EnKF large-ensemble scope and nonlinear limits

Assume the Axiom of Choice (AC), meaning every family of nonempty sets has a choice function, for the Gaussian-law construction. Fix a nonnegative integer horizon $T\in {\mathbb{Z}}_{\ge 0}$, positive integers $m_k$ for $0\le k\le T$ and $p_k$ for $1\le k\le T$, matrices $F_k\in {\mathbb{R}}^{m_k\times m_{k-1}}$ and $H_k\in {\mathbb{R}}^{p_k\times m_k}$ for $1\le k\le T$, and a finite linear-Gaussian coefficient model $a_k=F_k{a}_{k-1}+q_k,y_k=H_k{a}_k+r_k,$ where ${\mu }_0\in {\mathbb{R}}^{m_0}$, $a_0\sim N_{m_0}({\mu }_0,P_0)$, $q_k\sim N_{m_k}(0,Q_k)$, and $r_k\sim N_{p_k}(0,R_k)$ are mutually independent, $P_0,Q_k$ are positive semidefinite, and $R_k$ is positive definite. Fix arbitrary observation data $y_{1:T}$. For each integer ensemble size $N\ge 2$, draw the initial members $a_{N,i}^{a,0}$ independently from $N_{m_0}({\mu }_0,P_0)$. At each $k$, draw process perturbations $q_{N,i}^k\sim N_{m_k}(0,Q_k)$ and observation perturbations ${\eta }_{N,i}^k\sim N_{p_k}(0,R_k)$, with all these perturbations mutually independent across members and times and independent of the initial ensemble, and set $a_{N,i}^{f,k}=F_k{a}_{N,i}^{a,k-1}+q_{N,i}^k$. Define ${\bar{a}}_N^{f,k}=N^{-1}{\sum }_i{a}_{N,i}^{f,k}$ and the divisor-$N$ covariance $C_N^{f,k}=N^{-1}{\sum }_i(a_{N,i}^{f,k}-{\bar{a}}_N^{f,k})(a_{N,i}^{f,k}-{\bar{a}}_N^{f,k}{)}^T$. The common gain and perturbed-observation update are $K_N^k=C_N^{f,k}{H}_k^T(H_k{C}_N^{f,k}{H}_k^T+R_k{)}^{-1},a_{N,i}^{a,k}=a_{N,i}^{f,k}+K_N^k(y_k+{\eta }_{N,i}^k-H_k{a}_{N,i}^{f,k})\mathrm{.}$ Since $C_N^{f,k}$ is positive semidefinite and $R_k$ is positive definite, this inverse exists. Define the analysis empirical mean and covariance by ${\bar{a}}_N^{a,k}=N^{-1}{\sum }_i{a}_{N,i}^{a,k}$ and $C_N^{a,k}=N^{-1}{\sum }_i(a_{N,i}^{a,k}-{\bar{a}}_N^{a,k})(a_{N,i}^{a,k}-{\bar{a}}_N^{a,k}{)}^T$. Let $m_{k\mathrm{\mid }k}$ and $P_{k\mathrm{\mid }k}$ be the mean and covariance produced by the exact Kalman recursion in the preceding filtering proposition, which defines an everywhere-defined Gaussian filtering law at every fixed history $y_{1:k}$. At each fixed $k\le T$, ${\bar{a}}_N^{a,k}$ and $C_N^{a,k}$ converge in probability to $m_{k\mathrm{\mid }k}$ and $P_{k\mathrm{\mid }k}$ as $N\to \mathrm{\infty }$; moreover, for each fixed bounded Lipschitz function $\phi :{\mathbb{R}}^{m_k}\to \mathbb{R}$, $N^{-1}{\sum }_i\phi (a_{N,i}^{a,k})$ converges in probability to its integral under that filtering law. Here bounded Lipschitz means ${\sup }_x\mathrm{\mid }\phi (x)\mathrm{\mid }<\mathrm{\infty }$ and $\mathrm{\mid }\phi (x)-\phi (x^{\mathrm{\prime }})\mathrm{\mid }\le L\mathrm{\parallel }x-x^{\mathrm{\prime }}{\mathrm{\parallel }}_2$ for some finite $L$ and all $x,x^{\mathrm{\prime }}$. Convergence in probability means that for every $ϵ>0$, the probability that the error exceeds $ϵ$ tends to zero; use Euclidean norm for vectors, Frobenius norm for matrices, and absolute value for scalars. All probabilities refer to the ensemble sampling just described. The horizon is fixed as $N$ grows, with no uniform-in-time conclusion. At finite $N$, the common sample-covariance gain couples the update, and the theorem does not assert independent posterior samples or an exact posterior law for the ensemble. It covers the stated linear maps and perturbed-observation formula only. Covariance localization here means replacing the sample covariance by its entrywise product with a prescribed taper matrix; covariance inflation means rescaling the ensemble anomalies by a factor greater than one or adding a prescribed positive-semidefinite covariance; a transport update means applying a selected map to ensemble members instead of the displayed perturbed-observation formula. These modified procedures and nonlinear state or observation maps require their own statements and proofs. The claim is conditional on the chosen finite model; it is not a conclusion that a nonlinear fluid equation or an actual flow is described by it.

Finite coefficient observation and innovation interface

Exact filtering for the declared linear Gaussian model

Perturbed-observation ensemble Kalman filter

The perturbed-observation EnKF converges to the Kalman filter at fixed horizon

The Axiom of Choice

Law, Stuart, and Zygalakis, Chapter 4 §4.2.3, pp. 82–84

Mandel, Cobb, and Beezley, §§3–4, Theorem 1 and Corollary 1