Introduction

Migration imaging is based on the concept of dephasing. In poststack exploding reflector modeling, many subsurface point sources explode simultaneously at time t=0, and the generated wavefields propagate upward to the earth surface to be recorded. Migration is the process that transfers the surface recorded wavefields into the time t=0 wavefield, which is regarded as the subsurface reflectivity image, by removing the upward propagation phase accumulated by the modeling. For the purpose of dephasing, there are two categories of methods, downward continuation in depth and reverse propagation in time. The first category includes three subcategories, finite-difference (Claerbout, 1976, 1985), Kirchhoff integral (French, 1974, 1975; Schneider, 1978), and f-k phase-shift methods (Gazdag, 1978). The second category includes reverse time migration implemented in the space-time domain (McMechan, 1983) and in the f-k domain (Baysal et al., 1983). The dephasing of the f-k one-pass transformation method (Stolt, 1978) is accomplished by mapping from the temporal frequency into the spatial depth wavenumber. However, the idea of migration dephasing is best explained by examining the phase shift method (Gazdag, 1978).

In wave phenomena, the surface recorded wavefields, boundary conditions, are secondary point sources, according to Huygens' wavefront construction principle. And there are many point sources. If the clock is run forward in time, the boundary point source wavefields continue to expand upward; this is the modeling process. If the clock is run backward in time, the boundary point source wavefields focus downward; this is the migration process. The downward continuation in depth and reverse propagation in time of wavefields are equivalent in that they both result in downward focusing of the boundary source wavefields for propagation phase reduction.

The previous migration scheme that explicitly takes the surface recorded seismograms as sources is the reverse-time migration method (Levin, 1983). Reverse-time migration, taking the surface recorded wavefields as boundary conditions (Baysal et al., 1983; McMechan, 1983), actually treats them as a series of parallel point sources.

In common-shot forward modeling, seismic waves are generated by a surface point source, and propagate away; the history wavefields at the boundary receiver locations are the recorded wavefields. The mathematical methods that are used to do the common-shot forward modeling are ray-tracing-based methods (Cerveny et al., 1977), wave-equation-based methods [finite-difference (Kelly et al., 1976) and finite-element], and hybrid methods, such as the Gaussian beam approach (Cerveny et al., 1982).

Gaussian beam depth migration has been implemented by several researchers, Costa et al. (1989) through Gabor transform, Hill (1990) through f-k transform, and Lazaratos and Harris (1990) through $\tau-p$ transform. This paper describes the use of Gaussian beam wave propagation theory to migrate the surface recorded data as a series of individual point sources.

In Gaussian beam common-shot forward modeling, the propagation phase increases downward away from the source at the surface. In contrast, in Gaussian beam depth migration, the propagation phase decreases downward away from the source; that is, the surface recorded data are backprojected into the subsurface to image the high-wavenumber velocity band - the reflectivity. What is needed as additional input is a low-wavenumber velocity background. And in migration, every image point is a receiver.

The Gaussian beam depth migration combines the advantages of raytracing based Kirchhoff depth migration and wave equation migration. With Gaussian beam migration, as with Kirchhoff migration, the recording geometry need not be regular. And, as with wave equation migration, the wavefield is regular in caustic regions. Moreover, two-point ray tracing is unnecessary for Gaussian beam migration.

This paper first reviews the theory of Gaussian beam wave propagation. Then it demonstrates poststack Gaussian beam depth migration with synthetic data. Finally, it demonstrates prestack Gaussian beam migration, and outlines the procedures for doing Gaussian beam migration using earthquake seismic data.