Changeset 216

Timestamp:

12/17/15 22:16:47 (8 years ago)

Author:

leila_ocean

Message:

submitted, Leila

File:

• altifloat/doc/ocean_modelling/Draft1.tex (modified) (16 diffs)

altifloat/doc/ocean_modelling/Draft1.tex

@@ -149 +149 @@
 \begin{abstract}
 We present a new and fast method for blending altimetry and surface drifters data in the Eastern Levantine Mediterranean. The method is based on a variational assimilation approach for which the velocity is corrected 
-by matching real drifters positions with those predicted by a simple advection model, while taking into account the wind effect. The velocity correction is done in a time-continuous fashion by assimilating at once a whole trajectory of drifters in a time window, and by moving that window \textcolor{red}{to exploit correlations between observations}. Except for the wind component, the velocity is constrained to be divergence free. We show that with few drifters, our method improves the estimation of velocity in two typical situations: an eddy between the Lebanese coast and Cyprus, and velocities along the Lebanese coast. 
+by matching real drifters positions with those predicted by a simple advection model, while taking into account the wind effect. The velocity correction is done in a time-continuous fashion by assimilating at once a whole trajectory of drifters using a sliding time window. Except for the wind component, the velocity is constrained to be divergence free. We show that with few drifters, our method improves the the estimation of velocity in two typical situations: an eddy between the Lebanese coast and Cyprus, and velocities along the Lebanese coast. 
 \end{abstract}
 
@@ -279 +279 @@
 \subsection {\label{sec:model}Model data}
 Modeled surface velocity fields for September 2013 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,zodiatis2008} that covers the
-Northeast Levantine basin (1km resolution, west and south boundaries extended to 31$^o$00'E and 33$^o$00'N and north and east reach land).
+Northeast Levantine basin (1 km resolution, west and south boundaries extended to 31$^o$00'E and 33$^o$00'N and north and east reach land).
 %the North-East Levantin Bassin 
 %(31$^o$ 30’E - 36$^o$ 13’E  and 33$^o$ 30’N – 36$^o$ 55’N). 
@@ -380 +380 @@
 In order to produce a smooth time-dependent velocity field in $[0, T_f]$, a sliding window, of time shift $\sigma$, is used to obtain correction $\bo{\delta u}_k$ in \[[k\sigma, k\sigma +T_w], \, \, k=0, 1, 2 \cdots. \]
 
-\textcolor{red}{that accounts for temporal correlations in these various corrections $\delta u$} \\
-
-
-The reconstructed velocity is then obtained as a superposition of the time dependent background field and the weighted corrections   \[\bo{u}_{corrected}(t_i)=\bo{u^{b}}(t_i)+\sum_{k=0}^{N^i_w} w_k \bo{\delta u}_k.\]  A correction at a specific instant $t_i$ takes into account only $N^i_w$ windows sliding through $t_i$. The weight given to correction $\bo{\delta u}_k$ is chosen to be \[w_k=\frac{1}{\vectornorm{t_i-tc_k}},\] where $tc_k$ is the center of window $k,$ which means that the weight is inversely proportional to the ``distance" between time $t_i$ and the window's position.
+%\textcolor{red}{that accounts for temporal correlations in these various corrections $\delta u$} \\
+
+
+The reconstructed velocity is then obtained as a superposition of the time dependent background field and the weighted corrections   \[\bo{u}_{corrected}(t_i)=\bo{u^{b}}(t_i)+\sum_{k=0}^{N^i_w-1} w_k \bo{\delta u}_k.\]  A correction at a specific instant $t_i$ takes into account only $N^i_w$ windows sliding through $t_i$. The weight is inversely proportional to the ``distance" between time $t_i$ and the window's position according to
+\[w_k=\frac{1}{|k-k^*|+1},\] where $k^*$ corresponds to the window centered at $t_i.$ Note here that the weights are normalized to add to one. 
 
