This paper extends the 2024 study of iterative ensemble smoothers by Evensen et al., who used a sizeable 1000-member ensemble configuration, to now using smaller, more affordable ensemble sizes with localization. As is well known, localization is needed to increase the effective ensemble size and avoid degradation of the smoother solutions by spurious correlations. As an alternative to the standard distance-based localization, we propose a reformulation of an adaptive correlation-based localization method that, in a local update, considers only those observations for which the absolute value of the correlation to the model counterpart is larger than a user-defined threshold. In the standard distance-based localization, we update model variables using only nearby observations in physical distance. In correlation-based localization, we update variables using only observations with small correlation distances. We define the correlation distance as one minus the absolute value of the ensemble correlation between a predicted measurement and the variable we are updating. Using the same formulation and implementation as in the 2024 Evensen et al. study, we compare the performance of the two localization strategies in a coupled nonlinear multiscale model and demonstrate the better or at least comparable performance of the adaptive correlation-based localization. We attribute this to an additional measurement error variance inflation for the measurements with a correlation distance close to the truncation distance, effectively leading to smoother updates. Furthermore, it solves the problem of space–time localization that is hard to solve using localization based on physical distance in ensemble smoothers over longer time windows. We also discuss strategies for the efficient implementation of the correlation-based approach.

This paper identifies and explains particular differences and properties of adjoint-free iterative ensemble methods initially developed for parameter estimation in petroleum models. The aim is to demonstrate the methods’ potential for sequential data assimilation in coupled and multiscale unstable dynamical systems. For this study, we have introduced a new nonlinear and coupled multiscale model based on two Kuramoto–Sivashinsky equations operating on different scales where a coupling term relaxes the two model variables toward each other. This model provides a convenient testbed for studying data assimilation in highly nonlinear and coupled multiscale systems. We show that the model coupling leads to cross covariance between the two models’ variables, allowing for a combined update of both models. The measurements of one model’s variable will also influence the other and contribute to a more consistent estimate. Second, the new model allows us to examine the properties of iterative ensemble smoothers and assimilation updates over finite-length assimilation windows. We discuss the impact of varying the assimilation windows’ length relative to the model’s predictability time scale. Furthermore, we show that iterative ensemble smoothers significantly improve the solution’s accuracy compared to the standard ensemble Kalman filter update. Results and discussion provide an enhanced understanding of the ensemble methods’ potential implementation and use in operational weather- and climate-prediction systems.

This work demonstrates the efficiency of using iterative ensemble smoothers to estimate the parameters of an SEIR model. We have extended a standard SEIR model with age-classes and compartments of sick, hospitalized, and dead. The data conditioned on are the daily numbers of accumulated deaths and the number of hospitalized. Also, it is possible to condition the model on the number of cases obtained from testing. We start from a wide prior distribution for the model parameters; then, the ensemble conditioning leads to a posterior ensemble of estimated parameters yielding model predictions in close agreement with the observations. The updated ensemble of model simulations has predictive capabilities and include uncertainty estimates. In particular, we estimate the effective reproductive number as a function of time, and we can assess the impact of different intervention measures. By starting from the updated set of model parameters, we can make accurate short-term predictions of the epidemic development assuming knowledge of the future effective reproductive number. Also, the model system allows for the computation of long-term scenarios of the epidemic under different assumptions. We have applied the model system on data sets from several countries, i.e., the four European countries Norway, England, The Netherlands, and France; the province of Quebec in Canada; the South American countries Argentina and Brazil; and the four US states Alabama, North Carolina, California, and New York. These countries and states all have vastly different developments of the epidemic, and we could accurately model the SARS-CoV-2 outbreak in all of them. We realize that more complex models, e.g., with regional compartments, may be desirable, and we suggest that the approach used here should be applicable also for these models.