Bayesian Gravity Data Inversion Using a 3D Alpha Shape Source Representation

Gravity data inversion necessitates solving large-dimensional numerical ill-posed inverse problems. The nature of these problems being ill-posed requires the implementation of a regularization scheme or any approach that integrates previous knowledge of the solution. Typically, Tikhonov-like or Laplacian-based regularizing schemes are employed to mitigate the ill-posedness. Since these types of regularization strategies promote smoothness in the quantities of interest, their approximating solutions have limitations in interpreting the sharp boundaries of source bodies. As a result, the boundaries of the retrieved source bodies appear blurred. This study proposes focusing on the representation of the source as a regularization technique. With that objective in mind, we suggest a low-dimensional representation, constructed using the alpha shape algorithm, to allow the use of non-smooth three-dimensional bodies as inversion instruments. Subsequently, the inverse problem is formulated within a Bayesian framework, and a Markov Chain Monte Carlo approach is developed for simulating from the posterior distribution associated with the proposed representation. This methodology enables the retrieval of the position, shape, and density of 3D source bodies while establishing clear geometrical boundaries and obtaining estimates of uncertainty quantification. The results are evaluated in three challenging synthetic test cases.