Fractal heterogeneities in sonic logs and low-frequency scattering attenuation

Introduction

Propagation of waves in heterogeneous media involves attenuation and dispersion by scattering. Theoreticians are still challenged by the phenomenon of wave propagation in random media. The mean field theory (Karal and Keller, 1964; Uscinski, 1977; Chernov, 1960) is commonly used and provides both dispersion and attenuation, depending on scattering cross-sections of the heterogeneities (Kanaun and Levin, 2008; Wu and Aki, 1985; Waterman and Truell, 1961), which are described by their statistical spatial autocorrelation. Higher order correlations have been more recently incorporated in a frequency-dependent effective medium theory (Chesnokov et al., 1998). One major advance, pointed out by Wu and Aki (1985); Wu (1982), is the restriction of the validity of the mean field formalism to low frequency. The theory in fact includes destructive interferences, caused by averaging different realizations of the random medium, and overestimates attenuation at high frequencies. Alternative solutions have been proposed to remove this artificial decoherence of the phase. Two important examples are radiative transfer theory (Wu, 1993; Haney et al., 2005) and the Rytov approximation (Rytov et al., 1989), which is more adequate than the Born approximation when phase fluctuations are important.

In the area of seismic imaging, the layered structure of sediments led O'Doherty and Anstey (1971) to introduce the fundamental concept of stratigraphic filtering. The empirical formula postulated by O'Doherty and Anstey was demonstrated in 1D using mean field formalism (Resnick, 1990; Banik et al., 1985) and, alternately, using wave localization theory (Shapiro and Zien, 1993; Sheng et al., 1986; Shapiro and Hubral, 1999), with a recent extension to a larger frequency band for acoustic waves in 3D (Müller and Shapiro, 2001). The wave localization method utilizes phase, and logarithm of the amplitude, which have the property of self-averaging over some distance called the localization length, in a stationary random medium. These quantities are exactly the ones used in the Rytov method and avoid the phenomenon of artificial phase decoherence at high frequencies. Multiple scattering of seismic waves remains a complex and active research area.

The fractal property of subsurface heterogeneities was initially discussed by Hewett (1986) for hydrocarbon reservoirs and has been confirmed in both vertical and horizontal wells (Stefani and Gopa, 2001). The study of seismic scattering by 2D numerical wave propagation (Frankel and Clayton, 1986) demonstrates the necessity of self-similar heterogeneities for modeling both observations of coda waves and of traveltime anomalies. A $1/f$ spectrum of heterogeneities was modeled using the von Kármán spatial autocorrelation function (von Kármán, 1948) in order to obtain a constant quality factor at high frequencies. For scatterers larger than the wavelength, multipathing was observed, whereas 3D effects were revealed to be important for scattering loss at low frequency. Gist (1994) tried to explain seismic-wave attenuation in VSP surveys by 3D scattering from fractal heterogeneities. One of the first attempts to relate the statistics of well log data to seismic scattering used wave localization theory (White et al., 1990). A more detailed study of acoustic-wave localization effects in 1D fractal media (van der Baan, 2001) shows that a constant quality factor is possible only for the $1/f$ fractal spectrum and that localization can not occcur if the medium contains periodic layers involving resonance and violating the ergodicity assumption. Convergence of the localization effect in realistic 3D seismic surveys seems questionable. Presence of strong cycles in well log data is causing difficulties when the fractal exponent is being estimated (Dolan et al., 1998) and is commonly attributed to Milankovitch cycles (Anstey and O'Doherty, 2002a).

Further investigation of the relationship between cycles, fractal properties, and correlation lengths is necessary and low-frequency scattering theories in 3D fractal media can be appropriate for conventional seismic surveys. In this paper, we propose a nonlinear estimation method for fractal statistics of sonic-log heterogeneities using von Kármán's model. We attempt to identify different scales of the sedimentation process as proposed by O'Doherty and Anstey (1971) and Anstey and O'Doherty (2002a). The inversion captures small-scale heterogeneities while larger local cycles exist. We use the mean field theory to calculate analytical solution of low-frequency attenuation by scattering from 3D fractal heterogeneities and predict a shift of the dominant frequency with depth in seismic surveys.

The paper is organized into two parts. In the first part we present a description of cycles in sediments in connection with fractal statistics. The von Kármán spatial autocorrelation function is introduced, and we briefly review some features of fractal statistics. We present our estimation method, validate it on synthetic signals, apply it to our sonic-log data, and show that one can detect the high-frequency part of the superposition of different geological scales. The second part explains the derivation of scattering attenuation for low-frequency acoustic waves by 3D isotropic fractal heterogeneities using the mean field theory. Results comply with the Rayleigh regime and the Backus effective medium for very low frequency. We present analytical predictions of scattering attenuation and show the existence of a cutoff frequency for the penetration of waves. We use the frequency dependence of the penetration depth to calculate the shift of the dominant frequency of a Ricker wavelet. We conclude by suggesting further improvements.

Fractal heterogeneities in sonic logs and low-frequency scattering attenuation