Changeset 184 for altifloat


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Timestamp:
10/19/15 20:34:25 (9 years ago)
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jbrlod
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several small modifications

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  • altifloat/doc/ocean_modelling/Draft1.tex

    r183 r184  
    147147 
    148148To improve geostrophic velocities, especially near the coast, several types of data can be combined: specifically in situ observations, [Bouffard et al., 2010; Ruiz et al., 2009] provided by drifters, are useful. 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 can provide accurate information on their position and other 
    149 environmental parameters [Lumpkin and Pazos, 2007]. Figure 1. shows the real-time positions of three drifters launched south of Beirut on August 28 2013, in the context of the ALTIFLOAT* project. We observe from that figure that, unlike the corresponding positions simulated by the geostrophic field (provided here by AVISO), the drifters stay within 10-20 km from the coast. The background field shown in that figure is the geostrophic field, averaged over a period of 6 days. The drifters' data render a more precise image of the surface velocity than the altimetric one, because it includes geostrophic and non-geostrophic components; however, this only possible along the path following their trajectory. These types of data are therefore complementary and we propose to blend them in an optimal way in order to estimate the surface velocity field in the  
     149environmental parameters [Lumpkin and Pazos, 2007]. Figure~\ref{fig:cnrs} shows the real-time positions of three drifters launched south of Beirut on August 28 2013, in the context of the ALTIFLOAT* project. We observe from that figure that, unlike the corresponding positions simulated by the geostrophic field (provided here by AVISO), the drifters stay within 10-20 km from the coast. The background field shown in that figure is the geostrophic field, averaged over a period of 6 days. The drifters' data render a more precise image of the surface velocity than the altimetric one, because it includes geostrophic and non-geostrophic components; however, this only possible along the path following their trajectory. These types of data are therefore complementary and we propose to blend them in an optimal way in order to estimate the surface velocity field in the  
    150150Eastern Levantine basin, with a focus on the region close to the Lebanese coast. (not well studied?).  
    151151 
    … …  
    181181\vspace{-30mm} 
    182182\caption{CNRS drifters deployed in the context of the ALTIFLOAT project starting Aug 28 2013 (shown in  $-$x) , versus  trajectories simulated with AVISO (shown in $--$).  The velocity field shown is the  AVISO field, averaged over 6 days.} 
     183\label{fig:cnrs} 
    183184\end{center} 
    184185\end{figure} 
    … …  
    219220\subsection{Wind Data} 
    220221ECMW ERA-Interim wind products~\citep{Dee2011} were extracted in order to estimate wind-driven currents. Wind velocities closest to the surface (10 m) were extracted at a resolution of 1/8$^o$ at the same grid point as the \textit{Aviso} data. The data were resampled on a hourly time step. 
     222 
     223Wind velocities were used to estimate wind-driven effect on drifter velocity. 
     224The eulerian velocity field being used to estimate drifter succesive positions is the sum of the geostrophic velocity and the wind induced velocity given by the formula~\citep{poulain2009}: 
     225\begin{equation} 
     226\mathbf{U_{wind}} = 0.01exp(-28^oi)\times \mathbf{U_{10}} 
     227\end{equation} 
     228where $\mathbf{U_{wind}} = u_{wind}+iv_{wind}$ is the velocity induced by the wind and $ \mathbf{U_{10}} = u_{10}+iv_{10}$ is the wind velocity above the surface (10m) expressed as complex numbers. 
    221229 
    222230\subsection {Model data} 
    … …  
    342350 
    343351 
    344 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. 
     352The 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 (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.~\ref{fig:synth}. 
    345353 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].  
    346354  
    347355  
    348 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). 
     356Using 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~\ref{fig:wsize},\ref{fig:mwin},\ref{fig:numb}). 
    349357\begin{figure}[htbp] 
    350358\begin{center} 
    … …  
    352360\vspace{-30mm} 
    353361\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.} 
    354 \end{center} 
    355 \end{figure} 
    356 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 we consider corrections that are time independent in each window. 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 fact that the correction is fixed in a specific window, as mentioned before.  
