%% 
%% Copyright 2007, 2008, 2009 Elsevier Ltd
%% 
%% This file is part of the 'Elsarticle Bundle'.
%% ---------------------------------------------
%% 
%% It may be distributed under the conditions of the LaTeX Project Public
%% License, either version 1.2 of this license or (at your option) any
%% later version.  The latest version of this license is in
%%    http://www.latex-project.org/lppl.txt
%% and version 1.2 or later is part of all distributions of LaTeX
%% version 1999/12/01 or later.
%% 
%% The list of all files belonging to the 'Elsarticle Bundle' is
%% given in the file `manifest.txt'.
%% 
%% Template article for Elsevier's document class `elsarticle'
%% with harvard style bibliographic references
%% SP 2008/03/01

\documentclass[preprint,12pt,authoryear]{elsarticle}

%% Use the option review to obtain double line spacing
%% \documentclass[authoryear,preprint,review,12pt]{elsarticle}

%% Use the options 1p,twocolumn; 3p; 3p,twocolumn; 5p; or 5p,twocolumn
%% for a journal layout:
%% \documentclass[final,1p,times,authoryear]{elsarticle}
%% \documentclass[final,1p,times,twocolumn,authoryear]{elsarticle}
%% \documentclass[final,3p,times,authoryear]{elsarticle}
%% \documentclass[final,3p,times,twocolumn,authoryear]{elsarticle}
%% \documentclass[final,5p,times,authoryear]{elsarticle}
%% \documentclass[final,5p,times,twocolumn,authoryear]{elsarticle}

%% For including figures, graphicx.sty has been loaded in
%% elsarticle.cls. If you prefer to use the old commands
%% please give \usepackage{epsfig}

%% The amssymb package provides various useful mathematical symbols
\usepackage{amssymb}
\usepackage{multimedia}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{multimedia}
\usepackage{graphics}
\usepackage{bbm}
\usepackage{pstricks}
\usepackage{amsthm}
 \usepackage{graphics}
 \usepackage{pgf}
 \usepackage{wrapfig} 
 \usepackage{mathtools} 
\usepackage{caption}
\usepackage{subcaption}
%\usepackage{hyperref}



 
 
 
 
 
 \newcommand{\vectornorm}[1]{\left|\left|#1\right|\right|}
 \newcommand{\bo}[1]{\mathbf{#1}}
%\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}} 
\newcommand{\odiff}[2]{\frac{d^2 #1}{d #2^2}}

%% The amsthm package provides extended theorem environments
%% \usepackage{amsthm}

%% The lineno packages adds line numbers. Start line numbering with
%% \begin{linenumbers}, end it with \end{linenumbers}. Or switch it on
%% for the whole article with \linenumbers.
%% \usepackage{lineno}

\journal{Ocean Modellling}

\begin{document}

\begin{frontmatter}

%% Title, authors and addresses

%% use the tnoteref command within \title for footnotes;
%% use the tnotetext command for theassociated footnote;
%% use the fnref command within \author or \address for footnotes;
%% use the fntext command for theassociated footnote;
%% use the corref command within \author for corresponding author footnotes;
%% use the cortext command for theassociated footnote;
%% use the ead command for the email address,
%% and the form \ead[url] for the home page:
%% \title{Title\tnoteref{label1}}
%% \tnotetext[label1]{}
%% \author{Name\corref{cor1}\fnref{label2}}
%% \ead{email address}
%% \ead[url]{home page}
%% \fntext[label2]{}
%% \cortext[cor1]{}
%% \address{Address\fnref{label3}}
%% \fntext[label3]{}

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

%% use optional labels to link authors explicitly to addresses:
%% \author[label1,label2]{}
%% \address[label1]{}
%% \address[label2]{}

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

\address{}

\begin{abstract}
%% Text of abstract

\end{abstract}

\begin{keyword}
%% keywords here, in the form: keyword \sep keyword

%% PACS codes here, in the form: \PACS code \sep code

%% MSC codes here, in the form: \MSC code \sep code
%% or \MSC[2008] code \sep code (2000 is the default)

