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.

 

rocca
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).

 

• Derivation of the PS-DMO operator
• PS-DMO in the frequency-wavenumber log-stretch domain