Estimation of surface layer structure from Rayleigh wave dispersion. II. Sparse-data case - Analytical theory

The ill-posed nature of the estimation of subsurface properties from dispersion data necessitates the application of special approaches in which a priori information supplements the experimental data. Here we consider a particular approach involving the use of estimation theory combined with a nonparametric mathematical model of the measurement process. In a previous paper we considered the somewhat artificial case of continuous dispersion data (or, in practice, data that is sufficiently dense and extensive that interpolation and extrapolation can be carried out with impunity using standard techniques). Here we treat the sparse-data case using an estimator that is optimized for an input composed of a discrete data set. The treatment starts with a mathematical model giving a probabilistic description of the possible results of measurement including measurement errors. As usual, the estimator (a function providing the estimated profile of subsurface properties in terms of measured data) is optimized in a least-mean-square sense for its performance on the model. The theory also yields auxiliary measures pertaining to bias, data versus model dominance, resolution, and a posteriori variance.