The basis and challenge of strongly coupled data assimilation (CDA) is the accurate representation of cross-domain covariances between various coupled subsystems with disparate spatio-temporal scales, where often one or more subsystems are unobserved. In this study, we explore strong CDA using ensemble Kalman filtering methods applied to a conceptual multiscale chaotic model consisting of three coupled Lorenz attractors. We introduce the use of the local attractor dimension (i.e. the Kaplan–Yorke dimension,

As the world of climate modelling has moved towards coupled Earth system models which combine ocean, atmosphere, sea-ice, and biogeochemical effects, it is essential to understand how the respective domains of disparate spatio-temporal scales covary and influence each other. In the context of state estimation, strongly coupled data assimilation (CDA) in multiscale systems allows the quantification of cross-domain dynamics. Specifically, strong CDA uses the cross-covariances amongst all components to influence the analysis increment, meaning that unobserved subsystems are directly adjusted in the analysis step regardless of observation set and coupling strengths

One approach to reduce the requirement to adequately sample the background covariances is to perform CDA in the reduced subspace of the unstable modes, known as assimilation in the unstable subspace (AUS)

There is a computational cost accompanying the determination of the unstable and neutral subspaces which may or may not be less than the cost of adequately sampling the model covariances. We do not attempt to comment on numerical efficiency of AUS methods applied to high-dimensional systems. The exploration of cost-effective methods for determining the reduced subspace is left for future study.

The main motivation for this study comes from the conjecture that when applying ensemble Kalman filtering methods to high-dimensional nonlinear systems, the true time-dependent error covariance matrix collapses onto a subspace of the model domain which contains unstable, neutral, and sometimes weakly stable modes. While previous results prove the collapse of the error covariance matrix onto the unstable and neutral subspace for linear systems

Such transient error growth has previously been explored in ocean–atmosphere models of varying complexity. One way of quantifying local error growth is through finite-time Lyapunov exponents (FTLEs), i.e. rates of expansion and contraction over finite intervals of time.

In this study we utilize FTLEs and CLVs within the ensemble Kalman filtering framework applied to a low-dimensional chaotic model with spatio-temporal scale separation. The model was designed to represent the interaction between the ocean, extratropical atmosphere, and an ocean–atmosphere interface (referred to as the tropical atmosphere). The idea is that the ocean and extratropical atmosphere are only implicitly coupled through the interface, with the interface being strongly coupled to the ocean and weakly coupled to the extratropical atmosphere. We consider the performance of strong CDA on this paradigm model with different subsets of observations. We introduce the use of a varying number of CLVs to form the forecast error covariance matrix. The idea of AUS is incorporated through the use of the time-varying subspace defined by the local attractor dimension. The dimension is calculated through FTLEs and the error covariance matrix is constructed via a projection of the ensemble members onto a corresponding subset of the CLVs at each analysis step. We compare full-rank ensemble transform and square-root filters to “phase space” variants whose background covariances are defined in terms of either the finite-time or asymptotic attractor dimension. Another variant considered includes only the unstable, neutral, and weakest stable CLVs. We consider benchmark calculations compared to the recent work of

The paper is organized as follows. Section

This section describes a low-dimensional chaotic model representing a coupled ocean and atmosphere. It is a system of three coupled Lorenz '63 models introduced by

More generally, these choices lead to a tropical subsystem that is dominated by changes in the ocean subsystem, but has an extratropical atmosphere, representative of weather noise, whose behaviour is similar to the original Lorenz model in that the system exhibits chaotic behaviour with two distinct regimes observed in the

Example trajectories of the coupled Lorenz model (Eq.

Trajectories along attractors of the extratropical

We are interested in analysing both the short-term and asymptotic dynamics of the system (

We compute the Lyapunov exponents using a QR decomposition method (see e.g.

Asymptotic Lyapunov exponents of Eq. (

Given the approximated asymptotic Lyapunov exponents in Table

With the values in Table

Figure

Local Kaplan–Yorke dimension plotted along the trajectory in phase space for the extratropical atmosphere

The existence of Lyapunov vectors for a large class of dynamical systems was proven by

It should be mentioned that the push forward step need not be equal to

We examine the effectiveness of this algorithm on the

Local dynamical properties of a segment of an example model run.

