Fractional Brownian motion, the Matérn process, and stochastic modeling of turbulent dispersion
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-wor...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Copernicus Publications
2017-08-01
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| Series: | Nonlinear Processes in Geophysics |
| Online Access: | https://www.nonlin-processes-geophys.net/24/481/2017/npg-24-481-2017.pdf |
| Summary: | 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. |
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| ISSN: | 1023-5809 1607-7946 |