cloud and raong>inong> water emission, while high frequencychannels are very sensitive to ice scatterong>inong>g (37 GHz and 85 GHz).282.1.6 Polarization and Raong>inong> RateThe polarization ong>inong> the microwave channels are not only playong>inong>g a role ong>inong> distong>inong>guishong>inong>gearth surface backgrounds, but also are very important to reflect raong>inong>ong>inong>gfeatures.As is known, ocean background provides cold TMI brightness temperature andis strongly polarized observed by TMI sensors. When a storm is detected over theocean background, the optical thickness of the cloud and raong>inong> water would have atendency to warm up the brightness temperature as the cold ocean background isobscured by the precipitatong>inong>g layer (Petty, 2001). The ice aloft (if ice is ong>inong>volvedat all) would reduce the brightness temperature due to scatterong>inong>g, and also decreasethe polarization.the vertical gradients of brightness temperatures ong>inong> a storm iceparticles aloft. The polarization of ice particles ong>inong> TMI were addressed to depend onthe orientation and size distribution of the ice particles.Polarization is shown ong>inong> TMI 1B11 data.In Figure 2-7, polarization is long>inong>kedwith raong>inong> rate over an orbit of TMI data. Figure 2-7 shows that when raong>inong> rate issmall, typically, more radiation directly comes from the ocean, which appears to bestrongly polarized and cold (black dots ong>inong> Figure 2-7). As the raong>inong> rate ong>inong>creases,more and more ice aloft is ong>inong>volved, which is responsible for high frequency channels(85 GHz) to have brightness temperature depression (Figure 2-7 bottom, right). Inshort, high raong>inong> rate samples tends to have smaller polarization than low raong>inong> rateones. Besides, high frequency channels are more sensitive to ice scatterong>inong>g than lowfrequency channels ong>inong> TMI.


29Figure 2-6: Typical response of different TRMM channels to raong>inong> rate. Red dots are10 GHz. Blue dots are 19 GHz. Magenta dots are 37 GHz. Green dots are 85 GHz.The horizontal axis is retrieved raong>inong> rate, not directly measured PR raong>inong> rate. Theplots here serve to illustratively show the trends of brightness temperatures. Usong>inong>gretrieved raong>inong> rate would have more samples that represent trends of TMI brightnesstemperatures.


30Figure 2-7: Scatterong>inong>g plots of polarizations with raong>inong> rate under TMI channel: (upper,left) 10 GHz, (upper, right) 19 GHz, (bottom, left) 37 GHz, (bottom, right)85 GHz. Horizontal axes are brightness temperatures of vertically polarized channels,and vertical axes are brightness temperatures of horizontally polarized channels.


2.2 Bayesian Algorithm for Raong>inong> Rate Retrieval31To quantitatively retrieve the raong>inong> rate usong>inong>g the pre-mentioned features of eachmicrowave channels, the reduced-dimensional Bayesian algorithm (also known as UWalgorithm) was developed by Petty and Li (2013a). The goal for this algorithm isto take ong>inong> the nong>inong>e TMI channels and retrieve the surface raong>inong> rate at places wherethere is no PR available. The maong>inong> idea of UW algorithm is to create a year long‘look-up table’ that long>inong>ks the TMI brightness temperature to the PR raong>inong> rate, andretrieve raong>inong> rate. Compared with current GPROF (Kummerow et al., 2001), thereduced dimensional Bayesian algorithm features the high computational efficiencyand availability of posterior PDF of raong>inong> rate (Petty and Li, 2013b,a).As the first step, the dimension of variables are reduced.Often ong>inong> times, themost domong>inong>ant features (e.g.surface features) observed from microwave channelsare not necessarily the desired features (e.g. raong>inong> and cloud related). Through thedimensional reduction, the geophysical noise is filtered out, and the sensitivity ofprecipitation related features is captured.Thus, the dimensional reduction of nong>inong>e microwave physical channels consists oftwo parts. The first part is to reduce the geophysical noises from physical channels.With only no-raong>inong> pixels, the variance due to the non precipitatong>inong>g features is convertedto unit variance with zero cross-correlations (Petty, 2012). As a result, thegeophysical noises due to surface types or other non-precipitatong>inong>g-related features arefiltered out. The second part is to perform the second long>inong>ear-transformation. Whenaddong>inong>g the pixels of precipitatong>inong>g events, all the ong>inong>formation due to precipitation iscollected. The prong>inong>ciple component analysis is performed to collect the raong>inong> relatedong>inong>formation ong>inong>to smaller number of pseudo-channels.


32The three modes from prong>inong>ciple component analysis are ong>inong>dependent 1 from eachother, thus are called ong>inong>dependent channels (or ‘pseudo-channels’). Durong>inong>g the processof dimensional reduction, much of the background noise is filtered out (Petty and Li,2013b). In short, the dimensionality reduction serves to filter out the backgroundnoise, and reduce the number of channels from nong>inong>e to three for Bayesian raong>inong> rateretrieval. In addition, each of the scene was tagged with surface skong>inong> temperatureT skong>inong> and total precipitable water (TPW) σ water . Both T skong>inong> and σ water are ong>inong>putsto the algorithm, obtaong>inong>ed from the European Centre for Medium-range WeatherForecastong>inong>g (ECMWF) Reanalysis ERA-Interim (Dee et al., 2011).As the second step, the Bayesian ‘lookup table’ is created ong>inong>cludong>inong>g two environmentalvariables (T skong>inong> and σ water ) three ong>inong>dependent channels (CH1, CH2, andCH3) and PR raong>inong> rate. The background noise is majorly filtered out when ong>inong>dependentchannels are generated, so the three ong>inong>dependent channels can be used ong>inong> thetable with equal weights (Petty and Li, 2013b). Besides, to use the ‘lookup table’,the brightness temperatures for retrieval are required to be converted to the threeong>inong>dependent channels.As a result of the algorithm, posterior raong>inong> rate PDFs and related ong>inong>formation aregenerated for all the gridded ong>inong>dependent channels and environmental variables (columnwater vapor depth and surface temperature). The mean of each raong>inong> rate PDFis counted as the retrieved raong>inong> rate for each ong>inong>dependent channel grid. Comparisonbetween UW and 2A12 v7 was made (Petty and Li, 2013a) (Figure 2-8). The UWimproves both precision of raong>inong> rate retrieval and efficiency of fong>inong>dong>inong>g the match ong>inong>the ‘lookup’ tables.1 The three modes are ong>inong>dependent ong>inong> the long>inong>ear sense. The higher order correlation between the threecalculated modes (pseudo-channels) are possible to exist.


33Fig. 6. Bivariate histograms of the gridded annual raong>inong>fall totals depicted ong>inong> Fig. 3 for thetwoFigure TMI algorithms 2-8: Comparisons under evaluation of algorithms versus and thePR2A25 raong>inong>reference rate data. raong>inong>fall (Left) forcomparisonocean only. a)2A12, b) UW-M.between PR raong>inong>fall rate and raong>inong>fall retrieved ong>inong> 2A12; and (Right) comparisonbetween PR raong>inong>fall rate and raong>inong>fall retrieved with UW algorithm. (Petty and Li,2013a)37


Chapter 3Pseudo-channelsPseudo-channels were developed from the nong>inong>e TRMM microwave channels as the topthree ong>inong>dependent modes ong>inong> the prong>inong>ciple component analysis Petty and Li (2013a,b).To retrieve raong>inong> rate, the peudo-channels were the middle product for fong>inong>dong>inong>g thematch ong>inong> the traong>inong>ed ‘lookup’ table, representong>inong>g nong>inong>e TRMM microwave channels.However, the physical meanong>inong>g of each pseudo-channels are not known, ong>inong> order tofully corroborate the physical process of the UW algorithm. Gram-Schmidt processis employed to seek the essence of each pseudo-channel.By producong>inong>g pseudo-channels, the background noise such as land type, or otherfactors don’t necessarily reflect the raong>inong> rate features are filtered. More detailed algorithmof generatong>inong>g pseudo-channels were clearly addressed ong>inong> Petty and Li (2013b,a);Petty (2013).3.1 Over the OceanPrecipitatong>inong>g features over the ocean can be relatively easier to recognized throughmicrowave imagers than land. The ocean provides a colder and polarized backgroundong>inong> the microwave channels than land. Relative to the ocean background, liquid wateremission due to precipitatong>inong>g events appear to be warmer and unpolarized ong>inong>microwave channels. Therefore, ong>inong> lower frequency channels ong>inong> microwave sensors,precipitatong>inong>g events tend to have a warm image over cold background (Figure 3-334


35(a)). When ice aloft is ong>inong>volved, cold brightness temperature due to strong scatterong>inong>gand weak polarization are reflected ong>inong> high frequency microwave channels (Figure 3-3(b)).It was addressed that the relationship between brightness temperatures and variationsong>inong> different channels are nonlong>inong>ear Petty and Li (2013b). To long>inong>early decomposethe 9 observed microwave channels to pseudo-channels, the observed channels aretransformed ong>inong>to the followong>inong>g form Petty and Li (2013b):x i,o = log(T skong>inong> − T B,i ) (3.1)where, T B,i stands for brightness temperatures of different microwave channels, T skong>inong>is the skong>inong> temperature of the surface. Through the transform (Equation 3.1), thenonlong>inong>earity and background noise are elimong>inong>ated Petty and Li (2013b).Pseudochannelsover the ocean are generated with ⃗x usong>inong>g prong>inong>ciple component analysis.3.1.1 Gram-Schmidt ProcessThree pseudo-channels are calculated to be the key components to measure tropicalraong>inong>fall rate over the ocean.1Gram-Schmidt analysis is to convert three pseudochannelsto a set of observed channel vectors that are orthogonal to each other. Inthis way, the physical meanong>inong>g of the pseudo-channels can be ong>inong>ferred from observedTMI channels. Besides, Gram-Schmidt analysis serves to filter out the dependencebetween pseudo-channels due to errors caused by manual division on pseudo-channelscatter plots (Figure 3-1). Therefore, the ‘distong>inong>ct’ features of pseudo-channels canbe achieved as a result.1 It is worthwhile to notice that choosong>inong>g the top three modes from the result of PCA is decided concernong>inong>gtrade of reduction ong>inong> dimensionality from nong>inong>e physical channels and raong>inong> rate retrieval algorithm quality.Further ong>inong>formation regardong>inong>g the number of kept modes for pseudo-channels is illustrated ong>inong> Chapter 4Information Theory.


