Changeset 193 for altifloat/doc/ocean_modelling/Draft1.tex

Timestamp:

10/30/15 13:38:01 (9 years ago)

Author:

jbrlod

Message:

minor modifications

File:

• altifloat/doc/ocean_modelling/Draft1.tex (modified) (12 diffs)

altifloat/doc/ocean_modelling/Draft1.tex

@@ -139 +139 @@
 An accurate estimation of mesoscale to sub-mesoscale surface dynamics of the ocean is critical in several applications in the Eastern Levantine Mediterranean basin. For instance, this estimation can be used in the study of pollutant dispersion in this heavily populated region. A good knowledge of the surface velocity field is challenging, especially that direct observations are relatively sparse in this region. 
 
-Altimetry has been widely used to predict the large mesoscale features of the ocean resolving typically lengths on the order of $100$ km (\cite{chelton2007global}). There are however limitations to its usage. It is 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),
+Altimetry has been widely used to predict the large mesoscale features of the ocean resolving typically lengths on the order of $100$ km \citep{chelton2007global}. There are however limitations to its usage. It is 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),
 where satellite information is degraded; this is 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 (\cite{caballero2014validation}).
-
-To improve geostrophic velocities, especially near the coast, in situ observations provided by drifters, can be considered (e.g. \cite{bouffard2008}, \cite{ruiz2009mesoscale}). %[Bouffard et al., 2010; Ruiz et al., 2009] . 
+of high frequency atmospheric effects at the sea surface \citep{caballero2014validation}.
+
+To improve geostrophic velocities, especially near the coast, in situ observations provided by drifters, can be considered (e.g. \citet{bouffard2008, ruiz2009mesoscale}). %[Bouffard et al., 2010; Ruiz et al., 2009] . 
 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 provide accurate information on their position and other
-environmental parameters (\cite{lumpkin2007measuring}).
-
-To illustrate the information provided by drifters data, we show in Figure~\ref{fig:cnrs} the real-time positions of three drifters launched south of Beirut on August 28 2013. These positions can be compared to the positions that would have been obtained if the drifters were advected by the altimetric velocity field. We observe that unlike the corresponding positions simulated by the altimetric field (provided by AVISO), the drifters stay within 10-20 km from the coast. The background velocity field shown in that figure is the geostrophic field, averaged over a period of 6 days. The drifters' in situ data render a more precise image of the local surface velocity than the altimetric one; however, this only possible along the path following their trajectory. These types of data are therefore complementary. In this work, we propose a new algorithm that blends geostrophic and drifters data in an optimal way, taking into account the wind effect. The algorithm is then used to estimate the surface velocity field in the 
+environmental parameters \citep{lumpkin2007measuring}.
+
+To illustrate the information provided by drifters data, we show in Figure~\ref{fig:cnrs} the real-time positions of three drifters launched south of Beirut on August 28 2013. These positions can be compared to the positions that would have been obtained if the drifters were advected by the altimetric velocity field. We observe that unlike the corresponding positions simulated by the altimetric field provided by \textit{Aviso}(see section~\ref{sec:aviso}), the drifters stay within 10-20 km from the coast. The background velocity field shown in that figure is the geostrophic field, averaged over a period of 6 days. The drifters' in situ data render a more precise image of the local surface velocity than the altimetric one; however, this only possible along the path following their trajectory. These types of data are therefore complementary. In this work, we propose a new algorithm that blends geostrophic and drifters data in an optimal way, taking into account the wind effect. The algorithm is then used to estimate the surface velocity field in the 
 Eastern Levantine basin, in particular in the region between Cyprus and the Syrio-Lebanese coast, a part of the Mediterranean basin that has not been so well studied in the literature before. 
 
