5.1. LETKF Algorithm¶

The following is a brief conceptual overview from [Sluka2016] of how the LETKF algorithm operates, for a complete description see [Hunt2007].

The local ensemble transform Kalman filter (LETKF) is a type of ensemble Kalman filter (EnKF) which uses an ensemble of forecasts \(\left\{\mathbf{x}^{b(i)} : i = 1,2,...,k \right\}\) to determine the statistics of the background error covariance. This information is combined with new observations, \(\mathbf{y}^o\), to generate an analysis mean, \(\bar{\mathbf{x}}^a\), and a set of new ensemble members, \(\mathbf{x}^{a(i)}\). First, the model state is mapped to observation space by applying a nonlinear observation operator \(H\) to each background ensemble member

\[\mathbf{y}^{b(i)} = H\mathbf{x}^{b(i)}\]

note, that the application of the observation operator is applied outside this UMD-LETKF library.

A set of intermediate weights, \(\bar{\mathbf{w}}^{a}\) are calculated to find the analysis mean \(\bar{\mathbf{x}}^a\)

\[\tilde{\mathbf{P}}^a = \left [ \left( k-1 \right ) \mathbf{I} + \left( \mathbf{Y}^b \right )^T \mathbf{R}^{-1} \mathbf{Y}^b \right ]^{-1}\]

\[\bar{\mathbf{w}}^a = \tilde{\mathbf{P}}^a \left( \mathbf{Y}^b \right)^T \mathbf{R}^{-1} \left( \mathbf{y}^o - \bar{\mathbf{y}}^b \right)\]

\[\bar{\mathbf{x}}^a = \bar{\mathbf{x}}^b + \mathbf{X}^b \bar{\mathbf{w}}^a\]

where \(\bar{\mathbf{x}}^b\) and \(\bar{\mathbf{y}}^b\) are the ensemble mean of the background in model space and observation space, respectively. \(\mathbf{X}^b\) and \(\mathbf{Y}^b\) are the matrices whose columns represent the ensemble perturbations from those means, and \(\mathbf{R}\) is the observation error covariance matrix.

Last, the set of intermediate weights, \(\mathbf{W}^a\) are calculated to find the perturbations in model space for the analysis ensemble by

\[\mathbf{W}^a = \left[ \left( k-1 \right) \tilde{\mathbf{P}}^a \right]^{1/2}\]

\[\mathbf{X}^a = \mathbf{X}^b \mathbf{W}^a\]

the final analysis ensemble members, \(\mathbf{x}^{a(i)}\), are the result of adding each column of \(\mathbf{X}^a\) to \(\bar{\mathbf{x}}^a\)