Information Technology Reference
In-Depth Information
Chapter 2
The Impedance Eigenstate Problem
2.1 The Classical Formulation of the Tensor Eigenstate Problem
Rotating the impedance tensor [
Z
], one may obtain a variety of different apparent-
resistivity and impedance-phase curves,
yx
, sometimes drastically
conflicting in configuration. Such a great body of data is seemingly chaotic, but it
can be systematized by the methods concerned with the eigenstate problem.
Solving the eigenstate problem, we focus all the information, contained in the
components of the tensor, on its principal directions depending on geometry of the
target geoelectric structures.
Recall the classical formulation of the tensor eigenstate problem. Let a real-
valued symmetric tensor
xy
,
yx
and
xy
,
T
xx
T
xy
T
yx
T
yy
[
T
]
=
,
T
xy
=
T
yx
(2
.
1)
transform a real-valued vector
V
(
V
x
,
V
y
) to the collinear vector
t
V
(
tV
x
,
tV
y
):
[
T
]
V
=
t
V
,
(2
.
2)
where
t
is a scalar that characterizes the change in the vector modulus. The vector
V
satisfying (2.2) is the
eigenvector
of the tensor [
T
]. Its direction is the
principal
direction
of the tensor [
T
]. The scalar factor
t
is the
principal value
or
eigenvalue
of the tensor [
T
].
If (2.2) is valid, then
(
T
xx
−
t
)
V
x
+
T
xy
V
y
=
0
,
T
yx
V
x
+
(
T
yy
−
t
)
V
y
=
0
.
This uniform system of linear equations in
V
x
,
V
y
can have nonzero solutions if its
determinant is equal to zero:
