Characterization of heterogeneities from sonic velocity measurements using the wavelet transform

Abstract

In this paper we first show that the “multiple filter technique”, often used for analyzing the velocity dispersion of surface waves in seismology, is a type of the continuous wavelet transform (CWT). Given an analyzing Gaussian wavelet, which is supposed to be localized in some sense along the time axis (in this study along the depth axis), the choice of a scale parameter a in the CWT corresponds implicitly to an equivalent Gaussian filter whose dominant frequency (wavenumber) is roughly related to l/a. Second, a brief power spectral analysis of the measurements is presented and the wavelet transform is used as a mathematical microscope to study the local scaling properties of heterogeneities from sonic measurements made in the German KTB (Continental Deep Drilling Program)boreholes. The wide range of application of the WT shows that this transform turns out to be a natural tool for characterizing the structural properties of heterogeneities arising in the underground. It offers the possibility to separate the multiscale components. This paper is structured as follows. First, we give a short mathematical derivation to show that the multiple filter technique (Dziewonski et al., 1969; Li, 1994) is a special case of the CWT. Second, we use the wavelet transform to investigate the self-similar properties of the sonic measurements at different length scales. The aim is to illustrate the efficiency of this technique in capturing the local scaling properties of heterogeneities in general.