Changeset 175 for altifloat/doc

Timestamp:

09/23/15 10:06:49 (9 years ago)

Author:

leila_ocean

Message:

method 90%

File:

• altifloat/doc/ocean_modelling/Draft1.tex (modified) (5 diffs)

altifloat/doc/ocean_modelling/Draft1.tex

@@ -166 +166 @@
 \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 $[0,t_f]$.
+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
 \mathbf{u}(x,y,t)=(u(x,y,t),v(x,y,t)) 
@@ -172 +172 @@
 characterized by a length scale $R$ [Refs], 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.
+\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 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 3.5.
+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.
 
 
@@ -246 +246 @@
 \subsection{Algorithm for assimilation}
 
-
-
-We perform sequences of optimizations, where we minimize the following objective function with respect to the time independent control correction $\delta \bo{u},$ in a specific time window $[0,T_w].$
+We perform sequences of optimizations, where we minimize 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}) 
@@ -254 +252 @@
 
 The first component of the objective function quantifies the misfit between the model 
-obtained by iterations of \eqref{REquations}, to observations $\bo{r}^{\,obs}(m\Delta t)$. 
+obtained by iterations of \eqref{REquations}, and observations $\bo{r}^{\,obs}(m\Delta t)$. 
 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 the error covariance matrix  is $\mathbf{B}^{-1}.$ This term serves the dual purpose of regularization and information spreading or smoothing. We use the diffusion filter method of [Weaver and Courtier, 2001], where the Eulerian scale $R$ enters in ..
-The parameter $\alpha_1$ represents the relative weight of this regularization term with respect to the other terms, and it must be chosen carefully.
-The last component is a constraint on the geostrophic part of the velocity, required to stay divergence free. More here?.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 one fixed correction $\bo{\delta u}.$ We want to reconstruct 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 \[\bo{u_{corrected}}(t_i)=\bo{u^{b}}(t_i)+\sum_{k=0} w_k \bo{\delta u}_k.\]  A correction at at specific instant $t_i$ only takes into accounts windows sliding through $t_i$ and weight $w_i$ given to correction $\bo{\delta u}_i$ is inversely proportional to the ``distance" between time $t_i$ and the window's position. Here $i=1,2, \cdots T_f/\delta t$
+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.
+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 $\bo{\delta u}.$ 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. 
+%Here $i=1,2, \cdots T_f/\delta t$
 
 
@@ -270 +270 @@
 \subsection{Implementation of the assimilation method in YAO}
 
-\section{Twin Experiments}
-In the twin experiment approach, the observations are simulated using a dynamic model for the Eulerian velocity field. \textcolor{red}{Need details}.
+\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 
+\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. 
+
+
+
+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 $\sigma.$
+