Isotropic angle-domain elastic reverse-time migration

Discussion

Our presentation of the angle-domain reverse-time migration method outlined in the preceding sections deliberately ignores several practical challenges in order to maintain the focus of this paper to the actual elastic imaging condition. However, for completeness, we would like to briefly mention several complementary issues that need to be addressed in conjunction with the imaging condition in order to design a practical method for elastic reverse-time migration.

First, reconstruction of the receiver wavefield requires that the multicomponent recorded data be injected into the model in reverse-time. In other words, the recorded data act as a displacement sources at receiver positions. In elastic materials, displacement sources trigger both compressional and transverse wave modes, no matter what portion of the recorded elastic wavefield is used as a source. For example, injecting a recorded compressional mode triggers both a compressional (physical) mode and a transverse (non-physical) mode in the subsurface . Both modes propagate in the subsurface and might correlate with wave modes from the source side. There are several ways to address this problem, such as by imaging in the angle-domain where the non-physical modes appear as events with non-flat moveout. We can make an analogy between those non-physical waves and multiples that also lead to non-flat events in the angle-domain. Thus, the source injection artifacts might be eliminated by filtering the migrated images in the angle domain, similar to the technique employed by Sava and Guitton (2005) for suppressing multiples after imaging.

Second, the data recorded at a free surface contain both up-going and down-going waves. Ideally, we should use only the up-going waves as a source for reconstructing the elastic wavefields by time-reversal. In our examples, we assume an absorbing surface in order to avoid this additional complication and concentrate on the imaging condition. However, practical implementations require directional separation of waves at the surface (Hou and Marfurt, 2002; Wapenaar et al., 1990; Admundsen et al., 2001; Wapenaar and Haimé, 1990; Admundsen and Reitan, 1995). Furthermore, a free surface allows other wave modes to be generated in the process of wavefield reconstruction using the elastic wave-equation, e.g. Rayleigh and Love waves. Although those waves do not propagate deep into the model, they might interfere with the directional wavefield separation at the surface.

Third, we suggest in this paper that angle-dependent reflectivity constructed using extended imaging conditions might allow for elastic AVA analysis. This theoretical possibility requires that the wavefields are correctly reconstructed in the subsurface to account for accurate amplitude variation. For example, boundaries between regions with different material properties need to be reasonably located in the subsurface to generate correct mode conversions, and the radiation pattern of the source also needs to be known. Neither one of these aspects is part of our analyses, but they represent important considerations for practical elastic wavefield imaging.

Fourth, the wave-mode separation using divergence and curl operators, as required by Helmholtz decomposition, does not work well in elastic anisotropic media. Anisotropy requires that the separation operators take into account the local anisotropic parameters that may vary spatially (Yan and Sava, 2009). However, we do not discuss anisotropic wave-mode decomposition in this paper and restrict our attention to angle-domain imaging in isotropic models.

Isotropic angle-domain elastic reverse-time migration