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In-Depth Information
The principal
rotational invariants
of the magnetotelluric impedance tensor are
(Berdichevsky, 1968; Sharka and Menvielle, 1997):
I
1
=
tr [
Z
]
=
Z
xx
+
Z
yy
a
I
2
=
det [
Z
]
=
Z
xx
Z
yy
−
Z
xy
Z
yx
b
(1
.
29)
tr [
Z
]
Z
xx
+
Z
yy
=
I
3
=
=
tr [
Z
][
R
(
−
/
2)]
=
Z
xy
−
Z
yx
,
c
where tr [
Z
] and det [
Z
] are the trace and determinant of the impedance tensor [
Z
],
while tr [
Z
] is the trace of the Adam impedance tensor [
Z
].
Using (1.29), we introduce the effective impedance
Z
eff
and the Berdichevsky
impedance
Z
brd
:
Z
eff
=
Z
xx
Z
yy
−
Z
xy
Z
yx
,
(1
.
30)
Z
xy
−
Z
yx
Z
brd
=
Z
1
=
.
2
These three independent invariants can be supplemented with the quadratic
invariant derived from (1.29):
I
1
+
I
3
−
=
Z
xx
+
Z
xy
+
Z
yx
+
Z
yy
,
I
4
=
2I
2
=
tr [
C
]
(1
.
31)
[
Z
][
Z
]
T
, T denotes the transposition.
Since [
Z
] is determined by eight independent real-valued elements, the num-
ber of independent real rotational invariants should be less than eight. Sharka and
Menvielle (1997) proved that the maximum number of the real independent invari-
ants is equal to seven. They suggested the following standard set of independent
rotational invariants:
where tr [
C
] is the trace of the tensor [
C
]
=
J
1
=
2Re
Z
brd
=
Re I
3
=
Re
Z
xy
−
Re
Z
yx
a
J
2
=
2Im
Z
brd
=
Im I
3
=
Im
Z
xy
−
Im
Z
yx
b
J
3
=
tr [Re
Z
]
=
Re I
1
=
Re
Z
xx
+
Re
Z
yy
c
J
4
=
tr [Im
Z
]
=
Im I
1
=
Im
Z
xx
+
Im
Z
yy
d
(1
.
32)
J
5
=
det [Re
Z
]
=
Re
Z
xx
Re
Z
yy
−
Re
Z
xy
Re
Z
yx
e
J
6
=
det [Im
Z
]
=
Im
Z
xx
Im
Z
yy
−
Im
Z
xy
Im
Z
yx
f
J
7
=
Im det [
Z
]
=
Im (
Z
xx
Z
yy
−
Z
xy
Z
yx
)
,
g
where
Re
Z
xx
Im
Z
xx
Re
Z
xy
Im
Z
xy
=
,
=
.
[Re
Z
]
[Im
Z
]
Re
Z
yx
Re
Z
yy
Im
Z
yx
Im
Z
yy
