Information on physical systems is often available in two forms; on the one hand, it is available from observations and on the other hand through mathematical models describing the system's dynamics. The combination of both can lead to an improved description of the system. This is the aim of data assimilation, typically with a focus on state estimation.

Data assimilation has broad applications throughout the geosciences and can be already seen as an independent discipline . It is typically used to estimate states but is also used for parameters, i.e., in weather forecasting , for atmospheric chemical transport also coupled to meteorology , in oceanography, including biogeochemical processes , and in hydrology for flow, transport, and reaction in terrestrial surface and subsurface systems . Data assimilation is also increasingly applied in ecology, with applications ranging from the spread of infectious diseases and wildfires to population dynamics and the terrestrial carbon cycle .

In this study, we distinguish between geophysical systems that are divergent or convergent systems, depending on the development of two initial nearby states (Fig. ). In a divergent system, initially close states will inevitably drift apart, even if the system is described by a perfect model . This leads to an upper limit for the predictability in divergent systems . In a convergent system, nearby trajectories will coalesce. If the model to describe such a convergent system is perfect, this results in a high predictability . An error in the initial state will decay towards the truth after some transient phase. However, this is only true for perfect models, which is usually not the case for geophysical systems. This can lead to a bias with convergence to a wrong state.

Many recent advances in data assimilation methods have been developed in the context of weather forecasting . They are, therefore, designed to meet the challenges in the atmosphere – a predominantly divergent system. Due to the fundamental limit for long time predictions from uncertain initial conditions in divergent systems, data assimilation in operational weather forecasting primarily focuses on state estimation .

While weather forecasting or oceanography are divergent systems, several geophysical systems, such as soil hydrology or chemical transport, are convergent systems. In ensemble data assimilation methods, if uncertainties are only represented through an ensemble of states, convergent systems lead to decreasing uncertainties over time, which favors filter degeneration. This difference to divergent systems, where the ensemble spread increases exponentially, makes the direct transfer of data assimilation methods from divergent systems to convergent systems challenging and often requires adaptations to prevent filter degeneracy. The application of data assimilation when coupling a divergent and a convergent model, for example, in coupled chemistry meteorology models, may potentially lead to new difficulties .

The largest uncertainties in convergent systems typically do not reside in uncertain initial conditions but rather in boundary conditions, which include external forcings, the representation of sub-scale physics through parameterizations, and unrepresented physics in the model equations. These uncertainties should then be addressed in an integrated manner . Therefore, data assimilation methods have been used to not only estimate states but also parameters to reduce these uncertainties. The combined estimation of states and parameters is thought to be a solution for reducing the impact of model errors on parameter estimation . Estimating parameters in ensemble data assimilation methods through an augmented state requires a forward model for the parameters as well. This model is typically assumed to be constant, which is neither divergent nor convergent. However, the filter will gradually reduce the uncertainty in the parameters, which is not increased through a divergent forward model, and challenges similar to convergent systems can arise. This is sometimes alleviated by assigning a random walk as the forward model to the parameters, which then requires to determine an appropriate step size.

A challenge for sequential ensemble data assimilation in convergent systems is to maintain a sufficient ensemble spread. This would require an adequate representation of all uncertainties, including unrepresented physics in the model equations. In real-world systems, this is often difficult or impossible. Therefore, practical alternatives are necessary, which are often heuristic and may interfere with basic assumptions in the data assimilation methods. One possibility is inflation methods, which counteract the coalescing tendency. Unfortunately, there exists no universal method to accomplish this, and a range of approaches are followed. One example is the increase in the ensemble spread of parameters to a threshold value, as soon as the parameter uncertainty drops below this value. This approach was introduced for a sea breeze model and has been used in hydrology as well (e.g., ). A modification to this, also applied in hydrology, is to keep the parameter uncertainty entirely constant . This ensures a sufficient ensemble spread in the state itself but can impact the accuracy of the estimation. A widely used adjustment to limit the reduction in an ensemble spread in hydrology is the use of a damping factor . In soil hydrology, a multiplicative inflation method was proposed and specifically adjusted to the needs of the system . Similarly, showed that an atmospheric chemical transport model required much stronger inflation than reported in the meteorological literature and showed better results for a model-specific inflation, where the key parameters are perturbed to achieve an increased spread in the state. Consequently, a better understanding and control over errors has been recognized as a major challenge in chemical data assimilation as well .

