Changeset 195

Timestamp:

11/13/15 11:28:41 (9 years ago)

Author:

leila_ocean

Message:

Nov13

Location:

altifloat/doc/ocean_modelling

Files:

• Draft1.tex (modified) (11 diffs)
• fig/After_CORRANGLEpointw_win24_MEAN_color.pdf (modified) (previous)
• fig/After_L2pointw_win24_MEAN_color.pdf (modified) (previous)
• fig/Before_CORRANGLEpointw_win24_MEAN_color.pdf (modified) (previous)
• fig/Before_L2pointw_win24_MEAN_color.pdf (modified) (previous)
• fig/Dts_win24_f14_tf72.pdf (modified) (previous)
• fig/Nfs_win24_dt1_tf72_nofail.pdf (modified) (previous)
• fig/RealvsSimulatedTraj.pdf (modified) (previous)
• fig/ReconstructedCNRSExp_6days_average.pdf (modified) (previous)
• fig/ReconstructedCNRSExp_bmtogreen_2days_average.pdf (modified) (previous)
• fig/ReconstructedCNRSExp_bmtogreen_2days_average_zoom.pdf (modified) (previous)
• fig/RegionErroronAviso.pdf (modified) (previous)
• fig/Shifts_win24_dt1_f14_tf72.pdf (modified) (previous)
• fig/Wins_optshift_dt1_f14_tf72.pdf (modified) (previous)

altifloat/doc/ocean_modelling/Draft1.tex

@@ -104 +104 @@
 %% \fntext[label3]{}
 
-\title{Modeling Surface Currents in the Eastern Levantine Mediterranenan Using Surface Drifters and Satellite Altimetry Data}
+\title{Modelling Surface Currents in the Eastern Levantine Mediterranean Using Surface Drifters and Satellite Altimetry Data}
 
 %% use optional labels to link authors explicitly to addresses:
@@ -111 +111 @@
 %% \address[label2]{}
 
-\author{Leila Issa, Julien Brajard, Milad Fakhri, Dan Hayes, Laurent Mortier and Pierre-Marie Poulain. }
+\author{Leila Issa, Julien Brajard, Milad Fakhri, Daniel Hayes, Laurent Mortier and Pierre-Marie Poulain. }
 
 \address{}
 
 \begin{abstract}
-%% Text of abstract
+We present a new and fast method for blending surface drifters data and altimetry in the Eastern Levantine Mediterranean. The method is based on a variational assimilation approach where drifters data are matched  to a simple advection model for drifters' positions. This model takes into account the effect of the wind. The assimilation is continuous in time. This is done by assimilating a whole trajectory of drifters at once and by using a moving time window where observations are correlated. 
+We show that with few drifters, our method improves the estimation of an eddy between the Lebanese coast and Cyprus, and predicts real drifters trajectories along the Lebanese coast. 
 
 \end{abstract}
@@ -191 +192 @@
 \begin{figure}[htbp]
 \begin{center}
-\includegraphics[scale=0.7]{./fig/RealvsSimulatedTraj.pdf}
-\vspace{-30mm}
-\caption{CNRS drifters deployed in the context of the ALTIFLOAT project starting Aug 28 2013 (shown in  $-$x) , versus  trajectories simulated using the \textit{Aviso} field (shown in $--$).  The velocity field shown is the  \textit{Aviso} field, averaged over 6 days.}
+\includegraphics[scale=0.5]{./fig/RealvsSimulatedTraj.pdf}
+%\vspace{-30mm}
+\caption{CNRS drifters deployed in the context of the ALTIFLOAT project starting Aug 28 2013 (shown in  $-$x) , versus  trajectories simulated using the \textit{Aviso} field (shown in $\tiny{--}$).  The velocity field shown is the  \textit{Aviso} field, averaged over 6 days.}
 \label{fig:cnrs}
 \end{center}
@@ -203 +204 @@
 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 {\label{sec:aviso}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. The geostrophic absolute velocity fields were deduced from Maps of Absolute Dynamic Topography (MADT) using the regional Mediteranean Sea product.
+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. The geostrophic absolute velocity fields were deduced from Maps of Absolute Dynamic Topography (MADT) using the regional Mediterranean Sea product.
 
 Data were mapped daily at a resolution of 1/8$^o$. 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 second 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}.
+Drifters were deployed at the two target periods (2 drifters were selected for the first period in 2009 and 3 in the second 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 filtered 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
@@ -339 +340 @@
 
 
-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.
+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 account $N^i_w$ 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.
 
