HR: 09:45h
AN: NG31C-07    [Abstracts]
TI: Generalized Laplace Motion: A Multifractal That May Unite Turbulence, Subsurface Heterogeneity and Other Irregular Phenomena.
AU: Hyden, P D
EM: hyden@clemson.edu
AF: Mathematical Sciences, Clemson University, Clemson, SC 29632 United States
AU: * Molz, F J
EM: fredi@clemson.edu
AF: Environmental Engineering and Geology, Clemson University 342 Computer Court , Anderson, SC 29625 United States
AB: Stochastic fractals arise naturally from the theory of non-stationary stochastic processes with stationary increments. Examples of such processes that occur in nature include velocity distributions "V(x)" in inertial-range turbulence, tracer distributions "C(x)" in turbulent flow and log(hydraulic conductivity) distributions "ln(K(x))" in heterogeneous sediments. However, the increments (fluctuations over a distance "h") of such quantities ($\Delta$$_{h}$V = V(x + h) - V(x), $\Delta$$_{h}$C = C(x + h) - C(x), and $\Delta$$_{h}$ln(K) = ln(K(x + h)) - ln(K(x)) appear to be stationary (stochastic properties independent of position), and for some time this has served as a basis for study. Stationary increment distributions for a fixed h (lag) may be fitted empirically with probability density functions (PDFs), and if the selected PDF is a member of an infinitely divisible family, then some type of orderly scaling structure is built into the increments of the non-stationary process. A commonly observed scaling property is the occurrence of a power-law variogram [$<$($\Delta$$_{h}$V)$^{2}$$>$; $<$ $>$ = expected value], also called a second order structure function. Well-known PDFs that are infinitely divisible include the Gaussian and Levy distributions. If these PDFs are fitted to increment distributions, the resulting stochastic processes are known as fractional Brownian motion (fBm) and fractional Levy motion (fLm), respectively. However, studies over the past decade have shown that neither PDF is an ideal model for increment distributions. Such distributions are not always Gaussian or Levy, a requirement of fBm and fLm, and the infinite variance feature of the Levy PDF makes this distribution unrealistic physically. What has been observed for both $\Delta$$_{h}$V and $\Delta$$_{h}$ln(K) measurements, is a change from the so-called stretched Laplace PDF (tail decay slower than exponential) - through pure Laplace (double exponential PDF) - to Gaussian behavior as h changes from smaller values to larger values. It is not widely known that the Laplace PDF is infinitely divisible, and therefore can serve as the basis for a stochastic fractal (Meerschaert et al., GRL, 31, L08501, 2004). In order to derive the infinitely divisible family, one must develop the concept of a generalized Laplace PDF (Kotz et al., The Laplace Distributions and Generalizations, Birkhauser, Boston, 2001), hence the name generalized Laplace motion (gLam), having increments called generalized Laplace noise (gLan). In general, gLam is a multifractal (displays a nonlinear dependence of the structure function exponent on lag), but behaves as a monofractal in what might be called the scaling ranges of h. The most general form of the model allows for both positive and negative correlation of the increments, as well as skewed increment distributions. For small h, gLan exhibits highly intermittent behavior, a phenomenon that has been observed in turbulent flow. All of these various properties are presented, and we conclude that through the geometric central limit theorem that applies to the Laplace PDF, the generalized Laplace stochastic process may capture fundamental aspects of a variety of heterogeneous phenomena.
DE: 3379 Turbulence
DE: 4568 Turbulence, diffusion, and mixing processes
DE: 3250 Fractals and multifractals
DE: 1832 Groundwater transport
DE: 1869 Stochastic processes
SC: Nonlinear Geophysics [NG]
MN: 2004 AGU Fall Meeting