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Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union

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Nonlin. Processes Geophys., 24, 481-514, 2017
https://doi.org/10.5194/npg-24-481-2017
© Author(s) 2017. This work is distributed under
the Creative Commons Attribution 3.0 License.
Research article
21 Aug 2017
Fractional Brownian motion, the Matérn process, and stochastic modeling of turbulent dispersion
Jonathan M. Lilly1, Adam M. Sykulski2, Jeffrey J. Early1, and Sofia C. Olhede3 1NorthWest Research Associates, P.O. Box 3027, Bellevue, WA, USA
2Data Science Institute, Department of Mathematics and Statistics, Lancaster University, Lancaster, UK
3Department of Statistical Science, University College London, Gower Street, London, UK
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 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 O(NlogN) 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.

Citation: Lilly, J. M., Sykulski, A. M., Early, J. J., and Olhede, S. C.: Fractional Brownian motion, the Matérn process, and stochastic modeling of turbulent dispersion, Nonlin. Processes Geophys., 24, 481-514, https://doi.org/10.5194/npg-24-481-2017, 2017.
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Short summary
This work arose from a desire to understand the nature of particle motions in turbulence. We sought a simple conceptual model that could describe such motions, then realized that this model could be applicable to an array of other problems. The basic idea is to create a string of random numbers, called a stochastic process, that mimics the properties of particle trajectories. This model could be useful in making best use of data from freely drifting instruments tracking the ocean currents.
This work arose from a desire to understand the nature of particle motions in turbulence. We...
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