text:
\section{Introduction}
\label{intro}

Satellite altimetry is the main data source for the observation and reconstruction of sea surface dynamics on a global scale \citep{chelton_satellite_2001}. Current satellite altimeters only deliver along-track nadir observations. This results in a very scarce sampling of the ocean surface. Interpolation schemes are then key components of the operational processing of satellite altimetry data. Current operational products \citep{taburet_duacs_2019, cmems_2018} show however a limited ability to retrieve the full-range of mesoscale dynamics. Upcoming wide-swath altimetry SWOT mission, see e.g. \citep{gaultier_challenge_2015}, will provide for the first time two-dimensional observation of the sea surface height. The space-time sampling of satellite altimeters will however still remain scarce for a long time, which has motivated a recent research literature towards the improvement of the interpolation of satellite-derived SSH fields, see e.g. \citep{lopez-radcenco_analog_2019, lguensat_analog_2017, beauchamp_2021, Ballarotta_2019}. \\

Besides operational schemes based on optimal interpolation techniques \citep{taburet_duacs_2019} and data assimilation schemes for ocean circulation models \citep{Benkiran_2021}, we may sort the proposed SSH interpolation schemes into three main categories: extension of optimal interpolation approaches towards multi-scale schemes \citep{Ardhuin_2020}, data assimilation schemes using sea surface dynamical priors such as quasi-geostrophic (QG) dynamics \citep{LeGuillou_2020}, and data-driven interpolation methods. The latter comprises both EOF-based (Empirical Orthogonal Function) techniques \citep{beckers_eof_2003, alvera_2009}, analog approaches \citep{lguensat_analog_2017, tandeo_2020} and more recently deep learning schemes \citep{fablet2020joint, fablet_2022,Manucharyan_2021,beauchamp_2020} which relates to their recent application to computational imaging problems. \\

Here, we explore further this avenue and more specifically the 4DVarNet framework recently introduced in \citet{Fablet_2021}. As it relies on a variational data assimilation formulation, it appears particularly suited to the space-time interpolation of sea surface variables from irregularly-sampled observations. We propose a parametrization of the 4VarNet scheme dedicated to SSH interpolation from satellite altimeter data and report  OSSE (Observing System Simulation Experiment) results to support the relevance of the proposed scheme. Our main contributions are as follows:
\begin{itemize}
    \item The proposed 4DVarNet-SSH scheme delivers an end-to-end neural architecture using as inputs raw satellite altimeter data and optimally-interpolated fields. We also address uncertainty quantification issues using an ensemble method.
    \item For OSSE on two case-study regions, respectively along the GULFSTREAM and for an open ocean area dominated by mesoscale eddy dynamics, the 4DVarNet-SSH scheme outperforms previous work and significantly improves performance metrics with respect to the operational processing. We also support the relevance of wide-swath SWOT altimeter data to significantly improve the reconstruction of sea surface dynamics compared to nadir-only satellite altimeters.
    \item We deliver an open source code for the proposed 4DVarNet-SSH scheme. It relies on a Pytorch and associated state-of-the-art packages. As such, it supports multi-GPU configuration and can scale up to large-scale domains.
\end{itemize}
We believe these contributions to contribute to the development of deep learning approaches for satellite altimetry, and more broadly for operational oceanography. \\

This paper is organized as follows. Sect. \ref{prob_state} briefly reviews key methodological aspects and related work. We describe the proposed 4DVarNet-SSH approach in Sect. \ref{method} and Sect. \ref{data} presents the considered OSSE setting. We report our results in Sect. \ref{osse} and discuss further our main contributions in Sect. \ref{conclu}.

\section{Background and related work}
\label{prob_state}

From a methodological point of view, interpolation problems in geoscience are classically regarded as data assimilation issues \citep{asch_data_2016}. They aim at estimating the state $\x_t$ of a multi-dimensional dynamical system:
\begin{align}
\label{DA_eqs}
\begin{cases}
 \frac{d\mathbf{x}_t}{dt} & = \mathcal{M} (\mathbf{x}_t) + \boldsymbol {\eta}_t \\
 \mathbf{y}_t & = \mathcal{H}_t \big(\mathbf{x}_t\big) + \boldsymbol {\varepsilon}_t
\end{cases}.
\end{align}

The first equation relates to the forecast step which describes the evolution of the system from time $t$ to $t+dt$ according to the potentially non-linear model $\mathcal{M}$. The second equation introduces the observations $\mathbf{y}_t$ at time $t$ where $\mathcal{H}_t$ is the corresponding observation operator, usually known, but also potentially trainable. $\boldsymbol{\eta}(t)$ is the model error and  $\boldsymbol {\varepsilon}(t)$ the observation error. Both errors are generally assumed to be Gaussian, unbiased and uncorrelated over time. When discretized on a spatio-temporal grid where index $k=1,\cdots,T$ refers to time $t_k$, their associated covariance matrices write $\Q_k \in \mathbb{R}^{m \times m} $ and $\R_k \in \mathbb{R}^{p_k \times p_k}$.\\

Broadly speaking, a vast family of data assimilation methods stems from the minimization of some energy or functional which involves two terms, a dynamical prior and an observation term. We may distinguish two main categories of data assimilation approaches \citep{evensen_data_2009}: variational and statistical data assimilation. Specifically, within a variational data assimilation framework, the state analysis $\mathbf{x}^a$ results in a gradient-based minimization of the defined variational cost $\mathcal{J}(\mathbf{x})=\mathcal{J}_{\Phi}(\mathbf{x},\mathbf{y},\Omega)$ \citep{asch_data_2016}. The latter generally combines the sum of an observation term and a regularization term involving an operator $\Phi$:

\iffalse
where the variational cost function $\mathcal{J}(\mathbf{x})=\mathcal{J}_{\Phi}(\mathbf{x},\mathbf{y},\Omega)$ is generally the sum of an observation term and a regularization term:
\fi
\begin{align}
  \mathcal{J}_{\Phi}(\mathbf{x},\mathbf{y},\Omega)  & = \frac{1}{2}|| \mathbf{y} - \mathcal{H}(\mathbf{x}) ||^2_{\mathbf{R}^{-1}} +  \frac{1}{2} || \mathbf{x} - \Phi(\mathbf{x}) ||^2_{\mathbf{Q}^{-1}} \nonumber  \\
\end{align}
In a weak-constrained 4DVar scheme \citep{carassi_2018}, prior operator $\Phi$ is a time-stepping operator associated with dynamical model $\mathcal{M}$. $\mathcal{H}$ is the observation operator, $\Omega=\lbrace{ \Omega_k \rbrace}$ is the set of subdomains of $\mathcal{D}$ with observations at time $t_k$, $k=1,\cdots,T$. $\mathbf{Q}$ and $\mathbf{R}$ are, respectively, the
background and the observation error covariance matrices and $\Phi(\mathbf{x})$ denotes here the background estimation, i.e. the physical prior, more often noted as the deterministic forecast $\mathbf{x}^b$.

Regarding statistical data assimilation, many state-of-the-art methods rely on optimal formulation (OI),
Interestingly, the analyzed state obtained from OI matches the minimization of the 3DVar cost function, which relates to the stationary case of 4DVar formulation described above, see e.g. \citet{carassi_2018}. This establishes the formal link between the statistical DA frameworks and the optimal control theory used in the variational formulation. Optimal Interpolation (OI) has been used for decades \citep{taburet_duacs_2019} for the interpolation of along-track nadir altimeter datasets and is still used today for the  operational Marine (CMEMS) and Climate (C3S) production of the E.U. Copernicus program. It involves a significant smooting, solving spatial scales up to 150km. Extensions of OI schemes to multi-scale
to better account for mesoscale sea surface dynamics have recently been proposed \citep{Ardhuin_2020,Ubelmann_2016}.
 Variational DA schemes have also been widely explored for the assimilation of satellite altimeter data in ocean general circulation models, see e.g. \citet{Ngodock_2015}, \citet{Benkiran_2021} or \citet{Li_2021}. Previous works have also considered quasigeostrophic (QG) dynamics as an approximate and reduced-order dynamical prior for sea surface dynamics, leading to state-of-the-art performance \citep{Ubelmann_2016, LeGuillou_2020}. Overall, BOOST-SWOT 2020 data challenge (\url{https://github.com/ocean-data-challenges/2020a_SSH_mapping_NATL60}) provides a representative benchmarking framework to assess the performance of SSH mapping schemes for nadir-only and nadir+SWOT altimetry datasets. It further stresses the limited ability of the state-of-art to retrieve fine-scale dynamics below 1$^\circ$ and 10~days.\\

Whereas model-driven and optimal interpolation approaches are the state-of-the-art solutions for operational products, data-driven strategies have recently emerged as promising alternatives to improve the space-time resolution of interpolated products. We may cite among others DINEOF \citep {beckers_2003, Alvera-azcarate_reconstruction_2005, alvera_2009} and the Analog Data Assimilation, AnDA \citep{lguensat_analog_2017, tandeo_2020} and the recent developments of deep learning schemes \citep{barth_2019, beauchamp_2020} . \citet{beauchamp_2020} have reported a benchmarking experiment, which supported the relevance of data-driven schemes compared with the operational OI product. Here, we further explore deep learning approaches, and more particularly the 4DVarNet scheme \citep{fablet_2022}, which bridges variational data assimilation and deep learning. As detailed herefater, we introduce a parameterization of the 4DVarNet scheme dedicated to SSH interpolation issues and demonstrate its relevance in the context of the benchmarking settings introduced in BOOST-SWOT 2020 data challenge.

\section{Method}
\label{method}

This section details the proposed learning-based framework for the interpolation of satellite altimeter data. We first briefly review 4DVarNet framework recently introduced in \citet{Fablet_2021} in Sect. \ref{end-to-end} and present the proposed parameterization for SSH mapping from nadir and SWOT altimeter data in Sect. \ref{param}. We describe the resulting Pytorch package and associated implementation details in Sect. \ref{onthefly} and the proposed learning setting in Sect. \ref{learning_setting}.

