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

Summary We use a coupled surface wave version of the Born approximation to compute the3‐D sensitivity kernel K T ( r ) of a seismic body wave traveltime T measured by cross‐correlation 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 Δ T ( r ) of a differential traveltime Δ T measured by cross‐correlation 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 Δ T only if it is smooth.