Changeset 212 for altifloat


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Timestamp:
12/10/15 13:05:05 (8 years ago)
Author:
leila_ocean
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Dec 9, Laurent's comments

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altifloat/doc/ocean_modelling
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  • altifloat/doc/ocean_modelling/Draft1.tex

    r211 r212  
    5353\usepackage{caption} 
    5454\usepackage{subcaption} 
     55\usepackage{url} 
    5556%\usepackage{hyperref} 
    5657 
    … …  
    112113%% \address[label2]{} 
    113114 
    114 \author[lau]{Leila Issa}  
     115\author[lau]{Leila Issa\corref{cor1}}  
     116\cortext[cor1]{Corresponding author Tel: +9611786456, ext 1298, Email: leila.issa@lau.edu.lb} 
     117 
    115118\author[locean,inria]{Julien Brajard} 
    116119\author[cnrsl]{Milad Fakhri}  
    … …  
    120123 
    121124 
    122 \address[lau]{ Department of Computer Science and Mathematics\\ 
    123         Lebanese American University\\ 
     125\address[lau]{ Department of Computer Science and Mathematics, Lebanese American University\\ 
    124126        Beirut, Lebanon\\ 
    125     Email: leila.issa@lau.edu.lb} 
     127    } 
    126128\address[locean]{Sorbonne Universités 
    127129UPMC Univ Paris 06  
    … …  
    147149\begin{abstract} 
    148150We present a new and fast method for blending altimetry and surface drifters data in the Eastern Levantine Mediterranean. The method is based on a variational assimilation approach for which the velocity is corrected  
    149 by matching real drifters positions with a simple advection model simulation, 
    150 %after drifters data are matched to a simple advection model for their positions,  
    151 taking into account the effect of the wind. The velocity correction is done in a time-continuous fashion by assimilating at once a whole trajectory of drifters in a time window, and by moving that window to exploit correlations between observations. We show that with few drifters, our method improves the estimation of velocity in two typical situations: an eddy between the Lebanese coast and Cyprus, and velocities along the Lebanese coast.  
     151by matching real drifters positions with a simple advection model simulation that takes into account the wind effect. The velocity correction is done in a time-continuous fashion by assimilating at once a whole trajectory of drifters in a time window, and by moving that window to exploit correlations between observations. \textcolor{red}{The velocity correction is also constrained to be divergence free}. We show that with few drifters, our method improves the estimation of velocity in two typical situations: an eddy between the Lebanese coast and Cyprus, and velocities along the Lebanese coast.  
    152152\end{abstract} 
    153153 
    … …  
    169169\section{Introduction} 
    170170\label{} 
    171 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, which is important in this heavily populated region. A good knowledge of the surface velocity field is challenging, especially when direct observations are relatively sparse. 
    172  
    173 Altimetry has been widely used to predict the 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), 
     171An 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 emanating from heavily populated coastal areas. Small scale and accurate surface velocity estimation near coastal areas could also benefit the study of the paths of alien Lessepsian species. 
     172A good knowledge of the surface velocity field is thus important but can be challenging, especially when direct observations are relatively sparse. 
     173 
     174Altimetry has been widely used to predict the mesoscale features of the global 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 structures like eddies, fronts and filaments, which results in blurring these structures. Further errors and inaccuracies occur near the coastal areas (within 20-50 km from land), 
    174175where satellite information is degraded; this is due to various factors such as land contamination, inaccurate tidal and geophysical 
    175 corrections and incorrect removal 
     176corrections, inaccurate Mean Dynamic Topography and incorrect removal 
    176177of high frequency atmospheric effects at the sea surface \citep{caballero2014validation}. 
    177178 
    178 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] .  
     179To improve geostrophic velocities, especially near the coast, in situ observations provided by surface drifters can be considered (e.g. \citet{bouffard2008, ruiz2009mesoscale}). %[Bouffard et al., 2010; Ruiz et al., 2009] .  
    179180Drifters follow the currents and when numerous, they allow for an extensive spatial coverage of the region of interest. They are inexpensive, easily deployable and provide accurate information on their position and other 
    180181environmental parameters \citep{lumpkin2007measuring}. 
