Isotropic angle-domain elastic reverse-time migration

Elastic imaging vs. acoustic imaging

Multicomponent elastic data are often recorded in land or marine (ocean-bottom) seismic experiments. However, as mentioned earlier, elastic vector wavefields are not usually processed by specifically designed imaging procedures, but rather by extensions of techniques used for scalar wavefields. Thus, seismic data processing does not take full advantage of the information contained by elastic wavefields. In other words, it does not fully unravel reflections from complex geology or correctly preserve imaging amplitudes and estimate model parameters, etc.

Elastic wave propagation in a infinite homogeneous isotropic medium is characterized by the wave equation (Aki and Richards, 2002)

$$\rho\frac{\partial^2 {\bf u}}{\partial t^2} ={\bf f} +\lef... ...\bf u}} \right) -\mu\nabla \times {\nabla \times {{\bf u}}}$$ (1)

where ${\bf u}$ is the vector displacement wavefield, $t$ is time, $\rho$ is the density, ${\bf f}$ is the body source force, $\lambda$ and $\mu$ are the Lamé moduli. This wave equation assumes a slowly varying stiffness tensor over the imaging space. For isotropic media, one can process the elastic data either by separating wave-modes and migrating each mode using methods based on acoustic wave theory, or by migrating the whole elastic data set based on the elastic wave equation . The elastic wavefield extrapolation using equation  is usually performed in time by Kirchhoff migration or reverse-time migration.

Acoustic Kirchhoff migration is based on diffraction summation, which accumulates the data along diffraction curves in the data space and maps them onto the image space. For multicomponent elastic data, Kuo and Dai (1984a) discuss Kirchhoff migration for shot-record data. Here, identified PP and PS reflections can be migrated by computing source and receiver traveltimes using P-wave velocity for the source rays, and P- and S-wave velocities for the receiver rays. Hokstad (2000) performs multicomponent anisotropic Kirchhoff migration for multi-shot, multi-receiver experiments, where pure-mode and converted mode images are obtained by redatuming visco-elastic vector wavefields and application of a survey-sinking imaging condition to the reconstructed vector wavefields. The wavefield separation is effectively done by the Kirchhoff integral which handles both P- and S-waves, although this technique fails in areas of complex geology where ray theory breaks down.

Elastic reverse-time migration has the same two components as acoustic reverse-time migration: reconstruction of source and receiver wavefield and application of an imaging condition. The source and receiver wavefields are reconstructed by forward and backward propagation in time with various modeling approaches. For acoustic reverse-time migration, wavefield reconstruction is done with the acoustic wave-equation using the recorded scalar data as boundary condition. In contrast, for elastic reverse-time migration, wavefield reconstruction is done with the elastic wave-equation using the recorded vector data as boundary condition.

Since pure-mode and converted-mode reflections are mixed on all components of recorded data, images produced with reconstructed elastic wavefields are characterized by crosstalk due to the interference of various wave modes. In order to obtain images with clear physical meanings, most imaging conditions separate wave modes . There are two potential approaches to separate wavefields and image elastic seismic wavefields. The first option is to separate P and S modes on the acquisition surface from the recorded elastic wavefields. This procedure involves either approximations for the propagation path and polarization direction of the recorded data, or reconstruction of the seismic wavefields in the vicinity of the acquisition surface by a numerical solution of the elastic wave equation, followed by wavefield separation of scalar and vector potentials using Helmholtz decomposition (Zhe and Greenhalgh, 1997; Etgen, 1988). An alternative data decomposition using P and S potentials reconstructs wavefields in the subsurface using the elastic wave equation, then decomposes the wavefields into P- and S-wave modes. This is followed by forward extrapolation of the separated wavefields back to the surface using the acoustic wave equation with the appropriate propagation velocity for the various wave modes (Sun et al., 2006) by conventional procedures used for scalar wavefields.

The second option is to extrapolate wavefields in the subsurface using a numerical solution to the elastic wave equation and then apply an imaging condition that extracts reflectivity information from the source and receiver wavefields. In the case where extrapolation is implemented by finite-difference methods (Chang and McMechan, 1994,1986), this procedure is known as elastic reverse-time migration, and is conceptually similar to acoustic reverse-time migration (Baysal et al., 1983), which is more frequently used in seismic imaging.

Many imaging conditions can be used for reverse-time migration. Elastic imaging conditions are more complex than acoustic imaging conditions because both source and receiver wavefields are vector fields. Different elastic imaging conditions have been proposed for extracting reflectivity information from reconstructed elastic wavefields. Hokstad et al. (1998) use elastic reverse-time migration with Lamé potential methods. Chang and McMechan (1986) use the excitation-time imaging condition which extracts reflectivity information from extrapolated wavefields at traveltimes from the source to image positions computed by ray tracing, etc. Ultimately, these imaging conditions represent special cases of a more general type of imaging condition that involves time cross-correlation or deconvolution of source and receiver wavefields at every location in the subsurface.

Isotropic angle-domain elastic reverse-time migration