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 rst 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 aects the traveltime, in accordance with ray theory. We also use the Born approximation to compute the sensitivity kernel K *T (r) of a dierential traveltime *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 dierential SS^S kernel are extremely complicated, particularly in the vicinity of the surface reection point and the sourceto-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 inuence upon an SS^S dierential traveltime *T only if it is smooth.