Fractional Brownian motion, the Matérn process, and stochastic modeling of turbulent dispersion

• Main Authors: J. M. Lilly, A. M. Sykulski, J. J. Early, S. C. Olhede
• Format: Article
• Language: English
• Published: Copernicus Publications 2017-08-01
• Series: Nonlinear Processes in Geophysics
• Online Access: https://www.nonlin-processes-geophys.net/24/481/2017/npg-24-481-2017.pdf

Stochastic processes exhibiting power-law slopes in the frequency domain are frequently well modeled by fractional Brownian motion (fBm), with the spectral slope at high frequencies being associated with the degree of small-scale roughness or fractal dimension. However, a broad class of real-world signals have a high-frequency slope, like fBm, but a plateau in the vicinity of zero frequency. This low-frequency plateau, it is shown, implies that the temporal integral of the process exhibits diffusive behavior, dispersing from its initial location at a constant rate. Such processes are not well modeled by fBm, which has a singularity at zero frequency corresponding to an unbounded rate of dispersion. A more appropriate stochastic model is a much lesser-known random process called the Matérn process, which is shown herein to be a damped version of fractional Brownian motion. This article first provides a thorough introduction to fractional Brownian motion, then examines the details of the Matérn process and its relationship to fBm. An algorithm for the simulation of the Matérn process in <i>O</i>(<i>N</i><i>log</i><i>N</i>) operations is given. Unlike fBm, the Matérn process is found to provide an excellent match to modeling velocities from particle trajectories in an application to two-dimensional fluid turbulence.