Inversion of Solar Transmitted Radiance Profiles for the Atmospheric Optical Thickness
Sue0 Ueno
Kanmawa Institute of Technology P.O. Kanazawa-South lshikawa 921, Japan
ABSTRACT
In a series of papers (Ret%. [l-12]), quasilinearization and invariant imbedding have been successfully applied to an inversion of total spectral radiance profiles remotely sensed from space for dete rmining the atmospheric optical properties and the ground albedo, in a least square sense. In the present paper, with the aid of quasilinearization and invariant imbedding, a statistical estimation of the atmospheric optical thickness is made, using solar noisy transmitted radiance measurements on the ground.
1. INTRODUCTION
Recently, an estimation of unknown parameters in distributed parameter systems has been made from the noisy measurements. In other words, in an inverse problem, researchers tried to identify the coefficients of the operator with a known structure in terms of information yielded by some functionals of the solution. Such inverse problems, in which small perturbations in the observed functionals may result in large errors in the corresponding solu- tions, are often called ill-posed in the classical sense (cf. Nashed [13]). If certain additional restrictions are imposed on the admissible solutions, ill- posed problems may possess solutions that are stable against data perturba- tions. Such problems are called conditionally well posed. A need for ap- proximate solutions of conditionally well-posed problems with inaccurate data gave rise to an innovation: a regularization by which the identification problems are reduced to the minimization of performance (quadratic) func- tionals.
In recent years, with particular emphasis on temperature sounding, several kinds of such inversion methods (analytical regularization by addition of solution constraints, statistical regularization, iterated regularization, and
APPLIED MATHEMATlCS AND COMPUTATION 7:177-186 (1980) 177
0 Elsevier North Holland, Inc., 1980 52 Vanderbilt Ave., New York, NY 10017 ooQ6an3/80/060177+ 10$01.75
178 SUE0 UENO
others) have been discussed with regard to computational feasibility by several authors (Deepak [23]; Twomey [24]; Fymat and Zuev [25]). Further- more, the mathematical inversion of the equations of radiative transfer is a powerful tool for the remote sensing and probing of the earth-atmosphere system from space and air. Thus the determination of the composition and structure of the atmosphere (bounded by the ground or by the ocean) is fundamentally important not only in operational and predictive meteorology but also in the pattern recognition of digitized earth imagery. In a series of papers by several authors (Bellman, Kagiwada, Kalaba, and Ueno [l, 21; Kagiwada [22]; Bellman, Fymat, Ueno, and Vasudevan [5]; Kagiwada, Kalaba, and Ueno [S]), applying an invariant imbedding and quasilineariza- tion to the minimization of the regularizing functional, an inverse problem of estimating the atmospheric optical thickness, the optical properties such as the albedo for single scattering and the phase function, and other quantities has been solved numerically. Experience in the application of this method to multiple scattering has shown that the selection of a good initial approxima- tion is important, since in this case the method is rapidly convergent, with the number of correct digits approximately doubling with each additional step; if, however, the initial approximation is too poor, the method may be divergent.
In the present paper, with the aid of invariant imbedding and quasilin- earization, the atmospheric optical thickness of the earth-atmosphere system bounded by a diffuse reflector is determined, in the least square sense, from the noisy total spectral diffusely transmitted radiance at the bottom. It is assumed that the other optical properties of the atmosphere and the surface alhedo are known.
2. DIFFUSE REFLECTION AND TRANSMISSION
For the sake of simplicity, we consider a plane-parallel, hom*ogeneous, anisotropically scattering atmosphere of optical thickness 1c bounded by a diffuse reflector. From the analytical point of view, the introduction of optical inhom*ogeneity does not give rise to any difficulty (cf. Ueno and Wang [17]). Suppose that the top at t= 0 is uniformly illuminated from above in the direction 3, by a parallel beam of radiation of constant net flux TF per unit area normal to the direction of propagation. Let the upwelling intensity of radiation at level t in the direction Q be denoted by Z(t, + 52), and similarly, let the downwelling intensity of radiation at level t in the direction G be denoted by Z(t, -a). Here G stands for (2 u = cos8, +) (0 < 1) < 1,O < C#J < 27r), where 8 is the polar angle measured from the normal at the top, and + is the azimuthal angle. Let the phase function be denoted by P(G,&,) and be normalized to unity. Furthermore, the albedo for single scattering, denoted by X, is positive, being less than or equal to unity.
