Information Technology Reference
In-Depth Information
The parallel inversion summarizes all inversion criteria related to various res-
ponse functions. In solving a 2D inverse problem, the parallel inversion of
M
response functions reduces to the minimization of the Tikhonov's functional
containing the total misfit. For instanse,
M
inf
p
{
(
y
,
z
)
} =
inf
p
g
m
I
m
{
(
y
,
z
)
} + {
(
y
,
z
)
}
,
(12
.
20)
m
=
1
where
}
2
R
F
m
−
I
m
{
(
y
,
z
)
} =
F
m
{
y
,
z
=
0
,
,
(
y
,
z
)
2
{
(
y
,
z
)
} =
(
y
,
z
)
−
0
(
y
,
z
)
L
2
.
Here
p
is the vector of the sought-for parameters,
F
m
is the
m
th response function,
F
m
is an operator of the forward problem that calculates the
m
th response function
for a given electric conductivity
z
), I
m
is the misfit of the
m
th response function,
g
m
is a weight representing the significance of the
m
th response function,
(
y
,
0
(
y
,
z
)
is a hypothetical model.
At first glance, the parallel inversion seems to be the most attractive because it
incorporates simultaneously all response functions in use and considerably simpli-
fies the work of the geophysicist. One can even say that the parallel inversion opens
the way to the automatic inversion. Let us consider the situation more attentively.
If various response functions
F
m
have the same sensitivity to all parameters
p
(
p
1
,
) of the geoelectric medium and the same immunity to near-surface
distortions, their parallel inversion is not very advantageous because one, the most
reliably determined (the least susceptible to geoelectric noise), response function is
sufficient for a comprehensive inversion.
The use of several response functions can extend the inversion potentials if these
functions differ significantly in their sensitivity to various parameters of the geo-
electric medium and in their immunity to near-surface distortions. However, in this
case their joint inversion may become conflicting. Really the different functions
can bother each other, because they impose different constraints on the geoelectric
medium and demand different criteria for mimimizing the model misfit. Moreover
such a joint inversion increases the risk of falling into a local minimum.
True enough, it is possible that in some cases a fortunate choice of weights allows
one to construct a self-consistent meaningful model with a sufficiently small overall
misfit. However, the adequate selection of such weights is itself a complex problem
that is hard to solve rationally.
Apparently, the SPI method (successive partial inversions) is the best approach
to the solution of a multicriterion inverse problem.
Let a response function
F
m
be the most sensitive to the vector of parameters
p
(
m
)
. Then, the partial
m
th inversion in the multicriterion two-dimensonal problem
calls for the minimization of the following Tikhonov's functional on the set of the
parameters
p
(
m
)
, with the parameters
p
p
2
,...
−
p
(
m
)
being fixed:
