If the reflector we are interested in is not flat or if the velocity of the overburden medium has strong lateral variation, the surface-oriented wavefront synthesis does not produce a stack that has equally strong amplitude along the reflection from the reflector. The irregular amplitude of the reflected wavefield can be explained by the wavefront distortion resulting from the strong lateral velocity change or the variation of the incidence angle of the wavefront to the reflector from place to place.
Therefore, clear illumination of reflectors that are nonflat or located under a complex velocity overburden require target-oriented wavefront synthesis Claerbout (1976); Rietveld et al. (1992); Schultz and Claerbout (1978). In other words, we need a synthesis operator whose wavefront is a plane at a certain depth or along an irregular reflector, not at the surface. Using the property of the wavefield extrapolation operator that is close to unitary, we can easily generate a synthesis operator for a wave stack whose wavefront has a predefined shape at a certain subsurface location.
Suppose that we want to have a downgoing wavefield at a depth level zn. This wavefield can be obtained by propagating a wavefield at the surface, ,to the depth level zn, as follows:
Assuming that every shot wavefield, ,is an impulse source at xj, the wavefield obtained from equation () becomes the synthesis operator
For synthesizing a slanted wavefront at a certain depth, we only need to start with a slanted plane wave such as in equation (); that is
Figure is the stacked section obtained by synthesizing a plane wave with p = 0. (s/m) at 1200 m depth (Figure ) and Figure is obtained by synthesizing a plane wave with p = 0.0001 (s/m) (Figure ). In Figures and we can observe that some events that were not visible in Figures and are now visible.