Atwmpheric Optical Thickness 179
The equation of transfer appropriate to the nonconservative case takes the form
u~Z(t,u)=Z(t,u)- $1 P(P,O’)z(t,c)dS2’, 4?7
(1)
where da’ = du’d+‘. This equation is to be solved subject to the boundary conditions
Z(0, -!A) = 7rzqQ - a,), (2)
Z(X,~-Z)=~~~(S~,Q’)Z(X, -M)u’dQ’/u, (3)
where S is the Dirac delta function, &?a stands for (&=cos-‘u, +), and k(S2,Q’) is the bidirectional reflectance, which is the probability that a photon incident on the ground in the direction - Q’ will be reflected into the direction +Q within a unit solid angle. In the case of a perfect diffuse reflector, k is given by
k(Q,CY) =Au/a, (4)
where A is the ground albedo, in accordance with Lambert’s law. The total spectral diffuse radiances at the top and the bottom are expressed in the forms
Z(0, +a)= -$(r,A;Q,S2,), (5)
z(x,-a,=$ T(x,A :%a,), (6)
where S(x, A; iI, Q,,) and T(x, A; Q Q,) represent the scattering and transmis- sion functions for planetary atmosphere with nonzero ground reflectivity, respectively.
We expand the phase function in Legendre polynomials (cf. Chandrasek- har [18, p. 1771)
where P;“(u) is the associated Legendre function of degree 1 and order m,
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and
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I?,=1 if m=O, (9)
= 0 otherwise,
and the scattering and transmission functions are expressed in terms of the
Fourier components, respectively
S(x,A;Q,G,)= 5 S(m)(x,A;v,u)cosm(~-~‘o), m=O
(10)
Z’(x,A;Q,Qo)= 5 T(m)(x,A;u,u)cosm(+-+o). (11) m-0
In a manner similar to an invariant imbedding procedure (cf. Kagiwada and
Kalaba [19]; Bellman and Ueno 1201; Ueno and Wang [17J), the Fourier
components of the scattering and transmission functions, Scm)(x,A; u, u) and
T@)(x,A;u,u), are governed by
=(2- S,)h 5 (- l)‘+“C~~~(x,A,v)rCl;“(x,A,u) (12) l=m
and
& T(“)(r,A;u,u) + f T(“)(x,A;u,u)
=(2-&,Jh 5 C;“~;“(x,A,u)~~(r,A,u), (13) l=m
where
( -l)m+l &“‘(x,A,u)=Pr(v)+ 2(2_6 ) (‘Sc”‘(x,A;o,tu)P;“(~)~, (14)
om 0
Atmospheric Optical Thickness 181
and
+;“(x,A,v) = e-“/“P;“(v) + 2(2Js,) ~‘Tc-,(x,A;u,w,P;‘(~)~ . (15)
Equations (12) and (13) are to be solved subject to the initial conditions
S’“‘(0, A; u, u) = 4Auu6,, (16)
T(“+(O,A;u,u)=O, (17)
where 8, is the Kronecker delta function. Furthermore, the Fourier compo- nent S cm)- and T(m)-functions fulfill the reciprocity principles with respect to the angular arguments.
