Ensemble Kalman filter

Definition

Let the state of a system at time $t_k$ be a vector in $\R^n$. Let the model $M_k \colon \R^n \to \R^n$ map the state at $t_k$ to the state at $t_{k+1}$. Let the observation at $t_k$ be $y_k = H x_k + \varepsilon_k$, where $H$ is an $m \times n$ matrix. The noise $\varepsilon_k$ is Gaussian with mean $0$ and positive definite covariance matrix $R$ of size $m \times m$. Fix an ensemble size $N \ge 2$. The ensemble Kalman filter carries an ensemble $x^{(1)}, \dots, x^{(N)}$ of states and repeats two steps.

1. Forecast step: replace each member by $x^{f,(i)} = M_k(x^{(i)})$, possibly plus a sample of model noise.
2. Analysis step: compute the ensemble mean $\bar x^f = \frac{1}{N} \sum_i x^{f,(i)}$ and the sample covariance $$P^f = \frac{1}{N - 1} \sum_{i=1}^{N} (x^{f,(i)} - \bar x^f)(x^{f,(i)} - \bar x^f)^T.$$ Set the gain $K = P^f H^T (H P^f H^T + R)^{-1}$. Draw independent perturbations $\varepsilon^{(i)}$ with mean $0$ and covariance $R$, and replace each member by $$x^{a,(i)} = x^{f,(i)} + K \big( y_{k+1} + \varepsilon^{(i)} - H x^{f,(i)} \big).$$

The ensemble mean after the analysis step is the estimate of the state, and the spread of the ensemble is the estimate of its uncertainty.