Extending the multi-model ensemble Kalman filter to correlated inter-model forecast errors, per-model bias, and multi-attractor regimes

The multi-model ensemble Kalman filter (MM-EnKF) of Bach and Ghil (2023) combines forecasts from several dynamical models by treating each model's forecast as a noisy observation of the true state, but its derivation and implementation assume that the forecast errors of distinct models are uncorrelated and that per-model errors are unbiased apart from an additive model-error covariance $Q_m$. In the climate regime these assumptions fail: models share provenance and thus exhibit correlated errors, systematic biases dominate, and model ensembles relax onto different attractors so that an additive $Q_m$ loses its meaning. The open problem is to extend the multi-model data assimilation (MM-DA)/MM-EnKF framework to this regime — deriving and implementing the filter with estimated inter-model forecast-error cross-covariances, integrating per-model state-dependent bias correction, and clarifying the interpretation and estimation of $Q_m$ on different attractors — together with practical estimation methodology for the new error-statistics objects and demonstration on suitable multi-model testbeds.

Background

Data assimilation (DA) is the sequential estimation of the state of a dynamical system from noisy observations, combining a forecast model with incoming data; the Kalman filter is the optimal linear estimator for linear models with Gaussian errors, and the ensemble Kalman filter (EnKF) realizes it for nonlinear models by propagating an ensemble of state estimates and estimating error covariances from ensemble spread. Bach and Ghil (2023) formulated the MM-EnKF, an ensemble implementation of the previously proposed multi-model DA (MM-DA) framework, in which forecasts from $M$ distinct models are combined at each analysis step by treating model $m$'s forecast $x^f_m$ as an 'observation' $y_m$ of the true state: $y_m = G_m x + \epsilon_m$, where $G_m$ maps the reference-model (true) state space into model $m$'s state space (in practice a mapping operator between heterogeneous grids and variables) and $\epsilon_m$ has covariance $P^f_m$, model $m$'s forecast-error covariance, decomposed as $P^f_m(t_{i+1}) = P^p_m(t_{i+1}) + Q_m(t_i)$ into a predictability error $P^p_m$ arising from the dynamics acting on the initial-condition uncertainty and an additive, state-independent model-error covariance $Q_m$ (the source's $H_m$ is instead the observation operator mapping model $m$'s space to observation space, $H_m = G_m^{\dagger} H$). Under independence of the $\epsilon_m$, the multi-model estimate is a generalized-least-squares / best-linear-unbiased-estimate (GLS/BLUE) combination

$$\hat{x} = \Big(\sum_m G_m^{\top} (P^f_m)^{-1} G_m\Big)^{-1} \sum_m G_m^{\top} (P^f_m)^{-1} y_m$$

and the MM-EnKF implements this with ensemble-estimated covariances together with the standard EnKF remedies of localization (tapering estimated covariances with physical distance to suppress sampling error) and covariance inflation (multiplicatively or additively inflating the forecast covariance to compensate for underestimated ensemble spread); each model's forecast-error covariance $P^f_m$ plays the role of the observation-error covariance of that model's pseudo-observation, while the additive $Q_m$ enters through the forecast propagation — in the algorithm, as additive inflation of model $m$'s ensemble with samples $\eta \sim N(-b_m, Q_m)$, where $b_m$ is model $m$'s bias. Two statistical assumptions do the load-bearing work. First, forecast errors of distinct models are uncorrelated, so the stacked model-forecast error covariance, whose diagonal blocks are the $P^f_m$, is block diagonal. Second, each model's error is modeled as zero-mean plus an additive, state-independent $Q_m$. Both assumptions are documented to fail for multi-model climate ensembles: models in archives such as the Coupled Model Intercomparison Project (CMIP) share components and provenance — the 'climate model genealogy' effect — so the error of the multi-model mean does not shrink as $M^{-1/2}$ (Knutti et al. 2010; Christiansen 2020), and Bach and Ghil themselves must apply extra multiplicative inflation of the multi-model forecast covariance to compensate. The authors note that the formal GLS derivation in their appendix 'is easily modified' for correlated forecast errors — cross-covariance blocks simply enter the stacked model-forecast error covariance in place of its zero off-diagonal blocks — but this presumes the cross-covariances are known; the genuinely unresolved part is the ensemble implementation with estimated inter-model cross-covariances, which is severe because cross-covariances are harder to sample than variances at practical ensemble sizes and need localization. In addition, the climate regime is bias-dominated: state-dependent, systematic per-model biases require online bias estimation and correction, which interacts with the DA cycle. Moreover, when the models are imperfect enough that their ensembles relax onto different attractors (long-lived model climates distinct from truth), assimilating one model's states into another incurs representation error — the error of representing states from one attractor in the coordinate system of another (Hodyss and Nichols 2015) — and the meaning and estimation of an additive $Q_m$ become unclear; the authors state that whether and how the MM-EnKF can be extended to such climate problems must be addressed by future work. Related but distinct lines exist: Kalman filters with correlated system and observation errors have been developed for the single-model case (Berry and Sauer 2018), recently extended to an EnKF with forecast–observation error cross-correlation (Ohishi, Kobayashi and Miyoshi 2025) and to adaptive estimation of those cross-correlation parameters (Kobayashi, Ohishi and Miyoshi 2026); these treat a different correlation (forecast vs. observation error within one model), not correlations between the errors of two different models. Supermodeling (Schevenhoven et al. 2023), in which models are coupled to each other during the simulation itself rather than combined statistically afterwards, is an alternative rather than a resolution. Other successors of the source paper pursue different future-work threads: a hybrid EnKF blending data-driven and physics-based forecast models (Chattopadhyay, Nabizadeh, Bach and Hassanzadeh 2023), adaptive reduced-basis/multi-fidelity EnKF hierarchies (Silva, Pagliantini and Veroy 2025), and a consensus multi-model Kalman filter for vehicle state estimation that still assumes independent model errors (Mafi et al. 2025).

