Traveltime sensitivity kernels: Banana-doughnuts or just plain bananas?

The Banana-doughnut paradox

The important paradox is not the apparent contradiction between ray-theoretical and finite-frequency sensitivity kernels, since they are compatible in the high-frequency limit. Instead, the paradox is how do you reconcile the zero-sensitivity along the ray-path with your intuitive understanding of wave propagation?

A first potential resolution to the paradox is that the wavefront healing removes any effects of a slowness perturbation. This alone is a somewhat unsatisfactory explanation since it does not explain why traveltimes are sensitive to slowness perturbations just off the geometric ray-path.

A second potential resolution is that the hollowness of the banana is simply an artifact of modeling procedure. This is partially true. Both Born and Rytov are single scattering approximations, and a single scatterer located on the geometric ray-path may only contribute energy in-phase with the direct arrival. In contrast, if there are two scatterers on the geometric ray-path traveltimes may be affected. However, just because the paradox may appear to be an artifact of the modeling procedure does not mean it is not a real phenomenom. In the weak scattering limit, traveltimes will indeed be insensitive to a slowness perturbation situated on the geometric ray-path.

Traveltime sensitivity kernels: Banana-doughnuts or just plain bananas?