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Biorthogonality and radiative transfer in finite slab atmospheres
00224073/83 $3.W+ .I0 0 1983 Pergamon Press Ltd.
J. Quant. Speclrosc. Rodiot. Transfer Vol. 30, No. 2, pp 119-129, 1983 Pri:ded m Great Britain
BIORTHOGONALITY AND RADIATIVE TRANSFER IN FINITE SLAB ATMOSPHERES H. DOMKE Zentralinstitut
fiir Astrophysik,
DDR-15 Potsdam, DDR
(Received 18 October 1982)
Abstractlt is shown that the exact solution of transfer problems of polarized light in finite slab atmospheres can be obtained from an eigenmode expansion, if there is a known set of adjoints defined appropriately to treat two-point, half-range boundary-value problems. The adjoints must obey a half-range biorthogonality relation. The adjoints are obtained in terms of Case’s eigenvectors and the reflection or the transmission matrices. Half-range characteristic equations for the eigenvectors and their adjoints are derived, where the kernel functions of the integral operators are given by the boundary values of the source function matrix of the slab albedo problem. Spectral formulae are obtained for the surface Green’s functions. A relationship is noted between the biorthogonality concept and some half-range forms of the transfer equation for the surface Green’s functions and their adjoints. Linear and nonlinear functional equations that are well known from an invariance approach, are derived from a new point of view. The biorthogonality concept offers the opportunity for a better understanding of mathematical structures and the nonuniqueness problem for solutions of such functional equations.
I. INTRODUCTION
The eigenfunction expansion method introduced in the linear transport theory by Case’,’ offers a suitable way to treat radiative transfer problems in slab atmospheres. Complete sets of eigenmodes of the transfer equation can be found explicitly for very general models of anisotropic scattering,3 including polarization.4 If the eigenmodes are known, the solution of any radiative transfer problem amounts to the determination of the expansion coefficients by means of the boundary conditions. Typical boundary conditions for finite slab atmospheres prescribe the radiation field within the half range of incident directions at the boundaries T = 0 and T = b. The computation of eigenmode expansion coefficients from these boundary conditions can be performed by means of appropriately defined adjoints, which have to obey some biorthogonality relations on the halfrange of the angular variables. The concept of biorthogonality has been developed by McCormick and Kuscer’ for scalar radiative transfer in semi-infinite media. More recently, this work has been generalized to finite slabs by Gibbs and Seto.6 The concept of biorthogonality for finite slab atmospheres will be reconsidered and extended to polarization. It will be shown that the concept of biorthogonality can be developed in a rather genera1 form without any reference to the separation of angular variables. The adjoints of the eigenmodes obeying an appropriate half-range biorthogonality relation will be derived in terms of Case’s eigenmodes and the reflection or the transmission matrices of the finite slab atmosphere. An alternative way to define the adjoints will be noted and is based on the half-range form of the characteristic equation. The kernel functions appearing in the half-range integral operators have a well defined physical meaning as the boundary values of the source function matrix appropriate to the slab-albedo problem. For isotropic scattering, these kernel functions reduce to Chandrasekhar’s X- and Y-functions.7 The eigenfunction vectors and their adjoints will be utilized in order to derive the surface Green’s function matrices in spectral form. Although the spectral formulae quickly lose any computational advantage with increasing complexity of the phase matrix, they clearly reveal the mathematical structures of the solutions for transfer problems in finite slab atmospheres. They can be employed in order to derive linear and nonlinear functional equations, as well as linear singular integral equations, from a unified point of view. II9
120
H.DOMKE EXPANSIONS
2. EIGENMODE
The transfer of polarised radiation in a macroscopically described by means of independent transfer equations
pmw) -II
dr
-
UT,P) +;
isotropic
,d~‘Ps(w’M~,~‘) + B,‘0’h4 I_+I
slab medium
can be
s = 0, f 1,k 2,. . . ,
(2.1)
for each of the azimuthal Fourier components of the Stokes vector. The vectors I,(T,~) depend on the optical depth T and the cosine p of the polar angle. In the transport equation (2.1), the single scattering albedo is denoted by o, pS&,p’) is the azimuthal Fourier component of the phase matrix, and B,‘0’(r,p) describes primary sources. For a finite slab atmosphere of optical thickness b, the solution of the transfer equation has to be subjected to the half-range boundary conditions UOJJ) = Cl,(P),
CLE [O,ll, (2.2)
L(b,-/4=%b-4,
CLEW,~I,
which describe the incident radiation at the upper and lower boundaries of the slab. Eigenmodes are defined as independent solutions i,(r,p) of the homogeneous equation. For eigensolutions of the form iAr,~;;rl) = e-+W~,~), the eigenvector
(p&q)
obeys the characteristic
(77- CL)(P,(CL,~)) = ;q The eigenvectors corresponding written in the form4
(P&r))
I
(2.3)
equation
(2.4)
_;’ d~‘ps(~+‘)4Q~‘,n).
to the continuous
= ;VlJS(WJ)i(r,
transfer
part of the spectrum
- P) + A(+%
n E [- 1, + 11 can be
(2.5)
- q),
where
(2.6)
A(T)) =; [A(TJ+ i0) + h(TJ - iO)l,
A(z) = E -;
The characteristic equation
m-‘(z)
function
I
iI
-1
matrix
dwsbks(w)l(z - PL), ZE [-- 1,+ 11. +&,a)
is obtained
as the solution
(2.7)
of the integral
There is some arbitrariness in the choice of the matrix T,(P) appearing in Eqs. (2.7) and (2.8). One way to choose this matrix has been described earlier.4 Discrete eigenvalues ~i6f [- l,+ l] can be found as the roots of the transcendent equation det h(vi) = 0,
ail [- 1, + 11, i = + l,? 2, . . .
(2.9)
Biorthogonality
and radiative transfer in finite slab atmospheres
121
The corresponding regular eigensolutions for nondegenerated eigenvalues have the form (2.3) with (2.10) where the vectors M,’ are the nonzero solution vectors of the algebraic equations h(7)i)M,’ = 0, i = * 1, ? 2,. . . .
