Path-average kernels for long wavelength traveltime tomography

Abstract

SUMMARY Although much effort goes into improving the resolution of tomographic models, investigating their quality has only just started. Probabilistic tomography provides a framework for the quantitative assessment of uncertainties of long-wavelength tomographic models. So far, this technique has been used to invert maps of surface wave phase velocities and normal-mode splitting functions. Including body waves would substantially increase the depth resolution in the lowermost mantle. In surface wave tomography, the construction of phase velocity maps and splitting functions is a well-defined inverse problem, and the depth inversion is less well constrained but characterized by a small number of dimensions suitable for a Monte Carlo search. Traveltime tomography is mostly based on ray theory that covers the 3-D Earth, thus the dimension of the inverse problem is too large for a Monte Carlo search. The ray-mode duality suggests to apply the path-average approximation to body wave traveltimes. In this way the measured traveltime residual as a function of ray parameter can be inverted using path-average kernels, which depend on depth only, similar to surface wave tomography. We investigate the validity of the path-average approximation for delay times in both the forward and the inverse problem using the velocity model S20RTS as well as random models. We numerically illustrate the precision of such kernels compared with ray-theoretic and finitefrequency ones. We further invert traveltime residuals, calculated from Fermat rays, using the path-average kernels. We find that the agreement between classical ray theory and path-average theory is good for long wavelength structures. We suggest that for mapping long wavelength structures, body waves can be inverted in two steps, similar to surface waves, where the ray parameter and the vertical traveltime play the role of frequency and phase velocity, respectively.