    … …  
    183184 
    184185 
    185 Numerous studies aim at exploiting the information provided by drifters (Lagrangian data) to assess the Eulerian surface velocity. A large number of these rely on modifying a dynamical model of this velocity by minimising the distance between observed and model simulated drifters trajectories. This variational assimilation approach, which was classically used in weather predictions \citep{courtier1994strategy,dimet1986variational}, was tested successfully in this context, by using several types of models for the velocity, such as idealised point vortex models \citep{kuznetsov2003method}, General Circulation Models with simplified stratification (e.g. \cite{kamachi1995continuous};  \cite{molcard2005lagrangian}; \cite{ozgokmen2003assimilation}, \cite{nodet2006variational}). However, in a lot of applications involving pollutant spreading such as the ones we are interested in, a fast diagnosis of the velocity field is needed in areas which are not a priori known in details. This prompts the need for a simple model that is fast and easy to implement, but that keeps the essential physical features of the velocity. In this work, we propose a new algorithm that blends geostrophic and drifters data in an optimal way. The method is based on a simple advection model for the drifters, that takes into account the wind effect and that imposes a divergence free constraint on the geostrophic component.  
     186Numerous studies aim at exploiting the information provided by drifters (Lagrangian data) to assess the Eulerian surface velocity. A large number of these rely on modifying a dynamical model of this velocity by minimizing the distance between observed and model simulated drifters trajectories. This variational assimilation approach, which was classically used in weather predictions \citep{courtier1994strategy,dimet1986variational}, was tested successfully in this context, by using several types of models for the velocity, such as idealized point vortex models \citep{kuznetsov2003method}, General Circulation Models with simplified stratification (e.g. \cite{kamachi1995continuous};  \cite{molcard2005lagrangian}; \cite{ozgokmen2003assimilation}, \cite{nodet2006variational}). However, in a lot of applications involving pollutant spreading such as the ones we are interested in, a fast diagnosis of the velocity field is needed in areas which are not a priori known in details. This prompts the need for a simple model that is fast and easy to implement, but that keeps the essential physical features of the velocity. In this work, we propose a new algorithm that blends geostrophic and drifters data in an optimal way. The method is based on a simple advection model for the drifters, that takes into account the wind effect and that imposes a divergence free constraint \textcolor{red}{on the geostrophic component}.  
    186187The algorithm is used to estimate the surface velocity field in the  
    187188Eastern 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.  
    … …  
    189190 
    190191 
    191 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,stanichny2015parameterization}). Another approach relies on variational assimilation: the work of  \citet{taillandier2006variational} is based on a simple advection model for the drifters' positions that is matched to observations via optimisation. The implementation of this method first assumes the time-independent approximation of the velocity correction, then superimposes inertial oscillations on the mesoscale field.  
     192From 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,stanichny2015parameterization}). Another approach relies on variational assimilation: the work of  \citet{taillandier2006variational} is based on a simple advection model for the drifters' positions that is matched to observations via optimization. The implementation of this method first assumes the time-independent approximation of the velocity correction, then superimposes inertial oscillations on the mesoscale field.  
    192193These variational techniques had 
    193194led to the development of the so called ``LAgrangian Variational Analysis" (LAVA) algorithm,  initially tested and applied to correct model velocity fields using drifter trajectories \citep{taillandier2006assimilation,taillandier2008variational} and later  
    … …  
    199200 
    200201From 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,stanichny2015parameterization} the North Pacific \citep{uchida2003eulerian}, and the Mediterranean Sea \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.  
    201 The specific region which lies between the coasts of Lebanon, Syria and Cyprus is however characterised by a scarcity 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.   
    202  
    203  
    204 Our contribution focuses on the methodological aspect, and it can be considered an extension of the variational approach used in \citet{taillandier2006variational}. The purpose is to add physical considerations to the surface velocity estimation, without making the method too complex, in order to still allow for Near Real Time applications. We do that by constraining the geostrophic component of that velocity to be divergence-free, and by adding 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.  
     202The specific region which lies between the coasts of Lebanon, Syria and Cyprus is however characterised by a scarcity 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 the AltiFloat project) to study this particular region.   
     203 
     204 
     205Our contribution focuses on the methodological aspect, and it can be considered an extension of the variational approach used in \citet{taillandier2006variational}. The purpose is to add physical considerations to the surface velocity estimation, without making the method too complex, in order to still allow for Near Real Time applications. We 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. \textcolor{red}{We also constrain the velocity correction to be divergence-free, and add a component to the velocity due to the effect of the wind}, in the fashion done in \citet{poulain2009}.  
    205206 
    206207We show that with a few drifters, our method improves the estimation of an eddy between the Lebanese coast and Cyprus, and predicts real drifters trajectories along the Lebanese coast.  
    … …  
    213214 
    214215 
    215 This manuscript is organised as follows. We begin in section~\ref{sec:data} by describing the data sets used in the method and the validation process. In section~\ref{sec:method}, we  provide a thorough description of the method including the definition of parameters involved, the model, and the optimisation procedure. We validate the method by conducting a twin experiment and a set of sensitivity analysis in section~\ref{sec:twin}, followed by two real experiments in section~\ref{sec:real}, one in a coastal configuration  and another in an eddy.   