\subsection{4DVarNet framework}
\label{end-to-end}

4DVarNet framework introduced in \citet{Fablet_2021} provides a generic approach for the learning of 4DVar models and solvers. They have been shown to outperform classic 4DVar solver for toy case-studies such as Lorenz-63 and Lorenz-96 dynamics, when considering partially-observed systems. 4DVarNet framework can be regarded as an extension using trainable gradient-based solvers of the deep learning scheme, which led to the best SSH interpolation performance in our previous work \citep{beauchamp_2020}. \\

From a methodological point of view, 4DVarNet framework derives an end-to-end neural architecture from an underlying variational DA formulation:
\begin{align}
  \mathcal{J}_{\Phi}(\mathbf{x},\mathbf{y},\Omega)
& = \lambda_1 || \mathbf{y} - \mathcal{H}(\mathbf{x}) ||^2_\Omega + \lambda_2 || \mathbf{x} - \Phi(\mathbf{x}) ||^2,
\end{align}
where $\lambda_{1,2}$ are predefined or tunable scalar weights and we replaced the Mahalanobis norms $||.||_R^{-1}$ and $||.||_Q^{-1}$ by a standard mean-square norm for sake of simplicity. In the regularization term, we substitute to the traditional dynamical prior $\mathcal{M}$ a  neural operator $\Phi$ which convolutional architecture. Then, we can exploit the automatic differentiation tools embedded in deep learning framework to consider the following iterative gradient-based solver for the minimization of variational cost $\mathcal{J}_{\Phi}$ w.r.t. state $\x$:
\begin{equation}
\label{eq: lstm update}
\left \{\begin{array}{ccl}
     g^{(i+1)}& = &  LSTM \left[ \alpha \cdot \nabla_{\mathbf{x}}\mathcal{J}_{\Phi}(\mathbf{x}^{(i)} ,\mathbf{y},\Omega),  h(i) , c(i) \right ]  \\~\\
     x^{(i+1)}& = & x^{(i)} - {\cal{T}}  \left( g^{(i+1)} \right )  \\
\end{array}, \right.
\end{equation}
where ${\mathcal{L}}$ is a convolutional LSTM model, see e.g. \citep{Shi_2015}, $\alpha$ a normalization scalar and ${\cal{T}}$ a linear mapping. This iterative rule based on a trainable LSTM operator is similar to that classically used in meta-learning schemes \citep{andrychowicz2016learning}. Due to the ability of LSTM models to capture long-term dependencies, it results in a trainable gradient descent with momentum.

\begin{figure*}[t]
  \centering
  \input{GB.tex}
   \caption{Sketch of the gradient-based algorithm: the upper-left stack of images corresponds to an example of SSH observations temporal sequence with missing data used as inputs. The upper-right stack of images is an example of intermediate reconstruction of the SSH gradient at iteration $i$ while the bottom-left stack of images identifies the updated reconstruction fields used as new inputs after each iteration of the algorithm.}
   \label{sketch_GB}
\end{figure*}

Overall, a 4DVarNet scheme defines a neural architecture which runs a predefined number of iterative gradient-based update (see Eq. \ref{eq: lstm update}). The resulting neural architecture is referred as an end-to-end architecture in the sense as it uses as inputs raw observation data $y$ and an initial guess $\mathbf{x}^{(0)}$ and as outputs the reconstructed state $\widehat{\mathbf{x}}$. Let us denote by $\Psi_{\Phi,\Gamma}(\mathbf{x}^{(0)},\mathbf{y},\Omega )$ the output of the 4DVarNet architecture for given priors $\Phi$ and solver $\Gamma$,
see Fig. \ref{sketch_GB} and Algorithm below, the initialization $\mathbf{x}^{(0)}$ of state $\mathbf{x}$ and the observations $\mathbf{y}$ on domain $\Omega$.\\

\begin{figure*}[t]
  \centering
  \includegraphics[width=10cm]{algorithm.pdf}
  \label{algo9}
\end{figure*}

Then, the joint learning of operators $\lbrace\Phi,\Gamma\rbrace$ is stated as the minimization of a reconstruction cost, see Sect. \ref{learning_setting}:
\begin{equation}
\label{eq: E2E loss}
   \arg \min_{\Phi,\Gamma} \mathcal{L}(\mathbf{x},\mathbf{x}^\star) \mbox{  s.t.  }
   \mathbf{x}^\star = \Psi_{\Phi,\Gamma}  (\mathbf{x}^{(0)},\mathbf{y},\Omega).
\end{equation}

In Appendix \ref{appA}, a parameter-free fixed point version of the solver is also given, based on the previous results of \citep{beauchamp_2020}. In addition, \citet{beauchamp_2021} have already shown how the iterative gradient-based update is more efficient that the simpler fixed-point formulation.

\subsection{4DVarNet-SSH parameterization}
\label{param}

The proposed 4DVarNet-SSH framework aims at
exploiting and improving the mapping performance of current operational OI products. Given that OI products retrieve consistent large-scale dynamics, we rely on the following multiscale decomposition:
\begin{align}
\mathbf{x}=\mathbf{\overline{x}}+d\mathbf{x}+\epsilon
\end{align}
where the anomaly $d\mathbf{x}$ is seen as the difference between the true state $\mathbf{x}$ and the large-scale components $\mathbf{\overline{x}}$.
Regarding the observations data, let us denote by $\mathbf{y}(\Omega)=\lbrace \mathbf{y}_k(\Omega_k) \rbrace$  the partial and potentially noisy altimetry observations associated with masks $\Omega=\lbrace \Omega_k \rbrace \subset \mathcal{D}$, where $\overline{\Omega_k}$ corresponds to the gappy part of the field and index $k$ refers to time $t_k$.
We use the operational OI product as a gap-free obervation data, denoted as $\mathbf{\overline{y}}$, for state component $\mathbf{\overline{x}}$, whereas the observation data for anomaly $d\mathbf{x}$ is $\mathbf{y}-\mathbf{\overline{y}}$ over domain $\Omega$.

Numerical experiments showed that an augmented state formulation led to better interpolation performance regarding potential stripping artifacts due to the nadir along-track sampling. This results in the application of 4DVarNet model (see Eq. \ref{eq:  E2E loss}) to augmented states $\tilde{\mathbf{x}}$ and observations $\tilde{\mathbf{y}}$ defined as follows:
\begin{align}
\tilde{\mathbf{x}}=\begin{pmatrix} \mathbf{\overline{x}} \\ d\mathbf{x}_1 \\ d\mathbf{x}_2 \end{pmatrix},  \ \tilde{\mathbf{y}}=\begin{pmatrix} \mathbf{\overline{y}} \\ d\mathbf{y}_1 \\ d\mathbf{y}_2 \end{pmatrix}, \
\tilde{{\Omega}}=\begin{pmatrix} \mathbb{1} \\ {\Omega} \\ \mathbf{0} \end{pmatrix} \\
\end{align}
This augmented state parameterization introduces two anomaly components. While only the first one is actually observed, the reconstructed SSH state is given by $\mathbf{\overline{x}}+d\mathbf{x}_2$.

Following \citet{Fablet_2021}, the operator $\Phi$ follows a purely data-driven parameterization with a two-scale residual architectures involving  bilinear units \citep{fablet2020joint}. The number of residual blocks is set to 2 and the bilinear units are made of two hidden convolutional layers, respectively with linear and ReLU activations, followed by a linear scheme combining the outputs of the second layer. A final convolutional layer with linear activation is involved to bring the outputs back to the initial state dimension. In its current implementation, $\Phi$ contains about 500.000 parameters. In any case, the number of gradient iterations for the solver $\Gamma$ is fixed at 5. Complementary tests showed that a higher number of iterations leads to a large increase of the training time (because of the implicit number of parameters which grows linearly with this number of iterations) without a  significant gain in terms of 4DVarnet reconstruction skills.\\

Regarding the initial state for iterative gradient-based rule (\ref{eq: lstm update}), we consider the OI field
$\mathbf{\overline{y}}$ for state component $\mathbf{\overline{x}}$,  $\mathbf{y}-\mathbf{\overline{y}}$ for anomaly component $\mathbf{{dx_1}}$ and a zero state for  anomaly component $\mathbf{{dx_2}}$. For anomaly component $\mathbf{{dx_1}}$, gaps are initialized to 0.

\subsection{Learning setting}
\label{learning_setting}

We implement a classic supervised learning strategy using gap-free targets. The considered training loss $\mathcal{L}$ combines reconstruction losses and additional regularization terms:
\begin{align}
\mathcal{L}(\mathbf{x},\mathbf{x}^\star) &= \lambda_1 \sum_{i=1}^N \mathbf{w}_i ||\mathbf{x}-\mathbf{x}^\star||^2 + \lambda_2|\sum_{i=1}^N \mathbf{w}_i |\nabla_\mathbf{x}-\nabla_{\mathbf{x}^\star}||^2 \nonumber  \\
 & +\lambda_3 \sum_{i=1}^N \mathbf{w}_i ||\mathbf{x}-\Phi(\mathbf{x}^\star)||^2 +\lambda_4 \sum_{i=1}^N \mathbf{w}_i ||\mathbf{x}-\Phi(\mathbf{x})||^2
\end{align}
i.e., the L2-norm of the difference between state $\mathbf{x}$ and reconstruction $\mathbf{x}^\star$ as well as for the their gradients, and regularisation losses according to prior $\Phi$ to enforce that both the true states and the reconstructed ones are correctly encoded by prior $\Phi$. $\mathbf{w}= \lbrace \mathbf{w}_i \rbrace$, $i=1,\cdots,N$ denotes a weighting vector along the data assimilation window of size $N$ (=7 here). To give more importance to the center of the DAW, we use:
\begin{align}
\mathbf{w} = \begin{bmatrix} 0 & 0.25 & 0.75 & 1 & 0.75 & 0.25 & 0 \end{bmatrix}^{\mathrm{T}}
\end{align}
 This training loss is used in Sect. \ref{data} for the OSSE-based BOOST-SWOT data challenge framework.\\

\iffalse
In case of an unsupervised learning, given the observations $\mathbf{y}$ on domain $\Omega$ and hidden state $\mathbf{x}$, the 4DVar cost function may be used:
\begin{align*}
\mathcal{L}(\mathbf{x},\mathbf{x}^\star) = \lambda_1 || \mathbf{y} - \mathcal{H}(\mathbf{x}) ||^2_\Omega + \lambda_2 || \mathbf{x} - \Phi(\mathbf{x}) ||^2
\end{align*}
with weights $\lambda_1$ and $\lambda_2$ indicating how reliable the observations are.\\
\fi

Let precise that state sequences $\mathbf{x}_{k \pm l}=\mathbf{x}_{k-l:k+l}$ of length $N=2l+1$ are used in the training for the interpolation problem. The idea is to optimize the results for the time $t_k$ at the center of the window $[t_{k-l};t_{k+l}]$. The value of $N$ has to be chosen according to the dynamics of the geophysical field considered. In the following experiments, we use a value of $N=7$ which seems to be enough to describe the spatio-temporal correlations of the anomaly between the Ground Truth and the DUACS OI scheme

Regarding the training configuration, when the domain is small (see GULFSTREAM and OSMOSIS regions definition in Sect. \ref{data}), we use a single GPU and Adam optimizer with batch size of 2 over 200 epochs. The same set of parameters holds for larger domain (NATL and cNATL, see again Table \ref{table_domains}) but we use the 4DVarNet-distributed version of the code over 4 GPUs. The computational time of the training procedure lies between 4 and 5 hours for the small-domain setup and between 7 and 8 hours for the large-domain setup.

\subsection{Implementation aspects}

\label{onthefly}

We provide an open source PyTorch implementation of the 4dVarNet-SSH scheme\footnote{The code is available at https://doi.org/10.5281/zenodo.7186322}. Pytorch is a state-of-the-art deep learning framework. We benefit from associated packages such as lightning and hydra to provide a high-level environment and make easier the reproduction of the experiments as well as the development of other applications. Through lightning package, our implementation supports multi-GPU distributed learning configurations. This may be highly relevant to speed up the training process.