\end{keyword}

\end{frontmatter}

%% \linenumbers

%% main text
\section{Introduction}
\label{}
Recurrent marine pollution like the ones observed near the coastal regions of Lebanon is a major threat to the marine environment and halieutic resources. Polluting agents do not only have a immediate local effect but they are also transported through ocean currents to deep waters far away from the coast thereby having a long term, large scale effect. Modeling ocean currents is thus a major topic that mobilizes multidisciplinary researchers.  

The objective of this project is to combine two types of observations (drifters and satellite data) in order to reconstruct mesoscale features of the surface currents in the Levantine Mediterranean including coastal regions. The figure below shows positions of drifters and altimetry data in the Eastern Mediterranean from the 27th to the 29th of November 2009: the altimetric data gives the velocity field in the whole basin with a resolution of $10$ km smoothing considerably some mesoscale features with errors especially near the coastal regions. The buoys on the other hand allow a better reconstruction of the velocity field but only along their trajectories. These two types of data are therefore complimentary and we can then formulate the problem as follows: find the optimal velocity field from these two sources of information.   

The mathematical problem consists of ``inverting'' the Lagrangian trajectories (Drifters positions) to correct the Eulerian velocity field given my the altimetric data. The literature on the subject is abundant: ranging from optimal interpolation  \cite{ozgokmen2003assimilation}, to statistical approaches based on extended Kalman filter methods \cite{salman2006method}, to variational techniques \cite{kamachi1995continuous}, \cite{nodet2006variational}, \cite{taillandier2006variational}.
We propose to use a variational assimilation technique where we correct the Eulerian velocity field by minimizing the distance between a model solution and observations of the buoys. We use the incremental approach described in \cite{courtier1994strategy}, \cite{dimet2010variational}. The advantage of this method is that while being midway in complexity between OI (lower) and Kalman filter (higher), it permits a simple mathematical formulation of the dynamical constraints we wish to impose.



\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.5]{./fig/RealvsSimulatedTraj.pdf}
\vspace{-30mm}
\caption{Real CNRS observations of 3 drifters starting Aug 28 2013 -x , vs simulated -- with aviso same date. On top of averaged Aviso for 6 days starting aug 28  }
\end{center}
\end{figure}



\section{Data}
All the data detailed in this section were extracted for two target period : first from 25 August 2009 to 3 September 2009, and second from 28 August 2013 to 4 September 2013.
\subsection {Altimetry data}
Geostrophic surface velocity fields used as a background in the study were produced by Ssalt/\textit{Duacs} and distributed by \textit{Aviso}. Altimetric mission used were  Saral, Cryosat-2, Jason-1\&2. Data were mapped daily at a resolution of 1/8$^o$ over Mediterranean Sea. Data were linearly interpolated every hour at the advection model time step.

\subsection{\label{sec:drifters}Drifters data}
Drifters were deployed at the two target periods (2 drifters were selected for the first period in 2009 and 3 in the seconde period in 2013). Table~\ref{tab:drifters} present a summary of the 5 drifters used in this study. Drifter models were SVP with a drog at a depth of 15m. Drifter positions were fileterd with a low-pass filter in order to remove high-frequency current component especially inertial currents. The final time series were sampled every 6h. A more complete description of the drifters and the data processing procedure can be found in~\citet{poulain2009}.
\begin{table}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Deploy Date & Lat & Lon & Last Date & Lat & Lon \\
\hline
29 Jul. 2009 & 31.90 & 34.42 & 28 Oct. 2009 & 34.1 & 31.77 \\
\hline
03 Aug. 2009 & 32.59 & 32.63 & 26 Dec. 2009 & 32.92 & 34.28 \\
\hline
27 Aug. 2013 & 33.28 & 34.95 & 22 Sep. 2013 & 36.77 & 35.94 \\
\hline
27 Aug. 2013 & 33.28 & 34.98 & 04 Sep. 2013 & 34.13 & 35.64 \\
\hline
27. Aug. 2013 & 33.28 & 35.03 & 17 Sep. 2013 & 34.88 & 35.88 \\
\hline
\end{tabular}
\caption{\label{tab:drifters} List of drifters used to illustrate the methodology presented in this study, 
2 drifters deployed in 2009 (results were detailed in section~\ref{sec:cyprus}) and 
3 drifter were deployed in 2013 (results were detailed in sections~\ref{sec:lebanon})}
\end{table}

