Computation of traveltime covariances based on stochastic models of velocity heterogeneity

Abstract

SUMMARY We formulate the error covariance for calculated seismic traveltimes (traveltime covariance) along any two propagation paths as a double integral of a covariance function describing velocity-model error (velocity covariance) with sensitivity distributions for the paths. Two numerical techniques are presented for evaluating the traveltime covariance matrix for multiple paths. The first technique evaluates the covariance matrix directly. The second evaluates the inverse of the covariance matrix summed with a covariance matrix for observational errors, as is utilized in event locators. Our approach takes the velocity covariance to be the Green’s function of a differential operator, which can be specified in terms of physically meaningful parameters, such as spatially variable velocity variance and correlation lengths. Our numerical algorithms reduce to solving finite-difference equations based on the differential operator. As a demonstration, we compute traveltime covariance using ray-based sensitivity distributions and a suite of depth-dependent models of velocity covariance. We compare our theoretical calculationstoempiricalestimatesoftraveltimevarianceversusevent-stationdistance,derived from observed residuals relative to the ‘ak135’ velocity model. Our calculations predict and explain some key features of the distance dependence of observed residual statistics, such as abrupt changes in variance at crossover points separating branches of the first-arrival traveltime curve. We find that the observed traveltime variances in the distance range 2 ◦ ‐33 ◦ are well matched by assuming a velocity standard deviation (relative to ‘ak135’) of >10 percent in the crust and decaying from ∼2 percent in the uppermost mantle to near 0 percent below the 410-km discontinuity. These variance estimates hold over a wide range of assumed correlation lengths of velocity error, which are not well constrained by traveltime variance observations. By providing a physical understanding of traveltime covariance, our approach may help in the development of improved methods for locating seismic events, for estimating path-specific corrections to baseline traveltime models, and for constraining the statistics of velocity variations in the Earth.