Rejuvenation in particle filters is necessary to prevent the collapse of the weights when the number of particles is insufficient to properly sample the high-probability regions of the state space. Rejuvenation is often implemented in a heuristic manner by the addition of random noise that widens the support of the ensemble. This work aims at improving canonical rejuvenation methodology by the introduction of additional prior information obtained from climatological samples; the dynamical particles used for importance sampling are augmented with samples obtained from stochastic covariance shrinkage. A localized variant of the proposed method is developed. Numerical experiments with the Lorenz '63 model show that modified filters significantly improve the analyses for low dynamical ensemble sizes. Furthermore, localization experiments with the Lorenz '96 model show that the proposed methodology is extendable to larger systems.

Ensemble-based data assimilation

Previous work

This work explores a new approach to particle rejuvenation, which is necessary to prevent weight collapse in particle filters. Rejuvenation in particle filters is a particular type of stochastic regularization

This paper is organized as follows. Section

Bayesian inference

Classical particle filtering

The ensemble of weights is denoted by

Our goal is to find an analysis ensemble

The optimal coupling

A visual representation of the continuous optimal transport procedure. The probability distribution

The standard ETPF

In high-dimensional geophysical problems, spatial error correlations decrease with increasing spatial distance between states. Due to the undersampled nature of the ensemble, these correlations may not be accurately approximated. Localization allows us to strictly enforce the shrinking of correlations between distant states. For localization in the ETPF, we follow the

A visual representation of

Typically, the observation error distribution is assumed to be unbiased and Gaussian, with the probability density used to compute the weights in Eq. (

We assume that the observations are uncorrelated, making

A different transform matrix is computed for each state variable. Specifically, consider the ensembles of the

Particle and ensemble-based filters often underrepresent uncertainty

In order to avoid ensemble collapse, the ETPF employs a particle rejuvenation approach

Note that defining the matrix

In the context of ensemble methods, covariance shrinkage (

From a Bayesian perspective, covariance shrinkage seeks to incorporate additional prior information on error correlations into the analysis, in order to enhance the inference.
In many data assimilation models, climatological covariance information is often available; i.e., it is known prior information. Climatological covariances are typically precomputed or derived from climatological models and are often employed in variational data assimilation

Following

A representation of the mixing of the dynamical ensemble

Augmenting the dynamical ensembles with the synthetic ensembles leads to the total

Taking

In the covariance shrinkage approach, to ensure that the sample mean of the augmented ensemble is the same as that of the dynamic ensemble, the synthetic ensemble is constructed with a mean equal to the sample mean of the dynamic ensemble:

The weights

In this framework

Using the prior weight ensemble determined by Eq. (

Recall that in the traditional method of rejuvenation (

In this section we show that the FETPF reduces to (generalized) variants of the ETPF in two different ways: in the synthetic ensemble limit and in the synthetic distribution limit.

Assume that the synthetic sample distribution is inexact in the mean and covariance, violating the assumption made in Eq. (

Assume by contrast that the synthetic sample distribution is exact, meaning that the climatology produces samples indistinguishable from the forecast and that the assumption in Eq. (

This leaves a gap, however, as the shrinkage factor

It is conceivable that multiple climatological models give rise to multiple climatological covariances, or alternatively multiple candidates for the most “common” behavior of the model are to be chosen.

Given a collection of target covariances,

We start with a short introduction to test problem configurations and the numerical experiment setups.

In order to stay in line with other particle rejuvenation techniques, anomaly inflation is used as a heuristic to try and overcome deficiencies in the descriptive power of the synthetic ensemble. Formally,

In our experiments we report the error of the analysis mean with respect to the truth (reference), measured by the spatiotemporal root mean square error (RMSE):

We now describe the 3-variable Lorenz '63 model and the 40-variable Lorenz '96 model that are used in the experiments. We use the implementation of both these problems from the ODE test problem suite

For the first set of numerical experiments, we use the Lorenz '63 system

We perform 10 000 assimilation steps but discard the first 1000 that are used for spinup. The time interval between successive observations is

This problem setup is challenging for the ensemble Kalman filter, which does not converge even for larger ensemble sizes. Therefore, this is a relevant test for non-Gaussian algorithms.

As discussed previously, the canonical choice for the shrinkage covariance is the identity matrix. It has been the authors' experience that for most dynamical systems this choice is poor. Moreover, the sequential data assimilation problem typically provides ways to calculate climatological approximations to the covariance. We take advantage of such techniques in this paper.

