Volume 46 Number 5
  • E-ISSN: 1365-2478


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.


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  1. ArneodoA., GrasseauG.HolschneiderM.1989. Wavelet transform analysis of invariant measures of some dynamical system. In: Wavelet Time‐Frequency Methods and Phase Space (eds J.M. Combes, A. Grossmann and P. Tchamitchia), pp. 182–196. Springer‐Verlag Inc.
  2. BendatJ.S.PiersolA.G.1986. Random Data, Analysis and Measurement Procedures, 2nd edn (Revised and Expanded). John Wiley & Sons, Inc.
  3. BramK.1995. Logging under extreme conditions in the super‐deep borehole KTB‐Oberpfalz HB. In: KTB Report 94–1: Basic Research and Borehole Geophysics (Final Report), Borehole logging in the KTB‐Oberpfalz HB, Interval 6013.5–9101.0 m (eds K. Bram and J.K. Draxler), pp. 3–23.
  4. ChakrabortyA.OkayaD.1995. Frequency‐time decomposition of seismic data using wavelet‐based methods. Geophysics60,1906–1916.
    [Google Scholar]
  5. ChilcoatS.R.1977. Applications of the computer analysis of dispersed waves. MSc thesis, Colorado School of Mines.
  6. ChuiC.K.1992. An Introduction to Wavelets. Academic Press, Inc.
  7. DziewonskiA., BlochS.LandismanM.1969. A technique for the analysis of transient seismic signals. Bulletin of the Seismological Society of America59,427–444.
    [Google Scholar]
  8. FrankelA.1989. A review of numerical experiments on seismic wave scattering. PAGEOPH131,639–685.
    [Google Scholar]
  9. GoupillaudP., GrossmannA.MorletJ.1985. Cycle‐octave and related transforms in seismic signal analysis. Geoexploration23,85–102.
    [Google Scholar]
  10. GrossmannA., Kronland‐MartinetR.MorletJ.1989. Reading and understanding continuous wavelet transform. In: Wavelet Time‐Frequency Methods and Phase Space (eds J.M. Combes, A. Grossmann and P. Tchamitchia), pp. 2–20. Springer‐Verlag, Inc.
  11. HolligerK.1996. Upper crustal seismic velocity heterogeneity as derived from a variety of P‐wave sonic logs. Geophysical Journal International125,813–829.
    [Google Scholar]
  12. KatulG.G., AlbertsonJ.D., ChuC.R.ParlangeM.B.1994. Intermittency in atmospheric surface layer turbulence: the orthonormal wavelet representation. In: Wavelet in Geophysics (eds E. Foufoula‐Georgiou and P. Kumar), pp. 81–105. Academic Press, Inc.
  13. KneibG.1995. The statistical nature of the upper continental crystalline crust derived from in‐situ seismic measurements. Geophysical Journal International122,594–616.
    [Google Scholar]
  14. KumarP.Foufoula‐GeorgiouE.1994. Wavelet analysis in Geophysics: an introduction. In: Wavelet in Geophysics (eds E. Foufoula‐Georgiou and P. Kumar), pp. 1–34. Academic Press, Inc.
  15. LiX.‐P.1994. Decomposition of vibroseis data by multiple filter technique. 64th SEG meeting, Los Angeles, USA, Expanded Abstracts, 711–714.
  16. LiX.‐P.HauryJ.1995. Characterization of heterogeneities from sonic velocity measurements using the wavelet transform. 65th SEG meeting, Houston, USA, Expanded Abstracts, 488–491.
  17. LiX.‐P.WuR.‐S.1993. Investigation of random heterogeneities in the crust near the KTB. 55th EAEG meeting, Stavanger, Norway, Extended Abstracts, paper P72.
  18. PikeC.1994. Analysis of high resolution marine seismic data using the wavelet transform. In: Wavelet in Geophysics (eds E. Foufoula‐Georgiou and P. Kumar), pp. 183–211. Academic Press, Inc.
  19. WeberK.VollbrechtA.1989. The crustal structure at the KTB drill site, Oberpfalz. In: The German Continental Deep Drilling Program (KTB) (eds R. Emmermann and J. Wohlenberg), pp. 5–36. Springer‐Verlag, Inc.
  20. WuR.‐S., XuZ.LiX.‐P.1994. Heterogeneity spectrum and scale‐anisotropy in the upper crust revealed by the German Continental Deep‐Drilling (KTB) holes. Geophysical Research Letters21,883–886.
    [Google Scholar]

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