Changeset 196 for altifloat/doc

Timestamp:

11/18/15 11:18:51 (9 years ago)

Author:

leila_ocean

Message:

Nov 18, Dan's comments, Leila

File:

• altifloat/doc/ocean_modelling/Draft1.tex (modified) (14 diffs)

altifloat/doc/ocean_modelling/Draft1.tex

@@ -138 +138 @@
 \section{Introduction}
 \label{}
-An accurate estimation of mesoscale to sub-mesoscale surface dynamics of the ocean is critical in several applications in the Eastern Levantine Mediterranean basin. For instance, this estimation can be used in the study of pollutant dispersion in this heavily populated region. A good knowledge of the surface velocity field is challenging, especially that direct observations are relatively sparse in this region. 
-
-Altimetry has been widely used to predict the large mesoscale features of the ocean resolving typically lengths on the order of $100$ km \citep{chelton2007global}. There are however limitations to its usage. It is inaccurate in resolving short temporal and spatial scales of some physical processes, like eddies, which results in blurring these structures. Further errors and inaccuracies occur near the coastal areas (within 20-50 km from land),
+An accurate estimation of mesoscale to sub-mesoscale surface dynamics of the ocean is critical in several applications in the Eastern Levantine Mediterranean basin. For instance, this estimation can be used in the study of pollutant dispersion, which is important in this heavily populated region. A good knowledge of the surface velocity field is challenging, especially when direct observations are relatively sparse.
+
+Altimetry has been widely used to predict the mesoscale features of the ocean resolving typically lengths on the order of $100$ km \citep{chelton2007global}. There are however limitations to its usage. It is inaccurate in resolving short temporal and spatial scales of some physical processes, like eddies, which results in blurring these structures. Further errors and inaccuracies occur near the coastal areas (within 20-50 km from land),
 where satellite information is degraded; this is due to various factors such as land contamination, inaccurate tidal and geophysical
 corrections and incorrect removal
 of high frequency atmospheric effects at the sea surface \citep{caballero2014validation}.
 
-To improve geostrophic velocities, especially near the coast, in situ observations provided by drifters, can be considered (e.g. \citet{bouffard2008, ruiz2009mesoscale}). %[Bouffard et al., 2010; Ruiz et al., 2009] . 
-Drifters follow the currents and when numerous, they allow for an extensive spatial coverage of the region of interest. They are relatively not very expensive, easily deployable and provide accurate information on their position and other
+To improve geostrophic velocities, especially near the coast, in situ observations provided by drifters can be considered (e.g. \citet{bouffard2008, ruiz2009mesoscale}). %[Bouffard et al., 2010; Ruiz et al., 2009] . 
+Drifters follow the currents and when numerous, they allow for an extensive spatial coverage of the region of interest. They are not very expensive, easily deployable and provide accurate information on their position and other
 environmental parameters \citep{lumpkin2007measuring}.
 
-To illustrate the information provided by drifters data, we show in Figure~\ref{fig:cnrs} the real-time positions of three drifters launched south of Beirut on August 28 2013. These positions can be compared to the positions that would have been obtained if the drifters were advected by the altimetric velocity field. We observe that unlike the corresponding positions simulated by the altimetric field provided by \textit{Aviso}(see section~\ref{sec:aviso}), the drifters stay within 10-20 km from the coast. The background velocity field shown in that figure is the geostrophic field, averaged over a period of 6 days. The drifters' in situ data render a more precise image of the local surface velocity than the altimetric one; however, this only possible along the path following their trajectory. These types of data are therefore complementary. In this work, we propose a new algorithm that blends geostrophic and drifters data in an optimal way, taking into account the wind effect. The algorithm is then used to estimate the surface velocity field in the 
+To illustrate the information provided by drifters data, we show in Figure~\ref{fig:cnrs} the real-time positions of three drifters launched south of Beirut on August 28 2013. These positions can be compared to the positions that would have been obtained if the drifters were advected by the altimetric velocity field. We observe that unlike the corresponding positions simulated by the altimetric field provided by \textit{Aviso} (see section~\ref{sec:aviso}), the drifters stay within 10-20 km from the coast. The background velocity field shown in that figure is the geostrophic field, averaged over a period of 6 days. The drifters' in situ data render a more precise image of the local surface velocity than the altimetric one; however, this only possible along the path following their trajectory. These types of data are therefore complementary. In this work, we propose a new algorithm that blends geostrophic and drifters data in an optimal way, taking into account the wind effect. The algorithm is then used to estimate the surface velocity field in the 
 Eastern Levantine basin, in particular in the region between Cyprus and the Syrio-Lebanese coast, a part of the Mediterranean basin that has not been so well studied in the literature before. 
 