36The pesudo-channels are set to be zeros when there is no precipitatong>inong>g clouds atpresence. CH1 ranges always positive, CH2 and CH3 can be positive and negativevalues. In the scatter plots of CH1, CH2, CH3 (Figure 3-1), fives cases are bong>inong>ned tofurther ong>inong>vestigate each pseudo-channel (Table 3.1).⃗v i = T ⃗ i − T ⃗ 0 , i = 1, 2...5 (3.2)where T ⃗ i refers to the vector of 9 microwave brightness temperatures ong>inong> different caseswith case number i, and ⃗v is the vector of temperature difference of non-backgroundcases subtracted by background vector. T 0 is the brightness temperature vector ofbackground case (Table 3.1). The Gram-Schmidt analysis for pseudo-channels arecalculate:⃗u 1 = ⃗v 1 (3.3)⃗u 2 = ⃗v 2 − proj u1(⃗v 2 ) (3.4)⃗u 3 = ⃗v 3 − proj u1(⃗v 3 ) (3.5)⃗u 4 = ⃗v 4 − proj u1(⃗v 4 ) − proj u2(⃗v 4 ) − proj u3(⃗v 4 ) (3.6)⃗u 5 = ⃗v 5 − proj u1(⃗v 5 ) − proj u2(⃗v 5 ) − proj u3(⃗v 5 ) (3.7)where ⃗u i refers to the of orthogonal vectors between three pseudo-channels. In anotherwords, ⃗u 1 (CH1 positive) is orthogonal to ⃗u 2 (CH2 positive) and ⃗u 3 (CH2 negative),but ⃗u 2 and ⃗u 3 are not orthogonal. similarly, two cases ong>inong> CH3 are both orthogonal toCH1 and CH2 cases but are not necessarily orthogonal to each other.Equation 3.2∼3.7 provide the detailed procedure of calculatong>inong>g the orthogonalmodes for different cases, which are plotted ong>inong> Figure 3-2, demonstratong>inong>g three pseudochannelong>inong>dependent behaviors due to raong>inong> signal reflected on 9 observed microwavechannels.


37Figure 3-1: Scatter plots of three pseudo-channels over the ocean(1B11.20020911.27506.7.HDF): (a) CH1 and CH2, (b) CH3 and CH2, and (c)CH1 and CH3.The blue long>inong>e refers to the division of different cases for Gram-Schmidt analysis


38Case Range Description0 CH1 < 35, 10 > CH2 > −3, 3 > CH3 > −5 Background1 CH1 > 35, 10 > CH2 > −3, 3 > CH3 > −5 CH1 positive2 CH1 < 35, CH2 > 10, 3 > CH3 > −5 CH2 positive3 CH1 < 35, CH2 < −3, 3 > CH3 > −5 CH2 negative4 CH1 < 35, 10 > CH2 > −3, CH3 > 3 CH3 positive5 CH1 < 35, 10 > CH2 > −3, CH3 < −5 CH3 negativeTable 3.1: Case division for Gram-Schmidt analysis3.1.2 CH1In Figure 3-2 (a), the blue long>inong>e gives the positive CH1 feature when CH2 and CH3are close to zeros. Lower frequency microwave channels (10, 19, 37GHz) majorly contributeto positive CH1, while higher frequency channels (85GHz V and H) contributelittle (Figure 3-2). It is known that lower frequency channels ong>inong> TMI emphasize moreon cloud liquid water emission from the precipitatong>inong>g clouds, while the high frequencychannels have large contribution of ice scatterong>inong>g. Therefore, a positive CH1 ong>inong>dependentlydescribes the emission from cloud liquid water.Shown ong>inong> a storm (Figure 3-4 (a)) as an example, CH1 becomes positive at areaswhere 19GHz V shows warmer brightness temperature (19GHz V, ong>inong> Figure 3-3 (a)).A warmer brightness temperature ong>inong> 19GHz V reflects majorly emission from liquidprecipitatong>inong>g cloud.3.1.3 CH2In Figure 3-2 (b), the black triangle curve refers to negative CH2 case reflected on 9observed microwave channels by TMI. Figure 3-2 (b) shows ong>inong>significant contributions


39by lower frequency channels, but rather high and positive contributions by higherfrequency channels (85GHz V and H). 85GHz V and H feature ong>inong> a combong>inong>ation ofstrong emission from cloud liquid water and high sensitivity to ice scatterong>inong>g. Thenegative CH2 case majorly reflects events with strong emission from liquid cloud andsensitive reaction toward ice scatterong>inong>g, such as strong convective precipitatong>inong>g events.In Figure 3-2 (b), the green long>inong>e is the positive CH2 case, when CH1 and CH3are close to zeros.The positive CH2 case majorly reflects events that have lowerfrequency channels counted agaong>inong>st high frequency channels. In a precipitatong>inong>g event,It is known that lower frequency channels majorly observes the emissions from cloudliquid water, and high frequency channels reflects both cloud liquid water and icealoft. When high frequency channels are subtracted by lower frequency channels ong>inong> aprecipitatong>inong>g events, it is the ice aloft that is left. Therefore, a positive CH2 describesevents that only have ice aloft, which can be anvil of the storm, or cirrus clouds.Shown ong>inong> a storm (Figure 3-4 (b)) as an example, CH2 becomes negative at areaswhere 85GHz V (Figure 3-3 (b)) shows colder brightness temperature, which reflectsstrong ice scatterong>inong>g. The ice aloft overlaps with areas with strong emission between19GHz V and 85GHz V (Figure 3-3 (a) and (b)), which ong>inong>dicates the ice aloft ispart of the deep convective system.In Figure 3-4 (b), CH2 appears positive atareas surroundong>inong>g deep convective raong>inong> bands (Figure 3-3 (c) and 3-4 (d)). The areawhere CH2 appears positive corresponds with very little raong>inong> rate. The anvil of theconvective system is known to have thong>inong> ice aloft but doesn’t usually precipitate.Therefore positive CH2 ong>inong>dicates majorly the anvil of the convective system.3.1.4 CH3In Figure 3-2 (c), the red long>inong>e and magenta long>inong>es are negative and positive CH3 casesreceptively. Both red and magenta curves (Figure 3-2 (c)) are quite symmetric at axis


40y = 0 long>inong>e, with song>inong>usoid shapes, which resemble the higher order mode ong>inong> Fourierseries. In Figure 3-4 (c), strong positive and negative CH3 areas are coupled together,located at areas with strong horizontal raong>inong> rate gradient. At areas with very littlehorizontal precipitation gradient, there is little variation of CH3. It is ong>inong>ferred thatCH3 plays a role as ‘edge detector’ of precipitatong>inong>g events, although further researchis needed to analyze the dipole structures of CH3 to seek its association with scannong>inong>ggeometry of TMI.The 9 observed microwave channels are known to have different of field of views(FOVs). The different size of FOVs among microwave channels may cover differentfractures of non precipitatong>inong>g and precipitatong>inong>g areas over the sharp horizontal raong>inong>rate gradient, though all the FOVs are centered at same pixels. CH3 may capturesthe brightness temperature variations of nong>inong>e microwave channels due to brightnesstemperatures averaged ong>inong> different FOVs. The CH3 concernong>inong>g different FOVs wouldbe especially sensitive when the horizontal raong>inong> rate gradient is large.3.2 Probability Distribution FunctionsThe ‘lookup’ table has five dimensions, which are skong>inong> temperature, total precipitablewater (TPW), and three pseudo-channels. With ong>inong> each long>inong>e of the ‘lookup’ table, thesamples with different raong>inong> rates are bong>inong>ned to form a distribution. The retrieved raong>inong>rate withong>inong> each long>inong>e of the ‘lookup’ table is calculated as the mean of the distribution.It was addressed that the skong>inong>s temperature, TPW, and three pseudo-channelsneed to be bong>inong>ned ong>inong> an appropriate bong>inong> sizes, so that the samples ong>inong> each bong>inong>s aresufficiently densely populated but not too large Petty and Li (2013b). It was alsorealized that pseudo-channels range from 10 1 to 10 2 , yet most of the samples aredistributed around 10 1 , and very few get to 10 2or higher Petty and Li (2013b).


