Recent work of G. Beylkin helped set the stage for very general seismic inversions. We have combined these broad concepts for inversion with classical high-frequency asymptotics and perturbation methods to bring them closer to practically implementable algorithms. Applications include inversion schemes for both stacked and unstacked seismic data.
Basic assumptions are that the data have relative true amplitude, and that a reasonably accurate background velocity c(x, y, z) is available. The perturbation from this background is then sought. Since high-frequency approximations are used throughout, the resulting algorithms essentially locate discontinuities in velocity.
An expression for a full 3-D velocity inversion can be derived for a general data surface. In this degree of generality the formula does not represent a computationally feasible algorithm, primarily because a key Jacobian determinant is not expressed in practical terms. In several important cases, however, this shortcoming can be overcome and expressions can be obtained that lead to feasible computing schemes. Zero-offsets, common-sources, and common-receivers are examples of such cases.
Implementation of the final algorithms involves, first, processing the data by applying the FFT, making an amplitude adjustment and filtering, and applying an inverse FFT. Then, for each output point, a summation is performed over that portion of the processed data influencing the output point. This last summation involves an amplitude and traveltime along connecting rays. The resulting algorithms are computationally competitive with analogous migration schemes.