The theory of the continuous wavelet transform in the interpretation of potential fields: a review

Abstract

We consider the use of the continuous wavelet transform in the interpretation of potential field data. We report its development since the publication of the first paper by Moreauet al. in 1997. Basically, it consists in the interpretation in the upward continued domain since dilation of the wavelet transform is the upward continuation altitude. Thus within a range of altitudes, the wavelet transform of the noise is decreased faster than the wavelet transform of the potential field caused by underground sources; this means that the signal‐to‐noise ratio is much better than those involved in other enhancing methods (e.g., Euler deconvolution, gradient analysis, or the analytic signals). Similarly to the Euler deconvolution, its first target parameters were the source positions and shape. The method has then been developed to estimate size and directions of extended sources (e.g., faults and dikes of finite dimensions) and also the magnetization direction in the case of magnetic data. Latest developments show that when combined with a Radon transform, the continuous wavelet transform can help in the automatic detection of elongated structures in 3D, simultaneously to the estimation of their strike direction, shape and depth. Several applications to real case studies have been shown before; however for clarity's sake in the present paper, only synthetic cases have been reproduced to clearly sum up the development of the methodology.

Multiscale seismic attributes: source-corrected wavelet response and application to high-resolution seismic data

“…Such a result is analogous to what happens for the multiscale analysis of potential fields where the undistorted CWT obtained with wavelets belonging to the Poisson semi‐group is limited to dilations larger than the depth of the causative sources (Moreau et al. 1997, 1999; Sailhac et al. 2009).…”

“…In a way similar to the wavelet analysis of potential field performed with the Poisson wavelet semi‐group (Moreau et al. 1997, 1999; Sailhac et al. 2009), we work with Lévy alpha‐stable distributions to define an effective analysing wavelet ξ e ( t ) accounting for the source effects.…”

“…Our approach is similar to the one developed for the wavelet analysis of potential fields to characterize their causative sources by using analysing wavelet belonging to the Poisson semi‐group (Moreau et al. 1997, 1999; Sailhac et al. 2009).…”

Three-dimensional interpretation of magnetic and gravity anomalies using the finite-difference similarity transform

“….,v j ͒ is the full set of variables to which the function f shows the homogeneity and t is a coefficient, can be used directly for the same purpose. The wavelet transform ͑Boukerbout and Gibert, 2006;Fedi, 2007;Sailhac et al, 2009͒ and the similarity transform ͑Stavrev, 1997; Stavrev et al, 2006Stavrev et al, , 2009Gerovska et al, 2010͒ are examples of the successful application of this approach. Boukerbout and Gibert ͑2006͒ propose an inversion for locating the sources along profiles detected by the Radon transform of gridded data, based on calculating a series of complex analyzing Poisson wavelets ͑ridgelet transforms͒ -upward-continued analytic signals -for different dilation values.…”

“…Fedi et al ͑2009͒ propose a similar method; however, they form a redundant set of ridges of first-order derivatives of the field. The continuous wavelet transform ͑CWT͒ estimates size and direction of extended sources ͑for a review of CWT inversion methods, see Sailhac et al, 2009͒. Fedi ͑2007͒ introduces a scaling function as the derivative of the logarithm of the potential field with respect to the logarithm of the depth to form a transformation, the extreme points of which determine the source locations.…”