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

Fractional Brownian motion (fBm), introduced by

One important property that cannot be captured by fractional Brownian motion
is the tendency for a process to diffuse, or disperse from an initial
location at a uniform rate. In the fluid dynamics literature

A particular application is the stochastic modeling of velocities obtained
from particle trajectories in fluid flows. In the field of oceanography, one
of the main windows into studying the physics of the ocean circulation
consists of position data from instruments that drift freely with the
currents

One thread of research attempts to predict Lagrangian statistics based on
dynamical assumptions

A type of random process having a sloped spectrum that matches fBm at high
frequencies, but that takes on a constant value in the vicinity of zero
frequency, exists and is known as the Matérn process

The purpose of this paper is to investigate the theoretical properties of the
Matérn process, in particular its relationship to fractional Brownian
motion, and to establish the practical importance of this under-appreciated
process for modeling time series that exhibit the fundamental phenomenon of
diffusion. On the theoretical side, the Matérn process is seen to be a
damped version of fractional Brownian motion, in the same way that the
Ornstein–Uhlenbeck process is a damped version of standard Brownian motion. A
simple generalization of the Matérn process that incorporates a uniform
rotation rate is shown to describe a forced/damped fractional
oscillator. By “damped version”, we mean that the process is modified as
would be expected if a physical damping were introduced into its stochastic
differential or stochastic integral equation. This terminology, which draws
upon intuition for damped and undamped oscillators from elementary physics,
will be made more clear in Sect.

On the practical side, we find the Matérn process to be an excellent match
for Lagrangian velocity spectra from a numerical simulation of
two-dimensional turbulence, a classical system in fluid dynamics that has
been the subject of a large number of studies,

A transition of the spectrum to constant values at sufficiently low frequencies is expected to be a common feature of many physical systems. Systems are often characterized by a pressure to grow – represented by a forcing – together with some drag or resistance on that growth, represented by a damping. After a sufficiently long time, the forcing and the damping equilibrate and one reaches a bounded state. This leads to the speculation that many time series that are well described as fBm over relatively short timescales may be better matched by the Matérn process over longer timescales. More generally, the Matérn process adds a third parameter (damping) to the two parameters (amplitude together with spectral slope or the Hurst parameter) of fBm, thus permitting a wider range of spectral forms to be accommodated. It is therefore reasonable to think that the Matérn process could be of broad interest in many areas in which fBm has already proven itself useful.

Many of the results herein may be found somewhere in the literature; the
novelty and significance of this paper arise from placing these results in
context. The relevant literature is vast, and the results that form this
narrative are widely distributed within disparate communities. The concept of
diffusivity discussed in Sect.

The main contributions of this work are (i) to place the Matérn process in context by understanding its relationship to fractional Brownian motion; (ii) to establish why the Matérn process is important for stochastic modeling of time series, geophysical time series in particular, which is its ability to simultaneously capture the effects of long-timescale diffusivity and small-scale fractal dimensionality; (iii) to demonstrate its performance with an application to a classical physical system; and (iv) to accomplish these goals in a way that is accessible to a general audience.

This paper was inspired by the need to develop a stochastic model for a particular physical application. As such, we are cognizant of the need to make stochastic modeling tools accessible to a broad audience. We have therefore endeavored to present material in a manner that is grounded in concepts from signal analysis, as this is a common language shared by many fields. A priority is placed on being self-contained, in order to avoid referring the reader repeatedly to the literature. The use of stochastic differential equations, or other more mathematical tools, is avoided unless absolutely necessary. At the same time, we are aware of the need to maintain rigor, and have therefore sought to carefully qualify any approximate or informal statements. New results are denoted as such.

The structure of the paper is as follows. Section

All numerical software associated with this paper, including a script for
figure generation, is distributed as a part of a freely available MATLAB
toolbox, as described in Appendix

This section introduces background material on stochastic processes, and identifies the diffusivity as a fundamental second-order stochastic quantity. This importance of diffusivity is illustrated by briefly discussing an application to modeling particle velocities in fluid turbulence.

In this paper, we will work with continuous-time, complex-valued processes, a choice that deserves comment. The decision to use complex-valued processes stems from the fact that the main application, to fluid dynamics, consists of analyzing trajectories that may be regarded as positions on the complex plane. For the most part, the results all apply equally well to real-valued processes. The choice to work in continuous time reflects more than convenience, as physical phenomena are generally regarded as existing continuously in time. A discrete time series arises when a process, such as a fluid flow, happens to be sampled at discrete intervals, owing to the constraints of measurements with real-world instruments. For these reasons, we will work in continuous time, and discrete sampling effects will be addressed when relevant.

Let

It is well known that the autocovariance function of a complex-valued process
does not completely characterize its second-order statistics

It is considered standard that the covariance between two
zero-mean complex-valued time series

The statistical information contained in the autocovariance function of a
second-order stationary process,

The time integral of the velocity process

Why

If an ensemble of particles exhibits power-law dispersion near some time

The seminal work of

While the diffusivity is not a recognized quantity in time series analysis,
we will show that it is an essential second-order descriptor, on par with the
variance. If

The result that the diffusivity is the zero-frequency value of velocity
spectrum is not new. It is implicit in a result of

Because the diffusivity appears as a second-order descriptor of the velocity
process

A diffusive process in our
terminology is distinct from the idea of a Markov diffusion process,
which is the solution to a particular type of first-order stochastic
differential equation

The classification of a stochastic process as diffusive, subdiffusive, or
superdiffusive is related to a well-known property, the process

The process memory is therefore a classification based on the absolute
integrability of the autocovariance, whereas the diffusiveness is based on
its integrability, as seen in Eq. (

In this paper, we will be concerned with an application to the stochastic
modeling of particle trajectories, and the associated velocity time series,
from a numerical simulation of fluid turbulence. The system we will use,
known as forced-dissipative two-dimensional turbulence (see, e.g.,
Sect. 8.3 of

A doubly periodic domain means that the

A snapshot of current speed from the turbulence simulation (left) together with 1024 particle
trajectories (right). In the left panel, shading is the speed

A snapshot of the velocity field at the initial time, together with the
particle trajectories from the entire simulation, is shown in
Fig.

These 512 “eddy-free” trajectories are also displayed in
Fig.

Dispersion curves for the 3-year turbulence trajectories and the three
different stochastic models discussed in Sect.

The average estimated spectrum of the velocity signals

The velocity spectrum is observed to have three main features: an overall energy level, a high-frequency slope, and a low-frequency plateau. As shown in the preceding section, the low-frequency plateau of the velocity signals is a reflection of the diffusive behavior of the trajectories. The goal of this paper is to identify a stochastic model capable of reproducing these three features, and to thoroughly understand its properties.

Consider one-, two-, and three-parameter frequency spectra
having the forms

For the sake of brevity, we are glossing over the fact that the spectrum of white noise is defined only up to the Nyquist frequency, whereas the other two spectra are defined for all frequencies.

The second is a power-law spectrum that arises for fractional Brownian motionThe form of the Matérn spectrum is fit to the velocity spectra of the
turbulence trajectories, in a way that will be described in
Sect.

The Matérn spectral form is seen to provide an excellent match to the
observed Lagrangian velocity spectra over roughly eight decades of structure.
The high-frequency slope is seen to be roughly

Spectra for the trajectories shown in Fig.

These three different sets of random processes for the velocity time series
are then cumulatively summed to form trajectories, and are compared with the
original trajectories in Fig.

By contrast, the one-parameter and two-parameter spectral models provide poor
fits to the observed trajectories (see Fig.

Thus, the white noise model is able to correctly match the large-scale, low-frequency component of the velocity spectra that accounts for the diffusive behavior of the trajectories. The power-law model is able to correctly match the high-frequency component of the spectrum that sets the small-scale roughness. The Matérn spectrum allows one to match both. This provides a compelling example that motivates examining the Matérn process in more detail.

This section reviews the properties of fractional Brownian motion, focusing on the central importance of the spectrum. With a few noted exceptions, this section presents material that is already known in the literature. Readers already very familiar with this process may wish to skip to the description of the Matérn process in the next section.

