Fractal heterogeneities in sonic logs and low-frequency scattering attenuation

Discussion

The attenuation caused by anticorrelated 3D small-scale heterogeneities can be explained by a low-frequency scattering theory. The length scales that we estimated from sonic logs justify this approach for conventional seismic frequencies. The intensity of scattering attenuation and the value of frequency cutoff strongly depend on the size of the heterogeneities, and S-waves are more attenuated than P-waves at the same frequency. Using low-frequency P-waves provides a better depth of penetration. More reflectors can be detected and imaged, but, of course, with less resolution. This phenomenon was observed in sub-basalt imaging (Ziolkowski et al., 2003). Our analysis of sonic logs confirms the relevance of a fractal description for the high-frequency content of quasi-periodic geological layers. Because sediments are highly stratified, their layered structure has previously motivated use of 1D models for seismic scattering attenuation, but a realistic estimate needs to be conducted in 3D. More generally, description of geological heterogeneities and use of scattering theories should be as depicted in Figure 7.

Several limitations should be pointed out in our study. Well logs constitute 1D samples of the geological medium in the near vertical direction. Use of 3D isotropy is the simplest assumption consistent with our limited knowledge. We consequently ignore the anisotropic effect of layering, which undoubtly affects lateral scattering when $\lambda\sim b$. Also note that the relation between fractal exponent $\beta$ and parameter $H$ depends on spatial dimension $s$. We have therefore proposed different values of parameter $H$ for depth of penetration and evolution of dominant frequency in Figures 6 and 8 and have attempted to extract some general trends.

Our analysis is limited by the fact that we used variations of the seismic velocity, but not of density, in order to be consistent with the mean field theory. This theory retrieves the Backus limit and the Rayleigh diffusion regime. Fortunately, densities commonly exhibit fewer variations. Meanwhile, analysis of the logarithm of impedance $Z$ relates directly to the reflection coefficients (Shtatland, 1991). Backscattering is known to be related to impedance fluctuations (Wu, 1988; Banik et al., 1985). Modification of the Helmholtz equation (16) in 1D to incorporate $\ln\,Z$ was succesfully achieved by Banik et al. (1985), and they provided a proof of O'Doherty-Anstey formula. Impedance $Z(z)$, depending on depth $z$, and reflection coefficient series $R(z)$ are connected by

$$\lim_{dz\rightarrow 0}\,R(z+dz/2) = \lim_{dz\rightarrow 0}\,\frac{Z(z+dz)-Z(z)}{Z(z+dz)+Z(z)},$$ (39)

$$R(z) = \frac{1}{2}~{d\ln Z(z)}.$$ (40)

Reflection series can reasonably be considered to be Gaussian and stationary only inside blocky layers, using the segmentation method (Todoeschuck et al., 1990). Incrementation of the fractional Gaussian noise, corresponding to reflection series $R(z)$, produces the non-stationary and non-Gaussian fBm describing quantity $\ln Z(z)$ (Shtatland, 1991). A white spectrum of reflectivity coefficients occurs for $\beta=2$ and generates a Brownian walk describing $\ln Z(z)$ and involving an exponential autocorrelation. We advocate the use of a non-white reflectivity hypothesis, as previously recommended by several authors (Lancaster and Whitcombe, 2000; Todoeschuck et al., 1990; Anstey and O'Doherty, 2002b).

Fractal heterogeneities in sonic logs and low-frequency scattering attenuation