 
@@ -392 +393 @@
 \section{\label{sec:twin}Sensitivity analyses}
 
-To validate our method, we conduct a set of synthetic experiments where the observations are simulated using a known or ``true" velocity field, denoted by $\bo{u}_{true}$, and provided by the CYCOFOS-CYCOM model (see subsection~\ref{sec:model}). This allows us to assess the validity of our approach by comparing the corrected, $\bo{u}_{corrected}$, and true fields, based on the time-dependent RMS error 
+To validate our method, we conducted a set of synthetic experiments where the observations were simulated using a known or ``true" velocity field, denoted by $\bo{u}_{true}$, and provided by the CYCOFOS-CYCOM model (see subsection~\ref{sec:model}). This allows us to assess the validity of our approach by comparing the corrected, $\bo{u}_{corrected}$, and true fields, based on the time-dependent RMS error 
 \begin{equation} \label {RMSError}
 error (u, t)=\bigg(    \frac{\sum_{i,j} \big | \big |\bo{u}_{true} (i,j,t)-\bo{u} (i,j,t)\big | \big |^2}{\sum_{i,j} \big | \big |\bo{u}_{true} (i,j,t)\big | \big |^2 } \bigg)^{1/2}.
@@ -398 +399 @@
 Here, $\big | \big |. \big | \big |$ refers the the $L_2$ norm of a vector, and $\bo{u}$ could be the background velocity, $\bo{u}_b$, giving the error before assimilation or the corrected velocity, $\bo{u}_{corrected}$, giving the error after assimilation. The background velocity used is given by Eq.~\ref{euler_vel}, where the geostrophic component is provided by AVISO. Note that
 the CYCOFOS-CYCOM model was initialized by a large scale model having assimilated AVISO data.
-We shall also compare the trajectories of the drifters advected by the corrected velocity field with those advected by the true velocity field.
-
-The configuration of our experiment is the following: we put ourselves in the same context as that of the real drifter experiment conducted during the AltiFloat project, by the CNRS-L, the Lebanese national research council (refer to AltiFloat drifters in Table~\ref{tab:drifters}), where the drifters were 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. The initial positions of the drifters shown in red coincide with the positions of the drifters on 1 September 2013. The drifters' positions are simulated using a velocity field $\bo{u}_{true}$ obtained from the CYCOM model. 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 had hit land after 3 days, as shown in Fig.~\ref{fig:synth}, likely because of easterly winds.
+
+%We also compared the trajectories of the drifters advected by the corrected velocity field with those advected by the true velocity field.
+
+The configuration of our experiment was the following: we put ourselves in the same context as that of the real drifter experiment conducted during the AltiFloat project, by the CNRS-L, the Lebanese national research council (refer to AltiFloat drifters in Table~\ref{tab:drifters}), where the drifters were launched south of Beirut starting the end of August 2013. As shown in Fig.~\ref{fig:synth},  we deployed ``synthetic'' drifters in the region located  between 33.7 $^{\circ}$ and 34.25 $^{\circ}$ North and 34.9 $^{\circ}$ E and the coast. The initial positions of the two drifters shown in red coincide with the positions of two AltiFloat drifters on 1 September 2013 (by that time, the third AltiFloat drifter had left the region of interest). The drifters' positions were simulated using a velocity field $\bo{u}_{true}$ obtained from the CYCOM model. The experiment lasted for a duration of $T_f=3$ days. In principle, nothing forbids us of conducting longer experiments, but in this coastal region, the drifters had hit land after 3 days, as shown in Fig.~\ref{fig:synth}, likely because of easterly winds.
  
  %This is the same box in which the computation of the RMS error  Eq.~\ref{RMSError} is done.
  
-Using the relative RMS error before and after assimilation as a measure, we study the sensitivity of our method to the window size $T_w$, the time shift of the sliding window $\sigma$,  the number of drifters $N_f$ and to the sampling time $\Delta t$. We also assess the effect of the divergence free constraint term.   
+Using the relative RMS error before and after assimilation as a measure, we studied the sensitivity of our method to the window size $T_w$, the time shift of the sliding window $\sigma$,  the number of drifters $N_f$ and to the sampling time $\Delta t$. We also assessed the effect of the divergence free constraint term.   
 \begin{figure}[!htbp]
 \begin{center}
@@ -413 +415 @@
 \end{figure}
 
-A sensitivity analysis yields the optimal choice of $R=20$ km used in the diffusion filter, which is consistent with the range of values found in the Northwestern Mediterranean \citep{taillandier2006variational}.
+A sensitivity analysis yielded the optimal choice of $R=20$ km used in the diffusion filter, which is consistent with the range of values found in the Northwestern Mediterranean \citep{taillandier2006variational}.
 
