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 ${\bf \bar s} (r)$ along an irregular reflector r=z(x). This wavefield can be obtained by propagating a certain wavefield at the surface, ${\bf \bar s}(z_0)$,to the irregular reflector r=z(x), as follows:

$${\bf \bar s}(r) = W(r,z_0) {\bf \bar s}(z_0),$$ (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, ${\bf \bar s}(z_0)$ can be obtained by applying the adjoint operator to both sides of the above equation:

$$\begin{eqnarray} W^{\ast}(z_0,r) {\bf \bar s}(r) & = & W^{\ast}(z_0,r) W(r,z_0){\bf \bar s}(z_0) \nonumber \\ & = & {\bf \bar s}(z_0)\end{eqnarray}$$ (10)

where $W^{\ast}(z_0,r)$ 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,$W^{\ast}(z_0,r)$.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(z_n,r), and as well as define its adjoint operator, $W^{\ast}(r,z_n)$.These operators are easily derived by a modification of the algorithm shown above in this section.