A reduced-order strategy for 4D-Var data assimilation
Céline Robert, S. Durbiano, Eric Blayo, Jacques Verron, Jacques Blum, François-Xavier Le Dimet
aa r X i v : . [ phy s i c s . g e o - ph ] S e p A reduced-order strategy for 4D-Var dataassimilation
C. Robert ∗ , S. Durbiano, E. Blayo, J. Verron, J. Blum,F.-X. Le Dimet IDOPT Project, LMC-IMAG and INRIA Rhˆone-Alpes, BP 53X, 38041 Grenoblecedex, France
Abstract
This paper presents a reduced-order approach for four-dimensional variational dataassimilation, based on a prior EOF analysis of a model trajectory. This methodimplies two main advantages: a natural model-based definition of a multivariatebackground error covariance matrix B r , and an important decrease of the compu-tational burden of the method, due to the drastic reduction of the dimension of thecontrol space. An illustration of the feasibility and the effectiveness of this method isgiven in the academic framework of twin experiments for a model of the equatorialPacific ocean. It is shown that the multivariate aspect of B r brings additional in-formation which substantially improves the identification procedure. Moreover thecomputational cost can be decreased by one order of magnitude with regard to thefull-space 4D-Var method. The aim of this paper is to investigate a reduced-order approach for four-dimensional variational data assimilation (4D-Var), with an illustration in thecontext of ocean modelling, which is our main field of interest. 4D-Var isnow in use in numerical weather prediction centers (e.g. Rabier et al. ∗ Corresponding author. Email address : [email protected]
Preprint submitted to Elsevier Science 25 October 2018
Moore 1991; Schr¨oter et al. et al. et al. et al. et al. etal. et al. et al. et al. et al. , 2003).However, although considerable work and improvements have been performed,a number of difficulties remain, common to most applications (and also toother data assimilation methods). The first problem is the fact that oceanmodels are non-linear, while 4D-Var theory is established in a linear con-text. More precisely, variational approach can adapt in principle to non-linearmodels, but the cost function is no longer quadratic with regard to the initialcondition (which is the usual control parameter) which can lead to importantdifficulties in the minimization process and the occurence of multiple minima.Several strategies have been proposed to overcome these problems: Luong etal. (1998) and Blum et al. (1998) perform successive minimizations over in-creasing time periods; Courtier et al. (1994), with the so-called incrementalapproach, generate a succession of quadratic problems, which solutions shouldconverge (but with no general theoretical proof) towards the solution of theinitial minimization problem. A second major difficulty with variational prob-lem implementation lies in our poor knowledge of the background error, whosecovariance matrix plays an important role in the cost function and in the min-imization process. In the absence of statistical information, these covariancesare often approximated empirically by analytical (e.g. Gaussian) functions. Forinstance, the covariances, used in the “standard” 4D-Var experiment E F ULL described in section 3 are 3D but univariate. Moreover, as discussed in (Ler-musiaux, 1999), errors evolve with the dynamics of the system and thus theerror space should evolve in the same way. In realistic systems, it proves to bedifficult to catch correctly this evolution. The third major problem in the useof 4D-Var in realistic oceanic applications is probably the dimension of thecontrol space. In fact, this dimension is generally equal to the size of the modelstate variable (composed, in our case, by the two horizontal components of thevelocity, temperature and salinity), which is typically of the order of 10 -10 .This makes of course the minimization difficult and expensive (typically tensto hundreds times the cost of an integration of the model), even with the bestcurrent preconditioners.This last difficulty can be addressed by reducing the dimension of the min-imization space. This is for example the idea of the incremental approach(Courtier et al. etal. et al. et al. Let a model simply written as ∂ x ∂t = M ( x ) (1)with the state vector x in Ω × [ t , t N ], Ω being the physical domain. Supposethat we have some observations y o distributed over Ω × [ t , t N ], with an obser-vation operator H mapping x onto y . The classical 4D-Var approach consistsin minimizing a cost function J ( u ) = J o ( u ) + J b ( u )= N X i =0 ( H ( x i ) − y o i ) T R − i ( H ( x i ) − y o i ) + 12 ( u − u b ) T B − u ( u − u b ) (2)using the