Wavelet power spectrum analysis of heterogeneities from sonic velocity logs
Journal name: Geophysical Prospecting
Issue: Vol 46, No 5, September 1998 pp. 455 - 475
Info: Article, PDF ( 4.4Mb )
The continuous wavelet transform (CWT) is used to evaluate local variations in the power-law exponents of sonic log data. The resulting wavelet spectrum can be compared with the corresponding global estimates obtained by conventional Fourier transform methods. In Fourier analysis, the fundamental tool used to characterize a fluctuating velocity distribution is the power spectrum. It represents the energy contained in each wavenumber and thus provides information regarding the importance of each scale of heterogeneity. However, important spatial information regarding the location of events becomes implicit in the phase angle of the Fourier transform. In this paper, it is shown how the square of the amplitude of the wavelet transform is related to the Fourier spectrum and how spatial information can be expressed in an explicit manner. Using the conservation of energy, it is shown that the average wavelet power spectrum over the total depth range is equal to the global power spectrum. A Gaussian wavelet is chosen to realize the wavelet transform. Two synthetic sonic logs with exponential and von Karman correlation functions are used to demonstrate the potential of the suggested analysis. Furthermore, the wavelet transform is applied to the KTB (Continental Deep Drilling Program) sonic log data. The wide range of applications of the CWT shows that this transform is a natural tool for characterizing the structural properties of underground heterogeneities. It offers the possibility of separating the multiscale components of heterogeneities.