A Variational Formulation of the Euler Deconvolution Method for Gravity Data Inspired by Optical Flow

Abstract

The Euler deconvolution method has been widely used to estimate the structure and location of source bodies from measured geophysical data. The horizontal estimates for the source bodies, provided by the Euler deconvolution method, are excellent in most cases, and there are areas of opportunity for in-depth estimation when two or more sources are present. In the field of image processing, there is the problem of determining the optical flow within a sequence of images. A popular method for estimating the optical flow is the Lucas-Kanade method. Although geophysical data inversion and optical flow are different phenomena, the Euler deconvolution and Lucas-Kanade methods are very similar. This fact drives the research hypothesis of this work: Horn and Schunck proposed a variational formulation for the optical flow problem that improves the capabilities of the Lucas-Kanade method. Thus, a variational formulation for the Euler’s equation, in a sense similar to Horn and Schunck’s proposal, would improve the results offered by the classical Euler deconvolution method. This hypothesis is investigated in this work and the findings are reported in the light of challenging synthetic test cases.