(2.11)
For macroscopically isotropic media, the following symmetry properties hold:4 P&W’) = P-J-
CL,- IL’)=
PsT(E”‘,lr)
(2.12)
and
d/477) = cp-A-p, - 771,
(2.13)
if the Stokes parameters are given in CP-representation. The eigenvectors obey the full-range orthogonality relation
(2.14)
the normalization integrals are
For multiple roots of Eq. (2.9), the eigensolutions ik(T,P;q) (k = 1,2,. . . ,q) may belong to a higher dimensional irreducible representation of the translation group. Then, if Z’ik(T)is the representation matrix of the translation operator, ii(7 + a,/J,;qi) = k=l
Tjk(r)ik(W ;‘Vi),
(2.16)
where q is the multiplicity of the eigenvalue qi. As an example, we consider the degenerated eigenvalue I/~~, = 0 occurring for the zeroth azimuthal Fourier component in the case of conservative scattering w = 1. The two independent eigensolutions of the transfer equation are easily found to be 1 ir(7,1*;w) = 2 iO,
i-r(r+;m) = k (7i0- Pi,),
(2.17)
where i,, describes unpolarized light of unit intensity. The vector i, is defined as i, = [l - x,/3]-‘iO,
where xi/3 is the mean cosine of the single scattering angle. The representation matrix of the translation group corresponding to the basis (2.17) is given by {Tik} =
(t y), (i,k = * 1).
(2.18)
122
H.DOMKE
We suppose that we have found a complete set of independent eigensolutions. Then the radiation field of a source-free, finite-slab atmosphere can be written as a linear superposition of eigenmodes, viz.,
The spectral integral includes integration over the continuous spectrum n E [- 1, + 11, as well as summation over all eigenmodes corresponding to discrete eigenvalues qicE [- 1, + 11. The transfer problem then amounts to the determination of the expansion coefficients from the two-point, half-range boundary conditions (2.2). The concept of biorthogonality offers a systematic approach to performing this task. 3,BIORTHOGONALITY The finite slab boundary conditions (2.2) prescribe the radiation field on half-ranges at two different optical levels. The full-range orthogonality relations (2.14) are not applicable for computing the spectral expansion coefficients A(n) for this type of boundary conditions. However, we can solve this problem by using properly defined half-range adjoints. We introduce two sets of adjoints, a,(k,q) (n EC; p >O) and aT(~,q) (7 E a; p >O), defined on the positive half-range of the variable p such that the expansion coefficients may be written as linear functionals of the incident radiation in the form
(3.1) Taking into account the boundary condition (2.2), but also the eigenmode expansion find that Eq. (3.1) is consistent with the following biorthogonality relation: Eb(n -n’),
&j, 0,
o dCLCL[a,*(CL,B)i,(O,~;~‘) + ~$Thdi.v(~,- ~;q')l =
I’
(2.19) we
11; , 7 = rli,B = 9j; rl# 77’.
rl,g’E ]- 1, +
(3.2) If the eigensolutions
are of the form (2.3), it is easy to show from Eqs. (2.13) and (3.2) that
aS(~,q)= a-,(~, - q) F. The biorthogonality
relation (3.2) can then be written in the form
I’ 0
(3.3)
dw[asT(p,rlhhv’I + a%, - q)eh(“q-““‘)(ps(_ ~,r)‘)l=
E6(q-q’), &jt 0,
q,q’E[-l,+ll; 7)=7)iPl rl+
I
=r)j;
4.
(3.4) We consider now the eigenmode expansion defined as the solution of the transfer equation
of the Green’s function
matrix of a finite slab
+I
Es(r - ro)s(L‘ - PO), P&W ;~o,po)= - G(T,P;TO,PO)+; I_I dl.L’ps(CL,CL’)Gs(T,CLI;T0;CLO)+ (3.5) subject to the boundary
conditions
Gs(O+;TorPo) = 0,
F,/JO
E [%]I,
P,PO
E
(3.6)
Gs(b,-
CL;To+o) = 0,
[%]I.
Biorthogonality
and radiative transfer in finite slab atmospheres
123
By means of Eq. (3.5), the boundary conditions (3.6) and the symmetry properties (2.12) for the phase matrix, it may be shown that G~(w;To,Po)
=
G-,(b - T,- k;b - TO,- /JO).
(3.7)
Moreover, the reciprocity relation
WW;TO,PO) = G?s(~o,
- PO;T, -
IL)
(3.8)
can be derived in the same manner as for the scalar case outlined in the monography of Case and Zweifel.2 In particular, the surface Green’s function matrix G,(T,~;O,~~) (T > 0) appears as the solution of the homogeneous transfer equation subject to the boundary conditions
G,(O+,P;O,CLO) = ; ESb - PO), /APO E LOAl, (3.9) G,(b, - P;&PO) = 0,
CLIP0 E [OJI.
On the other hand, the surface Green’s function matrix is related to the albedo and transmission matrices by
P~(/-wo) = ; G,(O,- ~;O,po), /.woE [O,ll
(3.10)
and ~sbw~)
=
i
G,(b,p;O,po) - E;el -b’NNp -cLo) 3 I
(3.11)
respectively. Utilizing now Eqs. (2.19), (3.1), and (3.9) we find the eigenmode expansion of the surface Green’s function matrix in the form
G(T,P;O,P~) =
10 d7)i,(T,IL;rl)a,T(CLo,rl).
(3.12)
The same reasoning can be applied in order to derive the eigenmode expansion for the other surface Green’s function, Gs(7,p;b, - po), which obeys the homogeneous transfer equation as well as the boundary conditions
G,(O,cL;b, - PO)= 0,
EL,CLO E
tO,ll, (3.13)
We obtain the eigenmode expansion formula (3.14) Using Eqs. (3.12) and (3.14) for the surface Green’s f3nction matrices, we can easily derive also a spectral representation for the complete Green’s function matrix. For this purpose, we utilize the relation GS(~,~;~o,p,J = Gs’r’(~,p;~o,po)-
-
I 0
I,’
dCL’CL’G,(T,CL;O,CL’)G,(OC)(O,CL’;70,CLO)
(3.15)
I
dl*‘I*‘G,(7+;h, - cL’)G,‘“‘(b,-
CL’;‘TO,IL~),
124
H. DOMKE
in connection with the well known eigenmode expansion formula for the infinite medium Green’s function matrix GS(m’.2,4 Equation (3.15) can be derived formally by means of Eqs. (3.5) and (3.6) but also heuristically from invariant imbedding arguments. 4.
