SYSTEMS ENGINEERING ANALYSIS OF A TRMM-LIKE RETRIEVAL ALGORITHM: IMPLICATION FOR GPM DESIGN

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1 P5R. SYSTEMS ENGINEERING ANALYSIS OF A TRMM-LIKE RETRIEVAL ALGORITHM: IMPLICATION FOR GPM DESIGN C. R. Ros* and V. Chandasa Coloado Stat Univsity Fot Collins, CO INTRODUCTION Satllit systs such as th Topical Rainfall Masuing Mission (TRMM) a coplx and ly on th pop dsign and functioning of ach of its any subsysts satllit vhicl, gound validation (GV) sits, pcipitation ada (PR), calibation, and tival algoiths to nsu high quality data poducts. Th tival algoiths ployd in TRMM a basd on athatical odls and assuptions about th icophysics of pcipitation. Eos in th odl assuptions, vaiabls input to th odl, and odl paats can lad to os in th output, which in this cas, is th tivd ain at. In gnal, systs ngining consists of th thods and/o pocsss usd to optiiz th pfoanc of a syst givn liitd ti, tchnology and/o soucs. By analyzing th syst, th dsigns a abl to dtin which pats a ipotant and which pats contibut littl to th ovall succss o outco. Liwis, in th TRMM tival algoith, so input vaiabls, paats and odl assuptions a o ipotant than oths. Global Snsitivity analysis (SA) thods and tchniqus can b usd to assist in systs ngining analysis by poviding quantitativ insight as to th ipotanc of ach input facto. Saltlli (2, 24) dfind SA as th study of how th output of a odl (nuical o athatical) vais as a function of its input paats and how to lat th output unctainty (vaianc) bac to th unctainty in ach of th input paats. Th a at last two ain tchniqus fo iplnting global snsitivity analysis: Mont Calo (MC); and vaianc dcoposition. Each of ths tchniqus is sapling basd, aning that sapls a dawn fo th pobability dnsity functions (PDFs) of ach of th factos and th odl is xcutd onc fo ach st of sapld valus. A bloc diaga, adaptd fo Saltlli (999), of th y coponnts and flow of global SA using th vaianc dcoposition tchniqu is shown in Figu. Th odl bing studid, psntd by Y f(x i ), wh X i {X, X 2, X }, uss sapld valus fo th distibution spac of ach facto. * Cosponding autho addss: C. R. Ros, Coloado Stat Univsity -ail: chis..os@colostat.du Input Factos X X 2 X Th odl: y f (x, x 2,...x ) Analysis Mthod Vaianc Dcoposition X X X2 Fdbac Figu. Bloc diaga of th ipotant stps in global snsitivity analysis. Using SA th output vaianc can b dcoposd and th lativ ipotanc of ach of th input factos can b dtind. As shown in Figu, th piay output of SA (vaianc dcoposition) can b bst illustatd by a pi chat showing th lativ ipotanc of ach of th factos in lation to th total unconditional output vaianc. Th facto distibutions a sapld, th odl is xcutd, and th total vaianc is dcoposd. A facto that has a lag ipact on th total vaianc will b shown in th pi chat with a lag pcntag whil a facto that has inial influnc on th output vaianc will hav a cospondingly sall pcntag of th pi chat. Onc SA is copltd, fdbac can b povidd to th odl stuctu and to th undlying assuptions usd in th facto distibutions. This pap dscibs a systs ngining analysis of a TRMM-li (TL) tival algoith using th SA tchniqu of vaianc dcoposition to dtin and quantify th lativ ipotanc of ach of th odl input factos. In this cas, th odl output is stiatd ain at. Th t TL algoith is usd to convy th notion that th algoith studid in this pap is basd on th TRMM algoith but dviats in slct aspcts. Th pap is outlin is as follows:

2 sction 2 povids a bacgound and athatical basis of vaianc dcoposition; sction 3 dscibs th TL tival algoith that is studid and th assuptions ad in th odl; sction 4 shows th SA and UA sults fo thos analyss; sction 5 xains th iplications fo GPM; and sction 6 is a suay of this pap. 2. VARIANCE DECOMPOSITION Fo a givn odl with input factos and function f, th output Y can b xpssd as ( X X ) Y, 2,..., f X () wh X, X 2,, X a th input factos. Vaianc dcoposition thods includ th colation atio dscibd by McKay (998), th thod of Sobol, Sobol (2), Sobol (23), Saltlli (22), and th Foui Aplitud Snsitivity Tst (FAST) dvlopd by Cui (978). Th Mthod of