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

Lilly, JM; Sykulski, AM; Early, JJ; Olhede, SC; (2017) Fractional Brownian motion, the Matern process, and stochastic modeling of turbulent dispersion. Nonlinear Processes in Geophysics , 24 (3) pp. 481-514. 10.5194/npg-24-481-2017. Green open access

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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.

Type: Article
Title: Fractional Brownian motion, the Matern process, and stochastic modeling of turbulent dispersion
Open access status: An open access version is available from UCL Discovery
DOI: 10.5194/npg-24-481-2017
Publisher version: http://dx.doi.org/10.5194/npg-24-481-2017
Language: English
Additional information: © Author(s) 2017. This work is distributed under the Creative Commons Attribution 3.0 License.
Keywords: Science & Technology, Physical Sciences, Geochemistry & Geophysics, Meteorology & Atmospheric Sciences, TIME-SERIES MODELS, FRACTAL DIMENSION, 2-DIMENSIONAL TURBULENCE, DRIFTER TRAJECTORIES, 2ND-ORDER STATISTICS, EXACT SIMULATION, GAUSSIAN FIELDS, FREQUENCY, VORTICES, SPECTRUM
UCL classification: UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Statistical Science
URI: http://discovery.ucl.ac.uk/id/eprint/10027906
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