Wavelet based inversion of gravity data

Abstract

The Green's function of the Poisson equation, and its spatial derivatives, lead to a family of wavelets specifically tailored to potential fields. Upward continuation of the field is seen to be identical to these wavelets' scale change operation. The maxima at all heights of the field's horizontal gradients are termed the field's multiscale edges. A multiscale edge's field strength variation and geometry contain information about the geometry and type of discontinuity in the source. The assumptions that "rocks have edges" and that these discontinuities are represented in the field's multiscale edges appears to collapse much of the ambiguity inherent in the inversion of potential field data. One approach to inversion is purely visual, relying upon the way that multiscale edges for dipping fault blocks sometimes "mirror" the fault geometry. A second approach recovers the density contrast, the depths to top and bottom, and the dip angle of an isolated synthetic dipping fault block by performing a search for parameters that best recreate the observed multiscale edges. A third approach relies upon naïve downward continuation. When a field is downward continued below its actual source, a common assumption about the downward continuation operator is violated, introducing well-known oscillatory components to the previously smooth result. Such oscillatory components tend to arise first on multiscale edges. By following multiscale edges as we downward continue, we can pick the maximum depth to a source (provided our "rocks have edges" assumption is true). Finally, since the gravity field wavelet is (proportional to) a Green's function for mass dipoles, we can directly interpret the wavelet transform itself as being (proportional to) a distribution of sources composed entirely of horizontal mass dipoles. Thus, multiscale edges can be interpreted as the modes of the probability density of edges in the source distribution generated by the wavelet transform.