Pre-stack inversion of a 1-D medium

Assuming a one-dimensional (1-D) acoustic medium and a known density, we want to determine the velocity distribution from pre-stack surface seismic data. Basically, we have to identify a distributed parameter in the two-dimensional (2-D) wave equation. This inverse problem is set as an optimization problem. This allows us to expect that the information redundancy available in the data will provide an optimal signal-to-noise enhancement. Since the forward modeling we use gives a complete solution of the wave equation, all the events explained by acoustic wave propagation, such as guided waves or high-order multiples, will be used to infer information about the velocity distribution. From the above arguments we may expect an accurate result with high resolution, even with noise-corrupted data. Although the developed methodology is not optimal at all, the numerical experiments confirm this expectation and illustrate the power of the proposed inversion method.