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It remains to determine phases of the principal values. According to (2.74),
Z
xx
h
x
1
+
Z
xy
h
y
1
1
=
arg
1
=
arg
e
x
1
(2
.
83)
Z
xx
h
x
2
+
Z
xy
h
y
2
2
=
arg
2
=
arg
,
e
x
2
,
,
where the components of vectors
e
h
are obtained by (2.65) and (2.66) with
,
known from (2.62), (2.63), (2.69). It is easy to show that
1
and
2
are rotationally
invariant.
Thus, the La Torraca-Madden-Korringa method results in eight independent
eigenstate parameters:
|
1
|
,
1
=
arg
1
,
1
=
E
1
,
1
=
E
1
.
(2
84)
|
2
|
,
2
=
arg
2
,
2
=
H
1
,
2
=
H
1
,
which fill all eight degrees of freedom possessed by the matrix [
Z
].
There is a one-to-one correspondence between the impedance tensor and its prin-
cipal values, principal directions and eigenfield ellipticities. Given
1
,
1
,
1
and
2
,
2
,
2
, we can determine [
Z
] using (2.76).
Let us examine indications of the La Torraca-Madden-Korringa method in mod-
els of different dimensionality.
In the 1D-model, we have
Z
xx
=
Z
yy
=
0 and
Z
xy
=−
Z
yx
=
Z
where
Z
is the Tikhonov-Cagniard impedance. Here we get
|
1
| = |
2
| = |
Z
|
.
All other
definitions are not unique, since
P
E
1
=
0
/
0 and
P
H
1
=
0
/
0 (any orthogonal magnetic
fields are transformed to orthogonal electric fields).
Turn to the 2D-model with the strike along the x-axis. Here
Z
xx
=
Z
yy
=
0
Z
,
Z
⊥
. With (2.62), (2.63) and (2.81), (2.83) we get
and
Z
xy
=
Z
yx
=−
1
=
Z
,
2
Z
⊥
Z
⊥
,
2
Z
=
or
1
=
=
and
1
=
0
,
2
=
/
2or
1
=
/
2
,
2
0. The principal values of the tensor [
Z
] coincide with
the longitudinal and the transverse impedances, while the principal directions are
the longitudinal and transverse directions. The electric and magnetic eigenfields are
linearly polarized along the principal directions. Note that 2D indications, given by
the La Torraca-Madden-Korringa and Swift-Eggers methods, are the same. At a
single observation site, both methods cannot distinguish between the longitudinal
and transverse direction.
The similar situation is in the axially symmetric 3D-model. Here the principal
values of the tensor [
Z
] coincide with the tangential and radial impedances, while
the principal directions are the tangential and radial directions. The ellipticity
of eigenfields is zero. So, the eigenfields are linearly polarized along principal
directions.
In the asymmetric 3D-models, we observe the elliptic polarization of the elec-
tric and magnetic eigenfields (
=
0aswellas
1
,
2
=
=
0) and the violation of the perpendicularity of
