  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.
