To apply reverse‐time migration to prestack, finite‐offset data from variable‐velocity media, the standard (time zero) imaging condition must be generalized because each point in the image space has a different image time (or times). This generalization is the excitation‐time imaging condition, in which each point is imaged at the one‐way traveltime from the source to that point. Reverse‐time migration with the excitation‐time imaging condition consists of three elements: (1) computation of the imaging condition; (2) extrapolation of the recorder wave field; and (3) application of the imaging condition. Computation of the imaging condition for each point in the image is done by ray tracing from the source point; this is equivalent to extrapolation of the source wave field through the medium. Extrapolation of the recorded wave field is done by an acoustic finite‐difference algorithm. Imaging is performed at each step of the finite‐difference extrapolation by extracting, from the propagating wave field, the amplitude at each mesh point that is imaged at that time and adding these into the image space at the same spatial locations. The locus of all points imaged at one time step is a wavefront [a constant time (or phase) trajectory]. This prestack migration algorithm is very general. The excitation‐time imaging condition is applicable to all source‐receiver geometries and variable‐velocity media and reduces exactly to the usual time‐zero imaging condition when used with zero‐offset surface data. The algorithm is illustrated by application to both synthetic and real VSP data. The most interesting and potentially useful result in the processing of the synthetic data is imaging of the horizontal fluid interfaces within a reservoir even when the surrounding reservoir boundaries are not well imaged.
CHANG, W.F. and MCMECHAN, G. A. 1990. 3D acoustic prestack reverse-time migration. Geophysical Prospecting 38,737-755.A prestack reverse-time migration algorithm which operates on common-source gathers, recorded at the Earth's surface, from 3D structures, is conceived, implemented and tested. Reverse-time extrapolation of the recorded wavefield (a boundary-value problem), and computation of the excitation-time imaging condition for each point in a 3D volume (an initialvalue problem), are both performed using a second-order finite-difference solution of the full 3D scalar wave equation. The algorithm is illustrated by processing synthetic data for a point diffractor, an oblique wedge, and the French double dome and fault model.
S U M M A R Y Reverse-time imaging of earthquake source parameters is extended from two to three spatial dimensions and from two-to three-component recordings. Provided that the recording aperture is sufficiently large, and the data are not spatially aliased, source radiation patterns, in both time and space, can be reconstructed by elastic reverse-time propagation of body waves; such reconstructions are necessarily partial as only the energy that was recorded is available for reconstruction. For a point source, the origin time and the (3-D) spatial location of an event can be reconstructed by extracting the time and position, of the best focused energy from the backward propagating wavefield. For a spatially and temporally extended source, biased estimates of fault position and rupture/slip time history can be estimated if the recording aperture is sufficient; for smaller recording apertures, the time history can still be estimated if the fault location and geometry are known a priori. The latter is viable for a reasonable number (<200) of three-component recordings, and so is potentially applicable to real data. All these processes assume that a smoothed representation of the 3-D velocity distribution in the volume containing the source and receivers is available.
CHANG, W.F. and MCMECHAN, G.A. 1989. 3D acoustic reverse-time migration. Geophysical Prospecting 37,243-256.Acoustic reverse-time finite-difference migration for zero-offset data is extended from two-to three-dimensional media. The formulation is based on the full three-dimensional acoustic wave equation and so has no dip restrictions and it involves extrapolation in a velocity distribution variable in three dimensions. The algorithm is demonstrated by successful migration of synthetic data sets for three models: a point diffractor, an oblique pinchout, and a dome overlying a planar reflector.
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