M a t h e m a t i c a l B a c k g ro u n d o f DC3 D (P a rt 3 )
Derivation of Table 6 in Okada (1992)
[ I ] Integration for a finite rectangular source
.
Point source solutions given in Tables 2 through 5 have the form of
For a finite fault with a dislocation
, we can replace
to
using the concept of body
force equivalents. This operation yields the finite fault solution in the form of u
To get finite fault solutions, we need double integration with
source from
to
.
.
after replacing the location of point
Namely, after changing
in the point source solution, we need an operation
Here, for the sake of convenience, we change the integration
variables from
to
Then, we should change the variables in the point source solution to
and perform the integration
where
In the following, for the sake of simplicity, we will treat the displacement
For A- and B-parts of the displacement
and for the C-part of the displacement
The former
corresponds to the displacement parallel to up-dip
direction of the real fault, while the latter
corresponds to that
of the imaginary fault.
1
instead of
M a t h e m a t i c a l B a c k g ro u n d o f DC3 D (P a rt 3 )
(1) Strike slip
Displacement due to a point strike-slip at
are given in Table 2 as follows.
where,
Here, for the sake of simplicity, the term
term
in the z-component of
(see “Derivation of Table 2”).
was added to the z-component of
If we convert the displacement
,
to
For the integration, we substitute
So, integrand becomes
At first, let us integrate with
(refer Appendix : Table of Integration)
2
was restored to
and the
M a t h e m a t i c a l B a c k g ro u n d o f DC3 D (P a rt 3 )
Next, let us integrate with
(refer Appendix : Table of Integration)
Here,
and
The above three vectors correspond to the contents of the row of Strike-slip in Table 6.
( Evaluation of
et al. will be done in the later section )
(2) Dip slip
Displacement due to a point dip-slip at
are given in Table 2 as follows.
where,
Here, for the sake of simplicity, the term
term
was added to the z-component of
in the z-component of
(see “Derivation of Table 2”).
3
was restored to
and the
M a t h e m a t i c a l B a c k g ro u n d o f DC3 D (P a rt 3 )
If we convert the displacement
Here, since
,
to
,
For the integration, we substitute
So, integrand becomes
At first, let us integrate with
Next, let us integrate with
(refer Appendix : Table of Integration)
(refer Appendix : Table of Integration)
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M a t h e m a t i c a l B a c k g ro u n d o f DC3 D (P a rt 3 )
Here,
The above three vectors correspond to the contents of the row of Dip-slip in Table 6.
( Evaluation of
et al. will be done in the later section )
(3) Tensile
Displacement due to a point tensile fault at
are given in Table 2 as follows.
where,
Here, for the sake of simplicity, the term
(see “Derivation of Table 2”).
was added to the z-component of
If we convert the displacement
Here, since
in the z-component of
,
to
,
5
was restored to
and the term
M a t h e m a t i c a l B a c k g ro u n d o f DC3 D (P a rt 3 )
For the integration, we substitute
So, integrand becomes
At first, let us integrate with
Next, let us integrate with
(refer Appendix : Table of Integration)
(refer Appendix : Table of Integration)
Here,
The above three vectors correspond to the contents of the row of Tensile in Table 6.
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M a t h e m a t i c a l B a c k g ro u n d o f DC3 D (P a rt 3 )
(4) Evaluation of
～
For the integration, we substitute
to
So, the integrands and their integral with
Next, let us integrate with
through
of the point solution in Table 2.
become as follows (refer Appendix : Table of Integration)
(refer Appendix : Table of Integration)
< Case 1 >
< Case 2 >
So, as a whole,
Otherwise
takes either of (1) or (2) depending on
< Case 1 >
< Case 2 >
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or not.
M a t h e m a t i c a l B a c k g ro u n d o f DC3 D (P a rt 3 )
< Case 1 >
< Case 2 >
< Case 1 >
< Case 2 >
-----------------------------------------------------------------------------------------------------------------------------------As a conclusion, the latter part of
including
through
in Table 6 are given as follows (
).
(1) Strike-slip
(2) Dip-slip and Tensile
In case of
,
and
should be given as follows.
8
M a t h e m a t i c a l B a c k g ro u n d o f DC3 D (P a rt 3 )
Appendix : Table of Integration
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M a t h e m a t i c a l B a c k g ro u n d o f DC3 D (P a rt 3 )
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