(1) |

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) |

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.

4/27/2000