As described in the Introduction, many real-world processes are found to
exhibit power-law behavior over a broad range of frequencies. For a range of
spectral slopes, the power-law spectrum corresponds to that of a Gaussian
random process

A Gaussian random process is one for which every finite linear combination of samples has a jointly Gaussian distribution. For example, the distribution of the process at a fixed time is Gaussian, and the distribution between the process and itself at two different times is a jointly Gaussian function of two variables.

called fractional Brownian motion (fBm), introduced byExamples of complex-valued fractional Brownian motion are shown in
Fig.

Plan view of realizations of complex-valued fractional
Brownian motion

The main goal of this section is to utilize fBm to understand the
implications of the slope parameter

There are compelling reasons to work with the slope parameter

Fractional Brownian motion is defined in terms of a stochastic integral
equation, which will be presented later in this section. This stochastic
integral equation leads to a nonstationary autocovariance function given by

Observe that fractional Brownian motion is nonstationary: its
autocovariance is a function of “global” time

The normalizing constant in fBm, conventionally denoted

In addition to the autocovariance function, it is informative to also examine
a related second-order statistical quantity,

For fractional Brownian motion, cancellations in the variogram occur and one
obtains

Owing to its nonstationarity, the fBm autocovariance cannot be Fourier-transformed in the usual way to yield a spectrum that is independent of
global time

In general, the Fourier transform with respect to

For fractional Brownian motion, the Rihaczek distribution was stated by

This approach to proving that the power-law form is the correct spectrum to
associate with fBm may be critiqued on the grounds that taking the limit of
an average of the time-frequency spectral density, while mathematically
sensible, does not correspond well with a limiting action that occurs in
actual practice.

The most striking feature of fBm is that it is statistically identical to
rescaled versions of itself. To show this, we define a time- and
amplitude-rescaled version of

Because the original, unrescaled fBm process is Gaussian as well as zero
mean, its statistical behavior is completely characterized by its
autocovariance function. Thus, fBm is statistically identical to itself when
we “zoom in” in time, provided we also magnify the amplitude appropriately.
This property was referred to as self-similarity in the original work
of

The positive constant

An illustration of self-similarity is presented in
Fig.

A demonstration of self-similarity for fractional Brownian
motion, using the realizations presented in
Fig.

For stationary processes, self-similarity may also be seen in the frequency
domain. Apply the rescaling of Eq. (

Fractional Brownian motion is peculiar in that it has neither a well-defined
derivative nor a well-defined integral. Loosely speaking, one may say that a
derivative does not exist because the limiting action of taking a derivative
conflicts with the self-similarity. Because

The property of self-similarity, which is global in nature, was shown
in the previous section to be related to the spectral slope. The slope is
also related to two local properties, one associated with the slope at
small frequencies, or the behavior of the autocovariance at large time
offsets, and one associated with the slope at high frequencies, or the
autocovariance at small time offsets. The former property is the process
memory or long-range dependence discussed in
Sect.

Fractal dimension is a measure of the dimensionality of a curve (or some
higher-order surface) that accounts the effect of roughness

The dimension of the graph is closely related to the short-time behavior of
the autocovariance. As described by

As pointed out by

Fractional Brownian motion is defined via the stochastic integral equation

Note that standard Brownian motion, corresponding to

The stochastic integral equation of Eq. (

The weighting factors such as

While the left-hand side of the Cauchy integral formula is not interpretable
for non-integer

Returning to the definition of fBm in Eq. (

The previous section reviewed the properties of fractional Brownian motion, including its self-similarity and fractal dimension, and showed how these are related to the spectral slope. This section examines the Matérn process in detail, with a focus on its relationship to fBm. A simple extension, the inclusion of a “spin parameter”, generalizes the Matérn process to encompass a larger family of oscillatory processes that are shown to represent forced/damped fractional oscillators.

In Sect.

Plan view of realizations of the complex-valued Matérn
process, for 12 different values of the slope parameter

Examples of simulated Matérn processes are shown in Fig.

The damping parameter

The theoretical spectra corresponding to the realizations in
Fig.