 
-From the methodological point of view, combining altimetric and drifters data has been done using statistical approaches, with availability of extensive data sets. A common approach is to use regression models to combine geostrophic, wind and drifters components, with the drifters' velocity component being computed from drifters' positions using a pseudo-Lagrangian approach. When large data sets are available, this approach produces an unbiased refinement of the geostrophic circulation maps, with better spatial resolution. (e.g. \cite{poulain2012surface}, \cite{menna2012surface}, \cite{uchida2003eulerian}, \cite{maximenko2009mean}, \cite{niiler2003near}).
-Another approach relies on variational assimilation, a method classically used in weather predictions (\cite{courtier1994strategy}, \cite{dimet1986variational}).
-In the context of bending altimetric and drifters' data, the method was used by (\cite{taillandier2006variational}) and it is based on a simple advection model for the drifters' positions, matched to observations via optimisation. The implementation of the method first assumes the time-independent approximation of the velocity correction, then superimposes inertial oscillations on the mesoscale field. 
+From the methodological point of view, combining altimetric and drifters data has been done using statistical approaches, with availability of extensive data sets. A common approach is to use regression models to combine geostrophic, wind and drifters components, with the drifters' velocity component being computed from drifters' positions using a pseudo-Lagrangian approach. When large data sets are available, this approach produces an unbiased refinement of the geostrophic circulation maps, with better spatial resolution. (e.g. \citet{poulain2012surface,menna2012surface,uchida2003eulerian,maximenko2009mean,niiler2003near}).
+Another approach relies on variational assimilation, a method classically used in weather predictions \citep{courtier1994strategy,dimet1986variational}.
+In the context of bending altimetric and drifters' data, the method was used by \citet{taillandier2006variational} and it is based on a simple advection model for the drifters' positions, matched to observations via optimisation. The implementation of the method first assumes the time-independent approximation of the velocity correction, then superimposes inertial oscillations on the mesoscale field. 
 These variationnal techniques had
-led to the development of the so called ``LAgrangian Variational Analysis" (LAVA) ,  initially tested and applied to correct model velocity fields using drifter trajectories (\cite{taillandier2006assimilation}, \cite{taillandier2008variational}) and later 
- customised to several other applications such as model assimilation (\cite{chang2011enhanced},\cite{taillandier2010integration}). Recently,  \cite{berta2015improved} applied it to estimate surface currents in the Gulf of Mexico, where they also added a measure of performance consisting of skill scores, that compare 
+led to the development of the so called "LAgrangian Variational Analysis" (LAVA) ,  initially tested and applied to correct model velocity fields using drifter trajectories \citep{taillandier2006assimilation,taillandier2008variational} and later 
+ customised to several other applications such as model assimilation \citep{chang2011enhanced,taillandier2010integration}. Recently,  \citet{berta2015improved} applied it to estimate surface currents in the Gulf of Mexico, where they also added a measure of performance consisting of skill scores, that compare 
 the separation between observed and hindcast trajectories to the observed absolute dispersion.
 
 
-From the application point of view, blending drifters and altimetric data has been successfully applied to several basins, for example in: the gulf of Mexico (\cite{berta2015improved}), the black sea (\cite{kubryakov2011mean}), the North Pacific (\cite{uchida2003eulerian}), and the Mediterranean basin (\cite{taillandier2006assimilation}, \cite{poulain2012surface} and \cite{menna2012surface} ). In (\cite{menna2012surface}), there was a particular attention to the levantine sub-basin, where large historical data sets from 1992 to 2010 were used to characterise surface currents. 
-The specific region which lies between the coasts of Lebanon, Syria and Cyprus, is however characterised by sparsity of data. In the present work, we use in addition to the data sets used in \cite{menna2012surface}, more recent data from 2013 (in the context of Altifloat project) to study this particular region.  
-
-
-Our contribution focuses on the methodical aspect, and it can be considered an extension of the variational approach used in (\cite{taillandier2006variational}). The first purpose is to add more physical considerations to the surface velocity estimation, without making the method too complex, in order to still allow for Near Real Time applications. We constrain the geostrophic component of that velocity to be divergence-free, and we add a component due to the effect of the wind, in the fashion done in \cite{poulain2009}. We also provide a time-continuous correction by: (i) assimilating a whole trajectory of drifters at once and (ii) using a moving time window where observations are correlated. 
+From the application point of view, blending drifters and altimetric data has been successfully applied to several basins, for example in: the gulf of Mexico \citep{berta2015improved}, the black sea \citep{kubryakov2011mean}, the North Pacific \citep{uchida2003eulerian}, and the Mediterranean basin \citep{taillandier2006assimilation,poulain2012surface,menna2012surface}. In \citet{menna2012surface}, there was a particular attention to the levantine sub-basin, where large historical data sets from 1992 to 2010 were used to characterise surface currents. 
+The specific region which lies between the coasts of Lebanon, Syria and Cyprus, is however characterised by sparsity of data. In the present work, we use in addition to the data sets used in \citet{menna2012surface}, more recent data from 2013 (in the context of Altifloat project) to study this particular region.  
+
+
+Our contribution focuses on the methodical aspect, and it can be considered an extension of the variational approach used in \citet{taillandier2006variational}. The first purpose is to add more physical considerations to the surface velocity estimation, without making the method too complex, in order to still allow for Near Real Time applications. We constrain the geostrophic component of that velocity to be divergence-free, and we add a component due to the effect of the wind, in the fashion done in \citet{poulain2009}. We also provide a time-continuous correction by: (i) assimilating a whole trajectory of drifters at once and (ii) using a moving time window where observations are correlated. 
 