 \subsection{Sensitivity to the time window size}
-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 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 box shown in Fig.~\ref{fig:synth}, before and after the correction. \textcolor{green}{marker}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 due 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. 
+%Lower bound $T_w = \sigma = \Delta t$; in this case 
+We first show the effect of the window size, $T_w$. This parameter has to be within the Lagrangian time scale $T_L,$ estimated here to be $1-3$ days,  but it 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 box shown in Fig.~\ref{fig:synth}, before and after the correction. Note that for all the window sizes considered, the time shift of the sliding windows was selected to yield minimal error.  We first see that the error curves (after correction) in Fig.~\ref{fig:wsize} tend to increase generally as time increases. This  behavior may be attributed to the fact that, for this special coastal configuration,  the first three drifters hit the shore after $48$ h, and also due to the interaction of the spatial filter with land. We also 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 almost half of the error before correction. We mention here that for this coastal scenario, window sizes of three days or more caused the algorithm to become ill conditioned, which is expected due to the fact that the correction is fixed in a specific window, as mentioned before. 
+% we are correcting in one shot over a time period larger than the Lagrangian time scale.
+% that it was observed that for this coastal configuration, the departure of the background velocity from the co
 \begin{figure}[htbp]
 \begin{center}
@@ -426 +431 @@
 
 \subsection{Sensitivity to the time shift of the sliding window}
-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, sampling time 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. This is due to the fact that the moving window scheme exploits the correlation between the trajectories.
+We present here the effect of varying, $\sigma$, the time shift of the sliding window. The values considered were $\sigma=0, 6, 8$ and 12 h. Note that $\sigma = 0$ amounts to doing separate corrections. The window size, sampling time, and number of drifters were fixed to  $T_{w}=24$ h, $\Delta t=2$ h, and $N_f=14$ respectively.  In Fig.~\ref{fig:mwin}, we show the results by displaying the relative RMS error before and after the correction. 
+We observe here 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, especially close to the middle of the experiment's duration. This may be explained by the fact that the moving window is responsible for spreading the information smoothly in the domain. The improvement is also likely due to the weights that favor corrections by the nearest set of drifters at the given time. This is not the case when the windows are separate, for example the error in the velocity correction at the edge of the windows is larger, because the nearest adjacent window does not contribute to this correction. 
+
+%The improvement is also likely due to the fact that the weight is inversely proportional to the distance so that it favors correction by the drifters at the given time.
+
+%This may be explained as follows:  towards the middle of the experiment,, when the windows do not overlap the correction will only come from the region in the center of the domain. Whereas if the window is sliding, corrections will also be influenced by farther regions. Because the error is computed in the whole fixed region, this explains the improvement. Regarding the weights also, 
+%
+% 
+% 
+% 
+% The correction inat a given location at a given time, $t$ is the sum of weighted contributions from all the sliding time windows $k$ whose time span contains $t$, i.e. $t \in [t_k, t_k + T_w]$, where the weights are inversely proportional to the distance between $t$ and $t_k + \frac{T_w}{2}$. Noting that contribution from window $k$ is due to the part of drifter trajectory that falls inside the window.  
+%
+%
+%
+%This provides a correction that changes smoothly in time. It also implies that the drifter that spends most time in the region of interest contributes most to the correction. 
+%
+%
+%The error is computed over a fixed region.
+%
+%:
+%
+%the error in the velocity correction at points at the edge of each window is larger, because the nearest neighboring window does not contribute to this correction. We remind the reader here that the error is computed in a fixed region, while the drifters are moving. When the windows move, we have a better coverage of the region. Also the improvement is due to the fact the most weight is given to the correction in the nearest window.
+%\textcolor{blue}{
+%fixed box 
+%error in time 
+%middle time: more windows are contributing in the whole region \\
+%for a specific time, corrections are done in such a way that the most weight is given to the correction in the nearest window. This is not the case when the windows do not overlap. In this scenario, the error in the velocity correction at points at the edge of the window is larger, because the nearest neighboring window does not contribute to this correction.}
+
+%Different drifters positions in the region that needs to be corrected should contribute differently to the correction. A floater trajectory near the corner of the region is not expected to yield a correction as significant as a floater with trajectory spanning the entire area. In addition, the position on this trajectory that is closest to the center of the region contribute most to the correction. In terms of the sliding window, this position is the one that belongs to the largest number of sliding windows. Talk about impact of the weights. the "correct" or optimal weight is given to the window correction closest to the point of interest. } 
 \begin{figure}[htbp]
 \begin{center}
 \includegraphics[scale=0.55]{./fig/Shifts_win24_dt1_f14_tf72.pdf}
-\caption{The effect of the moving window: smaller shifts $\sigma$ yield smoother and better corrections.  $N_f=14$ and $\Delta t=2$ h.
+\caption{The effect of the moving window: smaller shifts $\sigma$ yield smoother and better corrections.  $N_f=14$, $\Delta t=2$ h, $T_w=24$ h.
  }
 \label{fig:mwin}
@@ -438 +470 @@
 
