Information Technology Reference
In-Depth Information
and
Z
xx
(
)
−
Z
yy
(
)
tan 2
=
)
.
(1
.
58)
+
Z
xy
(
)
Z
yx
(
One can see that the 2D-impedance has zero trace. But it should meet an addi-
tional condition
Z
xx
(
)
−
Z
yy
(
)
Im
)
=
0
Z
xy
(
)
+
Z
yx
(
to ensure
to be real. On rearrangement with account for (1.57), this condition can
be written as
Im (
Z
xy
Z
yy
+
Z
xx
Z
yx
)
=
J
12
=
0
,
(1
.
59)
where
J
12
is a rotational invariant defined by (1.34).
Generally the invariants I
1
and
J
12
characterize the geoelectric asymmetry (the
skewness ) of the medium. If I
1
=
0 and
J
12
=
0, then a model has a vertical plane of
mirror symmetry of
z
) that passes through the observation point. Following
Swift (1967) and Bahr (1988), we use these asymmetry parameters in normalized
form. The
Swift skew
is
(
x
,
y
,
=
Z
xx
+
I
1
I
3
Z
yy
w
=
.
ske
(1
60)
S
Z
xy
−
Z
yx
and the
Bahr skew
is
Im (
Z
xy
Z
yx
)
√
|
Z
yy
+
Z
xx
J
12
|
Z
xy
−
Z
yx
ske
w
B
=
=
.
(1
.
61)
|
I
3
|
Note that the parameter
ske
w
B
differs fro
m
the asymmetry parameter
η
initially
w
B
=
η/
√
2.
In the 2D-model, asymmetry parameters
ske
introduced by Bahr. We have
ske
w
S
and
ske
w
B
are equal to zero:
w
=
,
w
=
.
.
ske
0
ske
0
(1
62)
S
B
This is the necessary condition for the two-dimensionality. With this condition,
we can turn to (1.58) and determine an angle
defining the strike of the two-
dimensional structure (modulo
/2). Rotating the reference frame counterclock-
wise through the angle
, we obtain the impedance tensor (1.54) with components
Z
,
Z
⊥
on the secondary diagonal. Such a tensor with zero principal diagonal and
nonzero secondary diagonal will be specified as an
antidiagonal tensor
.
