Three-dimensional Frechet differential kernels for seismic delay times

Seismic traveltimes are the most widely exploited data in seismology. Their Frechet or sensitivity kernels are important tools in tomographic inversions based on the Born or single-scattering approximation. The current study is motivated by a paradox posed by two seemingly irreconcilable observations in the numerical calculations for the sensitivity kernels of the traveltime perturbations. Calculations of kernels for 2-D media by the normal-mode approach indicate that traveltimes are most sensitive to the structure on and around the geometrical ray paths corresponding to the seismic arrivals, whereas calculations for 3-D media by geometrical ray theory predict exactly zero sensitivities on the ray paths. In the current work, we employ these two completely different wave-propagation approaches, the more efficient geometrical ray theory and the more accurate normal-mode theory, to investigate the 3-D sensitivities of the delay times to shear-wave speed variations. Expressions for the delay-time Frechet kernels are presented for both methods, and extensive numerical experiments are conducted for various types of seismic phases as well as for different reference earth models. The results show that the contradictory observations are but two examples of a wide range of behaviours in the delay-time sensitivity. For most of the seismic phases in realistic reference models with multiple discontinuities, wave-speed gradients and low-velocity zones, the wavefields are highly complicated and ray theory, which describes the response by the contributions of a few geometrical rays between the source and receiver, produces qualitatively different delay-time kernels from those obtained by the normal-mode theory, which includes essentially all contributions.