Pseudo-algorithm

With a linear observation model \(\boldsymbol{H}\), the EnKF can be implemented as follows:

• Draw \(N\) samples from the prior \(\boldsymbol{X}_{1} \sim p(\boldsymbol{x}_{1})\)

• Set initial weights \(\boldsymbol{w}_{1} := [w_{1}^{1},\dots,w_{1}^{N}] = [1/N,\dots,1/N]\)

• For timestep \(t\) from \(1\) to \(T\):

  1. Assimilation step

     • Update the particle weights

     \[\begin{split} \begin{aligned} \widehat{w}_{t}^{n} &= w_{t-1}^{n} p(\boldsymbol{y}^{*}_{t}|\boldsymbol{X}_{t}) && \\ w_{t}^{n} &= \frac{\widehat{w}_{t}^{n}}{\sum_{i=1}^{N}\widehat{w}_{t}^{i}} && \forall n = 1,\dots,N \\ \end{aligned} \end{split}\]

     • Optional: resample the ensemble

     \[\begin{split} \begin{aligned} &i^{n} \sim \text{Resampling}(\boldsymbol{w}_{t}) && \\ &\boldsymbol{X}_{t}^{n} \leftarrow \boldsymbol{X}_{t}^{i^{n}}, \quad w_{t}^{n} = 1/N && \forall n = 1,\dots,N \end{aligned} \end{split}\]

  2. Forecast step

     • Predict the next states

     \(\boldsymbol{X}_{t+1} = \boldsymbol{M}( \boldsymbol{X}_{t}^{*}) + \boldsymbol{\epsilon}, \quad \boldsymbol{\epsilon} \sim \mathcal{N}(\boldsymbol{0},\boldsymbol{Q})\)