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.

Probabilistic forecasts are regarded as the highest achievable goal when predicting earthquakes, but limited information on stress, strength, and governing parameters of the seismogenic sources affects their accuracy. Ensemble data-assimilation methods, such as the Ensemble Kalman Filter (EnKF), estimate these variables by combining physics-based models and observations.While the EnKF has demonstrated potential in perfect model experiments using earthquake simulators governed by rate-and-state friction (RSF) laws, challenges arise from the non-Gaussian distribution of state variables during seismic cycle transitions. This study investigates the Adaptive Gaussian Mixture Filter (AGMF) and the Particle Flow Filter (PFF) as alternatives for improved stress and velocity estimation in earthquake sequences compared to Gaussian-based methods like the EnKF. We test the AGMF and the PFF's performance using Lorenz 96 and Burridge-Knopoff 1D models which are, respectively, standard simplified atmospheric and earthquake models. We test these models in periodic, and aperiodic conditions, and analyze the impact of assuming Gaussian priors on the estimates of the ensemble methods. The PFF demonstrated comparable performance in chaotic scenarios, yielding lower RMSE for the estimates of the Lorenz 96 models and stronger resilience to underdispersion for the Burridge-Knopoff 1D models. This is vital given the limited and sparse historical earthquake data, underscoring the PFF's potential in enhancing earthquake forecasting. These results emphasize the need for careful data assimilation method selection in seismological modeling

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.