Changeset 182 for altifloat/doc

Timestamp:

10/16/15 10:09:22 (9 years ago)

Author:

leila_ocean

Message:

Intro in progress

File:

• altifloat/doc/ocean_modelling/Draft1.tex (modified) (8 diffs)

altifloat/doc/ocean_modelling/Draft1.tex

@@ -137 +137 @@
 \section{Introduction}
 \label{}
-Recurrent marine pollution like the ones observed near the coastal regions of Lebanon is a major threat to the marine environment and halieutic resources. Polluting agents do not only have a immediate local effect but they are also transported through ocean currents to deep waters far away from the coast thereby having a long term, large scale effect. Modeling ocean currents is thus a major topic that mobilizes multidisciplinary researchers.  
-
-The objective of this project is to combine two types of observations (drifters and satellite data) in order to reconstruct mesoscale features of the surface currents in the Levantine Mediterranean including coastal regions. The figure below shows positions of drifters and altimetry data in the Eastern Mediterranean from the 27th to the 29th of November 2009: the altimetric data gives the velocity field in the whole basin with a resolution of $10$ km smoothing considerably some mesoscale features with errors especially near the coastal regions. The buoys on the other hand allow a better reconstruction of the velocity field but only along their trajectories. These two types of data are therefore complimentary and we can then formulate the problem as follows: find the optimal velocity field from these two sources of information.   
-
-The mathematical problem consists of ``inverting'' the Lagrangian trajectories (Drifters positions) to correct the Eulerian velocity field given my the altimetric data. The literature on the subject is abundant: ranging from optimal interpolation  \cite{ozgokmen2003assimilation}, to statistical approaches based on extended Kalman filter methods \cite{salman2006method}, to variational techniques \cite{kamachi1995continuous}, \cite{nodet2006variational}, \cite{taillandier2006variational}.
-We propose to use a variational assimilation technique where we correct the Eulerian velocity field by minimizing the distance between a model solution and observations of the buoys. We use the incremental approach described in \cite{courtier1994strategy}, \cite{dimet2010variational}. The advantage of this method is that while being midway in complexity between OI (lower) and Kalman filter (higher), it permits a simple mathematical formulation of the dynamical constraints we wish to impose.
-
-
-
-\begin{figure}[htbp]
-\begin{center}
-\includegraphics[scale=0.5]{./fig/RealvsSimulatedTraj.pdf}
-\vspace{-30mm}
-\caption{Real CNRS observations of 3 drifters starting Aug 28 2013 -x , vs simulated -- with aviso same date. On top of averaged Aviso for 6 days starting aug 28  }
+
+Recurrent marine pollution like the ones observed near the heavily populated coastal regions of the Eastern Levantine Mediterranean basin is a major threat to the marine environment and halieutic resources. Polluting agents, being transported through currents to either deep waters or to another part of the coast, have not only an immediate local effect, but also a long term, large scale one.
+It is clear that a good knowledge of the underlying surface velocity field is necessary to understand the dynamics of this transport process. 
+ Geostrophic velocities (measured by satellite altimeter) have been widely used to predict the large mesoscale features of the ocean resolving typically lengths on the order of $100$ km [Refs]. There are however limitations to their usage. They are 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); 
+in fact satellite information there is degraded 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. " [Caballero]. 
+
+
+To 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
+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 
+Eastern Levantine basin, with a focus on the region close to the Lebanese coast. (not well studied?). 
+
+
+The literature 
+\begin{itemize}
+\item From the methodological point of view, combining altimetric and drifters can be done using 
+\begin{enumerate}
+
+\item pseudo-Lagrangian + statistical approach (lots of data)  PPM et al. 
+\item The japanese [Uchida and Imawaki, 2003?
+\item Centurioni et. al.? what do they use
+\item Variational assimilation approach: [Thaillander, LAVA, etc..]. 
+\end{enumerate}
+\item From the application point of view, blending drifters and altimetric data has been applied to several ocean basins: gulf of mexico [LAVA], black sea [A. A. Kubryakov and S. V. Stanichny], japanese?, Mediterranean basin (ppm) but more west. 
+\end{itemize}
+Our approach focuses on the methodical aspect, and it can be considered as an extension of Thaillander's where we add the following: assimilate a whole trajectory, take into account the wind, impose a divergence free condition, and do a continuous time correction by doing a moving window. We consider two main regions of application and show that the method improves the estimation of an eddy, and predicts drifters trajectories along the coast.   
+Should we also talk about: One of the particularities of the Levantine bassin is an ensemble of 'tourbillons",? where?
+Plan of the paper goes here. 
+% le tourbillon au sud de Chypre et le tourbillon de Shikmona ˆ peu prs ˆ la mme latitude ˆ lÕouest de la c™te du Liban. Cet ensemble, parfois aussi appelŽ complexe tourbillonaire de Shikmona 12, est une structure permanente au sud de Chypre avec une variabilitŽ saisonnire. CÕest sur cet ensemble et sur son lien avec la topographie, notamment le mont sous-marin ƒratosthne sur lequel nous nous pencherons en particulier dans cette Žtude, comme dÕaprs la figure En effet, les monts sous-marins sont considŽrŽs comme une des causes des Žvolutions marines de mŽso-Žchelle.13. Dans le cas du mont ƒratosthne, il est possible que son influence sur la masse dÕeau environnante soit augmentŽe par son intŽraction avec les tourbillons quasi-permanents du complexe de Shikmona 14. La circulation dans le secteur du mont ƒratosthne est dominŽe par un anticyclone correspondant au tourbillon de Chypre. Here say that coastal configuration ??
+
+
+
+
+
+
+
+
+
+
+
+\begin{figure}[htbp]
+\begin{center}
+\includegraphics[scale=0.7]{./fig/RealvsSimulatedTraj.pdf}
+\vspace{-30mm}
+\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.}
 \end{center}
 \end{figure}
@@ -201 +233 @@
 \subsection{Statement of the problem}
 