On making use of Gaussian quadratures on the interval (0, l), the integrals in Eqs. (14) and (15) are replaced by sums, where the Fourier components S- and T-functions are given by
S$+,A) = S(“)(x,A;u,,ui), (13)
T;r’(x,A) = T@‘)(x,A;ui,ur). (19)
In these equations { ui} is the set of N roots of the shifted Legendre polynomials of degree N, P*(u) = PN( 1 - 2~). Then the functions S$“‘)(x,A) and I$“)( x, A) fulfill the system of ordinary differential equations
$ $“+,A) +
and
(20) l=m
2 T$“)(x,A) + $ T$‘+,A)=(2- 6,)X 2 C;“+ry(x,A)$“(x,A) (21) 1 l=m
for m=O,l ,..., M; i,i=1,2 ,..., N, where
+;(x,A)=P;+ (22)
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and
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In Eqs. (22) and (23) wI is the Christoffel weight corresponding to the value of ur. Equations (20) and (21) are to be solved subject to the initial conditions
S&m’(O,A) = 4Au c g i f onz* (24
T:;l")(O,A) =O, (W
respectively.
3. AN INVERSE PROBLEM
Suppose that the phase function P, the albedo h for single scattering, and the ground albedo A are together known. Let us assume that we have obtained N2 noisy measurements of the total transmitted diffuse radiance at the bottom, b+, where the subscript k denotes the Zcth component of the azimuth of the view angle. The optical thickness x is to be determined so that the theoretical diffuse transmittance pattern produced by using the estimated values in the differential equations (20) and (21) will agree as closely as possible, in the least square sense, with the observed field, {b+}. In other words, we wish to minimize the performance (quadratic) function
(26)
where
Z(x,A;i)i,&;uj,+,,)= 6 5 T(“)(x,A;o,,~~)cosm(~~-~,). (27) I m-o
4. QUASILINEARIZATION
With the aid of quasilinearization (cf. Bellman and Kalaba [21]; Kagiwada [22]; Kagiwada, Kalaba, and Ueno [S]), an inverse problem may be solved
Atmospheric Optical Thickness 183
iteratively. Then, writing the functions S$!m) and T.!“) as Smrf and Tmir, and similarly 4; = $&, g=+&, Pk”(ui) = PW, we Zave a system of linear differential equations for the (n + I)st approximation to Ttij and r:
ar”+l - 30, au w
where
x=ur (0 <u < 1.0) (31)
and
+(2- &,,)A : (- l)'+"C~~~~(x,A)~~"(x,A). (33) l-m
In Eq. (31) #A and c#& are given by Eqs. (22) and (23) in Gaussian quadrature form, respectively, where the nth approximations of the S- and T-functions satisfy the reciprocity principles
It should be mentioned that the reciprocity principle permits us to reduce the numerical computation of a system of differential equations. The solution
184 SUE0 UENO
is subject to the initial conditions Eqs. (24) and (25) and the boundary conditions (26) and (27).
Since Tz$’ is a solution of a system of linear differential equations, we express it in terms of a linear combination of a particular solution qmil and a hom*ogeneous solution h,,+
The system of differential equations for qmif is obtained by inserting the appropriate component of q wherever T”+ ‘, S”+l, or r”+i occurs in Eq. (28) or (29), with the initial conditions q(0) =O. Similarly, the system of equations for the hom*ogeneous solution is provided by dropping all terms not involving the (n + l)st approximation, whose initial vector h(0) has all components zero except for the last, which is unity. These integrations are readily performed on the interval 0 <t <r, since complete sets of initial conditions are given.
Then the initial conditions (25) are identically fulfilled. The solutions q(m) and h(m) are obtained by numerical integration on the interval (O,UT). By a simple differentiation, the value of T”+ ’ minimizing Eq. (26) is obtained from
7 4vibii~F-‘- ~~~q~iirCOSm(CC)
(36)
This is the required equation, which permits us to get an improved version of the optical thickness using the noisy total diffuse transmittance at the bottom.
5. CONCLUSION
In the present paper, with the aid of quasilinearization and invariant imbedding, we show how to solve analytically the inverse problem of the solar diffusely transmitted radiance pattern for the atmospheric optical thickness, provided that the other atmospheric optical properties and mean ground albedo are known. In a subsequent paper we shall present the result of a numerical computation of the optical thickness with the aid of Eq. (36).
Atmospheric Optical Thickness 185
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