Constrained inversion of reflection data using Gibbs' sampling

A procedure for the inversion of reflection data (zero offset section) is presented. The strategy is based on the generation of a 2D Markov Random Field (MRF) with a certain spatial continuity that is appropriate to represent the underlying velocity model. The MRF, which is generated using the Gibbs' sampler, is forced to fit the data and to satisfy a set of well constraints during its (iterative) generation process. The spatial continuity of the MRF is implemented by defining a neighborhood system with associated potentials that favor the formation of regions with similar velocity values. Thus, the resulting MRF exhibits spatial continuity, fits the data and honors the well constraints. The algorithm is specially suited for obtaining high resolution velocity images dominated by horizontal layers or other constant velocity blocks. Moderate dipping layers are also tolerated. We tested the algorithm using various velocity models, including the hard Marmousi model. The results show that realistic high resolution blocky images can be recovered accurately. Since the generation of a MRF is a stochastic process, the uncertainty of the estimated models can be analyzed by performing several realizations.