     216This manuscript is organized as follows. We begin in section~\ref{sec:data} by describing the data sets used in the method and the validation process. In section~\ref{sec:method}, we  provide a thorough description of the method including the definition of parameters involved, the model, and the optimization procedure. We validate the method by conducting a set of sensitivity analysis in section~\ref{sec:twin}, followed by two real experiments in section~\ref{sec:real}, one in a coastal area and another in an offshore eddy.   
    216217% 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 ?? 
    217218 
    … …  
    230231\includegraphics[scale=0.5]{./fig/RealvsSimulatedTraj.pdf} 
    231232%\vspace{-30mm} 
    232 \caption{Altifloat drifters deployed on 28 Aug. 2013 (shown in  $-$x) versus  trajectories simulated using the AVISO field (shown in $\tiny{--}$).  The velocity field shown is the AVISO field, averaged over 6 days from 28 Aug.  2013 to 3 Sept. 2013} 
     233\caption{AltiFloat drifters deployed on 28 Aug. 2013 (shown in  $-$x) versus  trajectories simulated using the AVISO field (shown in $\tiny{--}$).  The velocity field shown is the AVISO field, averaged over 6 days from 28 Aug.  2013 to 3 Sept. 2013} 
    233234\label{fig:cnrs} 
    234235\end{center} 
    … …  
    237238 
    238239\section{\label{sec:data}Data} 
    239 All the data detailed in this section were extracted from two target periods: first from 25 August 2009 to 3 September 2009, and second from 28 August 2013 to 4 September 2013. 
     240All the data detailed in this section were extracted from two target periods: the data associated with the NEMED project~\footnote{\url{http://nettuno.ogs.trieste.it/sire/drifter/nemed/nemed_main.html}} was from 25 August 2009 to 3 September 2009, and the data associated with the AltiFloat project was from 28 August 2013 to 4 September 2013. 
    240241\subsection {\label{sec:aviso}Altimetry data} 
    241 Geostrophic surface velocity fields used as a background in the study were produced by Ssalt/\textit{Duacs} and distributed by 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) of the regional Mediterranean Sea product~\footnote{www.aviso.altimetry.fr} using the recently released Mean Dynamic Topography by~\citet{rio2014}. 
     242Geostrophic surface velocity fields used as a background in the study were produced by Ssalt/\textit{Duacs} and distributed by AVISO~\footnote{www.aviso.altimetry.fr}. Altimetric mission used were  Saral, Cryosat-2, Jason-1\&2. The geostrophic absolute velocity fields were deduced from Maps of Absolute Dynamic Topography (MADT) of the regional Mediterranean Sea product using the recently released Mean Dynamic Topography by~\citet{rio2014}. 
    242243 
    243244Data were mapped daily at a resolution of 1/8$^o$. Data were linearly interpolated every hour at the advection model time step. 
    … …  
    255256NEMED & 03 Aug. 2009 & 32.59 & 32.63 & 26 Dec. 2009 & 32.92 & 34.28 \\ 
    256257\hline 
    257 Altifloat & 27 Aug. 2013 & 33.28 & 34.95 & 22 Sep. 2013 & 36.77 & 35.94 \\ 
     258AltiFloat & 27 Aug. 2013 & 33.28 & 34.95 & 22 Sep. 2013 & 36.77 & 35.94 \\ 
    258259\hline 
    259 Altifloat & 27 Aug. 2013 & 33.28 & 34.98 & 04 Sep. 2013 & 34.13 & 35.64 \\ 
     260AltiFloat & 27 Aug. 2013 & 33.28 & 34.98 & 04 Sep. 2013 & 34.13 & 35.64 \\ 
    260261\hline 
    261 Altifloat & 27. Aug. 2013 & 33.28 & 35.03 & 17 Sep. 2013 & 34.88 & 35.88 \\ 
     262AltiFloat & 27. Aug. 2013 & 33.28 & 35.03 & 17 Sep. 2013 & 34.88 & 35.88 \\ 
    262263\hline 
    263264\end{tabular} 
    … …  
    268269 
    269270\subsection{Wind Data} 
    270 ECMW ERA-Interim 6-hourly 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 AVISO data. The data were resampled on a hourly time step. 
    271  
    272 Wind velocities were used to estimate wind-driven effect on drifter velocity. 
     271ECMW ERA-Interim 6-hourly wind products~\citep{Dee2011} were extracted in order to estimate the  \textcolor{red}{effect of the wind and wind-driven currents on the drifters}. Wind velocities closest to the surface (10 m) were extracted at a resolution of 1/8$^o$ at the same grid point as the AVISO data. The data were resampled on a hourly time step. 