41Therefore, a scheme is used to convert the pseudo-channels for bong>inong> division.Y i ≡ ∆ arctan( P CH i∆ ) (3.8)where, P CH irefers to pseudo-channel values (i = 1, 2, 3), ∆ = 15 for pseudochannels.Similar conversion is applied to T skong>inong> with ∆ = 5K and TPW with∆ = 10mm. From here on ong>inong> statistic analysis, the pseudo-channels, skong>inong> temperaturesand TPWs take on the converted bong>inong>ned values, not origong>inong>al values, unlessstatement specifies.Probability distribution functions are generated, with raong>inong> rate bong>inong>ned exponentiallyong>inong> mm hr −1 :[0.0, 0.01, 0.02, 0.03, 0.05, 0.1, 0.2, 0.3, 0.5, 1.01.7, 3.05.2, 10.0, 17.0, 30.0]where, 0.0 has its own bong>inong>, and the rest goes from 0 < R ≤ 0.01 and so on. When noneof the three pseudo-channels are ong>inong>volved ong>inong> the retrieval, the probability distributionsof raong>inong> rate only vary with the two environmental variables (a.k.a. T skong>inong> and TPW),which are considered distributions of climatological data (Figure 4-3, 4-4, and 4-5 (a)).As each pseudo-channel is added to the ‘lookup’ table, the probability distributionsare altered, and retrieved raong>inong> rate becomes closer to the real value (Figure 4-3, 4-4,and 4-5 (b)(c)(d)). The three pseudo-channels are addong>inong>g ong>inong>formation ong>inong>to the raong>inong>rate retrieval process. It is very important to understand how much ong>inong>formation eachpseudo-channel provides to raong>inong> rate retrieval, which is discussed further ong>inong> the nextchapter.In short, Gram-Schmidt analysis revealed the ong>inong>dependent features each pseudochanneldescribes over the ocean. CH1 describes the emission of cloud and raong>inong> water.CH2 describes the ice aloft, with negative CH2 for ice on top of convective systemsand positive CH2 for ice aloft as anvils. CH3 is the ‘edge detector’, measurong>inong>g the


42horizontal raong>inong> rate gradient. However, it is still unknown about how important eachpseudo channel is to retrieve raong>inong> rate. The significance of each pseudo-channel forretrievong>inong>g raong>inong> rate ong>inong>cludes how important are the pseudo-channels between eachother and are three pseudo-channels enough to retrieve raong>inong> rate, and whether the‘forth’ (or higher) pseudo-channels are necessary for the purpose of improvong>inong>g theraong>inong> rate retrieval precision.


43Figure 3-2: Orthogonal modes of 9-microwave-channel distributions with: CH1 positiveong>inong> blue dotted long>inong>e, CH2 positive ong>inong> green triangles, CH2 negative ong>inong> blacktriangles, CH3 positive ong>inong> magenta squares, and CH3 negative ong>inong> red diamond.


44Figure 3-3: Microwave channels 19GHz V, 85GHz V, and retrieve raong>inong> rate of HurricaneLily ong>inong> October 2002, to the southwest of Florida. (a) 19GHz V brightnesstemperature (K), (b) 85GHz V brightness temperature (K), and (c) retrieved raong>inong>rate ong>inong> mm/hr.


45Figure 3-4: Pseudo-channels of Hurricane Lily ong>inong> October 2002, to the southwest ofFlorida. (a) CH1, (b) CH2, (c) CH3, and (d) RGB plot of CH1 (red), CH2 (green),and CH3 (blue).


Chapter 4Information TheoryIn previous chapters, the physical ong>inong>sight of three pseudo-channels are analyzed. Itwas observed that CH1 is related to column raong>inong> water, CH2 is related to ice scatterong>inong>galoft, and CH3 responds to edges and gradients of horizontal precipitation distribution.Besides, posterior probability densities are calculated based on one, two, or threepseudo-channels. The posterior PDFs based on one, two, or three pseudo-channelsallow us to quantitatively evaluate the ong>inong>formation added by each successive channel.Both Shannon entropy (SE) and relative entropy (RE) are employed to evaluatethe ong>inong>formation that is added by three successive pseudo-channels to the raong>inong> rateretrieval.4.1 Shannon Entropy and Relative EntropyIn this section, both SE and RE are calculated ong>inong> the toy models to reveal theirdistong>inong>ctions ong>inong> quantifyong>inong>g the ong>inong>formation content.4.1.1 Gaussian DistributionA Gaussian distribution f(x) is designed to represent the prior distribution.f(x) = 1σ √ − µ)2exp −(x (4.1)2π 2σ 246


47where σ = 2.0 is the standard deviation, µ = 0.0 is the mean. The bong>inong>s are unevenlydivided from −10 to 10, with exponentially divided spacong>inong>g. The posteriordistribution g(x) is designed to be Gaussian distribution with signal or dispersionchange. When signal is changed, the mean of Gaussian is shifted as: µ = −4.0, 0, 4.0.When the dispersion is changed, the standard deviation varies as σ = 1.0, 2.0, 4.0.The content of ong>inong>formation that changes distribution from f to g is evaluated foreach bong>inong>s ong>inong> terms of Shannon entropy change and relative entropy change. Accordong>inong>gto Equation 1.6 and 1.7, the relative entropies for both discrete and contong>inong>uousforms have identical results. Therefore, relative entropy RE(f → g), Discrete Shannonentropy difference ∆DSE(f → g), and Contong>inong>uous Shannon Entropy difference∆CSE(f → g) are calculated for each bong>inong>s, and the results for signal or dispersionvariations are shown ong>inong> Figure 4-1. In Figure 4-1, the blue shaded distribution ong>inong>the upper panel refers to the prior distribution f(x) and the pong>inong>k shaded distributionrefers to the posterior distribution g(x).In Figure 4-1(a), f(x) and g(x) are overlappong>inong>g with each other, which ong>inong>dicatesno ong>inong>formation is provided to change the distribution, and RE = ∆DSE = ∆CSE =0.0.Relative EntropyIn Figure 4-1 (b) and (c), signal is shifted to the right and leftrespectively to same distance. The REs are shown the same for signal shiftong>inong>gs toboth directions. It is noticed that RE has a large positive contribution at bong>inong>s wherethe prior probability is close to 0 (for example, less than 0.02), but the posteriorprobability is large (for example above 0.15) ong>inong> Figure 4-1 (b) or (c).RE has avery small and negative contribution at bong>inong>s where the prior probability is large (forexample, above 0.15), but the posterior probability is close to 0 (for example, lessthan 0.02) ong>inong> Figure 4-1 (b) or (c).


In Figure 4-1 (d) and (f), the RE values are small compared with RE ong>inong> Figure4-1 (e). It is noticed that the signals ong>inong> Figure 4-1 (d)–(f) are not shifted, only48the dispersions are changed.In Figure 4-1 (e), the dispersion is flattened, whenless possible area becomes more possible at the skew part, and more possible areabecomes less possible at the peak part. RE ong>inong> Figure 4-1 (e) is largely contributed bythe ong>inong>crease ong>inong> possibility at the skew parts. It is noticed that upon all the cases, REbecome more positive at places where it is not quite possible ong>inong> the prior distributionbut it becomes very possible ong>inong> the posterior distribution. This process that greatlyimproves the probability from a case not likely to happen at all to a case that quitelikely to happen can also be described ong>inong> the real life as surprise. Therefore, it isconfirmed that RE measures the signal shift through a piece of ong>inong>formation.Inanother word, RE quantifies the ‘surprise’ that a piece of ong>inong>formation is addong>inong>g tochange the prior distribution.Shannon EntropyIn Figure 4-1 (b) and (c), the signal shifts change very little onShannon Entropy, ∆CSE = 0.0 and ∆DSE = −0.28, 0.34 (−0.28 to the right and0.34 to the left respectively). The spacong>inong>g for bong>inong> division is uneven ong>inong> Figure 4-1.Accordong>inong>g to Equation 1.4 and 1.5, both DSE and CSE are identical when bong>inong>sare evenly divided for PDFs. The Gaussian distribution experiment with undividedbong>inong>s are designed to examong>inong>e the difference between ∆DSE and ∆CSE. In Bayesianstatistics, ∆DSE(f → g) and ∆CSE(f → g) are quantities to measure the ong>inong>formationcontent of the message that changes prior PDF f to posterior g. It wasaddressed that DSE is uniquely defong>inong>ed by the probability measure over the message,not dependent on bong>inong> sizes. And CSE measures the message with probability density(ong>inong>volvong>inong>g both probability and bong>inong> division) (Frigg and Werndl, 2011). By defong>inong>ition,DSE is uniquely defong>inong>ed for a discrete probability message (such as flippong>inong>g the


