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4.2 Vector Representation of the Wiese-Parkinson Matrix
The development of these graphical techniques was stimulated by pioneering works
of Parkinson (1959), Wiese (1962, 1965), Schmucker (1962, 1970), Everett and
Hyndman (1967), Jankovski (1972), Vozoff (1972), and Lilley (1974). The keystone
idea was to represent the complex-valued Wiese-Parkinson matrix as a complex
vector giving a pictorial image of the horizontal gradients of electric conductivity.
Such a vector received the name
tipper vector
or
induction arrow
. Complex tipper
(induction arrow) consists of real and imaginary tippers (real and imaginary induc-
tion arrows).
There exist two conventions concerning the real tipper direction. In the
Wiese
convention
, the real tipper points away from the zone of higher conductivity. In the
Parkinson convention
, the real tipper points toward the zone of higher conductivity.
In our topic we adopt the Wiese convention, which is popular in the Russian
magnetotelluric school.
In the following we consider three types of tippers: the Wiese-Parkinson tipper,
the Vozoff tipper, and the Schmucker tipper.
4.2.1 The Wiese-Parkinson Tipper Technique
Let us represent the Wiese-Parkinson matrix in the vector form
W
=
W
zx
1
x
+
W
zy
1
y
.
(4
.
13)
The vector
W
is given the name
Wiese-Parkinson tipper.
The complex tipper
W
falls into the real and imaginary tippers:
W
=
Re
W
+
i
Im
W
,
(4
.
14)
where
Re
W
=
Re
W
zx
1
x
+
Re
W
zy
1
y
,
.
(4
15)
Im
W
=
Im
W
zx
1
x
+
Im
W
zy
1
y
.
Note that the rotational invariants
P
1
and
P
2
defined by (4.9) have meaning of
the vector and scalar products of the real and imaginary tippers:
1
x
1
y
1
z
Re
W
zx
Re
W
zy
0
Im
W
zx
Im
W
zy
0
Re
W
×
Im
W
=
=
(Re
W
zx
Im
W
zy
−
Re
W
zy
Im
W
zx
)
1
z
=
P
1
1
z
Re
W
·
Im
W
=
Re
W
zx
Im
W
zx
+
Re
W
zy
Im
W
zy
=
P
2
.
.
(4
16)
