Information Technology Reference
In-Depth Information
RECORDED
E
x
PREDICTED
12
mV/km
E
x
DIFFERENCE
H
12
nT
D
Fig. 1.2
Comparison between recorded and predicted variations in the
E
x
-component (Wielondek
and Ernst, 1977)
Let us exemplify the stability of the impedance relation by experiments carried
out on the southeast of Poland (Wielondek and Ernst, 1977). Figure 1.2 shows the
observed variations in the
E
x
-component of electric field and the same variations
predicted from the variations in H- and D-components of magnetic field convolved
with the previously determined impedance [
Z
]
The predicted variations look like a
replica of the recorded variations including small details.
.
1.2 Rotation of the Impedance Tensor
Let us orient the components
Z
xx
,
Z
yy
of the impedance tensor [
Z
]inthe
direction corresponding to their first subscript. It means that
Z
xx
,
Z
xy
,
Z
yx
,
Z
xy
are oriented in
the
x
-direction, and
Z
yx
,
Z
yy
are oriented in the
y
-direction. The orientation of the
impedance components complies with the orientation of the corresponding electric
field components.
How do the components of the impedance tensor [
Z
] change as a frame of refer-
ence rotates? Let
be a clockwise rotation angle (Fig. 1.3). Consider the transition
y
and conversely.
The direct and converse rotation matrices are
y
to the new axes
x
,
from the old axes
x
,
cos
cos
sin
−
sin
)]
−
1
[
R
(
)]
=
,
[
R
(
=
.
(1
.
24)
−
sin
cos
sin
cos