\subsection{Wind Data}
ECMW ERA-Interim wind products~\citep{Dee2011} were extracted in order to estimate wind-driven currents. Wind velocities closest to the surface (10 m) were extracted at a resolution of 1/8$^o$ at the same grid point as the \textit{Aviso} data. The data were resampled on a hourly time step.

\subsection {Model data}
Model data of surface velocity fields were used to calibrate the assimilation method presented in section~\ref{sec:method}. The model selected was the CYCOFOS-CYCOM high resolution model~\citep{zodiatis2003} that covers the North-East Levantin Bassin 
(31$^o$ 30’E - 36$^o$ 13’E  and 33$^o$ 30’N – 36$^o$ 55’N). The model forecast were used without assimilation and were reinteroplated on a 1.8$^o$ grid point with an time step of one hour. The model forecast used for calibration purpose on September 2013.
%Wind + Dan


\section{\label{sec:method}Method}

%%%%%%%%%%%%%%%%
\subsection{Statement of the problem}

We consider $N_f$ Lagrangian drifters released at time $t=0$ at various locations. These drifters provide their positions each $\Delta t$, over a period $[0,T_f]$.
Our objective is to determine an estimate of the two-dimensional Eulerian velocity field  \begin{equation}\notag
\mathbf{u}(x,y,t)=(u(x,y,t),v(x,y,t)) 
\end{equation}
characterized by a length scale $R$ [Refs], given observations of the drifters' positions 
\begin{equation}
\bo{r}^{obs}_i(n\Delta t), \,\,\, i=1,2, \cdots, N_f, \,\,\, n=1, 2, \cdots N, \,\,\, \text{where}\,\,\, N \Delta t= T_f.
\end{equation}
The velocity shall be estimated on a specified grid with resolution $\Delta x,$ in the time frame $[0, T_f].$ 

The estimation is done following a variational assimilation approach [Refs], whereby the first guessed velocity, or background $\bo{u_b}$, is corrected by matching the observations with a model that simulates the drifters' trajectories. This correction is obtained using a sliding time window of size $T_w$, where we assume $\Delta t<T_w \leq T_L,$ and where $T_L$ is the Lagrangian time scale associated with the drifters in the concerned region [Refs]. The background field is considered to be the sum of a geostrophic component (provided by altimetry) on which we impose a divergence free constraint, and a velocity due the wind [Refs]. The details of this procedure are given in Section 3.3.




\subsection{Linearized model for Lagrangian data}



The position of a specific drifter $\mathbf{r}(t)=(x(t),y(t))$ is the solution of the non-linear advection equation
\begin{equation} \label{advection}
 \odif{\mathbf{r}}{t}=\mathbf{u}(\mathbf{r}(t),t), \,\,\,\,\bo{r}(0)=\bo{r}_0, \mathbf{u}(x,y,0)=\bo{u}_0.
 \end{equation}
This equation is integrated numerically using an Euler scheme for example. Since the drifters positions may not coincide with the Eulerian velocity's grid points, a spatial interpolation of $\mathbf{u}$ to these positions is also needed. 


