Geology Reference
In-Depth Information
since the volume element,
r
2
sinθ
dr d
θ
d
φ, is a scalar capacity. Taking the local
Cartesian focal co-ordinates to be oriented so that
x
1
,
x
2
and
x
3
are in the directions
of increasing θ, decreasing φ and decreasing
r
, respectively, for spherical polar
co-ordinates (
r
,θ,φ)inthis
epicentral
system of the field point, we have the dipole
force densities for the dip-slip system as
r
1
r
2
sinθ
1
r
sinθ
δ(
r
−
r
0
)δ(θ
−
θ
0
)δ
(φ)cos2α
+
δ
(
r
−
r
0
)δ(θ
−
θ
0
)δ(φ) sin 2α
(9.63)
φ
1
r
sinθ
δ(
r
r
0
)δ(θ
−
θ
0
)δ
(φ) sin 2α
+
−
−
+
δ
(
r
−
r
0
)δ(θ
−
θ
0
)δ(φ)cos2α
,
and the dipole force densities for the strike-slip system as
r
r
δ(
r
1
r
2
sinθ
r
0
)δ
(θ
−
θ
0
)δ(φ)cosα
−
−
θ
1
r
0
)δ(θ
−
θ
0
)δ
(φ)sinα
+
r
sinθ
δ(
r
−
(9.64)
−
δ
(
r
−
r
0
)δ(θ
−
θ
0
)δ(φ)cosα
φ
r
δ(
r
r
0
)δ
(θ
−
θ
0
)δ(φ)sinα
+
−
.
ects of the
transformation to spherical polar co-ordinates, there are additional terms arising
from co-ordinate curvature.
The spherical polar unit vectors
r
,
While expressions (9.63) and (9.64) take account of the direct e
ff
φ
in the epicentral system are related to
θ
,
the geocentric Cartesian unit vectors
e
1
,
e
2
,
e
3
in that system by
r
=
e
1
sinθcosφ
−
e
2
sinθsinφ
−
e
3
cosθ,
(9.65)
θ
=
e
1
cosθcosφ
−
e
2
cosθsinφ
+
e
3
sinθ,
(9.66)
φ
=−
e
1
sinφ
−
e
2
cosφ.
(9.67)
Now consider the double-couple force system arising from the strike-slip expres-
sion and illustrated in the upper left of Figure 9.7. Here, unit vectors
θ
have been
±
φ
have been transported in the
transported in the φ-direction and unit vectors
±
θ-direction. From (9.66) and (9.67), we find
θ
∂φ
=
φ
∂θ
=
∂
φ
cosθ and
∂
0.
(9.68)
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