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The inversion of all these curves is performed in the class of 1D layered models.
With regularized optimization, we get
m ( z )]
=
inf
[
m ( z )]
m
=
1
,
2
, ...
M
,
(12
.
4)
where ˜
m ( z )
=
˜
( x m ,
y m ,
z ) is an approximate solution of the 1D inverse problem,
m ( z )] is Tikhonov's functional involving
the misfit functional I and stabilizing functional
is a regularization parameter, and
[
:
[
m ( z )]
=
m ( z )]
+
m ( z )]
=
,
, ...
.
.
I[
[
m
1
2
M
(12
5)
The misfit functional assumes the form
˜
, m ( z )]
2
R
1 A [ x m ,
I[
m (z)]
=
brd ( x m ,
y m ,
T )
y m ,
T
m
=
1
,
2
, ...
M
,
(12
.
6)
1 A [ x m ,
where
, m ( z )] is an operator of the local one-dimensional direct prob-
lem that calculates the apparent resistivity
y m ,
T
A for a period T and a given electrical
conductivity
m ( z ).
The stabilizing functional
[
m ( z )] provides the proximity of the solution ˜
m ( z ),
obtained at a site O m , to the solution ˜
m 1 ( z ), obtained at a neighboring site O m 1 :
2
L 2
= m ( z )
[
m ( z )]
m 1 ( z )
˜
m
=
1
,
2
, ...
M
.
(12
.
7)
z ) within the area
under consideration. The initial normal conductivity distribution ˜
In this way we ensure the slow horizontal changes in
( x
,
y
,
= N ( z )is
taken at the site O 1 that belongs to the boundary C 1 of the normalized area S N
(Fig. 10.1). It can also be taken at any site O m , where a priori information (for
instance, well-log data) provides the reliable determination of ˜
1 ( z )
m ( z ).
Having solved the 1D inverse problems for all sites O m , we find a set of one-
dimensional approximate solutions ˜
m ( z ), m
=
1
,
2
,...
M . Next we synthesize
m ( z )
z ) and accomplish their spline-approximation in x , y with a min-
imum norm of horizontal derivatives of conductivity. So we construct a smoothed
quasi-one-dimensional solution ˜
˜
=
˜
( x m ,
y m ,
z ) of the three-dimensional inverse problem.
An operator of such quasi-one-dimensional inversion is designated as
( x
,
y
,
q 1 D :
q 1 D
˜
( x
,
y
,
z )
=
{
brd ( x m ,
˜
y m ,
T )
}
m
=
1
,
2
, ...
M
.
(12
.
8)
The misfit of the quasi-one-dimensional inversion is
M
T )
2
R ,
3 D
brd
I[ ˜
( x
,
y
,
z )]
=
[ x m ,
y m ,
T
,
˜
( x
,
y
,
z )]
brd ( x m ,
˜
y m ,
(12
.
9)
m = 1
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