Next: Helical factorization Up: Theory Previous: Theory

## Fourier finite-difference migration

Three-dimensional FFD extrapolation is based on the equation Ristow and Ruhl (1994),
 (1)
where v=v(x,y,z) is the medium velocity, c is a reference velocity (), and a and b are coefficients subject to optimization. The first term describes a simple Gazdag phase-shift that must be applied in the () domain; the second term describes the first-order split-step correction Stoffa et al. (1990), applied in (); and the third term describes an additional correction that can be applied as an implicit finite-difference operator Claerbout (1985), also applied in ().

In areas with strong lateral velocity variations (), FFD reduces to a finite-difference migration, while in areas of weak lateral velocity variations (), FFD retains the steep-dip accuracy advantages of phase-shift migration. As a full-wave migration method, FFD also correctly handles finite-frequency effects.

For constant lateral velocity, the finite-difference term in equation (1) can be rewritten as the following matrix equation,
 (2) (3)
where is a finite-difference representation of the x,y-plane Laplacian, , and and represent the diffraction wavefield at depths z and z+1 respectively . Scaling coefficients, and , are complex and depend both on the ratio, , and the ratio c/v.

The right-hand-side of equation (3) is known. The challenge is to find the vector by inverting the matrix, .For 2-D problems, only a tridiagonal matrix must be inverted; whereas, for 3-D problems the matrix becomes blocked tridiagonal. For most applications, direct inversion of such a matrix is prohibitively expensive, and so approximations are required for the algorithm to remain cost competitive with other migration methods.

A partial solution is to split the operator to act sequentially along the x and y axes. Unfortunately this leads to extensive azimuthal operator anisotropy, and necessitates expensive additional phase correction operators.

Next: Helical factorization Up: Theory Previous: Theory
Stanford Exploration Project
4/27/2000