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Kinematics of PS-DMO

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)
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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).