tion (R) and transmission (T) functions by one-dimensional 
R- and T-functions makes tractable the numerical computation. 
Let the observed diffuse radiance at top in the direction 
ft be denoted by I(z lf x,y;ft ,ft 0 ), whose optical thickness is 
x and the direction of incidence ('or ref lection) is denoted by 
fto = (arccosu, ()> 0 ) (or fi=(arccosv,^) )). In the above u(or v) is 
the zenith angle from the normal at top and 4> 0 (or <|>) is the 
azimuthal angle. Let the one-dimensional reflection and tran 
smission functions of the free atmosphere be denoted by R(x; 
ft ,ft 0 ) and T(t ; ft,fto), respectively, and furthermore, let 
the R- and T-functions for the incident radiation at bottom 
from below be denoted by R*( x;ft ,ft 0 ) and T*( X; ft,ft 0 ), al 
lowing for the polarity(cf.Ueno 1960). Additionally, let the 
ground albedo,i.e., diffuse reflectance, at horizontal rec 
tangular coordinates (x,y) of diffuse reflector be denoted 
by A(x,y),whose mean value of the background is designated 
by A(x,y). On making use of the diffuse radiance at top in 
the approximation scheme, the required ground albedo A(x,y) 
takes the form 
A (x, y) = [R° b ( ft, ft 0 ) ~R( T; ft, ft o) -4tt , ^T* ( X ; ft, ft ') U ( ft ' , ft 0 ) V ' dft ' ] 
x[ ¥(A(x,y),X ,ft ,fto)] -1 , (11) 
where . 
R ( ft,ft 0 )=I(z^,x,y;ft ,ft 0 )2TT/uF, (12) 
U( ft, ft 0 ) =A(x,y) [e _T/u +if'^( ft, 1 fte ) v' dft ' , (13) 
f(A(x,y), x , ft, fto) =e~ T (1/v+1/u) +TT _ e _T/v ^ TT D( ft;ft 0 )v'dft', (14) 
D ( ft, ft o ) =T ( x; ft, ft o ) +E ( ft,ft 0 ) e _T//U +Tr"^ ii E ( ft,ft')T(x ;ft',ft 0 ) 
xv' dft ' , (15 ) 
E ( ft, fto) = £ Q (n; ft ,ft 0 ) , (16 ) 
Q (n; ft,ft 0 )= tt ^Q(l; ft, ft ' ) Q (n-1; ft ' , ft 0 ) v ' dft ' , (17 ) 
Q (1; ft, ft o) = tt -, A (x, y) ^ n R* ( T;ft ^'Jv'dft'. (18) 
From physical aspects U-function represents the upwards in 
tensity of radiation diffusely reflected by Lambertian sur 
face, and D-function denotes the downwards intensity of ra 
diation at bottom, where E-function is due to multiple scat 
tering of radiation from below at bottom. Furthmore, f-fun- 
ction indicates the intensity of emergent radiation at top 
due to the direct and diffuse transmission of light diffusely 
reflected at bottom. After minor rearrangements, the reflec 
tion and transmission functions are converted into the scat 
tering and transmission functions, whose initial-value solu 
tions are reduced into the Riccati-type of non-linear inte- 
grodifferential equations being suitable for the numerical 
computation by the high-speed digital computers(cf.Ueno 1960; 
Busbridge 1960;Sobolev 1975). In the case of homogeneous at 
mospheres they are reduced to those given by Chandrasekhar 
(1960). Furthermore it should be mentioned that in the homo 
geneous case Eqs.(ll) ~(18) result in equations given by 
Odell and Weinman(1975). 
The subsequent computation depends on what kind of atmos 
pheric model is assumed. In the present study , the Elter-