THE ADJOINTS
We now investigate in greater detail the adjoints which we have introduced formally. Applying the full-range orthogonality (2.14) to the spectral formula (3.12) for the surface Green’s function, we find, after taking into account also Eqs. (3.10) and (3.1 l),
(4.la)
and
These formulae express the adjoints explicitly in terms Jf the albedo matrix or, alternatively, in terms of the transmission matrix. The adjoints of the positive continuous eigenspectrum are singular function vectors. Taking into account the explicit form for Case’s eigenvectors (2.5) we find from Eq. (4. la) after some partial fractional analysis,
with
%T(P,rl) = 9sThJ)+ 2rl
I
I’cW!?-$,
~)P,(P’>~1
0
I
-2/J o WhT(- P’,~>P&‘~P)(P’+ +)/(p’ + 77). In particular,
(4.3)
it follows from Eq. (4.3) that (4.4)
We now illustrate the use of Eq. (4.3) for the case of isotropic scattering,
$
z
1
&I
p)
=
$9 X(P’)X(cL) - Y(p’)Y@)
4
CL’SCL
where’ (4.5)
We obtain (4.6)
A more general form of Eq. (4.6) appropriate to scalar anisotropic scattering has been derived by Gibbs and Seto.” Equations (4.2) and (4.3) are well suited to performing a reduction of the adjoints to functions depending only on one variable. Although this is possible for any scattering model, a complete separation of variables would have practical advantages only for relatively simple phase matrices (e.g., Rayleigh or molecular scattering). The adjoints of the negative part of the eigenspectrum are regular function vectors. A
Biorthogonality
convenient
and radiative transfer in finite slab atmospheres
125
formula can be found from Eq. (4.lb) by means of partial fractional
u,~‘(P, - 17)= N,L’(-17)e-h/vwrl /(7)+ /.L) PT,(/&q), 77 E Khll,
I
2
analysis,
E PA117
P
i.e.,
(4.7)
with
(4.8)
In particular,
for isotropic scattering, a(p!
where
p)
=
Y(I*‘)X(/J)- X(cL’)Y(cL)
g
4
(4.9)
PI-P
we get
(4.10) For optically very thick layers, b B 1, we obtain the following asymptotic adjoints of the zeroth azimuthal Fourier component (s = 0):
relations for the
Nln e-W1
ooT(Wf) = unT(Wl) +
N,,-‘(n)(l_ n2 e-2b/q,)
‘d’l)UoT(Po,~),
77 >
0,
(4.11)
and a,Jr(P, _ ,,) = _ N,,~1(7))e-h”ill+lirl’)
where the vector
~~(7)
(l _
N,
n2e-2b/,,,)
Ko(17)~oT(PoJh)~
7 ‘0,
(4.12)
is given by
(4.13) In order to derive Eqs. (4.11) and (4.12), we have employed well known asymptotics for the reflection and transmission matrices.8 Equations (4.la,b) can be easily generalized to the case of multiple roots of Eq. (2.9). Taking into account the full-range orthogonality of the eigenmodes for different eigenvalues, we find from Eq. (3.12)
(4.14)
(4.15) where the coefficients
Ljk(7) are
L,k(T)
=
I
+I
-I
d/4iT(0,p;V)h(T,p ;T).
(4.16)
H. DOMKE
126
As an example, we consider the pair of degenerated equation. For conservative scattering. We obtain L&(T) = -
eigenvalues
(y i) i (ilT . io),
l/n*, = 0 of the transport
(4.17)
i,k = + 1.
Denoting no = (i,%)/6, we get from Eqs. (4.14), (4.15), and (4.17) the adjoints appropriate eigenmodes (2.1’1) in the form
to the
(4.18a)
(4.18b)
as functionals
of the albedo matrix and I aT,(r_~~,m)= &-
aT1(po,“) = --
1
[
(FoilT + bioT) embiN+ 2
1
I0
dF&ilT
+ bioT)&&
-b/m + 2 I
2nl
,
in terms of the transmission matrix. For optically thick atmospheres, b > 1, we employ the asymptotic ting the formulae of Ref. 8 by a factor of l/2)
1 ,
(4.19a)
(4.19b)
formulae* (after correc-
(4.20)
and p(b
(T -
=
m) -
CT,
(4.21)
with I
l.42 =
From Eqs. (4.19), we find the asymptotic
In order to obtain the adjoints can not employ Eq. (3.3). Applying the expansion formula (3.14) for albedo and transmission matrices. the form
I dw2ilTuW). o
(4.22)
relations
a$,(~,m) occurring in the biorthogonality relation (3.2), we the full-range orthogonality properties of the eigenmodes to G(T,p;b,- pa), it is easy to derive a$,(~,=) in terms of the The result can be expressed by means of the adjoints akl in
and
a~T(~0,4 =We note a relation between the biorthogonality
aT,(~o,~).
(4.25)
concept and the method of singular integral
Biorthogonality and radiative transfer in finite slab atmospheres
127
equations. Combining Eqs. (4,la) and (4.lb), we obtain
=
[e-b’rl- e-b’@qcp,T(po,?-j), q E (T.
(4.26)
Interpreting Eq. (4.26) for the continuous half spectra n E [O,l] and n E [- l,O] and taking into account Eq. (2.9, we obtain a system of two inhomogeneous linear singular integral equations for the reflection and the transmission matrices. Interpreted for the discrete eigenvalues, Eq. (4.26) yields a set of linear regular constraints. Moreover, it is obvious from Eq. (3.4) that the pair of adjoint vectors {a&,nJ, a_& -TJJ eb’*i}obeys the corresponding system of homogeneous linear singular integral equations. 5. HALF-RANGE
FORMS OF THE TRANSFER
EQUATIONS
We now show that the concept of biorthogonality can be related to some half-range formulations of the transfer equation. Let i,(r,~) denote any solution of the homogeneous transfer equation. Taking into account Eq. (3.5) for the Green’s function, but also the symmetry relation (2.12), we derive the functional relation
$-q_‘I dppG?dT', I
I
- p;~o,po)is(~+ +,p) = - i,(r + T~,~~)S(T’-
With r. = 0, we obtain after integrating Eq. (5.1) on the interval
is(T,- I*01=
I
I
0
(5.1)
TV).
I
7’ E
[O’,b],
1
4vG(O, - Po%L)~s(T4‘) + o dCLdLUwo;O,ph(b
+ 7, -
EL),
(5.2)
Here, we have employed also the symmetry relations (3.7) and (3.8) which hold for the Green’s function matrix. Equation (5.2) can be derived also heuristically by means of invariant imbedding arguments. The functional relation (5.2) can be employed in order to rewrite Eq. (2.1) in a half-range form. For this purpose, we introduce the function matrix (5.3) which represents the source function matrix of the finite-slab albedo problem. Inserting Eq. (5.2) in the transfer equation, we get
l-+shd=-k(W)+2
I
I dCL’B,(O,CL,CL’)is(T,CLI) 0
+ 2 o’dp’B_Jb, I
/L,/L')i,(T +
b, - /J’).
