Changeset 179 for altifloat/doc/ocean_modelling/Draft1.tex

Timestamp:

10/05/15 08:25:58 (9 years ago)

Author:

leila_ocean

Message:

twin progress

File:

• altifloat/doc/ocean_modelling/Draft1.tex (modified) (7 diffs)

altifloat/doc/ocean_modelling/Draft1.tex

@@ -190 +190 @@
 \subsection {Model data}
 Model data of surface velocity fields were used to calibrate the assimilation method presented in section~\ref{sec:method}. The model selected was the CYCOFOS-CYCOM high resolution model~\citep{zodiatis2003} that covers the North-East Levantin Bassin 
-(31$^o$ 30’E - 36$^o$ 13’E  and 33$^o$ 30’N – 36$^o$ 55’N). The model forecast were used withou assimilation and were reinteroplated on a 1.8$^o$ grid point with an time step of one hour. The model forecast used for calibration purpose on September 2013.
+(31$^o$ 30’E - 36$^o$ 13’E  and 33$^o$ 30’N – 36$^o$ 55’N). The model forecast were used without assimilation and were reinteroplated on a 1.8$^o$ grid point with an time step of one hour. The model forecast used for calibration purpose on September 2013.
 %Wind + Dan
 
@@ -209 +209 @@
 The velocity shall be estimated on a specified grid with resolution $\Delta x,$ in the time frame $[0, T_f].$ 
 
-The estimation is done following a variational assimilation approach [Refs], whereby the first guessed velocity, or background $\bo{u_b}$, is corrected by matching the observations with a model that simulates the drifters' trajectories. This correction is done using a sliding time window of size $T_w$, where we assume $\Delta t<T_w<T_L,$ and where $T_L$ is the Lagrangian time scale associated with the drifters in the concerned region [Refs]. The background field is considered to be the sum of a geostrophic component (provided by altimetry) on which we impose a divergence free constraint, and a velocity due the wind [Refs]. The details of this procedure are given in Section 3.3.
+The estimation is done following a variational assimilation approach [Refs], whereby the first guessed velocity, or background $\bo{u_b}$, is corrected by matching the observations with a model that simulates the drifters' trajectories. This correction is obtained using a sliding time window of size $T_w$, where we assume $\Delta t<T_w \leq T_L,$ and where $T_L$ is the Lagrangian time scale associated with the drifters in the concerned region [Refs]. The background field is considered to be the sum of a geostrophic component (provided by altimetry) on which we impose a divergence free constraint, and a velocity due the wind [Refs]. The details of this procedure are given in Section 3.3.
 
 
@@ -293 +293 @@
 
 
-Inside a time window $[0, T_w]$, we assimilate a whole trajectory of drifters, which gives a constant correction $\bo{\delta u}.$ 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 other corrections $\bo{\delta u}_k$ in $[k\sigma, k\sigma +T_w], \, \, k=0, 1, 2 \cdots$. 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 at specific instant $t_i$ only takes into accounts $N^i_w$ windows sliding through $t_i$ and weight $w_k$ given to correction $\bo{\delta u}_k$ 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 accounts $N^i_w$ windows sliding through $t_i$. The weight given to correction $\bo{\delta u}_k$ is defined as \[w_k=\frac{1}{\vectornorm{t_i-tc_k}},\] where $tc_k$ is the centre of window $k,$ which means that it is inversely proportional to the ``distance" between time $t_i$ and the window's position. then interpolation???
 %Here $i=1,2, \cdots T_f/\delta t$
 
-
-
-
-DIAGRAM IS USEFUL!
-
-\subsection{Implementation of the assimilation method in YAO}
+We end this section by pointing out that we implement the algorithm described above in YAO, [Refs.]
+a numerical tool very well adapted to variational assimilation problems that simplifies the computation and implementation of the adjoint needed in the optimization. 
+
+
+
+
 
 \section{Twin Experiment}
-In the twin experiment approach, the observations are simulated using a known velocity field referred to as the true velocity $\bo{u}_{true}.$ In this context, we are able to assess the validity of our approach by comparing the corrected $\bo{u}_{corrected}$ and the true fields. This will be based on the time-dependent RMS error 
+In the twin experiment approach, the observations are simulated using a known velocity field referred to as the true velocity $\bo{u}_{true}.$ In this context, we are able to assess the validity of our approach by comparing the corrected ($\bo{u}_{corrected}$) and true fields. This is based on the time-dependent RMS error 
 \begin{equation} \label {RMSError}
 error (u, t)=\bigg(    \frac{\sum_{i,j} |\bo{u}_{true} (i,j,t)-\bo{u} (i,j,t)|^2}{\sum_{i,j} |\bo{u}_{true} (i,j,t)|^2 } \bigg),
@@ -312 +312 @@
 