At this point we will comment on the non-trivial relationship between the CLVs and FTLEs calculated here. As discussed in

Comparison of the finite-time Kaplan–Yorke dimension calculated using the growth rates of CLVs (FTCLEs) and the QR method (FTBLEs).

In the following sections, we will utilize CLVs within the data assimilation framework of ensemble forecasting. The CLVs will be used to construct the forecast error covariance matrix, which informs the increment used on ensemble members to bring them closer to observations. Using CLVs in this context suggests a more accurate method of forming the forecast error covariance matrix when the true covariance is undersampled due to an insufficient number of ensemble members (see e.g.

We now summarize the Kalman filter equations. For detailed derivations we refer the interested reader to the reviews by

In the Kalman filter method, Eqs. (

Ensemble Kalman filtering methods use Monte Carlo sampling to form the approximate error statistics of a model. An ensemble of model states

Another deterministic scheme for ensemble Kalman filtering which uses a left-multiplied transform matrix was shown by

When using ensemble Kalman filtering methods like the ones introduced here, sampling errors can often occur. For nonlinear models in particular, there is a systematic underestimation of analysis error covariances which eventually leads to filter divergence

Here we define the error covariance matrix

To determine the number of CLVs required to form the basis for

It is important that we span the local dimension of the attractor within the ensemble DA framework in order to avoid ensemble collapse. As mentioned in Sect.

We compare the alignment of the CLVs at two time steps of a model run. The two time steps have a similar Kaplan–Yorke measure with the distribution of the leading FTLEs, shown in the bottom panels. We observe that the behaviour of the alignment can be vastly different for similar FTLE behaviour; however, the method of retaining CLVs based on the Kaplan–Yorke measure gives a reliable way of reflecting the true local dimension regardless of alignment.

We perform a collection of data assimilation experiments for the system (

The initialization settings for the DA experiments are as follows, unless otherwise stated. We use the settings from

The dynamical properties of the experiments are calculated with respect to the ensemble mean trajectory. The FTLEs are computed using the QR decomposition over the previous 400 time steps leading up to the assimilation time step. The local Kaplan–Yorke dimension is then calculated from the FTLEs. The CLVs are then calculated using a slight modification to Algorithm 1 – due to the absence of an accurate future trajectory of the ensemble mean, we do not perform the reorthogonalization to the eigenvectors of

In the Kalman filtering method introduced in Sect.

The main differences in our subsequent experiments are in the subset of observations used and their corresponding observational errors. We aim to assess the performance of the reduced-rank strong CDA within the different configurations. We first present a benchmark test on the CLV method which is identical to an experiment presented in

The first DA experiment we consider is a benchmark case with observations (

The error statistics of all the experiments are listed in Table

Summary metrics of DA experiments using a right-transform matrix (Eq.

We observe that all experiments generally succeed at constraining the full system. The trajectories, spread, increments, and error metrics of the variable CLV experiments are shown in Fig.

Segment of DA using the variable CLV method and benchmark observations (

To take a closer inspection of the dynamics during the assimilation, Fig.

Local attractor properties of DA using the variable CLV method and benchmark observations (

We now consider the case where only the two atmosphere subsystems are observed; however, the observations are less sparse within each subsystem in that we take both the

Summary metrics of DA experiments using a right-transform matrix (Eq.

Table

The strong coupling and low-frequency variation in the ocean and tropical atmosphere subsystems represent an ENSO-like variability. We therefore refer to the case of observing the

Since the extratropical subsystem is likely to be unconstrained with these observations, the ensemble mean will not be accurately estimated. In such a case, the variable CLV method fails due to the fact that the first CLV (which corresponds to the directions of fastest error growth) is inaccurately calculated. Therefore the true directions of unstable growth are inaccurately sampled in the reduced space experiments. This is amplified by the fact that we are using uncorrelated observation errors; if the observation errors have a temporal correlation, the dominant direction of nonlinear unstable growth can be ascertained even without tracking the extratropical subsystem (see the following section on shadowed observations). The inaccurate dimension reduction leads to exponential growth in the system and numerical instability. For this reason we turn our focus only on the full-rank (nine CLVs) method and the accuracy of the observations.

