Fractal heterogeneities in sonic logs and low-frequency scattering attenuation

Fractal statistics

Among different concepts introduced by the theory of fractals (Mandelbrot, 1983), self-affine property accounts for invariance of roughness of a curve observed at different scales. Self-affine fractals can be characterized by the power-law dependence of their energy spectrum $E(f)$ on frequency $f$:

$$E(f) \propto f^{-\beta}.$$ (9)

The exponent $\beta$, in the energy spectrum $E^{(s)}_{H,b}(\mathbf {k})$ from equation 7, is

$$\beta = 2H+s.$$ (10)

For $\beta=0$, energy spectrum is constant and describes the familiar white noise. Causal integration of Gaussian white noise produces the classical Brownian motion, or random walk, characteristic of diffusion processes, and results in an energy spectrum with $\beta=2$. The autocorrelation function of Brownian motion signals is a decreasing exponential and the autocorrelation in equation 6 properly reduces to $\exp{[-r/b]}$ for $H=0.5$. Another interesting form of spectrum is for $\beta =1$. The associated signal is called Flicker noise (Schottky, 1926; Dolan et al., 1998) and can be interpreted as the superposition of different relaxation processes. For geological layers, such form of spectrum was interpreted as the expression of quasi-cyclicity and blocky layering (Shtatland, 1991). Generalization, including Gaussian white noise and Brownian motion, leads to two types of fractal signals (Li, 2003; Shtatland, 1991; Turcotte, 1997), namely

• fractional Gaussian noise (fGn) defined as filtered Gaussian white noise with $-1 \leq \beta \leq 1$,
• fractional Brownian motion (fBm) built by causal integration of fGn above and resulting in $1 \leq \beta \leq 3$.

The fGn is stationary and Gaussian, whereas the fBm is neither stationary nor Gaussian. Exponent $\beta_{fBm}$ of the fBm is related to exponent $\beta_{fGn}$ of the fGn, used for integration, by $\beta_{fBm}=\beta_{fGn}+2$.

The significance of parameter $H$ in equation 10 is delicate and connected to the Hurst exponent $Hu$, which measures the correlation of time series (Hurst, 1951) by

$$Hu = \frac{\log\left(R/\sqrt{S}\right)}{\log(T)}\,,$$ (11)

where $R$ and $S$ are respectively range of variations and variance calculated for the length $T$ of the signal. The meaning of the value of $Hu$ is

• antipersistence for $0\leq Hu\leq 0.5$,
• random process for $Hu=0.5$, and
• persistence for $0.5\leq Hu\leq 1$.

Estimation of $Hu$ using formula 11 is relevant only for fGn signals (Turcotte, 1997). For example, Gaussian white noise produces $Hu=0.5$. Parameter $H$ defined in equation 10 is associated with the Hurst exponent by

• $H=Hu-1$ for fGn with $-1 \leq \beta \leq 1$;
• $H$ equals the Hurst exponent of incremented fGn (Li, 2003) for fBm with $1 \leq \beta \leq 3$.

The different self-affine 1D fractal models are presented in Table 1 according to the nature and persistency of the signal. A previous analysis of the logarithm of acoustic impedance from well data by Walden and Hosken (1985) shows $1/2 \leq \beta \leq 3/2$, promoting the so-called $1/f$ geology. An improved solution, reproducing the complete well log sequence in sedimentary rocks, uses a similar random process based on fractional Lévy motion (Painter and Paterson, 1994), but fBm can be adequate at small scales, inside different facies (Lu et al., 2002).

Fractal exponent	Von Kármán exponent $H$	Description	Geology
$\beta=0$	$-0.5$	Gaussian white noise	Random process
$0<\beta<1$	$-0.5<H<0$	Persistent fGn
$\beta =1$	$0$	Flicker noise	Blocky layers
$1<\beta<2$	$0<H<0.5$	Antipersistent fBm	Quasi-cyclic deposition
$\beta=2$	$0.5$	Brownian walk	Random deposition
$2<\beta<3$	$0.5<H<1$	Persistent fBm	Transitional deposition

Table 1. Classification of 1D fractal statistics according to the exponent $\beta$ and geological interpretation.

Fractal heterogeneities in sonic logs and low-frequency scattering attenuation