Chapter 2 Sensitivity Analysis: Differential Calculus of Models

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1 Chpter 2 Sensitivity Anlysis: Differentil Clculus of Models Abstrct Models in remote sensing nd in science nd engineering, in generl re, essentilly, functions of discrete model input prmeters, nd/or functionls of continuous model input prmeters. In this sense, the sensitivities of model output prmeters re, essentilly, derivtives prtil derivtives with respect to discrete input prmeters, nd vritionl derivtives with respect to continuous input prmeters. Specificity of models, s compred to functions nd functionls in generl, is due to the fct tht they hve two mndtory components. The first component describes the object or process of study, s is. It is system of differentil eqution, or equtions, with initil nd/or boundry conditions, which is referred to s forwrd problem. The second component describes the procedure of deriving the output prmeters of the model, which simulte the observed quntities, observbles, from the solution of the forwrd problem. In contrst to the first component, it is just n nlytic epression, which is referred to s n observbles procedure. In rre prcticl cses, the forwrd problem hs n nlytic solution, nd the output prmeters re nlytic functions or nlytic functionls of the input prmeters. Correspondingly, nlytic evlution of sensitivities is possible. In most prcticl cses, only numericl forwrd solutions re vilble, nd specific pproches of sensitivity nlysis considered in this monogrph become indispensble. Keywords Model input nd output prmeters Forwrd problem Observbles Observbles procedure Sensitivities 2.1 Generl Considertions In the contet of this book, we define forwrd models s quntittive tools, which return the desired output prmeters for given input prmeters. These models re ssumed to provide n dequte quntittive description of the objects under study, nd this description is specified by given vlues of the input prmeters (model prmeters). The output prmeters of these models simulte the results of The Author(s) 2015 E.A. Ustinov, Sensitivity Anlysis in Remote Sensing, SpringerBriefs in Erth Sciences, DOI / _2 3

2 4 2 Sensitivity Anlysis: Differentil Clculus of Models observtions (observbles). Thus, in nutshell, we del with functions, whose rguments re the model prmeters nd the vlues re the observbles. Indeed, s ws pointed out in the Introduction, the forwrd models used in remote sensing consist of two bsic components. The first component, the forwrd problem, specifies the quntittive description of the object under study s is, bsed on relevnt lws of physics, which govern the sptil structure nd temporl behvior of this object. This description the forwrd problem is provided by corresponding differentil equtions nd initil nd/or boundry conditions. The second component, the observbles procedure, specifies the quntittive recipe of drwing the observbles from the solution of the forwrd problem the forwrd solution. This recipe is provided in the form of closed-form nlytic epression, which converts the forwrd solution into the observbles. Thus, bstrcting from the inner workings of the forwrd models, they represent essentilly functions whose rguments re the model prmeters nd vlues re the observbles. There is cvet though. In mny prcticl cses, the models involve the continuous prmeters, which re functions themselves functions of spce nd/or time. Then the forwrd models become the functionls defined on these functions. From the viewpoint of prcticl implementtion, there is little or no difference between functionls nd functions of mny vribles, becuse the prcticl implementtion requires representtion of those functions on n dequte grid of their rguments. But, s we will see in relevnt emples below, from the viewpoint of nlytic work tht is necessry to conduct the sensitivity nlysis in ech specific cse, it is instructive to tret the models with continuous prmeters s functionls, nd to pply corresponding tools of vritionl nlysis. In reltively smll number of prcticl cses, the forwrd problems used in prcticl forwrd models of remote sensing hve nlytic solutions. For emple, this is the cse in remote sensing of plnetry tmospheres in the therml spectrl region, when tmospheric scttering cn be neglected (see Sects. 4.3 nd 5.3). In such cses, the forwrd solution cn be represented s closed-form nlytic epression, which fter the ppliction of the observbles procedure results in n nlytic epression of observbles directly through the model input prmeters. Then, depending upon whether the given input prmeter is just constnt or function of spce nd/or time vribles, the observbles re functions or functionls. Accordingly, the sensitivities cn be found using stndrd techniques of the differentil clculus or vritionl clculus. In most prcticl cses, though, the corresponding forwrd problems cn be solved only numericlly. This mens tht for given numericl vlues of the input prmeters, the forwrd solution is obtined using pproprite numericl methods. This mens tht the nlytic reltion between the model input prmeters nd observbles does not eist. This is where the methods of sensitivity nlysis become indispensble. At this point we need to tke more detiled look t sensitivities with respect to different types of model prmeters nd observbles.

