Computing traveltime and amplitude sensitivity kernels in finite-frequency tomography

The efficient computation of finite-frequency traveltime and amplitude sensitivity kernels for velocity and attenuation perturbations in global seismic tomography poses problems both of numerical precision and of validity of the paraxial approximation used. We investigate these aspects, using a local model parameterization in the form of a tetrahedral grid with linear interpolation in between grid nodes. The matrix coefficients of the linear inverse problem involve a volume integral of the product of the finite-frequency kernel with the basis functions that represent the linear interpolation. We use local and global tests as well as analytical expressions to test the numerical precision of the frequency and spatial quadrature. There is a trade-off between narrowing the bandpass filter and quadrature accuracy and efficiency. Using a minimum step size of 10km for S waves and 30km for SS waves, relative errors in the quadrature are of the order of 1% for direct waves such as S, and a few percent for SS waves, which are below data uncertainties in delay time or amplitude anomaly observations in global seismology. Larger errors may occur wherever the sensitivity extends over a large volume and the paraxial approximation breaks down at large distance from the ray. This is especially noticeable for minimax phases such as SS waves with periods >20s, when kernels become hyperbolic near the reflection point and appreciable sensitivity extends over thousands of km. Errors becomes intolerable at epicentral distance near the antipode when sensitivity extends over all azimuths in the mantle. Effects of such errors may become noticeable at epicentral distances140?. We conclude that the paraxial approximation offers an efficient method for computing the matrix system for finite-frequency inversions in global tomography, though care should be taken near reflection points, and alternative methods are needed to compute sensitivity near the antipode.