Remark on an earlier proposed iterative tomographic algorithm

Abstract

The problem underlying the discussion is that of estimating the distribution of velocity for seismic waves in a cross-hole section using first arrival travel times for explosion-generated waves. We will give a new look at an earlier tomographic algorithm (Dines & Lytle 1979) and discuss its convergence properties. But first, some more general introductory words. In the simplified linear model discussed below we assume straight ray paths neglecting refraction and reflection. The plane region between the boreholes and the surface is decomposed into small cells in which the seismic velocity is assumed constant and isotropic. The nature 06 this decomposition is irrelevant for the following discussion, but in practice all the ‘cells’ are rectangular and with parallel boundaries. Let mi be the slowness in cellj, di the measured travel time for ray i, and Gii the distance that ray i covers across cell j . Let N denote the number of rays and M the number of cells. Usually N is the product of the number of shots and the number of geophones. Ideally then, N linear equations ought t o be fulfilled which can be written with matrix notation d = G . m . These equations do not hold exactly in practice, not only because of the simplified model but also because of errors in measurement. This last type of error can be accounted for by introducing the following statistical model: d = G . m + E where E is a random N-vector variable with expectation 0 and covariance matrix usually proportional to the identity matrix Z. Our problem has now turned into a well-known estimation problem of mathematical statistics: we have N observations, the di, and we want to estimate the M unknown parameters mi. A convenient solution to this problem is furnished by the least-squares method which (in the full-rank case), according to the Gauss-Markov theorem (Silvey 1970, chapter 3), gives estimators with desirable statistical properties (minimum-variance unbiased estimators). Standard deviations of the estimators are also obtained.