Iterative seismic inversion

Abstract

Two well‐known iterative algorithms, steepest descent and conjugate‐gradient descent, are used to solve the non‐linear seismic inversion problem from reflection data by minimizing the mean‐square error between the measured data and data generated by the estimated model. This approach offers great flexibility, both in incorporating a priori information and in its ability to weight the measurements in an optimal fashion based on error statistics, which serves to regularize an otherwise ill‐conditioned inversion. In minimizing the mean‐square error, many authors have proposed Newton‐like methods, but the above descent algorithms are generally more stable and avoid the need to invert a large matrix containing second‐derivative information. The problem can be formulated so that the first iterate gives the Born inversion. In most iterative approaches the gradient of the measurements with respect to the model plays a fundamental role in minimizing the mean‐square error. The gradient is usually derived by linearizing the measurement‐model relationship (the Lippmann‐Schwinger equation) and then ‘differentiating’ with respect to the model. This procedure gives the correct gradient only when the Green's function is updated after each iteration on the basis of the latest model estimate. A new approach is used here for deriving the gradient based on a generalization of a formula due to Weston (1979) which results in explicit expressions for the updated Green's function in both the time and frequency domains for arbitrary illumination and near‐field detection. A formula is also derived for the finite (i.e. non‐infinitesimal) change in the measured field due to a finite change in the model. This formula could be useful in formulating descent algorithms using exact finite differences without derivatives. Copyright © 1988, Wiley Blackwell. All rights reserved