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Since S first arrivals are clearer and stronger than P first arrivals,
we use S arrivals to estimate a vector in the general direction of P arrivals.
The idea is to find the direction most perpendicular to the strongest
(direct) S arrivals.
We setup a
matrix of shear first arrivals,
, by windowing around the first shear arrival.
As we seek a vector
that is perpendicular
to the shear arrival direction (which is a P-arrival) we want
. |
(1) |
We solve this by minimizing the objective function
|
(2) |
subject to the constraint
. |
(3) |
Let
and
be two spherical surface coordinate parameters
over which we will minimize.
Taking partial derivatives of the constraints yields:
|
(4) |
and
, |
(5) |
which says that
is perpendicular to the two partial derivatives.
Next, taking partial
derivatives of the sum of squares expression gives
|
(6) |
and
. |
(7) |
Therefore
is also perpendicular to both
partial derivatives and consequently must
be parallel to
. This means that
, |
(8) |
where
is the eigenvalue of the matrix
that will make the least squares expression a minimum.
Transposing we get
, |
(9) |
which is a classic eigenvector problem for the matrix
.
Since the right singular vectors of
are the same as the eigenvectors of
, we
used the LAPACK routine SGESVD to find our desired P-wave direction
vector.
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| Hunting for microseismic reflections using multiplets | |
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Next: Bibliography
Up: Appendix A
Previous: Appendix A
2012-05-10