3 2.2 Input nd Output Prmeters of Models Input nd Output Prmeters of Models Without losing ny generlity, the model input prmeters cn be divided into two brod groups: discrete prmeters nd continuous prmeters. Discrete prmeters re constnts, which do not depend on ny rguments, such s spce, time, etc. Continuous prmeters re functions with essentilly the sme domin of rguments, s the forwrd solutions. Consider couple of simplified emples. Motion of mteril point in the plnetry grvity field. If the finite size of the mteril object in the grvity field cn be neglected s compred to the scle of sptil vrition of the grvity field of the plnet, then this object cn be considered s mteril point, nd its motion, ssumed here to be non-reltivistic, is described by forwrd problem in the form: 8 d 2 r >< dt 2 ¼ gr; ð tþ dr ð2:1þ dt ¼ v 0 t¼0 >: rj t¼0 ¼ r 0 The model input prmeters here consist of two discrete prmeters, v 0 nd r 0, nd one continuous prmeter gr; ð tþ. The forwrd solution rðþis t function of time t. Trnsfer of therml rdition in the non-scttering plnetry tmosphere. Neglecting the verticl spn of the plnetry tmosphere s compred to the rdius of the plnet, the rditive trnsfer in the non-scttering tmosphere cn be described by forwrd problem in the form: 8 u di >< dz þ ðþiz; z ð uþ ¼ ðþbz z Iz; ð uþ ¼ 0; for z ¼ 0; u [ 0 ð2:2þ >: Iz; ð uþ ¼ 2A R1 Iz; ð u 0 Þu 0 du 0 þ B s ; for z ¼ z 0 ; u\0 0 Here we hve two discrete prmeters, lbedo A nd source function B s of the underlying surfce, nd two continuous prmeters, etinction coefficient ðþ, z nd source function Bz ðþof the tmosphere itself. The forwrd solution Iz; ð uþ is function of the verticl coordinte z nd of cosine u of the ndir ngle of propgtion. The prcticl implementtion of the retrievl lgorithms in the form of computer progrms intended for the interprettion of prcticl dt involves the representtion of the continuous prmeters s finite-dimensionl rrys with vlues corresponding to n pproprite grid of rgument vlues of those continuous prmeters. But in mny prcticl cses, the nlytic work, which is necessry to be done in order to derive the nlytic bckground of the retrievl lgorithms, is esier to perform in

4 6 2 Sensitivity Anlysis: Differentil Clculus of Models terms of continuous prmeters considered s functions, using pproprite tools of vritionl nlysis. This will be demonstrted in the sections to follow. On the other hnd, the output prmeters, the observbles R, re lwys discrete prmeters due to the nture of mesurements themselves. In the first emple bove, such observbles my be reltive distnces nd velocities of the mteril point s mesured from some known loction of the observer. At ech instnt of observtion: R ¼ jr r i j S ¼ jv V i j ð2:3þ In prctice, the mesured vlues re lwys integrted over some finite spn of rguments of the forwrd solution: time, spce, viewing ngle, etc. But the integrtion results re lwys benchmrked by some instnt vlues of these rguments. More informtion will be provided in the sections to follow. 2.3 Sensitivities: Just Derivtives of Output Prmeters with Respect to Input Prmeters As mentioned in the Introduction, sensitivity of ny output prmeter with respect to ny input prmeter is merely derivtive of suitble type. Sensitivity of the observble R with respect to discrete input prmeter p is prtil derivtive: ð2:4þ Sensitivity of the observble R with respect to continuous input prmeter pðþis vritionl derivtive: K ðþ¼ dpðþ ð2:5þ There re few different definitions of the vritionl derivtive, which sometimes is lso clled the functionl derivtive. For ll prcticl purposes in this book, the vritionl derivtive with respect to the continuous prmeter p ðþis defined s kernel of the liner integrl epression Z d p R ¼ dpðþ dp ðþd ð2:6þ D