     272 
     273Wind velocities were used to estimate the wind-driven effect on drifters' velocity. 
    273274The Eulerian velocity field in the advection model (Eq.~\ref{advection}) is the sum of the geostrophic velocity and the wind induced velocity (Eq.~\ref{euler_vel}) given by the formula~\citep{poulain2009} (for SVP drifter with drogue attached): 
    274275\begin{equation} 
    275276\mathbf{U_{wind}} = 0.007exp(-27^oi)\times \mathbf{U_{10}} 
    276277\end{equation} 
    277 where $\mathbf{U_{wind}} = u_{wind}+iv_{wind}$ 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. 
     278where $\mathbf{U_{wind}} = u_{wind}+iv_{wind}$ is the drifter's velocity induced by the overall effect of the wind and $ \mathbf{U_{10}} = u_{10}+iv_{10}$ is the wind velocity above the surface (10m) expressed as complex numbers. 
    278279 
    279280\subsection {\label{sec:model}Model data} 
    … …  
    282283%the North-East Levantin Bassin  
    283284%(31$^o$ 30’E - 36$^o$ 13’E  and 33$^o$ 30’N – 36$^o$ 55’N).  
    284 The model forecasts were used without assimilation and were reinterpolated on a 1/8$^o$ grid point with a time step of one hour. 
     285The model forecasts were used without assimilation and were re-interpolated on a 1/8$^o$ grid point with a time step of one hour. 
    285286% The model forecast used for calibration purpose on September 2013. 
    286287 
    … …  
    301302The velocity shall be estimated on a specified grid with resolution of $1/8^{\circ}$ in both longitude and latitude, and in the time frame $[0, T_f].$  
    302303 
    303 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. 
    304  
    305  
    306  
    307  
    308 \subsection{Linearised model for Lagrangian data} 
     304The 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. \textcolor{red}{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. 
     305 
     306 
     307 
     308 
     309\subsection{Linearized model for Lagrangian data} 
    309310 
    310311The position of a specific drifter $\mathbf{r}(t)=(x(t),y(t))$ is the solution of the non-linear advection equation 
    … …  
    337338 
    338339 
    339 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 
     340Using the incremental approach \citep{courtier1994strategy}, the nonlinear observation operator $\mathcal{M}$ is linearized 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 
    340341\begin{align}\label{totalR} 
    341342\bo{r}&=\bo{r^b}+\delta \bo{r} \\ \notag 
    … …  
    361362\subsection{Algorithm for velocity correction} 
    362363 
    363 We perform sequences of optimisations, where we minimise the following objective function with respect to the time independent correction $\delta \bo{u},$ in a specific time window $[0,T_w]$ 
     364We perform sequences of optimizations, where we minimise the following objective function with respect to the time independent correction $\delta \bo{u},$ in a specific time window $[0,T_w]$ 
    364365\begin{equation} 
    365 \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}).  
     366\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 \,\sum_{i,j} (\nabla \cdot \bo{\delta u})^2.  
    366367\end{equation} 
    367368 
    … …  
    372373Here 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. 
    373374The parameter $\alpha_1$ represents the relative weight of this regularisation term with respect to the other terms. 
    374 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.  
     375The 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. \textcolor{red}{This term is added to ensure a physical correction, avoiding artefacts especially near the coasts}.  
    375376 
    376377 
    … …  
    382383 
    383384We end this section by pointing out that we implement the algorithm described above in YAO~\citep{badran2008}, 
    384 a numerical tool very well adapted to variational assimilation problems that simplifies the computation and implementation of the adjoint needed in the optimisation.  
    385  
    386 The solution was found by using the M1QN3 minimiser linked with the YAO tool. The convergence of the assimilation in a typical time window of $24$ h takes $20$ seconds on a sequential code compiled on a CPU Intel(R) Core(TM) at 3.40GHz. 
    387  
    388  
    389 \section{\label{sec:twin}Twin Experiment} 
    390  
    391 In the twin experiment approach, the observations are simulated using a known velocity field, the ``true" velocity $\bo{u}_{true}$ given by the CYCOFOS-CYCOM model (see section~\ref{sec:model}). 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  
     385a numerical tool very well adapted to variational assimilation problems that simplifies the computation and implementation of the adjoint needed in the optimization.  
     386 
     387The solution was found by using the M1QN3 minimiser \citep{gilbert1989some} linked with the YAO tool. The convergence of the assimilation in a typical time window of $24$ h takes $20$ seconds on a sequential code compiled on a CPU Intel(R) Core(TM) at 3.40GHz. 