Regarding computational issues, the OSSE-based applications, see Sect. \ref{data}, involves the processing of 7x240x240 tensors (i.e., 7-day time series over a $12^\circ \times 12 ^\circ$ domain with a 1/20$^\circ$ resolution). GPU with a significant RAM (typically above 30Go), such as NVidia V100, A40, A100, can process such tensors through the proposed 4DVarNet architecture. The direct training 4DVarNet models over larger spatial domains is however limited by the GPU memory.
To address this issue, we develop a specific data management module, through the so-called \textit{dataloaders}. Our \textit{dataloader} module automatically extracts patches of a predefined size (typically, 7x240x240 in the reported experiments) from the considered training dataset according to  stride parameters as sketched in Fig.\ref{patch_decomposition}. One can exploit the same approach to apply a learnt model to a large domain during the evaluation or production stage. In both cases, we benefit from the fully-convolutional feature of the considered neural architecture. This guarantees that, up to border effects, the 4DVarNet processing is translation-invariant.

\begin{figure}[H]
\begin{center}
\includegraphics[width=7.5cm]{figs/patch_strategy.png}
\end{center}
\caption{Patch-based strategy: the whole spatio-temporal dataset is split into small patches. The temporal size of the patches corresponds to the data assimilation window. The spatial size of the patches is chosen to match the maximal distance with spatial autocorrelation of the SSH}
\label{patch_decomposition}
\end{figure}

\iffalse
We show below the whole domain NATL, see Fig. \ref{NATL_4DVarNet}, as a demonstrator for assessing the efficiency of our new patch-based strategy, deployed on the fly during the training phase:

\begin{figure}[H]
\centering
\includegraphics[width=8cm, trim= 90 0 90 70, clip]{figs/xp_NATL/4DVarNet_xp_BOOST_SWOT_Train_NATL_Test_NATL.png}
\caption{SSH Ground Truth, 1 swot + 4 nadirs observations, DUACS OI \& 4DVarNet reconstruction on the 2012-10-25 for the NATL domain}
\label{NATL_4DVarNet}
\end{figure}

\fi

\section{Observation System Simulation Experiments}
\label{data}

This Section details the experimental setup considered in this study for the quantitative evaluation of the proposed framework. We first introduce the simulation dataset used in our experiments as well as the case-study regions. Sect. \ref{pseudo-obs} reviews the simulation satellite altimetry datasets and Sect. \ref{eval_frame} describes our evaluation framework.

\subsection{NATL60 dataset and case-study regions}

In our study, the Nature Run (NR) corresponds to the NATL60 configuration \citep{jean-marc_molines_meom-configurations} of the NEMO (Nucleus for European Modeling of the Ocean) model. It is one of the most advanced state-of-the-art basin-scale high-resolution (1/60$^\circ$) simulation available today, whose surface field effective resolution is about 7km.\\

\begin{figure}[H]
\includegraphics[width=7.5cm,trim=30 30 30 30,clip=true]{figs/domain.pdf}
\caption{Extents of the GULFSTREAM, GULFSTREAM2, OSMOSIS, OSMOSIS2 and cNATL domains used in this work, all part of the North Atlantic (NATL) basin used in the BOOST-SWOT data challenge}
\label{domains}
\end{figure}

In this work, we will use five different subdomains of the North Atlantic basin (see Fig. \ref{domains}):
\begin{itemize}
\item two $10\degree \times 10\degree$ GULFSTREAM and GULFSTREAM2 domains,
\item two $8\degree \times 10\degree$ OSMOSIS and OSMOSIS2 "open ocean" domains,
\item a large $20\degree \times 40\degree$ cNATL domain, at the center North Atlantic basin, used to assess 4DVarNet training on large domains, without any pieces of land inside to avoid any issues in the learning process.
\end{itemize}

The GULFSTREAM and OSMOSIS domains (blue and red solid lines in Fig. \ref{domains}) are the domains used by the BOOST-SWOT project in the framework of the NATL60 OSSE throughout the different related studies, see their 2020 ocean data challenges and \citet{LeGuillou_2020}. Because we aim at exploring the capabilities of 4DVarNet to deploy at the basin scale, we also propose the two alternate GULFSTREAM2 and OSMOSIS2 domains (blue and red dashed lines in Fig. \ref{domains}), with similar dynamical properties than the two initial domains, as well as a larger domain centered in the North Atlantic basin (cNATL, purple dashed lines). The full extent of the subdomains are summarized in Table \ref{table_domains}:
\begin{table}[H]
    \centering
    \caption{Description of the NATL subdomains used for assessing 4DVarNet capabilites generalization}
\scalebox{0.8}{
\begin{tabular}{ l l l l}
\hline
  Domain & longitude & latitude & extent \\ \hline
  GULFSTREAM     & [-65 $^\circ$, - 55 $^\circ$] & [33 $^\circ$, 43 $^\circ$] & 10°$\times$10°  \\
  GULFSTREAM2     & [-45 $^\circ$, - 35 $^\circ$] & [42 $^\circ$, 52 $^\circ$] & 10°$\times$10°  \\  \hline
  OSMOSIS     & [ -19.5 $^\circ$, -11.5 $^\circ$] & [45 $^\circ$, 55 $^\circ$] & 8°$\times$10°  \\
  OSMOSIS2     & [-28.5 $^\circ$, -20.5 $^\circ$] & [50 $^\circ$, 60 $^\circ$] & 8°$\times$10°  \\  \hline
  cNATL     & [-50 $^\circ$, -10 $^\circ$] & [33 $^\circ$, 53 $^\circ$] & 20° $\times$ 40°  \\
  NATL     & [-79 $^\circ$, 7 $^\circ$] & [27 $^\circ$, 65 $^\circ$] & 38° $\times$ 88°  \\  \hline
\end{tabular}
}
\label{table_domains}
\end{table}

The GULFSTREAM regions display physical processes 100 times more energetic at scales larger than 100km with a greater temporal variability than the OSMOSIS regions. As a consequence, the SSH spatial gradient at scales above 100km is lower for OSMOSIS regions which explains why we can see more small scales related structures on such domains. In addition of their intrinsic differences in terms of dynamical regimes, the latitudes of GF-based and OSMOSIS-based regions implies different SWOT temporal samplings.  For OSMOSIS regions, one SWOT observation is available every day, while over the low-latitude GULFSTREAM domains, the SWOT sampling is irregular leading to sequences of several days with only pseudo-nadir observations.\\

Over these regions, the Sea Surface Height (SSH) resolution of the nature run is downgraded to $1/20^\circ$, which is enough to capture both mesoscale dynamical regimes and the OSMOSIS-related smaller scales, while avoiding unnecessary heavy computation time.\\

The NATL60 nature run will then be used as the reference Ground Truth (GT) in an observing system simulation experiments (OSSE). The pseudo-altimetric nadir and SWOT observational datasets will be generated by a realistic sub-sampling of satellite constellations.\\

\subsection{Simulated altimetry datasets}
\label{pseudo-obs}

Regarding the pseudo-nadir altimetry dataset, representative of the current pre-SWOT observational altimetric dataset, we use the groundtracks of 4 altimetric missions (TOPEX/Poseidon, Geosat, Jason-1 and Envisat) picked up from the 2003 constellation to interpolate the NATL60 simulation from October 1st, 2012 to September 29th, 2013. A Gaussian white noise with variance $\sigma^2=(4 \cdots 9)$cm$^2$ is added to the interpolated NATL60 simulation by the SWOTsimulator tool to mimic a noise with a spectrum of error consistent with global estimates from the Jason-2 altimeter \citep{dufau_mesoscale_2016}.
We aggregate the nadir pseudo-observations on a daily basis to procude the gappy daily fields used as inputs by 4DVarNet-SSH.  Fig. \ref{fig_ssh_data_GULFSTREAM}(c) vs Fig. \ref{fig_ssh_data_GULFSTREAM}(d) and Fig. \ref{fig_ssh_data_OSMOSIS}(c) vs Fig. \ref{fig_ssh_data_OSMOSIS}(d) illustrate the resulting nadir altimetry data on 2012, October 25.\\

\vspace{-.5cm}
\begin{figure}[H]
\centering
  \includegraphics[width=9cm, trim= 150 150 0 150, clip]{plot_data.pdf}
  \caption{NATL60 Ground Truth (a) and its gradient (b) ; one day accumulated along-track 4 nadirs (c) and wide-swath SSH pseudo-observations + 4 nadirs (d) on 2012, October 25 (domain: GULFSTREAM)}
  \label{fig_ssh_data_GULFSTREAM}
\end{figure}

\begin{figure}[H]
\centering
  \includegraphics[width=9cm, trim= 150 150 0 150, clip]{plot_data_OSMOSIS.pdf}
  \caption{NATL60 Ground Truth (a) and its gradient (b) ; one day accumulated along-track 4 nadirs (c) and wide-swath SSH pseudo-observations + 4 nadirs (d) on 2012, October 25 (domain: OSMOSIS}
  \label{fig_ssh_data_OSMOSIS}
\end{figure}

We proceed similarly to simulate SWOT pseudo observations using  the swotsimulator tool \citep{gaultier_challenge_2015} in its swath mode with an along-track and across-track 2km spatial resolution (the same theoretical resolution that the upcoming SWOT mission derived products is expected to provide). Let us note that we consider error-free SWOT pseudo-observations.

\subsection{Evaluation framework}
\label{eval_frame}
Our evaluation framework exploits and extends the one introduced in \cite{LeGuillou_2020} as follows:\\

{\bf Training and evaluation setting:} We train all learning-based models using the time period from 2013, February 4 to September 30 as training period. During the training procedure, we select the best model according to metrics computed over the validation period from 2013, January 1 to February 2. Overall, we evaluate performance metrics over the test period from 2012, October 22 to December 2 for intercomparison purposes.

{\bf Evaluation metrics:} We use BOOST-SWOT DC metrics to benchmark 4DVarnet-SSH scheme with respect to the state-of-the-art SSH interpolation schemes. They comprise:  RMSE-scores, in terms of mean -$\mu$(RMSE)- and standard deviation -$\sigma$(RMSE)-, and minimal spatial and temporal scales resolved ($\lambda$x and $\lambda$t). We refer the reader to \cite{LeGuillou_2020} for the detailed description of these metrics.  Besides this quantitative metrics, we analyse the space-time distribution of the interpolation error. We also explore the impact of the interpolaion onto the characterization of mesoscale eddy dynamics.
Based on the work of \citet{mason_2014}, we detect anticyclonic and cyclonic eddies in the Ground Truth NATL60 outputs and interpolated SSH fields using py-eddy-tracker toolbox  (\url{https://py-eddy-tracker.readthedocs.io}) and analyze how key features of matching eddies, such as  speed radius (km), outter radius (km), amplitude (cm) and speed max (cm/s), are retrieved.

\section{Results}
\label{osse}

This section presents the considered OSSE for the evaluation of the 4DVarNet-SSH scheme. We first report the benchmarking experiments with respect to the state-of-the-art (Sect. \ref{perf_GF_OSMOSIS}). Sect. \ref{swot_contrib} studies the impact of wide-swath SWOT data to improve the reconstruction of finer-scale SSH pattern. Last, we analyze generalization issues and uncertainy quantitication in Sect. \ref{perf_gen} and \ref{perf_var}.