Sobol is usd to pfo vaianc dcoposition o coputationally fficint than th colation atio whil yilding o infoation, Chan (997). It is basd on th dcoposition of total vaianc givn as i ij i i j> i V V + V + + V wh V is th total unconditional vaianc, V i is th vaianc du to ach input facto X i by itslf. V ij is th vaianc du to ach pai of input factos, X ij. Additionally, high-od intaction ts a includd all th way up to V 2 which is th vaianc du to th intaction of all th input factos. A asu of snsitivity S i fo a facto can b divd by dividing ach t in (2) by th total unconditional output vaianc V giving, (2) Si + Sij S2... (3) i i j> i with th usful popty that th su of all th snsitivity indics and intactions su to on. Th S i vaiabls a calld th fist-od snsitivity indics. Th S ij a calld th scond od indics. Liwis, S ij a calld thid od and so on up to th highst od, S 2.., which accounts fo th intaction of all input factos. Fo a odl that is additiv, Si (4) i aning th su of th fist-od snsitivity indics is on. If th S i do not su to on, thn high-od intactions a psnt. Rath than dictly coput all th fist and high-od snsitivity indics, it is oftn sufficint to coput only th S i and total snsitivity indics, S Ti fo ach input facto X i. Th S Ti indics asu th contibution to th total vaianc as a sult of th fist and high-od intactions of ach facto X i. Fo xapl, assuing 3, th total snsitivity indx S T fo facto X would b S T + S + S2 + S3 S23 (5) which accounts fo all th intactions of input facto X with X 2 and X 3. Siila xpssions can b gnatd fo factos X 2 and X TL RETRIEVAL ALGORITHM 3.. Bacgound Th tival algoith usd to stiat th TRMM ainfall at is dsignatd as 2A25 in th TRMM data poducts and is thooughly dscibd in Iguchi (2). Th undlying assuption of th tival thod is that spcific attnuation () in db/, at so ang fo th satllit, can b odlld as ( ) α ( ) (6) and that th ainfall at in /h can b found fo R b ( ) a ( ) (7) wh α,, a and b a piically-divd cofficints and th tu ada flctivity facto is dsignatd as (). Bcaus of attnuation, () is asd and ust b stiatd. Th obsvd o asud ada flctivity facto () ( 6-3 ) at ang is latd to () by th two-way attnuation facto A(), and A() is givn by ( ) ( ) A( ) (8) ().2 ln( ) () s ds A xp (9) Using th assuption of (6), (8) can b solvd fo () and wittn as () AHB wh A HB () is th Hitschfld-Bodan (HB) divd attnuation facto, ds q α() s () s / A () HB with q.2 ln. Not that log log and ln log. Using a hybid of th HB and an indpndnt sufacfnc tchniqu (SRT), () and () can b wittn as 2

3 ( ) ε q α() s () s ds / (2) Th cofficint ε is found fo th path-intgatdattnuation fo HB (PIA HB ) and SRT ( σ ) valus so that can b calculatd at ach ang bin, Mnghini (2). In this wo, ε is calculatd using th siplistic lina function fo Iguchi (994) instad of th o dtaild axiu-lilihood thod dscibd by Iguchi (2). Noally, aft th final ε valu is calculatd, th a and b ts in (7) a calculatd (adjustd) using scond-od logaithic polynoials to povid th final -R lationship, b' ( ) ( ) R a (3) with R() in /h such that ε is lind to th DSD via th α-coction pocss, Kozu (2). Th adjustnt fo non-unifo ba filling (NUBF) is not iplntd in this wo Modl Using quations (6) though (3), a odl fo th TL algoith can b foulatd accoding to th snsitivity analysis guidlins of th fo f (, α,, a, b, σ ) R (4) wh th ain-at R() is a function of th listd input factos and ang. Although th input pofil,, is any bins high, th snsitivity analysis is pfod using only th botto bin nast th gound Input Factos Each of th input factos listd in (4) ust hav a spcifid PDF that can b sapld as pat of th SA pocss. Fo ocan tival, th TL-odl facto distibutions a shown in Tabl. Tabl. List of th input factos and thi distibutions fo th TL algoith ainfall-at snsitivity analyss ov ocan. Th noinal valus fo ths factos w obtaind fo Iguchi (2), assuing 2 C statifo ain. Facto Na Pobability Distibution α U{ ± 2% } constant a U{ ±.4 } b U{ ±.36} σ N{ σ tu, db } (o in ach bin) N{, db} statifo. is constant and is fixd at its dfind valu fo ain, 2 C, statifo, Iguchi (2). As this pap focuss on