     388 
     389 
     390\section{\label{sec:twin}Sensitivity analyses} 
     391 
     392To validate our method, we conduct a set of synthetic experiments where the observations are simulated using a known velocity field, the ``true" velocity $\bo{u}_{true}$ given by the CYCOFOS-CYCOM model (see section~\ref{sec:model}). 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  
    392393\begin{equation} \label {RMSError} 
    393394error (u, t)=\bigg(    \frac{\sum_{i,j} \big | \big |\bo{u}_{true} (i,j,t)-\bo{u} (i,j,t)\big | \big |^2}{\sum_{i,j} \big | \big |\bo{u}_{true} (i,j,t)\big | \big |^2 } \bigg)^{1/2}, 
    … …  
    397398 
    398399 
    399 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 Altifloat drifters in Table~\ref{tab:drifters}), 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  Eq.~\ref{RMSError} is done. The initial positions of the drifters shown in red coincide with the positions of the drifters on 1 September 2013.  The drifters' positions are simulated using a velocity field $\bo{u}_{true}$ obtained from the CYCOM model described in section 2.4. The experiment 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}. 
     400The configuration of our synthetic experiment is the following: we put ourselves in the same context as that of the real drifter experiment conducted during the AltiFloat project, by the CNRS-L, the Lebanese national research council (refer to AltiFloat drifters in Table~\ref{tab:drifters}), where the drifters were 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  Eq.~\ref{RMSError} is done. The initial positions of the drifters shown in red coincide with the positions of the drifters on 1 September 2013.  The drifters' positions are simulated using a velocity field $\bo{u}_{true}$ obtained from the CYCOM model. The experiment 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 had hit land after 3 days, as shown in Fig.~\ref{fig:synth}, because of easterly winds. 
    400401 The background velocity field is composed of the geostrophic component obtained from AVISO and the wind component as described in the method section. These fields are interpolated to $\delta t$. A sensitivity analysis yields the optimal choice of $R=20$ km, which is consistent with the range of values found in the Northwestern Mediterranean \citep{taillandier2006variational}. 
    401402  
    402403  
    403 Using the relative RMS error before and after assimilation as a measure, we study the sensitivity of our method to the number of drifters $N_f$, the time sampling $\Delta t,$ the window size $T_w$ and to the moving parameter $\sigma.$ (Figures~\ref{fig:wsize},\ref{fig:mwin} and \ref{fig:numb}). 
     404Using the relative RMS error before and after assimilation as a measure, we study the sensitivity of our method to the number of drifters $N_f$, the time sampling $\Delta t,$ the window size $T_w$ and to the moving parameter $\sigma.$ We also asses the effect of the divergence free constraint term.    
    404405\begin{figure}[htbp] 
    405406\begin{center} 
    406407\includegraphics[scale=0.5]{./fig/RegionErroronAviso.pdf} 
    407408%\vspace{-30mm} 
    408 \caption{Region of RMS error computation for the twin experiment. Observations generated by CYCOM model starting on 1 Sept. 2013 (for 3 days) are shown on top of the background field. The red locations correspond to Altifloat drifters' locations.} 
     409\caption{Region of RMS error computation for the sensitivity experiments. Observations generated by CYCOM model starting on 1 Sept. 2013 (for 3 days) are shown on top of the background field. The red locations correspond to AltiFloat drifters' locations.} 
    409410\label{fig:synth} 
    410411\end{center} 
    411412\end{figure} 
     413 
     414\subsection{Sensitivity to the time window size} 
    412415We first show the effect of the window size $T_w.$ On one hand, this parameter has to be within the Lagrangian time scale $T_L,$ estimated here to be $1-3$ days, but it 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 $\Delta_t=2$ h), 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 $48$ h 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 $24$ h, which is within the range mentioned above. The error in this case is reduced by almost a half from the original one. We mention here 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.  
    413416\begin{figure}[htbp] 
    … …  
    419422\end{center} 
    420423\end{figure} 
     424 
     425\subsection{Sensitivity to the shifting parametre} 
    421426We then show the effect of shifting the window by varying the parameter $\sigma.$ The window size here is fixed to $T_{w}=24$ h, time sampling to $\Delta t=2$ h, and $N_f=14.$ We start with $\sigma=0$, which amounts to doing separate corrections, then slide every $\sigma=12, 8, 6$ h. In Fig.~\ref{fig:mwin}, we show the results by displaying the relative RMS error before and after the correction.  
    422427We 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. This is due to the fact that the moving window scheme exploits the correlation between the trajectories. 
