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These representations exhibit three inherent properties of the two-dimensional
impedance tensor [
Z
]:
1. Tensor [
Z
], defined on its principal directions, has zero principal diagonal. By
virtue of (2.29):
Z
xx
=
Z
yy
=
0
(2
.
33)
2. Eigenfields
E
τ
1
,
H
τ
1
as well as eigenfields
E
τ
2
,
H
τ
2
are quasi-orthogonal. At
arbitrary orientation of the
x,y
-axes
E
x
1
H
x
1
+
E
y
1
H
y
1
=
0
(2
.
34)
E
x
2
H
x
2
+
E
y
2
H
y
2
=
0
,
which is in agreement with (2.26). The tensor
E
τ
2
,
H
τ
2
is characterized by the
EH
quasi-orthogonality of electric and magnetic eigenfields.
3. Eigenfields
E
τ
1
,
E
τ
2
as well as eigenfields
H
τ
1
,
H
τ
2
are orthogonal. At arbi-
trary orientation of the
x,y
-axes
E
x
1
E
x
2
+
E
y
1
E
y
2
=
0
(2
.
35)
H
x
2
+
H
y
2
=
H
x
1
H
y
1
0
,
which is in agreement with (2.20) and (2.23). The tensor [
Z
] is characterized by the
EE
and
HH
orthogonality of electric and magnetic eigenfields.
The above properties of the two-dimensional impedance give a clue to gener-
alization of the magnetotelluric eigenstate problem to the 3D model. Solving the
3D eigenstate problem, we attribute one of these properties to the three-dimensional
impedance tensor. In the following we consider three methods of this kind. All these
methods share the common property that in the case of a 2D medium they define
the longitudinal and transverse directions as principal directions as well as the lon-
gitudinal and transverse impedances as principal impedances. In the general case of
a 3D asymmetric medium they offer principal directions and principal impedances
of an equivalent 2D tensor [
Z
].
2.4 The Swift-Sims-Bostick Method
This method has been proposed by Swift (1967) and advanced by Sims and Bostick
(1967). The Swift-Sims-Bostick method (
SSB-method
) comes from (2.33) and
reduces to the reference frame rotation that minimizes the principal diagonal com-
ponents
Z
xx
,
Z
yy
of the impedance tensor [
Z
] obtained at the surface of a three-
dimensional Earth. Sometimes the Swift-Sims-Bostick method (SSB method) is
referred to as
rotation method
.
The parameter
M
(
) is introduced to characterize the relation between princi-
pal and secondary diagonal components,
Z
xx
,
Z
yy
and
Z
xy
,
Z
yx
, of the impedance
tensor:
