The classical one-dimensional (1-D) inverse problem consists of estimating reflection coefficients from surface seismic data using the 1-D wave equation. Several authors have found stable solutions to this problem using least-squares model-fitting methods. We show that the application of these plane-wave solutions to seismic data generated with a point source can lead to errors in estimating reflection coefficients. This difficulty is avoided by using a least-squares model fitting scheme describing vertically traveling waves originating from a point source. It is shown that this method is roughly equivalent to deterministic deconvolution with built-in multiple removal and compensation for spherical spreading. A true zero-offset field data set from a specially designed seismic experiment is then used as input to estimate reflection coefficients. Stacking velocities from a conventional seismic survey were used to estimate spherical spreading. The resulting reflection coefficients are shown to correlate well with an available well log.

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