Finite coefficient observation and innovation interface

Fix a nonnegative integer horizon $T\in {\mathbb{Z}}_{\ge 0}$. For each integer $k$ with $0\le k\le T$, let $V_k$ be a finite-dimensional real vector space with an ordered basis ${\mathrm{\Phi }}_k=({\varphi }_{k,1},\dots ,{\varphi }_{k,m_k})$, where $m_k\ge 1$. An ordered basis is a linearly independent spanning list. A coefficient vector $a_k=(a_{k,1},\dots ,a_{k,m_k}{)}^T\in {\mathbb{R}}^{m_k}$ represents the trial field $u_k={\sum }_{j=1}^{m_k}{a}_{k,j}{\varphi }_{k,j}\in V_k$. This is a finite-dimensional representation; it does not assert that a fluid field in an untruncated model belongs to $V_k$.

For each $1\le k\le T$, choose a real linear observation map $L_k:V_k\to {\mathbb{R}}^{p_k}$, with $p_k\ge 1$, and a real matrix $F_k\in {\mathbb{R}}^{m_k\times m_{k-1}}$ for the coefficient transition. The observation matrix $H_k\in {\mathbb{R}}^{p_k\times m_k}$ is defined by $[H_k{]}_{ij}=[L_k({\varphi }_{k,j}){]}_i$; hence $L_k(u_k)=H_k{a}_k$. A symmetric matrix $C$ is positive semidefinite, written $C⪰0$, when $v^{T}Cv\ge 0$ for every vector $v$ of matching dimension, and is positive definite, written $C\succ 0$, when this inequality is strict for every nonzero $v$.

For stochastic assimilation, assume the Axiom of Choice (AC), meaning every family of nonempty sets has a choice function, for the Gaussian-law construction. Choose an instance of the finite linear-Gaussian state-space model with independent noises: $a_0\sim N_{m_0}({\mu }_0,P_0)$, $q_k\sim N_{m_k}(0,Q_k)$, and $r_k\sim N_{p_k}(0,R_k)$ are mutually independent, with $P_0,Q_k,R_k⪰0$, and $a_k=F_k{a}_{k-1}+q_k,y_k=H_k{a}_k+r_k\mathrm{.}$ Here $N_d(\mu ,C)$ denotes the $d$-dimensional Gaussian law with mean $\mu$ and symmetric positive-semidefinite covariance $C$, including singular $C$. For a Kalman-filter run, write $m_{k-1}^a$ and $P_{k-1}^a$ for its previous analysis mean and covariance, and define its forecast mean and covariance, innovation, and innovation covariance by $m_k^f=F_k{m}_{k-1}^a,P_k^f=F_k{P}_{k-1}^a{F}_k^T+Q_k,d_k=y_k-H_k{m}_k^f,S_k=H_k{P}_k^f{H}_k^T+R_k\mathrm{.}$ The admissible filter inputs are those for which each innovation covariance to be solved is positive definite. The gain is the solution $K_k=P_k^f{H}_k^T{S}_k^{-1}$, computed by the supplied Cholesky solves, and the analysis outputs are $m_k^a=m_k^f+K_k{d}_k$ and $P_k^a=P_k^f-K_k{S}_k{K}_k^T$. The formula describes the exact target of the supplied recursion; it is not an instruction to form a matrix inverse numerically.

For a one-step perturbed-observation ensemble Kalman update, choose dimensions $m,p\ge 1$, a linear observation matrix $H\in {\mathbb{R}}^{p\times m}$, a datum $y\in {\mathbb{R}}^p$, an integer ensemble size $N\ge 2$, and a positive-definite observation covariance $\mathrm{\Gamma }\in {\mathbb{R}}^{p\times p}$. Given forecast coefficients ${\widehat{a}}^1,\dots ,{\widehat{a}}^N\in {\mathbb{R}}^m$, take perturbations ${\eta }^i\sim N_p(0,\mathrm{\Gamma })$ independent across members and independent of the forecast ensemble. Define $\bar{a}=N^{-1}{\sum }_i{\widehat{a}}^i$ and the divisor-$N$ covariance $C_N=N^{-1}{\sum }_i({\widehat{a}}^i-\bar{a})({\widehat{a}}^i-\bar{a}{)}^T$. The gain and perturbed-data update are $K_N=C_N{H}^T(H{C}_N{H}^T+\mathrm{\Gamma }{)}^{-1}$ and $a^{a,i}={\widehat{a}}^i+K_N(y+{\eta }^i-H{\widehat{a}}^i)$. Since $C_N⪰0$, $H{C}_N{H}^T+\mathrm{\Gamma }\succ 0$. This is the specified finite-ensemble procedure; the definition alone gives no finite-ensemble posterior-exactness claim.

In a single-window finite inverse view, a real matrix $G\in {\mathbb{R}}^{p\times m}$ maps coefficients $a\in {\mathbb{R}}^m$ to predicted observations $Ga$. A datum has the form $y=Ga+e$, where $e\in {\mathbb{R}}^p$ is its realized error and a symmetric positive-definite matrix $R\in {\mathbb{R}}^{p\times p}$ is the declared error covariance and weighting matrix. This deterministic inverse interface does not assume a probability distribution for the realized vector $e$.

Choosing a finite trial space, a linearization, or process-noise covariances is part of adopting the finite model. These choices do not show that a nonlinear fluid equation or an actual sensor is described exactly by that model.

Linear-Gaussian state-space models with independent noises

Exact linear-Gaussian Kalman filter

Perturbed-observation ensemble Kalman filter

The Axiom of Choice

Law, Stuart, and Zygalakis, Chapter 2 §§2.1, 2.4 and Chapter 4 §§4.1, 4.2.3