Fractal heterogeneities in sonic logs and low-frequency scattering attenuation |

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 :

The exponent , in the energy spectrum from equation 7, is

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

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 .

- for fGn with ;
- equals the Hurst exponent of incremented fGn (Li, 2003) for fBm with .

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 |

Fractal heterogeneities in sonic logs and low-frequency scattering attenuation |

2013-07-26