notations of Ide et al. (1997). u b is a background value for thecontrol vector u , and B u is its associated error covariance matrix. In mostapplications, the control variable u is the state variable at the initial time : u = x ( t ), and the background state u b = x b is typically a forecast from a3revious analysis given by the data assimilation system. In this case, once themodel is discretized, the size of u (i.e. the dimension of the control space U )is equal to the size of x , denoted by n . x i stands for the state variable at time t i . In equation (2), x i is propagated by M , the fully non-linear model.In the incremental formulation which is used here, the cost function J is writ-ten as a function of δ x = x − x b and the J o term is calculated using thelinearized model M : J ( δ x ) = ( δ x ) t B − δ x + N X i =1 ( H i M t i , t δ x − d i ) t R − ( H i M t i , t δ x − d i ) (3)where d i stands for the innovation vector: d i = y i − H ( x b ( t i )) and M t i ,t is thetemporal evolution performed by the model M between the instants t and t i .The basic idea then, for constructing a reduced-order approach, consists indefining a convenient mapping M from W ≡ IR r into U ≡ IR n , with r ≪ n ,and in replacing the control variable u by the new control variable w with u = M ( w ). Since we want to preserve a good solution while having only a rathersmall number r of degrees of freedom on the choice of w , the subspace M ( W )of U must be chosen in order to contain only the “most pertinent” admissiblevalues for u . More precisely, in the case of the control of the initial condition u = x ( t ), we decide to define the mapping M by an affine relationship of theform : x ( t ) = M ( w ) = ˆ x + r X i =1 w i L i with w = ( w , . . . , w r ) ∈ W ≡ IR r (4)In order to let w span a wide range of physically possible states, ˆ x representsan estimate of the state of the system, and L , . . . , L r are vectors containingthe main directions of variability of the system (the w i are scalars). Such a def-inition relies on the fact that most of the variability of an oceanic system canbe described by a low dimensional space. Even if it is only rigorously provedfor very simplified models (Lions et al. , 1992), it is often expected that, awayfrom the equator, ocean circulation can be seen as a dynamical system havinga strange attractor. This means that the system trajectories are attracted to-wards a (low dimension) manifold. In the vicinity of this attractor, orthogonalperturbations will be naturally damped, while tangent perturbations will not(they can even be greatly amplified, due to the chaotic character of the sys-tem). To retrieve a system trajectory over of period of time [ t , t N ], it seemsthus necessary to propose an initial condition x ( t ) containing such variabilitymodes tangent to the attractor, but not necessarily variability modes orthog-onal to it. Thus, in definition (4), ˆ x should ideally be located on the attractor,4nd L , . . . , L r should correspond to the main directions of variability tangentto it. In the tropical ocean, the rationale is different, and even simpler since thetropical ocean dynamics is mostly linear, and can be represented by a ratherlimited number of linear, and possibly non-linear, modes (e.g. De Witte et al. x = x b , i.e. the background state that would beused in the corresponding classical 4D-Var approach. With this choice, theincrement δ x = x ( t ) − x b is equal to δ x = r X i =1 w i L i = Lw . In this reduced-space approach, we define a new expression for the background term J b of thecost function J : J b ( w ) = 12 w T B − w w (5)where B w is the background error covariance matrix in the reduced space. Thenatural representation of B w in the full space is the singular matrix B r = LB w L T (6)Minimization is performed using a quasi-Newton descent method with an ex-act line search (algorithm M1QN3, Gilbert and Lemar´echal 1989). As in theclassical 4D-Var method, the problem is preconditionned by defining a newcontrol variable δ v = B − / δ x , which implies J b ( δ v ) = δ v T δ v . From a pro-gramming point of view, this approach implies nearly no modification to theoriginal code, since we only have to add a mapping procedure correspondingto M , and the adjoint of this procedure.It is important to point out that the choice of the subspace M ( W ) of U isperformed using additional information (the information leading to the con-struction of the L i s) with regard to usual 4D-Var with no order reduction.This is done of course in order to make the choice of M effective, but it willalso automatically introduce this extra information into the assimilation pro-cedure (through L and B w ), and thus possibly help making the assimilationefficient.Concerning the actual choice of ( L , . . . , L r ), different families of vectors canbe proposed : • The variability of the system can be defined in a statistical sense, whichmeans that we seek directions maximizing the variance around a mean stateof the system. This is actually the definition of Empirical Orthogonal Func-tions (EOFs), which can be computed from a sampling of a model trajectory(see section 3.1). 5
