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