49coong>inong>s), not dependent on the bong>inong> sizes. CSE is uniquely defong>inong>ed for a contong>inong>uousprobability density message (PDFs) not dependent on the bong>inong> sizes.In short, to describe ong>inong>formation content of PDFs, the CSE and DSE would beidentical if the bong>inong>s are evenly divided. Otherwise CSE for quantifyong>inong>g the ong>inong>formationmeasure PDFs would ong>inong>clude the coordong>inong>ates ong>inong>fluence. This conclusion is alsoconfirmed by the numerical experiment ong>inong>volvong>inong>g Gaussian distribution.Accordong>inong>g to Equation 1.4, ∆DSE counts each bong>inong> equally ong>inong> significance overthe Gaussian PDFs (for example the Gaussian PDF f(x) ong>inong> blue shades). Yet ong>inong>the experiment the Gaussian PDFs have unevenly divided bong>inong>s. Therefore calculated∆DSE is a combong>inong>ation of ong>inong>formation content of the PDF message and the bong>inong>coordong>inong>ates that is chosen to describe the message. The signal shift to the left and tothe right ong>inong> Figure 4-1 (b) and (c) are quantified by ∆DSE differently when the bong>inong>sare not evenly divided. It is observed that Shannon entropy (∆CSE) doesn’t changewhen there is a signal shift (Figure 4-1 (b) and (c)).When the bong>inong>s are evenly divided ong>inong> 4-2 (a) and (b), ∆DSE counts each bong>inong>with equal significance. The calculated ∆DSE, under evenly divided bong>inong>s, is ableto exclusively quantify the Shannon entropy ong>inong>formation content solely due to signalshift, without the ong>inong>fluence of bong>inong>nong>inong>g. As shown ong>inong> 4-2, signal shifts on both positiveand negative side of the axis correspond with zero change ong>inong> ∆DSE, but a same REmeasure. As comparison, for same Gaussian function signal shift, ∆CSE correctlymeasures the ong>inong>formation content regardless of bong>inong> division (4-1 (b) and (c)). ∆CSEhas limitations, compared with ∆CSE. ∆CSE is used for contong>inong>uous distributions.When used ong>inong> discretized fashion, ∆CSE becomes ong>inong>valid when bong>inong> width is zero(Equation 1.5).The posterior data for three successive pseudo-channels is made ong>inong>to discrete commulativedistributions, with unevenly discretized bong>inong>nong>inong>g. Accordong>inong>g to the distribu-


50tion, more samples are fallong>inong>g ong>inong>to the lower or zero raong>inong> rate range than high raong>inong> raterange. Thus, the bong>inong>s for raong>inong> rate is set exponentially so that narrower bong>inong>s at lowerend (for example, 0.1 ∼ 0.2 mm hr −1 ) is equally important as wider bong>inong>s at higherend (for example, 10 ∼ 20 mm hr −1 ). Besides, the first bong>inong> is non-raong>inong>ong>inong>g bong>inong>, whichis counted as bong>inong> with zero bong>inong> width. Due to the discrete form and the non-raong>inong>ong>inong>gbong>inong>, ∆DSE is more suitable than ∆CSE to take measurements for Shannon entropyof three successive pseudo-channels.A couple of reversed processes for PDF change ong>inong> dispersion are depicted ong>inong> Figure4-1 (e) and (f). Figure 4-1 (e) shows the less dispersion ong>inong> blue turnong>inong>g to moredispersion ong>inong> pong>inong>k. Figure 4-1 (f) shows the opposite. It is reflected ong>inong> entropydistribution that equal and opposite Shannon entropies changes (∆CSE and ∆DSE)are shown for 4-1 (e) and (f). When the posterior PDF becomes more spread out(flattened ong>inong> PDF curve shape), ∆DSE and ∆CSE are positive. When the posteriorPDF becomes less spread out (sharpened ong>inong> PDF curve shape) ∆DSE and ∆CSE arenegative. In short, the Shannon entropies measure exclusively the dispersion changeof a distribution, and cannot reflect the signal change at all.4.2 Pseudo-channelsThe raong>inong> rate probability distribution are generated for retrievals with one, two, orthree pseudo-channels.The raong>inong>ong>inong>g samples are observed to be more clustered atthe lower raong>inong> rate range. Much fewer samples falls ong>inong>to higher raong>inong> rate range morescattered than lower raong>inong> rate samples. The raong>inong> rate samples are observed to bedistributed ong>inong> an exponential fashion.In another words, light raong>inong> appears to bemuch more likely than heavier raong>inong>. Therefore, bong>inong>s are designed to be ong>inong> exponential,where, for example, raong>inong> rate bong>inong> rangong>inong>g from 0.01 to 0.02mm hr −1 is considered as


51important as the bong>inong> from 10 to 20mm hr −1 ong>inong> the ong>inong>formation content analysis.The discrete Shannon entropy difference and relative entropy are calculated tomeasure the ong>inong>formation content of each pseudo channels to the raong>inong> rate retrieval.Three cases are picked out to demonstrate the probability distribution changes byaddong>inong>g three successive pseudo-channels ong>inong> Figure 4-3, 4-4, and 4-5.In Figure 4-3(a), the probability distribution is based on raong>inong> rate samples collectedaccordong>inong>g to the environmental variables (skong>inong> temperature and total precipitablewater 1 ). When CH1 is added (CH1 = 41 ong>inong> Figure 4-3(b)), the posterior probabilitydistribution has a signal shift, which is reflected by positive RE. RE = 2.26 ong>inong> Figure4-3(b) is bigger than RE = 1.03 ong>inong> (c) and RE = 0.01 ong>inong> (d). Clearly reflected by RE,the signal change between prior and posterior distributions by addong>inong>g CH1 ong>inong> Figure4-3 (a) and (b) is more significant than shifts by addong>inong>g CH2 or CH3 ong>inong> this presentedcase. It is worthwhile to notice that CH1 CH2 and CH3 are added ong>inong> order to test theirong>inong>formation contributions to raong>inong> retrieval. The ong>inong>formation contributions of thesethree successive pseudo-channels depend on the order that they are added successivelydurong>inong>g the retrievong>inong>g process and their own ong>inong>dependent ong>inong>formation contents. In thischapter of ong>inong>formation theory, our focus is maong>inong>ly on evaluatong>inong>g withong>inong> a set order (i.e.CH1, CH2, and then CH3), how much ong>inong>formation each pseudo-channel is addong>inong>g tothe retrieval ong>inong> addition to the priorly added channels and environmental variables.For example, CH3’s ong>inong>formation content ong>inong> our ong>inong>formation analysis refers to howmuch additional ong>inong>formation CH3 contributes more than environmental variables,CH1, and CH2. It is noticed that ong>inong> climate data Figure 4-3 (a), zero raong>inong> rate bong>inong>has the largest amount of samples. The dispersion change between Figure 4-3 (a)and (b) are not large compared with Figure 4-3 (b) and (c), which is reflected by1 All the values of T skong>inong> , σ water , CH1, CH2, CH3 ong>inong> the Information theory chapter are all scaled valuesfor statistical purposes, not the physical values. The scalong>inong>g process is mentioned ong>inong> Petty and Li (2013a).


52∆SE = −0.12 ong>inong> (b), much smaller than ∆SE = −0.92 ong>inong> (c). In short, ∆SE andRE reflect that addong>inong>g CH1 ong>inong> Figure 4-3 brong>inong>gs significant signal change of raong>inong> rateprobability distribution, but not strong dispersion change, compared with addong>inong>gCH2.Between 4-3 (b) and (c), the signal shift between prior and posterior distributionsis not large compared with (a) and (b), which is reflected by RE value. RE = 1.03for addong>inong>g CH2 ong>inong> Figure 4-3 (c) is only half of RE = 2.26 for addong>inong>g CH1 ong>inong> Figure4-3 (b). The dispersion is elimong>inong>ated by addong>inong>g CH2 from Figure 4-3 (b) to (c). Itis observed that Figure 4-3 (b) has a wider dispersion than (c), which is reflected bynegative ∆SE value ong>inong> (c), greater ong>inong> magnitude than ∆SE = −0.12 ong>inong> Figure 4-3(b). In short ong>inong> Figure 4-3 (c), the RE shows that addong>inong>g CH2 brong>inong>gs about a signalshift on raong>inong> rate probability distribution, not as much as addong>inong>g CH1 ong>inong> (b). Besides,ong>inong> Figure 4-3 (c), ∆SE shows that addong>inong>g CH2 decreases dispersion on posterior raong>inong>rate probability distribution, more than addong>inong>g CH1 does ong>inong> (b).Between Figure 4-3(c) and (d) probability dispersion, the signal shift between priorand posterior distributions is very little, so is the dispersion change. ∆SE = −0.04and RE = 0.01 ong>inong> Figure 4-3(d) are much closer to zero than those ong>inong> (b) and (c). Inshort, CH3 ong>inong> 4-3(d) adds little ong>inong>formation to the probability distribution change,compared with CH1 or CH2. As a summary for Figure 4-3, CH1 brong>inong>gs significantsignal change of raong>inong> rate probability distribution, compared with CH2 and CH3. Inanother word, CH1 ong>inong> Figure 4-3 provides majority of ong>inong>formation content for raong>inong>rate retrieval algorithm compared with CH2 and CH3.In Figure 4-4, another set of raong>inong> rate probability distributions are shown todemonstrate the ong>inong>formation contents provided by three successive pseudo-channels.In Figure 4-4(b), little is changed ong>inong> the probability distribution signal shiftong>inong>g byaddong>inong>g CH1, which is reflected by small value RE = 0.03 (small compared with


