Kirchhoff inversion for incident waves synthesized from common-shot data gathers

Synthesis is a process for producing reflection responses from more general sources or from prescribed incident waves by combining common-shot data gathers. Synthesis can provide survey-wide data sets, similar in that regard to common-offset data gathers, but with the added advantage that each synthesized data set is a solution to a single wave equation. A common-offset data set does not have this last feature. Thus, synthesized data sets can be processed by true amplitude wave equation migration. The output is then known to be true amplitude, as well, in the same sense as is the output of Kirchhoff inversion. Alternatively, the Kirchhoff inversion of synthesized data has a Beylkin determinant that is expressed in terms of the WKBJ Green’s function amplitude. This is in contrast to 3D common offset inversion where the Beylkin determinant is most difficult to compute. We present here a theory of data synthesis based on application of Green’s theorem to the ensemble of common-shot gathers and prescribed more general sources or prescribed incident waves. Specific examples include delayed-shot line sources and incident dipping plane waves at the upper surface. We also discuss two cases in which waves are prescribed at depth, back-projected to the upper surface and then used to generate a synthesized data set.