To examine the role of

The variance and diffusivity of the Matérn process are both finite, and are
found to be given by

The Matérn spectrum can be rewritten in terms of the variance

The autocovariance function corresponding to the Matérn spectrum, Eq. (

Examples of theoretical Matérn autocovariance functions are presented in
Fig.

Theoretical spectra

The asymptotic behavior of the Matérn covariance for large and small times
is as follows. For

The expression Eq. (

From this small-

A very simple modification can expand the range of possibilities of the
Matérn process, and also aid in the development of physical intuition. We
add a deterministic tendency for the process to spin on the complex plane at
rate

Thus, the oscillatory Matérn process subsumes the Matérn process and the complex Ornstein–Uhlenbeck process into a larger family. In this next section we will determine the stochastic integral equation of this oscillatory Matérn process.

Unlike fractional Brownian motion, the Matérn process is not generally
defined in terms of a stochastic integral equation or a stochastic
differential equation. A stochastic integral equation that will generate an
oscillatory Matérn process is

The Fourier transform of a Green's function

As an aside, we point out that
this result implies that with

Examples of the Green's functions for

Identifying this stochastic integral equation sheds light on the nature of
the Matérn process itself. The Green's function

Note that here we have avoided attempting to write the Matérn process as a
stochastic differential equation, as there are mathematical
difficulties in ensuring that the fractional-order derivatives
exist.

The expansion of the fractional-order operator in
Eq. (

We can also now understand why

All of these factors support interpreting

As shown in the next section, if the damping vanishes, the stochastic integral equation for the Matérn process becomes identical to that for fractional Brownian motion, apart from a modification that sets the initial condition for fBm.

Having identified the stochastic integral equation for the Matérn process,
we now examine its relationship with fractional Brownian motion. The Green's
function of the oscillatory Matérn process, Eq. (

If fractional Brownian motion and the standard Matérn processes are
essentially facets of the same process, one should be able to see this
directly from their autocovariances. This is indeed the case. For time shifts

The intuitive interpretation of this result is that a Matérn process has a
second-order structure that behaves for small time offsets

To look at this another way, imagine that a modified Matérn process were
constructed with an integral matching the form of that for fractional
Brownian motion in Eq. (

The qualitatively significant difference between the Matérn process and fBm – that the former is stationary,
while the latter is nonstationary – can be seen as a consequence of the lack of damping in the latter case.
In applications, we believe it would be unphysical to observe a process that remains nonstationary for all timescales.
Rather, for sufficiently long observational periods, it is more likely that the process will eventually settle into
stationary behavior. For the Matérn process, this occurs when the observational window is sufficiently long compared
with the decay timescale

This section addresses means to simulate realizations of fractional Brownian
motion and the Matérn process numerically. The main contribution is a new
approach to simulating a diffusive process such as the Matérn in

The standard approach to simulating a Gaussian random process with a known
covariance matrix is a method called the Cholesky decomposition, which we
discuss here. In this section, as we will be dealing with vectors and
matrices, a change of notation is called for. We now let

This sequence is arranged into a length

The Cholesky decomposition factorizes the covariance matrix as

Thus, to simulate a length

A limitation of this approach is that the Cholesky decomposition requires, in
its most straightforward implementation,

To devise our generation method, we will first renormalize the Green's
function so that we may use

For each of these integrals over a short segment, we approximate the Green's
function by a constant, namely the value of the Green's function at the
segment midpoint, which occurs when

The numerical evaluation of the oversampled Green's function can be
simplified by noting the behavior of

The autocovariance function of

In practice, the summations over the duration of the Green's function must be truncated at some point. It is tempting to truncate the Green's function after a relatively short time. However, for spectra having a large dynamic range, this truncation leads to undesirable leakage effects, just as in spectral analysis, that degrade the spectrum of the generated sequences. Instead, we will utilize a Green's function that is longer than entire length of the time series.

Firstly we need to determine a suitable cutoff for limiting the long-term
influence of the Green's function. We denote by

Anticipating transforming to the Fourier domain, we will define sequences that are periodized. Because we intend to employ
a periodic convolution, yet wish to prevent noise values at the end of the time series from influencing the beginning, we
will create a longer sequence of length

The

The advantage to the Green's function approach is that
Eq. (

If desired, the matrix

As an example, in Fig.