 We show that with few drifters, our method (i) improves the estimation of an eddy between the Lebanese coast and Cyprus, and (ii) predicts real drifters trajectories along the Lebanese coast. 
@@ -193 +193 @@
 \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.}
+\caption{CNRS drifters deployed in the context of the ALTIFLOAT project starting Aug 28 2013 (shown in  $-$x) , versus  trajectories simulated using the \textit{Aviso} field (shown in $--$).  The velocity field shown is the  \textit{Aviso} field, averaged over 6 days.}
 \label{fig:cnrs}
 \end{center}
@@ -202 +202 @@
 \section{Data}
 All the data detailed in this section were extracted for two target period : first from 25 August 2009 to 3 September 2009, and second from 28 August 2013 to 4 September 2013.
-\subsection {Altimetry data}
+\subsection {\label{sec:aviso}Altimetry data}
 Geostrophic surface velocity fields used as a background in the study were produced by Ssalt/\textit{Duacs} and distributed by \textit{Aviso}. Altimetric mission used were  Saral, Cryosat-2, Jason-1\&2. The geostrophic absolute velocity fields were deduced from Maps of Absolute Dynamic Topography (MADT) using the regional Mediteranean Sea product.
 
@@ -262 +262 @@
 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 (\cite{courtier1994strategy}, \cite{dimet1986variational}), 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. The details of this procedure are given in Section 3.3.
+The estimation is done following a variational assimilation approach \citep{courtier1994strategy,dimet1986variational}, 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. The details of this procedure are given in Section 3.3.
 
 
@@ -299 +299 @@
 
 
-Using the incremental approach (\cite{courtier1994strategy}), the nonlinear observation operator $\mathcal{M}$ is linearised around a reference state. In a specific time window, we consider time independent perturbations $\delta \bo{u}$ on top of the background velocity field, that is
+Using the incremental approach \citep{courtier1994strategy}, the nonlinear observation operator $\mathcal{M}$ is linearised around a reference state. In a specific time window, we consider time independent perturbations $\delta \bo{u}$ on top of the background velocity field, that is
 \begin{align}\label{totalR}
 \bo{r}&=\bo{r^b}+\delta \bo{r} \\ \notag
@@ -334 +334 @@
 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 regularisation and information spreading or smoothing. To obtain $\bo{B}$, we use the diffusion filter method of \cite{weaver2001correlation}, 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 regularisation and information spreading or smoothing. To obtain $\bo{B}$, we use the diffusion filter method of \citet{weaver2001correlation}, 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 regularisation 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. We note here that the total velocity may have a divergent component due to the wind. 
@@ -342 +342 @@
 
 
-We end this section by pointing out that we implement the algorithm described above in YAO, \textcolor{red}{Julien [Refs.]}
+We end this section by pointing out that we implement the algorithm described above in YAO~\citep{badran2008},
 a numerical tool very well adapted to variational assimilation problems that simplifies the computation and implementation of the adjoint needed in the optimisation. 
 
@@ -359 +359 @@
 
 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 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, \textcolor{red}{which is consistent with the range of values found in the wider region? in \citep{taillandier2006variational}.}
  
  
@@ -465 +465 @@
 Fig.~\ref{fig:leb1}, shows the trajectories simulated with corrected field on top of the observed ones, 
 in good agreement, even for small scale structures near the coast.
+Average correction over 6 days are shown on the figure, but the actual corrections are time-dependent.
 
 As expected, the velocity field is modified in the neighbourhood of the drifters trajectories. It can be noticed that the main effect of the correction is to increase the velocity parallel to the coast, and decrease the velocity normal to the coast. The background field was determined using altimetric data and is expected to have significant bias close to the coast~\citep{bouffard2008}, and the consequence is that the corrected field is able to correct some of these bias.
@@ -470 +471 @@
 To validate more quantitatively the corrected velocities, another sensitivity study was considered. Only two drifters (the easternmost magenta drifter and the westernmost black drifter) were assimilated in order to correct the velocity field. The third drifter is used only to validate the corrected field by comparing its actual trajectory with the simulated trajectory using the velocity field.
 