We can also define the variability in a harmonical sense. In that case, thevectors can be defined by a Fourier or wavelets analysis of a model trajectory.Note however that, with regard to a rectangular domain, the presence ofcontinental boundaries makes the analysis more difficult. • If we consider the notion of variability within the framework of dynam-ical systems, we look for vectors maximizing a ratio of the form k x ( t = T ) k / k x ( t = T ) k , for some norm k . k . The problem can be simplified bymaking a tangent linear approximation, which leads to the computation ofsingular vectors (SVs). In the limit case where T − T becomes large (in-finite), SVs converge towards Lyapunov vectors (LVs). Properties of SVsand LVs can be found for instance in Legras and Vautard (1995). The tan-gent linear assumption can also be relaxed, and vectors corresponding toSVs and LVs can be computed with the fully non-linear model. They arecalled respectively non-linear singular vectors (NSVs, Mu 2000) and bredmodes (BVs, Toth and Kalnay 1997). Note that, to our knowledge, these“non-linear” vectors have been introduced in an empirical way, with nearlyno related properties established theoretically.Durbiano (2001) performed a thorough study of these families of vectors(EOFs, SVs, LVs, NSVs and BVs) in the perspective of their use as reducedbasis for several data assimilation problem. In particular, she compared theirperformances for the present problem of the control of the initial condition ina reduced space, in the case of a 2-D shallow water model. She concluded inthis case to the clear superiority of EOFs with regard to the other familiesof vectors. This is probably due to the fact that EOFs take into account thenonlinearity of the model (while SVs and LVs do not), and also that theircovariance matrix B w is quite accurately known, which is not the case forthe other families of vectors. That is why we used EOFs in the realistic 3-Dexperiment described in section 3. Note that this way of approximating thevariability of the system in a data assimilation process by a low dimensionspace generated by the first r EOFs is similar to the method used in theSEEK filter, or in the reduced order filter proposed by Cane et al. (1996).
The model used in our tests is the primitive equation ocean general circulationmodel OPA (Madec et al. z -coordinate rigid-lid version. Theregion of interest is the equatorial Pacific ocean, from 30 ◦ S to 30 ◦ N. Thehorizontal resolution is set to 1 ◦ zonally, and varies meridionally from 1/2 ◦ atthe equator to 2 ◦ at 30 ◦ . Vertically the ocean is discretized using 25 levels.6he state vector consists of temperature, salinity and horizontal velocity, andhas a size slightly greater than 10 .A one-year simulation was performed, starting from a previous restart builtwith the ECMWF wind stresses and heat fluxes and using ERS-TAO dailywind stresses and ECMWF heat fluxes to force the model. In a 10 ◦ -wide bandnear the northern and southern boundaries, buffer zones are prescribed wherethe model solution is relaxed towards Levitus climatology. This version ofthe model has been used previously in a number of studies, and details canbe found therein (e.g. Vialard et al. et al. et al. X = ( X , . . . , X p ), which leads to diagonalizing the covariance matrix X T X , with X j = 1 σ i [ x ( t j ) − ¯ x ] and ¯ x = 1 p p X j =1 x ( t j ). The inner product is theusual one for a state vector containing several physical quantities expressed indifferent units : < X j , X k > = n X i =1 σ i ( x ( t j ) − ¯ x ) i ( x ( t k ) − ¯ x ) i (7)where σ i is the empirical variance of the i -th component : σ i = 1 p p X j =1 ( X ij ) .This diagonalization leads to a set of orthonormal eigenvectors ( L , . . . , L p )corresponding to eigenvalues λ > . . . > λ p >
