Analysis of potential field data in the wavelet domain

Abstract

Various Green’s functions occurring in Poisson potential field theory can be used to construct non‐orthogonal, non‐compact, continuous wavelets. Such a construction leads to relations between the horizontal derivatives of geophysical field measurements at all heights, and the wavelet transform of the zero height field. The resulting theory lends itself to a number of applications in the processing of potential field data. Some simple, synthetic examples in two dimensions illustrate one inversion approach based upon the maxima of the wavelet transform (multiscale edges). These examples are presented to illustrate, by way of explicit demonstration, the information content of the multiscale edges. We do not suggest that the methods used in these examples be taken literally as a practical algorithm or inversion technique. Rather, we feel that the real thrust of the method is towards physically based, spatially local filtering of geophysical data images using Green’s function wavelets, or compact approximations thereto. To illustrate our first steps in this direction, we present some preliminary results of a 3‐D analysis of an aeromagnetic survey.

Identification of sources of potential fields with the continuous wavelet transform: Two‐dimensional wavelets and multipolar approximations

“…[28] Let us first illustrate the multipolar series from equation (20) in case of the wavelet transform, W y 1 G jgz (b, a), of the vertical component of the gravity field, g z (h, z) = À@V(h, z)/@z, measured at level z = 0 as introduced by Hornby et al [1999]:…”

“…Here, we have B 2 0 6 ¼0 and B 2 2 = 0. The horizontal gradient wavelet coefficients as introduced by Hornby et al [1999] are given in equation (25) with series expansion given in equation (B3). From cylindrical symmetry, which implies B 2 2 = 0, this can be simplified at large distance r = ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 þ z 2 a p , where r = ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 þ y 2 p and z a = z 0 + a:…”

Inferring Geological Structures using Wavelet-Based Multiscale Edge Analysis and Forward Models

“…Mallat & Zhong, 1992). By applying this approach to potential field theory, Hornby et al (1999) have shown that, with the choice of an appropriate wavelet, derived from Green's function of gravitational acceleration, the measured potential field or its spatial derivatives can be treated as a wavelet transform of the source distribution.…”

“…The detection and visualisation of these irregularities at different scales (multiscale edges) can greatly enhance the interpretation of potential field data in terms of the 3D geology (Hornby, et al, 1999).…”

“…Multiscale edge mapping is an automatic process of picking edges in potential field data at a variety of different scales (Hornby, et al, 1999;Archibald, et al, 1999). On a horizontal plane these edges form isolated 'worms' which are similar to isolated lines drawn by a geoscientist manually interpreting a potential field dataset (e.g.…”