\subsection{Benchmarking experiments}
\label{perf_GF_OSMOSIS}

Regarding the BOOST-SWOT OSSE data challenge on the GULFSTREAM domain, we provide both performance with 4 nadirs and 1 swot + 4nadirs in Table \ref{table_scores_GF_OSSE}. For both settings, the improvement is quite significant with respect to all benchmarked schemes, {\em i.e.}
not only compared to DUACS OI \citep{taburet_duacs_2019}, but also with respect to the recently proposed SSH interpolation schemes  \citet{LeGuillou_2020}, DYMOST (Dynamic OI accounting for the SSH non-linear temporal propagation), see e.g. \citet{Ballarotta_2020} and MIOST (Multi-scale OI) of \citet{Ardhuin_2020}. While DUACS OI has minimal spatial and temporal resolution of 1.42°(4 nadirs)/1.22°(1 swot + 4 nadirs) and 12 days (4 nadirs)/11.15 days (1 swot + 4 nadirs), 4DVarNet-SSH reaches 0.83°(4 nadirs)/0.62°(1 swot + 4 nadirs) and 8.01 days (4 nadirs)/5.29 days (1 swot + 4 nadirs). It amounts to a gain up to \textbf{33\%} in the \textbf{4 nadirs} setup and \textbf{50\%} in the \textbf{1 swot + 4 nadirs} configuration.

\begin{table*}[t]
    \centering
    \caption{4DVarNet-SSH performance on the GULFSTREAM domain compared to DUACS OI (traditional covariance-based Optimal Interpolation), BFN (Back and Forth Nudging of A QG model), MIOST (Multi-scale OI), DYMOST (Dynamic OI accounting for non-linear temporal propagation of the SSH fields), and Fixed-point versions with 10 and a single iteration  of the 4DVarNet solver, over the period from 2012-10-22 to 2012-12-02 (42 days)}
    \begin{tabular}{l l l l l l}
    \hline
        Method & Description & $\mu$(RMSE) & $\sigma$(RMSE) & $\lambda$x (degree) & $\lambda$t (days) \\ \hline
        DUACS 4 nadirs & OI & 0.92 & 0.01 & 1.42 & 12.0  \\
        BFN 4 nadirs & QG-based DA (nudging) & 0.92 & 0.02 & 1.23 & 10.6  \\
        DYMOST 4 nadirs & Dynamic OI & 0.91 & 0.01 & 1.36 & 11.79  \\
        MIOST 4 nadirs & Multi-scale OI & 0.93 & 0.01 & 1.35 & 10.19 \\
        4DVarNet 4 nadirs & Fixed-Point solver, $N_i$=10 & 0.92 & 0.01 & 1.22 & 11.51 \\
        4DVarNet 4 nadirs & NN-based 4DVar (ours) & \textbf{0.94} & \textbf{0.01} & \textbf{0.83} & \textbf{8.01} \\ \hline
        DUACS 1 swot + 4 nadirs & OI & 0.92 & 0.01 & 1.22 & 11.15  \\
        BFN 1 swot + 4 nadirs & QG-based DA (nudging) & 0.93 & 0.02 & 0.8 & 10.09 \\
        DYMOST 1 swot + 4 nadirs & Dynamic OI & 0.93 & 0.02 & 1.2 & 10.07  \\
        MIOST 1 swot + 4 nadirs & Multi-scale OI & 0.94 & 0.01 & 1.18 & 10.14  \\
        4DVarNet 1 swot + 4 nadirs & Fixed-Point solver, $N_i$=10  & 0.94 & 0.01 &  1.18 & 9.65 \\
        4DVarNet 1 swot + 4 nadirs & NN-based 4DVar (ours) & \textbf{0.95} & \textbf{0.01} & \textbf{0.62} & \textbf{5.29} \\ \hline
    \end{tabular}
    \label{table_scores_GF_OSSE}
\end{table*}

Fig. \ref{plot_OSSE_GF} displays the SSH gradient field of DUACS OI and 4DVarNet-SSH interpolations on October, 25.  The comparison to the associated Groundtruth  displayed in Fig. \ref{fig_ssh_data_GULFSTREAM}(b) clearly reveals the improvement brought by 4DVarNet-SSH, in particular along the main meandrum of the GULFSTREAM.

\begin{figure}[H]
\centering
    \includegraphics[width=8cm]{plot_OSSE_GF}
    \caption{SSH Gradient (DUACS OI \& 4DVarNet reconstruction) on the 2012-10-25 for the GULFSTREAM domain}
\label{plot_OSSE_GF}
\end{figure}

We can draw similar conclusions from the experiments reported in Table \ref{table_scores_OSMOSIS_OSSE} and Fig. \ref{plot_OSSE_OSMOSIS} for the OSMOSIS domain. We may emphasize that 4DVarNet-SSH interpolation for the 1 swot + 4 nadirs configuration, see e.g. Fig.\ref{plot_OSSE_OSMOSIS}, retrieves most of the fine-scale features of the SSH fields, which are smoothed out by the optimal interpolation.

\begin{table*}[t]
    \centering
    \caption{4DVarNet-SSH performance on the OSMOSIS domain compared to DUACS OI over the period from 2012-10-22 to 2012-12-02 (42 days)}
    \begin{tabular}{l l l l l }
    \hline
        Method & $\mu$(RMSE) & $\sigma$(RMSE) & $\lambda$x (degree) & $\lambda$t (days) \\ \hline
             duacs 4 nadirs &     0.78 &      0.02 &             \textbf{1.10} &          18.80 \\
          4DVarNet 4 nadirs (ours) &     0.80 &      0.01 &             1.18 &          \textbf{14.51} \\ \hline
    duacs 1 swot + 4 nadirs &     0.81 &      0.02 &             1.03 &          17.50 \\
 4DVarNet 1 swot + 4 nadirs (ours) &     0.87 &      0.02 &             \textbf{0.35} &           \textbf{6.84} \\ \hline
    \end{tabular}
    \label{table_scores_OSMOSIS_OSSE}
\end{table*}

\begin{figure}[H]
\centering
    \includegraphics[width=8cm]{plot_OSSE_OSMOSIS.pdf}
\caption{SSH Gradient (DUACS OI \& 4DVarNet-SSH reconstruction) on the 2012-10-25 for the OSMOSIS domain}
\label{plot_OSSE_OSMOSIS}
\end{figure}

\subsection{Impact of SWOT data on the interpolation performance}
\label{swot_contrib}

Thanks to its ability to reconstruct finer-scale patterns, 4DVarNet-SSH complements the assessment of the potential impact of SWOT data onto the reconstruction of mesocale sea surface dynamics. Though the interpolation performance (Tab. \ref{table_scores_GF_OSSE} and \ref{table_scores_OSMOSIS_OSSE}) improves with the use of SWOT data for all the interpolation methods, the relative improvement strongly depends on the interpolation method. Interestingly, contrary to OI DUACS scheme, we report a significant improvement when using SWOT data with 4DVarNet-SSH for both GULFSTREAM and OSMOSIS regions. These results emphasize the ability of our scheme to exploit irregularly-sampled high-resolution data. For instance, for the OSMOSIS region, we truly benefit from SWOT data to reconstruct mesoscale dynamics up to 0.4$^\circ$ and 7~days, whereas OI DUACS smooths out the altimetry signals in the mesoscale range below 1$^\circ$ and 14~days.

While we report relative gains of $20-25\%$ for the GULFSTREAM region for the different evaluation metrics, it reaches $40-60\%$ for the OSMOSIS domain. We interpret these results as a direct consequence of differences in the space-time sampling  of SWOT data for these two regions. As revealed by Fig. \ref{perf_GF_OSSE}(b) and \ref{perf_OSMOSIS_OSSE}(b) no SWOT data may be available over 4 (resp. 1) consecutive days for the GULFSTREAM (resp. OSMOSIS) domain, This time variability of the sampling pattern translates for the GULFSTREAM region into a periodic variability of the MSE time series. By contrast, the OSMOSIS region leads to a much lower time variability of the interpolation performance. The PSD-based analysis reported in Fig.\ref{perf_GF_OSSE}(c) and \ref{perf_OSMOSIS_OSSE}(c) further supports these conclusions.

\begin{figure*}
\centering
\includegraphics[width=11cm]{perf_GF_OSSE.pdf}
\caption{(a) Spatial performance: RMSE time series are computed for each spatial position of the GF domain (left: 4 nadirs ; right: 1 swot + 4 nadirs) ; (b) Temporal performance: RMSE daily GF maps are computed along the BOOST-SWOT DC evaluation period (left: 4 nadirs ; right: 1 swot + 4 nadirs) ; (c) Spectral performance: the PSD-based score evaluates the spatio-temporal scales resolved in GF mapping (yellow area) (top: 4 nadirs ; bottom: 1 swot + 4 nadirs)}
\label{perf_GF_OSSE}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[width=11cm]{perf_OSMOSIS_OSSE.pdf}
\caption{(a) Spatial performance: RMSE time series are computed for each spatial position of the GF domain (left: 4 nadirs ; right: 1 swot + 4 nadirs) ; (b) Temporal performance: RMSE daily GF maps are computed along the BOOST-SWOT DC evaluation period (left: 4 nadirs ; right: 1 swot + 4 nadirs) ; (c) Spectral performance: the PSD-based score evaluates the spatio-temporal scales resolved in GF mapping (yellow area) (top: 4 nadirs ; bottom: 1 swot + 4 nadirs)}
\label{perf_OSMOSIS_OSSE}
\end{figure*}

To complement with the analysis of contribution of SWOT altimetry on the interpolation performance, Fig. \ref{eddy_id_GF_1swot_4nadirs} displays eddy identification results on 2022, October 25 after application of a 200km high pass filter when using 1 swot + 4 nadirs configuration. Additional Figures are given in Appendix \ref{appB} for illustrations for both the GULFSTREAM and OSMOSIS domains in the two observational configurations (4 nadirs and 1 swot + 4 nadirs). Clearly, 4DVarNet-SSH improves the matching between true and interpolated eddies (39 vs 35), and the features of the matching eddies are also more similar to those of the true eddies,
in terms of speed radius (km), outter radius (km), amplitude (cm) and speed max (cm/s), with respect to their true values. Again, the interpolation of eddy-related dynamics significantly improves with the exploitation of SWOT data.\\

\begin{figure}[H]
 \centering
\includegraphics[width=5cm]{eddies/Eddy_identification_GF_1swot_4nadirs.png}
\caption{Eddies detected on the GULFSTREAM domain (2012-10-25) over SSH (1 swot + 4 nadirs)}
\label{eddy_id_GF_1swot_4nadirs}
\end{figure}

\begin{figure}[H]
 \centering
\includegraphics[width=8cm]{eddies/Scatterplot_Eddies_GF_1swot_4nadirs.png}
\caption{Speed radius (km), outter radius (km), amplitude (cm), speed max (cm/s) scatterplots of 4DVarNet/DUACS OI (1 swot + 4 nadirs) vs Ground truth on the GULFSTREAM domain (2012-10-25) for matching eddies}
\label{scatt_eddy_GF_1swot_4nadirs}
\end{figure}

\subsection{Generalization performance}
\label{perf_gen}

Whereas the results reported in the previous sections involve 4DVarNet-SSH models evaluated on the same domain as the training one, we assess how 4DVarNet-SSH schemes trained for a specific domain may also apply to another one. Besides the GULFSTREAM and OSMOSIS regions, we consider three additional domains:
\begin{itemize}
    \item cNATL domain: a larger 20$^\circ \times$40$^\circ$ North Atlantic domain, which involves a variety of dynamical regimes;
    \item GF2 domain: a domain similar to the reference GULFSTREAM domain in terms of upper ocean dynamics, but with a disjoint spatial extent;
    \item OSMOSIS2 domain: a domain similar to the reference OSMOSIS domain in terms of upper ocean dynamics, but with a disjoint spatial extent;
\end{itemize}
For the 1 swot + 4 nadirs configuration, we train 4DVarNet-SSH schemes on these three domains. We then evaluate how these models compare with the models reported in Sect. \ref{perf_GF_OSMOSIS} for the GULFSTREAM and OSMOSIS domains. We also evaluate how the different models apply to the cNATL domain.  Table \ref{table_scores_gen} summarizes the resulting performance metrics.

