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where pv means that integral is taken in the sense of the Cauchy principal value, and
Z
Z
R
=
Re
o
,
X
=
Im
o
.
i
i
These relations exist if the impedance
Z
has no poles in the upper half-plane of
the complex frequency
.
2. The dispersion relations of the second kind. They relate the apparent resistivi-
ties and impedance phases. These relations are in the form:
= +
i
λ
=−
4
−
o
pv
∞
d
(
o
)
ln
A
(
)
o
,
π
2
−
0
(1
.
99)
4
+
)
0
o
)
pv
∞
ln
A
(
4
π
d
)
=
o
,
(
2
A
(
∞
−
where
) is the high-frequency asymptotic value of the apparent resistivity.
These relations exist if the impedance
Z
satisfies the condition of the minimum
phase, that is, if it has neither poles nor zeros in the upper half-plane of the complex
frequency
A
(
∞
.
The existence of the dispersion relations in the two- and three-dimensional mod-
els is among the most controversial subjects of magnetotellurics.
Weidelt and Kaikkonen (1994) gave rigorous proof to the validity of the disper-
sion relations of both kinds in the
H
-polarized 2D-models. We examined numer-
ically a lot of the
E
-polarized 2D-models with different characteristic structures
(dyke, ledge, horst, graben, canyon) and revealed that all these models met the dis-
persion relations of both kinds.
Yee and Paulson (1988) considered the impedance tensor of the heterogeneous
Earth as a linear casual operator and on this ground state that the dispersion relations
of both kinds hold good in all models, including 3D ones. But this consideration
is vulnerable to criticism since the electrical and magnetic fields interact with each
other and we hardly can say that one of these fields is a cause and another is an effect
(Svetov, 1991). The magnetelluric system is casual in the sense that the electrical
and magnetic fields are effects of the same cause, for instance, of ionospheric or
magnetospheric currents.
Many people were involved in this discussion (Fischer and Schnegg, 1980, 1993;
Egbert, 1990; Svetov, 1991; Berdichevsky and Pokhotelov, 1997a, b). Nowadays
it is evident that we have to leave room for the possibility of violation of disper-
sion relations in the
E
-polarized 2D-models and 3D-models. The discussion makes
a clear practical sense: if the Kramers-Kronig relations are violated, our philoso-
phy of amplitude-phase inversion of MT-data should be revised. This is seen from
the following example. Take a regional elongated (quasi 2D) depression with local
near-surface 3D inhomogeneities that violate the dispersion relations. Here the sep-
arate inversions of transverse apparent resistivity and phase curves in the class of
H
-polarized 2D-models may yield conflicting geoelectric structures.
= +
i
λ