(5.4)
This is a half-range functional integro-differential equation valid for any solution of the original homogeneous transfer equation. As an application, let us consider eigensolutions of the form (2.3). From Eq. (5.4) we find the following characteristic equation for the eigenvectors: I (P&V) = 2 1’ dCL’B,(O,CLlr.L’)(PS(~‘,~) + 2 emb”’ dp’B_,(b, - ~,~‘)(p,(- ~‘,a). 0
(5.5)
By means of purely formal arguments, it is easy to find an adjoint form of Eq. (5.5) consistent
128
H. DOMKE
with the biorthogonality
relation (3.4) i.e.,
(1-k)~sT(/.vt)=
2 ’ d I_L‘as T(~‘,n)B,(O,~‘,~)
dp’a’&‘,
- 2 eb”
- n)B,(b, - IL’+).
(5.6)
Equations (5.5) and (5.6) are remarkable. They show that the eigenvectors, as well as their adjoints, can be reduced to the kernel functions B,(O,p’,p) and B,(b, - p’,~). In other words, the solution of any transfer problem in finite slab media can be expressed as a functional of the surface values of the source function matrix for the slab albedo problem. Equation (5.6) can be interpreted as the eigenequation for eigenmodes
is+(7,cL;q) = asTh4q) e+
(5.7)
of the following transfer equation:
-
2
Io’
dcL'i+,(b
- T,p’; - q)B,(b,
- F’,p).
(5.8)
Moreover, the spectral form (3.12) of the surface Green’s function with eigenmodes (2.3) can be rewritten in terms of the adjoint eigenmodes (5.7) as
of the form
(5.9) It is obvious from Eqs. (5.8) and (5.9) that the surface Green’s function half-range transport equation:
PO-&(W ;O,PO) = - G(~,cL;O,PO) + 2I
obeys the following
I
dCC’G,(T,CL;O,CL’)Bs(O,CL’,CLo)
0
- 2 o’ dp’G-,(b I Taking into account the symmetry
(5.10)
- 7, - p;O+‘)B,(b,
- CL’+,,), 7 > 0.
relations (3.7) and (3.8), we rewrite Eq. (5.10) in the form I
~o-$dO,
-
PO;T,P)
=
-
GsKJ,
-
2
-
PO;~>P)
+
Io’G’B %, -
2
G’B
%W>~o)G(O,
-
P’;T,FL)
I 0
(5.11)
p’,poWs(b,l*‘;w).
The function matrix G,(O, - EL.~;T,~)describes the escape of radiation through the surface at r = 0 of the finite slab. A corresponding equation for the escape function G,(b,Fo;T,p) can be derived from Eq. (5.11) by means of Eq. (3.7). Equations equivalent to Eq. (5.11) were first derived by SobolevgX” within the concept of quantum escape probability introduced by this author into radiative transfer theory. Comparing Eq. (3.5) for the surface Green’s function with its adjoint form (5.10), we obtain the functional relation
I:’
(PO - PW~(~,CL;O,CLO) = ; PO
dCL’Ps(l*.,CL’)G,(?,~‘;O,CLo)
-
I
I
‘& o dCL’Gs(T,tL;O,~‘)Bs(O,CL’,CLo)
I
+2P
I G’G-s(b 0
7, -
v-;O,p’Mb, - E”‘,cLo).
(5.12)
Biorthogonality
and radiative transfer in finite slab atmospheres
129
For scalar anisotropic scattering, this equation has been derived earlier by Yanovitskij.” Taking the limits T = O+ and T = b, we obtain from Eq. (5.12) a system of two nonlinear integral equations for the reflection and transmission matrices that are well known from the principle of invariance approach.7”0X’2 If the kernel functions Bs(O,~,~O) and B,(b, - p,po) are known, the solution of Eq. (5.12) would not be unique. Indeed, it is obvious from the approach outlined by us, that any pair of bilinear combinations of eigenvectors with their adjoints in the form {~S(~,~)e-‘” aST(~,q); cp_,(~, - n) e”-‘“” a?,(~, - n)}, (7 E a) inserted instead of {G,(T); G_,(b - 7)) will be also a solution of Eq. (5.12). Apparently, the conception of biorthogonality as developed by us offers a convenient approach to the nonuniqueness problem for the reflection and transmission matrices. This problem will be considered in a forthcoming paper. 6. CONCLUSIONS
It has been shown that, without complete separation of angular variables, a concept of biorthogonality can be developed in order to treat transfer problems of polarized light in finite slab atmospheres. The appropriate adjoints appear as eigenmode expansion coefficients of the surface Green’s function matrix. Explicit expressions of the regular and singular adjoints have been derived in terms of the diffuse reflection or the transmission matrices. The extension of the theory to the treatment of degenerate discrete eigenvalues has been demonstrated explicitly for the case of conservative scattering. A relation between the biorthogonality approach and the singular integral equation method has been established, as well as the relation to an adjoint half-range form of the transfer equation that is valid for the surface Green’s function matrix. This equation is fundamental to the well known quantum escape probability approach. Half-range forms of the characteristic equation for Case’s eigenvectors and their adjoints have been obtained, where the source function matrix of the slab albedo problem, taken at the surfaces of the slab, appear as kernel functions of the half-range integral operators. As a consequence, any solution of a transfer problem in a finite slab can be expressed in terms of the two function matrices B(O,~JL~), B(b, - p,po). For the surface Green’s function matrix, a functional equation has been derived, from which emerge the well known nonlinear integral equations for the reflection and transmission matrices by suitable specialisations of the parameters. It is apparent from our analysis that one cannot immediately expect to obtain from the biorthogonality conception new effective methods for practical computations. However, the use of biorthogonality may contribute to a better understanding of linear and nonlinear functional equations in radiative transfer theory. This may be helpful for the analysis of theoretical problems, as well as for the choice of practical solution strategies in radiative transfer. REFERENCES I. 2. 3. 4. 5. 6.
7. 8. 9.
10. I I. I?.
K. M. Case, Ann. Phys. (N.Y.) 9, I (1960). K. M. Case and P. F. Zweifel, Linear Transport Theory. Addison-Wesley, Reading, Mass. (1967). J. Mika. Nucl. Sci. Eag111: Il. 415 (1961). H. Domke, JQSRT 15. 669 (197.0. N. J. McCormick and 1. Kuscer. J. Math. Phys. 7, 2036 (1966). A. G. Gibbs and R. Seto. J. Mafh. Phps. 19, 2591 (1978). S. Chandrasekhar, Radiatire Transfer. Oxford University Press, London (1950). H. Domke, Astrm. Nachrichtetl 299. 95 (1978). V. V. Sobolev, Asfran. Sh. 28, 3S5 (1951). V. V. Sobolev, A Treatise au Radiative Transfer. Van Nostrand, Princeton, New Jersey (1963). E. G. Yanovitskij. Asfrm. Sh. 56, 833 (1979). J. E. Hansen and L. D. Travis, Spare Sci. Rev. 16. 527 (1974).