The observation operator, denoted it schematically by  $\bo{r}=\mathcal{M} (\bo{u}, \bo{r}),$ consists then of numerical advection and interpolation, and it is given by  
\begin{equation} \label{euler_advection}
\bo{r}(k\delta t)=\bo{r}((k-1)\delta t)+\delta t \, interp(\bo{u}((k-1)\delta t), \bo{r}((k-1)\delta t)), \,\,\,\,\, k=1,2, \cdots
\end{equation}
%Let us denote by $\bo{r}_k = (x_k ,y_k)$ the position of the drifter at time $t_k=k\delta t$ , $\mathbf{u}_k$ the Eulerian velocity at position $\mathbf{r}_k$ and time $t_k$, and
where $\delta t$ the time step of the scheme, typically a fraction of $\Delta t$. We choose bilinear interpolation 
\begin{align}
interp(\mathbf{u}, (x , y )) &= \mathbf{u}_1 + (\mathbf{u}_2 -\mathbf{u}_1 )\frac{(x - x_1)}{\Delta x} + (\mathbf{u}_3 -\mathbf{u}_1 )\frac{(y -y_1 )}{ \Delta y} \\ \notag
&+ (\mathbf{u}_1-\mathbf{u}_2-\mathbf{u}_3 + \mathbf{u}_4 )\frac{(x-x_1 )(y -y_1 )}{\Delta x \, \Delta y}, 
\end{align}
where
\begin{align} \notag
\mathbf{u}_1 &= \mathbf{u} (x_1 , y_1 ), \\  \notag
\mathbf{u}_2 &= \mathbf{u} (x_1 + 1, y_1 ), \\ \notag
\mathbf{u}_3 &=\mathbf{u}(x_1 , y_1 + 1), \\  \notag
 \mathbf{u}_4 &= \mathbf{u}(x_1 + 1, y_1 + 1). \notag
\end{align}
Here, $x_1=\left \lfloor{x}\right \rfloor $ is the floor function and $(x_1, y_1), (x_1 + 1, y_1), (x_1, y_1 + 1)$ and $(x_1 + 1, y_1 + 1)$ are the grid points which are nearest to $(x, y)$. 



%This has the advantage that the cost function becomes quadratical; it has a unique minimum and this minimum is assumed to be close to that of the full non-quadratical cost function (below). 

Using the incremental approach [Refs], the nonlinear observation operator $\mathcal{M}$ is linearized around a reference state. In a specific time window, we consider time independent perturbations $\delta \bo{u}$ on top of the background velocity field, that is
\begin{align}\label{totalR}
\bo{r}&=\bo{r^b}+\delta \bo{r} \\ \notag
\bo{u}&=\bo{u^b}+\delta \bo{u}. 
\end{align}
The linearized equations become 
\begin{align} \label{REquations}
&\bo{r^b}(k\delta t)=\bo{r^b}((k-1)\delta t)+\delta t \, interp\bigl(   \bo{u^b}((k-1)\delta t)), \bo{r^b}((k-1)\delta t     \bigr) ,\,\,\,\,\, \text{background}  \\  \notag
&\bo{\delta r}(k\delta t) = \bo{\delta r}((k-1)\delta t) + \delta t \, \{ interp(\bo{\delta u},\bo{r^b}((k-1)\delta t)) \\ \notag 
&+ \bo{\delta r}((k-1)\delta t) \cdot \partial _{(x,y)} interp \bigl(\bo{u^b}((k-1)\delta t),\bo{r^b}((k-1)\delta t)\bigr)\},  \,\,\, \text{tangent} 
\end{align}
where the drifters' positions are initialized with observations, and where $k=1,2,3, \cdots    \left \lfloor{T_w/\delta t}\right \rfloor .$ 
Here, $ \partial _{(x,y)} interp$ is the derivative of the interpolation function with respect to $(x,y)$. 


The background velocity used in the advection of the drifters is the aggregate of a geostrophic component $\bo{u}_{geo}$ provided by altimetry and a component driven by the wind $\bo{u}_{wind}$, which is parametrized by two parameters as described in PPM. So we have  
\[\bo{u}^b=\bo{u}_{geo}+\bo{u}_{wind}
\]
The data for both velocities are provided daily as described in the Data Section. This means that an interpolation in time of these velocities may be needed.