0. Since trajectories are com-puted with the fully non-linear model, these modes represent non-linear vari-ability around the mean state over the whole period.The first level ( z = 5m) of the first EOF is displayed on Fig. 1. As can be seen,it is mostly representative of the variability of the equatorial zonal currents,of the north-south temperature oscillation and of the mean structure of thesea surface salinity.The fraction of variability (or “inertia”) which is conserved when retainingonly the r first vectors is r X j =1 λ j / p X j =1 λ j . Its variation as a function of r isdisplayed in Fig. 2. We can see that a large part of the total variance can berepresented by a very few EOFs : 80% for the first 13 EOFs, 92% for the first30 EOFs.Finally, let us emphasize that a natural estimate for the covariance matrix of7he first r eigenvectors ( L , . . . , L r ), i.e. B w in our reduced-order 4D-Var, issimply the diagonal matrix Diag( λ , . . . , λ r ). A 4D-Var assimilation scheme, based on the incremental formulation of Courtier et al. (1994), has been developped for the OPA model (Weaver et al. et al. J ( x ( t )) isexpressed in terms of the increment δ x , and that its minimization is replacedby a sequence of minimizations of simplified quadratic cost functions. The ba-sic state-trajectory used in the tangent linear model is regularly updated inan outer loop of the assimilation algorithm, while the iterations of the actualminimizations are performed within an inner loop.Different statistical models can be chosen for representing the correlations ofbackground error. In the present study, we used a Laplacian-based correla-tion model, which is implemented by numerical integration of a generalizeddiffusion-type equation (Weaver and Courtier, 2001). The horizontal correla-tion lengths for the gaussian functions are equal to 8 o in longitude and 2 o inlatitude near the equator and 4 o in longitude/latitude outside the area situ-ated between 20 o N/S. The vertical correlation lengths depend on the depth. B is thus block diagonal : covariances are spatially varying but remain mono-variate. Such a choice for B leads to significantly better results than thosegiven by a simple diagonal representation of this matrix. However, since B remains univariate, the links between the model variables come only fromthe action of the model dynamics. The development of a multivariate modelfor B is presently under way in research groups. Ricci et al. (2004) includea state-dependent temperature-salinity constraint, which works quite well inthe 3D-Var case but is not yet operational for the 4D-Var case.The observation error covariance matrices R i depend of course of the assimi-lated data. We will consider in the present case only temperature observations,which are assumed independent with a standard error equal to σ T . The R i arethus taken equal to σ T Id .We have used for our experiments the classical framework of twin experiments.A one-year simulation of the model was performed, starting at the beginningof 1993. This simulation (further denoted E REF ) will be the reference exper-iment. Pseudo-observations of the temperature field were then generated, byextraction from this one-year solution at the locations of the 70 TAO moor-ings (Fig. 3), with a periodicity of 6 hours, on the first 19 levels of the model(i.e. the first 500 meters of the ocean). This corresponds to observing 0.17%8f the model state vector every 6 hours. Those temperature values have beenperturbed by the addition of a gaussian noise, with a standard error set to σ T = 0 . ◦ C, which is an upper bound for the standard error of the real TAOtemperature dataset.A 4D-Var assimilation of these pseudo-observations (i.e. with full control vari-able δ x , built from the state vector (u,v,T,S) in the whole space) was thenperformed, using an independent field x b (a solution of the model three monthslater) as the first guess (background field) for the minimization process. Thisfirst assimilation experiment will be denoted E F ULL , since it uses the full con-trol space. In order to improve the validity of the tangent linear approximation,the assimilation time window was divided into successive one-month windows.Then an additional simulation was performed, using the reduced-space ap-proach described in section 2 with r = 30 EOFs (which represent 92% ofthe total inertia - Fig. 2). This second assimilation experiment will be de-noted E REDUC . As detailed previously, the control variable in this case is w = ( w , . . . , w r ), with the mapping δ x = Lw and the preconditionning δ v = B − / w w = B − / w L T δ x . As explained in section 2, the reduced-space assimilation algorithm presentstwo main differences with regard to the full-space algorithm, which are themultivariate nature of the background error covariance matrix, and the smalldimension of the control space. Both aspects are expected to improve theefficiency of the assimilation, and we will now illustrate their respective impact.