@@ -155 +155 @@
 From the methodological point of view, combining altimetric and drifters data has been done using statistical approaches, with availability of extensive data sets. A common approach is to use regression models to combine geostrophic, wind and drifters components, with the drifters' velocity component being computed from drifters' positions using a pseudo-Lagrangian approach. When large data sets are available, this approach produces an unbiased refinement of the geostrophic circulation maps, with better spatial resolution. (e.g. \citet{poulain2012surface,menna2012surface,uchida2003eulerian,maximenko2009mean,niiler2003near}).
 Another approach relies on variational assimilation, a method classically used in weather predictions \citep{courtier1994strategy,dimet1986variational}.
-In the context of bending altimetric and drifters' data, the method was used by \citet{taillandier2006variational} and it is based on a simple advection model for the drifters' positions, matched to observations via optimisation. The implementation of the method first assumes the time-independent approximation of the velocity correction, then superimposes inertial oscillations on the mesoscale field. 
-These variationnal techniques had
-led to the development of the so called "LAgrangian Variational Analysis" (LAVA) ,  initially tested and applied to correct model velocity fields using drifter trajectories \citep{taillandier2006assimilation,taillandier2008variational} and later 
- customised to several other applications such as model assimilation \citep{chang2011enhanced,taillandier2010integration}. Recently,  \citet{berta2015improved} applied it to estimate surface currents in the Gulf of Mexico, where they also added a measure of performance consisting of skill scores, that compare 
-the separation between observed and hindcast trajectories to the observed absolute dispersion.
-
-
-From the application point of view, blending drifters and altimetric data has been successfully applied to several basins, for example in: the gulf of Mexico \citep{berta2015improved}, the black sea \citep{kubryakov2011mean}, the North Pacific \citep{uchida2003eulerian}, and the Mediterranean basin \citep{taillandier2006assimilation,poulain2012surface,menna2012surface}. In \citet{menna2012surface}, there was a particular attention to the levantine sub-basin, where large historical data sets from 1992 to 2010 were used to characterise surface currents. 
-The specific region which lies between the coasts of Lebanon, Syria and Cyprus, is however characterised by sparsity of data. In the present work, we use in addition to the data sets used in \citet{menna2012surface}, more recent data from 2013 (in the context of Altifloat project) to study this particular region.  
-
-
-Our contribution focuses on the methodical aspect, and it can be considered an extension of the variational approach used in \citet{taillandier2006variational}. The first purpose is to add more physical considerations to the surface velocity estimation, without making the method too complex, in order to still allow for Near Real Time applications. We constrain the geostrophic component of that velocity to be divergence-free, and we add a component due to the effect of the wind, in the fashion done in \citet{poulain2009}. We also provide a time-continuous correction by: (i) assimilating a whole trajectory of drifters at once and (ii) using a moving time window where observations are correlated. 
-
-We show that with few drifters, our method (i) improves the estimation of an eddy between the Lebanese coast and Cyprus, and (ii) predicts real drifters trajectories along the Lebanese coast. 
+In the context of bending altimetric and drifters' data, the method was used by \citet{taillandier2006variational} and it is based on a simple advection model for the drifters' positions, matched to observations via optimisation. In the work of \citet{taillandier2006variational}, the implementation of the method first assumes the time-independent approximation of the velocity correction, then superimposes inertial oscillations on the mesoscale field. 
+These variational techniques had
+led to the development of the so called ``LAgrangian Variational Analysis" (LAVA),  initially tested and applied to correct model velocity fields using drifter trajectories \citep{taillandier2006assimilation,taillandier2008variational} and later 
+ customised to several other applications such as model assimilation \citep{chang2011enhanced,taillandier2010integration} and more recently blending drifters and altimetry to estimate surface currents in the Gulf of Mexico \citet{berta2015improved}. 
+ %applied it 
+ %, where they also added a measure of performance consisting of skill scores, that compare 
+%the separation between observed and hindcast trajectories to the observed absolute dispersion.
+
+
+From the application point of view, blending drifters and altimetric data has been successfully applied to several basins, for example in: the Gulf of Mexico \citep{berta2015improved}, the Black Sea \citep{kubryakov2011mean}, the North Pacific \citep{uchida2003eulerian}, and the Mediterranean Sea \citep{taillandier2006assimilation,poulain2012surface,menna2012surface}. In \citet{menna2012surface}, there was a particular attention to the levantine sub-basin, where large historical data sets from 1992 to 2010 were used to characterise surface currents. 
+The specific region which lies between the coasts of Lebanon, Syria and Cyprus, is however characterised by a scarcity of data. In the present work, we use in addition to the data sets used in \citet{menna2012surface}, more recent data from 2013 (in the context of Altifloat project) to study this particular region.  
+
+
+Our contribution focuses on the methodological aspect, and it can be considered an extension of the variational approach used in \citet{taillandier2006variational}. The first purpose is to add more physical considerations to the surface velocity estimation, without making the method too complex, in order to still allow for Near Real Time applications. We constrain the geostrophic component of that velocity to be divergence-free, and we add a component due to the effect of the wind, in the fashion done in \citet{poulain2009}. We also provide a time-continuous correction by: (i) assimilating a whole trajectory of drifters at once and (ii) 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. 
 