 
@@ -358 +359 @@
 
 
-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.~\ref{fig:synth},  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 Section 2.4. 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.~\ref{fig:synth}.
- The background velocity field is composed of the geostrophic component obtained from \textit{Aviso} and the wind component as described in the method section. Starting September first 2013, these fields are interpolated to $\delta t$. The optimal choice of parameter $R$ is found to be $20$ km, which is consistent with the range of values found in the Northwestern Mediterranean \citep{taillandier2006variational}.
- 
- 
-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 $N_f$, the time sampling $\Delta t,$ the window size $T_w$ and to the moving parameter $\sigma.$ (Figures~\ref{fig:wsize},\ref{fig:mwin},\ref{fig:numb}).
+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 Altifloat drifters in Table~\ref{tab:drifters}), where the drifters are launched south of Beirut starting the end of August 2013. As shown in Fig.~\ref{fig:synth},  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 Altifloat drifters on September 1 2013.  The drifters' positions are simulated using a velocity field $\bo{u}_{true}$ obtained from the CYCOM model described in Section 2.4. The experiment 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.~\ref{fig:synth}.
+ The background velocity field is composed of the geostrophic component obtained from \textit{Aviso} and the wind component as described in the method section. These fields are interpolated to $\delta t$. A sensitivity analysis yields the optimal choice of $R=20$ km, which is consistent with the range of values found in the Northwestern Mediterranean \citep{taillandier2006variational}.
+ 
+ 
+Using the relative RMS error before and after assimilation as a measure, we study the sensitivity of our method to the number of drifters $N_f$, the time sampling $\Delta t,$ the window size $T_w$ and to the moving parameter $\sigma.$ (Figures~\ref{fig:wsize},\ref{fig:mwin},\ref{fig:numb}).
 \begin{figure}[htbp]
 \begin{center}
 \includegraphics[scale=0.5]{./fig/RegionErroronAviso.pdf}
-\vspace{-30mm}
+%\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.}
 \label{fig:synth}
 \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 $T_L,$ estimated here to be $1-3$ days, but it also cannot be too large because we consider corrections that are time independent in each window. In Fig.~\ref{fig:wsize}, 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. We mention here that with window sizes close to three days, the algorithm becomes ill conditioned, which is expected due to the fact that the correction is fixed in a specific window, as mentioned before. 
+We first show the effect of the window size $T_w.$ On one hand, this parameter has to be within the Lagrangian time scale $T_L,$ estimated here to be $1-3$ days, but it also cannot be too large because we consider corrections that are time independent in each window. In Fig.~\ref{fig:wsize}, we show the results corresponding to various window sizes (fixing $N_f=14$ and $\Delta_t=2$ h), 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 $48$ h 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 $24$ h, which is within the range mentioned above. The error in this case is reduced by almost a half from the original one. We mention here that with window sizes close to three days, the algorithm becomes ill conditioned, which is expected due to the fact that the correction is fixed in a specific window, as mentioned before. 
 \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}
+%\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 $\Delta t=2$ h}
 \label{fig:wsize}
 \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.~\ref{fig:mwin}, we show the results by displaying the relative RMS error before and after the correction. 
+We then show the effect of shifting the window by varying the parameter $\sigma.$ The window size here is fixed to $T_{w}=24$ h, time sampling to $\Delta t=2$ h, and $N_f=14.$ We start with $\sigma=0$, which amounts to doing separate corrections, then slide every $\sigma=12, 8, 6$ h. In Fig.~\ref{fig:mwin}, 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.
+%\vspace{-25mm}
+\caption{The effect of the moving window: smaller shifts $\sigma$ yield smoother and better corrections. Here $N_f=14$ and $\Delta t=2$ h.
  }
 \label{fig:mwin}
 \end{center}
 \end{figure}
-The effect of the number of drifters is shown next in Fig.~\ref{fig:numb}. Respecting coverage, we start with $N_f=14$ drifters (positioned as shown in Fig.~\ref{fig:synth}), 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 three days, we get an error curve that is evenly distributed, as shown in the dashed curve in Fig.~\ref{fig:numb}. Finally, we show the effect of the time sampling parameter $\Delta t$ of the observations in Fig.~\ref{fig:time}. Curves after correction correspond to $\Delta t=6, 4$ and $2$ hours and as we see from the figure, the difference between these cases is not too large. The realistic scenario of  $\Delta t=6$ hs still yields a good correction.
+The effect of the number of drifters is shown next in Fig.~\ref{fig:numb}. Respecting coverage, we start with $N_f=14$ drifters (positioned as shown in Fig.~\ref{fig:synth}), 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 three days, we get an error curve that is evenly distributed, as shown in the dashed curve in Fig.~\ref{fig:numb}. Finally, we show the effect of the time sampling parameter $\Delta t$ of the observations in Fig.~\ref{fig:time}. Curves after correction correspond to $\Delta t=6, 4$ and $2$ hours and as we see from the figure, the difference between these cases is not too large. The realistic scenario of  $\Delta t=6$ h still yields a good correction.
 \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.}
+%\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 $T_w=24$ h and $\Delta t=2$ h.}
 \label{fig:numb}
 \end{center}
@@ -404 +405 @@
 \begin{center}
 \includegraphics[scale=0.4]{./fig/Dts_win24_f14_tf72.pdf}
-\vspace{-30mm}
-\caption{The effect of the time sampling $\Delta t$ of the observations. Here $T_w=24$ hs, and $N_f=14.$}
+%\vspace{-30mm}
+\caption{The effect of the time sampling $\Delta t$ of the observations. Here $T_w=24$ h, and $N_f=14.$ The realistic scenario of  $\Delta t=6$ h is not too far from the smallest $\Delta t$.}
 \label{fig:time}
 \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. In Fig.~\ref{fig:lerror}, we show the point-wise $L_2$ error, defined as 
+For the twin experiment with the optimal choice of parameters ( $T_w=24$ h, $\sigma=6$ h, $N_f=14$ and $\Delta t=2$), we now show the trajectories of the drifters simulated with the corrected velocity field on top of the actual observations. We also compare background and corrected fields in the region of interest. In Fig.~\ref{fig:lerror}, we display the point-wise $L_2$ error, defined as 
 \begin{equation} \label {L2Error}
-error_{L_2} (u, t)=\big | \big |\bo{u}_{true} (i,j,t)-\bo{u} (i,j,t) \big| \big|,
-\end{equation}
-between the true field and either the background or corrected fields. In the left panel, we display that error between the background and true fields in the region of interest, averaged in time over the duration of the experiment. The right side shows this same error after correction. On top of that, we observe the excellent 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.~\ref{fig:cerror}.  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. 
+error (u,i,j,t)=\big | \big |\bo{u}_{true} (i,j,t)-\bo{u} (i,j,t) \big| \big|,
+\end{equation}
+between the true field and either the background or corrected fields. In the left panel, we show that error between the background and true fields, averaged in time over the duration of the experiment. The right side shows this same error after correction. On top of that, we observe the excellent agreement between the positions of the drifters simulated with the corrected field and the actual observations. Next, the correction in terms of direction is shown in Fig.~\ref{fig:cerror}: we display the cosine of the angle between the background and true field 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. We see this strong correlation between true and corrected fields by observing how the blue color (left pannel of Fig.\ref{fig:cerror} ) turns into deep red (right pannel of Fig.\ref{fig:cerror} ) in the region where the drifters were deployed. Finally, in Fig. \ref{fig:summary}, we show the actual current maps before and after correction. We clearly see that the drifters 'fixed' the poorly simulated coastal meander.
 
