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- The following topics will be covered in this lecture:
- The iterative ensemble Kalman smoother
- The single-iteration ensemble Kalman smoother

In the last lecture, we saw how the

**ensemble Kalman filter (EnKF)**can be used to**propagate covariance estimates in the nonlinear model**, while making the linear-Gaussian approximation at first order.Rather than analytically computing a large covariance matrix, and its evolution using the tangent-linear approximation, a

**nonlinear sample is drawn**and a**sample-based estimate is performed for the covariance**.This sample-based covariance thus forms the

**background weights**for the optimization of the**nonlinear filtering cost function**.This can be extended, like the smoothing problem in 4D-VAR, to a global analysis over a time series.

- Such an approach is what is often known as an
**ensemble-variational (EnVAR)**technique, in which a 4D iterative optimization is made over an initial condition, - but the
**weights and the estimate are constructed with the ensemble**.

- Such an approach is what is often known as an
This approach can be considered to thus extend 4D-VAR to include a time-varying background covariance using the ensemble-based estimates.

- Often, however, due to the small feasible ensemble size that can be simulated in the nonlinear model, the
**ensemble-based background weights**are**interpolated with a climatological covariance to regularize the problem**.

- Often, however, due to the small feasible ensemble size that can be simulated in the nonlinear model, the

The ensemble-variational approach is at the basis of the

**iterative ensemble Kalman filter / smoother (IEnKF/S)**.This technique seeks to perform an ensemble analysis like the square root ETKF by

**defining the ensemble estimates and the weight vector in the ensemble span**\[ \begin{alignat}{2} & & {\color{#d95f02} {\widetilde{\mathcal{J}} (\pmb{w})} } &= {\color{#d95f02} {\frac{1}{2} \parallel \hat{\pmb{x}}_{0|L-S}^\mathrm{smth} - \mathbf{X}^\mathrm{smth}_{0|L-S} \pmb{w}- \hat{\pmb{x}}^\mathrm{smth}_{0|L-S} \parallel_{\mathbf{P}^\mathrm{smth}_{0|L-S}}^2} } + {\color{#7570b3} {\sum_{k=L-S+1}^L \frac{1}{2} \parallel \pmb{y}_k - \mathcal{H}_k\circ {\color{#1b9e77} { \mathcal{M}_{k:1}\left( {\color{#d95f02} { \hat{\pmb{x}}^\mathrm{smth}_{0|L-S} + \mathbf{X}^\mathrm{smth}_{0|L-S} \pmb{w} } } \right)}}\parallel_{\mathbf{R}_k}^2 } }\\ \Leftrightarrow & & {\color{#d95f02} {\widetilde{\mathcal{J}} (\pmb{w})} } &= {\color{#d95f02} { \frac{1}{2} \parallel \pmb{w}\parallel^2} } + {\color{#7570b3} {\sum_{k=L-S+1}^L \frac{1}{2} \parallel \pmb{y}_k - \mathcal{H}_k\circ {\color{#1b9e77} { \mathcal{M}_{k:1}\left( {\color{#d95f02} { \hat{\pmb{x}}^\mathrm{smth}_{0|L-S} + \mathbf{X}^\mathrm{smth}_{0|L-S} \pmb{w} } } \right)}}\parallel_{\mathbf{R}_k}^2 } } \end{alignat} \]

One measures the cost as the

**discrepancy from the observations**with the nonlinear evolution of the perturbation to the ensemble mean,\[ \begin{align} \hat{\pmb{x}}^\mathrm{smth}_{0|L-S} + \mathbf{X}^\mathrm{smth}_{0|L-S} \pmb{w} \end{align} \] combined with the

**size of the perturbation relative to the ensemble spread**.The key is again, how the gradient is computed for the above cost function.

The

**gradient of the ensemble-based cost function**is given by,\[ \begin{align} {\color{#d95f02} {\nabla_{\pmb{w}} \widetilde{\mathcal{J}} } }:= {\color{#d95f02} {\pmb{w}}} - {\color{#7570b3} {\sum_{k=L-S+1}^L \widetilde{\mathbf{Y}}_k^\top \mathbf{R}^{-1}_k\left[\pmb{y}_k - \mathcal{H}_k \circ {\color{#1b9e77} { \mathcal{M}_{k:1}\left({\color{#d95f02} {\hat{\pmb{x}}_{0|L-S}^\mathrm{smth} + \mathbf{X}_{0|L-S}^\mathrm{smth} \pmb{w}} }\right) } } \right]}}, \end{align} \]

where \( {\color{#7570b3} { \widetilde{\mathbf{Y}}_k } } \) represents a

**directional derivative**of the observation and state models,\[ \begin{align} {\color{#7570b3} { \widetilde{\mathbf{Y}}_k } }:= {\color{#d95f02} {\nabla\vert_{\hat{\pmb{x}}^\mathrm{smth}_{0|L-S}} } } {\color{#7570b3} {\left[\mathcal{H}_k \circ {\color{#1b9e77} {\mathcal{M}_{k:1} } } \right] } } {\color{#d95f02} {\mathbf{X}^\mathrm{smth}_{0|L-S}} }. \end{align} \]

In order to

**avoid the construction of the tangent-linear and adjoint models**, the “bundle” version makes an explicit**approximation of finite differences with the ensemble**\[ \begin{align} {\color{#7570b3} { \widetilde{\mathbf{Y}}_k } }\approx& {\color{#7570b3} { \frac{1}{\epsilon} \mathcal{H}_k \circ {\color{#1b9e77} {\mathcal{M}_{k:1} \left( {\color{#d95f02} { \pmb{x}_{0|L-S}^\mathrm{smth} \pmb{1}^\top + \epsilon \mathbf{X}_{0|L-S}^\mathrm{smth} } }\right) } } \left(\mathbf{I}_{N_e} - \pmb{1}\pmb{1}^\top / N_e \right)} }, \end{align} \] for a small constant \( \epsilon \).

The scheme produces an

**iterative estimate**using a**Gauss-Newton**- or, e.g.,**Levenberg-Marquardt**-based optimization.A similar scheme used more commonly in reservoir modeling is the

**ensemble randomized maximum likelihood estimator (EnRML)**.

While accuracy increases with

**iterations in the 4D-MAP estimate**, every iteration comes at the**cost of the model forecast**\( {\color{#1b9e77} { \mathcal{M}_{L:1} } } \).In synoptic meteorology the

**linear-Gaussian approximation**of the evolution of the densities is actually an**adequate approximation**;**iterating over the nonlinear dynamics****may not be justified by the improvement in the forecast statistics**.

However, the

**iterative optimization over a nonlinear observation operator \( \mathcal{H}_k \) or hyper-parameters in the filtering step**of the classical EnKS can be run**without the additional cost of model forecasts**.- This can be performed similarly to the IEnKS with the
**maximum likelihood ensemble filter (MELF)**analysis.

- This can be performed similarly to the IEnKS with the
Subsequently, the

**retrospective analysis**in the form of the filtering right-transform can be applied to**condition the initial ensemble**\[ \begin{align} \mathbf{E}^\mathrm{smth}_{0|L} = \mathbf{E}_{0:L-1}^\mathrm{smth} \boldsymbol{\Psi}_L \end{align} \]

As with the 4D cost function, one can

**initialize the next DA cycle in terms of the retrospective analysis**, and gain the benefit of the improved initial estimate.This scheme, is the

**single-iteration ensemble Kalman smoother (SIEnKS)**.

- Compared to the classical EnKS, this
**adds an outer loop**to the**filtering cycle**to produce the**posterior analysis**.