The background error covariance matrix used in the reduced-space approachis defined empirically by the EOF analysis and is expressed in the full-spaceas B r = LB w L T . It integrates statistical information on the consistency be-tween the different model variables, and is naturally multivariate. On the otherhand, the matrix B used in the full-space 4D-Var is univariate, since provid-ing a multivariate model for this matrix remains challenging. This aspect isof course very important, and should lead to significant changes in the assim-ilation results. Note that Buehner et al. (1999) have proposed a similar wayof representing error covariances with EOF analysis in the context of 3D-Var.However they consider that the reduced basis is not sufficient to span theanalysis increment space and blend this EOF basis with the prior B projectedinto the sub-space orthogonal to the EOFs.9n interesting way to illustrate these differences between the full-space B andthe reduced-space B r is to perform preliminary assimilation experiments witha single observation. For that purpose, we use a single temperature observationlocated within the thermocline at 160 ◦ W on the equator, and specified at theend of a one-month assimilation time window. The innovation is set to 1 ◦ K.The analysis increment at the initial time in such an experiment is proportionalto the column of BM Tt n ,t corresponding to the location of the observation. Ascan be seen in Fig. 4, the reduced-space method performs, as expected, a ratherweak correction over the whole basin, while the full-space method generates amuch stronger and local increment. The structure of the increment is indeedmuch more elaborate in the reduced-space experiment, with scales larger thanin the full-space experiment. Note that the input from the first EOF (shownon Fig. 1) is quite clear in the horizontal pattern of the increment, since w / k w k = 0 .
86 in this particular case. The maximum value of the incrementhowever is only 0.06 ◦ C for the reduced-space 4D-Var, while it is 0.94 ◦ C in thefull-space 4D-Var.The interest of the naturally multivariate aspect of B r is also clear in theresults of our twin experiments. Two different types of diagnostics were per-formed, the first one concerning only the assimilated variables (i.e. tempera-ture in the present case), while the second one relates to all other variablesthat are not assimilated. This second type of diagnostic is of course the mostsignificant, since it evaluates the capability of the assimilation procedure topropagate information over the whole model state vector.An example of the first type of diagnostic is given in Fig. 5a, which displaysthe temperature rms error defined byrms T ( z, t ) = (cid:18)Z ( T ( λ, θ, z, t ) − T REF ( λ, θ, z, t )) dλ dθ (cid:19) / (8)The discretized formula becomes :rms T ( z, t ) = k x − x ref k = N x × N y N x X i =1 N y X j =1 ( T ( i, j, z, t ) − T ref ( i, j, z, t )) / (9)where N x and N y are the number of grid points in x and y. This error is sig-nificantly weaker in E REDUC than in E F ULL , although the assimilation systemin E REDUC has much less degrees of freedom to adjust the model trajectoryto these data.An example of the second type of diagnostic is shown in Fig. 5b,c. In our testcase, these results are clearly in favour of the reduced-space approach. Theerrors on the salinity S and the zonal component of the velocity u for the10olution provided by E F ULL are systematically greater than for E REDUC .The interest of this approach can also be illustrated by the results in the lowerlevels. It is well-known that the time-scale for the information to penetratefrom the upper ocean into the deep ocean within an assimilation process maybe quite long. However, in experiment E REDUC the EOFs add informationon the vertical structure of the flow (see Fig. 4) and then make the verticaladjustment easier. We have plotted for example in Fig. 6 the errors of the dif-ferent solutions at level 20 (depth : 750 m, ie below the observations). E REDUC performs a very good identification of the solution due to the propagation ofthe information in depth.These results are only part of what should be shown in terms of diagnosticanalyses. But all of them clearly prove that the results of E REDUC vs E F ULL are significantly improved for all, assimilated or not, variables.Finally, it must be mentioned that we have also illustrated the fundamentalrole of the multivariate nature of B r by performing an additional reduced-orderexperiment (not shown) using univariate EOFs. In this case, the directionsproposed for the minimization were not relevant, and the assimilation failed. The second important difference brought by the reduced-space approach withregard to the full-space approach is the dimension of the minimization space,which is decreased by several orders of magnitude. This should reduce thenumber of iterations necessary for the minimization, i.e. reduce the cost of thedata assimilation algorithm, which is an important practical issue.The evolution of the