 \subsection{Sensitivity to the number of drifters}
-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. We also show in this figure the effect of removing the drifters that fail before the end of the experiment: the corresponding error curve is shown in the dashed curve of Fig.~\ref{fig:numb}, and it is evenly distributed in time as expected. 
+The effect of the number of drifters, $N_f$, is shown next in Fig.~\ref{fig:numb}. Respecting coverage, we started with $N_f=14$ (positioned as shown in Fig.~\ref{fig:synth}), then reduced it to 10, 6, and 3. Naturally more drifters yielded a better correction but we notice that even with three drifters, the error was still reduced by $20\%$ and much more so close to the beginning of the experiment. We also show in this figure the effect of removing the drifters that fail before the end of the experiment: the corresponding error is shown in the dashed curve of Fig.~\ref{fig:numb}, and it is evenly distributed in time as expected. 
 \begin{figure}[!htbp]
 \begin{center}
@@ -449 +481 @@
 
 \subsection{Sensitivity to the sampling time}
-We show the effect of the sampling time $\Delta t$ 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 very good correction.
+We show the effect of the sampling time $\Delta t$ 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 yielded a very good correction.
 
 \begin{figure}[!htbp]
@@ -460 +492 @@
 
 \subsection{Sensitivity to the effect of the divergence constraint}
-
-The role of the divergence constraint in the optimization is determined by a delicate balance between the various terms. This term should be non negligible because as mentioned earlier, it forces the correction to be in the direction tangent to the coast, making the component perpendicular to the coast small. However, it cannot be too strong as to interfere with the regularization term, because that would make the optimization ill-conditionned. To show its effect on the correction, we conduct a sensitivity experiment where we compare the results (in the same setting as the previous experiments) with and without this term. As seen from Fig.~\ref{fig:div}, we obtain an improvement of about $10\%$ in the overall error if this term is present in the cost function. This is expected because we are correcting the velocity in a region close to the coast.
-
+The role of the divergence constraint in the optimization is determined by a delicate balance between the various terms. This term should be non negligible because as mentioned earlier, it forces the correction to be in the direction tangent to the coast, making the component perpendicular to the coast small. However, it cannot be too strong as to interfere with the regularization term, because that would make the optimization ill-conditioned. To show its effect on the correction, we conducted a sensitivity experiment where we compared the results (in the same setting as the previous experiments) with and without this term. As seen from Fig.~\ref{fig:div}, we obtained an improvement of about $10\%$ in the overall error if this term was present in the cost function. This is expected because we are correcting the velocity in a region close to the coast.
+%\textcolor{blue}{It is equally important, if not more important than the reduction in error, is the satisfaction of the no-through flow boundary at the coast. Without it, the field near the coast is useless. Choice of $\alpha_2$ ideally should be zero away from the coast, and become larger as we get closer to the coast.}
+% For the spatial resolution of the altimetric grid, numerical estimate of the divergence of the velocity is in large error , (for central differencing, error is of order Delta^2). Therefore, it is meaningless 
 
