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

Abstract

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.

On Temporal Scale Separation in Coupled Data Assimilation with the Ensemble Kalman Filter

“…Although asymptotically neutral or weakly-stable, these directions may display high variance in the local error growth rate, thus be often intermittently unstable. As rigorously proved by [23,24] this situation is known to drive the error upwell from the unfiltered to the filtered subspace, eventually leading to divergence. It seems thus paramount that the EnKF ensemble subspace encompasses at least all of these near-neutral directions, preferably also the asymptotically weakly stable..…”

A detectability criterion and data assimilation for nonlinear differential equations

“…Recall from Remark 3.2 that this is as good as can be hoped for linear systems in the presence of noisy observations. We stress that the observational noise introduces an additional non-linear term pQ Q ⊺ H ⊺ η in the error equation (28); hence, large p > 0 not only makes the discretized equation stiff but, importantly amplifies the noise! On the other hand, Q ⊺ acts as a projection onto the range of H ⊺ HQ, and thus Q ⊺ H ⊺ η in fact represents the projection of η onto the range of HQ.…”

“…Now, define g(τ ; s, t) := ξ ⊺ (t)∇f i (t, τ sξ(t) + x(t)). We get that f i (t, ξ(t) + x(t)) − f i (t, x(t)) − ξ ⊺ (t)∇f i (t, x(t)) = and so the estimation error ξ solves (28). Now, S1 follows 10 from [7, p.27, Thm.…”

“…The importance of Lyapunov exponents and Lyapunov vectors for analysis of DA methods has also been stressed in [27]. Very recently, robusteness of the LEs to model errors were studied in [28] for discrete-time systems.…”

“…small perturbation.Indeed, one may interpret ξ(t) as a perturbation about a particular solution x, i.e. z(t) = x+ξ, and the dynamics of this perturbation is defined by the "perturbed" equation(28): here N is considered as a perturbation of the linear unperturbed equation X = (A − LH)X. By Theorem 4.2 it follows that the exponential decay of solutions of the "perturbed" equation(28)is equivalent to the exponential decay of the solutions of unperturbed linear equation provided the LEs are forward-regular.…”