Three-dimensional sensitivity kernels for finite-frequency traveltimes: the banana-doughnut paradox

We use a coupled surface wave version of the Born approximation to compute the 3-D sensitivity kernel K-T(r) of a seismic body wave traveltime T measured by crosscorrelation of a broad-band waveform with a spherical earth synthetic seismogram. The geometry of a teleseismic S wave kernel is, at first sight, extremely paradoxical: the sensitivity is zero everywhere along the geometrical ray! The shape of the kernel resembles that of a hollow banana; in a cross-section perpendicular to the ray, the shape resembles a doughnut. The cross-path extent of such a banana-doughnut kernel depends upon the frequency content of the wave. The kernel for a very high-frequency wave is a very skinny hollow banana; wave-speed heterogeneity wider than this banana affects the traveltime, in accordance with ray theory. We also use the Born approximation to compute the sensitivity kernel K-Delta T(T) Of a differential traveltime Delta T measured by crosscorrelation of two phases, such as SS and S, at the same receiver. The geometries of both an absolute SS wave kernel and a differential SS-S kernel are extremely complicated, particularly in the vicinity of the surface reflection point and the source-to-receiver and receiver-to-source caustics, because of the minimax character of the SS wave. Heterogeneity in the vicinity of the source and receiver exerts a negligible influence upon an SS-S differential traveltime Delta T only if it is smooth.