3-D Cauchy-type integrals for terrain correction of gravity and gravity gradiometry data

Fundamental to complex analysis is the Cauchy integral theorem, and the derivation of Cauchy-type integrals. For over 40 yr, Cauchy-type integrals have been used to describe analytical continuation, establish the location of singular points, and study non-single-valued solutions of inverse problems in 2-D potential field theory. In this paper, we revive this interesting and fundamental area of potential field theory to introduce Cauchy-type integrals for 3-D potential fields. In particular, we show how one can evaluate the gravity and gravity gradiometry responses of 3-D bodies as surface integrals over arbitrary volumes that may contain spatially variable densities. This method of 3-D spatial-domain potential field modelling has never been realized before, and we show how it is particularly suited to the terrain correction of airborne gravity and gravity gradiometry data. The surface integrals are evaluated numerically on a topographically conforming grid with a resolution equal to the digital elevation model. Thus, our method directly avoids issues related to prismatic discretization of the digital elevation model and their associated volume integration which may result in inappropriate discretization of the earth model, particularly for regions of rugged topography. We demonstrate our method with a model study for airborne gravity gradiometry data simulated for a next-generation 1 Eo system over the Kauring test site in Western Australia.