The parametric Kalman filter (PKF) is a computationally efficient alternative method to the ensemble Kalman filter. The PKF relies on an approximation of the error covariance matrix by a covariance model with a space–time evolving set of parameters. This study extends the PKF to nonlinear dynamics using the diffusive Burgers equation as an application, focusing on the forecast step of the assimilation cycle. The covariance model considered is based on the diffusion equation, with the diffusion tensor and the error variance as evolving parameters. An analytical derivation of the parameter dynamics highlights a closure issue. Therefore, a closure model is proposed based on the kurtosis of the local correlation functions. Numerical experiments compare the PKF forecast with the statistics obtained from a large ensemble of nonlinear forecasts. These experiments strengthen the closure model and demonstrate the ability of the PKF to reproduce the tangent linear covariance dynamics, at a low numerical cost.
Covariance functions in geophysical flows are known to evolve in
both time and space
Another route can be investigated that relies on analytical derivation of
covariance tensor dynamics
An intermediate formulation, between the approximation by an ensemble
and the theoretical formulation by analytic derivation, has
recently been introduced by
As mentioned earlier, the PKF formulation has been tested so far on linear dynamics. It is thus interesting for more general applications to consider extension to a nonlinear setting. The goal of the present work is to formulate and illustrate the forecast step of the PKF for the nonlinear dynamics given by the Burgers equation.
The Burgers equation is a nonlinear advection-diffusion model that usually involves
one variable in a one-dimensional space –
However, preliminary numerical tests have
shown that the treatment of the physical diffusion, as proposed in
In Sect.
Geophysical flow dynamics can be represented as a nonlinear system of the form
Due to the lack of precise knowledge of the initial condition,
The two-point covariance function
The numerical cost of solving Eq. (
The ensemble Kalman filter is a robust algorithm that applies
to low-order dynamical systems as well as to large dimension
systems encountered in geophysical applications.
The main difference for geophysical applications is that the
covariance matrix is closely related to the continuous covariance
function, which may not be the case for all discrete
low-order models. Thereafter, it is assumed that a discrete model
results from the discretization of a continuous model, making a
clear connection between the discrete and the continuous covariance
representations. This offers simplifications in the following derivations.
To that end, in what follows, the covariance function
We now give details about another approximation, which relies on the continuum, namely the parametric formulation.
The diffusion covariance model factorizes the covariance matrix as
Knowing the dynamics of the variance field
The dynamics of the variance field
However, the dynamics of the diffusion tensor is not as obvious to derive. A possible way to describe its dynamics is to consider some approximations that we will describe in the next section.
The dynamical equations of the local diffusion can be obtained taking advantage of approximations used in data assimilation for the estimation of the local diffusion tensor from ensemble data.
Following
The importance of the metric tensor comes from
its direct connection with the error field.
In dimension one, the metric is the scalar
In dimension two (three), the metric is a
Consequently, an approximation for the dynamics of the parametric
formulation based on the diffusion equation is given by
Equation (
Here, we consider the dynamics associated with the Burgers equation in 1-D:
For any smooth function
The random field
The dynamics of the fluctuation
Hence, the dynamics of the mean flow and of the fluctuations are described
by the coupled system (Eq.
Note that the term
If the magnitude of the perturbation
The aim is now to determine the dynamics of the two-point error covariance
function,
Since the offset (iv) modifies the mean but not the higher statistical moments
of
The effect of each process in Eq. (
The contribution of the production term (Eq.
The production term describes the amplification of the error due to the gradient of the mean field
The time evolution of the variance and the diffusion fields due to the transport
term (Eq.
The dynamics of error variance fields, deduced from Eq. (
From the commutation of the ensemble average and the partial derivative,
this simplifies to
Since
The dynamics of the metric tensor is deduced from Eq. (
With the normalized error
From the identity
Hence, the variance and the local diffusion
These equations represent the transport of the variance and of the diffusion by the mean flow: the variance is conserved, while the diffusion tensor is warped by the mean flow.
Following the same procedure, the dynamics of
As is expected while dealing with Reynolds equations, a closure
problem appears since the term
To proceed further, we take advantage of the link between
the unknown quantity
The quantities
Two particular cases are interesting to discuss: when the random field is
statistically homogeneous and, moreover, when the correlation function is
a Gaussian function.
In the case where the error random field is homogeneous,
the error correlation function is homogeneous too:
We propose to use these results to formulate a closure model:
for a general smooth error random field of the metric field
With this Gaussian closure (Eq.
In the one dimensional case, the dynamics of the local diffusion tensor is deduced
from the dynamics of the metric from
Contrary to the production (Eq.
The parametric covariance dynamics for the Burgers equation is now expressed collecting all these results.
From Eq. (
Equation (
A numerical experiment is now proposed to illustrate and assess these theoretical results.