 
 \begin{figure}[htdp]
 
-\begin{subfigure}{0.55\textwidth}
-
-  \includegraphics[width=1.25\linewidth]{./fig/Before_L2pointw_win24_MEAN_color.pdf}
-
+\begin{subfigure}{0.6\textwidth}
+\includegraphics[width=1.08\linewidth]{./fig/Before_L2pointw_win24_MEAN_color.pdf}
 \end{subfigure}%
-\hspace{-10mm}
-\begin{subfigure}{0.55\textwidth}
+%\hspace{-20mm}
+\begin{subfigure}{0.6\textwidth}
  % \centering
-  \includegraphics[width=1.25\linewidth]{./fig/After_L2pointw_win24_MEAN_color.pdf}
-
+ \includegraphics[width=1\linewidth]{./fig/After_L2pointw_win24_MEAN_color.pdf}
 \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.}
+\vspace{-25mm}
+\caption{Point-wise $L_2$ error averaged over time, 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{fig:lerror}
 \end{figure}
@@ -438 +436 @@
 \begin{figure}[htdp]
 %\centering
-\begin{subfigure}{0.55\textwidth}
+\begin{subfigure}{0.6\textwidth}
  % \entering
-  \includegraphics[width=1.25\linewidth]{./fig/Before_CORRANGLEpointw_win24_MEAN_color.pdf}
-
+  \includegraphics[width=1.05\linewidth]{./fig/Before_CORRANGLEpointw_win24_MEAN_color.pdf}
 %  \label{bla}
 \end{subfigure}%
-\hspace{-10mm}
-\begin{subfigure}{0.55\textwidth}
+%\hspace{-10mm}
+\begin{subfigure}{0.6\textwidth}
  % \centering
-  \includegraphics[width=1.25\linewidth]{./fig/After_CORRANGLEpointw_win24_MEAN_color.pdf}
-
+  \includegraphics[width=1.05\linewidth]{./fig/After_CORRANGLEpointw_win24_MEAN_color.pdf}
 %  \label{blo}
 \end{subfigure}
-\vspace{-30mm}
-\caption{Averaged correction in terms of direction. Left: $\cos{(\bo{u}_{b},\bo{u}_{true})}$, right: $\cos{(\bo{u}_{corrected},\bo{u}_{true})}.$ }
+\vspace{-5mm}
+\caption{Correction in terms of direction. Left: $\cos{(\bo{u}_{b},\bo{u}_{true})}$, right: $\cos{(\bo{u}_{corrected},\bo{u}_{true})}.$ }
 \label{fig:cerror}
 \end{figure}
 
