Optimal 3-D Geophysical Tomography

Abstract

Acoustic and electromagnetic tomographic methods attempt to provide accurate and highly resolved estimates of spatially varying parameters. High resolution is obtained by discretizing the domain into a large number of parameters to be estimated. Because the resulting problem size is large, most implementations rely on iterative methods that attempt to minimize the output least-squares criterion through the repetitive application of relatively simple parameter updates. The accuracy and efficiency of such methods depends on the quality of the initial estimates and the parameterization employed, as well as on the update mechanism. Conversely, optimal filtering methods enable the computation of non-iterative, minimum-variance updates and can be used to yield estimates of both parameter values and covariances. Despite the statistical superiority of optimal methods, they have seldom been exploited in geophysical applications due to their computational demands for large systems. We have devised an approximate extended Kalman filter, which is a recursive, Bayesian, minimum variance estimator for nonlinear dynamic systems. The inverse of an a priori estimate of the parameter covariance matrix is used to damp the system and the weighting matrix includes the inverse of the measurement error covariance matrix. While the recursive nature of the filter is designed to handle time-series data, it can also be exploited to assimilate batch data iteratively and/or recursively in smaller batches. We have built efficient approximations into the filter that, in conjunction with our domain-decomposition strategy and dynamic parameterization scheme, enable application of the method to very large domains. The integrated method simultaneously estimates the number, geometry, value, and covariance of spatially distributed parameters. Threedimensional tomographic results are presented. METHOD The parameter estimate is constructed incrementally, from the top to the bottom of the 3-D domain, by conditioning successive overlapping 3-D subdomains, as depicted in Figure 1. iz ‘“sliding 3-D conditioning window” (W), which spans the entire length and breadth of the domain but includes only a small contiguous portion of the domain in the vertical dimension, defines one 3-D subdomain. This “window” starts at the top of the domain and is moved down through the domain, one depth node at a time. Conditioning of the ln(sZozuness) fields within each subdomain is achieved via the approximate extended Kalman filter (AEKF) [Eppstein. and Dougherty, 1996, 1998a, 1998b]. Transmission measurements used for conditioning zones within a subdomain are limited to those from sources within a sliding “source window” (S, which is usually the same as the