A numerical experiment is proposed to illustrate the ability of the PKF forecast to reproduce the statistical evolution of the errors in the diffusive Burgers model. The numerical setting is first introduced, followed by an evaluation of the kurtosis closure. Then, the PKF is assessed using a large ensemble of nonlinear forecasts (6400 members). A sensitivity test on the different terms in the PKF concludes the section.
For the numerical validation, a front-like situation is considered
on a periodic domain of length
The random perturbation at initial time,
Nonlinear solution of the diffusive Burgers equation for the times
The covariance function is then defined as
The time evolution of the true error covariance functions is computed
considering a large ensemble of
Since the parametric covariance dynamics (Eq.
Figure
These ensembles are first used to tackle the closure of kurtosis, as discussed now.
Solutions from the initial perturbations of magnitude
Diagnoses of the length scale
The aim of this section is to compare the kurtosis diagnosed from the true
error covariance (Eq.
The local metric and kurtosis can be computed from the
ensemble considering Eq. (
For each position
Figure
At
Note that all the previous results are similar for the smaller initial
uncertainty magnitude
The parametric setting is based on the time integration of the nonlinear
coupled system (Eq.
The mean, the error variance, and length-scale fields are reproduced in
Figs.
Mean state at time
Parametric (dashed line) vs. ensemble estimated (continuous line)
variance fields for initial perturbations of standard
deviation magnitude
In order to appreciate the differences between the ensemble and the
parametric means,
the discussion is focused on the results at the final time
The variance (Fig.
The case of the error magnitude of
Beyond the variance attenuation, a maximum at
In order to assess the role of the nonlinear term,
Parametric (dashed line) vs.
ensemble estimated (continuous line)
length-scale field for initial standard deviations
Verification of the parametric variance
The case of the larger initial error magnitudes of
The increase of the PKF variance prediction might be a side effect due to the
tangent-linear-like derivation of the PKF which could fail to predict the
saturation of the error magnitude. In order to tackle the long-term
behaviour, a comparison is conducted with a longer time window of
Parametric (dashed line) vs. numerical (continuous line)
time series of the variance fields' maximum, from
From these results, we can conclude that the PKF forecast, as implemented by
Eq. (
This study focused on the forecast step of the parametric Kalman filter (PKF) applied to the nonlinear dynamics of the diffusive Burgers equation. The parametric approach consists in approximating the error covariance matrix by a covariance model with evolving parameter fields. Here the covariance model considered is based on the diffusion equation, parameterized by the error variance and local diffusion fields. Hence, the forecast of the error covariance matrix, which is computationally very demanding in real applications with high-dimensional systems, amounts to the forecast of the error variance and local-diffusion fields, whose numerical cost is of the order of a single nonlinear forecast. In comparison, ensemble methods need dozens of members for the covariance forecast (which could be parallelized though), as well as localization to address the rank deficiency.
The derivation of the PKF dynamics was first rigorously deduced from the dynamics of the perturbation under a small error magnitude assumption. However, a closure problem appears due to the physical diffusion process. This closure issue has been related to the fourth term in the Taylor expansion of the correlation function, the kurtosis, and a closure has been proposed based on a homogeneous Gaussian approximation for the kurtosis.
Numerical experiments in which the true covariance evolution has been diagnosed from an ensemble forecast were performed. First, a comparison with the PKF prediction showed the relevance of the closure, even for large error magnitudes. Moreover, these experiments have demonstrated the ability of the parametric formulation to reproduce the main features of the error dynamics when the tangent linear approximation is valid. When the tangent linear dynamics is not valid anymore, the PKF can only reproduce a part of the error statistics evolution, at least for mid-term forecasts.
This contribution is a step toward the PKF formulation of more complex dynamics in geophysics. From the present study, we learned the difficulties of handling the higher order derivatives, since the coupling between the error variance and diffusion fields has been due to the physical diffusion. The Gaussian closure, similar to the one introduced in the kurtosis' treatment, will be useful in providing prognostic dynamics. But we expect that the main difficulties will be encountered in the forecast of multivariate statistics that govern the balance between geophysical fields. Theoretically, the PKF formulation enables the forecast of covariance matrices in high dimensions. Hence, it might offer new theoretical tools to approximate and to investigate important aspects of the dynamics of errors, such as the unstable subspace of chaotic dynamics. These points will be investigated in further developments.
No data sets were used in this article.
The aim of this section is to show the following theorem
(given here for d
Multiplication by
The aim is to reformulate terms of the form
The variance
The term
For the term
Identifying with Eq. (
The third-order term
For the last fourth-order term
With the ensemble average, it follows that
The authors declare that they have no conflict of interest.
This article is part of the special issue “Numerical modeling, predictability and data assimilation in weather, ocean and climate: A special issue honoring the legacy of Anna Trevisan (1946–2016)”. It is not associated with a conference.
The authors are thankful to Stephen E. Cohn and an anonymous reviewer for their useful comments, suggestions, and corrections.
CEREA is a member of the Institut Pierre-Simon Laplace (IPSL). This work was supported by the French national program LEFE/INSU (Étude
du filtre de KAlman PAramétrique, KAPA).