In ensemble-based sequential data assimilation, model error is often modeled as random forcing with a prescribed temporal autocorrelation, yet the true decorrelation time-scale is unknown and mis-specifying it degrades both the actual posterior error and the error the system reports. Ren, Amezcua and van Leeuwen (2021) quantified this degradation analytically for a Kalman smoother on a scalar linear model and numerically for an ensemble Kalman smoother (EnKS) on a higher-dimensional linear system, and showed that the model-error decorrelation time-scale can be estimated online via state augmentation — adding the unknown time-scale to the state vector being estimated — when observations are dense. They explicitly left open the extension to nonlinear dynamics and nonlinear observation operators and robust estimation under sparse observations, and they noted that in reality the spatial structure of the model error is unknown as well; further restrictions of their analysis — an exponentially decaying memory chosen for simplicity and finite-ensemble algebra in the scalar case only — leave correlation families beyond the exponential memory, higher-dimensional and realistic models, and finite-ensemble sampling effects unaddressed. Their own follow-up (Amezcua, Ren and van Leeuwen 2023, Tellus A) made partial progress on the estimation thread with an iterative EnKS (IEnKS), a variant that repeats the smoother update within each assimilation window to cope with nonlinearity, on low-dimensional systems but concluded the problem remains 'highly nonlinear, and more difficult than expected even in zero-dimensional systems', deferring realistic models and operational online updating to future work. The open problem is therefore: characterize how mis-specified temporal correlation of model error affects nonlinear (ensemble) Kalman smoothing, and develop robust online estimation of time-correlated-model-error parameters that works beyond dense-observation, low-dimensional, linear settings.

Background

Data assimilation (DA) combines noisy observations with a dynamical model to estimate the state of a geophysical system, as in numerical weather prediction. The Kalman smoother (KS) is the exact linear-Gaussian solution that estimates states over a whole assimilation window using all observations in it; the ensemble Kalman smoother (EnKS) is its Monte-Carlo implementation, in which an explicit ensemble of forecasts carries the error covariances and which extends heuristically to weakly nonlinear dynamics. Classical DA assumes a perfect model and attributes all error to the initial condition and the observations, but real forecast models contain systematic error from unresolved or simplified physical processes. A standard remedy is weak-constraint DA, in which a random model-error forcing is added at each time step. If this forcing is taken white in time the model error accumulates as a random walk, but true model errors are typically correlated in time; a common parameterization is Cov⁡(νi,νj)=ϕ(∣i−j∣,ω)Q\operatorname{Cov}(\nu_i,\nu_j)=\phi(|i-j|,\omega)Q, where QQ is a spatial covariance matrix, ϕ\phi a memory function and ω\omega a decorrelation (memory) time-scale. Amezcua and van Leeuwen (2018) formulated such time-correlated model error for ensemble smoothers and showed, in the linear case, that prescribing the correlation time-scale incorrectly can degrade solutions substantially, interacting with finite-ensemble-size effects. Ren, Amezcua and van Leeuwen (2021) then quantified the effect of mis-specification: in a scalar linear model, the posterior variance declared by the smoother depends only on the guessed time-scale ωg\omega_g, while the actual posterior mean-square error (MSE) also depends on the true time-scale ωr\omega_r and grows with the mismatch ∣ωg−ωr∣|\omega_g-\omega_r|; the mismatch matters most when observations are sparse and shrinks as observation density grows. Because ωr\omega_r is unknown in practice, they further estimated it online: augmenting the EnKS state vector with the (log) memory time-scale and updating it from observations works over several assimilation windows when the observation density is high. The analysis was restricted to linear dynamics, a linear observation operator, an exponentially decaying memory ϕ(∣i−j∣,ω)=exp⁡(−∣i−j∣/ω)\phi(|i-j|,\omega)=\exp(-|i-j|/\omega), prescribed spatial structure QQ, and algebraic finite-ensemble characterizations in the scalar case only. The immediate practical setting is geophysical DA, where model error is large and significantly time-correlated, e.g. Bonavita (2021) diagnosed time-correlated systematic model error in the ECMWF operational analysis system, and several threads (variational and machine-learning model-error covariance estimation) estimate model-error statistics offline; none of them characterizes mis-specified time correlation within ensemble smoothers online.

Problem Statement

For an (ensemble) Kalman smoother that assimilates observations of a dynamical system whose model error is temporally correlated with an unknown true decorrelation time-scale ωr\omega_r (and more generally an unknown correlation structure), while the smoother assumes a guessed time-scale ωg\omega_g and memory function ϕ(∣i−j∣,ω)=exp⁡(−∣i−j∣/ω)\phi(|i-j|,\omega)=\exp(-|i-j|/\omega): (1) characterize, beyond the linear scalar and linear higher-dimensional settings already analyzed, how the mis-specification ωg≠ωr\omega_g \neq \omega_r (and mis-specified correlation family or spatial structure) affects the actual posterior mean-square error, the declared posterior variance, and their ratio, when the dynamics and/or the observation operator are nonlinear and when observation density within the assimilation window ranges from sparse (a single observation) to dense; and (2) develop a robust online estimation method, compatible with ensemble Kalman smoothing, for the time-correlation parameters of the model error (decorrelation time-scale, and correlation families beyond the pure exponential, jointly with spatial structure where relevant) that accumulates information across successive assimilation windows and converges, in regimes where these parameters are identifiable from the available observations, without dense within-window observations and without problem-specific tuning, while clarifying what is estimable in regimes that are genuinely too sparse for identification, and demonstrate it on nonlinear and higher-dimensional models. A related analytic gap noted but not pursued by Ren et al. (2021) is a closed-form or tractable expression for the posterior moments in the multi-observation linear scalar case (the ratios of integrals over double sums they were unable to evaluate analytically).

References

  1. Ren, H., Amezcua, J. and van Leeuwen, P. J. (2021). Effects of mis-specified time-correlated model error in the (ensemble) Kalman Smoother. Quarterly Journal of the Royal Meteorological Society, 147(734), 573-588. doi:10.1002/qj.3934, https://doi.org/10.1002/qj.3934
  2. Amezcua, J., Ren, H. and van Leeuwen, P. J. (2023). Using the (Iterative) Ensemble Kalman Smoother to Estimate the Time Correlation in Model Error. Tellus A: Dynamic Meteorology and Oceanography, 75(1), 108-128. doi:10.16993/tellusa.55, https://doi.org/10.16993/tellusa.55
  3. Amezcua, J. and van Leeuwen, P. J. (2018). Time-correlated model error in the (ensemble) Kalman smoother. Quarterly Journal of the Royal Meteorological Society, 144(717), 2650-2665. doi:10.1002/qj.3378, https://doi.org/10.1002/qj.3378
  4. Bonavita, M. (2021). Exploring the structure of time-correlated model errors in the ECMWF data assimilation system. Quarterly Journal of the Royal Meteorological Society, 147(739), 3454-3471. doi:10.1002/qj.4137, https://doi.org/10.1002/qj.4137
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