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Introduction to shot-profile migration

The two most widely-used downward-continuation algorithms for imaging prestack seismic data are shot-profile migration and shot-geophone migration.

Shot-profile migration is based an imaging condition proposed by Claerbout (1971) as part of his theory of reflector mapping. He proposed obtaining reflector maps by crosscorrelating upgoing and downgoing wavefields in the earth. The upgoing and downgoing wavefields can be obtained by downward continuing into the earth the recorded wavefield and source function respectively.

An alternative to shot-profile migration is shot-geophone migration, so-called because the entire dataset, parameterized by shot and geophone (or equivalently by midpoint and offset) is downward continued one frequency at a time. This is achieved with a ``survey sinking'' operator derived from the double-square-root (DSR) equation Claerbout (1985).

Both algorithms may also be adapted to image offset-dependent (or angle-dependent) reflectivity. Appendix [*] addresses this subject in more detail.

For a given dataset any differences between migration images produced by the two methods depend on small implementation details. Therefore the choice of whether shot-geophone or shot-profile migration is more appropriate depends on the cost of the method, which in turn depends principally on the acquisition geometry of the dataset.

The cost of shot-geophone migration in the midpoint and offset domain is proportional to the number of midpoints times the number of offsets: in 2-D this is $N_{x} \times N_{h}$, and in 3-D this is $N_{x} \times N_{h_x} \times N_{y} \times N_{h_y}$. The cost of shot-profile migration proportional to the number of shots times the number of midpoints: $N_{x} \times N_{y} \times N_{s}$.

Therefore, for geometries with large numbers of shots, such as typical 3-D marine streamer geometries, shot-geophone migration is a more attractive choice, especially when the dimensionality of the problem can be reduced by a common-azimuth approximation Biondi and Palacharla (1996). However, for wide-azimuth geometries where the number of shots (or reciprocal receivers) is small compared to the number of offsets, shot-profile migration may be preferable. 3-D data collected with technologies such as vertical cables, borehole seismometers, and ocean bottom seismometers, may be efficiently migrated with shot-profile methods.

Like many other industrial strength geophysical processes, shot-profile migration is the adjoint of a linear forward modeling operator that mimics wave propagation in the earth. By breaking down the shot-profile modeling and migration operators into their constituent components, I am able to show that for some sparse-shot geometries, the least-squares inverse of the forward modeling operator is a chain of conventional migration operator followed by a model-space weighting function that can be calculated cheaply during the migration process.

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