No substantial difference between spectra computed with the two different
algorithms is seen over many decades of structure, indicating that fast
algorithm is able to simulate the Matérn process to a very high degree of
accuracy. In generating this plot for time series of length 1000, 2000, 4000,
and 8000 points (as shown here), the Green's function method executes
respectively 3, 7, 11, and 45 times faster than the Cholesky algorithm on a
Mac desktop. Note that the Green's function method does not depend on any
special properties of the Matérn process, apart from the particular
definition of the cutoff time

A comparison of the spectra of simulated unit-variance Matérn processes having 25 different

This section presents the details of an application of the Matérn process
to modeling particle velocities in a numerical simulation of two-dimensional
fluid turbulence, a preview of which was presented in
Sect.

A system called forced-dissipative quasigeostrophic turbulence is
created by integrating an equation for the streamfunction

An integration of Eq. (

A snapshot of current speed from the first day of the simulation after the
end of the spin-up period is shown in the left panel of Fig.

The analysis here is based on a set of 1024 particle trajectories that are
tracked throughout this experiment, shown in the right panel of
Fig.

The simplest way to remove the effects of vortices is simply to discard those
trajectories which conspicuously exhibit the effects of vortex trapping. A
common measure of the impact of vortices on a trajectory is the so-called
spin parameter

We take the modulus of the time-averaged spin,

This section describes the method by which the Matérn parameters are
estimated from a finite data sample, which necessitates some new notation. In
reality one only observes a random process

A standard approach would be to form a parametric estimate using the maximum
likelihood method implemented in the time domain. However, this method
involves a computationally expensive matrix inversion, which becomes a
limiting factor when analyzing large datasets. An alternative approach to
estimating the parameters is to do so in the frequency domain using a method
called the Whittle likelihood

The model parameters are estimated by finding the value of

In turns out to be the case that in the inference of parameters for a steep
spectrum, such as we are dealing with here, this approach is inadequate as it
ignores potentially significant effects associated with the finite sample
size. In particular, spectral blurring associated with the periodogram
can lead to quite incorrect slopes at high frequencies. Instead we use the
de-biased Whittle likelihood method recently developed by

Here we give details on how the realizations shown in
Fig.

After forming the tapered spectral estimate for each of the 512 velocity time series, we apply the de-biased Whittle likelihood to infer the
best fit Matérn parameters for each time series. Here the frequency set

To generate the trajectories shown in Fig.

For the power-law realizations, we cannot employ fractional Brownian motion
because the observed slopes – which in this simulation are steeper than those
found in the ocean – are outside the fBm range. Instead we use the implied
spectral amplitudes

The point of the application is to show that Matérn process provides an excellent match to the turbulence data. This opens the door to investigating a number of interesting physical questions regarding the distributions and interpretations of those parameters, which must, however, be left to the future.

This paper has examined the Matérn process as a stochastic model for time series, which we have shown to be equivalent to damped fractional Brownian motion (fBm). The damping is shown to be essential for permitting the phenomenon of diffusivity to arise in the temporal integral of the process, referred to here as the trajectory, which disperses from its initial location at a constant rate. The rate of diffusion of the trajectory is given by the value of the spectrum of the process at zero frequency. At higher frequencies, the spectrum transitions to a power-law slope, like fBm, with the location of this transition being controlled by the damping parameter.

Because damping is a common feature in physical systems, the Matérn process
is expected to be valuable in describing time series which, when observed
over shorter time intervals, appear to consist of fractional Brownian
motion. The addition of a spin parameter leads to a still more general
process that satisfies the stochastic integral equation for a damped
fractional oscillator forced by continuous-time white noise, and that
encompasses the standard Matérn process as well as the complex

A categorization of stochastic processes as diffusive,
subdiffusive, and superdiffusive was proposed, depending upon
their value at zero frequency. These categorizations refer to the nature of
the dispersion experienced by the trajectory associated with the process,
assuming that the integral of the process is well defined. This
categorization is related to, yet distinct from, the conventional designation
of a random process as short-memory or long-memory

