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%% Template article for Elsevier's document class `elsarticle'
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\documentclass[preprint,12pt,authoryear]{elsarticle}

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 \newcommand{\vectornorm}[1]{\left|\left|#1\right|\right|}
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%\newcommand{\vectornorm}[1]{\left|\left|#1\right|\right|}
\newcommand{\dif}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\diff}[2]{\frac{\partial^2 #1}{\partial {#2}^2}}
\newcommand{\odif}[2]{\frac{d #1}{d #2}} 
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\journal{Ocean Modellling}

\begin{document}

\begin{frontmatter}

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\title{Modeling Surface Currents in the Eastern Levantine Mediterranenan Using Surface Drifters and Satellite Altimetry Data}

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\author{Leila Issa, Julien Brajard, Pierre-Marie Poulain, Laurent Mortier, Dan Hayes, Milad Fakhri}

\address{}

\begin{abstract}
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\end{abstract}

\begin{keyword}
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\end{frontmatter}

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\section{Introduction}
\label{}
\section{Data}
\section{Method}
\begin{itemize} 

\item $\vec{U_0}=\vec{U}_{b}+\vec{\delta U},$ $\vec{U}_b=\textcolor{red}{\vec{U}_{alt}}$ and $\vec{r}=\vec{r}^{\,b}+\vec{\delta r}$
\item Model and Linear tangent (forward Euler for the numerical integration, and bilinear Lagrange interpolation in space for instance):  
\small{
\begin{align} \notag
&\vec{r}^{\,b}_i(k\delta t)=\vec{r}^{\,b}_i((k-1)\delta t)+2\delta t \, interp(\vec{U}^b, \vec{r}^{\,b}_i((k-1)\delta t)),\,\,\,\,\, \text{\textcolor{blue}{model}}  \\  \notag
&\vec{\delta r}_i(k\delta t) = \vec{\delta r}_i((k-1)\delta t) + 2 \delta t \, \{ interp(\textcolor{green}{\vec{\delta U}},\vec{r}^{\,b}_i((k-1)\delta t)) \\ \notag 
&+ \vec{\delta {r}}_i((k-1)\delta t) \cdot \partial _{(x,y)} interp (\vec{U}^b,\vec{r}^{\,b}_i((k-1)\delta t))\}  \,\,\, \text{\textcolor{blue}{tangent}} \notag
\end{align}
initialized with observations, for $k=1,2,3, \cdots $ and $i=1, \cdots N_{drift}$ 
}
%\vec{r}(0)= \vec{r}^{o}(0)$
\item Perform sequences of optimization
\[\mathcal{J}(\textcolor{green}{\delta \vec{U}})= \frac{1}{2} \left \{ \sum _i \sum_{m=1}^{P} \vectornorm{\vec{r}^{\,b}_i+\delta \vec{r}_i(\textcolor{green}{\delta \vec{U}}) -\vec{r}_i^{\,o}(m\Delta t) }^2 +\vectornorm{ \textcolor{green}{\vec{\delta U}} }^2_{\bo{B}}  \right\}   \]
$P$ depends on $T_{L}/\Delta t$. $T_L$ characteristic of autocorrelation of drifters (typical 1-3 days) 

%+\sum_{i,m=1}^{m=P} \vectornorm{\vec{r}^{\,b}+\delta \vec{r}(\textcolor{green}{\delta \vec{u}}) -\vec{r}_i^{\,o}(m\Delta t) }^2   \right]   \]
\end{itemize}




\section{Twin Experiments}
\section{Experiment with Real Data}
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