    … …  
    430435\end{center} 
    431436\end{figure} 
    432 The 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. We also show in this figure the effect of removing the drifters that fail before the end of the experiment: the corresponding error curve is shown in the dashed curve of Fig.~\ref{fig:numb}, and it is evenly distributed in time as expected. 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  $\Delta t=6$ h still yields a very good correction. 
     437 
     438\subsection{Sensitivity to the number of drifters} 
     439The 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. We also show in this figure the effect of removing the drifters that fail before the end of the experiment: the corresponding error curve is shown in the dashed curve of Fig.~\ref{fig:numb}, and it is evenly distributed in time as expected.  
    433440\begin{figure}[htbp] 
    434441\begin{center} 
    … …  
    440447\end{figure} 
    441448 
     449 
     450\subsection{Sensitivity to the time sampling size} 
     451Finally, 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  $\Delta t=6$ h still yields a very good correction. 
     452 
    442453\begin{figure}[htbp] 
    443454\begin{center} 
    … …  
    449460\end{figure} 
    450461 
    451 For the twin experiment with the optimal choice of parameters ( $T_w=24$ h, $\sigma=6$ h, $N_f=14$ and $\Delta t=2$), we now show the trajectories of the drifters simulated with the corrected velocity field on top of the ``true" observations. We also compare background and corrected fields in the region of interest. In Fig.~\ref{fig:lerror}, we display the point-wise $L_2$ error, defined as  
     462\subsection{Sensitivity to the effect of the divergence constraint} 
     463 
     464\textcolor{red}{blabla } 
     465\begin{figure}[htbp] 
     466\begin{center} 
     467\includegraphics[scale=0.4]{./fig/Div_win24_dt1_f14_tf72.pdf} 
     468%\vspace{-30mm} 
     469\caption{The effect of divergence constraint. Here $T_w=24$ h, and $N_f=14.$ } 
     470\label{fig:div} 
     471\end{center} 
     472\end{figure} 
     473 
     474 
     475\subsection{Summary of results} 
     476 
     477\textcolor{red}{where should we put that the effect of the wind was found to be weak?} 
     478 
     479For the experiment with the optimal choice of parameters ( $T_w=24$ h, $\sigma=6$ h, $N_f=14$ and $\Delta t=2$), we now show the trajectories of the drifters simulated with the corrected velocity field on top of the ``true" observations. We also compare background and corrected fields in the region of interest. In Fig.~\ref{fig:lerror}, we display the point-wise $L_2$ error, defined as  
    452480\begin{equation} \label {L2Error} 
    453481error (u,i,j,t)=\big | \big |\bo{u}_{true} (i,j,t)-\bo{u} (i,j,t) \big| \big|, 
    454482\end{equation} 
    455 between the true field and either the background or corrected fields. In the left panel, we show that error between the background and true fields, averaged in time over the duration of the experiment. The right side shows this same error after correction. On top of that, we observe the excellent agreement between the positions of the drifters simulated with the corrected field and the ``true" observations. Next, the correction in terms of direction is shown in Fig.~\ref{fig:cerror}: we display the cosine of the angle between the background and true field 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. We see this strong correlation between true and corrected fields by observing how the blue color (left pannel of Fig.\ref{fig:cerror} ) turns into deep red (right pannel of Fig.\ref{fig:cerror} ) in the region where the drifters were deployed. Finally, in Fig. \ref{fig:summary}, we show the actual current maps before and after correction. We clearly see that the drifters corrected the poorly simulated coastal meander. 
     483between the true field and either the background or corrected fields. In the left panel, we show that error between the background and true fields, averaged in time over the duration of the experiment. The right side shows this same error after correction. On top of that, we observe the excellent agreement between the positions of the drifters simulated with the corrected field and the ``true" observations. Next, the correction in terms of direction is shown in Fig.~\ref{fig:cerror}: we display the cosine of the angle between the background and true field 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. We see this strong correlation between true and corrected fields by observing how the blue color (left pannel of Fig.\ref{fig:cerror}) turns into deep red (right pannel of Fig.\ref{fig:cerror}) in the region where the drifters were deployed. Finally, in Fig. \ref{fig:summary}, we show the actual current maps before and after correction. We clearly see that the drifters corrected the poorly represented coastal meander in the AVISO altimetric velocity field. 
    456484 
    457485 
    … …  
    493521\begin{center} 
    494522\includegraphics[scale=0.4]{./fig/summary_twin.pdf} 
    495 \caption{Background velocity field (blue) versus corrected velocity field (red) for the twin experiment with the optimal choice of parameters.} 
     523\caption{Background velocity field (blue) versus corrected velocity field (red) for the sensitivity experiment with the optimal choice of parameters.} 
    496524\label{fig:summary} 
    497525\end{center} 
    … …  
    518546 
    519547It 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.  
