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So, we have the similar polarization ellipses with perendicular major axes, but
with the same sense of field rotation (Fig. 2.2b). Relationships of this kind will be
referred to as the
EH quasi-orthogonality.
With these definitions, we can enter into a discussion of basic approaches to the
impedance eigenstate problem.
2.3 Basic Approaches to the Impedance Eigenstate Problem
The impedance eigenstate problem has been advanced by Sims and Bostick (1967),
Eggers (1982), Spitz (1985), LaTorraca et al. (1986), Counil et al. (1986) and
Yee and Paulson (1987). Entire gamut of different methods has been thoroughly
reviewed by Yee and Paulson (1987), Groom and Bailey (1991) and Vozoff (1991).
In our topic we will restrict ourselves to a close look at three most-used methods:
(1) the Swift-Sims-Bostick method (the rotation approach), (2) the Swift-Eggers
method (the modified classical approach), (3) the LaTorraca-Madden-Korringa
method (the modified SVD approach). These methods determine the eigenstate of
the three-dimensional impedance tensor so that the solution obtained satisfies one
of the inherent properties of the two-dimensional impedance tensor.
Let us take a 2D-model with the strike along the
x
-axis. Here, according to (1.54),
0
Z
[
Z
]
=
,
(2
.
29)
Z
⊥
−
0
whence
Z
H
y
,
Z
⊥
H
x
.
E
x
=
E
y
=−
(2
.
30)
It is natural to consider the longitudinal and transverse directions that run along
and across the model strike as principal directions of [
Z
]. In this context, the fields
E
τ
and
H
τ
linearly polarized along and across the model strike should be consid-
ered as eigenfields of [
Z
]. The two-dimensional tensor [
Z
] has two pairs of the
eigenfields:
E
x
1
0
0
E
y
2
⎧
⎨
⎧
⎨
E
τ
1
=
E
τ
2
=
0
H
y
1
H
x
2
0
.
TE
−
mode
TM
−
mode
(2
.
31)
⎩
⎩
H
τ
1
=
H
τ
2
=
In each pair, the electric eigenfield is the transform of the magnetic eigenfield:
E
τ
1
=
1
01
−
H
τ
1
E
τ
2
=
2
01
−
H
τ
2
,
(2
.
32)
10
10
ς
1
=
Z
ς
2
=
Z
⊥
where
and
are the principal values (eigenvalues) of [
Z
].
