next up previous [pdf]

Next: Fractal statistics Up: Browaeys & Fomel: Fractals Previous: Introduction

Statistical model of heterogeneities

Let us consider the spatial fluctuations of seismic velocities to be small and to constitute a second-order stochastic process. We describe the fluctuations by using different realizations of the random function $f(\mathbf {x})$ with the expectation value $\langle f \rangle = 0$ and with the spatial covariance depending on the relative distance $\mathbf{r}$ defined by

$\displaystyle \langle f(\mathbf{x})f(\mathbf{x}+\mathbf{r})\rangle$ $\textstyle =$ $\displaystyle \sigma^2 N(\mathbf{r}),$ (1)

where $\sigma$ is the standard deviation and $N(\mathbf{r})$ is the spatial autocorrelation function with $N(\mathbf{0})=1$. The energy spectrum $E^{(s)}(\mathbf{k})$ of the fluctuations in $s$ dimensions ($s=1,2,3$) is related to the autocorrelation by the Wiener-Khintchine theorem (Born and Wolf, 1964):
$\displaystyle E^{(s)}(\mathbf{k}) = \vert F(\mathbf{k})\vert^2$ $\textstyle =$ $\displaystyle \sigma^2\int N(\mathbf{r})e^{-i\mathbf{k}\cdot\mathbf{r}}d\mathbf{r},$ (2)
$\displaystyle F(\mathbf{k})$ $\textstyle =$ $\displaystyle \int f(\mathbf{x})e^{-i\mathbf{k}\cdot\mathbf{x}}d\mathbf{x},$ (3)

where $\mathbf{k}$ is the spatial wave vector and $F(\mathbf {k})$ is the Fourier transform of $f(\mathbf {x})$. The energy spectrum in equation 2 can be simplified, for an isotropic correlation function, to
$\displaystyle E^{(1)}(k)$ $\textstyle =$ $\displaystyle 2~\sigma^2\int_{0}^{\infty}N(r)\cos(kr)dr,$ (4)
$\displaystyle E^{(3)}(k)$ $\textstyle =$ $\displaystyle \frac{4\pi}{k}~\sigma^2\int_{0}^{\infty} rN(r)\sin(kr)dr,$ (5)

where $k=\vert\mathbf{k}\vert$. The von Kármán autocorrelation function $N_{H,b}(\mathbf{r})$ describes a self-affine medium relevant for geological structures (Goff and Jordan, 1988; Klimes, 2002; Dolan et al., 1998; Goff and Holliger, 2003; Sato and Fehler, 1998; Holliger and Levander, 1992). This function was initially derived by von Kármán (1948) while studying the velocity field in a turbulent fluid and has been used to describe heterogeneous media (Frankel and Clayton, 1986; Tatarski, 1961). The Fourier transform of $N_{H,b}(\mathbf{r})$ was given by Lord (1954). The statistical autocorrelation $N_{H,b}(\mathbf{r})$ and the energy spectrum $E^{(s)}_{H,b}(\mathbf {k})$ in the Fourier domain are
$\displaystyle N_{H,b}(\mathbf{r})$ $\textstyle =$ $\displaystyle \frac{2^{(1-H)}}{\Gamma(H)}~(r/b)^{H}K_{H}(r/b),$ (6)
$\displaystyle E^{(s)}_{H,b}(\mathbf{k})$ $\textstyle =$ $\displaystyle \sigma^2~C^{(s)}_{H}
\frac{\left(2b\right)^s}{\left(1+b^2k^2\right)^{H+\frac{s}{2}}},$ (7)
$\displaystyle \mbox{with}$   $\displaystyle C^{(s)}_{H}=\left\vert\frac{\Gamma(H+\frac{s}{2})}{\Gamma(H)}\right\vert\pi^{\frac{s}{2}},$ (8)

where $r=\vert\mathbf{r}\vert$, $K_{H}$ is the modified Bessel function of the second kind with order $H$, and $\Gamma$ is the Gamma function. Parameters describing the heterogeneities are characteristic distance $b$, below which the distribution is fractal, and exponent $H$, characterizing the roughness of the medium. We use the energy spectrum in equation 7 with $s=1$ to analyze sonic logs and with $s=3$ to predict 3D scattering attenuation.

next up previous [pdf]

Next: Fractal statistics Up: Browaeys & Fomel: Fractals Previous: Introduction