-We consider $N_f$ Lagrangian drifters released at time $t=0$ at various locations. These drifters provide their positions each $\Delta t$, over a period $[0,T_f]$.
-Our objective is to determine an estimate of the two-dimensional Eulerian velocity field  \begin{equation}\notag
+We consider $N_f$ Lagrangian drifters released at time $t=0$ at various locations. These drifters provide their positions every $\Delta t$, over a period $[0,T_f]$.
+Our objective is to determine an estimate of the two-dimensional Eulerian surface velocity field  \begin{equation}\notag
 \mathbf{u}(x,y,t)=(u(x,y,t),v(x,y,t)) 
 \end{equation}
-characterized by a length scale $R$ [Refs], given observations of the drifters' positions 
+characterized by a typical length scale $R$, given observations of the drifters' positions 
 \begin{equation}
 \bo{r}^{obs}_i(n\Delta t), \,\,\, i=1,2, \cdots, N_f, \,\,\, n=1, 2, \cdots N, \,\,\, \text{where}\,\,\, N \Delta t= T_f.
 \end{equation}
-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 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.
+The velocity shall be estimated on a specified grid with resolution of $1/8^{\circ}$ in both longitude and latitude, and 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 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. 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 component due to the wind [Refs]. The details of this procedure are given in Section 3.3.
 
 
@@ -217 +249 @@
 
 \subsection{Linearized 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
@@ -224 +254 @@
  \odif{\mathbf{r}}{t}=\mathbf{u}(\mathbf{r}(t),t), \,\,\,\,\bo{r}(0)=\bo{r}_0, \mathbf{u}(x,y,0)=\bo{u}_0.
  \end{equation}
-This equation is integrated numerically using an Euler scheme for example. Since the drifters positions may not coincide with the Eulerian velocity's grid points, a spatial interpolation of $\mathbf{u}$ to these positions is also needed. 
+This equation is integrated numerically, for example, using an Euler scheme. Since the drifters positions do not coincide with the Eulerian velocity's grid points, a spatial interpolation of $\mathbf{u}$ to these positions is needed. 
 
 
@@ -290 +320 @@
 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 regularization and information spreading or smoothing. To obtain $\bo{B}$. we use the diffusion filter method of [Weaver and Courtier, 2001], 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 regularization and information spreading or smoothing. To obtain $\bo{B}$, we use the diffusion filter method of [Weaver and Courtier, 2001], 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 regularization 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. More here?.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 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???
+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 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.
 %Here $i=1,2, \cdots T_f/\delta t$
 
@@ -306 +336 @@
 
 \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 true fields. This is based on the time-dependent RMS error 
+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),
@@ -326 +356 @@
 \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.
+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. 
 \begin{figure}[htbp]
 \begin{center}
@@ -344 +374 @@
 \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. 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.
+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.
 \begin{figure}[htbp]
 \begin{center}