As expected for each evaluation domain, we retrieve the best performance for the model trained on this domain. For the GULFSTREAM regions, the difference in terms of minimal temporal scales is negligible while the minimal spatial scales may exhibit an increase of 30\% using the model trained on the GULFSTREAM2 domain. This does not hold in the other way when applying on the GULFSTREAM2 domain a model learnt on GULFSTREAM, with similar spatial scales resolved in the end. The same conclusions hold for the OSMOSIS regions, except that the minimal resolved temporal scales also display a slight increase (lower than 20\%) over OSMOSIS. These results are consistent with the dynamical properties given in Sect. \ref{data} and support the generalization capabilities of 4DVarNet-SSH schemes. The comparison with the performance metrics reported for the model trained on the cNATL domain suggests that the considered 4DVarNet-SSH parameterization applies to a regional scale. This training configuration only leads to a relatively marginal gain w.r.t. OI DUACS when applied to the GULFSTREAM region. We report a slightly better performance for the OSMOSIS domain. We expect future work to explore new 4DVarNet-SSH parameterizations, which could better account for basin-scale variabilities.

\begin{table*}[t]
    \centering
    \caption{4DVarNet performance on the GULFSTREAM and OSMOSIS domain compared to DUACS OI over the period from 2012-10-22 to 2012-12-02 (42 days)}
\scalebox{0.85}{
\begin{tabular}{ l l l l l l l l}
\hline
  Domain & Method & $\mu$(RMSE) & $\sigma$(RMSE) & $\lambda$x (degree) & $\lambda$t (days) & Train/Test \\ \hline
  \multirow{3}{*}{GF}       & DUACS OI 1 swot + 4 nadirs &     0.92 &      0.01 &             1.22 &          11.31 & - \\
           & 4DVarNet 1 swot + 4 nadirs &     0.96 &      0.01 &             0.62 &          5.29 &     GF/GF \\
           & 4DVarNet 1 swot + 4 nadirs &      0.95 &      0.01 &             0.86 &           5.67 &      GF2/GF \\
           & 4DVarNet 1 swot + 4 nadirs &     0.92 &      0.02 &             1.25 &          10.93 &     cNATL/GF \\ \hline
  \multirow{3}{*}{OSMOSIS}  & DUACS OI 1 swot + 4 nadirs &     0.81 &      0.02 &             1.04 &          17.80  & - \\
           & 4DVarNet 1 swot + 4 nadirs &     0.89 &      0.02 &             0.35 &           6.84 &     OSMOSIS/OSMOSIS \\
           & 4DVarNet 1 swot + 4 nadirs &      0.88 &      0.02 &             0.41 &           8.05 &     OSMOSIS2/OSMOSIS \\
           & 4DVarNet 1 swot + 4 nadirs &     0.84 &      0.02 &             0.93 &           9.59 &     cNATL/OSMOSIS \\ \hline
  \multirow{3}{*}{cNATL}  & DUACS OI 1 swot + 4 nadirs &      &       &              &            & - \\
           & 4DVarNet 1 swot + 4 nadirs &      &       &              &            &     cNATL/cNATL \\
           & 4DVarNet 1 swot + 4 nadirs &       &       &             &           &     cNATL/GF \\
           & 4DVarNet 1 swot + 4 nadirs &     &       &              &           &     cNATL/OSMOSIS \\ \hline
\end{tabular}
}
\label{table_scores_gen}
\end{table*}

\subsection{Uncertainty Quantification for 4DVarNet-SSH interpolations}
\label{perf_var}

\begin{table}[H]
    \centering
    \caption{4DVarNet performance on the GULFSTREAM domain based on nine different trainings with random initialization of both $\Phi$ and $\Gamma$ weights but similar training parameters (number of epochs, learning rates, optimizers, gradient steps, etc.) over the period from 2012-10-22 to 2012-12-02 (42 days)}
\scalebox{0.85}{
\begin{tabular}{ l l l l l l l l}
\hline
  Members & $\mu$(RMSE) & $\sigma$(RMSE) & $\lambda$x (degree) & $\lambda$t (days) \\ \hline
4DVarNet (\#1) &     0.96 &      0.01 &             0.68 &           5.16 \\
4DVarNet (\#2) &     0.96 &      0.01 &             0.66 &           4.52 \\
4DVarNet (\#3) &     0.96 &      0.01 &             0.62 &           4.66 \\
4DVarNet (\#4) &     0.96 &      0.01 &             0.63 &           4.12 \\
4DVarNet (\#5) &     0.96 &      0.01 &             0.87 &           4.92 \\
4DVarNet (\#6) &     0.96 &      0.01 &             0.86 &           5.07 \\
4DVarNet (\#7) &     0.96 &      0.01 &             0.68 &           5.18 \\
4DVarNet (\#8) &     0.96 &      0.01 &             0.85 &           4.99 \\
4DVarNet (\#9) &     0.96 &      0.01 &             0.62 &           5.29 \\
4DVarNet (median) &     0.96 &      0.01 &             0.67 &           4.62 \\
\hline
\end{tabular}
}
\label{table_int_var}
\end{table}

Besides gap-free fields, operational interpolation products generally require to provide some evaluation of the reconstruction uncertainty. While this is a built-in feature of OI and statistical DA methods, uncertainty quantification may involve specific methodological or computational methods for other data assimilation schemes, among which ensemble methods represent a widely-considered family of approaches, see e.g. \citet{asch_data_2016}. Their common feature is to generate an ensemble of solutions, generally through some randomization process.

Here, we benefit from the stochastic nature of the training procedure of 4DVarNet-SSH schemes \citep{Goodfellow_2016}. Similarly to most deep learning schemes, we exploit a stochastic gradient descent during the learning stage and a random initilization of model parameters. As such, for a given training configuration, we can learn an ensemble of 4DVarNet-SSH schemes by running multiple training procedures.

\begin{figure}[H]
 \centering
 \includegraphics[width=6cm]{4DVarNet_int_var.pdf}
\caption{Interpolation performance of an ensemble of nine 4DVarNet-SSH models trained using similar training parameters (number of epochs, learning rates, optimizers, gradient steps, etc.) but different random initialization of both $\Phi$ and $\Gamma$ weights. (a) : spatial RMSE time series on the BOOST-SWOT DC evaluation period ; (b) 4DVarNet median run (2012-10-25, GF domain) and (c) its spatial standard deviation}
\label{4DVarNet_int_var}
\end{figure}

We apply this approach to build an ensemble of nine 4DVarNet-SSH schemes for a given training configuration, which comprises a training dataset, the considered 4DVarNet-SSH parameterization and given training hyperparameters ({\em i.e.}, number of epochs, learning rates, optimizers). For a given observation time window, we then retrieve nine interpolations, from which we can compute a median field and the associated standard deviation. We report in Tab. \ref{table_int_var} the performance metrics for the GULFSTREAM domain of the nine trained models as well as the median model. It reveals the internal variability of the training process. Though it does not reach the best performance, the median model combines a resolved spatial scale below $0.7^\circ$ and a resolved time scale below 5~days, which is only the case of 6 over 9 of the trained models. Fig. \ref{4DVarNet_int_var}(a) further illustrates this aspect. Interestingly, the standard deviation of the ensemble of 4DVarNet-SSH schemes correlates to the interpolation error, with an R2 coefficient of determination equals to 0.86, see Fig. \ref{4DVarNet_UQ2}. As such, it can be regarded as an indicator of the interpolation error.

\begin{figure}[H]
 \centering
 \includegraphics[width=9cm,trim = 0 190 0 220,clip]{4DVarNet_xp_BOOST_SWOT_Train_GF_Test_GF_UQ2.png}
\caption{Standard deviation of the 4DVarNet-SSH median ensemble interpolation error (left) and average of the daily standard deviation interpolation errors on the test period (right)}
\label{4DVarNet_UQ2}
\end{figure}

\section{Conclusion and Discussion}
\label{conclu}

This paper introduced 4DVarNet-SSH scheme, an end-to-end neural architecture for the space-time interpolation of SSH fields from nadir and wide-swath satellite altimetry data. 4DVarNet-SSH scheme draws from recent methodological development to bridge data assimilation and deep learning with a view to learning 4DVar DA models and solvers from data. Numerical experiments within an OSSE setting support the relevance of 4DVarNet-SSH scheme with respect to the state-of-the-art.

We discuss further our main contributions according to three aspects: the added value of deep learning scheme for satellite altimetry and operational oceanography, the exploitation of upcoming SWOT data and the ability to scale up learning approaches from regional case-studies to the global scale.

~\\

{\bf Deep learning for satellite altimetry and operational oceanography:} This study contributes to a growing research effort regarding the potential benefit of deep learning schemes for space and operational oceanography challenges, see e.g. \cite{Ballarotta_2020}. Given the sampling of available satellite and in situ data sources, interpolation problems naturally arise as critical challenges. This study brings an additional evidence of the potential of deep learning schemes to outperform the state-of-the-art operational techniques, generally based on optimal interpolation and data assimilation. Importantly, we do not rely on the off-the-shelf application of some reference deep learning architectures. The considered class of neural architectures relates to a variational DA formulation, such that it can be regarded as the implementation of a neural and trainable version of a DA model and solver. Our results for satellite altimetry are in line with other recent studies for other ocean parameters, such as sea surface temperature \citep{barth_2019}, suspended sediments \citep{vient_2022} and 3D temperature and salinity fields \citep{Pauthenet_2022}. All these studies support the potential of neural approaches to retrieve finer-scale variabilities from available satellite and/or in situ observations. Regarding satellite altimetry, future challenge includes the application to real altimetry datasets, see e.g. the 2021 Observation System Experiment (OSE) BOOST-SWOT data challenge \url{https://github.com/ocean-data-challenges/2021a_SSH_mapping_OSE}, as well as the exploitation of multimodal synergies \citep{fablet_2022}.\\

{\bf Making the most of SWOT data:} Our study brings new evidence that the wide-swath space-time sampling of upcoming SWOT mission could lead to a very significant improvement of the reconstruction of mesoscale sea surface dynamics. For the considered case-study regions, with contrasted dynamical regimes in play and revisit times of SWOT orbits, we report relative gains from $20\%$ to $60\%$ compared to nadir altimetry data only in terms of RMSE and resolved space-time scales. These results assume an error-free SWOT product. Therefore, exploring further how these results could generalize to error-prone \citep{esteban_2014,gaultier_2010} and uncalibrated SWOT data \citep{febvre_2022} is a critical challenge. Preliminary preprocessing of the pseudo-SWOT observations \citep{metref_2020} to filter out its correlated components and avoid major issues in the assimilation and/or learning process of the interpolation methods may also be considered.  The extension of the considered OSSE to multi-swot configurations could also provide new means to optimize the deployment of multi-satellite configurations in coming years. \\

{\bf Scaling up to a global scale with learning-based scheme:} Our numerical experiments focused mainly on a regional- scale, typically $10^\circ \times 10^\circ$ domains as illustrated by the GULFSTREAM and OSMOSIS regions.
The reported results support the relevance of the proposed 4DVarNet-SSH parameterization to account for such regional space-time variabilities. Scaling up to a basin scale or even the global scale naturally arises as a key challenge for future work. Through the built-in features of Pytorch framework and associated packages, our open-source code can leverage multi-GPU distribution learning schemes and on-the-fly mini-batch generation tools to deal with larger-scale dataset from a computational point of view. To account for a greater diversity of dynamical regimes in play on the global scale, or even on a basin scale, it also seems necessary to explore more complex 4DVarNet-SSH parameterizations, especially regarding dynamical prior $\Phi$. This could benefit from the variety of neural architectures recently introduced in computational imaging \citep{Barbastathis_2019}, especially using attention mechanisms \citep{Vaswani_2017} to achieve some decomposition of the underlying space-time variabilities.