     362\label{fig:synth} 
     363\end{center} 
     364\end{figure} 
     365We 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 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. Finally, we mention 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.  
    357366\begin{figure}[htbp] 
    358367\begin{center} 
    … …  
    360369\vspace{-30mm} 
    361370\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} 
    362 \end{center} 
    363 \end{figure} 
    364 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.  
     371\label{fig:wsize} 
     372\end{center} 
     373\end{figure} 
     374We then show the effect of shifting the window by varying the parameter $\sigma.$ The window size here is fixed to $T_{w}=24$ hs, time sampling to $\Delta t=2$ hs, and $N_f=14.$ We start with $\sigma=0$, which amounts to doing separate corrections, then slide every $\sigma=12, 8, 6$ hs. In Fig.~\ref{fig:mwin}, we show the results by displaying the relative RMS error before and after the correction.  
    365375We 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.  
    366376\begin{figure}[htbp] 
    … …  
    370380\caption{The effect of the moving window: smaller shifts $\sigma$ yield smoother and better corrections. Here $N_f=14$ and $\Delta t=2$ hs. 
    371381 } 
    372 \end{center} 
    373 \end{figure} 
    374 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. 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 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. 
     382\label{fig:mwin} 
     383\end{center} 
     384\end{figure} 
     385The 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 3 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. 
    375386\begin{figure}[htbp] 
    376387\begin{center} 
    … …  
    378389\vspace{-30mm} 
    379390\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.} 
    380 \end{center} 
    381 \end{figure} 
     391\label{fig:numb} 
     392\end{center} 
     393\end{figure} 
     394 
    382395\begin{figure}[htbp] 
    383396\begin{center} 
    … …  
    385398\vspace{-30mm} 
    386399\caption{The effect of the time sampling} 
    387 \end{center} 
    388 \end{figure} 
    389  
    390 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. 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.  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.  
     400\label{fig:time} 
     401\end{center} 
     402\end{figure} 
     403 
     404As 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.  
    391405 
    392406 
    … …  
    397411  \includegraphics[width=1.3\linewidth]{./fig/Before_L2pointw_win24_MEAN_color.pdf} 
    398412%  \caption{Pressure drop.} 
    399   \label{bla} 
     413%  \label{bla} 
    400414\end{subfigure}% 
    401415\hspace{-10mm} 
    … …  
    404418  \includegraphics[width=1.3\linewidth]{./fig/After_L2pointw_win24_MEAN_color.pdf} 
    405419%  \caption{Mass flowrate.} 
    406   \label{blo} 
     420%  \label{blo} 
    407421\end{subfigure} 
    408422\vspace{-30mm} 
    409423\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.} 
    410 \label{fblo} 
     424\label{fig:lerror} 
    411425\end{figure} 
    412426 
    … …  
    418432  \includegraphics[width=1.2\linewidth]{./fig/Before_CORRANGLEpointw_win24_MEAN_color.pdf} 
    419433%  \caption{Pressure drop.} 
    420   \label{bla} 
     434%  \label{bla} 
    421435\end{subfigure}% 
    422436\hspace{-10mm} 
    … …  
    425439  \includegraphics[width=1.2\linewidth]{./fig/After_CORRANGLEpointw_win24_MEAN_color.pdf} 
    426440%  \caption{Mass flowrate.} 
    427   \label{blo} 
     441%  \label{blo} 
    428442\end{subfigure} 
    429443\vspace{-30mm} 
    430444\caption{Averaged correction in terms of direction. Left: $\cos{(\bo{u}_{b},\bo{u}_{true})}$, right: $\cos{(\bo{u}_{corrected},\bo{u}_{true})}.$ } 
    431 \label{fblo} 
     445\label{fig:cerror} 
    432446\end{figure} 
    433447 
    … …  
    452466\caption{Real Exp starting Aug 28 for 6 days, window=24h, move=6hs, } 
    453467\end{center} 
     468\label{fig:lvel} 
    454469\end{figure} 
    455470 
    … …  
    461476\caption{Real 2-1 Exp starting Aug 28 for 2 days, window=24h, move=24hs, } 
    462477\end{center} 
     478\label{fig:lzoom} 
    463479\end{figure} 
    464480 
    465481\subsection{\label{sec:cyprus}Improvement of velocity field in an eddy} 
    466482In the context of the Nemed deployment (see section ~\ref{sec:drifters}), we selected two drifters trajectories from 25 August 2009 to 3 September 2009. Assimilating the successive positions of the drifters every six hours, the AVISO velocity field was corrected. 
     483 
     484In 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. 
    467485 
    468486In 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.  
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