53RE = 2.1 ong>inong> Figure 4-4(c)). In Figure 4-4(c), the zero raong>inong> rate bong>inong> sample numberis reduced, and numbers of samples ong>inong> bong>inong>s with ong>inong>termediate/light raong>inong> rate0.05 ∼ 1.0mm/hr are ong>inong>creased by addong>inong>g CH2. In correspondence, RE = 2.1 reflectsthe distribution signal ong>inong> Figure 4-4c is shifted towards the light raong>inong> rate bong>inong>s relativeto (b), and ∆SE = 2.37 reflects the dispersion of the probability distribution ong>inong> (c)is ong>inong>creased relative to (b). From Figure 4-4(c) to (d), the distribution changes verylittle, which is reflected by small RE and ∆SE ong>inong> (d) compared to those ong>inong> (c). Inshort, CH2 ong>inong> Figure 4-4 contribute majorly to the raong>inong> rate distribution, by providong>inong>gthe largest ong>inong>formation content RE of all three successive pseudo-channels.In Figure 4-5, another set of raong>inong> rate probability distributions are shown todemonstrate the ong>inong>formation contents provided by three successive pseudo-channels.In Figure 4-5 (a), (b), and (c), small RE and ∆SE compared with those ong>inong> (d) reflectsmall ong>inong>formation contents CH1 and CH2 provide for raong>inong> rate retrieval ong>inong> thisspecific case. In Figure 4-5(d), RE = 1.81 reflects significantly large signal shift ofraong>inong> rate distribution when CH3 is added.In Figure 4-5(d), ∆SE = 2.3 reflectssignificantly large dispersion of raong>inong> rate distribution when CH3 is added. From theraong>inong> rate probability distribution ong>inong> Figure 4-5, probability of ong>inong>termediate/light raong>inong>rate (0.05 ∼ 1.0mm/hr) is ong>inong>creased and zero raong>inong> rate probability is decreased whenCH3 is added. In short, CH3 ong>inong> case demonstrated by Figure 4-5 has major ong>inong>fluenceon raong>inong> rate retrieval, by havong>inong>g the largest ong>inong>formation content of all threepseudo-channels.As a sum-up, three over-ocean cases ong>inong> correspondence of Figure 4-3, 4-4, and 4-5are chosen to confirm the physical ong>inong>dication of RE and ∆SE. Based on the analysisong>inong> this section, three successive pseudo-channels can all provide large ong>inong>formationcontent, alterong>inong>g the posterior raong>inong> rate probability distributions ong>inong> the Bayesianalgorithm.


54However, the ong>inong>formation content of three successive pseudo-channels over otherland types are still unknown. Also, it is unknown of how much keepong>inong>g first one,two or three pseudo-channels would alter the raong>inong> rate retrieval results. Moreover,it is important to understand what ranges of retrieved raong>inong> rate are ong>inong>fluenced mostby each pseudo-channels. Are higher order pseudo-channels (ex, CH2, CH3) alwaysnecessary to keep to retrieve raong>inong> rates of all samples? Is the higher order pseudochannel(ex, CH4, CH5, etc) necessary to ong>inong>crease the precision of the presentedBayesian algorithm? Further analysis is performed ong>inong> the next section to associatepseudo-channels with retrieved raong>inong> rate over each surface type, usong>inong>g RE and ∆SEto quantify the raong>inong> rate distribution change.


56Figure 4-1: Information entropy changes between two Gaussian distributions overunevenly divided bong>inong>s. Blue blocks are normal distribution before, and pong>inong>k blocksare normal distribution afterwards. All upper panels are probability density functions(PDFs), and lower panels are changes of ong>inong>formational entropy of different kong>inong>ds. Thegreen long>inong>es are contong>inong>uous Shannon entropy difference (∆CSE). The magenta long>inong>esare discrete Shannon entropy difference (∆DSE). The cyan long>inong>es are relative entropy(RE).


57Figure 4-2: Information entropy changes between two Gaussian distributions overevenly divided bong>inong>s. Blue blocks are normal distribution before, and pong>inong>k blocks arenormal distribution afterwards. All upper panels are probability density functions(PDFs), and lower panels are changes of ong>inong>formational entropy of different kong>inong>ds.The magenta long>inong>es are discrete Shannon entropy difference (∆DSE). The cyan long>inong>esare relative entropy (RE).


58Figure 4-3: Probability distribution changes and ong>inong>formation entropy changes withrestrictions of CH1, CH2, and CH3. The land type is 0w. After scalong>inong>g, T skong>inong> = 6,σ water = 7, CH1 = 41, CH2 = 38, CH3 = 23. Color blocks are probability withong>inong> eachbong>inong>, usong>inong>g the vertical axis to the left. The black triangle long>inong>es are relative entropy(RE). The black dotted long>inong>es are change of shannon entropy ∆SE. Both triangleand dotted long>inong>es follow the vertical coordong>inong>ates to the right. Shown ong>inong> each panelsare: (a) only climate data, (b) climate data with CH1 restriction, (c) climate datawith restrictions of CH1 and CH2, and (d) climate data with restrictions of CH1,CH2, and CH3.


59Figure 4-4: Probability distribution changes and ong>inong>formation entropy changes withrestrictions of CH1, CH2, and CH3. The land type is 0w. After scalong>inong>g, T skong>inong> = 3,σ water = 2, CH1 = 23, CH2 = 27, CH3 = 23. Color blocks are probability withong>inong> eachbong>inong>, usong>inong>g the vertical axis to the left. The black triangle long>inong>es are relative entropy(RE). The black dotted long>inong>es are change of shannon entropy ∆SE. Both triangleand dotted long>inong>es follow the vertical coordong>inong>ates to the right. Shown ong>inong> each panelsare: (a) only climate data, (b) climate data with CH1 restriction, (c) climate datawith restrictions of CH1 and CH2, and (d) climate data with restrictions of CH1,CH2, and CH3.


60Figure 4-5: Probability distribution changes and ong>inong>formation entropy changes withrestrictions of CH1, CH2, and CH3. The land type is 0w. After scalong>inong>g, T skong>inong> = 4,σ water = 1, CH1 = 23, CH2 = 23, CH3 = 26. Color blocks are probability withong>inong> eachbong>inong>, usong>inong>g the vertical axis to the left. The black triangle long>inong>es are relative entropy(RE). The black dotted long>inong>es are change of shannon entropy ∆SE. Both triangleand dotted long>inong>es follow the vertical coordong>inong>ates to the right. Shown ong>inong> each panelsare: (a) only climate data, (b) climate data with CH1 restriction, (c) climate datawith restrictions of CH1 and CH2, and (d) climate data with restrictions of CH1,CH2, and CH3.


4.3 Necessity61In this section, both relative entropy and Shannon entroy are calculated under chosenenvironment variables and land types. When the prior retrieved results are alteredto posterior ones by addong>inong>g each pseudo-channel, both RE and ∆SE are evaluatedto measure the ong>inong>formation content of each pseudo-channel. Retrieved raong>inong> rate distributionsare compared between posterior and prior to verify the quantification ofong>inong>formation for each pseudo-channel. The goal of this section is to demonstrate thenecessary condition of three successive pseudo-channels ong>inong> terms of raong>inong> rate retrieval.In another words this section is to answer to what extent does each pseudo-channelcontribute to the best to raong>inong> rate retrievals over different surface types.4.3.1 CH1CH1 is added to alter the retrieved raong>inong> rate from climate data that solely depends onthe environmental variables (T skong>inong> and σ water ) (Figure 4-6 4-7 and Table 4.1). Foreach surface type, the environmental variables that ong>inong>clude the highest RE by addong>inong>gCH1 is selected for the plots to show the CH1 ong>inong>formation content. For example,Figure 4-6 with environmental variables of T skong>inong> = 6 and σ water = 6 has the largestRelative entropy (RE = 5.79 ong>inong> Table 4.1) among other environmental variable valuesover the ocean.In Figure 4-6, higher CH1 values corresponds with higher posterior 2 retrieved raong>inong>rates, which is depicted ong>inong> (c) and (d), and higher CH1 values are responsible fora higher prior-posterior retrieved raong>inong> rate difference, which is depicted ong>inong> (a) and(b). Reflected ong>inong> the ong>inong>formation content, generally lower CH1 samples have smaller2 posterior and prior are relative terms. In this section, retrievals before and after addong>inong>g CH1 areconsidered prior and posterior respectively.


62relative entropy values, higher CH1 samples have larger relative entropy. When CH1ong>inong> Figure 4-6 is small (ex. 15 ∼ 25), both prior and posterior raong>inong> rate probabilitydistribution peak at zero or low raong>inong> rate bong>inong>s (c) and (d). As CH1 ong>inong>creases (ex,CH1 > 25), more samples ong>inong> the prior distributions are recognized by CH1 to havesignificantly larger raong>inong> rate. At the same time, more non-raong>inong>ong>inong>g samples ong>inong> the priordistributions are filtered by CH1. As a result, ong>inong> Figure 4-6, peak of the raong>inong> ratedistribution is shifted to higher raong>inong> rate bong>inong>s as CH1 ong>inong>creases.When CH1 is small (ex. CH1 < 25), negative ∆SE ong>inong> Figure 4-6(f) depicts thedispersion reduction ong>inong> the posterior retrieved raong>inong> rate distribution. Low CH1 valuesitself ong>inong> Figure 4-6 (c) and (d) correspond with posterior low raong>inong> rates. It can beong>inong>ferred that prior raong>inong> rate distribution before addong>inong>g CH1 should peak at low ornon-raong>inong> rate bong>inong>s, so that the mean of prior raong>inong> rate distribution, as retrieved raong>inong>rate, can be close to zero ong>inong> value. This ong>inong>ference of prior raong>inong> rate distribution is tocertaong>inong> extent reflected by Figure 4-3(a), 4-4(a), 4-5(a), where majority of samplesare located at zero raong>inong> rate bong>inong>, although the environmental variables are not thesame. In the posterior raong>inong> rate distribution, reduction of dispersion through CH1further reflect the samples low or non raong>inong> rate features. In this way, a small CH1value is considered as confirmation of little raong>inong> rate.As CH1 > 25, ∆SE is positive (Figure 4-6(f)), which ong>inong>dicates an ong>inong>crease ong>inong>dispersion of posterior raong>inong> rate distributions compared with prior ones.At largeCH1 values ong>inong> Figure 4-6 (ex. 25 ∼ 40 ong>inong> (f)) the possibility of non-raong>inong> or little raong>inong>rate is reduced, and the samples with moderate or large raong>inong> rates are left and appeardiverse ong>inong> raong>inong> rate probabilities despite the fact they share similar CH1 values.It is also ong>inong>terestong>inong>g to notice that as CH1 > 27 ong>inong> Figure 4-6(f), ∆SE turns lesspositive as CH1 ong>inong>crease ong>inong> value. As CH1 ong>inong>creases beyond CH1 = 27, more samplesof high raong>inong> rates are left from the prior distribution due to their high CH1 value, and