-Figure~\ref{fig:lebzoom} shows the results of this experiment. The real drifter trajectory (empty circle with thin line) was compared to the simulated trajectory using either the background field (bold cyan line) or the corrected field (bold green line). It can be noticed that the trajectory was greatly improved using the corrected field. It shows that the corrected field can be used to simulate realistic trajectories in the neighbourhood of the assimilation positions, even in a coastal region. It can be a decisive point for application such as pollutant transport estimation.
-
-
-\textcolor{red}{Julien: Even though we show averages of the correction, stress that corrections are done in time dependent manner, can show a movie?}
-
-
+Figure~\ref{fig:lebzoom} shows the results of this experiment. The real drifter trajectory (empty circle with thin line) was compared to the simulated trajectory using either the background field (bold cyan line) or the corrected field (bold green line). 
+
+It can be noticed that the trajectory was greatly improved using the corrected field. It shows that the corrected field can be used to simulate realistic trajectories in the neighbourhood of the assimilation positions, even in a coastal region. 
+It can be a decisive point for application such as pollutant transport estimation.
 
 \begin{figure}[htbp]
@@ -499 +498 @@
 
 \subsection{\label{sec:cyprus}Improvement of velocity field in an eddy}
-In 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.
+In 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 \textit{Aviso} velocity field was corrected.
 
 In 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.
 
-In 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. 
+In Fig.~\ref{fig:eddy-velocity}, the trajectory of the drifters were represented in gray, the mean \textit{Aviso} surface geostrophic velocity field in blue and the mean corrected geostrophic field in red. 
 
 The real trajectory of the drifters and the simulated trajectory using the total corrected field (sum of corrected field in red and the wind-induced velocity) are very close. 
@@ -513 +512 @@
 \centering
 \includegraphics[scale =0.6]{./fig/Eddy_velocity.png}
-\caption{\label{fig:eddy-velocity} Corrected surface velocity field (in red) compared to AVISO background field (in blue). The assimilated drifter trajectories are represented in gray. The North-West coast in the figure is Cyprus.}
-\end{figure}
-
-In this case, it can be seen that the drifter trajectories were situated in an eddy. The AVISO field is produced by an interpolation method which tends to overestimate the spatial extent of the eddy and underestimate the intensity. In order to estimate the effect of the assimilation on the eddy characteristics, we computed the Okubo-Weiss parameter~\citep{isern2004} on the mean velocity fields before correction (background) and after correction. Eddies are characterized by a negative Okubo-Weiss parameter, the value of the parameter is an indicator of the intensity of the eddy. Results are shown in Fig.~\ref{fig:okubo-weiss}. As expected, it can be noticed that the Okubo-Weiss parameter had greater absolute values and a slightly smaller spatial extent which indicated a improvement of the Aviso processing bias. This results constitutes a validation of the assimilation method presented in this paper showing that eddies were better resolved after assimilating drifter trajectories.
+\caption{\label{fig:eddy-velocity} Corrected surface velocity field (in red) compared to \textit{Aviso} background field (in blue). The assimilated drifter trajectories are represented in gray. The North-West coast in the figure is Cyprus.}
+\end{figure}
+
+In this case, it can be seen that the drifter trajectories were situated in an eddy. The \textit{Aviso} field is produced by an interpolation method which tends to overestimate the spatial extent of the eddy and underestimate the intensity. In order to estimate the effect of the assimilation on the eddy characteristics, we computed the Okubo-Weiss parameter~\citep{isern2004} on the mean velocity fields before correction (background) and after correction. Eddies are characterized by a negative Okubo-Weiss parameter, the value of the parameter is an indicator of the intensity of the eddy. Results are shown in Fig.~\ref{fig:okubo-weiss}. As expected, it can be noticed that the Okubo-Weiss parameter had greater absolute values and a slightly smaller spatial extent which indicated a improvement of the Aviso processing bias. This results constitutes a validation of the assimilation method presented in this paper showing that eddies were better resolved after assimilating drifter trajectories.
 
 \begin{figure}[h]