    520 It can be a decisive point for application such as pollutant transport estimation. 
     548This can be a decisive point for applications such as pollutant transport estimation. 
    521549 
    522550\begin{figure}[htbp] 
    … …  
    524552\includegraphics[scale=0.5]{./fig/ReconstructedCNRSExp_6days_average.pdf} 
    525553%\vspace{-30mm} 
    526 \caption{\label{fig:leb1} Prediction of the positions of 3 Altifloat drifters, launched on 28 Aug. 2013. $T_f=6$ days.  $T_w=24$ h and $\sigma=6$ h. Positions of drifters simulated with corrected field (cross markers) are shown on top of observed positions (circle markers). Corrected field is shown in red whereas background field is shown in blue. } 
     554\caption{\label{fig:leb1} Prediction of the positions of 3 AltiFloat drifters, launched on 28 Aug. 2013. $T_f=6$ days.  $T_w=24$ h and $\sigma=6$ h. Positions of drifters simulated with corrected field (cross markers) are shown on top of observed positions (circle markers). Corrected field is shown in red whereas background field is shown in blue. } 
    527555\end{center} 
    528556\end{figure} 
    … …  
    540568 
    541569\subsection{\label{sec:cyprus}Improvement of velocity field in an eddy} 
    542 In the context of the NEMED deployment (see section ~\ref{sec:drifters}), we selected two drifters trajectories from 25 Aug. 2009 to 3 Sept. 2009. Assimilating the successive positions of the drifters every six hours, the AVISO velocity field was corrected. 
     570In the context of the NEMED deployment (see section ~\ref{sec:drifters}), we selected 2 drifters trajectories from 25 August 2009 to 3 September 2009. The AVISO velocity field was corrected by assimilating successive positions of the drifters every six hours. 
    543571 
    544572In 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. 
    … …  
    550578%(expressed in arc length)  
    551579is  
    552 0.96 km 
     580$0.96$ km 
    553581%$8.6\times 10^{-3}$ degrees  
    554582with a maximum of  
    555583%$0.06$ degrees.  
    556 6.7km. 
    557 The real trajectory and simulated trajectory would be indiscernible in Fig.~\ref{fig:eddy-velocity}.  
     584$6.7$ km. 
     585The real and simulated trajectory would be indiscernible in Fig.~\ref{fig:eddy-velocity}.  
    558586 
    559587 
    … …  
    565593\end{figure} 
    566594 
    567 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 characterised 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. 
     595In 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 its 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 characterised 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 an 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. 
    568596 
    569597\begin{figure}[h] 
    … …  
    577605We presented a simple and efficient algorithm to blend drifter Lagrangian data with altimetry Eulerian velocities in the Eastern Levantine Mediterranean. The method has a cheap implementation and is quick to converge, so it is well fitted for near-real time applications. Assimilating two successive drifter positions produces a correction of the velocity field within a radius of 20km and for approximatively 24h before and after the measurement. 
    578606 
    579 This algorithm was able to correct some typical weaknesses of altimetry fields, in particular the estimation of velocity near the coast and accurate estimations of eddies dimensions and intensity. 
     607This algorithm was able to correct some typical weaknesses of altimetric fields, in particular the estimation of velocity near the coast and accurate estimations of eddies dimensions and intensity. 
    580608 
    581609\section{Acknowledgement} 
    582 The altimeter products were produced by Ssalto/Duacs and distributed by Aviso, with support from Cnes (http://www.aviso.altimetry.fr/duacs/). 
     610The altimeter products were produced by Ssalto/Duacs and distributed by AVISO, with support from CNES (http://www.aviso.altimetry.fr/duacs/). 
    583611 
    584612Wind data were produced by ECMWF and downloaded from\\ 
    585613 (http://apps.ecmwf.int/datasets/data/interim-full-daily/). 
    586614 
    587 This work was partially funded by the ENVI-MED program in the framework of the Altifloat project and by the U.S. Office of Naval Reasearch under grant N00014081094. We thank A. Bussani and M. Menna for processing the drifter data. 
    588  
    589  The Lebanese CNRS funded through ``CANA project", the campaigns of drifters' deployment using the platform of the Lebanese research vessel ``CANA-CNRS". These MetOcean Iridium drifters (SVP) were provided through the Istituto Nazionale di Oceanografia e di Feofisica Sperimentale (OGS), Italy and LOCEAN institute of ``Pierre et Marie Curie University", France 
     615This work was partially funded by the ENVI-Med program in the framework of the AltiFloat project and by the U.S. Office of Naval Research under grant N00014081094.  