63more samples of moderate raong>inong> rates are filtered due to their low CH1 value, and theraong>inong> rate distribution is shifted towards the higher raong>inong> rate , as reflected ong>inong> Figure4-6(c) and (d). In another words, high raong>inong> rates samples are less diverse when theyshare similar CH1 value (ex, CH1 > 40), and samples with moderate and low raong>inong>rates are more diverse when they share similar CH1 value (ex,25 < CH1 < 30).As a sum-up, to retrieve raong>inong> rate, CH1 serves to correct the climate-data basedBayesian algorithm more at higher raong>inong> rate bong>inong>s. For lower raong>inong> rate samples, addong>inong>gCH1 provides confirmation to the retrieval based on climate-data, simply becauselower raong>inong> rate samples are highly likely to have low CH1 values. Besides, CH1 isbetter to exclusively determong>inong>e the high raong>inong> rate samples than it does to do moderateor low raong>inong> rate samples. Lower or moderate raong>inong> rate samples sharong>inong>g same CH1have more diverse raong>inong> rate distribution than high raong>inong> rate ones do sharong>inong>g sameCH1.Figure 4-7 is plotted to depict the ong>inong>formation content associated with priorposteriorretrieved raong>inong> rate difference over the warm vegetable land. Similar trendof CH1 over warm vegetable land is found as it is ong>inong> ocean surface.For some other surfaces types such as 1w, 1c, 2w, and 4w, ong>inong> Table4.1, similartrends are found as 0w, that CH1 provides more ong>inong>formation on higher raong>inong> rate bong>inong>s.For some surface types, the numbers of chosen samples are so limited, not enoughto present a trend, such as 2c, 3w, 3c, 4c, 5w, 5c, 6w, and 6c. Table 4.1 collectsall the surface types with the largest relative entropy values ong>inong> each surface, to moreefficiently illustrate the necessary value of CH1 quantified ong>inong> ong>inong>formation content. InTable 4.1 for retrievals ong>inong> surface types 2c, 3c, 5w, 5c, 6w, and 6c, CH1 plays a roleas confirmation to the low raong>inong> rates, reflected by little RE values. In Table 4.1 forretrievals ong>inong> surface types 1w, 1c, 2w, 3w, and 4w, CH1 appears to alter the posteriorraong>inong> rate distributions ong>inong> terms of shiftong>inong>g the signal, and ong>inong>creasong>inong>g the dispersion


64Figure 4-6: Raong>inong> rate ong>inong>formation change by addong>inong>g CH1 restriction.(a) averageraong>inong> rate (R) difference, (b) avaerage raong>inong> rate ratio, (c) probability of R = 0, (d)probability of R > 1mm hr −1 , (e) relative entropy after addong>inong>g CH1, and (f) Shannonentropy difference after addong>inong>g CH1. The land type is 0w. Environmental variables:T skong>inong> = 6, σ water = 6 after scalong>inong>g.


65from prior distributions. In Table 4.1 for retrievals ong>inong> surface types 1w, 1c, 2w, 3w,and 4w, the retrieved raong>inong> rates after addong>inong>g CH1 appear to be small or moderateong>inong> quantities. Reflected by ∆SE, addong>inong>g CH1 ong>inong> surface type 1w, 1c, 2w, 3w, and4w ong>inong> Table 4.1 ong>inong>creases the diversity of samples with low or moderate raong>inong> rate. Itis worth to notice that Table 4.1 displays the case of addong>inong>g CH1 with the highestrelative entropy ong>inong> each surface. Lower RE values of CH1 ong>inong> different surface types,similar to the previous analysis on Figure 4-6, ong>inong>dicate CH1 as confirmations of lowerraong>inong> rate feature.


66Figure 4-7: Same as above. The land type is 1w. Environmental variables: T skong>inong> = 5,σ water = 6 after scalong>inong>g.


67Surface Tskong>inong> σwater CH1 R Diff. R Ratio P(R = 0 mm hr −1 ) P(R > 1 mm hr −1 ) RE ∆SE0w 6 6 44 12.27 39.14 0.00 1.00 5.79 0.071w 5 6 36 7.04 20.60 0.00 0.92 3.65 1.161c 11 2 25 0.20 7.97 0.48 0.06 1.09 2.062w 6 6 32 2.82 9.92 0.01 0.69 2.61 1.612c 11 1 21 -0.01 0.04 0.99 0.00 0.02 -0.163w 6 3 26 0.49 27.60 0.29 0.13 2.78 3.043c 11 1 25 0.01 4.89 0.93 0.00 0.08 0.434w 5 6 27 0.55 2.41 0.14 0.25 1.22 1.914c - - - - - - - - -5w 1 1 24 0.02 2.22 0.89 0.00 0.05 0.435c 11 1 25 0.02 5.05 0.88 0.00 0.14 0.636w 3 2 22 -0.04 0.28 0.94 0.00 0.04 -0.436c 11 1 21 -0.01 0.56 0.97 0.00 0.01 -0.12Table 4.1: Maximum retrieved raong>inong> rate changes ong>inong> posterior probability distributions by addong>inong>g CH1 under various surfacetypes. The columns are denoted as (from left to right): surface categories, surface skong>inong> temperature, total precipitable water,CH1, raong>inong> rate difference (posterior - prior), raong>inong> rate ratio (posterior/prior), probability of posterior raong>inong> rate equal to zeromm/hr, probability of posterior raong>inong> rate greater than 1 mm/hr, relative entropy, and Shannon entropy difference. Thethreshold of sample numbers ong>inong> this analysis is 1000 samples. All cases ong>inong> 4c have less than the threshold, therefore, the 4ccases cannot be considered statistically representative.


684.3.2 CH2CH2 is added to alter the retrieved raong>inong> rate from ‘climate+CH1 data’ that depends onboth environmental variables and CH1 (Figure 4-8, 4-9 and Table 4.2). Alike previoussection about CH1, for each surface type, the environmental variables that ong>inong>cludethe highest RE by addong>inong>g CH2 is selected to show the CH2 ong>inong>formation content. Forexample, Figure 4-8 with environmental variables of T skong>inong> = 6 and σ water = 6 hasthe largest Relative entropy (RE = 1.22 ong>inong> Table 4.2) among other environmentalvariable values over the ocean. It is worth noticong>inong>g that the scatter plots for CH2 aretwo dimensional, because both CH1 and CH2 are variables for ong>inong>formation contenteven though environmental variables are settled.In Figure 4-8, higher CH1 values correspond with higher retrieved raong>inong> rate distributionsaccordong>inong>g to (c) and (d). The raong>inong> rate varies with CH2 quite weakly ong>inong>Figure 4-8(c) and (d) compared with it does with CH1.When CH1 is large (ex.30 < CH1 < 40 ong>inong> Figure 4-8(d)), CH2 decreases as more samples have raong>inong> rateslarger than 1mm hr −1 . When CH1 is small (ex. CH1 < 25 ong>inong> Figure 4-8(c)), CH2ong>inong>creases as less samples have zero raong>inong> rates, which is opposite ong>inong> trend from largeCH1 mentioned previously.In Figure 4-8(a) and (b), when CH1 is large (ex. 30 < CH1 < 40), CH2 alters theretrieved raong>inong> rate from prior distribution 3 . For example, at CH1 = 40 and CH2 = 35ong>inong> Figure 4-8(a) and (b), the posterior distribution estimates a lower raong>inong> rate thanprior distribution does by addong>inong>g CH2. At CH1 = 40 and CH2 = 17 ong>inong> Figure 4-8(a)and (b), the posterior distribution estimates a higher raong>inong> rate than prior distributiondoes by addong>inong>g CH2. Reflected ong>inong> ong>inong>formation content (Figure 4-8(e)), positive RE at3 To calculate retrieved raong>inong> rate, the mean is taken from the raong>inong> rate distribution. In this section, priorand posterior are relative terms ong>inong>dicatong>inong>g the distributions before and after addong>inong>g CH2 respectively.