cost functions for experiments E F ULL and E REDUC aredisplayed on Fig. 7. Since we use different covariance matrices B and B r inthese two experiments, the curves are not quantitatively comparable. How-ever, it is clear in Fig. 7 that the number of iterations required to stabilizethe cost function is reduced by nearly one order of magnitude between thefull-space 4D-Var approach (which needs typically several tens of iterations)and the reduced-space approach (which needs eight to ten iterations). In thepresent experiments, we have kept the same number of iterations (2 outerloops of ten iterations each) in the two experiments to strictly compare theresults. But having a look at the cost function, it is clear that the minimum isquickly reached by E REDUC experiment. Considering the low number of free-dom degrees, the computational cost can be thus divided by a factor of 4 or 5between the two methods. 11
Conclusion
This paper presents a reduced-space approach for 4D-Var data assimilation.A new control space of low dimension is defined, in which the minimization isperformed. An illustration of the method is given in the case of twin experi-ments with a primitive equation model of the equatorial Pacific ocean.This method presents two important features, which make the assimilation al-gorithm effective. First the background error covariance matrix B r is builtusing statistical information (an EOF analysis) on a previous model run.This introduces relevant additional information in the assimilation processand makes B r naturally multivariate, while providing an analytical multivari-ate model for B is still challenging. This improves the identification of thesolution, both on observed and non-observed variables, and at all depths inthe model. Secondly the reduction of the dimension of the control space limitsthe number of iterations for the minimization, which results in a decrease ofthe computational cost by roughly one order of magnitude.However the results presented in this work are only a first (but necessary) step,since they concern twin experiments. They need of course to be confirmed byadditional experiments in other contexts, in particular experiments with realdata and in other geographical areas. As a matter of fact, the efficiency ofthe method is closely related to the fact that the reduced basis does containpertinent information on the variability of the true system. That is why, in thecontext of real observations (i.e. in the case of an imperfect model), the controlspace must probably not be limited to model-based variability. Therefore, wecan imagine either compute EOFs from results of previous data assimilationusing for example full-space 4D-Var (Durbiano 2001), and/or improve theassimilation results by performing a few full-space iterations at the end of thereduced-space minimization (Hoteit et al. et al. et al. Acknowledgments
References [1] Bennett, A. F., 1992: Inverse methods in physical oceanography.
CambridgeMonographs on Mechanics and Applied Mathematics . Cambridge University Press.[2] Blayo, E., Blum, J. and Verron, J., 1998: Assimilation variationnelle de donn´eesen oc´eanographie et r´eduction de la dimension de l’espace de contrˆole. Pp. 199–219in
Equations aux D´eriv´ees Partielles et Applications . Gauthier-Villars.[3] Blum, J., Luong, B. and Verron, J., 1998: Variational assimilation of altimeterdata into a non-linear ocean model: temporal strategies.
ESAIM Proceedings , ,21–57.[4] Brasseur, P., Ballabrera-Poy, J. and Verron, J., 1999: Assimilation of altimetricdata in the mid-latitude oceans using the Singular Evolutive Extended Kalmanfilter with an eddy-resolving primitive equation model. J. Mar. Syst. , , 269–294.[5] Buehner, M., Brunet, G. and Gauthier, P., 1999: Empirical orthogonal functionsfor modeling 3D-Var forecast error statistics. Pp. 324–327 in Proceedings ofthe Third WMO International Symposium on Assimilation of Observations inMeteorology and Oceanography, 7–11 June 1999, Quebec City, Canada.[6] Gauthier, P., Buehner, M. and Fillion, L., 1998: Background-error statisticsmodelling in a 3D variational data assimilation scheme: estimation and impacton the analyses. P-p 131–145 in Proceedings of the ECMWF Workshop on thediagnostics of assimilation systems, 2-4 November 1998, Reading, U.K.[7] Cane, M. A., Kaplan, A., Miller, R. N., Tang, B., Hackert, E. C. andBusalacchi, A. J., 1996: Mapping tropical Pacific sea level: Data assimilation viaa reduced state Kalman filter. J Geophys. Res. , , 22599–22617.[8] Courtier, P., Th´epaut, J.-N. and Hollingsworth, A., 1994: A strategy foroperational implementation of 4D-Var, using an incremental approach. Q. J. R.Meteorol. Soc. , , 1367–1388.[9] D’Andr´ea, F. and Vautard, R., 2001: Reducing systematic errors by empiricallycorrecting model errors. Tellus , , 21–41.[10] Devenon, J.-L., Dekeyser, I., Leredde, Y. and Lellouche, J.-M., 2001: Dataassimilation method by a variational methodology using the adjoint of a 3-Dcoastal circulation primitive equation model. Oceanol. Acta , , 395–407.
