Changeset 194 for altifloat/doc

Timestamp:

11/09/15 10:32:04 (9 years ago)

Author:

leila_ocean

Message:

Nov9

Location:

altifloat/doc/ocean_modelling

Files:

• Draft1.tex (modified) (9 diffs)
• mybib.bib (modified) (1 diff)

altifloat/doc/ocean_modelling/Draft1.tex

@@ -352 +352 @@
 In the twin experiment approach, the observations are simulated using a known velocity field, the ``true" velocity $\bo{u}_{true}.$ In this context, we are able to assess the validity of our approach by comparing the corrected, $\bo{u}_{corrected}$, and true fields. This is based on the time-dependent RMS error 
 \begin{equation} \label {RMSError}
-error (u, t)=\bigg(    \frac{\sum_{i,j} |\bo{u}_{true} (i,j,t)-\bo{u} (i,j,t)|^2}{\sum_{i,j} |\bo{u}_{true} (i,j,t)|^2 } \bigg),
-\end{equation}
-where $\bo{u}$ could be $\bo{u}_b$, giving the error before assimilation or $\bo{u}_{corrected}$, giving the error after assimilation. We shall also compare the trajectories of the drifters obtained by simulation with the corrected velocity field and the observed trajectories. 
+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.
 
 
 
 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, \textcolor{red}{which is consistent with the range of values found in the wider region? in \citep{taillandier2006variational}.}
+ 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}.
  
  
@@ -371 +371 @@
 \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 $T_L,$ \textcolor{red}{which is estimated as $1-3$ days in this region,? ,} 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 $\Delta_t=2$ 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. 
 \begin{figure}[htbp]
 \begin{center}
@@ -391 +391 @@
 \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 a more averaged error curve 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  $\Delta t=6$ hs still yields a good correction.
 \begin{figure}[htbp]
 \begin{center}
@@ -410 +410 @@
 \end{figure}
 
-As an additional assessment of our method, we show the trajectories of the drifters reconstructed with the corrected velocity field on top of the actual observations, for the same experiment described above with the following parameters: a window of size $T_w=24$ hs, a shift of $\sigma=6$ hs, $N_f=14$ and $\Delta t=2$ hs. The left side of Fig.~\ref{fig:lerror} shows the point-wise $L_2$ error between the background and true fields in the region of interest, averaged over the duration of the experiment. The right side shows this error after correction as well as the agreement between the positions of the drifters simulated with the corrected field and the actual observations. Finally, the correction in terms of direction is shown in Fig.~\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. 
+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 
+\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. 
 
 
@@ -417 +421 @@
 \begin{subfigure}{0.55\textwidth}
 
-  \includegraphics[width=1.3\linewidth]{./fig/Before_L2pointw_win24_MEAN_color.pdf}
+  \includegraphics[width=1.25\linewidth]{./fig/Before_L2pointw_win24_MEAN_color.pdf}
 
 \end{subfigure}%
@@ -423 +427 @@
 \begin{subfigure}{0.55\textwidth}
  % \centering
-  \includegraphics[width=1.3\linewidth]{./fig/After_L2pointw_win24_MEAN_color.pdf}
+  \includegraphics[width=1.25\linewidth]{./fig/After_L2pointw_win24_MEAN_color.pdf}
 
 \end{subfigure}
@@ -436 +440 @@
 \begin{subfigure}{0.55\textwidth}
  % \entering
-  \includegraphics[width=1.2\linewidth]{./fig/Before_CORRANGLEpointw_win24_MEAN_color.pdf}
+  \includegraphics[width=1.25\linewidth]{./fig/Before_CORRANGLEpointw_win24_MEAN_color.pdf}
 
 %  \label{bla}
@@ -443 +447 @@
 \begin{subfigure}{0.55\textwidth}
  % \centering
-  \includegraphics[width=1.2\linewidth]{./fig/After_CORRANGLEpointw_win24_MEAN_color.pdf}
+  \includegraphics[width=1.25\linewidth]{./fig/After_CORRANGLEpointw_win24_MEAN_color.pdf}
 
 %  \label{blo}
@@ -490 +494 @@
 \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 $-\tiny{\square}$ green, on top of observed position shown in light green $-o$. 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.}
+\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}
 

altifloat/doc/ocean_modelling/mybib.bib

@@ -3 +3 @@
 
 
-%% Created for Leila Issa at 2015-10-30 10:05:12 +0200 
+%% Created for Leila Issa at 2015-10-30 11:43:18 +0200