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In-Depth Information
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