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their ellipses ( E 1 H 1 = /
2). However, a quasi-symmetric 3D structure with
ske
w S
=
0 and ske
w B
=
0 is a special case. If Z xx +
Z yy
=
0, then it follows
P H 1
from (2.82) that P E 1
=−
1. Therefore the electric and magnetic eigenfields are
orthogonal and E 1 H 1 = /
2. Here the La Torraca-Madden-Korringa method
displays the same effect of quasi-symmetry as the Swift-Eggers method.
Similar to the Swift-Eggers method, the La Torraca-Madden-Korringa method
offers three characteristic parameters which reveal the lateral inhomogeneities and
indicate their dimensionality. Using (2.84) and going back to (2.53), (2.54) and
(2.55), we can specify the inhomogeneity parameter N , the angular parameter of
asymmetry ske
w pol .
Let us find relations between the principal values of the impedance tensor [ Z ]
and scalar rotationally invariant impedances Z eff and
w ang , and the polarization parameter of asymmetry ske
Z
introduced by (1.30) and
(1.33d). According to (2.76), (2.81), we get
Z eff = det [ Z ]
det [ U e ] det [
=
] det [ U h ]
= det [
= 1 2 = Z xx Z yy
]
Z xy Z yx ,
Z xy
Z yx
Z yy
2
2
2
2
2
2
| Z xx |
+
+
+
| 1 |
+ | 2 |
1
2
Z rms =
=
Z
=
,
2
2
(2
.
85)
where Z eff and Z rms are the effective and root-mean-square impedances.
Using rotationally invariant principal values
2 , we apply (2.58) and arrive
at the principal apparent-resistivity and phase curves,
1 and
1 , 2 and
1 , 2 , oriented
along principal directions.
Furthermore, we can turn to (2.59) and plot the effective apparent-resistivity
and impedance-phase curves
eff , eff as well as the root-mean-square apparent-
resistivity curve
Z rms
o
rms =
(2
.
86)
calculated from rotational invariant Z rms .
2.7 Final Remarks on the Impedance Eigenstate Problem
We have considered three basic methods to solve the magnetotelluric eigenstate
problem: (1) the Swift-Sims-Bostick, SSB method (the rotation approach), (2) the
Swift-Eggers, SE method (the modified classical approach), (3) the LaTorraca-
Madden-Korringa, LMK method (the modified SVD approach).
It is self-evident that the SE method and the LMK method are most informa-
tive, since, contrary to the rotation approach, they fill all eight degrees of freedom
possessed by the matrix [ Z ] and provide a one-to-one correspondence between the
impedance tensor [ Z ] and its eigenstate parameters.
 
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