Information Technology Reference
In-Depth Information
here
Z
and
Z
⊥
are the longitudinal and transverse impedances. Substituting (1.54)
into (1.89) and (1.92), we find
Z
H
+
Z
⊥
2
sin
2
2
cos
2
Z
H
(
H
)
=
H
1
(1
.
97)
Z
E
(
E
)
=
.
sin
2
cos
2
E
E
+
Z
⊥
Z
2
2
Polar diagrams of the
H-
and
E-
polarized impedances are a regular oval with a
waist and an ellipse, respectively. Their principal diameters 2
Z
and 2
Z
⊥
are
oriented along and across the strike of the model. The
Z
H
and
Z
E
diagrams in an
axisymmetric 3D-model have similar shape, with their principal diameters 2
|
Z
r
|
and 2
being oriented along the radial and tangential directions.
In a quasi-symmetric 3D-model with
ske
|
Z
t
|
0, the diagrams of
Z
H
and
Z
E
retain a regular shape and are elongated in perpendicular directions (3D,a).
In an asymmetric 3D-model with
ske
w
S
=
ske
w
CLM
=
0, a regular shape of the
Z
H
and
Z
E
diagrams is preserved but the angle between their elongation directions can deviate
significantly from the right angle (3D,b). This is the only feature of the
Z
H
and
Z
E
diagrams that can be used as an indicator distinguishing an asymmetric 3D medium
from a 2D or an axisymmetric 3D medium.
w
CLM
=
1.5 Dispersion Relations in the Impedance Tensor
The dispersion relations were first derived by Kramers and Kronig in the theory of
dispersion of optic rays (Mathews and Walker, 1964). These integral relations are
the direct consequence of the principle of causality.
The Kramers-Kronig dispersion relations were introduced in geoelectrics by
Kaufman (1960) and Vanyan et al. (1961). These authors used the dispersion rela-
tions to convert the apparent resistivity curves of frequency sounding into the phase
curves.
In 1972, Weidelt published his famous paper that laid the groundwork for the
mathematical theory of magnetotelluric sounding (Weidelt, 1972). In this paper he
gave rigorous analytical proof for the existence of the dispersion relations in the
Tikhonov-Cagniard one-dimensional model.
Following Weidelt, we will consider the dispersion relations of two kinds.
1. The dispersion relations of the first kind. These relations connect the real and
imaginary parts of the normalized impedance. They assume the form
0
2
pv
∞
X
(
)
R
(
o
)
=
o
d
,
2
−
(1
.
98)
2
o
pv
∞
R
(
)
X
(
o
)
=−
d
,
2
−
o
0
