next up previous [pdf]

Next: Synthetic realizations Up: Statistical model of heterogeneities Previous: Statistical model of heterogeneities

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 $f$:

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

The exponent $\beta $, in the energy spectrum $E^{(s)}_{H,b}(\mathbf {k})$ from equation 7, is
$\displaystyle \beta$ $\textstyle =$ $\displaystyle 2H+s.$ (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 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

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

next up previous [pdf]

Next: Synthetic realizations Up: Statistical model of heterogeneities Previous: Statistical model of heterogeneities