Information Technology Reference
In-Depth Information
ant
log
1
Z
brd
=
2
{
Z
brd
(
y
=−
c
0
)
+
Z
brd
(
y
=
c
0
)
}
(10
.
5)
Z
(
Z
⊥
(
Z
(
c
0
)
Z
⊥
(
c
0
)
−
c
0
)
+
−
c
0
)
+
+
=
ant
log
.
4
Using a spline or linear extrapolation, the values of
Z
brd
are taken beyond
the observation profile
P
0
in such a way that the conditions
Z
brd
Z
brd
and
=
Z
brd
y
=
0 be valid at the points
y
=−
c
1
and
y
=
c
1
. The extrapolation
c
0
) by the transition zones
P
t
(
frames the profile
P
0
(
−
c
0
≤
y
≤
−
c
1
<
y
<
−
P
t
(
c
0
<
y
<
c
1
) and the infinite normalized profiles
P
N
(
y
c
0
)
,
≤−
c
1
)
,
P
N
(
y
≥
c
1
)
Z
brd
. On the one-dimensional inversion of the
with the normal impedance
Z
N
≈
Z
brd
we get a model with a symmetric normal background:
normal impedance
Z
N
≈
⎧
⎨
N
(
z
)
y
≤−
c
1
t
(
y
,
−
c
1
<
y
<
−
z
)
c
0
,
−
c
0
≤
≤
(
y
,
z
)
=
(
y
z
)
y
c
0
(10
.
6)
⎩
t
(
y
,
z
)
c
0
<
y
<
c
1
N
(
z
)
y
≥
c
1
.
Alternatively, we can determine a symmetric normal background by separate
extrapollation of the longitudinal or transverse impedances,
Z
or
Z
⊥
.
Introduction of a two-dimensional symmetric homogeneous background is quite
understandable, if the impedance values measured at the edges of the observation
profile
P
0
do not greatly differ from each other. In the regions with strongly pro-
nounced asymmetry (for instance, on the ocean coast or at foothills), geophysicists
give usually preference to an asymmetric background characterized by the different
normal impedances
Z
N
Z
N
and the different normal conductivities ˙
N
(
z
)
which provide the best adjustment to the real geoelectric structures bordering the
observation profile. Evidently this approach is tolerable provided that a sufficient
information on the areas adjacent to edges of the profile is available.
Note that any two-dimensional asymmetric model by means of mirror-image can
be reduced to a symmetric model with a homogeneous background.
,
N
(
z
)
,
¨
10.1.2 On Detailedness of the Multi-Dimensional Inversion
Compared to the one-dimensional inversion, the two-dimensional and three-
dimensional inversions are less stable since they require a much greater number of
free parameters for constructing adequate models. Solution of the two-dimensional
or three-dimensional inverse problem is meaningful provided it is sought within a
sufficiently narrow set of plausible models forming an interpretation model. But
here we come up against the
paradox of instability
. The more restricted the inter-
pretation model, the more stable the inverse problem and the poorer the detailedness
of its solution. On the other hand, the more stable the inverse problem, the higher
