Constraining the shape of a gravity anomalous body using reversible jump Markov chain Monte Carlo

Typical geophysical inversion problems are ill-posed, non-linear and non-unique. Sometimes the problem is trans-dimensional, where the number of unknown parameters is one of the unknowns, which makes the inverse problem even more challenging. Detecting the shape of a geophysical object underneath the earth surface from gravity anomaly is one of such complex problems, where the number of geometrical parameters is one of the unknowns. To deal with the difficulties of non-uniqueness, ill-conditioning and non-linearity, a statistical Bayesian model inference approach is adopted. A reversible jump Markov chain Monte Carlo (RJMCMC) algorithm is proposed to overcome the difficulty of trans-dimensionality. Carefully designed within-model and between-model Markov chain moves are implemented to reduce the rate of generating inadmissible geometries, thus achieving good overall efficiency in the Monte Carlo sampler. Numerical experiments on a 2-D problem show that the proposed algorithm is capable of obtaining satisfactory solutions with quantifiable uncertainty to a challenging trans-dimensional geophysical inverse problem. Solutions from RJMCMC appear to be parsimonious for the given prior, in the sense that among the models satisfactorily represent the true model, models with higher posterior probabilities tend to have fewer number of parameters. The proposed numerical algorithm can be readily adapted to other similar trans-dimensional geophysical inverse applications.