rade 
the 
;er- 
lat 
ions. 
of 
‚ion 
  
  
represents a well known problem in estimation ME 1276). 
A suitable set of L observations(dE(r,)+ E(r,));....(0E Et, ) 
together with the corresponding. set of Kernels Clr 2,007" Yu 
G(r; ,z,m(z)), are required for the solution. The existance of a 
unique solution for a given set of surface measurements is depend- 
ant on measurement errors as well as on the corresponding set of 
kernels. There are several possible linear(Rodgers,1976) and non - 
linear (Smith,1970;Chahine,1972) techniques for inversion of the 
atmospheric remote sensing integral equation. We shall consider 
Chahine's nonlinear relaxation method because of its stability 
{Yong and Goulard,1975) and insensitivity to measurement errors, 
Chahine, 1977). The only requirement of this method is that kernels 
have maxima and are suitably related to each other. They must be 
neither completely overlapped nor widely separated(Weinreb and 
Crosby,1972). Fig.(2) shows a set of kernels used for satellite 
sensing of atmospheric microwave emissions at different frequencies 
(Staelin,1969). Comparison of Fig(1) and Fig(2) reveals analogy of 
the two problems. A solution of the resistivity problem could, 
therefore,be obtained assuming weighting functions for a uniform 
resistivity half space as an initial inpyt kernel. The A 
technique could then be used to obtain NE hence /7(z), 
where the suffix denotes the order of the kernel modification. The 
obtained resistivity-depth profile could then be used to modify 
kernels for the next relaxation according to the boundary condition 
in the stratified earth(Telford et 81,1975); 
Jo (z) 
Ga 102m 4(2)) = € 
1-1 
90000 
! G,(r,z,m (2)) (5) 
where n is the order of modification. Relaxations could then be 
terminated when the rms error is less than a prescribed error 
limit. The result is a resistivity profile and its depth investig- 
ation characteristic estimated-in the rms sense- by surface 
observations. 
INFORMATION CONTENT: 
Study and extraction of information content of kernels is 
an important part of the remote sensing technique. According to 
Mateer,1965 and Twomey,1977; the number of useful pieces of infor- 
mation are given by the number of eigenvalues of the covariance 
matrix of kernels which exceed a given value that depends on the 
accuracy with which measurements could be done. In other words, 
eigenanalysis of our covariance matrix 
oo 
Ci; = REO Gr, , 2,m(2)) dz (6) 
Ó 
where s is the final modification index;should give information 
about the number of measurements necessary to retreive the 
resistivity-depth profile,for a given expected measurement accuracy 
It should be noted that the elements of the above covariance matrix 
are obtained from kernels modified by the resistivity profile 
Structure,i.e.actual kernels. The difference between actual and 
model kernels may be large in the presence of large profile varia- 
tions. 
Since the covariance matrix is symmetric,its eigenvalues are 
983