J. Quant. Speclrosc. Rodiot. Transfer Vol. 30, No. 2, pp 119-129, 1983 Pri:ded m Great Britain
BIORTHOGONALITY AND RADIATIVE TRANSFER IN FINITE SLAB ATMOSPHERES H. DOMKE Zentralinstitut
fiir Astrophysik,
DDR-15 Potsdam, DDR
(Received 18 October 1982)
Abstractlt is shown that the exact solution of transfer problems of polarized light in finite slab atmospheres can be obtained from an eigenmode expansion, if there is a known set of adjoints defined appropriately to treat two-point, half-range boundary-value problems. The adjoints must obey a half-range biorthogonality relation. The adjoints are obtained in terms of Case’s eigenvectors and the reflection or the transmission matrices. Half-range characteristic equations for the eigenvectors and their adjoints are derived, where the kernel functions of the integral operators are given by the boundary values of the source function matrix of the slab albedo problem. Spectral formulae are obtained for the surface Green’s functions. A relationship is noted between the biorthogonality concept and some half-range forms of the transfer equation for the surface Green’s functions and their adjoints. Linear and nonlinear functional equations that are well known from an invariance approach, are derived from a new point of view. The biorthogonality concept offers the opportunity for a better understanding of mathematical structures and the nonuniqueness problem for solutions of such functional equations.
I. INTRODUCTION
The eigenfunction expansion method introduced in the linear transport theory by Case’,’ offers a suitable way to treat radiative transfer problems in slab atmospheres. Complete sets of eigenmodes of the transfer equation can be found explicitly for very general models of anisotropic scattering,3 including polarization.4 If the eigenmodes are known, the solution of any radiative transfer problem amounts to the determination of the expansion coefficients by means of the boundary conditions. Typical boundary conditions for finite slab atmospheres prescribe the radiation field within the half range of incident directions at the boundaries T = 0 and T = b. The computation of eigenmode expansion coefficients from these boundary conditions can be performed by means of appropriately defined adjoints, which have to obey some biorthogonality relations on the halfrange of the angular variables. The concept of biorthogonality has been developed by McCormick and Kuscer’ for scalar radiative transfer in semi-infinite media. More recently, this work has been generalized to finite slabs by Gibbs and Seto.6 The concept of biorthogonality for finite slab atmospheres will be reconsidered and extended to polarization. It will be shown that the concept of biorthogonality can be developed in a rather genera1 form without any reference to the separation of angular variables. The adjoints of the eigenmodes obeying an appropriate half-range biorthogonality relation will be derived in terms of Case’s eigenmodes and the reflection or the transmission matrices of the finite slab atmosphere. An alternative way to define the adjoints will be noted and is based on the half-range form of the characteristic equation. The kernel functions appearing in the half-range integral operators have a well defined physical meaning as the boundary values of the source function matrix appropriate to the slab-albedo problem. For isotropic scattering, these kernel functions reduce to Chandrasekhar’s X- and Y-functions.7 The eigenfunction vectors and their adjoints will be utilized in order to derive the surface Green’s function matrices in spectral form. Although the spectral formulae quickly lose any computational advantage with increasing complexity of the phase matrix, they clearly reveal the mathematical structures of the solutions for transfer problems in finite slab atmospheres. They can be employed in order to derive linear and nonlinear functional equations, as well as linear singular integral equations, from a unified point of view. II9
120
H.DOMKE EXPANSIONS
2. EIGENMODE
The transfer of polarised radiation in a macroscopically described by means of independent transfer equations
pmw) -II
dr
-
UT,P) +;
isotropic
,d~‘Ps(w’M~,~‘) + B,‘0’h4 I_+I
slab medium
can be
s = 0, f 1,k 2,. . . ,
(2.1)
for each of the azimuthal Fourier components of the Stokes vector. The vectors I,(T,~) depend on the optical depth T and the cosine p of the polar angle. In the transport equation (2.1), the single scattering albedo is denoted by o, pS&,p’) is the azimuthal Fourier component of the phase matrix, and B,‘0’(r,p) describes primary sources. For a finite slab atmosphere of optical thickness b, the solution of the transfer equation has to be subjected to the half-range boundary conditions UOJJ) = Cl,(P),
CLE [O,ll, (2.2)
L(b,-/4=%b-4,
CLEW,~I,
which describe the incident radiation at the upper and lower boundaries of the slab. Eigenmodes are defined as independent solutions i,(r,p) of the homogeneous equation. For eigensolutions of the form iAr,~;;rl) = e-+W~,~), the eigenvector
(p&q)
obeys the characteristic
(77- CL)(P,(CL,~)) = ;q The eigenvectors corresponding written in the form4
(P&r))
I
(2.3)
equation
(2.4)
_;’ d~‘ps(~+‘)4Q~‘,n).
to the continuous
= ;VlJS(WJ)i(r,
transfer
part of the spectrum
- P) + A(+%
n E [- 1, + 11 can be
(2.5)
- q),
where
(2.6)
A(T)) =; [A(TJ+ i0) + h(TJ - iO)l,
A(z) = E -;
The characteristic equation
m-‘(z)
function
I
iI
-1
matrix
dwsbks(w)l(z - PL), ZE [-- 1,+ 11. +&,a)
is obtained
as the solution
(2.7)
of the integral
There is some arbitrariness in the choice of the matrix T,(P) appearing in Eqs. (2.7) and (2.8). One way to choose this matrix has been described earlier.4 Discrete eigenvalues ~i6f [- l,+ l] can be found as the roots of the transcendent equation det h(vi) = 0,
ail [- 1, + 11, i = + l,? 2, . . .
(2.9)
Biorthogonality
and radiative transfer in finite slab atmospheres
121
The corresponding regular eigensolutions for nondegenerated eigenvalues have the form (2.3) with (2.10) where the vectors M,’ are the nonzero solution vectors of the algebraic equations h(7)i)M,’ = 0, i = * 1, ? 2,. . . .