a TL algoith, th calculation and usag of a and b a handld diffntly than th thod in th TRMM algoith. Fo low ainfall ats, wh ε is unity (o clos to unity), and α- coction is not pfod, th -R lationship of (7) is usd to calculat ain at using stiatd PDFs (shown in Tabl ) fo a and b. Fo high ainfall ats, th a and b thod givn in th TRMM algoith (log polynoials basd on α-coction) is usd to calculat ain at instad of sapling th PDFs. Th TL algoith autoatically tansitions fo on thod to th oth. Th unifo PDFs, usd in this pap, fo a and b w divd fo global disdot data fo Bingi (23) using a tchniqu basd on th wo of Kwiatowsi (22) which pfos -R cuv fits and gnats ultipl a, b pais, lind to α-coction, which can thn b statistically analyzd. Bingi (2) showd that an altnativ pocdu is to p b fixd and lat all vaiabilitis of -R though th concpt of noalizd DSD. At low ain ats, statistical vaiation (o) fo th a, b cofficints popagats though th odl, and at high ain ats, vaiation (o) in and σ popagat though th odl and a iplicit in th calculation of ε and in th valus of a and b. Not that vn sall vaiations in a and b in (7) (o a and b in (3)) can caus lag swings in stiatd ainfall at, and consquntly, th distibutions of a and b can b vy ipotant to accuat ainfall stiation. Th valu of σ dpnds on whth it is ov land o ocan. Fo ocan, it is assud to b -db, standad dviation, an valu qual to tu PIA fo th siulatd data sts. Th input facto,, is a vtical pofil (vcto) of flctivity valus. It is undstood that th is asunt o in ach bin, and that th o is noally distibutd, zo an, with a -db standad dviation. Bcaus th snsitivity and unctainty analysis thods don t allow fo a dict apping of an o vcto in th function (4), th pofils ust fist b appd to anoth ando vaiabl calld a tigg facto o tigg vaiabl. Saisana (25) showd that th tigg facto svs as an indx to slct spcific pofils fo SA. In this way, th o in th pofils can b intgatd into th odl valuations. 4. TL-ALGORITHM OCEAN RESULTS SA was pfod on th TL function (4) fo th DSD pais listd in Tabl 2 fo ocan conditions. Each pofil is basd on a vtical ain colun, 3- high assuing 2 bins, ach with a ada ang solution of.25. Th fist-od indics, S α, S a, S b, S σ, S, vsus ain at a shown in Figu 2. In this wo, α is assud to b unifo and vay ± 2% about its noinal valu spcifid fo ain, 2 C, 3

4 Tabl 2. List of th N w, D o pais and ain at that w usd to cat pofils. Vaianc Dcoposition % Do () Nw Rain Rat (/h) 9% 8% 7% 6% 5% 4% 3% 2% %.65 33, , , , , , , , , , , S b S a S S σ o % Rain Rat (/h) Figu 2. Gaph showing th fist-od vaianc dcoposition of th ipotant factos in th TRMM-li algoith ov ocan. Th factos a and b hav significant ipact in th low ainfall gion but diinish as th algoith tansitions to high gion wh th log polynoials a usd to calculat a and b. At high ain ats, th o in σ doinats. Fo th SA sults, th is a cla pa in th contibution of α to th total output vaianc aound th 3 /h ang. At low ain ats, th S a t doinats whil th contibution fo S α and S σ a ngligibl. In th low ain-at gion, th TL algoith uss th HB solution which dos not us th σ input valus. In th high ain-at gion, th σ input is usd o significantly in th tival algoith. A tansition btwn HB and σ thods occus btwn about 5-8 /h. In both th low and upp gions, th su of th th, fist-od snsitivity factos is appoxiatly % at ach ain-at indicating that th TL odl of (4) is additiv. Howv, in th tansition gion, th odl is not puly additiv and high-od snsitivity intactions occu. Abov th tansition, S σ doinats th ain at vaianc. As xpctd, th contibution fo th o in S α σ dclins with incasing ain at bcaus th tu- σ valus incas and th o associatd with this vaiabl dcass as a pcntag of its tu valu and hnc its contibution to output vaianc dcass. 