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10
01
[
I
]
=
.
Thus, in the measurement coordinates
M
=
S
τ
+
M
=
S
τ
+
[
I
]
,
[
I
]
(4
.
100)
and
˜
W
=
S
z
M
−
1
˜
W
=
S
z
M
−
1
,
,
.
(4
101)
whence
1
S
yy
sin
2
S
yy
sin
cos
+
−
M
=
S
yy
sin
cos
S
yy
cos
2
−
1
+
(4
.
102)
1
S
yy
sin
2
S
yy
sin
cos
+
−
M
=
S
yy
sin
cos
S
yy
cos
2
−
1
+
and
S
zy
˜
W
=
S
yy
−
sin
cos
1
+
(4
.
103)
S
zy
˜
W
=
S
yy
−
.
sin
cos
+
1
Applying this approach to the superimposition problem, we can decompose a three-
dimensional magnetovariational anomaly into two independent two-dimensional
anomalies and reduce 3D inversion to two self-contained 2D inversions.
Figure 4.9 presents an example of such a decomposition. Consider a three-
dimensional superimposition model consisting of the sedimentary cover (
1
=
10 Ohm
·
m
,
h
1
=
0
.
1km
,
2
=
100 Ohm
·
m
,
h
2
=
0
.
9km)
,
the resistive
lithosphere (
3
=
1000 Ohm
·
m
,
h
3
=
99 km) and the conductive mantle (
4
=
20 Ohm
m). The lithosphere contains the crustal two-dimensional conductive prism
CP of 6 Ohm
·
·
m resistivity, its thickness, width and azimuth being 3 km, 100 km
and 135
o
respectively, and the mantle two-dimensional conductive prism MP of 5
m resistivity, its thickness, width and azimuth being 50 km, 300 km and 0
o
respectively. Azimuths of the crustal and mantle prisms
are determined f
rom the
pseudo-topographies of the 3D tipper invariants
Ohm
·
W
zy
2
, plotted
2
=
W
|
W
zx
|
+
on periods of 1 and 10,000 s. Decomposing 3D invariant
W
into two 2D invari-
ants
W
and
W
, we separate partial two-dimensional effects of the crustal and
mantle prisms, CP and MP, with reasonable accuracy.
