Interferometric imaging condition for wave-equation migration

Discussion

The strategies described in the preceding section have notable similarities and differences. The imaging procedures 4-6-7 and 5-8 are similar in that they employ wavefields reconstructed from the surface data in similar ways. Neither method uses the surface recorded data directly, but they use wavefields reconstructed from those data as boundary conditions to numerical solutions of the acoustic wave-equation. The actual wavefield reconstruction procedure is identical in both cases.

The techniques are different because imaging with equations 4-6-7 employs independent wavefield reconstruction from receiver locations ${ \mathbf{x} }$ to image locations ${ \mathbf{y} }$. In practice, this requires separately solving the acoustic wave-equation, e.g. by time-domain finite-differences, from all receiver locations on the surface. Such computational effort is often prohibitive in practice. In contrast, imaging with equations 5-8 is similar to conventional imaging because it requires only one wavefield reconstruction using all recorded data at once, i.e. only one solution to the acoustic wave-equation, similar to conventional shot-record migration.

The techniques 4-6-7 and 5-8 are similar in that they both employ noise suppression using pseudo Wigner distribution functions. However, the methods are parametrized differently, the former relative to data coordinates with 2D local space averaging and the later relative to image coordinates with 3D local space averaging.

The imaging functionals presented in this paper are described as functions of space coordinates, ${ \mathbf{x} }$ or ${ \mathbf{y} }$, and time, ${ t }$. As suggested in Appendix C, pseudo WDFs can be implemented either in time or frequency, so potentially the imaging conditions discussed in this paper can also be implemented in the frequency-domain. However, we restrict our attention in this paper to the time-domain implementation and leave the frequency-domain implementation subject to future study.

Equations 4-6-7 can be collected into the zero-offset imaging functional

$$R_{CINT} \left ( { \mathbf{y} } \right) = \delta\left (t\rig... ...{ \textcolor{darkgreen}{ {{ t }_h}} }{2} \right) \right]\;,$$ (11)

where the temporal $\delta$ function implements the zero time imaging condition. A similar form can be written for the multi-offset case. Equation 11 corresponds to the time-domain version of the coherent interferometric functional proposed by Borcea et al. (2006b,a,c). Consistent with the preceding discussion, the cost required to implement this imaging functional is often prohibitive for practical application to seismic imaging problems.

Interferometric imaging condition for wave-equation migration