WEMVA sensitivity kernels

Depth imaging of complex structures depends on the quality of the velocity model. However, conventional Migration Velocity Analysis (MVA) procedures often fail when the wavefield is severely distorted by lateral velocity variations and thus complex multipathing occurs. In the preceding sections, I introduce a method of migration velocity analysis using wave-equation techniques (WEMVA), which aims to improve the quality of migrated images, mainly by correcting moveout inaccuracies of specular energy. WEMVA finds a slowness perturbation which corresponds to an image perturbation. It is thus similar to ray-based migration tomography (104; 14; 3; 33; 55), where the slowness perturbation is derived from depth errors, and to wave-equation inversion (108) or tomography (114; 28; 72) where the slowness perturbation is derived from measured wavefield perturbations.

WEMVA has the potential of improving velocity estimation when complex wave propagation makes conventional ray-based MVA methods less reliable. Imaging under rugged salt bodies is an important case where WEMVA has the potential of making a difference in the imaging results. Here, I analyze the characteristics of the tomographic operator inverted in WEMVA to update the velocity model, and contrast these characteristics with the well-known characteristics of ray-based tomographic operators.

One way of characterizing integral operators, e.g. tomography operators, is through sensitivity kernels, which describe the sensitivity of a component of a member of the data space to a change of a component of a member of the model space. In this section, I formally introduce the sensitivity kernels for wave-equation migration velocity analysis and show 2D and 3D examples.

The analysis of WEMVA sensitivity kernels provides an intuition on WEMVA's potential for overcoming limitations of ray-based MVA. Some of these limitations are intrinsic, other are practical. An important practical difficulty encountered when using rays to estimate velocity below salt bodies with rough boundaries is the instability of ray tracing. Rough salt topographies create 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 the rays that connect 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 (i.e. sensitivity kernels) usable for velocity inversion.

Ray-tracing has intrinsic limitations when modeling wave-propagation through salt bodies with complex geometry, because of the asymptotic assumption on which it is based. This intrinsic limitation prevent ray-tracing from modeling the frequency-dependency of full-bandwidth wave propagation. The comparison of sensitivity kernels computed assuming different frequency bandwidths illustrates clearly the drawbacks of the asymptotic assumptions. Top-salt rugosity causes the WEMVA sensitivity kernels to be strongly dependent on the bandwidth. Furthermore, in these conditions, sensitivity kernels are drastically different from simple ``fat rays''. Therefore, they cannot be approximated by kernels computed by a bandwidth-dependent fattening of geometric rays (57).

I compute the sensitivity kernels for perturbations in the phase as well as perturbations in the amplitude. It is interesting to notice that the 3D kernels for phase perturbations are hollow in the middle, exactly where the geometric rays would be. This result is consistent with the observations first made by (113); then extensively discussed in the global seismology community (28; 58), and further analyzed by (76).

 

fat2d.Tray2a
Figure 10 The dependence of sensitivity kernels to frequency and image perturbation. From top to bottom, the frequency range is 1-4 Hz, 1-8 Hz, 1-16 Hz and 1-32 Hz. The left column corresponds to kinematic image perturbations, and the right column corresponds to dynamic image perturbations. The wavefield is produced from a point source.

 

fat2d.Tray2b
Figure 11 The dependence of sensitivity kernels to frequency and image perturbation. From top to bottom, the frequency range is 1-4 Hz, 1-8 Hz, 1-16 Hz and 1-32 Hz. The left column corresponds to kinematic image perturbations, and the right column corresponds to dynamic image perturbations. The wavefield is produced by a horizontal incident plane-wave.

 

• Fréchet derivative integral kernels
• Sensitivity kernels examples