(2.11)
For macroscopically isotropic media, the following symmetry properties hold:4 P&W’) = P-J-
CL,- IL’)=
PsT(E”‘,lr)
(2.12)
and
d/477) = cp-A-p, - 771,
(2.13)
if the Stokes parameters are given in CP-representation. The eigenvectors obey the full-range orthogonality relation
(2.14)
the normalization integrals are
For multiple roots of Eq. (2.9), the eigensolutions ik(T,P;q) (k = 1,2,. . . ,q) may belong to a higher dimensional irreducible representation of the translation group. Then, if Z’ik(T)is the representation matrix of the translation operator, ii(7 + a,/J,;qi) = k=l
Tjk(r)ik(W ;‘Vi),
(2.16)
where q is the multiplicity of the eigenvalue qi. As an example, we consider the degenerated eigenvalue I/~~, = 0 occurring for the zeroth azimuthal Fourier component in the case of conservative scattering w = 1. The two independent eigensolutions of the transfer equation are easily found to be 1 ir(7,1*;w) = 2 iO,
i-r(r+;m) = k (7i0- Pi,),
(2.17)
where i,, describes unpolarized light of unit intensity. The vector i, is defined as i, = [l - x,/3]-‘iO,
where xi/3 is the mean cosine of the single scattering angle. The representation matrix of the translation group corresponding to the basis (2.17) is given by {Tik} =
(t y), (i,k = * 1).
(2.18)
122
H.DOMKE
We suppose that we have found a complete set of independent eigensolutions. Then the radiation field of a source-free, finite-slab atmosphere can be written as a linear superposition of eigenmodes, viz.,
The spectral integral includes integration over the continuous spectrum n E [- 1, + 11, as well as summation over all eigenmodes corresponding to discrete eigenvalues qicE [- 1, + 11. The transfer problem then amounts to the determination of the expansion coefficients from the two-point, half-range boundary conditions (2.2). The concept of biorthogonality offers a systematic approach to performing this task. 3,BIORTHOGONALITY The finite slab boundary conditions (2.2) prescribe the radiation field on half-ranges at two different optical levels. The full-range orthogonality relations (2.14) are not applicable for computing the spectral expansion coefficients A(n) for this type of boundary conditions. However, we can solve this problem by using properly defined half-range adjoints. We introduce two sets of adjoints, a,(k,q) (n EC; p >O) and aT(~,q) (7 E a; p >O), defined on the positive half-range of the variable p such that the expansion coefficients may be written as linear functionals of the incident radiation in the form
(3.1) Taking into account the boundary condition (2.2), but also the eigenmode expansion find that Eq. (3.1) is consistent with the following biorthogonality relation: Eb(n -n’),
&j, 0,
o dCLCL[a,*(CL,B)i,(O,~;~‘) + ~$Thdi.v(~,- ~;q')l =
I’
(2.19) we
11; , 7 = rli,B = 9j; rl# 77’.
rl,g’E ]- 1, +
(3.2) If the eigensolutions
are of the form (2.3), it is easy to show from Eqs. (2.13) and (3.2) that
aS(~,q)= a-,(~, - q) F. The biorthogonality
relation (3.2) can then be written in the form
I’ 0
(3.3)
dw[asT(p,rlhhv’I + a%, - q)eh(“q-““‘)(ps(_ ~,r)‘)l=
E6(q-q’), &jt 0,
q,q’E[-l,+ll; 7)=7)iPl rl+
I
=r)j;
4.
(3.4) We consider now the eigenmode expansion defined as the solution of the transfer equation
of the Green’s function
matrix of a finite slab
+I
Es(r - ro)s(L‘ - PO), P&W ;~o,po)= - G(T,P;TO,PO)+; I_I dl.L’ps(CL,CL’)Gs(T,CLI;T0;CLO)+ (3.5) subject to the boundary
conditions
Gs(O+;TorPo) = 0,
F,/JO
E [%]I,
P,PO
E
(3.6)
Gs(b,-
CL;To+o) = 0,
[%]I.
Biorthogonality
and radiative transfer in finite slab atmospheres
123
By means of Eq. (3.5), the boundary conditions (3.6) and the symmetry properties (2.12) for the phase matrix, it may be shown that G~(w;To,Po)
=
G-,(b - T,- k;b - TO,- /JO).
(3.7)
Moreover, the reciprocity relation
WW;TO,PO) = G?s(~o,
- PO;T, -
IL)
(3.8)
can be derived in the same manner as for the scalar case outlined in the monography of Case and Zweifel.2 In particular, the surface Green’s function matrix G,(T,~;O,~~) (T > 0) appears as the solution of the homogeneous transfer equation subject to the boundary conditions
G,(O+,P;O,CLO) = ; ESb - PO), /APO E LOAl, (3.9) G,(b, - P;&PO) = 0,
CLIP0 E [OJI.
On the other hand, the surface Green’s function matrix is related to the albedo and transmission matrices by
P~(/-wo) = ; G,(O,- ~;O,po), /.woE [O,ll
(3.10)
and ~sbw~)
=
i
G,(b,p;O,po) - E;el -b’NNp -cLo) 3 I
(3.11)
respectively. Utilizing now Eqs. (2.19), (3.1), and (3.9) we find the eigenmode expansion of the surface Green’s function matrix in the form
G(T,P;O,P~) =
10 d7)i,(T,IL;rl)a,T(CLo,rl).
(3.12)
The same reasoning can be applied in order to derive the eigenmode expansion for the other surface Green’s function, Gs(7,p;b, - po), which obeys the homogeneous transfer equation as well as the boundary conditions
G,(O,cL;b, - PO)= 0,
EL,CLO E
tO,ll, (3.13)
We obtain the eigenmode expansion formula (3.14) Using Eqs. (3.12) and (3.14) for the surface Green’s f3nction matrices, we can easily derive also a spectral representation for the complete Green’s function matrix. For this purpose, we utilize the relation GS(~,~;~o,p,J = Gs’r’(~,p;~o,po)-
-
I 0
I,’
dCL’CL’G,(T,CL;O,CL’)G,(OC)(O,CL’;70,CLO)
(3.15)
I
dl*‘I*‘G,(7+;h, - cL’)G,‘“‘(b,-
CL’;‘TO,IL~),
124
H. DOMKE
in connection with the well known eigenmode expansion formula for the infinite medium Green’s function matrix GS(m’.2,4 Equation (3.15) can be derived formally by means of Eqs. (3.5) and (3.6) but also heuristically from invariant imbedding arguments. 4.