\iffalse
Because we use the NATL60 dataset, we also inherit from a set of metrics provided by the BOOST-SWOT project, in particular in the framework of the related 2020 ocean data challenge (\url{https://github.com/ocean-data-challenges/2020a_SSH_mapping_NATL60}). They are used to confront 4DVarNet to state-of-the-art DUACS Optimal Interpolation, but also to other on-going research methods used in this data challenge. When trained on small regions of the North-Atlantic basin, driven by a unique and easily identifiable dynamical regime, we demonstrate how 4DVarNet outperforms the state-of-the art approaches.
Compared to DUACS OI minimal spatial and temporal scales resolved, 4DVarNet brings a significant gain up to \textbf{50\% in the 1 swot + 4 nadirs} configuration, and \textbf{33\% in the 4 nadirs setup} on the GULFSTREAM domain. On the OSMOSIS region, the improvement may even exceeds \textbf{60\% in both configurations}, due to the regular SWOT sampling and the less energetic spatio-temporal dynamics on this region. In any case, 4DVarNet also outperforms the other state-of-the-art reconstruction techniques participating to the BOOST-SWOT data challenge.\\

The preliminary tests initiated at the basin scale and using the on-the-fly patch-based decomposition strategy are also encouraging for the future developments of 4DVarNet: the next step to make it fully operational, in terms of both computational aspects and metrics performance, will be to move to a multi-regime formulation of the variational cost to embed all the dynamics existing at the basin-scale in a single model, leading to the following variational cost:
\begin{align*}
\mathcal{J}_{\lbrace{\Phi_k\rbrace}}(\mathbf{x},\mathbf{y},\Omega) & = \mathcal{J}^o(\mathbf{x},\mathbf{y},\Omega) + \mathcal{J}_{\lbrace{\Phi_k\rbrace}}^b(\mathbf{x})  \nonumber \\
& = \lambda_1 || \mathbf{y} - \mathcal{H}(\mathbf{x}) ||^2_\Omega \\ &+ \lambda_2 \sum_{k=1}^N \mathbf{w}_k(\x,\y) || \mathbf{x} - \Phi_k(\mathbf{x}) ||^2
\end{align*}
where $\Phi_k$, $k=1,\cdots,N$ are the N dynamical regimes (modes), potentially orthogonal, but with continuous interactions across the ocean, and $\mathbf{w}_k(\x,\y)$ would be a spatio-temporal weighting of $\Phi_k$ depending of the state and the observations. For example, let suppose that the GULFSTREAM domain is driven by one regime and OSMOSIS by an other, we would like these two regimes progressively interact between the two domains, as it is probably the case over the cNATL domain, mainly driven by large mesoscale processes on its northwestern part but with OSMOSIS-related dynamics in the left three quarters of the domain. Going back to the simpler GP formulation with covariance-based optimal interpolation used in large operational system like DUACS, this corresponds to define region adaptative covariance kernels to deal with different dynamical regimes over large domains. Here, it would be straightforward in the near-future of the 4DVarNet developments to propose such a formulation, but with higher potential since we would be able to have a continuous adaptation of the neural network through the spatio-temporal weighting $\mathbf{w}_k(\x,\y)$ in the variational cost, instead of discontinuous patch-based covariance approaches. Along this line, this also paves the way to infer not only the best recontruction of the state but also to have both estimation and multi-regime segmentation embedded in the 4DVarNet scheme. This would be of particular interest to reach and outperform state-of-the art algorithms with a single 4DVarNet model and to be able to handle specific regions such as equatorial, tropical and polar regions over the globe, driven by specific dynamics. On the latter, training the classic 4DVarNet formulation will smooth out all the dynamical mode properties into a single one, thus excluding the regimes related to small specific subdomains when large training domain are considered. \\

Based on the results provided by the BOOST-SWOT OSSE framework, 4DVarNet is now ready for operational uses in the experimental setup assessed until there. For DUACS-based products, providing real altimetric datasets with the DUACS interpolation as inputs for 4DVarNet ensures to improve the quality of the end product in terms of RMSE and both spatial and temporal scales resolved.\\

In terms of methodological development, we plan to apply 4DVarNet on real nadir altimetric datasets, then continue to exploit the BOOST-SWOT data challenges framework and its Observation System experiment (OSE) application (\url{https://github.com/ocean-data-challenges/2021a_SSH_mapping_OSE}) using 7 along-track nadirs in 2017 over the entire Noth-Atlantic basin. Preliminary tests suggest that applying the OSSE-trained 4DVarNet model is straightforward and also outperforms the state-of-the-art approaches on this dataset. Further developments may be considered if it is possible to learn something more from actual data. Thanks to its plug and play formulation, 4DVarNet should know many flavours on the near-future to embed complementary developments focusing on specific questions such as the learning of optimal sampling, calibration operator or multimodal synergies. On the latter, as already mentioned for the multi-regime strategy considered, it is possible to embed in the variational cost non-linear relationships between multiple state variables. The first tests carried out when adding the NATL60 sea surface temperature \citep{fablet_2022}, as a way of generalizing surface quasi-geostrophic formulation, bring an additional improvement to the already satisfying OSSE performance. These experiments will be extented in the OSE framework, by making use of L4 SST CMEMS products datasets. The plug-and-play implementation of 4DVarNet also allows for the reconstruction of unobserved variables (e.g. geostrophic velocities), including in the supervised training loss function differential operators (somehow related to PINNs, xPINNs), and also extends the variational framework to learn additional observation NN-based operator $\mathcal{H}(\x)$  with sparsity constraints, thus drawing a path for optimal sampling design, or calibration operator to filter out noisy observations \citep{febvre_2022}. Last, the develoment a stochastic 4DVarNet flavor would be valuable: because most of altimetric DUACS-related products are provided with an associated uncertainty, this leads us to  move forward a stochastic version of the algorithm for uncertainty quantification of the high resolution 4DVarNet-based products.
\fi
\paragraph{Code and data availability} The open-source 4DVarNet version of the code is available at (\url{https://doi.org/10.5281/zenodo.7186322}). The datasets is shared through the BOOST-SWOT data challenge also available on Github (\url{https://github.com/ocean-data-challenges/2020a_SSH_mapping_NATL60})

\paragraph{Video supplement} The animations corresponding to the 4DVarNet comparison to DUACS OI on the BOOST-SWOT DC  test period are given for both GULFSTREAM and OSMOSIS domains in the 4 nadirs and 1 swot + 4 nadirs configuration. They can be found on the AI CHair OceaniX Youtube channel:
\begin{itemize}
\item GF (4 nadirs): \url{https://youtube.com/shorts/QKXukB_Rd5E}
\item GF (1 swot + 4 nadirs): \url{https://youtube.com/shorts/i91Z1pMm4gY}
\item OSMOSIS (4 nadirs): \url{https://youtube.com/shorts/Pxcsd0Afco0}
\item OSMOSIS (1 swot + 4 nadirs): \url{https://youtube.com/shorts/HbVSJFtdG6Q}
\end{itemize}

\paragraph{Author contributions} Maxime Beauchamp designed the experiments, ran the analysis of the results and wrote the paper. Ronan Fablet is the principal investigator of the 4DVarNet methodology. Quentin Febvre led the developments of the 4DVarNet implementation on large domains. Hugo Georgenthum ran the experiments used in this paper. All the authors actively participate to the open-source 4DVarNet version of the code available at (https://doi.org/10.5281/zenodo.7186322)

\paragraph{Acknowledgements} This work was supported by LEFE program (LEFE MANU project IA-OAC), CNES (grant OSTST DUACS-HR) and ANR Projects Melody and OceaniX. It benefited from HPC and GPU resources from Azure (Microsoft Azure grant) and from GENCI-IDRIS (Grant 2020-101030).