     616 
     617 The Lebanese CNRS funded the campaign of drifters' deployment using the research vessel ``CANA". The AltiFloat MetOcean Iridium drifters (SVP) were provided by the Istituto Nazionale di Oceanografia e di Geofisica Sperimentale (OGS), Italy and LOCEAN institute of ``Pierre et Marie Curie University", France. 
     618 The drifter data are distributed by the MedSVP portal of OGS. We thank A. Bussani and M. Menna for processing the drifter data. 
     619 
    590620%% The Appendices part is started with the command \appendix; 
    591621%% appendix sections are then done as normal sections 

  • altifloat/doc/ocean_modelling/mybib.bib

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    33 
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     5%% Created for Leila Issa at 2015-12-09 11:32:32 +0200  
    66 
    77 
    … …  
    99 
    1010 
     11 
     12@article{gilbert1989some, 
     13        Author = {Gilbert, Jean Charles and Lemar{\'e}chal, Claude}, 
     14        Date-Added = {2015-12-09 08:12:24 +0000}, 
     15        Date-Modified = {2015-12-09 08:12:24 +0000}, 
     16        Journal = {Mathematical programming}, 
     17        Number = {1-3}, 
     18        Pages = {407--435}, 
     19        Publisher = {Springer}, 
     20        Title = {Some numerical experiments with variable-storage quasi-Newton algorithms}, 
     21        Volume = {45}, 
     22        Year = {1989}} 
    1123 
    1224@article{stanichny2015parameterization, 
    … …  
    300312 
    301313@article{zodiatis2008, 
    302   title={Operational ocean forecasting in the Eastern Mediterranean: implementation and evaluation}, 
    303   author={Zodiatis, G and Lardner, R and Hayes, DR and Georgiou, G and Sofianos, S and Skliris, N and Lascaratos, A}, 
    304   journal={Ocean Science}, 
    305   volume={4}, 
    306   pages={31--47}, 
    307   year={2008} 
    308 } 
     314        Author = {Zodiatis, G and Lardner, R and Hayes, DR and Georgiou, G and Sofianos, S and Skliris, N and Lascaratos, A}, 
     315        Journal = {Ocean Science}, 
     316        Pages = {31--47}, 
     317        Title = {Operational ocean forecasting in the Eastern Mediterranean: implementation and evaluation}, 
     318        Volume = {4}, 
     319        Year = {2008}} 
    309320 
    310321@inproceedings{zodiatis2003, 
    … …  
    361372        Year = {2008}} 
    362373 
    363  
    364  
    365 @TechReport{altimetry2009, 
    366   author =       {Aviso}, 
    367   title =        {SSALTO/DUACS user handbook:(M) SLA and (M) ADT near-real time and delayed time products}, 
    368   institution =  {CNES}, 
    369   year =         {2015}, 
    370   key =          { CLS-DOS-NT-06-034}, 
    371   OPTtype =      {}, 
    372   OPTnumber =    {}, 
    373   OPTaddress =   {}, 
    374   OPTmonth =     {}, 
    375   OPTnote =      {}, 
    376   OPTannote =    {} 
    377 } 
     374@techreport{altimetry2009, 
     375        Author = {Aviso}, 
     376        Institution = {CNES}, 
     377        Key = {CLS-DOS-NT-06-034}, 
     378        Title = {SSALTO/DUACS user handbook:(M) SLA and (M) ADT near-real time and delayed time products}, 
     379        Year = {2015}} 
    378380 
    379381@article{rio2014, 
    380   title={Computation of a new mean dynamic topography for the Mediterranean Sea from model outputs, altimeter measurements and oceanographic in situ data}, 
    381   author={Rio, Marie H{\'e}l{\`e}ne and Pascual, Ananda and Poulain, Pierre-Marie and Menna, Milena and Barcel{\'o}-Llull, B{\`a}rbara and Tintor{\'e}, Joaqu{\'\i}n and others}, 
    382   journal={Ocean Science}, 
    383   volume={10}, 
    384   number={4}, 
    385   pages={731--744}, 
    386   year={2014} 
    387 } 
     382        Author = {Rio, Marie H{\'e}l{\`e}ne and Pascual, Ananda and Poulain, Pierre-Marie and Menna, Milena and Barcel{\'o}-Llull, B{\`a}rbara and Tintor{\'e}, Joaqu{\'\i}n and others}, 
     383        Journal = {Ocean Science}, 
     384        Number = {4}, 
     385        Pages = {731--744}, 
     386        Title = {Computation of a new mean dynamic topography for the Mediterranean Sea from model outputs, altimeter measurements and oceanographic in situ data}, 
     387        Volume = {10}, 
     388        Year = {2014}} 
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