Vertical seismic profile (VSP) data may be partitioned in a variety of ways by application of wave‐field transformations. These transformations provide insights into the nature of the data and aid in the design of processing operations. Transformations are implemented in a reversible sequence that takes the observed VSP data from the depth‐time (z-t) domain through the slowness‐time intercept (p-τ) domain (by a slant stack), to the slowness‐frequency (p-ω) domain (by a 1-D Fourier transform over τ), to the wavenumber‐frequency (k-ω) domain (by resampling using the Fourier central‐slice theorem), and finally back to the z-t domain (by an inverse 2-D Fourier transform). Multidimensional wave‐field transformations, combined with k-ω, p-ω, and p-τ filtering, can be applied to wave‐field resampling, interpolation, and extrapolation; separation of P-waves and S-waves; separation of upgoing and downgoing waves; and wave‐field decomposition for isolation, identification, and analysis of arrivals.

Summary 2‐D modelling and imaging of line double‐couple and spatially extended earthquake sources is demonstrated. Extrapolation of two‐component (elastic) wavefields is implemented using finite differences. Running the extrapolatioin forward in time produces synthetic seismograms; running it backward in time produces an image of the source. Provided that the recording aperture is adequate, the source location, extent, orientation and radiation pattern may be recovered from the data. Synthetic examples include a normal fault at the edge of a rift, a reverse fault in a subduction zone and a line double couple in a cylindrical, whole‐earth model.

Subsurface imaging with common‐source cross‐hole data can be achieved using prestack reverse‐time migration. The algorithm consists of extrapolation of the recorded wave field, application of the excitation‐time imaging condition, and postprocessing of the resulting image with a low‐pass wavenumber filter. The wavenumber filter removes the artifact associated with the direct arrival; this artifact is not separable from the scattered data before migration because, in the cross‐hole geometry, they significantly overlap in time, space, and wavenumber. Migration of synthetic data produces the best possible results, but images produced by migration of scale‐model data are not greatly inferior. Apparently, acceptable images can be obtained from a surprisingly few sources, if these sources are located sufficiently far apart to give independent information and the recording aperture is sufficiently wide.

This note addresses two methods of computing the imaging condition for prestack migration of common‐shot seismic data; our work is based on the ideas from reverse‐time migration for both poststack (Loewenthal and Mufti, 1983; McMechan, 1983) and prestack data (Chang and McMechan, 1986). In reverse‐time migration of poststack data, the whole stacked section is backward‐extrapolated in time, with half of the medium velocity to time zero. All exploding reflectors are imaged at once at time zero. The time zero is referred to as the imaging condition. In prestack migration, the imaging condition is more involved. Each spatial grid point (treated as a point diffractor) has a different excitation time, which is equal to the one‐way traveltime from the source to that grid point. Each point diffractor is imaged separately at its excitation (the “imaging time”).

Prestack reverse‐time migration for acoustic waves has recently been developed for vertical seismic profile (VSP) data (Chang and McMechan, 1986) and for cross‐hole (CH) data (Hu et al., 1988). Both sets of authors use the same migration software and produce images from the scattered (reflected and diffracted) energy in the recorded wave fields.