The Matérn process was found to provide an excellent match to velocity time
series from particle trajectories in forced/dissipative two-dimensional fluid
turbulence that are not directly influenced by the presence of vortices. This
is an important contribution, since we show that a power-law process such as
fBm cannot hope to capture the diffusive behavior. Despite its simple
three-parameter form, trajectories associated with the Matérn process were
seen to be visually virtually indistinguishable from those from the numerical
model. This suggests that the Matérn form may prove useful for describing
similar trajectories taken by instruments tracking the actual ocean currents.
Such “Lagrangian data” comprise one of the main windows into observing the ocean
circulation, yet surprisingly little work has been done to analyze the
velocity spectra in major Lagrangian datasets

In this paper, we have taken essentially an observational approach, and
sought to fit a parametric model to the trajectories as a descriptive
analysis, without requiring a physical justification. A next step is to
attempt to understand this model on physical grounds. A number of researchers
have attempted to derive forms for the Lagrangian velocity spectrum (or,
equivalently, the autocovariance function) under simplified dynamical
assumptions

The data analyzed in this paper consist of model output from a simulation
of two-dimensional turbulence. Because the trajectory data are rather large (about 1 GB) and consist of model output
rather than real-world measurements, a suitable data repository has not been found.
Instead, the results can be reproduced in three ways.
First, two-dimensional turbulence models of this type are quite common in the community, e.g.

All software needed to carry out the analyses described in this paper, and to
generate all figures, is distributed as a part of a freely available toolbox
of MATLAB functions. This toolbox, called

Here we show that for a second-order stationary process, the diffusivity

The time-dependent diffusivity can be understood in several different ways;
see also

In the case that

In this appendix we examine the relationship between the properties of memory
and diffusiveness by constructing examples of processes with different
combinations of these two properties through modifying the Matérn process.
Here we will make use of a number of quantities that are not defined until
the Matérn process is examined in Sect.

Spectra of stationary processes corresponding to different combinations of
memory and diffusiveness are given in Table

Examples of spectra for short- and long-memory processes of
subdiffusive, diffusive, and superdiffusive types. The term in the box is the
spectrum of the Matérn process, as given in Eq. (

Multiplying the Matérn transfer function given by Eq. (

These results show that diffusiveness and memory, while related, are distinct
properties that can be varied independently. In this table we have also noted
the parameter ranges required for the process spectrum to integrate to a
finite variance, and therefore for the process to be stationarity. In general
for a spectrum of the form

Here we derive Eq. (

The coefficient of the integral in Eq. (

The usual form of the coefficient for fractional Brownian motion, in terms of
the Hurst parameter

Now, using the reflection formula Eq. (

Define the difference of a fractional Brownian motion process at one time and
itself a different time as

The autocovariance function for continuous-time fractional Gaussian noise
will be denoted as

The expression Eq. (

The normalized fGn covariance function

The behavior of the fractional Gaussian noise covariance function allows us
to discuss the property of persistence. For

The fractional Gaussian noise autocovariance function

The persistence transition in fractional Gaussian noise at

The memory of fractional Gaussian noise may be determined as follows. The fGn
autocovariance Eq. (

In this appendix we derive the form of the small-

Provided that

The Green's function Eq. (

The authors declare that they have no conflict of interest.

The work of Jonathan M. Lilly and Jeffrey J. Early was supported by award #1235310 from the Physical Oceanography program of the United States National Science Foundation. The work of Adam M. Sykulski was supported by a Marie Curie International Outgoing Fellowship. The work of Sofia C. Olhede was supported by awards #EP/I005250/1 and #EP/L025744/1 from the Engineering and Physical Sciences Research Council of the United Kingdom, and by award #682172 from the European Research Council.

The authors are grateful to an anonymous referee and to Peter Ditlevsen for their comments during the review process, which led to an improved paper. Helpful and inspiring interactions with Alfred Hanssen, Tim Garrett, Joe LaCasce, Shane Elipot, Rick Lumpkin, and Brendon Lai at various stages in the preparation of this work are also gratefully acknowledged. Edited by: Stéphane Vannitsem Reviewed by: Peter Ditlevsen and one anonymous referee