69large CH1 (ex. 30 < CH1 < 40) shows signal shifts ong>inong> raong>inong> rate distribution, where theupper positive anomaly (CH2 = 35) ong>inong>dicates a shift to a lower raong>inong> rate, and bottompositive anomaly (CH2 = 17) ong>inong>dicates a shift to a higher raong>inong> rate accordong>inong>g to (c)and (d). In Figure 4-8(f), negative ∆SE at large CH1 (ex. 30 < CH1 < 40) showsless dispersion ong>inong> distribution after addong>inong>g CH2 compared with prior distribution.When CH1 is small (ex, CH1 < 25), CH2 changes the raong>inong> rate distribution.In Figure 4-8(e), when CH1 = 20, CH2 = 27, a significant positive RE ong>inong>dicates asignal shift of raong>inong> rate from the prior distribution. The change of distribution ofCH1 = 20, CH2 = 27 can be reflected by ong>inong>creased raong>inong> rate ratio at same spot ong>inong>Figure 4-8(b). ∆SE ong>inong> Figure 4-8(f) shows positive anomaly at CH1 = 20, CH2 = 27,which ong>inong>dicates a larger dispersion ong>inong> the posterior distribution than the prior one.Negative ∆SE at CH1 = 25, CH2 = 20 ong>inong> Figure 4-8(f) ong>inong>dicates the posteriordistribution, the raong>inong> rate dispersion is less spread out, where higher probability ofnon-raong>inong> samples are left ong>inong> the posterior distribution after addong>inong>g CH2. In Figure 4-8(f), the abrupt ∆SE anomaly at 20 < CH1 < 25, which is reflected correspondong>inong>glyong>inong> (b) and (e), suggests an ong>inong>dependent raong>inong>ong>inong>g mechanism that ong>inong>volves envil cloudsand produce little raong>inong> fall.In short, Figure 4-8 shows the raong>inong> rate distribution change by addong>inong>g CH2 overthe ocean. CH2 over the ocean is observed to change the signal of raong>inong> rate PDF athigher retrieved raong>inong> rate samples. For lower raong>inong> rate samples, CH2 seems to be ableto reflect different low raong>inong> rate precipitation types by ice.Figure 4-9 shows the precipitation change by addong>inong>g CH2 over warm vegitatedland (1w) for a set of chosen environmental variables. The environmental variablesT skong>inong> = 5 and σ water = 6 were chosen for the plot because the case under suchenvironment variables contaong>inong>s the largest RE value for CH2 to have the biggestsignal change on posterior raong>inong> rate probability distribution. In Figure 4-9(c), for low


70raong>inong> rate samples (CH1 < 25), little variation of raong>inong> rate is shown along CH2 axis.In another words, CH2 reflect little on low raong>inong> rate variation. In Figure 4-9(d), forhigh raong>inong> rate samples (CH1 > 25), CH2 shows to reflect raong>inong> rate probability densityvariation for a chosen CH1. For example, when CH1 = 30 and CH2 = 20, moresamples have raong>inong> rate over 1mm hr −1 than they do when CH1 = 30 and CH2 = 27ong>inong> Figure 4-9(d). Figure 4-9(d) shows that CH2 is responsible for more variation ofmoderate and high raong>inong> rates over warm vegitated land than it is for ocean.Addong>inong>g CH2 is shown to have larger RE at areas ong>inong> Figure 4-9(e) when CH1 > 25where lower than prior raong>inong> rate is evaluate due to higher CH2 value and higher thanprior raong>inong> rate is evaluated due to smaller CH2 for the same CH1 values. In Figure4-9(f), raong>inong> rate distribution has significant less dispersion to retrieve high raong>inong> rate.Besides, data of other land types with largest RE ong>inong> each case are listed ong>inong> Table4.2, which shows that CH2 gives the signal shift more ong>inong> warm surfaces (0w,1w, 2w, 3w, 4w), reflected by higher RE than cold surfaces (1c, 2c, 3c) and Tibetan/Himalayanrange (5w, 5c, 6w, 6c). On average, the maximum RE by addong>inong>gCH2 appears to be half or less than half as much as the maximum RE by addong>inong>gCH1 (Table 4.1).


71Figure 4-8: Raong>inong> rate ong>inong>formation change by addong>inong>g CH2 restriction.(a) averageraong>inong> rate (R) difference, (b) avaerage raong>inong> rate ratio, (c) probability of R = 0, (d)probability of R > 1mm hr −1 , (e) relative entropy after addong>inong>g CH2, and (f) Shannon


72Figure 4-9: Same as above. The land type is 1w. Environmental variables: T skong>inong> = 5,σ water = 6.


73Surface Tskong>inong> σwater CH1 CH2 R Diff. R Ratio P(R = 0 mm hr −1 ) P(R > 0.1 mm hr −1 ) RE ∆SE0w 6 6 42 37 -3.36 0.38 0.00 1.00 1.22 -0.701w 5 6 26 20 2.30 3.48 0.01 0.99 1.32 -0.861c 11 2 23 21 0.05 4.65 0.81 0.12 0.17 0.792w 6 6 23 25 0.19 7.07 0.55 0.35 0.78 1.712c 11 1 24 23 -0.01 0.36 0.98 0.01 0.02 -0.193w 3 2 25 25 0.29 3.66 0.33 0.51 0.81 1.743c 11 1 25 22 0.01 1.67 0.90 0.06 0.01 0.194w 5 5 22 25 0.06 8.85 0.77 0.15 0.42 1.184c - - - - - - - - - -5w 2 1 24 24 0.03 2.21 0.81 0.12 0.06 0.515c 11 1 24 21 0.00 1.53 0.95 0.03 0.01 0.066w 5 2 23 23 -0.01 0.77 0.90 0.06 0.01 -0.056c 10 1 23 21 0.00 1.06 0.96 0.03 0.01 0.07Table 4.2: Maximum retrieved raong>inong> rate changes ong>inong> posterior probability distributions by addong>inong>g CH2 under various surfacetypes. The columns are denoted as (from left to right): surface categories, surface skong>inong> temperature, total precipitable water,CH1, CH2, raong>inong> rate difference (posterior - prior), raong>inong> rate ratio (posterior/prior), probability of posterior raong>inong> rate equalto zero mm/hr, probability of posterior raong>inong> rate greater than 0.1 mm/hr, relative entropy, and Shannon entropy difference.The threshold of sample numbers ong>inong> this analysis is 1000 samples. All cases ong>inong> 4c have less than the threshold, therefore,the 4c cases cannot be considered statistically representative.


744.3.3 CH3Figure 4-10 (over the ocean), Figure 4-11 (over the warm vegitated land), and Table4.3 (maximum RE over all land types) are shown to evaluate the ong>inong>formation CH3is added towards the raong>inong> rate retrievals. Figure 4-10 (over the ocean) and Figure4-11 (over the warm vegetated land) are chosen because they both contaong>inong> the largestRE values ong>inong> each case, ong>inong>dicatong>inong>g the biggest signal changes from prior raong>inong> ratedistribution to posterior one due to CH3. The prior raong>inong> rate probability distributionong>inong> this section refers to the raong>inong> rate distribution retrieved by environment variables(T skong>inong> and σ water ), CH1, and CH2. The posterior raong>inong> rate probability distributionong>inong> this section refers to the raong>inong> rate distribution retrieved by environment variables(T skong>inong> and σ water ), CH1, CH2, and CH3.Over the ocean ong>inong> Figure 4-10, large RE values ong>inong> (e) (for example, CH3 = 26and CH2 = 22) correspond with large raong>inong> ratio change ong>inong> (b), where the raong>inong> ratesare low or moderately low, accordong>inong>g to (c) and (d). The ∆SE reflects an ong>inong>crease ong>inong>dispersion ong>inong> Figure 4-10(f) on the posterior distribution at places where RE is largeong>inong> (e).Over the land ong>inong> Figure 4-11, large RE values ong>inong> (e) (for example, CH3 = 26 andCH2 = 23) correspond with low raong>inong> rate signal shifted further towards lower endcompared with prior distribution, which is reflected ong>inong> small raong>inong> rate difference ong>inong>(a) and small raong>inong> rate ratio (b) and high possibility of non-raong>inong> samples (c). Thenegative ∆SE ong>inong> Figure 4-11(f) shows a shrong>inong>k ong>inong> dispersion on raong>inong> rate distribution,for example at CH3 = 26 and CH2 = 23. Both RE and ∆SE around CH3 = 26 andCH2 = 23 ong>inong> Figure 4-11(e) and (f) ong>inong>dicates a confirmation of low raong>inong> rate samples.In Table 4.3, the maximum RE values over all surface types are smaller than 1.0,not significant on alterong>inong>g the raong>inong> rate distribution signal shift.


75Figure 4-10: Raong>inong> rate ong>inong>formation change by addong>inong>g CH3 restriction. (a) averageraong>inong> rate (R) difference, (b) avaerage raong>inong> rate ratio, (c) probability of R = 0, (d)probability of R > 1mm hr −1 , (e) relative entropy after addong>inong>g CH3, and (f) Shannonentropy difference after addong>inong>g CH3. The land type is 0w. CH1 = 24, Environmentalvariables: T skong>inong> = 4, σ water = 3 after scalong>inong>g.


76In short, CH3 doesn’t appear to ong>inong>fluence as significantly to raong>inong> rate retrievals asCH1 or CH2. The contribution of CH3 to raong>inong> rate retrieval is shown ong>inong> this sectionto provide confirmation to the samples with low/zero raong>inong> rate.


77Figure 4-11: Same as above. The land type is 1w. CH1 = 24, Environmentalvariables: T skong>inong> = 6, σ water = 6 after scalong>inong>g.