%\subsection{Discussion on the choice of $\delta t$ }
%The observations $\bo{r}^{obs}$ are available to us for each $\Delta t$, that is we have $\bo{r}^{obs}(i \Delta t), \,\,  i=0,1,2, \cdots $. Let us denote by $\Delta x$ the typical distance traveled by a buoys during $\Delta t$ and let $\Delta s$ denote the spatial resolution of the correction $\delta \bo{u}$. We have two scenarios: 
%\begin{enumerate}[(i)]
%\item If $\Delta s >> \Delta x$, then we can take $\delta t \sim \Delta t$ (that should simplify equation \eqref{Evolve}) 
%\item If $\Delta s << \Delta x$, we need to iterate many $\delta t$ to arrive at an observation. We take $N \delta t=\Delta t$
%\end{enumerate}

\subsection{Algorithm for assimilation}

We perform sequences of optimizations, where we minimize the following objective function with respect to the time independent correction $\delta \bo{u},$ in a specific time window $[0,T_w].$
\begin{equation}
\mathcal{J}(\delta \bo{u})=  \sum _{i=1}^{N_f} \sum_{m=1}^{\left \lfloor{T_w/\Delta t}\right \rfloor} \vectornorm{\bo{r}^{\,b}_{i}(\bo{u^b})+\delta \bo{r}_i(\delta \bo{u}) -\bo{r}_i^{\,obs}(m\Delta t) }^2 +\alpha_1 \vectornorm{ \bo{\delta u} }^2_{\bo{B}} +\alpha_2 \,div (\bo{u}_{geo}) 
\end{equation}

The first component of the objective function quantifies the misfit between the model 
obtained by iterations of \eqref{REquations}, and observations $\bo{r}^{\,obs}(m\Delta t)$. 
We highlight the dependence of $\bo{r}^b$ on the background velocity only, whereas $\bo{\delta r}$ depends on both background and correction.
The second component states that the corrected field is required to stay close to the background velocity. 
Here, the $B$-norm is defined as $\vectornorm{\psi}^2_{\bo{B}} \equiv \psi^T \mathbf{B}^{-1} \psi,$ where $\bo{B}$ is the error covariance matrix.  This term serves the dual purpose of regularization and information spreading or smoothing. To obtain $\bo{B}$. we use the diffusion filter method of [Weaver and Courtier, 2001], where a priori information on the typical length scale $R$ of the Eulerian velocity can be inserted.
The parameter $\alpha_1$ represents the relative weight of this regularization term with respect to the other terms.
The last component is a constraint on the geostrophic part of the velocity, required to stay divergence free. More here?.We note here that the total velocity may have a divergent component due to the wind. 


Inside a time window $[0, T_w]$, we assimilate a whole trajectory of drifters, which gives a constant correction in time $\bo{\delta u}.$ We refine the method by reconstructing a smooth time dependent velocity inside $[0, T_f]$: we achieve this by sliding the window by a time shift $\sigma$ to obtain corrections $\bo{\delta u}_k$ in a window $[k\sigma, k\sigma +T_w], \, \, k=0, 1, 2 \cdots$. The reconstructed velocity is then obtained as \[\bo{u}_{corrected}(t_i)=\bo{u^{b}}(t_i)+\sum_{k=0}^{N^i_w} w_k \bo{\delta u}_k.\]  A correction at a specific instant $t_i$ only takes into accounts $N^i_w$ windows sliding through $t_i$. The weight given to correction $\bo{\delta u}_k$ is defined as \[w_k=\frac{1}{\vectornorm{t_i-tc_k}},\] where $tc_k$ is the centre of window $k,$ which means that it is inversely proportional to the ``distance" between time $t_i$ and the window's position. then interpolation???
%Here $i=1,2, \cdots T_f/\delta t$

We end this section by pointing out that we implement the algorithm described above in YAO, [Refs.]
a numerical tool very well adapted to variational assimilation problems that simplifies the computation and implementation of the adjoint needed in the optimization. 