The first type of climatological covariance that we investigate is that of the distribution over the whole manifold of the dynamics. The trace-state-normalized

Our first round of experiments compares the canonical method of rejuvenation in the ETPF and the ETPF2 with a rejuvenation factor of

The results for the first round of experiments are shown in Fig.

For the Lorenz '63 model, analysis RMSE versus dynamic ensemble size (

The FETPF without synthetic inflation performs worse than both the ETPF and the ETPF2 for Gaussian synthetic samples, while it performs better when equipped with Laplacian synthetic samples.
When the synthetic samples are inflated with inflation factor

The second round of experiments uses multiple values of the climatological covariance

The second round of experiments is reported in Fig.

For the Lorenz '63 model, analysis RMSE versus dynamic ensemble size (

The results empirically show that supplementing the ensemble with additional synthetic information during assimilation is more effective than randomly perturbing the ensemble post-assimilation, for a small problem.
The authors hypothesize that the results point strongly towards the need to intelligently and adaptively choose the target covariance matrices and to the need for better operational calculation of the covariance shrinkage factor

Our third round of experiments with the Lorenz '63 system seeks to understand the effect of selecting the two free parameters, i.e., the synthetic ensemble size

Figure

For the Lorenz '63 model, analysis RMSE of the covariance shrinkage approach to particle rejuvenation (FETPF) for different values of the synthetic ensemble size

An interesting effect is that very large synthetic ensemble sizes are required to correspond to a noticeable decrease in error relative to the dimension of the system. This might pose a challenge when this algorithm is utilized without further corrections such as localization.

Our fourth experiment with the Lorenz '63 equations looks at the distribution of the values of the shrinkage parameter

Figure

The distribution of the

For numerical experiments with localization, we use the Lorenz '96 system

We test with two observation operators. First, we consider a standard linear observation operator

The matrix

Trace-normalized climatological

For localization, we take the Gaspari–Cohn

For the Lorenz '96 experiments we aim to compare LFETPF to the LETPF and to the localized ensemble transform Kalman filter (LETKF). We set the synthetic ensemble size to a constant

For the linear observation operator (

For the Lorenz '96 model with the linear observation operator (

For the nonlinear observation operator (

For the Lorenz '96 model with the nonlinear observation operator (

This paper introduces a stochastic covariance shrinkage-based particle rejuvenation technique for the ensemble transport particle filter. Instead of incorporating synthetic noise as an attempt to regularize the distribution of the ensemble, we attempt to incorporate an ensemble derived from some known prior information. This is done through the use of synthetic anomalies. These synthetic anomalies are sampled from any chosen distribution family, such that they are consistent with the climatological covariance information. We provide a philosophical justification for why we believe our approach is more in line with the assumptions underlying Bayesian inference.

Numerical experiments with the simple three-variable Lorenz system show that the use of climatological prior information to perform rejuvenation leads to reduced analysis errors than the typical rejuvenation approach. Additionally, the FETPF methodology seems to be much more stable for smaller dynamical ensemble sizes than the original rejuvenation approach. This leads us to believe that the stochastic shrinkage approach augments the original ensemble in a meaningful way.

Numerical experiments with localization techniques show that the LFETPF is comparable in performance to the LETPF; however, a large synthetic ensemble size is likely needed. Future work combining the LFETPF and the LFETKF

One limitation of this work is the focus on synthetic Gaussian samples. Methods such as generative adversarial networks

By far the largest limitation of this work is common to research on particle filters: the large-dimensional setting. While localization methods have begun the foray of particle filters into the medium-dimensional setting, alternatives to

It is the authors' belief that future research into all the limitations that have been identified might significantly improve the performance of the FETPF and create a method that can be applied to operational problems.

The version of ODE test problems used to generate all the experiments in this work is available at the persistent link

All data for this work were generated through the software packages above and can be made available by a request to the authors.

All the authors contributed to the final draft of the work. Additionally, AAP wrote the first draft and contributed to the numerical experiments, ANS contributed to the numerical experiments, and AS contributed to the formulation of the initial idea.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We would like to thank the rest of the members of the Computational Science Laboratory at Virginia Tech.

This research has been supported by the Division of Mathematical Sciences (grant no. NSF CDS&E-MSS–1953113) and the Advanced Scientific Computing Research (grant no. DOE ASCR DE–SC0021313).

This paper was edited by Olivier Talagrand and reviewed by Manuel Pulido and Alban Farchi.