 
-In our experiment, we obtain the true velocity field from a dynamic model \textcolor{red}{Need details from Dan}. We start the optimisation with the background velocity field as described in the method section. The region we study is located off the shore of Beirut Explain that CNRS drifters were launched south of Beirut, so we put drifters close to there. As shown in Fig. 2, we compute the error in a box between 33.7 and 34.25 North and 34.9 E and the coast. There we ``deploy" a maximum of 14 drifters. The duration of the experiment is 3 days (nothing forbids of going longer, but some drifters hit the coast). We shall study the density with respect to the number of drifters (respecting coverage), to the time sampling $\Delta t,$ to the window size $T_w$ and to the moving parameter $\sigma.$
-
-
-
+The configuration of our twin experiment is the following: we put ourselves in the same context as that of the real experiment conducted by the CNRS (refer to data section above), where the drifters are launched south of Beirut starting the end of August 2013. As shown in Fig. 2,  we deploy ``synthetic'' drifters in the region located  between 33.7 $^{\circ}$ and 34.25 $^{\circ}$ North and 34.9 $^{\circ}$ E and the coast. This is the same box in which the computation of the RMS error \eqref{RMSError} is done. The initial positions of the drifters shown in red coincide with the positions of the CNRS drifters on September first 2013.  The drifters' positions are simulated using a velocity field $\bo{u}_{true}$ obtained from the dynamic model described in (Refer to data section). The experiment starts on September first  2013, and lasts for a duration of $T_f=3$ days. In principle, nothing forbids us of conducting longer experiments, but in this coastal region, the drifters hit land after 3 days, as shown in Fig. 2.
+ The background velocity field is composed of the geostrophic component obtained from AVISO and the wind component as described in the method section. Starting September first 2013, these fields are interpolated to $\delta t$. The parameter $R$ is chosen to be $20$km, [Refs]. 
+ 
+ 
+Using the relative RMS error before and after assimilation as a measure, we first study the sensitivity of our method to the number of drifters $N_f$, time sampling $\Delta t,$ the window size $T_w$ and to the moving parameter $\sigma.$ (Figures 3 to 6).
 \begin{figure}[htbp]
 \begin{center}
 \includegraphics[scale=0.5]{./fig/RegionErroronAviso.pdf}
 \vspace{-30mm}
-\caption{Region for RMS Error Computation surrounding the observations generated by Dan's model of Sept 1st 2013. on top of the average over 3 days Aviso Field starting  Sept 1st 2013, }
-\end{center}
-\end{figure}
-
-
-
-\begin{figure}[htbp]
-\begin{center}
-\includegraphics[scale=0.5]{./fig/Shifts_win24_dt1_f14_tf72.pdf}
-\vspace{-30mm}
-\caption{The effect of the moving window for a window of size $24$hs with 14 drifters and $dt=2hs$ }
-\end{center}
-\end{figure}
-
-
-\begin{figure}[htbp]
-\begin{center}
-\includegraphics[scale=0.5]{./fig/Wins_optshift_dt1_f14_tf72.pdf}
-\vspace{-30mm}
-\caption{The effect of the window size for 14 drifters and $dt=2$hs}
-\end{center}
-\end{figure}
-
-\begin{figure}[htbp]
-\begin{center}
-\includegraphics[scale=0.5]{./fig/Nfs_win24_dt1_tf72.pdf}
-\vspace{-30mm}
-\caption{The effect of the number of drifters}
-\end{center}
-\end{figure}
-
-\begin{figure}[htbp]
-\begin{center}
-\includegraphics[scale=0.5]{./fig/Dts_win24_f14_tf72.pdf}
+\caption{Region of RMS error computation for the twin experiment. Observations generated by model starting on Sept 1st 2013, for 3 days, are shown on top of averaged background field. The red locations correspond to CNRS drifters locations.}
+\end{center}
+\end{figure}
+We first show the effect of the window size $T_w.$ On one hand, this parameter has to be within the Lagrangian time scale $T_L,$ which is estimated to $1-3$ days in this region, according to [Refs], but it also cannot be too large because of the linearisation approach. In Fig. 3, we show the results corresponding to various window sizes (fixing $N_f=14$ and $\Delta_t=2$ hs), by displaying the relative RMS error (computed in the aforementioned box), before and after the correction. We first see that the error curve (after correction) tends to increase generally as time goes by. This is due to this special coastal configuration where the first three drifters hit the shore after $48$ hs and also to the effect of the spatial filter, correcting the region behind the initial positions of the drifters. We then observe that the optimal window size for this configuration is $24$ hs, which is within the range mentioned above. The error in this case is reduced by almost a half from the original one. Finally, we mention that with window sizes close to three days, the algorithm becomes ill conditioned, which is expected due to the linear tangent hypothesis.