Although plenty of knowledge and experience are available in the different communities on how to handle data assimilation methods in their specific models, we are not aware of a fundamental analysis of the difference between divergent and convergent models with respect to their utilization within ensemble data assimilation frameworks. We investigate and demonstrate the different challenges that illustrate, for example, the different requirements for inflation methods, using the ensemble Kalman filter . The divergent case is illustrated using the Lorenz 96 model , while for the convergent case, a soil hydrological system described by Richards' equation is used. Naturally, these are highly simplified models compared to real-world applications. Still, they demonstrate key aspects that also have to be addressed in more complicated situations. The specific adjustments applied there depend on the particular model. Our focus here is on the fundamentals and not on the wide range of specifics.

For this study we chose the ensemble Kalman filter (EnKF), a sequential data assimilation method, based on Bayes' theorem and the assumption of unbiased Gaussian error distributions. It was introduced as an extension of the Kalman filter for nonlinear models and approximates the Gaussian distributions by an ensemble of states ψi, where the subscript i denotes the ensemble member.

To sequentially assimilate new observations, the EnKF alternates between a forecast (subscript “f”) and an analysis step (subscript “a”). In the forecast, the ensemble is propagated using the nonlinear model f(⋅) as follows: 1ψf,ik=fψf,ik-1+βk, where βk is a stochastic model error and k is a discrete time.

In the following, the analysis state ψa is calculated by applying the Kalman gain: 2Kk=PfkH⊺HPfkH⊺+Rk-1, to every ensemble member. The Kalman gain weights the forecast error covariance P with the observation error covariance R. The observation operator H maps the state from state space to observation space. The forecast error covariance P is calculated using the forecast ensemble as follows: 3Pfk=ψfk-ψfk‾ψfk-ψfk‾⊺‾.

It is necessary to add a realization of the observation error to the observation for each ensemble member . The resulting analysis state is as follows: 4ψa,ik=ψf,ik+Kkdk-Hψf,ik+ϵik,ϵik∝N0,Rk. By combining the information from the measurement and the model, the uncertainty in the analysis ensemble is decreased. For sequential data assimilation, the process of forecast and analysis is iterated for every new observation.

The joint estimation of states and parameters can be realized through state augmentation. The original state ψ is extended with the parameters p to an augmented state, as follows: 5u=ψp. The model equation (Eq. ) changes to the following: 6uk=ψkpk=fψψk-1,pk-1+βψkfppk-1+βp, with the models for the state fψ(⋅) and for the parameters fp(⋅), as well as the corresponding model errors βψ and βp. The model for the state is typically nonlinear, while the parameters are often assumed constant fp(pk-1)=pk-1.

In this study, we set both stochastic model errors in Eq. () to zero. The EnKF is used without any extensions, and an ensemble size of N=100 was chosen for all cases.

This section demonstrates the data assimilation behavior for a divergent system on the example of the 40-dimensional Lorenz 96 model, which has been widely used to test data assimilation methods in atmospheric sciences e.g.,. We first introduce the model before we look into four characteristically different cases.

This section demonstrates the data assimilation behavior for a convergent system on the example of soil water flow. We, again, first introduce the model before we look into four characteristic cases.

For the divergent Lorenz 96 system, the EnKF is able to estimate the state for the true model as well as for the case with a wrong parameter. In a divergent system, the volume of the prior distribution in the state space increases during forward propagation . For the EnKF, this is directly connected to the ensemble spread, which increases rapidly between the observations. This prevents a collapse of the ensemble even in the presence of an unrepresented parameter error. However, if the observation interval is too small relative to the characteristic time for divergence, the EnKF leads to a decrease in the ensemble spread such that the filter degenerates in the case of an unrepresented parameter error. Nevertheless, the divergent behavior of the model prevents a complete collapse of the ensemble so that the filter is still able to improve the state in a limited way.

In the convergent soil hydrological system, the volume of the prior distribution decreases during forward propagation such that the prior becomes more certain, even without an observation and data assimilation. For a perfect model, the predictability and state estimation in a convergent model are trivial. The initial ensemble will converge to the truth after some time, even with a rough initial approximation. In this case, data assimilation is not necessary.

