  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 nonlinearity, 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.
  Keywords: trans-dimensional geophysical inversion, reversible jump Markov
chain Monte Carlo; gravity anomaly.
