Extremal inversion of static earth displacements due to volume sources

Summary. The inverse problem of using static displacements observed at the surface to infer volume changes within the Earth is considered. This problem can be put in a form such that the method of ideal bodies and the method of positivity constraints may both be applied. Thus all of the techniques previously developed for the gravity inverse problem can be extended to the static displacement problem. Given bounds on the depth, the greatest lower bound on the fractional volume change can be estimated, or, given bounds on the fractional volume change, the least upper bound on the depth can be estimated. Methods of placing bounds on generalized moments of the perturbing body are also developed, and techniques of handling errors in the data are discussed. Examples are given for both two- and three-dimensional problems. The ideal body method is suited for both 2- and 3-D problems when only two data points are considered, but is unwieldy for more data points. The method of positivity constraints is more versatile and can be used when there are many data points in the case of 2-D problems, but it may lead to an excessive amount of computation in 3-D problems.

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