    Fractal heterogeneities in sonic logs and low-frequency scattering attenuation  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 on frequency :   (9)

The exponent , in the energy spectrum from equation 7, is   (10)

For , 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 . The autocorrelation function of Brownian motion signals is a decreasing exponential and the autocorrelation in equation 6 properly reduces to for . Another interesting form of spectrum is for . 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 ,
• fractional Brownian motion (fBm) built by causal integration of fGn above and resulting in .
The fGn is stationary and Gaussian, whereas the fBm is neither stationary nor Gaussian. Exponent of the fBm is related to exponent of the fGn, used for integration, by .

The significance of parameter in equation 10 is delicate and connected to the Hurst exponent , which measures the correlation of time series (Hurst, 1951) by   (11)

where and are respectively range of variations and variance calculated for the length of the signal. The meaning of the value of is
• antipersistence for ,
• random process for , and
• persistence for .
Estimation of using formula 11 is relevant only for fGn signals (Turcotte, 1997). For example, Gaussian white noise produces . Parameter defined in equation 10 is associated with the Hurst exponent by
• for fGn with ;
• equals the Hurst exponent of incremented fGn (Li, 2003) for fBm with .
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 , promoting the so-called 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 Description Geology  Gaussian white noise Random process  Persistent fGn  Flicker noise Blocky layers  Antipersistent fBm Quasi-cyclic deposition  Brownian walk Random deposition  Persistent fBm Transitional deposition

Table 1. Classification of 1D fractal statistics according to the exponent and geological interpretation.    Fractal heterogeneities in sonic logs and low-frequency scattering attenuation  Next: Synthetic realizations Up: Statistical model of heterogeneities Previous: Statistical model of heterogeneities

2013-03-02