Computational aspects of finite-frequency traveltime inversion kernels

Finite-frequency traveltime inversion offers higher accuracy for velocity model building than ray-based traveltime inversion. The adjoint force is the key for the computation of inversion kernels. Starting at the definition of inversion kernels for the acoustic wave equation, we have derived the explicit formula for the spectral distribution density function used in the adjoint force computation. Two formulations are developed for the computation of adjoint forces for receiver-side extrapolation, frequency-domain representation, and time-domain representation. The accuracy of finite-frequency traveltime inversion kernels is benchmarked with the analytical solutions for homogeneous isotropic media. We use wavefront construction to compute the first Fresnel zones for kernel conditioning. Based on dynamic ray tracing, we design a processing procedure guided by synthetic data tests to extract the desired events for wavefield backward extrapolation from the data. Unlike ray-based velocity tomography, finite-frequency inversion can resolve the velocity structures comparable with the size of Fresnel zones as we demonstrate on a marine salt model using ocean-bottom node acquisition geometry. Despite the fact that the inversion kernels are based on Born approximation, velocities with errors up to 20% can be well-resolved. For practical purposes, a simple formulation is given for the determination of the shot spacing. Our workflow for finite-frequency inversion is efficient and converges in only very few iterations.