BACKUS-GILBERT APPROACH TO INVERSION OF TRAVEL-TIME DATA FOR 3-DIMENSIONAL VELOCITY STRUCTURE

Summary. Considerable effort has recently been directed towards constructing three‐dimensional velocity models to fit seismic traveltime data. Most studies have involved parameterizing the model by a limited number of boxes within which the velocity is assumed constant. An alternative viewpoint discussed in this paper is that the structure is an unknown function which can be viewed through a smoothing window. For a limited data set, the resolution length scale of this window will never be zero. Although formulation of the inverse problem for one‐dimensional velocity structure is a straightforward application of the ideas of Backus and Gilbert, it does not generalize directly to three dimensions. This is because rays in three dimensions do not sample the structure in the volume between the rays and thus the data kernels are not suitable for approximating a delta function whose peak does not lie at the intersection of many rays. We solve this problem by replacing the delta function by an ideal averaging volume with sufficiently large dimensions that it is always intersected by rays. Non‐square‐integrable singularities in the data kernels are quelled by integration. The mean‐square misfit between the transformed ideal averaging volume and its approximation (the actual averaging volume) constructed from the quelled data kernels leads to two natural measures of the resolution length scale of the actual averaging volume. For a simple synthetic example, one of these resolution measures converges rapidly to the scale of the ideal averaging volume as the size of the ideal volume increases. The other measure converges more slowly. We discuss strategy for finding the minimum permissible size of the ideal volume for which the average in the actual volume is a reliable estimate of the average in the ideal volume. In our synthetic example, this always requires in excess of 25 rays penetrating the averaging volume, but this result is not general and may be pessimistic for many real problems. For inaccurate data, the evaluation of the trade‐off of resolution versus the variance of the smoothed model can proceed by relaxing the fit of the approximate averaging volume to the ideal volume in a manner exactly analogous to that described by Backus and Gilbert. Another strategy with considerable computational advantage which may sometimes be useful is to increase the size of the ideal averaging volume above its permissible minimum without relaxing the fit. Copyright © 1979, Wiley Blackwell. All rights reserved