5 2.3 Sensitivities: Just Derivtives of Output Prmeters 7 Here, the integrtion is conducted over the domin D of rguments of the prmeter p ðþ; dpðþis the vrition of the input prmeter p ðþ, nd d p R is the vrition of the output prmeter R cused by the vrition dpðþ. For continuous prmeter specified on grid of its rguments, there eists simple reltionship between vlues of the vritionl derivtive on this grid nd vlues of prtil derivtives with respect to grid vlues of the continuous prmeter. The ccurcy of this reltionship depends on the mesh width of this grid. As n emple, ssume tht p ðþis defined on n intervl 2 ½; bšrepresented by set of grid vlues p j ¼ pð j Þ; ðj ¼ 1;...nÞ. Then, the output prmeter R becomes function of n vribles, nd Eq. (2.6) cn be pproimted in the form: Z d p R ¼ D j dpðþ dp ðþd X j dp j D j dpðþ j ð2:7þ From Eq. (2.7) we j D j ; or vice vers, dpðþ j dpðþ j D j ð2:8þ The definition of vritionl derivtive, Eq. (2.6), hs n importnt prcticl vlue. Throughout this book we will derive the epressions for sensitivities to continuous prmeters which re vritionl derivtives by trnsforming the epressions for vritions of the output prmeters to the form of Eq. (2.6) nd obtining the sensitivities s kernels of resulting integrl epressions. Equtions (2.5) nd (2.6) will serve s definition of sensitivities to continuous prmeters. Consider specil kind of functionl: the vlue of function f ðnþ defined on the intervl ½; bš t the specified vlue of its rgument n ¼. Representing fðþin the form fðþ¼ dn ð Þf ðnþ dn ð2:9þ we tke the vritions of both sides of Eq. (2.9): dfðþ¼ dn ð Þdf ðnþ dn ð2:10þ

6 8 2 Sensitivity Anlysis: Differentil Clculus of Models Compring Eq. (2.10) with the definition of vritionl derivtive, Eqs. (2.5) nd (2.6), we obtin: dfðþ df ðnþ ¼ dn ð Þ ð2:11þ In similr fshion, one cn derive n epression for the vritionl derivtive of the ordinry derivtive of the function f 0 ðþwith respect to the function fðþitself: df 0 ðþ df ðnþ ¼ d0 ðn Þ ð2:12þ where d 0 ðþ¼dd ðþ=d is the derivtive of the d-function. We hve: df 0 ðþ¼ dn ð Þdf ðnþ dn ð2:13þ Integrting by prts we obtin: df 0 ðþ¼dn ð Þdf ðnþj b d 0 ðn Þdf ðnþ dn ð2:14þ The off-integrl term in Eq. (2.14) equls zero for ll vlues of within the intervl ½; bš. Compring the resulting Eq. (2.14) with the definition of vritionl derivtive, Eqs. (2.5) nd (2.6), we see tht Eq. (2.12) is vlid everywhere within this intervl. In number of pplictions considered in chpters tht follow, we will need to convert the liner vritionl eqution LdX ¼ ds ð2:15þ into n eqution for corresponding vritionl derivtives with respect to some continuous prmeter: L dx d ¼ ds d ð2:16þ Here, d is the vrition of the prmeter ðþ, which results in corresponding vrition of the right-hnd term S ðþ, which in turn, results in corresponding vrition of the solution X ðþ. The liner opertor my, in generl, be function of, too.

7 2.3 Sensitivities: Just Derivtives of Output Prmeters 9 In the pplictions below, t ech given vlue of the rgument ¼ n, the function S ðþdepends on the prmeter ðþonly t the sme vlue of the rgument ¼ n. This mens tht S is function of ðþ, not functionl of ðnþ. Accordingly, dsðþ¼ dn dðnþ dn ¼ dn ð dðnþ dn ð2:17þ Compring with the definition of the vritionl derivtive, Eqs. (2.5) nd (2.6), we hve: dsðþ dðnþ ¼ dn ð Þ@S ð2:18þ Thus, the right-hnd term of Eq. (2.12) hs the form: d ðþ ð2:19þ On the other hnd, the solution dxðþof Eq. (2.15) depends on the vrition of the prmeter ðnþ in the whole intervl n 2 ½ 0 ; 1 Š. In other words, X ðþis functionl of ðnþ, while still being function of : X ðþ¼xn ½ ð Þ; Š ð2:20þ Thus, in the vritionl eqution, Eq. (2.15) dxðþ¼ dxðþ dðnþ dðnþdn ð2:21þ Substituting Eqs. (2.17) nd (2.21) in Eq. (2.15) nd moving the right-hnd term into the left side, we hve: L Z dxðþ b dðnþ dðþdn n dn ð dðnþdn ¼ 0 ð2:22þ Reclling tht the opertor L my be function of, nd observing tht is not n integrtion vrible in Eq. (2.22), we cn rewrite this eqution s L dx ðþ dðnþ dn ð Þ@S dðþdn n ¼ 0 ð2:23þ

8 10 2 Sensitivity Anlysis: Differentil Clculus of Models Finlly, demnding tht Eq. (2.23) be stisfied for n rbitrry vrition d, we obtin the eqution for vritionl derivtives, Eq. (2.16), which, in detiled form cn be written s L dx ðþ dðnþ ¼ dn ð Þ@S ð2:24þ We will use this result in pplictions of the lineriztion pproch of sensitivity nlysis to vrious forwrd problems considered in the chpters tht follow.

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