Kinematics of PS-DMO

The downgoing and upgoing waves travel at different velocities for converted wave data. This difference makes the kinematics more complicated than that of single mode data.

The kinematics of PS-DMO have been widely discussed in the literature. () is the first one to derive the zero-offset mapping equation for converted waves. He uses an integral-summation approach, similar to () in order to apply DMO to converted waves data. He also applies the PS zero-offset mapping and the adjoint of his operator in order to obtain the PS Rocca's operator.

I present a review of the kinematics for PS-DMO. The impulse response 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 offset to zero, to produce hyperbolas. This operation creates the impulse response that represents the Rocca's DMO+NMO operator ().

Figure  shows a comparison between the Rocca's smear operator for single mode P data and for converted mode PS data. It is possible to observe, kinematically, that the Rocca's operator for converted waves is shifted laterally toward the receiver. This is the expected result, since the upgoing wave path is slower than the downgoing wave path.

 

rocca
Figure 1 Rocca's operator, for single mode P data (left) and converted mode PS data (right)

() shows the amplitude distribution for a PS-DMO operator. This amplitude distribution is also shown in Figure (). The denser the dots, the higher the amplitude should be.

One of the challenges is to obtain an independent expression for the DMO operator, but this task has already been solved in the literature. Here I present a review of this process, since the result is used by () and later in this paper.