+\begin{figure}[htbp]
+\begin{center}
+\includegraphics[scale=0.4]{./fig/Wins_optshift_dt1_f14_tf72.pdf}
+\vspace{-30mm}
+\caption{The effect of the window size. Error before correction is shown with a solid line. Errors after are shown with symbols for several window sizes. Here $N_f=14$ and $\Delta t=2$ hs}
+\end{center}
+\end{figure}
+We then show the effect of shifting the window by varying the parameter $\sigma.$ The window size here is fixed to $T_{w}=24$ hs, time sampling to $\Delta t=2$ hs, and $N_f=14.$ We start with $\sigma=0$, which amounts to doing separate corrections, then slide every $\sigma=12, 8, 6$ hs. In Fig. 4, we show the results by displaying the relative RMS error before and after the correction. 
+We observe that if the corrections are done separately, the correction is not smooth; in fact smaller values of $\sigma$ yield not only smoother, but better corrections. 
+\begin{figure}[htbp]
+\begin{center}
+\includegraphics[scale=0.4]{./fig/Shifts_win24_dt1_f14_tf72.pdf}
+\vspace{-25mm}
+\caption{The effect of the moving window: smaller shifts $\sigma$ yield smoother and better corrections. Here $N_f=14$ and $\Delta t=2$ hs.
+ }
+\end{center}
+\end{figure}
+The effect of the number of drifters is shown next in Fig. 5. Respecting coverage, we start with $N_f=14$ drifters (positioned as shown in Fig. 2), then reduce to $N_f=10,6,3.$ Naturally more drifters yield a better correction but we notice that even with three drifters, the error is still reduced by $20\%$ and much more so close to the beginning of the experiment. If we took only the drifters that do not hit the coast before 3 days, we get a more averaged error curve as shown in the dashed curve in Fig. 5. 
+
+\begin{figure}[htbp]
+\begin{center}
+\includegraphics[scale=0.4]{./fig/Nfs_win24_dt1_tf72_nofail.pdf}
+\vspace{-30mm}
+\caption{The effect of the number of drifters. More drifters yield better corrections but corrections are possible with only 3 drifters. The dashed line shows the effect of just taking drifters that do not hit the shore before the end of the experiment. Here $T_w=24$ hs and $\Delta t=2$ hs.}
+\end{center}
+\end{figure}
+
+Finally, we show the effect of the time sampling parameter $\Delta t$ of the observations in Fig. 6. Curves after correction correspond to $\Delta t=6, 4$ and $2$ hours. There is not a large difference between these cases and the realistic scenario of  $\Delta t=6$ hs still yields a good correction.  too small means we can do a Lagrangian method.
+\begin{figure}[htbp]
+\begin{center}
+\includegraphics[scale=0.4]{./fig/Dts_win24_f14_tf72.pdf}
 \vspace{-30mm}
 \caption{The effect of the time sampling}
@@ -359 +361 @@
 \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. The left side of Fig. 7 shows the point-wise $L_2$ error between the background and true fields in the region of interest, averaged over the duration of the experiment. The right side shows this error after correction as well as the agreement between the positions of the drifters simulated with the corrected field and the actual observations. Finally, the correction in terms of direction is shown in Fig. 8.  show the angle correlation  corrections ... These are shown in Figures 7 to 8.
 
 \begin{figure}[htdp]
@@ -367 +367 @@
 \begin{subfigure}{0.55\textwidth}
  % \entering
-  \includegraphics[width=1.2\linewidth]{./fig/Before_L2pointw_win24_MEAN_color.pdf}
+  \includegraphics[width=1.3\linewidth]{./fig/Before_L2pointw_win24_MEAN_color.pdf}
 %  \caption{Pressure drop.}
   \label{bla}
@@ -374 +374 @@
 \begin{subfigure}{0.55\textwidth}
  % \centering
-  \includegraphics[width=1.2\linewidth]{./fig/After_L2pointw_win24_MEAN_color.pdf}
+  \includegraphics[width=1.3\linewidth]{./fig/After_L2pointw_win24_MEAN_color.pdf}
 %  \caption{Mass flowrate.}
   \label{blo}
 \end{subfigure}
 \vspace{-30mm}
-\caption{the point-wise L2 error before and after in m/s}
+\caption{Averaged point-wise $L_2$ error before (left) and after (right) correction. In the right frame, drifters' positions obtained by simulation with corrected field (magenta) versus actual observations(black) are shown on top of the error.}
 \label{fblo}
 \end{figure}