78Surface Tskong>inong> σwater CH1 CH2 CH3 R Diff. R Ratio P(R = 0 mm hr −1 ) P(R > 0.01 mm hr −1 ) RE ∆SE0w 4 3 24 22 27 0.08 9.31 0.61 0.38 0.85 1.801w 6 6 24 22 27 -0.41 0.02 0.99 0.01 0.90 -2.591c 11 1 25 22 26 -0.05 0.09 0.98 0.02 0.15 -0.862w 6 6 25 22 24 0.81 4.25 0.30 0.69 0.53 1.102c 11 1 24 22 22 -0.00 0.54 0.98 0.02 0.01 -0.133w 4 2 23 24 25 0.07 11.04 0.78 0.22 0.41 1.193c 11 1 24 22 20 -0.00 0.70 0.99 0.01 0.02 -0.154w 5 5 23 22 25 0.07 3.39 0.84 0.15 0.09 0.604c - - - - - - - - - - -5w 1 1 23 23 21 -0.01 0.17 0.98 0.02 0.04 -0.345c 11 1 23 23 25 0.00 2.55 0.98 0.02 0.01 0.076w 2 2 22 23 21 0.01 1.58 0.94 0.06 0.01 0.036c 11 1 22 21 22 -0.00 0.69 0.97 0.03 0.01 -0.09Table 4.3: Maximum retrieved raong>inong> rate changes ong>inong> posterior probability distributions by addong>inong>g CH3 under various surfacetypes. The columns are denoted as (from left to right): surface categories, surface skong>inong> temperature, total precipitable water,CH1, CH2, CH3, raong>inong> rate difference (posterior - prior), raong>inong> rate ratio (posterior/prior), probability of posterior raong>inong> rateequal to zero mm/hr, probability of posterior raong>inong> rate greater than 0.01 mm/hr, relative entropy, and Shannon entropydifference. The threshold of sample numbers ong>inong> this analysis is 1000 samples. All cases ong>inong> 4c have less than the threshold,therefore, the 4c cases cannot be considered statistically representative.


4.4 Sufficiency79In this section, the sufficiency of three successive pseudo-channels is evaluated byusong>inong>g bivariant histograms to compare the prior and posterior raong>inong> rate distributionswhen each pseudo-channel is successively added to the algorithm.Representativebivariate histograms are displayed ong>inong> Figure 4-12 (over the ocean), Figure 4-13 (overthe vegetated land), Figure 4-14 (over the land/water mix, coast), and Figure 4-15(over all surface types). The goal of this section is to show whether three successivepseudo-channels are sufficient to retrieve the raong>inong> rate, is the fourth pseudo-channelnecessary ong>inong> terms of improvong>inong>g the precision of retrieved raong>inong> rates.Over the ocean ong>inong> Figure 4-12(a), from climatology to climatology and CH1, thesamples are completely redistributed ong>inong> the posterior bong>inong>s. In Figure 4-12(b), fromclimatology + CH1 to climatology + Channel 1 and 2, addong>inong>g CH2 doesn’t changethe raong>inong> rate distribution much when retrieved raong>inong> rate is above 0.2mm hr −1 , butCH2 changes the raong>inong> rate distribution significantly at among low raong>inong> rate samples.In Figure 4-12(c), the majority samples O(10 8 ) are not significantly redistributed byaddong>inong>g CH3, which appears to be y = x long>inong>ear relationship ong>inong> the diagram. Thereare some samples ong>inong> the lower raong>inong> rate bong>inong>s that are redistributed by CH3 ong>inong> Figure4-12(c), with around O(10 3 ) or O(10 4 ) samples.Over the warm vegetated land ong>inong> Figure 4-13(a), from climatology to climatologyand CH1, the samples are largely redistributed ong>inong> the posterior bong>inong>s. In Figure 4-13(b), CH2 alters the distribution to an even spread almost along all the bong>inong>s. InFigure 4-13(c), addong>inong>g CH3 doesn’t appear to change much on the higher end of thebong>inong>s (ex. raong>inong> rate above 1.0mm hr −1 ). At lower raong>inong> rate bong>inong>s, majority (O(10 8 )) ofsamples are not altered by addong>inong>g CH3 ong>inong> Figure 4-13(c), and some mong>inong>or samplesO(10 3 ) are redistributed from moderate raong>inong> rate bong>inong>s like 0.3mm hr −1 ong>inong> prior to low


80raong>inong> rate bong>inong>s like 0.01mm hr −1 ong>inong> posterior.Over the warm land/water mix (coast) ong>inong> Figure 4-14(a), like previous two cases,CH1 alters the distribution greatly. In Figure 4-14(b) and (c) CH2 and CH3 don’tappear to alter the distribution much, as majority (O(10 5 )) are on the y = x long>inong>e.The bivariate for all surfaces classes are plotted ong>inong> Figure 4-15. It is shown that ong>inong>Figure 4-15(a), CH1 alters the distribution greatly from climatology to climatologyand CH1. In Figure 4-15(b) and (c), CH2 and CH3 appear to alter little the majority(O(10 8 )) of the samples. Samples of O(10 3 ) are redistributed by CH3 among lowerand moderate raong>inong> rate bong>inong>s.As a sum-up, ong>inong> most of the cases examong>inong>ed so far over ocean and different landtypes, it appears that the majority of ong>inong>formation affectong>inong>g the posterior probabilitydistribution of raong>inong> rate is found ong>inong> the first two pseudo-channels. Droppong>inong>g the thirdchannel from the retrieval usually has only a mong>inong>or effect on the retrieved probabilitydistributions. But there are exceptions, especially for less common combong>inong>ations ofthe first two channels.


81Figure 4-12: Bivariate histograms comparong>inong>g distributions of raong>inong> rates prior to (horizontalaxis) and followong>inong>g (vertical axis) the successive addition of channels, startong>inong>gwith Channel 1 relative to climatology (left), Channel 1+2 relative to Channel 1 only(center), Channels 1-3 relative to 1+2 (right). The surface type here is ocean (0w).Figure 4-13: Same as above. The surface type here is warm vegetated land (1w).


82Figure 4-14: Same as above. The surface type here is warm land/water mix, coast(2w).Figure 4-15: Same as above. It ong>inong>cludes all surface types (0w, 1w, 1c, 2w, 2c, 3w, 3c,4w, 4c, 5w, 5c, 6w, 6c), where surface 4c has zero selected samples for the statisticsthat are statistically representative.


Chapter 5ConclusionsThe UW algorithm was developed to retrieve raong>inong> rate from TMI, usong>inong>g reduceddimensionalBayesian algorithm. Three pseudo-channels are generated mathematicallyusong>inong>g prong>inong>ciple component analysis, presumably takong>inong>g three most importantroles to retrieve raong>inong> rate.In this project, the ong>inong>formation that three pseudo-channels provide are analyzedboth qualitatively and quantitatively.Firstly, the physical meanong>inong>g of the threepseudo-channels is demonstrated by usong>inong>g the Gram-Schmidt Process. The ong>inong>itialresults suggest that over the ocean:• CH1 is related to column raong>inong> water;• CH2 is related to ice scatterong>inong>g aloft: negative CH2 is related to the ice ong>inong> aconvective storm, and positive CH2 is related to the ice ong>inong> anvil clouds;• CH3 responds to edges and gradients.As part of the UW algorithm, raong>inong> rate probability distributions are calculatedfor chosen environmental variables and successive pseudo-channels. With this advantage,each of the three successive pseudo-channels are evaluated for the ong>inong>formationcontribution to the UW algorithm by keepong>inong>g one pseudo-channels, two, or three andcomparong>inong>g the probability distributions between each other.83


84From Climatology to Climatology + CH1, addong>inong>g CH1 provides the most domong>inong>antong>inong>formation to the raong>inong> rate distributions among all three pseudo-channels. CH1itself ong>inong>creases as the posterior raong>inong> rate ong>inong>creases. CH1 provides more signal shiftto the posterior raong>inong> rate distribution at larger CH1 values. The signal shift of theposterior raong>inong> rate distribution can be reflected by a ong>inong>crease or decrease on retrievedraong>inong> rate from posterior distribution ong>inong> Bayesian algorithm, which is calculated asthe average of the probability distribution. CH1 provides an ong>inong>crease ong>inong> dispersion ofposterior probability distribution at small CH1 values and a decrease ong>inong> dispersion ofposterior probability distribution at large CH1 values.From Climatology + CH1 to Climatology + CH1 +CH2, addong>inong>g CH2 providessome ong>inong>formation to the raong>inong> rate distributions. For low CH1 values (low retrieved raong>inong>rate from posterior distribution), CH2 would correspond little variation with retrievedraong>inong> rate. For large CH1 values (high retrieved raong>inong> rate from posterior distribution),larger CH2 would correspond with a smaller retrieved raong>inong> rate. Addong>inong>g CH2 wouldcause signal shift of raong>inong> rate distributions over all ranges of raong>inong> rate. CH2 can bothong>inong>crease the retrieved raong>inong> rate from prior distribution when CH2 is towards the highend and decrease the retrieved raong>inong> rate from prior distribution when CH2 is towardsthe low end.From Climatology + CH1 + CH2 to Climatology + three successive pseudochannels,most of the time, CH3 provides little ong>inong>formation to the raong>inong> rate distributions.However, for certaong>inong> cases with rare combong>inong>ations of CH1 and CH2, CH3may appear to provide significant ong>inong>formation content.As for evaluation on various surface types, all three successive pseudo-channels providelargest ong>inong>formation content 1 over ocean, warm vegetated land, warm land/water1 Over each surface type, the maximum relative entropy is ranked to evaluate ong>inong>formation content eachpseudo-channel provides. For CH1, the threshold for RE is set to be 1.0. For CH2, the threshold for RE is


85mix (coast) and warm desert. Only CH1 and CH2 but CH3 provide significant ong>inong>formationcontent over warm raong>inong> forest. Only CH1 but CH2 or CH3 provides significantong>inong>formation content over cold vegetated land. None of the three pseudo-channels appearto provide significant ong>inong>formation content over cold land/water mix (coast), colddesert, Tibetan Plateau and similar, and Himalayan range and similar.set to be 0.4. For CH3, the threshold for RE is set to be 0.2.


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