5. GPM IMPLICATIONS SA thods hav bn applid to a GPM dualfquncy tival algoith basd on pofiloptiization tchniqus, Ros (25). This thod assus that th pofils fo D o and log(n w ) a lina and that an optiization outin can find appopiat top and botto D o, N w valus such that intnally calculatd, 2 pofils atch th input pofils. This GPM tival algoith has bn adaptd to th fo, ( ) (, ) R f (5) 2 wh and 2 a th input, asud-flctivity pofils at 3.6 and 35.6 GHz, spctivly, and R() is th ain at. In this cas, th o associatd with th and 2 pofils is zo an,.5-db standad dviation, noally distibutd, in ach bin. Fo SA, th fist and scond-od snsitivity indics, S, S 2, S 2, a shown in Figu 3. Vaianc Dcoposition % 9% 8% 7% 6% 5% 4% 3% 2% % S S 2 S 2 % Rain Rat (/h) Figu 3. Gaph showing th fist and high-od vaianc dcoposition of th two input factos fo th GPM pofiloptiization thod. At low ain ats, th odl is not additiv wh significant high-od intactions a psnt. At high ain ats, th odl is additiv. Th o in 2 is th piay contibuto to total output vaianc thoughout th nti ain at gion. Th analysis shows that at low ain ats, a significant high-od intaction btwn th input factos is psnt, and at high ain ats, that th o associatd with 2 (S M2 ) doinats. As with th TL algoith, tigg vaiabls w usd with both and 2 to ap th to th SA pocss. Th iplication fo GPM is that th o in 2 doinats th output vaianc and o phasis should b applid to its asunt and pocssing than to that of. 6. SUMMARY This pap has bifly shown how global snsitivity analysis can b applid to a TL algoith indicating which input factos a ost ipotant in contibuting to output vaianc in th ain-at tival. W hav also 4

5 shown pliinay SA sults and iplications fo a GPM tival algoith basd on pofil optiization. ACKNOWLEDGEMENT This sach was jointly suppotd by th NASA Pcipitation Poga and Los Alaos National Laboatoy. REFERENCES Bingi, V. N., V. Chandasa, 2: Polaitic Doppl Wath Rada, Cabidg Univsity Pss. Bingi, V. N., V. Chandasa, J. Hubbt, 23: Raindop Siz Distibution in Diffnt Cliatic Rgis fo Disdot and Dual-Polaizd Rada Analysis. J. Atos. Sci., Vol. 6, Januay, Chan, K., A. Saltlli, S. Taantola, 997: Snsitivity Analysis of Modl Output: Vaianc-Basd Mthods Ma th Diffnc. Pocdings Wint Siulation Confnc. Cui, R. I., H. B. Lvin, K. E. Shul, 978: Nonlina Snsitivity Analysis of Multipaat Modl Systs. Jounal of Coputational Physics, 26, -42. Iguchi, T., R. Mnghini, 994: Intcopaison of Singl-Fquncy Mthods fo Rtiving a Vtical Rain Pofil fo Ai-bon o Spac bon Rada Data. J. Atos. Ocanic Tchnol., Vol., Iguchi, T., T. Kozu, R. Mnghini, J. Awaa, K. Oaoto, 2: Rain-Pofiling Algoith fo th TRMM Pcipitation Rada. Jounal of Applid Mtoology, Vol. 39, Dcb, Kozu, T, T. Iguchi, 2: Dop Siz Distibution (DSD) Modls fo TRMM 2A25. TRMM PR Us s Manual. Kwiatowsi, J., R. Mnghini, A. Toay, 22: On Incopoating Dop Siz Masunts into th TRMM Satllit Rada Rain Rtival Algoith. TRMM Intnational Scinc Confnc, July, Honolulu, Hawaii 2P.2 p. 23. McKay, M. D., J. D. Moison, S. C. Upton, 998: Evaluating Pdiction Unctainty in Siulation Modls. Los Alaos National Laboatoy, LA-UR Mnghini, R., T. Iguchi, T. Kozu, L. Liao, K. Oaoto, J. A. Jons, and J. Kwiatowsi, 2: Us of th Sufac Rfnc Tchniqu fo Path Attnuation Estiats fo th TRMM Pcipitation Rada. J. Appl. Mto., 39, Ros, C. R., V. Chandasa, 25: A GPM Dual Fquncy DSD Rtival Mthod Basd on Lina Modl fo DSD Vtical Pofil. Invitd Tal, IGARSS 5, July 25-29, Soul, Koa. Saisana, M., A. Saltlli, S. Taantola, 25: Unctainty and Snsitivity Analysis Tchniqus as Tools fo th Quality Assssnt of Coposit Indicatos. Jounal of th Royal Statistical Socity A, Vol. 68, Issu 2, pp 37, Mach. Saltlli, A., S. Taantola, K. Chan, 999: A Quantitativ Modl-Indpndnt Mthod fo Global Snsitivity Analysis of Modl Output. Tchnotics, Vol. 4, No.. Saltlli, A., S. Taantola, F. Capolongo, 2: Snsitivity Analysis as an Ingdint of Modling. Statistical Scinc, Vol. 5, No. 4, Saltlli, A., 22: Maing Bst Us of Modl Evaluations to Coput Snsitivity Indics. Coput Physics Counications, Vol. 45, pp Saltlli, A., S. Taantola, F. Capolongo, M. Ratto, 24: Snsitivity Analysis in Pactic: A Guid to Assssing Scintific Modls, J. Wily & Sons. Sobol, I., 993: Snsitivity Analysis fo Nonlina Mathatical Modls. Math. Modlling Coput. Exp Sobol, I. M., 2: Global Snsitivity Indics fo Nonlina Mathatical Modls and Thi Mont Calo Estiats. Mathatics and Coputs in Siulation, Vol. 55, Issu 3. 5