THE ADJOINTS
We now investigate in greater detail the adjoints which we have introduced formally. Applying the full-range orthogonality (2.14) to the spectral formula (3.12) for the surface Green’s function, we find, after taking into account also Eqs. (3.10) and (3.1 l),
(4.la)
and
These formulae express the adjoints explicitly in terms Jf the albedo matrix or, alternatively, in terms of the transmission matrix. The adjoints of the positive continuous eigenspectrum are singular function vectors. Taking into account the explicit form for Case’s eigenvectors (2.5) we find from Eq. (4. la) after some partial fractional analysis,
with
%T(P,rl) = 9sThJ)+ 2rl
I
I’cW!?-$,
~)P,(P’>~1
0
I
-2/J o WhT(- P’,~>P&‘~P)(P’+ +)/(p’ + 77). In particular,
(4.3)
it follows from Eq. (4.3) that (4.4)
We now illustrate the use of Eq. (4.3) for the case of isotropic scattering,
$
z
1
&I
p)
=
$9 X(P’)X(cL) - Y(p’)Y@)
4
CL’SCL
where’ (4.5)
We obtain (4.6)
A more general form of Eq. (4.6) appropriate to scalar anisotropic scattering has been derived by Gibbs and Seto.” Equations (4.2) and (4.3) are well suited to performing a reduction of the adjoints to functions depending only on one variable. Although this is possible for any scattering model, a complete separation of variables would have practical advantages only for relatively simple phase matrices (e.g., Rayleigh or molecular scattering). The adjoints of the negative part of the eigenspectrum are regular function vectors. A
Biorthogonality
convenient
and radiative transfer in finite slab atmospheres
125
formula can be found from Eq. (4.lb) by means of partial fractional
u,~‘(P, - 17)= N,L’(-17)e-h/vwrl /(7)+ /.L) PT,(/&q), 77 E Khll,
I
2
analysis,
E PA117
P
i.e.,
(4.7)
with
(4.8)
In particular,
for isotropic scattering, a(p!
where
p)
=
Y(I*‘)X(/J)- X(cL’)Y(cL)
g
4
(4.9)
PI-P
we get
(4.10) For optically very thick layers, b B 1, we obtain the following asymptotic adjoints of the zeroth azimuthal Fourier component (s = 0):
relations for the
Nln e-W1
ooT(Wf) = unT(Wl) +
N,,-‘(n)(l_ n2 e-2b/q,)
‘d’l)UoT(Po,~),
77 >
0,
(4.11)
and a,Jr(P, _ ,,) = _ N,,~1(7))e-h”ill+lirl’)
where the vector
~~(7)
(l _
N,
n2e-2b/,,,)
Ko(17)~oT(PoJh)~
7 ‘0,
(4.12)
is given by
(4.13) In order to derive Eqs. (4.11) and (4.12), we have employed well known asymptotics for the reflection and transmission matrices.8 Equations (4.la,b) can be easily generalized to the case of multiple roots of Eq. (2.9). Taking into account the full-range orthogonality of the eigenmodes for different eigenvalues, we find from Eq. (3.12)
(4.14)
(4.15) where the coefficients
Ljk(7) are
L,k(T)
=
I
+I
-I
d/4iT(0,p;V)h(T,p ;T).
(4.16)
H. DOMKE
126
As an example, we consider the pair of degenerated equation. For conservative scattering. We obtain L&(T) = -
eigenvalues
(y i) i (ilT . io),
l/n*, = 0 of the transport
(4.17)
i,k = + 1.
Denoting no = (i,%)/6, we get from Eqs. (4.14), (4.15), and (4.17) the adjoints appropriate eigenmodes (2.1’1) in the form
to the
(4.18a)
(4.18b)
as functionals
of the albedo matrix and I aT,(r_~~,m)= &-
aT1(po,“) = --
1
[
(FoilT + bioT) embiN+ 2
1
I0
dF&ilT
+ bioT)&&
-b/m + 2 I
2nl
,
in terms of the transmission matrix. For optically thick atmospheres, b > 1, we employ the asymptotic ting the formulae of Ref. 8 by a factor of l/2)
1 ,
(4.19a)
(4.19b)
formulae* (after correc-
(4.20)
and p(b
(T -
=
m) -
CT,
(4.21)
with I
l.42 =
From Eqs. (4.19), we find the asymptotic
In order to obtain the adjoints can not employ Eq. (3.3). Applying the expansion formula (3.14) for albedo and transmission matrices. the form
I dw2ilTuW). o
(4.22)
relations
a$,(~,m) occurring in the biorthogonality relation (3.2), we the full-range orthogonality properties of the eigenmodes to G(T,p;b,- pa), it is easy to derive a$,(~,=) in terms of the The result can be expressed by means of the adjoints akl in
and
a~T(~0,4 =We note a relation between the biorthogonality
aT,(~o,~).
(4.25)
concept and the method of singular integral
Biorthogonality and radiative transfer in finite slab atmospheres
127
equations. Combining Eqs. (4,la) and (4.lb), we obtain
=
[e-b’rl- e-b’@qcp,T(po,?-j), q E (T.
(4.26)
Interpreting Eq. (4.26) for the continuous half spectra n E [O,l] and n E [- l,O] and taking into account Eq. (2.9, we obtain a system of two inhomogeneous linear singular integral equations for the reflection and the transmission matrices. Interpreted for the discrete eigenvalues, Eq. (4.26) yields a set of linear regular constraints. Moreover, it is obvious from Eq. (3.4) that the pair of adjoint vectors {a&,nJ, a_& -TJJ eb’*i}obeys the corresponding system of homogeneous linear singular integral equations. 5. HALF-RANGE
FORMS OF THE TRANSFER
EQUATIONS
We now show that the concept of biorthogonality can be related to some half-range formulations of the transfer equation. Let i,(r,~) denote any solution of the homogeneous transfer equation. Taking into account Eq. (3.5) for the Green’s function, but also the symmetry relation (2.12), we derive the functional relation
$-q_‘I dppG?dT', I
I
- p;~o,po)is(~+ +,p) = - i,(r + T~,~~)S(T’-
With r. = 0, we obtain after integrating Eq. (5.1) on the interval
is(T,- I*01=
I
I
0
(5.1)
TV).
I
7’ E
[O’,b],
1
4vG(O, - Po%L)~s(T4‘) + o dCLdLUwo;O,ph(b
+ 7, -
EL),
(5.2)
Here, we have employed also the symmetry relations (3.7) and (3.8) which hold for the Green’s function matrix. Equation (5.2) can be derived also heuristically by means of invariant imbedding arguments. The functional relation (5.2) can be employed in order to rewrite Eq. (2.1) in a half-range form. For this purpose, we introduce the function matrix (5.3) which represents the source function matrix of the finite-slab albedo problem. Inserting Eq. (5.2) in the transfer equation, we get
l-+shd=-k(W)+2
I
I dCL’B,(O,CL,CL’)is(T,CLI) 0
+ 2 o’dp’B_Jb, I
/L,/L')i,(T +
b, - /J’).
