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(1)
(
x
˜
,
y
,
z
)
q
1
D
3
D
1
A
[
x
m
,
=
{
˜
brd
(
x
m
,
y
m
,
T
)
−
brd
[
x
m
,
y
m
,
T
,
˜
(
x
,
y
,
z
)]
+
y
m
,
T
,
m
(
z
)]
˜
}
m
=
1
,
2
, ...
M
,
(12
.
16)
1
D
A
are operators of the quasi-one-dimensional inversion,
three-dimensional inversion, and local one-dimensional inversion respectively.
At the second iteration
q
1
D
,
3D
brd
,
where
and
˜
(2)
(
x
,
y
,
z
)
=
q
1
D
{
brd
(
x
m
,
y
m
,
−
3
D
brd
[
x
m
,
y
m
,
,
(1)
(
x
,
,
+
1
A
[
x
m
,
y
m
,
,
(1
m
(
z
)]
}
˜
T
)
T
˜
y
z
)]
T
˜
=
,
, ...
.
m
1
2
M
(12
.
17)
At the
k
th iteration
(
k
)
(
x
˜
,
y
,
z
)
3
D
=
q
1
D
{
brd
(
x
m
,
˜
y
m
,
T
)
−
brd
[
x
m
,
y
m
,
T
,
˜
(
k
−
1)
(
x
,
y
,
z
)]
+
1
A
[
x
m
,
y
m
,
T
,
˜
(
k
−
1)
(
z
)]
}
m
m
=
1
,
2
, ...
M
.
(12
.
18)
Its misfit is
M
T
)
2
R
.
(
k
)
(
x
3
D
brd
(
k
)
(
x
,
,
=
[
x
m
,
y
m
,
,
,
,
−
brd
(
x
m
,
y
m
,
.
I[ ˜
y
z
)]
T
˜
y
z
)]
˜
(12
19)
m
=
1
3
D
q
1
D
1
D
A
.
It means that we near a solution where apparent resistivities at each observation site
coincide with their locally normal values.
The quasi-one-dimensional inversion can be considered as an analog of the
smoothing inversion. Both the methods define models with a smooth conductivity
distribution. The difference is that in the smoothing inversion stabilizer (12.1) is
minimized over the entire inhomogeneity domain (integral smoothing), while in the
quasi-one-dimensional inversion we use stabilizer (12.7) and minimize the differ-
ence between conductivities at neighboring sites (local smoothing).
Convergence of this iterative procedure for sufficiently slow horizontal change
Each iteration brings
brd
closer to ˜
brd
. Hence, ˜
(
x
,
y
,
z
) approaches
{
}
in
z
) has been proved by Barashkov (1983). In actual practice, we usually
observe rather fast convergence of iterative procedure (3-5 iterations).
From the practical standpoint, the quasi-one-dimensional inversion is very con-
venient, since the multi-dimensional direct problem is solved only several times.
Figure 12.2 presents examples of quasi-one-dimensional inversion of the transverse
apparent-resistivity
(
x
,
y
,
⊥
−
curves calculated for a model with a two-dimensional
graben and horst. Here the second and the third iterations approximate fairly well
the topograhy of a resistive basement.