 \begin{figure}[!htbp]
@@ -475 +507 @@
 \subsection{Summary of results}
 
-For the 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 ``true" 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 
+For the experiment with the optimal choice of parameters ($T_w=24$ h, $\sigma=6$ h, $N_f=14$ and $\Delta t=2$), we compared the trajectories of the drifters simulated with the corrected velocity field with the ``true" observations. We also compared background and corrected fields in the region of interest. In Fig.~\ref{fig:lerror}, we display the point-wise $L_2$ error between the true field and either the background or corrected fields. This error is defined as the time average of 
 \begin{equation} \label {L2Error}
-error (u,i,j,t)=\big | \big |\bo{u}_{true} (i,j,t)-\bo{u} (i,j,t) \big| \big|,
+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 ``true" 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 corrected the poorly represented coastal meander in the AVISO altimetric velocity field.
+ The left panel corresponds to the ``before" picture, where the error is between the background and true fields and the right one corresponds to the ``after" picture, where the error is between the corrected and true fields. On top of that, we observe the excellent agreement between the positions of the drifters simulated with the corrected field and the true observations. Next, the correction in terms of the velocity 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 (dark red) 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 corrected the poorly represented coastal meander in the AVISO altimetric velocity field.
 
 
 \begin{figure}[htbp]
 \begin{subfigure}{0.5\textwidth}
-\includegraphics[width=1.04\linewidth]{./fig/Before_L2pointw_win24_MEAN_color.pdf}
+\includegraphics[width=1\linewidth]{./fig/Before_L2pointw_win24_MEAN_color.pdf}
 \end{subfigure}%
 %\hspace{-20mm}
@@ -529 +561 @@
 
 Three drifters were launched on 28 August 2013 from the South of Beirut, at the positions shown in circles in Fig.~\ref{fig:leb1}. They provide their position every $\Delta t= 6$ 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 the corrected field on top of the observed ones, 
-in very good agreement, even for small scale structures near the coast.
-Averaged corrections over 6 days are shown on the figure, but the actual corrections are time-dependent.
-
-As expected, the velocity field is modified in the neighbourhood of the drifters trajectories. It can be noticed that the main effect of the correction is to increase the velocity parallel to the coast, and decrease the velocity normal to the coast. The background field was determined using altimetric data and is expected to have significant bias close to the coast~\citep{bouffard2008}, and the consequence is that the method is able to correct some of this bias.
-
-To validate more quantitatively the corrected velocities, a sensitivity study was carried out. Only two drifters (the easternmost magenta drifter and the westernmost black drifter) were assimilated in order to correct the velocity field. The third drifter is used only to validate the corrected field by comparing its actual trajectory with the simulated trajectory using the velocity field.
-
-Figure~\ref{fig:lebzoom} shows the results of this experiment. The real drifter trajectory (empty circle with thin line) was compared to the simulated trajectory using either the background field (bold cyan line) or the corrected field (bold green line). 
-
-It can be noticed that the trajectory was greatly improved using the corrected field. It shows that the corrected field can be used to simulate realistic trajectories in the neighbourhood of the assimilation positions, even in a coastal region. 
-This can be a decisive point for applications such as pollutant transport estimation.
+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 and the time shift of the sliding window were chosen to be $T_w=24$ h and  $\sigma=6$ h respectively. 
+%These choices are consistent with the sensitivity study carried out in the previous study.
+
+Fig.~\ref{fig:leb1} shows that the trajectories simulated with the corrected field and the observed ones are in very good agreement, even for small scale structures near the coast.
+Note that the correction presented in the figure is the time average of the instantaneous corrections, over a period of 6 days. %Note that although time-averaged corrections over 6 days are shown on the figure, the actual corrections are time-dependent. 
+As expected, the velocity field is modified in the neighborhood of the drifters trajectories. It can be noticed that the main effect of the correction is to increase the velocity parallel to the coast, and decrease the velocity normal to the coast. The background field was determined using altimetric data and is expected to have significant bias close to the coast~\citep{bouffard2008}, and the consequence is that the method is able to correct some of this bias.
+%It seems in this case that the floater is emulating the divergence-free condition, i.e. forcing the flow to be parallel to the coast. This may explain why enforcing the divergence free-condition by setting large $\alpha_2$ did not change anything.
+
+To validate more quantitatively the corrected velocities, a sensitivity study was carried out. Only two drifters (the eastern-most magenta drifter and the western-most black drifter) were assimilated in order to correct the velocity field. The third drifter is used only to validate the corrected field by comparing its actual trajectory with the simulated trajectory using the velocity field. Figure~\ref{fig:lebzoom} shows the results of this experiment. The real drifter trajectory (empty circle with thin line) is compared to the simulated trajectory using either the background field (bold cyan line) or the corrected field (bold green line). It can be noticed that the trajectory is greatly improved using the corrected field. It shows that the corrected field can be used to simulate realistic trajectories in the neighborhood of the assimilation positions, even in a coastal region.  This can be a decisive point for applications such as pollutant transport estimation.
 