 
@@ -268 +270 @@
 
 
-\subsection{Linearized model for Lagrangian data}
+\subsection{Linearised 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
@@ -277 +279 @@
 
 
-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  
+The observation operator, denoted it schematically by  $\bo{r}=\mathcal{M} (\bo{u}, \bo{r}),$ consists then of numerical advection and interpolation $\mathcal{I}$, 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}
-
+\bo{r}(k\delta t)=\bo{r}((k-1)\delta t)+\delta t \, \mathcal{I}(\bo{u}((k-1)\delta t), \bo{r}((k-1)\delta t)), \,\,\,\,\, k=1,2, \cdots
+\end{equation}
 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
+\mathcal{I}(\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}
@@ -305 +306 @@
 \bo{u}&=\bo{u^b}+\delta \bo{u}. 
 \end{align}
-The linearized equations become 
+The linearised 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} 
+&\bo{r^b}(k\delta t)=\bo{r^b}((k-1)\delta t)+\delta t \, \mathcal{I}\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 \, \{ \mathcal{I}(\bo{\delta u},\bo{r^b}((k-1)\delta t)) \\ \notag 
+&+ \bo{\delta r}((k-1)\delta t) \cdot \partial _{(x,y)} \mathcal{I} \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 initialised 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 $(x,y)$. 
+Here, $ \partial _{(x,y)} \mathcal{I}$ is the derivative of the interpolation operator with respect to $(x,y)$. 
 
 
@@ -324 +325 @@
 
 
-\subsection{Algorithm for assimilation}
-
-We perform sequences of optimisations, where we minimise the following objective function with respect to the time independent correction $\delta \bo{u},$ in a specific time window $[0,T_w].$
+\subsection{Algorithm for velocity correction}
+
+We perform sequences of optimisations, where we minimise 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}) 
+\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}
 
@@ -335 +336 @@
 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 regularisation and information spreading or smoothing. To obtain $\bo{B}$, we use the diffusion filter method of \citet{weaver2001correlation}, where a priori information on the typical length scale $R$ of the Eulerian velocity can be inserted.
+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 regularisation and information spreading or smoothing. To obtain $\bo{B}$, we use the diffusion filter method of \citet{weaver2001correlation}, 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 regularisation 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. 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 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.
+Inside a specific time window $[0, T_w]$, a whole trajectory of drifters contribute to give a constant correction in time $\bo{\delta u}.$ 
+We now refine the method in a way to produce a smooth time-dependent velocity field in $[0, T_f]$, that accounts for temporal correlations in these various corrections $\delta u.$
+One way to achieve this is to slide 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 a superposition of the time dependent background field and 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 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.
 
 
 We end this section by pointing out that we implement the algorithm described above in YAO~\citep{badran2008},
-a numerical tool very well adapted to variational assimilation problems that simplifies the computation and implementation of the adjoint needed in the optimisation. 
+a numerical tool very well adapted to variational assimilation problems that simplifies the computation and implementation of the adjoint needed in the optimisation. \textcolor{red}{Give CPU time}
 
 
@@ -355 +359 @@
 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},
 \end{equation}
-where $\big | \big |. \big | \big |$ refers the the $L_2$ norm of a vector, and $\bo{u}$ 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.
+where $\big | \big |. \big | \big |$ refers the the $L_2$ norm of a vector, and $\bo{u}$ 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 corrected and true velocity fields.
 
 
@@ -368 +372 @@
 \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.}
+\caption{Region of RMS error computation for the twin experiment. Observations generated by CYCOM model starting on Sept 1st 2013 (for 3 days) are shown on top of the background field. The red locations correspond to Altifloat drifters' locations.}
 \label{fig:synth}
 \end{center}
@@ -382 +386 @@
 \end{figure}
 We then show the effect of shifting the window by varying the parameter $\sigma.$ The window size here is fixed to $T_{w}=24$ 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. 
+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 \textcolor{red}{put an interpretation, we exploit the correlation...}. 
 \begin{figure}[htbp]
 \begin{center}
@@ -392 +396 @@
 \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  $\Delta t=6$ h 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. 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. 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 $T_w=24$ h and $\Delta t=2$ h.}
+\caption{The effect of the number of drifters. More drifters yield better corrections but corrections are possible with 3 drifters only. 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}
@@ -406 +410 @@
 \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 $T_w=24$ h, and $N_f=14.$ The realistic scenario of  $\Delta t=6$ h is not too far from the smallest $\Delta t$.}
+\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 $\Delta t=2$ h.}
 \label{fig:time}
 \end{center}
 \end{figure}
 
-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 
+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 ``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 
 \begin{equation} \label {L2Error}
 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.
+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 'fixed' the poorly simulated coastal meander.
 
 
@@ -429 +433 @@
 \end{subfigure}
 \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.}
+\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 ``true" observations(black) are shown on top of the error.}
 \label{fig:lerror}
 \end{figure}
@@ -486 +490 @@
 \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.  $T_w=24$ 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. }
+\caption{\label{fig:leb1} Prediction of the positions of 3 Altifloat 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}