Optimal 3-D Geophysical Tomography

Acoustic and electromagnetic tomographic methods attempt to provide accurate and<br>highly resolved estimates of spatially varying parameters. High resolution is obtained by<br>discretizing the domain into a large number of parameters to be estimated. Because the<br>resulting problem size is large, most implementations rely on iterative methods that attempt<br>to minimize the output least-squares criterion through the repetitive application of relatively<br>simple parameter updates. The accuracy and efficiency of such methods depends on the<br>quality of the initial estimates and the parameterization employed, as well as on the update<br>mechanism. Conversely, optimal filtering methods enable the computation of non-iterative,<br>minimum-variance updates and can be used to yield estimates of both parameter values and<br>covariances. Despite the statistical superiority of optimal methods, they have seldom been<br>exploited in geophysical applications due to their computational demands for large systems.<br>We have devised an approximate extended Kalman filter, which is a recursive, Bayesian,<br>minimum variance estimator for nonlinear dynamic systems. The inverse of an a priori<br>estimate of the parameter covariance matrix is used to damp the system and the weighting<br>matrix includes the inverse of the measurement error covariance matrix. While the recursive<br>nature of the filter is designed to handle time-series data, it can also be exploited to assimilate<br>batch data iteratively and/or recursively in smaller batches.<br>We have built efficient approximations into the filter that, in conjunction with our<br>domain-decomposition strategy and dynamic parameterization scheme, enable application<br>of the method to very large domains. The integrated method simultaneously estimates<br>the number, geometry, value, and covariance of spatially distributed parameters. Threedimensional<br>tomographic results are presented.