Next: Derivation of the PS-DMO
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The kinematics of PS-DMO have been widely discussed in the literature.
Harrison (1990) is the first to derive the zero-offset mapping
equation for converted waves. He uses an
integral-summation approach, similar to Deregowski and Rocca (1981) to
apply DMO to converted-wave data. He also applies the PS zero-offset mapping
and the adjoint of his operator in order to obtain the operator that
describes the cascade of PS-NMO plus PS-DMO, also known as the Rocca's PS
operator. The impulse response for the Rocca's PS operator
is produced by taking an impulse on a constant offset section and
migrating it to produce ellipses. Each element or point along the
ellipses is then diffracted, setting the offset to zero, to produce
hyperbolas Claerbout (1999).
Figure shows a comparison between the Rocca's smear
operator for single-mode PP data and for converted-mode PS data.
Kinematically, Rocca's operator
for converted waves is both laterally shifted and non-symmetric.
This is an expected result, since the upgoing wave path is slower
than the downgoing wave path, so that the inflection point of the PS impulse response
is at a later time than the PP impulse response. This is because the
S-wave velocity is slower than the P-wave velocity.
Figure 5 Rocca's operator, for
single-mode PP data (left) and converted-mode PS data (right)
This thesis presents a more accurate converted-wave DMO operator.
This operator also accounts for lateral movement of the midpoint location
after doing DMO and follows the amplitude distribution presented
by Jaramillo (1997).