(5.4)
This is a half-range functional integro-differential equation valid for any solution of the original homogeneous transfer equation. As an application, let us consider eigensolutions of the form (2.3). From Eq. (5.4) we find the following characteristic equation for the eigenvectors: I (P&V) = 2 1’ dCL’B,(O,CLlr.L’)(PS(~‘,~) + 2 emb”’ dp’B_,(b, - ~,~‘)(p,(- ~‘,a). 0
(5.5)
By means of purely formal arguments, it is easy to find an adjoint form of Eq. (5.5) consistent
128
H. DOMKE
with the biorthogonality
relation (3.4) i.e.,
(1-k)~sT(/.vt)=
2 ’ d I_L‘as T(~‘,n)B,(O,~‘,~)
dp’a’&‘,
- 2 eb”
- n)B,(b, - IL’+).
(5.6)
Equations (5.5) and (5.6) are remarkable. They show that the eigenvectors, as well as their adjoints, can be reduced to the kernel functions B,(O,p’,p) and B,(b, - p’,~). In other words, the solution of any transfer problem in finite slab media can be expressed as a functional of the surface values of the source function matrix for the slab albedo problem. Equation (5.6) can be interpreted as the eigenequation for eigenmodes
is+(7,cL;q) = asTh4q) e+
(5.7)
of the following transfer equation:
-
2
Io’
dcL'i+,(b
- T,p’; - q)B,(b,
- F’,p).
(5.8)
Moreover, the spectral form (3.12) of the surface Green’s function with eigenmodes (2.3) can be rewritten in terms of the adjoint eigenmodes (5.7) as
of the form
(5.9) It is obvious from Eqs. (5.8) and (5.9) that the surface Green’s function half-range transport equation:
PO-&(W ;O,PO) = - G(~,cL;O,PO) + 2I
obeys the following
I
dCC’G,(T,CL;O,CL’)Bs(O,CL’,CLo)
0
- 2 o’ dp’G-,(b I Taking into account the symmetry
(5.10)
- 7, - p;O+‘)B,(b,
- CL’+,,), 7 > 0.
relations (3.7) and (3.8), we rewrite Eq. (5.10) in the form I
~o-$dO,
-
PO;T,P)
=
-
GsKJ,
-
2
-
PO;~>P)
+
Io’G’B %, -
2
G’B
%W>~o)G(O,
-
P’;T,FL)
I 0
(5.11)
p’,poWs(b,l*‘;w).
The function matrix G,(O, - EL.~;T,~)describes the escape of radiation through the surface at r = 0 of the finite slab. A corresponding equation for the escape function G,(b,Fo;T,p) can be derived from Eq. (5.11) by means of Eq. (3.7). Equations equivalent to Eq. (5.11) were first derived by SobolevgX” within the concept of quantum escape probability introduced by this author into radiative transfer theory. Comparing Eq. (3.5) for the surface Green’s function with its adjoint form (5.10), we obtain the functional relation
I:’
(PO - PW~(~,CL;O,CLO) = ; PO
dCL’Ps(l*.,CL’)G,(?,~‘;O,CLo)
-
I
I
‘& o dCL’Gs(T,tL;O,~‘)Bs(O,CL’,CLo)
I
+2P
I G’G-s(b 0
7, -
v-;O,p’Mb, - E”‘,cLo).
(5.12)
Biorthogonality
and radiative transfer in finite slab atmospheres
129
For scalar anisotropic scattering, this equation has been derived earlier by Yanovitskij.” Taking the limits T = O+ and T = b, we obtain from Eq. (5.12) a system of two nonlinear integral equations for the reflection and transmission matrices that are well known from the principle of invariance approach.7”0X’2 If the kernel functions Bs(O,~,~O) and B,(b, - p,po) are known, the solution of Eq. (5.12) would not be unique. Indeed, it is obvious from the approach outlined by us, that any pair of bilinear combinations of eigenvectors with their adjoints in the form {~S(~,~)e-‘” aST(~,q); cp_,(~, - n) e”-‘“” a?,(~, - n)}, (7 E a) inserted instead of {G,(T); G_,(b - 7)) will be also a solution of Eq. (5.12). Apparently, the conception of biorthogonality as developed by us offers a convenient approach to the nonuniqueness problem for the reflection and transmission matrices. This problem will be considered in a forthcoming paper. 6. CONCLUSIONS
It has been shown that, without complete separation of angular variables, a concept of biorthogonality can be developed in order to treat transfer problems of polarized light in finite slab atmospheres. The appropriate adjoints appear as eigenmode expansion coefficients of the surface Green’s function matrix. Explicit expressions of the regular and singular adjoints have been derived in terms of the diffuse reflection or the transmission matrices. The extension of the theory to the treatment of degenerate discrete eigenvalues has been demonstrated explicitly for the case of conservative scattering. A relation between the biorthogonality approach and the singular integral equation method has been established, as well as the relation to an adjoint half-range form of the transfer equation that is valid for the surface Green’s function matrix. This equation is fundamental to the well known quantum escape probability approach. Half-range forms of the characteristic equation for Case’s eigenvectors and their adjoints have been obtained, where the source function matrix of the slab albedo problem, taken at the surfaces of the slab, appear as kernel functions of the half-range integral operators. As a consequence, any solution of a transfer problem in a finite slab can be expressed in terms of the two function matrices B(O,~JL~), B(b, - p,po). For the surface Green’s function matrix, a functional equation has been derived, from which emerge the well known nonlinear integral equations for the reflection and transmission matrices by suitable specialisations of the parameters. It is apparent from our analysis that one cannot immediately expect to obtain from the biorthogonality conception new effective methods for practical computations. However, the use of biorthogonality may contribute to a better understanding of linear and nonlinear functional equations in radiative transfer theory. This may be helpful for the analysis of theoretical problems, as well as for the choice of practical solution strategies in radiative transfer. REFERENCES I. 2. 3. 4. 5. 6.
7. 8. 9.
10. I I. I?.
K. M. Case, Ann. Phys. (N.Y.) 9, I (1960). K. M. Case and P. F. Zweifel, Linear Transport Theory. Addison-Wesley, Reading, Mass. (1967). J. Mika. Nucl. Sci. Eag111: Il. 415 (1961). H. Domke, JQSRT 15. 669 (197.0. N. J. McCormick and 1. Kuscer. J. Math. Phys. 7, 2036 (1966). A. G. Gibbs and R. Seto. J. Mafh. Phps. 19, 2591 (1978). S. Chandrasekhar, Radiatire Transfer. Oxford University Press, London (1950). H. Domke, Astrm. Nachrichtetl 299. 95 (1978). V. V. Sobolev, Asfran. Sh. 28, 3S5 (1951). V. V. Sobolev, A Treatise au Radiative Transfer. Van Nostrand, Princeton, New Jersey (1963). E. G. Yanovitskij. Asfrm. Sh. 56, 833 (1979). J. E. Hansen and L. D. Travis, Spare Sci. Rev. 16. 527 (1974).