We use the following settings for all the DA experiments with ENSO observations: assimilation window 0.08, inflation factor 1 %, and 10 ensemble members. The model is run for the same amount of time as in the benchmark observation experiments. We study the effect of reducing the observation error variances in

Summary metrics of DA experiments using right-transform matrix (

We find that for the standard observation errors, there is a collapse in the variance of the ensemble mean for the extratropical subsystem (Fig.

Trajectories of DA using nine CLVs,

In this section we explore a different type of observation error. Rather than randomly perturbing the control run to form our observation points, we use a trajectory that shadows the control run which produces correlated observational errors. In other words, we construct a new trajectory with a relaxation to the control run. This is implemented in the model as follows:

We initialize the shadowed trajectory with a small perturbation to the control trajectory initial condition. We then propagate the shadowed trajectory along with the control trajectory, taking the observations from the shadowed trajectory at each assimilation cycle. Figure

Comparison of the two different types of observations used in the data assimilation experiments. We show the observation space for the benchmark observations case (

We repeat the observation experiments of the previous three sections: benchmark observations, atmospheric observations, and ENSO observations. We only focus on the full-rank and variable CLV methods. When using correlated observational errors in any ensemble Kalman filtering method, a larger inflation value and ensemble size are needed to avoid ensemble collapse. We find that for the standard ETKF method increasing the ensemble size to 11 members is sufficient. To facilitate a direct comparison, we therefore also use this ensemble size for the CLV experiments. The inflation value varies slightly with the different observation cases. The setup and results of the experiments are shown in Table

Summary metrics of DA experiments using a right-transform matrix (Eq.

When using the benchmark observation set from the shadowed trajectory, the variable CLV method outperforms the full-rank method. This shows improvement over the case with random observation errors discussed in Sect.

We finally consider the case where one subsystem is fully observed and the others are completely unobserved. We choose to observe the extratropical subsystem, as it is the most extreme case with weakly coupled fast dynamics. Due to the difficulty of constraining the full system with such minimal observations, the assimilation window is reduced to 0.02 and we use

It is also clear from the previous experiments that the inability to constrain unobserved subsystems leads to a collapse in dimension and correspondingly a collapse in the covariances. A collapse in covariance is commonly avoided through the use of inflation

Here we introduce a new method for adaptive scaling of the Kalman gain. Rather than explicitly calculating higher moments of the anomalies, we account for the underestimation through a spread-dependent factor which balances our forecast error covariance

We scale the Kalman gain in the following way:

Due to the fact that only the Kalman gain is being adjusted, for ease of implementation we use the ESRF method introduced in Sect.

Trajectories of DA experiments using variable CLVs, left-transform matrix (Eq.

Summary metrics of DA experiments using a left-transform matrix (Eq.

We see from Fig.

This study presents an initial understanding of the transient dynamics associated with the Kalman filter forecast error covariance matrix for nonlinear multiscale coupled systems. We have explored the varying rank of the error covariance matrix related to the transient growth in the stable modes of the system, and in particular the applicability of this varying rank to different configurations of strong CDA. Additionally, we have shown the large impact of using isolated observations and cross-domain covariances in such a coupled system. The cross-covariances are significantly underestimated when the observed subsystems are weakly coupled to the unobserved ones; however, this can be compensated through either reduced observational error or the use of an adaptive scaling of the Kalman gain.

The dynamical properties of strongly coupled DA in a multiscale system were investigated through a low-dimensional nonlinear chaotic model to represent the interactions between the extratropical atmosphere, ocean, and tropical atmosphere–ocean interface. The model contains significant spatio-temporal scale separations between the subsystems, as well as varying coupling strengths. We introduced a local dimension measure, namely the Kaplan–Yorke dimension calculated using FTLEs, to specify the appropriate rank of the forecast error covariance matrix at each analysis step. We have shown that through using time-varying CLVs to form a reduced-rank forecast error covariance matrix, comparable results to the full-rank ETKF and ESRF schemes are achievable.

