Migration/inversion: think image point coordinates, process in acquisition surface coordinates

Abstract

We state a general principle for seismic migration/inversion processes: think image point coordinates; compute in surface coordinates. This principle allows the natural separation of multiple travel paths of energy from a source to a reflector to a receiver. Further, the Beylkin determinant (Jacobian of transformation between processing parameters and acquisition surface coordinates) is particularly simple in stark contrast to the common-offset Beylkin determinant in standard single arrival Kirchhoff .A feature of this type of processing is that it changes the deconvolution structure of Kirchhoff operators or the deconvolution imaging operator of wave equation migration into convolution operators; that is, division by Green's functions is replaced by multiplications by adjoint Green's functions.This transformation from image point coordinates to surface coordinates is also applied to a recently developed extension of the standard Kirchhoff inversion method. The standard method uses Green's functions in the integration process and tends to produce more imaging artefacts than alternatives, such as methods using Gaussian beam representations of Green's functions in the inversion formula. These methods point to the need for a true-amplitude Kirchhoff technique that uses more general Green's functions: Gaussian beams, true-amplitude one-way Green's functions, or Green's functions from the two-way wave equation. Here, we present a derivation of a true-amplitude Kirchhoff that uses these more general Green's functions. When this inversion is recast as an integral over all sources and receivers, the formula is surprisingly simple.