 \begin{figure}[!h]
@@ -561 +589 @@
 
 \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 2 drifters trajectories from 25 August 2009 to 3 September 2009. The AVISO velocity field was corrected by assimilating successive positions of the drifters every six hours.
-
-In this experiment the window size $T_w$ was chosen to be $72$ hours as the velocity field was more stable in this case than in coastal areas. The shifting of the time window was of $18$ hours.
-
-In Fig.~\ref{fig:eddy-velocity}, the trajectory of the drifters were represented in gray, the mean AVISO surface geostrophic velocity field in blue and the mean corrected geostrophic field in red. 
-
-The real trajectory of the drifters and the simulated trajectory using the total corrected field (sum of corrected field in red and the wind-induced velocity) are very close. 
+In the context of the NEMED deployment (see section ~\ref{sec:drifters}), we selected 2 drifters trajectories from 25 August 2009 to 3 September 2009. The AVISO velocity field was corrected by assimilating successive positions of the drifters every six hours. In this experiment the window size $T_w$ was chosen to be $72$ h as the velocity field was more stable in this case than in coastal areas. The shifting of the time window was chosen to be $\sigma = 18$ h.
+
+In Fig.~\ref{fig:eddy-velocity}, the trajectory of the drifters are represented in gray, the mean AVISO surface geostrophic velocity field in blue and the mean corrected geostrophic field in red. It can be observed that the real trajectory of the drifters and the simulated trajectory using the total corrected field (sum of corrected field in red and the wind-induced velocity) are indiscernible. 
 The mean position error 
 %(expressed in arc length) 
@@ -576 +600 @@
 %$0.06$ degrees. 
 $6.7$ km.
-The real and simulated trajectory would be indiscernible in Fig.~\ref{fig:eddy-velocity}. 
-
-
-
 \begin{figure}[h]
 \centering
@@ -586 +606 @@
 \end{figure}
 
-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 its intensity. In order to estimate the effect of the assimilation on the eddy characteristics, we computed the Okubo-Weiss parameter~\citep{isern2004} on the mean velocity fields before correction (background) and after correction. Eddies are characterized by a negative Okubo-Weiss parameter, the value of the parameter is an indicator of the intensity of the eddy. Results are shown in Fig.~\ref{fig:okubo-weiss}. As expected, it can be noticed that the Okubo-Weiss parameter had greater absolute values and a slightly smaller spatial extent which indicated an improvement of the AVISO processing bias. This results constitutes a validation of the assimilation method presented in this paper showing that eddies were better resolved after assimilating drifter trajectories.
+In this case, the drifter trajectories are chosen to be 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 its intensity. In order to estimate the effect of the assimilation on the eddy characteristics, we computed the Okubo-Weiss parameter~\citep{isern2004} on the mean velocity fields before correction (background) and after correction. Eddies are characterized by a negative Okubo-Weiss parameter, the value of the parameter is an indicator of the intensity of the eddy. Colored distributions of the Okubo-Weiss parameter before and after correction are shown in Fig.~\ref{fig:okubo-weiss}. After correction, the Okubo-Weiss parameter has greater absolute values and a slightly smaller spatial extent (bottom figure) which is an improvement to the AVISO processing bias (top figure). This result constitutes a validation of the assimilation method presented in this paper showing that eddies were better resolved after assimilating drifter trajectories.
 