\section{Twin Experiment}
In the twin experiment approach, the observations are simulated using a known velocity field referred to as the true velocity $\bo{u}_{true}.$ In this context, we are able to assess the validity of our approach by comparing the corrected ($\bo{u}_{corrected}$) and true fields. This is based on the time-dependent RMS error 
\begin{equation} \label {RMSError}
error (u, t)=\bigg(    \frac{\sum_{i,j} |\bo{u}_{true} (i,j,t)-\bo{u} (i,j,t)|^2}{\sum_{i,j} |\bo{u}_{true} (i,j,t)|^2 } \bigg),
\end{equation}
where $\bo{u}$ could be $\bo{u}_b$, giving the error before assimilation or $\bo{u}_{corrected}$, giving the error after assimilation. We shall also compare the trajectories of the drifters obtained by simulation with the corrected velocity field and the observed trajectories. 



The configuration of our twin experiment is the following: we put ourselves in the same context as that of the real experiment conducted by the CNRS (refer to data section above), where the drifters are launched south of Beirut starting the end of August 2013. As shown in Fig. 2,  we deploy ``synthetic'' drifters in the region located  between 33.7 $^{\circ}$ and 34.25 $^{\circ}$ North and 34.9 $^{\circ}$ E and the coast. This is the same box in which the computation of the RMS error \eqref{RMSError} is done. The initial positions of the drifters shown in red coincide with the positions of the CNRS drifters on September first 2013.  The drifters' positions are simulated using a velocity field $\bo{u}_{true}$ obtained from the dynamic model described in (Refer to data section). The experiment starts on September first  2013, and lasts for a duration of $T_f=3$ days. In principle, nothing forbids us of conducting longer experiments, but in this coastal region, the drifters hit land after 3 days, as shown in Fig. 2.
 The background velocity field is composed of the geostrophic component obtained from AVISO and the wind component as described in the method section. Starting September first 2013, these fields are interpolated to $\delta t$. The parameter $R$ is chosen to be $20$km, [Refs]. 
 
