Fractal heterogeneities in sonic logs and low-frequency scattering attenuation

Seismic-scale heterogeneities

Inside a main facies, we can not recover the short wavelengths of type 1 in the O'Doherty-Anstey classification because of well-logging tool limitation. Type 3 of the classification includes Milankovitch cycles of about 10 to 20 m and third-order sea-level cycles from 15 to 300 m (Anstey and O'Doherty, 2002a). Seismic reflectors are conventionally identified as being chronostratigraphic horizons separating different geological units. They correspond to the wavelength of conventional seismic surveys, up to 100 Hz, and should induce some resonant scattering with ``friendly'' multiples. Multipathing is observed for this ratio of seismic wavelength to the size of heterogeneities, as shown by numerical experiments (Frankel and Clayton, 1986). This domain should be treated using wave-localization theory and the Rytov method. These could nevertheless fail to explain the data because of quasi-periodicity of the medium at this scale, violating the ergodicity assumption required by wave-localization theory. Statistical methods using the autocorrelation function seem to be adapted to describe quasi-periodic media when the ratio of $b$ over the seismic wavelength is small. When the wavelength is of the same size, local quasi-cyclicity of the sedimentary sequence should not be ignored (Morlet et al., 1982; Stovas and Ursin, 2007).

The non-Gaussian nature and non-stationarity of sedimentary layers call for more sophisticated methods to be used, especially in order to capture larger scale pseudo-cyclic heterogeneities, as, for example, a multifractal analysis (Marsan and Bean, 2003) or a local cyclicity detection by wavelet analysis (Rivera et al., 2004). The wavelet transform was indeed introduced so that seismic signals in locally cyclic sedimentary layers could be analyzed (Morlet et al., 1982).

Fractal heterogeneities in sonic logs and low-frequency scattering attenuation