Introduction

Cross-well seismic survey has proven to be an effective, high-resolution method of reservoir characterization because of its high, wide frequency band (Beydoun et al., 1989). Several researchers (McMechan et al., 1987; Harris, 1988; Bregman et al., 1989; Chen et al., 1990; Luo and Schuster, 1991; Lines, 1991) have developed methods of transmission tomography to produce tomographic velocity cross-sections of the subsurface between the two wells. But transmission traveltime tomography uses only the traveltimes of the first arrivals, which represent only a small fraction of the total information contained in the data. Scattered events in seismic sections, such as reflections and diffractions, carry rich lithologic information. Full realization of the potential of crosswell seismic data for detailed delineation of the geological structure requires the imaging of these scattered events, which promises resolution in the scale of wavelength. For this purpose, Pratt and Worthington (1988) applied diffraction tomography and Beydoun and Mende (1989) applied ray-Born migration/inversion to crosswell data. Recently, Lazaratos et al. (1991) applied VSP-CDP mapping to crosswell reflection imaging with success. But a more robust method would have to be based on migration that can honor arbitrary velocity structures. This paper describes such a method, reverse-time migration.

In a crosswell seismic survey geometry, the shots and receivers are located in separate wells. Thus crosswell migration is best considered as a problem of prestack common-shot profile migration. A profile consists of the seismograms of one shot and many receivers. Because it is a true physical experiment, processing of profiles is more fundamental than that of other geometries. Methods of prestack profile migration were first presented by Claerbout (1970, 1976). According to his U/D imaging concept, reflectors exist in the earth at places where the onset of the downgoing wave is time-coincident with an upcoming wave. Several authors have since extended Claerbout's theorem of wavefield extrapolation and the imaging condition (Jacobs, 1982; Reshef and Kosloff, 1986; Chang and McMechan, 1986). Chang and McMechan (1986) generalized Claerbout's imaging condition, and stated the excitation-time imaging condition for prestack reverse-time migration, by which each reflector point is imaged at the one-way direct-wave traveltime from the source to the point at which it is excited. Reverse-time profile migration with the excitation-time imaging condition consists of three elements: (1) computation of the imaging condition, that is, computation of the direct wave traveltime from the source point to every image point, (2) extrapolation backward in time of the common-shot profile wavefields from the receivers into the background velocity model by the two-way wave equation finite-difference method, and (3) imaging. Imaging is performed at each step of the wavefield extrapolation process by extracting, from the propagating wavefields, the values at the particular grid points that are excited and thus imaged at that time and adding them as reflectivity into the image plane at the same spatial locations. The locus of all points imaged at one time step corresponds to the direct wave exciting wavefront from the source.

Hu et al. (1988) and Zhu et al. (1988) presented reverse-time migration of crosswell seismic data by second-order finite differencing in space and time. Because of its strict requirements of fine spatial sampling, second-order finite differencing often is not sufficient for modeling surface seismic data that has a frequency band much lower than that of crosswell seismic data. The success of modeling high frequency crosswell seismic data depends on efficient implementation of high-order finite differencing, which requires much fewer spatial samples per wavelength for accurate numerical computation (Dablain, 1986). Gray and Lines (1992) introduced Kirchhoff depth migration of crosswell seismic data by two separate VSP migrations. However, their method is difficult to implement because it requires first decomposing the already complex wavefields into upgoing and downgoing reflections.

This paper first reviews modeling and migration, showing how they are interrelated. It then describes the application of fourth-order-in-time, tenth-order-in-space finite differencing to perform crosswell seismic modeling of the two-way acoustic wave equation, on velocity models that contain high-wavenumber velocity reflectors superimposed in a low-wavenumber smooth velocity background. Finally, it shows how reverse-time wavefield extrapolation coupled with the excitation-time imaging condition can be applied to perform migration on the scattered wavefields of crosswell seismic common-shot profiles. The modeling and migration are based on the geometry of a true field survey. At the end, the paper presents some suggestions to facilitate field data migration, which include removal of tube waves and wavefield separation.