Perturbed-observation ensemble Kalman filter

Fix an ensemble size $N\ge 2$, forecast map $\mathrm{\Psi }$, observation map $G$, datum $y\in {\mathbb{R}}^p$, and observation covariance $\mathrm{\Gamma }\succ 0$. The input is a state ensemble $x^1,\dots ,x^N\in {\mathbb{R}}^d$, forecast-error draws ${\xi }^i$ that are independent across members and of the input ensemble, each with the caller-specified additive model-error law, and observation perturbations ${\eta }^i\sim N_p(0,\mathrm{\Gamma })$ under Multivariate normal law, including singular covariance. The observation perturbations are independent across members and independent of the forecast ensemble and forecast-error draws.

1. Forecast each member: ${\widehat{x}}^i=\mathrm{\Psi }(x^i)+{\xi }^i$.

2. Form predicted observations $z^i=G({\widehat{x}}^i)$, means $\bar{x}=N^{-1}{\sum }_i{\widehat{x}}^i$, $\bar{z}=N^{-1}{\sum }_i{z}^i$, and empirical covariance blocks $C_{xz}=N^{-1}{\sum }_i({\widehat{x}}^i-\bar{x})(z^i-\bar{z}{)}^T,C_{zz}=N^{-1}{\sum }_i(z^i-\bar{z})(z^i-\bar{z}{)}^T\mathrm{.}$ The divisor-$N$ convention is part of this algorithm.

3. Form $S_N=C_{zz}+\mathrm{\Gamma }$. Since $C_{zz}$ is positive semidefinite and $\mathrm{\Gamma }$ is positive definite, $S_N$ is positive definite. Factor it by Cholesky and solve $S_N{K}_N^T=C_{xz}^T$ by triangular solves.

4. Update every member using its perturbed datum: $x^{a,i}={\widehat{x}}^i+K_N(y+{\eta }^i-z^i)\mathrm{.}$

Return the updated ensemble and its empirical mean and covariance. When $G(x)=Hx$, $C_{xz}=C_x{H}^T$ and $C_{zz}=H{C}_x{H}^T$, giving the usual sample-covariance Kalman gain. This item specifies the perturbed-observation variant; deterministic square-root updates are different finite-ensemble procedures. Finite ensembles in nonlinear models are approximations and carry no general posterior-exactness claim.