Problem Statement

Extend the multi-model data assimilation framework and MM-EnKF of Bach and Ghil (2023) to the regime in which its two statistical assumptions fail. Specifically: (i) derive and implement the multi-model Kalman filter and MM-EnKF when the forecast errors of distinct models are correlated, i.e. with nonzero inter-model error cross-covariance blocks in the stacked model-forecast error covariance whose diagonal blocks are the $P^f_m$ — going beyond the formal GLS modification with known cross-covariances that the authors note is easily modified, to an ensemble implementation in which the cross-covariances are estimated from finite ensembles, with sampling-error control such as localization or shrinkage (tapering or regularizing the sample-covariance estimates) that still leaves the estimated inter-model cross-covariance blocks nonzero, so that the resulting filter genuinely accounts for inter-model error correlation instead of reducing to the independence-assuming original MM-EnKF; (ii) integrate per-model, state-dependent bias estimation and correction into the MM-EnKF cycle for bias-dominated climate applications; and (iii) clarify the meaning and estimation of the additive per-model model-error covariance $Q_m$ in the decomposition $P^f_m = P^p_m + Q_m$ when the model ensembles relax onto different attractors, where representation error invalidates that additive decomposition, and establish whether and how the MM-EnKF can be extended to such climate problems. The three parts share one objective — making the multi-model filter's statistically optimal model weighting valid for ensembles of related, biased, structurally imperfect models — and correspond to distinct future-work items of the source paper. Parts (i)-(iii) are independently posed: each is a self-contained question that can be resolved on its own. The extension is to be developed together with practical estimation methodology for the newly introduced error-statistics objects and demonstrated on suitable multi-model testbeds.

References

1. Bach, Eviatar; Ghil, Michael (2023). A Multi-Model Ensemble Kalman Filter for Data Assimilation and Forecasting. Journal of Advances in Modeling Earth Systems 15(1), e2022MS003123. DOI: 10.1029/2022MS003123. https://doi.org/10.1029/2022MS003123 (also arXiv:2202.02272, DOI 10.48550/arXiv.2202.02272, https://arxiv.org/abs/2202.02272)
2. Mafi, Farid; Khoshnevisan, Ladan; Pirani, Mohammad; Khajepour, Amir (2025). Bridging Data-Driven and Physics-Based Models: A Consensus Multi-Model Kalman Filter for Robust Vehicle State Estimation. arXiv:2506.12862. https://arxiv.org/abs/2506.12862
3. Silva, Francesco A. B.; Pagliantini, Cecilia; Veroy, Karen (2025). An Adaptive Hierarchical Ensemble Kalman Filter with Reduced Basis Models. SIAM/ASA Journal on Uncertainty Quantification 13(1), 140–170. DOI: 10.1137/24m1653690. https://doi.org/10.1137/24m1653690
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