\renewcommand{\bibname}{References}
\bibliographystyle{copernicus}

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MarylandReporter.com (http://marylandreporter.com/2018/03/25/opinion-time-for-supreme-county-to-limit-gerrymandering/)
Opinion: Time for Supreme Court to limit gerrymandering
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U.S. Supreme Court facade by Kevin H. with Flickr Creative Commons License
By Michael Collins
For MarylandReporter.com
The U.S. Supreme Court on Wednesday is scheduled to hear Benisek v. Lamone in what could be a landmark decision for — or against — political gerrymandering. The decision will either inflame or temper partisan passions, but in either case, will shape the nation for generations to come.
Benisek v. Lamone is similar to the Whitford v. Gill case heard by the court last October. While each case makes technical arguments about whether inferior courts applied precedents or interpreted laws properly, both cases turn on the question of whether redistricting based on party affiliation unfairly tramples a citizen's First Amendment right to freedom of association.
Maryland case
Benisek stems from the 2012 redistricting which specifically targeted Maryland's 6th Congressional District in an attempt to flip it from Republican to Democrat. In order to do that, hundreds of thousands of voters were moved into and out of the 6th District in order to make it more winnable by Democrats.
The plaintiff sued but the U.S. District court refused to hear the case, and refused to let it be heard by a three-judge panel. The Fourth Circuit also refused to hear the case. The Supreme Court (SCOTUS) ruled that they at least deserved a hearing and that led to some interesting discovery.
Wisconsin case
Whitford v. Gill was similarly intriguing but referred to state legislative districts. Republicans took control of the Wisconsin legislature for the first time in generations and controlled redistricting in 2011. They did so with a vengeance, moving district lines such that in 2012, even though Republicans legislators only got 48% of the vote, they took 60 of 99 seats in the legislature, while in 2014, they got 63 of 99 seats while winning 52% of the vote.
Maryland Democrats are equally insufferable in redrawing state legislative districts. They routinely mix single- and multi-member districts to advantage Democrats. For example, after legislative District 31 in the Pasadena area of Anne Arundel County became reliably Republican, it was split into districts 31a and 31b in order to carve a Democratic seat from a Republican district. In neighboring District 32, however, all members run at large in a majority Democratic district, thereby claiming all three delegates seats for the Democrats.
In addition, Maryland allows a 10% population difference in state districts, and most Democratic-leaning districts have fewer voters than Republican ones.
This helps give Democrats a 3-to-1 advantage in the legislature despite only a 2-to-1-voter registration advantage. That power was then used to redraw congressional districts to give Democrats a 7-to-1 advantage over Republicans.
Redistricting challenges
Federal District and Appellate Courts have played an interesting role. Federal courts typically do not hear challenges to redistricting based upon claims of discriminating against party affiliation. Since Elbridge Gerry first carved up Massachusetts for partisan advantage in 1812, courts have recognized partisan redistricting as a power granted state legislatures by the Constitution.
The need to break up Jim Crow offered a path for judicial intervention in redistricting. The Voting Rights Act of 1965 outlawed racial gerrymandering that diluted minority voting power. However, this gave legislative mapmakers both a sword and a shield for more partisanship—especially when combined with Big Data and computers.
Republicans often packed as many minorities into a district—ostensibly to comply with the Voting Rights Act. Not coincidentally, this also made districts solidly Democrat, thereby weakening Democrat power in other districts while preserving and advancing minority representation.
Democrats were equally cynical in racial gerrymandering. They packed as many minorities into the heart of Republican districts to split it—making the district or multiple districts, vulnerable to Democratic challengers. These tactics are called "packing and cracking."
Partisan courts
During his two terms in office, President Obama sought to reorient the nation by shifting the balance of power in appellate courts. The traditionally conservative Fourth Circuit Court of Appeals, which hears cases from Maryland, Virginia, and North Carolina, was packed with liberal jurists. Thus, the Fourth Circuit became more aggressive in redistricting cases—but only in Republican controlled states.
The Fourth Circuit court, for example, ordered Virginia and North Carolina to redraw districts to help Democrats. At the same time, it refused to hear the Benisek case or challenges to Maryland's Third Congressional District, which has been called the most gerrymandered district in the nation.
More recently, the Pennsylvania Supreme Court made up of five Democrats and two Republicans—all elected—voted along party lines to order that Pennsylvania's congressional map be redrawn to the advantage of Democrats.
With such judicial activism in left-leaning courts, judicial restraint by conservative jurists is a suicide pact. While conservative jurists defer to gerrymandering in states like Maryland, liberal jurists show no such restraint and order new district boundaries in states like North Carolina to help out Democrats—heads, we win, tails, you lose.
SCOTUS has a rare opportunity to create a blueprint for the nation to follow on both federal and state legislative districts. The court can use the Whitford and Benisek cases to denounce partisan gerrymandering by Republicans and Democrats in both state legislative districts and congressional districts.
By setting strict parameters on compactness, contiguousness, minimal allowable differences in district population, etc., the court can erect guardrails to help steer our politics away from the gutters and more to the center—perhaps restoring some civility to our body politic.
While the court rightfully should be wary that such rules would enmesh the courts in future redistricting fights, it needs to be clear-eyed—they are already there. Kicking the can on this issue is a fool's errand. The Whitford and Benisek cases give SCOTUS the choice either to act now, or react forever. The choice is clear.
Michael Collins can be reached at michaelcollins.capital@gmail.com
"Lawsuit forces Maryland Democrats to acknowledge the obvious: Redistricting was motivated by politics"
http://www.baltimoresun.com/news/maryland/politics/bs-md-redistricting-case-20170601-story.html
Benisek v. Lamone
https://www.brennancenter.org/legal-work/benisek-v-lamone-amicus-brief
Whitford v. Gill
https://www.brennancenter.org/legal-work/whitford-v-gill
Virginia Gerrymandering
https://www.washingtonpost.com/local/virginia-politics/circuit-judge-tosses-virginia-gerrymandering-case-says-districts-debatable/2017/03/31/21b6fc40-1646-11e7-ada0-1489b735b3a3_story.html?utm_term=.527468fa888c
North Carolina Gerrymandering
http://www.richmond.com/news/virginia/government-politics/judges-impose-new-va-congressional-map-redrawing-rd-th-districts/article_0ad5053b-6818-5d7e-b96e-c9ce02ad45cb.html
Pennsylvania Gerrymandering
https://www.ydr.com/story/news/2018/02/22/pennsylvania-gop-congressmen-file-challenge-district-map/364881002/
Opinion: School funding workgroup is stacked against counties
The Blueprint for Maryland's Future Funding Formula Workgroup, has now been created to make recommendations for the distribution of funds by local school districts and between state and local governments. This workgroup will also make recommendations for specific funding formulas. I am greatly troubled by the make-up of the workgroup, which has only two members to represent interests of the county governments that pay the local costs of schools.
State Roundup, July 18, 2019
State Roundup Archives
MarylandReporter.com is a daily news website produced by journalists committed to making state government as open, transparent, accountable and responsive as possible.
We depend on your support. A generous gift in any amount helps us continue to bring you this service.
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Disaster Response Haiti Project #53396
Haiti Earthquake Relief Fund
by GlobalGiving
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Your donation is fueling response efforts across Haiti
By Victoria Mendez - Disaster Response Fellow
Photo: CORE
Since August 2021, the people of Haiti have been grappling with the impacts of a massive earthquake that was quickly followed by a devastating tropical storm. At the same time, political unrest and rising gang violence have significantly challenged response efforts. However, with generous support from people like you, GlobalGiving's trusted nonprofit partners in Haiti are at the forefront of providing critical humanitarian assistance to those in need.
Your donation to the Haiti Earthquake Relief Fund has already supported 15 emergency grants to 10 community-led nonprofit organizations facing the challenges of earthquake recovery head-on. Here is what Marie-Rose Romain Murphy, an incredible nonprofit leader at Economic Stimulus Projects for Work and Action (ESPWA) shared about the impact of your donation:
"The needs are overwhelming. I am trying hard not to think of all the devastation to focus on what we can do. We're working at all hours and I have to remind people that we need to sleep. THANK YOU!!"
We are excited to share a short description of the critical work you've supported since our last update in early September:
Foundation Tous Ensemble has donated more than 100 prostheses to people injured by the earthquake. The organization operates the only rehabilitation clinic in Les Cayes that specializes in physical and occupational therapy, fabrication of prosthetics, and orthotic and rehabilitation services.
Economic Stimulus Projects for Work and Action (ESPWA) is a Haitian-led organization working in collaboration with the Haiti Community Foundation. Together, the organizations continue to provide emergency relief to communities in the Grand'Anse department.
Lambi Fund of Haiti is another Haitian-led organization that has been a GlobalGiving partner for more than 10 years! The fund has been supporting its network of more than 60 community-based organizations with response strategies and critical resources.
IsraAID immediately activated its Emergency Response Team following the earthquake. The team includes many of the organization's former Haitian team members who were critical in IsraAID's response to Hurricane Matthew in 2016. It has partnered with Les Amis d'Haiti - LADHA project Medishare for Haiti to distribute hygiene and resilience kits.
Agua Pura Para El Pueblo has two teams of youth volunteer leaders in southern Haiti who are reaching several remote villages with emergency supplies and water purification materials. The teams will also provide training in safe water practices and distribute clean water storage containers.
CORE (Community Organized Relief Effort) is leveraging its expertise and experience in responding to the 2010 Haiti earthquake. Its teams are on the ground in the hardest-hit areas of Grand'Anse, Sud, and Nippes. Its local Haitian leadership allows the organization to respond most effectively to the most pressing needs on the ground as they evolve throughout its response.
Airlink Inc - is supporting various humanitarian organizations with airfreight service for emergency cargo into Port-au-Prince airport.
Build Health International - has partnered with Health Equity International and Partners In Health to repair and construct additional emergency operating rooms in the St. Boniface Hospital. Their team is also facilitating shipments of medical supplies, deploying surgeons, and ensuring the existing medical facilities are safe to use.
Health Equity International - In addition to leading the region's COVID-19 response, HEI is supplying the St. Boniface hospital in Fond des Blancs with essential mediations, medical supplies, and medical transportation for earthquake survivors.
Partners in Health (PIH)- is collaborating with multiple organizations including Zanmi Lasante, PIH's sister organization in Haiti, to assess earthquake damage and meet hospitals' needs across southern Haiti. Last month, PIH deepened its support of mobile clinics in hard-to-reach areas, where many survivors had yet to receive medical care.
Earthquake recovery in Haiti is far from over. Your support of the Haiti Earthquake Relief Fund powers community-led responses, and it means local leaders like Marie-Rose Romain Murphy get the resources they need to rebuild their communities from within.
Thank you again for your generosity. I will be back in your inbox in early 2022 with more updates on the impact of your donation!
Victoria + the GlobalGiving Team
An update on your donation to the Haiti Earthquake Relief Fund
I want to start by saying thank you. In times of tragic disasters like the earthquake in Haiti last month, our team becomes focused on getting funds into the hands of locally-led nonprofits as fast as possible. You and hundreds of other supporters of the Haiti Earthquake Relief Fund make this possible. For that, we cannot thank you enough. Within one week of the earthquake, we sent emergency grants to five incredible partner organizations on the front lines of delivering critical aid to communities across southern Haiti.
Here is a brief overview of our vetted nonprofit partners in Haiti that have received our first round of emergency grants:
Economic Stimulus Projects for Work and Action (ESPWA) is a Haitian-led organization that uses a strong network of community leaders to respond to SOS calls and provide water, food, medical supplies, and temporary housing.
Lambi Fund of Haiti is another Haitian-led organization providing water, food, and other essential supplies to communities in the South of Haiti. The Lambi Fund of Haiti has been a trusted GlobalGiving partner since 2008 and was also heavily involved in responding to the 2010 Haiti earthquake!
Health Equity International runs the St. Boniface hospital in Fond des Blancs, a hospital run and led by Haitian people. Thankfully, the hospital did not experience severe damage from the earthquake and could treat critically injured patients transferred to the hospital from surrounding cities and towns right away.
Build Health International is rapidly deploying healthcare infrastructure in areas most affected by the earthquake. The nonprofit's primarily Haitian team of engineers, architects, logisticians, contractors, electricians, and plumbers have mobilized to ensure hospitals' structural safety and send supplies, specialists, and surgeons to people as quickly as possible.
Partners in Health sent a team of doctors to Les Cayes to treat patients and assess surgical needs. The doctors and nurses are part of a team of more than 6,500 people—99% Haitian. With a history of working in Haiti dating back to 1986, the organization is committed to long-term disaster recovery after this recent earthquake.
In the coming months, GlobalGiving will continue to send more specific updates, photos, and stories from the organizations powering recovery from within their communities.
Thank you again for your generosity and for choosing to support community-led disaster relief and recovery.
Victoria Méndez + the GlobalGiving Team
Location: Washington, D.C. - USA
Website: https://www.globalgiving.org
Twitter: @GlobalGiving
About GlobalGiving's Disaster Response
When a disaster strikes, recovery efforts led by people who live and work in affected communities are often overlooked and underfunded. GlobalGiving is changing this reality. Since 2004, we've been shifting decision-making power to crises-affected communities through trust-based grantmaking and support.
We make it easy, quick, and safe to support people on the ground who understand needs in their communities better than anyone else.
They were there long before the news cameras arrived, and they'll be there long after the cameras leave. They know how to make their communities more resilient to future disasters, and they're already hard at work. GlobalGiving puts donations and grants directly into their hands. Because the status quo—which gives the vast majority of funding to a few large organizations—doesn't make sense.
Questions about this project? Contact us
$521,219 raised of $3,000,000 goal
2,633 donations
$2,478,781 to go
emergency earthquake relief and long-term recovery support
Monthly giving is as easy, safe, and as inexpensive as a Netflix subscription. Start a monthly donation to GlobalGiving today and get matched at 100%. Terms and conditions apply.


