Accommodating lateral velocity changes in Kirchhoff migration by means of Fermat’s principle

Abstract

When velocity varies laterally as well as with depth, an exact Kirchhoff depth migration requires that rays be traced from each depth point in the section to each source/receiver location. Because such a procedure is prohibitively expensive, Kirchhoff migration is usually carried out by using a velocity function that depends only on depth. This paper introduces a new method, based on Fermat’s principle, which is a compromise between these two extremes. The slowness (reciprocal velocity) function is written as the sum of two functions, the first of which is large and depends only on depth, while the other is small and varies both with depth and position along the line. Raypaths are traced for the first slowness function and are used to calculate migration curves. For each depth point these same raypaths are used to calculate traveltime perturbations due to the laterally varying part of the slowness. The traveltime perturbations are added to the migration curve to obtain an approximation to the exact migration curve. More precisely, suppose that the slowness function can be written in the form [Formula: see text] where [Formula: see text] Using a(z), we generate a table giving two‐way traveltime T as a function of scatterer depth z and surface offset Δ, and a raypath table which gives the ray offset ξ as a function of scatterer depth, surface offset, and ray depth [Formula: see text] For fixed z, T(Δ, z) is a migration curve and conventional Kirchhoff migration of zero‐offset reflection data ψ(x, t) is performed by summation along such curves: [Formula: see text] by [Formula: see text] where W is a weighting factor. The raypath table is used to calculate the traveltime perturbation [Formula: see text] where the integral is taken over the unperturbed raypath. For fixed x and z the new migration curve is [Formula: see text] and migration is per formed by [Formula: see text] This new migration scheme is much less expensive than the exact Kirchhoff scheme because only one set of rays need be traced. Numerical tests have shown that this scheme works surprisingly well even when the lateral variation of velocity is large.

A ray-Kirchhoff method for body-wave calculations in inhomogeneous media: theory

“…Perturbation methods based on velocity model (4.1) have been used in inverse scattering (e.g. Cohen & Hagin 1985) and seismic migration (Carter & Frazer 1984). Their results show that these methods work well even when the 'small' perturbation assumption is violated.…”

“…Contribution from this distant hemisphere is ignored. Mathematically, wave extrapolation amounts to computing the wave field at a depth point x1 from the boundary value at the surface z = 0 (Schneider 1978;Carter & Frazer 1984).…”

Comparison of seismic processing and interpretation tradeoffs between Kirchoff and finite-difference migrations, using poststack migrated data in the west Niger Delta, Nigeria

“…However, Kirchoff's pitfalls include difficulty in handling lateral velocity variation. The method is expensive and can also enhance noise (Jain and deFigueiredo, 1982;Carter and Frazier, 1984). Finite-difference migration model: The migration is based on an algorithm of the type "finite difference" which solves the acoustic wave equation in the spacefrequency (x, f) domain.…”

The Kirchhoff–helmholtz Integral Pair

“…Several approaches to solve the migration problem have been described in the geophysical literature. [2][3][4][5][6] Among the many migration schemes that have been studied so far, there exists one operation that is traditionally accepted as an approximate inverse to the Kirchhoff-Helmholtz integral. It is called Kirchhoff depth migration.…”