Sensitivity kernels examples

In the first example (fat2d.Tray2a and fat2d.Tray2b), I compare the shapes of sensitivity kernels when changing the type of source for the background wavefield, its frequency content and the method used to generate an image perturbation in the subsurface. I show the results as a superposition of the velocity model, the background wavefield and the sensitivity kernels from a fixed point in the subsurface.

fat2d.Tray2a shows the sensitivity kernels for a point source on the surface, and fat2d.Tray2b 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 kin, and the right column corresponds to amplitude image perturbations of amp 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, I show wave-equation MVA sensitivity kernels for a 3D velocity model fat3.sC corresponding to a salt environment. I consider the case of a point source on the surface and data with a frequency range of 1-16 Hz. fat3.fp3 shows the sensitivity kernelfor a kinematic image perturbation, while fat3.fq3 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 (28) in the context of finite-frequency traveltime tomography. I illustrate it for WEMVA in fat3.svty with two horizontal slices in the sensitivity kernels shown in fat3.fp3 and fat3.fq3.