text:
Current: Reform of the NHS procurement rules: Long-Term Plan, or long-term pipe dream?Reform of the NHS procurement rules: Long-Term Plan, or long-term pipe dream?
Reform of the NHS procurement rules: Long-Term Plan, or long-term pipe dream?
Published on 1 October 2019
The ten-year NHS Long Term Plan sets out some ambitious reform proposals for NHS procurement. But with Brexit dominating parliamentary time and sapping resource, how likely are these proposals to materialise? And when?
The NHS Long Term Plan puts forward two major proposals for reforming the procurement of NHS health services.
The first is the proposal to repeal what are sometimes referred to as the "section 75 regulations" – so called because they were made under section 75 of the Health and Social Care Act 2012. These regulations, the National Health Service (Procurement, Patient Choice and Competition (No. 2) Regulations 2013 were made as part of the Lansley reforms and have always been controversial.
Their aim is to ensure that NHS commissioners (they apply only to commissioners and not to providers) commission health services with a view to securing the needs of service users and at the same time improving quality and efficiency, including through integration. They are also aimed at driving value for money, in part by requiring commissioners to allow providers to express interest in providing services where appropriate. The controversy has stemmed from what many see as a set of rules setting up a slippery slope towards a market-driven, fragmented NHS increasingly open to the private sector.
The second is the proposal to remove health services from the scope of the broader public procurement regime (in England, the Public Contracts Regulations 2015). This would be the case for whoever happened to be buying a health service – certainly a commissioner, and also potentially a provider. Currently, EU law requires health services to be advertised whenever the value of a contract exceeds a threshold – currently just over £600,000. Thereafter, fair and proportionate award procedures must follow where expressions of interest are received. These requirements apply in all EU member countries.
These are ambitious plans and, as and when implemented, would radically alter the shape of NHS procurement and the statutory framework in which it operates.
Towards a new horizon: a new "targeted" NHS Bill
At last week's joint board meeting of NHS England and Improvement saw the publication of a set of recommendations to government and Parliament for an NHS Bill. Its purpose is to foster collaboration and facilitate the actual implementation of the NHS Long Term Plan. Key changes include the "scrapping" of section 75 of the 2012 Act with its "burdensome and wasteful" procurement processes.
There is a desire to find a better name for the so-called "best value" test, which would replace the "section 75" regime, and to set out details of how the new procurement regime would be intended to operate. It is understood that the new approach will be subject to a separate dedicated public consultation, at the same time as the Bill is published.
But how open is it to any government to carry these plans into fruition?
That is a big question, because the ability to do so depends - in part - on the outcome of the UK's tortuous path out of the EU.
As a member state of the EU, we are subject to EU public procurement law. Our domestic procurement regime – the Public Contracts Regulations – implement that law, and we must keep them in place for use as long as we remain part of the EU. Once out of the EU, it is not actually certain whether we will have new-found flexibility to amend our procurement regime; any ability to do that (and to extract health services from its scope as part of the job) will depend on the nature of any withdrawal agreement which the UK might strike with the EU.
Here's why.
If we were to leave with a withdrawal agreement this would be likely to entail a transition period during which time we would continue to be subject to the EU procurement regime. In that event, we would not be at liberty to remove health services from the ambit of the procurement regime, because EU law requires them to be part of that regime. After the end of any transition period, whether removal might be possible would depend on the substance of any future trade agreement we might subsequently strike with the EU. A deal might require us to have a procurement regime that mirrored that of the EU, including covering health services in much the same way as today.
If, by contrast, we leave the EU without a deal, then from exit day we will (technically) be in a position to extract health services from the ambit of our procurement regime. However, the precise mechanics and timing of this would be entirely dependent on the policy of the Government of the day and the will of Parliament.
What we do know (because we already have procurement policy notes from the current Government describing it) is that in the immediate term there are no plans for significant changes to the Public Contracts Regulations. Some minor amendments will be necessary from exit day but those are simply to ensure a correctly-functioning statute book in the absence of any more sweeping changes in the longer term. Probably the biggest immediate change in a no-deal situation is that contract notices (OJEU advertisements) and other forms of procurement notice would no longer be published in the Official Journal of the European Union; instead we would resort to a UK e-notification service. We understand that the content of such notices would remain similar to their present form.
The current "section 75" regulations are not the product of EU law, and repealing them does not depend at all on our status vis-à-vis the EU, nor on the detail of any subsequent trading relationship with the EU. Repealing them (which might be very welcome in many quarters, as they do not exactly make for an easy fit with the public procurement rules) will depend only on the preferences of the government of the day. Unlike the Public Contracts Regulations – which are there to promote cross-border trade and competition in an EU of which we will no longer be part – the section 75 regulations aim to foster the interests of patients by driving up quality, efficiency and value in healthcare.
There are many who might say that the sooner the time can be found to rid the statute-book of those regulations, thereby taking the focus away from competition, the better.
If you would like to find out more about this topic or you if need legal advice, please contact our legal experts on health and care, procurement and procurement disputes.
Healthcare regulation under review: when is statutory regulation appropriate?
The Department of Health and Social Care is consulting on how the powers to introduce and remove professions from regulation might be used in the future...
Latest judgment in Worcestershire case concerning section 117 duties and the meaning of ordinarily resident
In this article, we consider the Court of Appeal's decision and its implications.
Contact us or register for updates to stay up-to-date
Procurement disputes


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Q: AWS CloudFormation: Apply RDS DB Snapshot without recreation DB Instance Is it possible to apply DB Snapshot to existing DB using CloudFormation in one step?
I know it is possible if you run your CloudFormation twice - first time to rename existing resource and second time reverting the name back and set DBSnapshotIdentifier: !Ref DBSnapshot variable.
But it take to much time, is it possible to do it in one step?
The main goal - update database using snapshot from another database (for example apply STG db to DEV to have similar data)

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TV Shows – Movie News – Renewed Cancelled TV Series
Home TV Shows The War of the Worlds
The War of the Worlds is Becoming a TV Series
2:07 AM - 10/14/2016
Aliens invading the Earth has always been a popular subject matter used by both Hollywood and TV producers. Sometimes the aliens were the bad guys and sometimes the friendly ones. Some great movies were made like Alien franchise while some projects turned out to be complete flukes.
There are some TV series reviving the genre on the small screen right now Colony and The X-Files. Another one to draw attention is MTV's new project, the TV adaptation of The War of the Worlds, a sci-fi novel by H.G. Wells. Andrew Cochran will be writing the script for the series which will take place in a modern setting. Some of you might know him from another MTV TV series, Teen Wolf.
Apart from that, there really is not much news yet about the TV series whether about the premiere date or the members of the cast. We will keep you udpated about The War of the Worlds TV series, follow us on social media to not to miss anything!
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Mr. Bernanke gets his fair share of criticism for pretty much every move he makes, some of which is warranted, others not as much. In his defense, he finds himself in a unique place in history, having made the decision that what a sick economy needed in order to avoid catastrophe was an ultra low interest rate environment and enough government provided liquidity to weather a recessionary storm until it could grow itself back to health. Several years into this process, the Fed is now working out the logistics of their exit strategy.
That's one of the unique things about the financial markets, there is no definitive user's manual with step by step instructions, like something you buy from IKEA. Whether you are Ben Bernanke trying to monitor the U.S. economy or an individual trying to keep a handle on your investment portfolio, everyone is forced to navigate according to their own philosophy, strategy, and instinct. Of course, even if there were a manual, there's no guarantee people would use it. I once charred a meatloaf that I was supposed to be broiling because I didn't read the manual which told me to leave the oven door ajar so as to avoid having the oven automatically entering in to self-cleaning mode.
In his ruckus causing speech in June, Bernanke revealed that the Fed will continue to let the economic data be the primary driver of when they will begin to "taper" (the next financial buzzword of which you will grow weary) their asset purchases and eventually raise interest rates. His statement characterized the U.S. economy as growing at a moderate pace with labor market conditions showing further improvement. The current unemployment rate sits at 7.6% as the following chart shows. The Fed believes that we are on track to have that drop to 6.5% - 6.8% by the end of 2014.
Last year at this time, the unemployment rate was at 8.2%, so the current 7.6% is obviously an improvement but the downside is the percentage of people participating in the labor force dropped from 63.8% to 63.4%.
The pace of these moves will accelerate if economic conditions prove better than expected or will be extended if economic conditions deteriorate.
Federal Reserve assistance to one that is capable of standing on its own. At Castle, we have begun the process of reducing some of the interest rate sensitivity in our portfolios. We're not giving up on holding bonds, just reducing the amount of bonds (particularly bond funds) that are likely to react negatively when interest rates rise and increasing our emphasis on owning ladders of individual bonds.
The following chart, which is not a performance chart, shows how some classes of bonds correlate to U. S. Treasuries. As conditions warrant, we want to hold more of the kinds of bonds that are less correlated to Treasuries and thereby less interest rate sensitive.
So far this year, the economy has been a mixed bag with some areas such as housing showing a welcome improvement while other areas like manufacturing showing weakness. Overall the final GDP number for the first quarter came in at 1.8%. While we are glad this number is positive, showing growth, the 20 year average is 2.5% demonstrating the continued sluggishness of our economic recovery.
At Castle, we will continue to be true to our philosophy of managing diversified portfolios while always keeping our clients' best interests in mind. In closing, we are including a possible timetable, again from our friends at Capital Economics, which specifically shows what actions the Fed might take during the next couple of years. This is not a prediction, the actual events are likely to be different but it does give you a general idea of what a path toward normal interest rates might look like. Thanks again for allowing us to be your partners.
July: QE3 continues – Fed buys $85bn of assets a month.
September: Start to taper QE3 – Fed buys $75bn of assets a month.
October: QE3 tapering continues – Fed buys $65bn of assets a month.
December: QE3 tapering continues – Fed buys $55bn of assets a month.
January: QE3 tapering continues – Fed buys $45bn of assets a month.
March: QE3 tapering continues – Fed buys $30bn of assets a month.
April: QE3 tapering continues – Fed buys $15bn of assets a month.
June: QE3 ends. Language in statement says low rates for a "considerable time".
September: Fed stops reinvesting maturing assets.
October: Language in statement changed to Fed will be "patient" in removing accommodation.
January: Language in statement changed to accommodation will be removed at a "measured" pace.
March: Rates raised to 0.50%. Fed implements measures to drain liquidity from banking system.
June: Rates raised to 0.75%.
September: Rates raised to 1.00%.
January: Rates raised to 1.25%.
March: Rates raised to 1.50%.
April: Rates raised to 1.75%.
June: Rates raised to 2.00%.
July: Rates raised to 2.25%.
September: Rates raised to 2.50%.
October: Rates raised to 2.75%.
December: Rates raised to 3.00%.
March: Rates raised to 3.25%. If Fed decides to sell Treasuries it may start now.
June: Rates raised to 3.50%.
September: Rates raised to 3.75%.
December: Rates raised to 4.00%.


text:
2022-23 Statistics (.pdf)
Coach Todd Chapman
Live Video (Home)
Jayhawks Tennis Center
🎾 Kansas Set to Compete in the 2022 NCAA Tournament
LAWRENCE, Kan. – The No. 26 ranked Kansas Jayhawks are set to square off against No. 24 UC Santa Barbara Friday, May 6, at 12 p.m. CT, in the opening round of the NCAA Tournament. This is the 12th time in program history Kansas has made it to the NCAA tournament, and the fifth time under Head Coach Todd Chapman. The match will begin with doubles, followed by singles play.
""I think this time of year, you've put in a season's worth of work and now is the time you hopefully get to go out and enjoy the fruits of your labor. We had a great practice yesterday, the weather was great and I think the energy is good for the girls. I feel like we're in a good spot.""
Todd Chapman
The doubles pairing of Sonia Smagina and Raphaelle Lacasse has been a formidable force all season, coming in at No. 79 in the most recent ITA doubles rankings. They are currently 1-1, with one win over a nationally ranked opponent. Redshirt freshman Maria Titova comes in at #No. 52 in the latest ITA singles rankings, with a record of 21-8 on the year.
Titova was also selected for NCAA Singles Championships, the first Kansas freshman to compete in that event since 1996. Along with Titova being named a first-team All-Big 12 selection and to the Big 12 All-Freshman Team, Malkia Ngounoue was named a second-team All-Big 12 selection.
""This is what it's about. Once again, you work hard all year, especially after missing the tournament last year for the first time in a while. There's a lot of joy in the process right now and I'm excited for what we have in front of us. I think that's the mindset we have to have, just to go out and lay it all on the line.""
Fans and supporters will be able to stream the match from the TennisONE app from a computer or cellular device.


Quite a few years ago we cross stitched the Beatitudes. They hang right outside my bathroom and I look at them each day. Recently, Tracy at Its A T-Sweets Day posted an article about her version of the Beatitudes. I made a comment about our little pillows and she said she would like to see them. Since I cannot post the pictures on her blog or our blog without a post . . . you all get to see them!
Our friend, Karen Olson, finished these cute samplers into pillows for us when we owned a cross stitch shop. They have made it through two moves and survived. Our little beatitude pillows are housed in a wooden shadow box that has a few other nicknack pieces in it as well.
They do make me smile each time I pass them.
If you get a chance, check out Tracy's version in the link I shared above. I'd love to hear if you have incorporated the Beatitudes in any of your quilting projects. Let us know in the comments below.
Today is Day 2 in the Put A Little Love in Your Quilt blog hop. Here are today's participates – check them out!
Oh these are so beautiful! All of your stitching and then making them into pillows and the display just make me smile😊❤ Thank you so much for sharing them!
Next story I Love Thursday #19 – On Wednesday!