A fundamental problem with ray-based MVA is that rays are poor approximations of the actual wavepaths when a band-limited seismic wave propagates through a rugose top of the salt. Figure 1 illustrates this issue quite clearly. It shows three sensitivity kernels for frequencies of 1-26 Hz. The top panel in Figure 1 shows a wavepath that could be reasonably approximated using the method introduced by Lomax (1994) to trace fat rays using asymptotic methods. In contrast, the wavepaths shown in both the middle and bottom panels cannot be well approximated using Lomax' method. The amplitude and shapes of these wavepaths are much more complex than a simple fattening of a geometrical ray could ever describe. The bottom panel illustrates the worst situation for ray-based tomography because the rugosity of the top of the salt has the same scale as the spatial wavelength of the seismic wave.
The fundamental reason why the true wavepaths cannot be approximated using fattened geometrical rays is that they are frequency dependent. Figure 2 illustrates this dependency by depicting the wavepath shown in the bottom panel of Figure 1 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 and defocussing of the energy varies with the frequency bandwidth.
In the next example (Figures 3 and 4) we compare the shapes of sensitivity kernels when we change the type of source for the background wavefield, its frequency content and the method used to generate an image perturbation in the subsurface. As for the preceding example, we show the results as a superposition of the velocity model, the background wavefield and the sensitivity kernel from a fixed point in the subsurface.
Figure 3 shows the sensitivity kernels for a point source on the surface, and Figure 4 shows the sensitivity kernels for a plane-wave propagating vertically at the surface. In both pictures, the left column corresponds to kinematic image perturbations of equation (6), and the right column corresponds to amplitude image perturbations of equation (7) obtained by scaling of the background image by an arbitrary number. From top to bottom, we show sensitivity kernels of increasing frequency range: 1-4 Hz, 1-8 Hz, 1-16 Hz and 1-32 Hz. Once again, we can see the large frequency dependence of the sensitivity kernels. The area of sensitivity reduces with increased frequency which is a clear indication that a frequency dependent migration velocity analysis method like WEMVA can better handle subsalt environments with patchy illumination and that illumination itself is a frequency dependent phenomenon which needs to be addressed in this way.
Finally, we show wave-equation MVA sensitivity kernels for a 3D velocity model Figure 5 corresponding to a salt environment. We consider the case of a point source on the surface and data with a frequency range of 1-16 Hz. Figure 6 shows the sensitivity kernel for a kinematic image perturbation, while Figure 7 for a amplitude image perturbation. In both cases, the shapes of the kernels are complicated, which is an expression of the multipathing occurring as waves propagate through rough salt bodies. The horizontal slice indicates multiple paths linking the source point on the surface with the image perturbation in the subsurface.
One noticeable characteristic is that the sensitivity kernels constructed from amplitude image perturbations show the largest sensitivity in the center of the kernel, as opposed the kinematic kernels which show the largest sensitivity away from the central path. This phenomenon was discussed by Dahlen et al. (2000) in the context of finite-frequency traveltime tomography. We illustrate it for WEMVA in Figure 8 with two horizontal slices in the sensitivity kernels shown in Figures 6 and 7.