In the case of model errors, in our study realized through a wrong parameter, the situation is different. The ensemble converges to a wrong state, the filter degenerates, and data assimilation fails. Increasing the ensemble size can only improve the performance marginally since all ensemble members converge to the same fixed point.

Representing the parameter error by assigning each ensemble member a different parameter increases the divergence of the system, and the filter is able to estimate the state again. Between the observations, the ensemble spread increases rapidly because the ensemble members diverge to different fixed points apart from the truth. This results in a finite predictability. By representing the parameter error, the Richards equation gains a divergent part similar to the Lorenz 96 model. In the case of a convergent system, it is necessary to represent the parameter error, otherwise the ensemble collapses.

To increase the predictability of the system again, it is necessary to not only represent but also to reduce the parameter uncertainty. In synthetic cases without model structural errors, the convergent property of the system supports the state estimation, and the predictability increases if the parameter estimation is successful. This shows the importance of parameter estimation for convergent systems. For divergent systems, parameter estimation also increases the predictability but only up to a point because predictability is limited by the system's dynamics.

For the application of data assimilation to real data, model errors typically cannot be attributed to unknown parameters or uncertainties in boundary conditions alone but also stem from model structural errors, like a simplified representation of sub-scale physics or unrepresented processes in the dynamics. Uncertainties in parameters and boundary conditions can be represented in an ensemble, but representing model structural errors is challenging or can be impossible. For example, in hydrology, the model errors are typically ill-known and can vary both in space and time, which then can lead to filter degeneracy and biased parameter estimation . While divergent models can alleviate the effect of an unrepresented error within certain bounds, in a convergent system it is necessary to represent all relevant model errors sufficiently to prevent filter degeneracy and enable an optimal state and parameter estimation. Since this is challenging or even impossible, heuristic ways to address filter degeneracy are necessary.

A practical alternative to increase the ensemble spread and avoid filter degeneracy is to use inflation methods. For example, keeping a constant ensemble spread for the parameters can provide a sufficient spread in state space . The advantage is that this approach is a model-specific inflation and avoids the over-amplification of spurious correlations . The disadvantage is that the EnKF is prevented from reducing the prediction uncertainty. This behavior is also shown in CC3. In this case, the ensemble spread in the parameter space adds a divergent component to the ensemble that results in an increased ensemble spread in state space and prevents filter degeneracy.

Multiplicative inflation increases the ensemble spread through an inflation factor . In a divergent system a small multiplicative inflation is sufficient to increase the existing ensemble spread. In a convergent system this requires a larger inflation and can lead to the over-inflation of spurious correlations . To avoid this and to cope with spatial and temporal varying model errors, the use of more sophisticated adaptive inflation methods e.g. may be necessary in convergent systems.

Heuristic inflation methods cannot fully replace the representation of model errors and, hence, must be used judiciously. They can lead to the over-inflation of spurious correlations and can lead to biases in the estimation of parameters. If unrepresented model errors can be identified and are limited in space or time, these biases can be prevented by using a closed-eye period for the extent of the unrepresented model errors. In the closed-eye period, parameters are kept constant, and only the state is estimated, which can require inflation methods for a successful state estimation. This enables an improved parameter estimation without compensating for the unrepresented model dynamics though biased parameters outside of the closed-eye period, when and where uncertainties can be represented. The use of the closed-eye period in combination with the representation of relevant uncertainties has already been demonstrated in hydrology .

We demonstrated the differences in and challenges of ensemble data assimilation for divergent and convergent systems on the example of the EnKF applied to the divergent Lorenz 96 model and a convergent soil water movement model based on the Richards equation.

Sequential ensemble data assimilation methods require a sufficient divergent part in the ensemble to maintain an adequate ensemble spread and prevent filter degeneration. In divergent systems, this is inherent to the system, provided that observation intervals and divergence times match. In convergent systems, relevant model errors must be represented to increase the ensemble spread and to, thereby, add a divergent part to the ensemble to avoid filter degeneracy. If errors stem from unknown parameters, estimating the parameters improves state estimation. However, this will reduce the ensemble spread again and require the remaining relevant model errors to be represented. Since this can be challenging, increasing the model errors artificially, by limiting the reduction of parameter uncertainty or through inflation methods, can be required in convergent systems.

This paper highlights the challenges when transferring sequential ensemble data assimilation methods from divergent systems to convergent systems, which must be considered when applying data assimilation.