-
+\begin{figure}[htbp]
+\begin{center}
+\includegraphics[scale=0.4]{./fig/summary_twin.pdf}
+\caption{Background velocity field (blue) versus corrected velocity field (red) for the twin experiment with the optimal choice of parameters.}
+\label{fig:summary}
+\end{center}
+\end{figure}
 
 
@@ -465 +467 @@
 \subsection{\label{sec:lebanon}Improvement of velocity field near the coast}
 
-Three drifters were launched on August 28 2013 from the South of Beirut, at the positions shown in circles in In Fig.~\ref{fig:leb1}, they provide their position very $\Delta t= 6$ hs and stay within 20 km of the coast for the duration of the experiment. 
-The experiment considered here lasts for six days (a time frame where the three drifters are still spatially close before two of them hit the shore). The window size is $T_w=24$ hs. The smoothing parameter $\sigma=6$ hs. 
+Three drifters were launched on August 28 2013 from the South of Beirut, at the positions shown in circles in In Fig.~\ref{fig:leb1}, they provide their position very $\Delta t= 6$ h and stay within 20 km of the coast for the duration of the experiment. 
+The experiment considered here lasts for six days (a time frame where the three drifters are still spatially close before two of them hit the shore). The window size is $T_w=24$ h. The smoothing parameter $\sigma=6$ h. 
 Fig.~\ref{fig:leb1}, shows the trajectories simulated with corrected field on top of the observed ones, 
 in good agreement, even for small scale structures near the coast.
@@ -483 +485 @@
 \begin{center}
 \includegraphics[scale=0.5]{./fig/ReconstructedCNRSExp_6days_average.pdf}
-\vspace{-30mm}
-\caption{\label{fig:leb1} Prediction of the positions of 3 CNRS Drifters, launched on August 28 2013. $T_f=6$ days.  $T_w=24$ hs and $\sigma=6$ hs. Positions of drifters simulated with corrected field (cross markers) are shown on top of observed positions (circle markers). Corrected field is shown in red whereas background field is shown in blue. }
-\end{center}
-
+%\vspace{-30mm}
+\caption{\label{fig:leb1} Prediction of the positions of 3 CNRS Drifters, launched on August 28 2013. $T_f=6$ days.  $T_w=24$ h and $\sigma=6$ h. Positions of drifters simulated with corrected field (cross markers) are shown on top of observed positions (circle markers). Corrected field is shown in red whereas background field is shown in blue. }
+\end{center}
 \end{figure}
 
@@ -493 +494 @@
 \begin{center}
 \includegraphics[scale=0.5]{./fig/ReconstructedCNRSExp_bmtogreen_2days_average_zoom.pdf}
-\vspace{-30mm}
-\caption{\label{fig:lebzoom} Prediction of the position of the green drifter using the observed black and magenta drifters. $T_f=2$ days.  $T_w=24$ hs and $\sigma=6$ hs. Position of the green drifter simulated with corrected field is shown in green squares, on top of observed position shown in light green circles. Compare to the position of the drifter obtained with background field only, shown in cyan. Corrected field is shown in red whereas background field is shown in blue.}
-\end{center}
-
+%\vspace{-30mm}
+\caption{\label{fig:lebzoom} Prediction of the position of the green drifter using the observed black and magenta drifters. $T_f=2$ days.  $T_w=24$ h and $\sigma=6$ h. Position of the green drifter simulated with corrected field is shown in green squares, on top of observed position shown in light green circles. Compare to the position of the drifter obtained with background field only, shown in cyan. Corrected field is shown in red whereas background field is shown in blue.}
+\end{center}
 \end{figure}