 \begin{figure}[!h]
@@ -592 +612 @@
 \includegraphics[scale=0.525]{./fig/okubo_weiss_aviso.png}
 \includegraphics[scale=0.525]{./fig/okubo_weiss_analyse.png}
-\caption{\label{fig:okubo-weiss} Okubo-Weiss parameter calculated on background field (upper panel) and corrected field (lower panel). The negativity of this parameter characterized eddies, and the absolute value corresponds to the intensity of the eddy. It can be noticed that eddy is smaller in size and more intense after the correction process.}
+\caption{\label{fig:okubo-weiss} Okubo-Weiss parameter calculated on background field (upper panel) and corrected field (lower panel). The negativity of this parameter characterizes eddies, and the absolute value corresponds to the intensity of the eddy. It can be noticed that eddy is smaller in size and more intense after the correction process.}
 \end{figure}
 
 \section{Conclusion}
-We presented a simple and efficient algorithm to blend drifter Lagrangian data with altimetry Eulerian velocities in the Eastern Levantine Mediterranean. After implementation, the method needs very few computing resources and is quick to converge, so it is well fitted for near-real time applications. Assimilating two successive drifter positions produces a correction of the velocity field within a radius of 20km and for approximatively 24h before and after the measurement.
-
-This algorithm was able to correct some typical weaknesses of altimetric fields, in particular the estimation of velocity near the coast and accurate estimations of eddies dimensions and intensity.
+
+A novel and efficient method for blending altimetry and surface drifters data was presented. The method is based on a variational assimilation approach for which the velocity is corrected 
+by matching observed drifters positions with those predicted by a simple advection model, taking into account the wind effect and imposing a divergence free condition on the geostrophic part of the velocity. The velocity correction is done in a time-continuous fashion by assimilating at once a whole trajectory of drifters using a sliding time window. %Sensitivity analysis showed that proper choice of the window size and time shift resulted in significant improvement in estimation of the velocity field. 
+Sensitivity analyses showed that significant improvement in the estimation of the velocity field can be achieved for a proper choice of the window size and time shift, even when few drifters are used. We found that assimilating two successive drifter positions produces a correction of the velocity field within a radius of 20 km and for approximatively 24 h before and after the measurement.
+The method was applied to two real experiments, one close to the Lebanese coast and one in a off-shore eddy between Lebanon and Cyprus. In these two scenarios, the method was able to correct some typical weaknesses of altimetric fields, in particular the estimation of velocity near the coast and accurate estimation of eddies dimensions and intensity.
+The algorithm needed very few computational resources and was quick to converge, rendering it well fitted for near-real time applications. 
+
+%The method also constrains the velocity field, except for the wind component, is to be divergence free. Imposing the divergence free condition corrected the velocity field near at the coast.  
+
+
+%We show that with few drifters, our method improves the estimation of velocity in two typical situations: an eddy between the Lebanese coast and Cyprus, and velocities along the Lebanese coast. 
+
+
+
+%The first experiment used data from three drifters in the Eastern Levantine Mediterranean (off the Lebanese coast) to correct the altimetric field. 
+
+
+
+%I would split this into statements: 
+% first for the novelty of the method
+%        - Moving window with overlap
+%        - minimization over all trajectories (within the window) at once
+%        - divergence-free constraint
+% second: application in the eastern mediterranean.
+%        - coastal region (this particular region is not well studied before)
+%        - Eddy: ... 
+%We presented a simple and efficient algorithm to blend drifter Lagrangian data with altimetric Eulerian velocities in the Eastern Levantine Mediterranean. 
+%
+%
+%
+%Assimilating two successive drifter positions produces a correction of the velocity field within a radius of 20 km and for approximatively 24 h before and after the measurement.
+%This algorithm was able to correct some typical weaknesses of altimetric fields, in particular the estimation of velocity near the coast and accurate estimations of eddies dimensions and intensity.
 
 \section{Acknowledgement}