 
Using the relative RMS error before and after assimilation as a measure, we first study the sensitivity of our method to the number of drifters $N_f$, time sampling $\Delta t,$ the window size $T_w$ and to the moving parameter $\sigma.$ (Figures 3 to 6).
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.5]{./fig/RegionErroronAviso.pdf}
\vspace{-30mm}
\caption{Region of RMS error computation for the twin experiment. Observations generated by model starting on Sept 1st 2013, for 3 days, are shown on top of averaged background field. The red locations correspond to CNRS drifters locations.}
\end{center}
\end{figure}
We first show the effect of the window size $T_w.$ On one hand, this parameter has to be within the Lagrangian time scale $T_L,$ which is estimated to $1-3$ days in this region, according to [Refs], but it also cannot be too large because of the linearisation approach. In Fig. 3, we show the results corresponding to various window sizes (fixing $N_f=14$ and $\Delta_t=2$ hs), by displaying the relative RMS error (computed in the aforementioned box), before and after the correction. We first see that the error curve (after correction) tends to increase generally as time goes by. This is due to this special coastal configuration where the first three drifters hit the shore after $48$ hs and also to the effect of the spatial filter, correcting the region behind the initial positions of the drifters. We then observe that the optimal window size for this configuration is $24$ hs, which is within the range mentioned above. The error in this case is reduced by almost a half from the original one. Finally, we mention that with window sizes close to three days, the algorithm becomes ill conditioned, which is expected due to the linear tangent hypothesis.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.4]{./fig/Wins_optshift_dt1_f14_tf72.pdf}
\vspace{-30mm}
\caption{The effect of the window size. Error before correction is shown with a solid line. Errors after are shown with symbols for several window sizes. Here $N_f=14$ and $\Delta t=2$ hs}
\end{center}
\end{figure}
We then show the effect of shifting the window by varying the parameter $\sigma.$ The window size here is fixed to $T_{w}=24$ hs, time sampling to $\Delta t=2$ hs, and $N_f=14.$ We start with $\sigma=0$, which amounts to doing separate corrections, then slide every $\sigma=12, 8, 6$ hs. In Fig. 4, we show the results by displaying the relative RMS error before and after the correction. 
We observe that if the corrections are done separately, the correction is not smooth; in fact smaller values of $\sigma$ yield not only smoother, but better corrections. 
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.4]{./fig/Shifts_win24_dt1_f14_tf72.pdf}
\vspace{-25mm}
\caption{The effect of the moving window: smaller shifts $\sigma$ yield smoother and better corrections. Here $N_f=14$ and $\Delta t=2$ hs.
 }
\end{center}
\end{figure}
The effect of the number of drifters is shown next in Fig. 5. Respecting coverage, we start with $N_f=14$ drifters (positioned as shown in Fig. 2), then reduce to $N_f=10,6,3.$ Naturally more drifters yield a better correction but we notice that even with three drifters, the error is still reduced by $20\%$ and much more so close to the beginning of the experiment. If we took only the drifters that do not hit the coast before 3 days, we get a more averaged error curve as shown in the dashed curve in Fig. 5. Finally, we show the effect of the time sampling parameter $\Delta t$ of the observations in Fig. 6. Curves after correction correspond to $\Delta t=6, 4$ and $2$ hours. There is not a large difference between these cases and the realistic scenario of  $\Delta t=6$ hs still yields a good correction.  too small means we can do a Lagrangian method.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.4]{./fig/Nfs_win24_dt1_tf72_nofail.pdf}
\vspace{-30mm}
\caption{The effect of the number of drifters. More drifters yield better corrections but corrections are possible with only 3 drifters. The dashed line shows the effect of just taking drifters that do not hit the shore before the end of the experiment. Here $T_w=24$ hs and $\Delta t=2$ hs.}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.4]{./fig/Dts_win24_f14_tf72.pdf}
\vspace{-30mm}
\caption{The effect of the time sampling}
\end{center}
\end{figure}

As an additional assessment of our method, we show the trajectories of the drifters reconstructed with the corrected velocity field on top of the actual observations, for the same experiment described above with the following parameters: a window of size $T_w=24$ hs, a shift of $\sigma=6$ hs, $N_f=14$ and $\Delta t=2$ hs. The left side of Fig. 7 shows the point-wise $L_2$ error between the background and true fields in the region of interest, averaged over the duration of the experiment. The right side shows this error after correction as well as the agreement between the positions of the drifters simulated with the corrected field and the actual observations. Finally, the correction in terms of direction is shown in Fig. 8.  where we display the cosine of the angle between the background and true field (averaged) on the left side versus the  cosine of the angle between the corrected and true fields on the right. Note that a cosine of one indicates a strong correlation in direction between the two fields. 


\begin{figure}[htdp]
%\centering
\begin{subfigure}{0.55\textwidth}
 % \entering
  \includegraphics[width=1.3\linewidth]{./fig/Before_L2pointw_win24_MEAN_color.pdf}
%  \caption{Pressure drop.}
  \label{bla}
\end{subfigure}%
\hspace{-10mm}
\begin{subfigure}{0.55\textwidth}
 % \centering
  \includegraphics[width=1.3\linewidth]{./fig/After_L2pointw_win24_MEAN_color.pdf}
%  \caption{Mass flowrate.}
  \label{blo}
\end{subfigure}
\vspace{-30mm}
\caption{Averaged point-wise $L_2$ error before (left) and after (right) correction. In the right frame, drifters' positions obtained by simulation with corrected field (magenta) versus actual observations(black) are shown on top of the error.}
\label{fblo}
\end{figure}


