Chaotic dynamics and the role of covariance inflation for reduced rank Kalman filters with model error

Grudzien, Colin; Carrassi, Alberto; Bocquet, Marc

doi:10.5194/npg-25-633-2018

Cited by 25 publications

(54 citation statements)

References 45 publications

(64 reference statements)

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“…However, our results suggest the need to further increase the rank of ensemble-based gains. The significance of this result for ensemble-based Kalman filters and their divergence is further elaborated on in the sequel [26].…”

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confidence: 80%

“…The present study is concerned with extending the limits of the results developed in deterministic dynamics (perfect models), now to the presence of stochastic model errors. This manuscript and its sequel [26] seek to: (i) determine the extent to which stable dynamics confine the uncertainty in the sequential state estimation problem in models with additive noise, and (ii) to use these results to interpret the properties, and suggest design, of ensemble-based Kalman filters with model error. This manuscript studies the asymptotic properties of the full rank, theoretical Kalman filter [33], and the unfiltered errors in the stable BLVs.…”

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confidence: 99%

“…This manuscript studies the asymptotic properties of the full rank, theoretical Kalman filter [33], and the unfiltered errors in the stable BLVs. The sequel [26] utilizes these results to interpret filter divergence for reduced rank, ensemble-based Kalman filters.…”

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confidence: 99%

“…This analysis allows us to evaluate how many of the stable BLVs can remain unfiltered while achieving an acceptable forecast performance. This comparison has a special significance when considering reduced rank, sub-optimal filters, which is the subject of the sequel [26].…”

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confidence: 99%

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Asymptotic Forecast Uncertainty and the Unstable Subspace in the Presence of Additive Model Error

Grudzien¹,

Carrassi²,

Bocquet³

2018

SIAM/ASA J. Uncertainty Quantification

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It is well understood that dynamic instability is among the primary drivers of forecast uncertainty in chaotic, physical systems. Data assimilation techniques have been designed to exploit this phenomena, reducing the effective dimension of the data assimilation problem to the directions of rapidly growing errors. Recent mathematical work has, moreover, provided formal proofs of the central hypothesis of the Assimilation in the Unstable Subspace methodology of Anna Trevisan and her collaborators: for filters and smoothers in perfect, linear, Gaussian models, the distribution of forecast errors asymptotically conforms to the unstable-neutral subspace. Specifically, the column span of the forecast and posterior error covariances asymptotically align with the span of backward Lyapunov vectors with non-negative exponents.Earlier mathematical studies have focused on perfect models, and this current work now explores the relationship between dynamical instability, the precision of observations and the evolution of forecast error in linear models with additive model error. We prove bounds for the asymptotic uncertainty, explicitly relating the rate of dynamical expansion, model precision and observational accuracy. Formalizing this relationship, we provide a novel, necessary criterion for the boundedness of forecast errors. Furthermore, we numerically explore the relationship between observational design, dynamical instability and filter boundedness. Additionally, we include a detailed introduction to the Multiplicative Ergodic Theorem and to the theory and construction of Lyapunov vectors.While forecast error in the stable subspace may not generically vanish, we show that even without filtering, uncertainty remains uniformly bounded due its dynamical dissipation. However, the continuous re-injection of uncertainty from model errors may be excited by transient instabilities in the stable modes of high variance, rendering forecast uncertainty impractically large. In the context of ensemble data assimilation, this requires rectifying the rank of the ensemble-based gain to account for the growth of uncertainty beyond the unstable and neutral subspace, additionally correcting stable modes with frequent occurrences of positive local Lyapunov exponents that excite model errors.

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confidence: 80%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Asymptotic Forecast Uncertainty and the Unstable Subspace in the Presence of Additive Model Error

Grudzien¹,

Carrassi²,

Bocquet³

2018

SIAM/ASA J. Uncertainty Quantification

Self Cite

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“…Trevisan and Uboldi 2004;Trevisan et al 2010;Bocquet and Carrassi 2017). Indeed, the need of performing assimilation in a state space that includes also some stable modes has been recently discussed in Grudzien et al (2018). We believe that the reason for this is that, in some regions of the phase space, some (or even many) dimensions of the stable space (defined by the CLVs featuring negative LEs) might feature a (finite-time) positive growth rate for the perturbations initialised on the corresponding CLVs.…”

Section: Discussionmentioning

confidence: 99%

A new mathematical framework for atmospheric blocking events

Lucarini

Gritsun

2019

Clim Dyn

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We use a simple yet Earth-like hemispheric atmospheric model to propose a new framework for the mathematical properties of blocking events. Using finite-time Lyapunov exponents, we show that the occurrence of blockings is associated with conditions featuring anomalously high instability. Longer-lived blockings are very rare and have typically higher instability. In the case of Atlantic blockings, predictability is especially reduced at the onset and decay of the blocking event, while a relative increase of predictability is found in the mature phase. The opposite holds for Pacific blockings, for which predictability is lowest in the mature phase. Blockings are realised when the trajectory of the system is in the neighbourhood of a specific class of unstable periodic orbits (UPOs), natural modes of variability that cover the attractor the system. UPOs corresponding to blockings have, indeed, a higher degree of instability compared to UPOs associated with zonal flow. Our results provide a rigorous justification for the classical Markov chains-based analysis of transitions between weather regimes. The analysis of UPOs elucidates that the model features a very severe violation of hyperbolicity, due to the presence of a substantial variability in the number of unstable dimensions, which explains why atmospheric states can differ a lot in term of their predictability. The resulting lack of robustness might be a fundamental cause contributing to the difficulty in representing blockings in numerical models and in predicting how their statistics will change as a result of climate change. This corresponds to fundamental issues limiting our ability to construct very accurate numerical models of the atmosphere, in term of predictability of the both the first and of the second kind in the sense of Lorenz.

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Adaptive covariance inflation in the ensemble Kalman filter by Gaussian scale mixtures

Raanes¹,

Bocquet

Carrassi³

2019

Quart J Royal Meteoro Soc

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This paper studies multiplicative inflation: the complementary scaling of the state covariance in the ensemble Kalman filter (EnKF). Firstly, error sources in the EnKF are catalogued and discussed in relation to inflation; nonlinearity is given particular attention as a source of sampling error. In response, the “finite‐size” refinement known as the EnKF‐N is re‐derived via a Gaussian scale mixture, again demonstrating how it yields adaptive inflation. Existing methods for adaptive inflation estimation are reviewed, and several insights are gained from a comparative analysis. One such adaptive inflation method is selected to complement the EnKF‐N to make a hybrid that is suitable for contexts where model error is present and imperfectly parametrized. Benchmarks are obtained from experiments with the two‐scale Lorenz model and its slow‐scale truncation. The proposed hybrid EnKF‐N method of adaptive inflation is found to yield systematic accuracy improvements in comparison with the existing methods, albeit to a moderate degree.

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Chaotic dynamics and the role of covariance inflation for reduced rank Kalman filters with model error

Cited by 25 publications

References 45 publications

Asymptotic Forecast Uncertainty and the Unstable Subspace in the Presence of Additive Model Error

Asymptotic Forecast Uncertainty and the Unstable Subspace in the Presence of Additive Model Error

A new mathematical framework for atmospheric blocking events

Adaptive covariance inflation in the ensemble Kalman filter by Gaussian scale mixtures

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