Many factors determine the failure of ray-based MVA in a sub-salt environment. Some of them are successfully addressed by the wave-equation MVA (WEMVA) method, whereas others, for example the problems that are caused by essential limitations of the recorded reflection data, are only partially solved by WEMVA.
An important practical difficulty encountered when using rays to estimate velocity below rugose salt bodies is the instability of ray tracing. Rough salt topology creates poorly illuminated areas, or even shadow zones, in the subsalt region. The spatial distribution of these poorly illuminated areas is very sensitive to the velocity function. Therefore, it is often extremely difficult to trace rays connecting a given point in the poorly illuminated areas with a given point at the surface (two-point ray-tracing). Wavefield extrapolation methods are robust with respect to shadow zones and they always provide wavepaths usable for velocity inversion.
A related and more fundamental problem with ray-based MVA, is that rays poorly approximate actual wavepaths when a band-limited seismic wave propagates through a rugose top of the salt. zifat illustrates this issue by showing three band-limited (1-26 Hz) wavepaths, also known in in the literature as fat rays or sensitivity kernels (114; 28; 72). Each of these three wavepaths is associated with the same point source located at the surface but corresponds to a different sub-salt ``event''. The top panel in zifat shows a wavepath that could be reasonably approximated using the method introduced by (57) to trace fat rays using asymptotic methods. In contrast, the wavepaths shown in both the middle and bottom panels in zifat cannot be well approximated using Lomax' method. The amplitude and shapes of these wavepaths are significantly more complex than a simple fattening of a geometrical ray could ever describe. The bottom panel illustrates the worst-case-scenario situation for ray-based tomography because the variability of the top salt topology is at the same scale as the spatial wavelength of the seismic wave. The fundamental reason why true wavepaths cannot be approximated using fattened geometrical ray is that they are frequency dependent. zifrq2 illustrates this dependency by depicting the wavepath shown in the bottom panel of zifat as a function of the temporal bandwidth: 1-5 Hz (top), 1-16 Hz (middle), and 1-64 Hz (bottom). The width of the wavepath decreases as the frequency bandwidth increases, and the focusing/defocussing of energy varies with the frequency bandwidth.
The limited and uneven ``illumination'' of both the reflectivity model and the velocity model in the subsalt region is a challenging problem for both WEMVA and conventional ray-based MVA (see SIG.imgC for an example of this problem). For the reflectors under salt, the angular bandwidth is drastically reduced in the angle-domain common image gathers (adcig). This phenomenon is caused by a lack of oblique wavepaths in the subsalt, which deteriorates the ``sampling'' of the velocity variations in the subsalt. Consequently, the velocity inversion is more poorly constrained in the subsalt sediments than in the sediments on the side of the salt body.
Uneven illumination of subsalt reflectors is even more of a challenge than reduced angular coverage. It makes the velocity information present in the ADCIGs less reliable by causing discontinuities in the reflection events and creating artifacts. MVA methods assume that when the migration velocity is correct, events are flat in ADCIGs along the aperture-angle axis. Velocity updates are estimated by minimizing curvature of events in ADCIGs. MVA methods may provide biased estimates where uneven illumination creates events that are bending along the aperture-angle axis, even where the image is created with correct velocity. I address this issue by weighting the image perturbations before inverting them into velocity perturbations, as described in wemva. The weights are function of the ``reliability'' of the moveout measurements in the ADCIGs.
I demonstrate the WEMVA method using synthetic and real datasets corresponding to subsalt environments.