Second-order smoothness prior over the Delaunay Tessellation in Bayesian geophysical inversion

Abstract

Prior information is always used to form up additional restrictions in geophysical inversions to solve the non-uniqueness problem of the solution. A commonly used restriction is on the smoothness (the second-order derivative) of the inverted model. The smoothness is usually calculated through interpolation over regular grids for easy implementation in numerical calculations. When observed data are irregularly distributed, such as in geodetic inversions, interpolation based on a Delaunay Tessellation (DT) over the observation locations is popularly used to avoid additional interpolations and to maintain the flexibility in the resolution of the model solution. However, the numerical calculation of the second-order derivatives (smoothness) of a function based on the DT interpolation is more difficult than differentiating over regular grids. In this paper, we compare the performance of two methods in calculating the smoothness of the DT based interpolators: a previous method that is called the double linear interpolation (DLI), and a newly proposed quadratic interpolation (QI) method. Their efficiency is verified through numerical experiments in the framework of Bayesian inversion and applied to a gravity inversion problem. The results show that the QI method is slightly better than the DLI method.