Finite-difference migration derived from the Kirchhoff-Helmholtz integral

摘要

In this paper, FD migration algorithms like the well-known "15°" and "45°" equations were derived from the Kirchhoff-Helmholtz theory. In the case of inhomogeneous media, there also exists, in addition to the usual surface integral, a volume integral over arbitrary slowness variations that acts as secondary sources for the wavefield. Usually the derivation of FD migration schemes starts with the variable-slowness, one-way wave equation with the exact square root operator that is approximated by a truncated series expansion. Here a more fundamental and more physical approach is given. Wavefield back propagation is performed in a constant or depth-only dependent slowness background field. Lateral variations are taken into account by considering them as secondary sources of the wavefield. These secondary sources are initialized by the incident downward-propagated wavefield that penetrates the regions of velocity variations. The wavefield contributions generated by these secondary sources also are downward propagated by the Green's function for the constant-slowness background. The Kirchhoff integral is the mathematical representation of this Huygens principle. At the beginning of the derivation, quadratic terms of the slowness variations in the source term must be neglected. To linearize the problem in this way the relative slowness perturbation Δu/u_0 must remain small compared to 2. If we choose the maximum occurring slowness as the reference slowness u_0, this bounded slowness perturbation condition holds even in the limit of infinitely strong slowness variations Δu because there also the reference slowness becomes infinite. This clearly shows that in hybrid migration schemes involving a phase-shift operator the minimum velocity (or maximum slowness) must be used. Further academic research should investigate the role of the neglected quadratic term (Δu)~2. It is expected that, carrying out the cumbersome algebraic manipulations, we obtain again the variable-slowness, one-way wave equation. Otherwise, all FD depth migration algorithms would be wrong.