We considered a benchmark experiment previously explored in

We then tested the effectiveness of the reduced-rank forecast error covariance matrix in strong CDA when a subsystem is completely unobserved, i.e. using only cross-covariances to determine the increment of the unobserved system. The first set of these experiments used observations from the two atmosphere subsystems, extratropical and tropical, while the ocean was left completely unobserved. In this case we found that the DA succeeded in constraining the system to the observations when using the full rank, asymptotic dimension, and local dimension to determine the rank of the covariance matrix; however, the variable-rank method performed better than the fixed-rank methods. The second set of experiments consisted of ENSO observations or observations from the strongly coupled tropical and ocean subsystems only. In this case, the observational errors and weak coupling to the extratropical subsystem caused the reduced-rank experiment to fail. The full-rank experiment succeeded in tracking the tropical and ocean subsystems but left the extratropical subsystem unconstrained. This resulted in a collapse of the variance in the ensemble mean and a subsequent reduction in the average dimension. However, we found that reducing the observational error variance of the tropical subsystem provided an increase in ensemble mean variance of the extratropical subsystem and therefore an increase in the average dimension. Reducing the observational error variance of the ocean did not provide a significant improvement since it is only indirectly coupled to the extratropical subsystem.

The effect of correlated observational errors was also explored. We constructed a trajectory which shadowed the control run and used this as our observational set, repeating all the previous experiments with different observation subsets. Since the correlated errors preserve the underlying dynamical structure of the system, we found that the reduced-rank method based on local dimension was the most successful in all the experiments when compared to those using random observational error. This included the ENSO observations case, where the extratropical subsystem remained unconstrained.

Finally, we showed that when only observing the extratropical subsystem, the unobserved subsystems could not be constrained due to their weak or indirect coupling to the observations. This manifested as an overall reduction in dimension as well as a collapse in the cross-domain covariances. In order to counter the covariance and dimension collapse, we introduced a novel scheme for adaptive Kalman gain scaling. This adaptive scaling is based on a measure of the overall spread of the system, therefore accounting for unobserved subsystems that have become unconstrained. Through use of the adaptive scaling the weakly coupled unobserved subsystems were able to be relatively constrained, and moreover the ensemble mean of the unobserved subsystems was able to track the control run. The adaptive scaling introduced here should be tested on additional systems with weak coupling in order to assess its general applicability, although care may need to be taken in the choice of the norm.

We now turn to the implications for more realistic high-dimensional systems. It has been shown that when using a finer model resolution (increasing dimension) there is an increase in near-zero asymptotic Lyapunov exponents

The adaptive gain result presented here highlights the utility of ensemble filtering methods. While the ensemble mean of the subsystems manages to track the control run, the individual members are not so constrained. The variability in the spread of the ensemble members can provide a measure for uncertainty of the corresponding subsystem at a given region in phase space. Additionally, the ability to constrain the ensemble mean of the system from only observations of the weakly coupled fast subsystem is a new result for strong CDA. While dynamically this is intuitive (accurate knowledge of the fast dynamics of a system is sufficient to reconstruct the full attractor), this has not previously been shown to be achievable in DA experiments. If such a scheme could be shown to scale to high-dimensional climate models, then accurate and frequent atmospheric observations could potentially be sufficient to constrain the full system. For this reason it is important that the scheme be analysed for general applicability and tested on a variety of coupled dynamical systems.

We address the implication of the adaptive Kalman gain scaling for the two extreme cases: ensemble collapse (

All simulation data are available from the corresponding author by request.

All the authors designed the study. The schemes for the calculation of Lyapunov exponents and CLVs were adapted and implemented by CQ in both Python and Matlab. The Python codes for the ensemble Kalman filtering methods were produced by VK, with modifications by CQ. All the figures were produced by CQ. All the authors contributed to the direction of the study, discussion of results, and the writing and approval of the manuscript.

The authors declare that they have no conflicts of interest.

The authors would like to thank Dylan Harries, Pavel Sakov, and Paul Sandery for their valuable input throughout the preparation of this paper, as well as the two anonymous reviewers for their thorough and insightful comments.

The authors were supported by the Australian Commonwealth Scientific and Industrial Research Organisation (CSIRO) Decadal Climate Forecasting Project (

This paper was edited by Alberto Carrassi and reviewed by two anonymous referees.