Information Technology Reference
In-Depth Information
3
D
where
z
)] is an operator of the three-dimensional direct prob-
lem that calculates the apparent resistivity
brd
[
x
m
,
y
m
,
T
,
˜
(
x
,
y
,
brd
for a period
T
and a given electric
conductivity ˜
(
x
,
y
,
z
).
12.3.2 Using the
S
-Method
The foregoing technique of one-dimensional successive inversions is convenient if
the number of geoelectric layers does not change within the area under consider-
ation. But in the case that the sedimentary strata contain pinching-out layers, we
have to solve the 1D problems with the maximum number of layers. Evidently this
impairs the solution stability, because false thin layers may appear. In that event
we can take advantage of the
S
method which allows the Earth stratification to be
made at the last stage of the inversion (Dmitriev, 1987).
The
S
−
−
method consists in solving an unstable nonlinear 1D inverse problem in
two steps.
The first step reduces to determining conductance distributions
z
S
m
(
z
)
=
S
(
x
m
,
y
m
,
z
)
=
m
(
z
)
dz
m
=
1
,
2
, ...
M
(12
.
10)
0
from ˜
T
). This is a stable nonlinear problem (recall the Dmitriev theo-
rem of stability of the
S
- distribution). The second step is that of determining
brd
(
x
m
,
y
m
,
m
(
z
)
from
S
m
(
z
). This is an unstable linear problem. Thus, instead of solving a single
complicated problem, we successively solve two simpler problems.
The conductance distributions,
S
m
(
z
), are evaluated by minimizing the Tikhonov
functional
z
[
S
m
(
z
)]
=
I[
S
m
(
z
)]
+
[
S
m
(
z
)]
,
S
m
(
z
)
=
m
(
z
)
dz
m
=
1
,
2
, ...
M
,
0
(12
.
11)
where the misfit functional is
=
˜
,
m
(
z
)]
2
R
1
A
[
x
m
,
I[
S
m
(z)]
brd
(
x
m
,
y
m
,
T
)
−
y
m
,
T
m
=
1
,
2
, ...
M
and the stabilizing functional is
S
m
(
z
)
S
m
−
1
(
z
)
2
L
2
[
S
m
(
z
)]
=
−
m
=
1
,
2
, ...
M
.
m
(
z
) is a piecewise-constant function digitized on a large number of steps.
When integrating unstable
Here
S
m
(
z
). This stage in the 1D
m
(
z
), we get stable