\begin{figure}[htdp]
%\centering
\begin{subfigure}{0.55\textwidth}
 % \entering
  \includegraphics[width=1.2\linewidth]{./fig/Before_CORRANGLEpointw_win24_MEAN_color.pdf}
%  \caption{Pressure drop.}
  \label{bla}
\end{subfigure}%
\hspace{-10mm}
\begin{subfigure}{0.55\textwidth}
 % \centering
  \includegraphics[width=1.2\linewidth]{./fig/After_CORRANGLEpointw_win24_MEAN_color.pdf}
%  \caption{Mass flowrate.}
  \label{blo}
\end{subfigure}
\vspace{-30mm}
\caption{Averaged correction in terms of direction. Left: $\cos{(\bo{u}_{b},\bo{u}_{true})}$, right: $\cos{(\bo{u}_{corrected},\bo{u}_{true})}.$ }
\label{fblo}
\end{figure}




\section{Experiment with Real Data}







The methodology described in section~\ref{sec:method} was applied to two case studies : one along the lebanese coast and one in an eddy south-east of Cyprus.
\subsection{\label{sec:lebanon}Improvement of velocity field near the coast}
%lebanese drifters
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.5]{./fig/ReconstructedCNRSExp_6days_average.pdf}
\vspace{-30mm}
\caption{Real Exp starting Aug 28 for 6 days, window=24h, move=6hs, }
\end{center}
\end{figure}


\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.5]{./fig/ReconstructedCNRSExp_bmtogreen_2days_average_zoom.pdf}
\vspace{-30mm}
\caption{Real 2-1 Exp starting Aug 28 for 2 days, window=24h, move=24hs, }
\end{center}
\end{figure}

\subsection{\label{sec:cyprus}Improvement of velocity field in an eddy}
In the context of the Nemed deployment (see section ~\ref{sec:drifters}), we selected two drifters trajectories from 25 August 2009 to 3 September 2009. Assimilating the successive positions of the drifters every six hours, the AVISO velocity field was corrected.

In Fig.~\ref{fig:eddy-velocity}, the trajectory of the drifters were represented in gray, the AVISO surface geostrophic velocity field in blue and the corrected geostrophic field in red. In this case, it can be seen that the drifter trajectories were situated in an eddy. The AVISO field is produced by an interpolation method which tends to overestimate the spatial extent of the eddy and underestimate the intensity.
\begin{figure}[h]
\centering
\includegraphics[scale =0.6]{./fig/Eddy_velocity.png}
\caption{\label{fig:eddy-velocity} Corrected field (in red) compared to AVISO background fields (in blue). The assimilated drifter trajectories are represented in gray. The North-West coast in the figure is Cyprus.}
\end{figure}

\section{Acknowledgement}
The altimeter products were produced by Ssalto/Duacs and distributed by Aviso, with support from Cnes (http://www.aviso.altimetry.fr/duacs/).

Wind data were produced by ECMWF and downloaded from (http://apps.ecmwf.int/datasets/data/interim-full-daily/).

This work was funded by the ENVI-MED program in the framework of the Altifloat project.
\textcolor{red}{Laurent, Pierre-Marie, Milad, des choses à ajouter pour Cana et les drifters ?}


%% The Appendices part is started with the command \appendix;
%% appendix sections are then done as normal sections
%% \appendix

%% \section{}
%% \label{}

%% If you have bibdatabase file and want bibtex to generate the
%% bibitems, please use
%%
\section{Bibliography}

  \bibliographystyle{elsarticle-harv} 
  \bibliography{mybib.bib}

%% else use the following coding to input the bibitems directly in the
%% TeX file.

%\begin{thebibliography}{00}

%% \bibitem[Author(year)]{label}
%% Text of bibliographic item

%\bibitem[ ()]{}

%\end{thebibliography}
\end{document}

\endinput
%%
%% End of file `elsarticle-template-harv.tex'.