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

TL;DR

This paper compares fractional Brownian motion and the Matern process, showing that the Matern process better models turbulent dispersion with low-frequency plateaus, and provides an efficient simulation algorithm.

Contribution

It introduces the Matern process as a damped version of fBm, demonstrating its suitability for modeling turbulent velocities with low-frequency plateaus.

Findings

Matern process matches turbulent velocity data better than fBm.

An O(N log N) algorithm for simulating the Matern process is developed.

The Matern process captures diffusive behavior in stochastic modeling.

Abstract

Stochastic process exhibiting power-law slopes in the frequency domain are frequently well modeled by fractional Brownian motion (fBm). In particular, the spectral slope at high frequencies is 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 Matern 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 Matern process and its relationship to fBm. An algorithm for the simulation of the Matern process in O(N log N) operations is given. Unlike fBm, the Matern process is found to provide an excellent match to modeling velocities from particle trajectories in an application to two-dimensional fluid turbulence.