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

Abstract

Abstract. 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 lowfrequency 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 O(N log N ) 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.

Smoothing and Interpolating Noisy GPS Data with Smoothing Splines

“…Throughout this manuscript we generate synthetic data for both the signal and the noise. The velocity of the signal is generated from a Gaussian process known as the Matérn (Lilly et al, 2017). The spectrum of the Matérn is given by…”

Surface Drift and Dispersion in a Multiply Connected Fjord System

“…A brief overview of the concepts that are relevant to the analysis in this paper are given. For a more thorough treatment of the topic the reader may refer to works such as Mandelbrot and Van Ness (1968); Molz et al (1997), and Lilly et al (2017).…”

“…Here H is related to the fractal dimension D of the two-dimensional drifter motion by D = min[1∕H, 2], assuming isotropic statistics (Sanderson & Booth, 1991); A is an amplitude coefficient after Lilly et al (2017); and V H is the weighting kernel as defined by Mandelbrot and Van Ness (1968).…”

Element analysis: a wavelet-based method for analysing time-localized events in noisy time series

“…The power-law spectrum (34) corresponds for 1 2 < α < 3 2 to a random process x (t) consisting of fractional Brownian motion [29], see [25] and references therein. Although fractional Brownian motion is itself not stationary, as we have assumed above, a damped version of fractional Brownian motion known as the Matérn process is stationary [25].…”