Next: Controlling incidence angle Up: Wavefront synthesis using wave Previous: Target-oriented wavefront synthesis

## Wavefront synthesis along an irregular surface

So far I have described plane wave synthesis along a flat datum either at the surface or at a depth level. In order to synthesize a plane-wave along an irregular datum, we need an extrapolation algorithm that extrapolates wavefield from an irregular datum. Suppose that we want to have a downgoing wavefield along an irregular reflector r=z(x). This wavefield can be obtained by propagating a certain wavefield at the surface, ,to the irregular reflector r=z(x), as follows:
 (9)
where W(r,z0) represents a wave propagation operator from z0 to r in forward time. If we assume the propagation operator, W(r,z0), is an unitary operator, can be obtained by applying the adjoint operator to both sides of the above equation:
 (10)
where is the adjoint to W(r,z0), which represents a wave propagation operator from r to z0 and backward in time. Then the wavefield obtained by equation () becomes the synthesis operator, as explained in the previous section.

Therefore, to synthesize a plane-wave along an irregular reflector, we need to formulate the wavefield extrapolation from a flat surface to an irregular surface as a forward operator, W(r,z0), and find its adjoint,.Those extrapolations are formulated by modifying a standard depth extrapolation operator and are explained in Appendix .

Another case that requires an extrapolation similar to that described above is one in which we synthesize a plane-wave at a depth level using shot gathers collected along a nonflat surface. In this case, we need to define the forward operator as an extrapolation from a flat datum to an irregular datum, W(zn,r), and as well as define its adjoint operator, .These operators are easily derived by a modification of the algorithm shown above in this section.

Next: Controlling incidence angle Up: Wavefront synthesis using wave Previous: Target-oriented wavefront synthesis
Stanford Exploration Project
2/5/2001