11] De Witte, B., Reverdin, G. and Maes, C., 1998: Vertical structures of an OGCMsimulation of the equatorial Pacific ocean in 1985-1994.
J. Phys. Oceanogr. , ,1542–1570.[12] Durbiano, S., 2001: Vecteurs caract´eristiques de mod`eles oc´eaniques pour lar´eduction d’ordre en assimilation de donn´ees. PhD thesis, University of Grenoble.[13] Faugeras, B., Levy, M., Memery, L., Verron, J., Blum, J. and Charpentier, I.,2003: Can biogeochemical fluxes be recovered from nitrate and chlorophyll data? A case study assimilating data in the Northwestern Mediteranean Sea at theJGOFS-DYFAMED station. Jour. of Mar. Sys. , , 99–125.[14] Gilbert, J.-C. and Lemar´echal, C., 1989: Some numerical experiments withvariable storage quasi-Newton algorithms. Mathematical programming , , 407–435.[15] Greiner, E., Arnault, S. and Morli`ere, A., 1998: Twelve monthly experiments of4D-variational assimilation in the tropical Atlantic during 1987. Part 1: methodand statistical results. Progr. Oceanogr. , , 141–202.[16] Greiner, E. and Arnault, S., 2000: Comparing the results of a 4D-variationalassimilation of satellite and in situ data with WOCE CITHER hydrographicmeasurements in the tropical Atlantic. Prog. Oceanogr. , , 1–68.[17] Hoang, H.S., Baraille, R. and Talagrand, O., 2001: On the design of a stableadaptive filter for state estimation in high dimensional systems. Automatica , ,341–359.[18] Hoteit, I., Khol, A., Stammer, D. and Heimbach, P., 2003: A reduced-orderoptimization strategy for four dimensional variational data assimilation. InProceedings of EGS-AGU-EUG Joint Assembly, 6-11 April 2003, Nice, France.[19] Ide, K., Courtier, P., Ghil, M. and Lorenc, A. C., 1997: Unified notation fordata assimilation: operational, sequential and variational. J. Meteorol. Soc. Jpn , , 181–189.[20] Lawson, L. M., Spitz, Y. H., Hofman, B. E. and Long, R. B., 1995: A dataassimilation technique applied to a predator-prey model. Bull. Math. Biol. , ,593–617.[21] Legras, B. and Vautard, R., 1995: A guide to Liapunov vectors. Pp. 143–156 inProceedings of the ECMWF seminar on Predictability.[22] Lellouche, J.-M., Ouberdous, M. and Eifler, W., 2000: 4D-Var data assimilationsystem for a coupled physical-biological model. Earth and Planetary Science , ,491–502.[23] Lermusiaux, P. J. F. and Robinson, A. R., 1999: Data Assimilation via ErrorSubspace Statistical Estimation. Part I: Theory and Schemes. Mon. Wea. Rev. , , 1385–1407.[24] Leredde, Y., Lellouche, J.-M., Devenon, J.-L. and Dekeyser, I., 1998: On initial,boundary conditions, and viscosity coefficient control for Burger’s equation. Int.J. Num. Meth. Fluids , , 113–128.
25] Lions, J.L., Temam, R. and Wang, S., 1992: On the equation of large scaleOcean.
Nonlinearity , , 1007–1053.[26] Luong, B., Blum, J. and Verron, J., 1998: A variational method for theresolution of a data assimilation problem in oceanography. Inverse Problems , ,979–997.[27] Madec, G., Delecluse, P., Imbard, M. and Levy, C., 1999: OPA release 8.1, Oceangeneral circulation maodel reference manual. Internal report, LODYC/IPSL ,France.[28] Moore, A. M., 1991: Data assimilation in a quasigeostrophic open-ocean modelof the Gulf-Stream region using the adjoint model.
J. Phys. Oceanogr. , , 398–427.[29] Mu, M., 2000: Nonlinear Singular Vectors and Nonlinear Singular Values. Science in China , , 375–385.[30] Pham, D. T., Verron, J. and Roubaud, M.-C., 1998: A singular evolutiveextended Kalman filter for data assimilation in oceanography. J. Mar. Syst. , ,323–340.[31] Rabier, F., Jarvinen, H., Klinker, E., Mantouf, J.-F. and Simmons, A., 2000: TheECMWF operational implementation of 4D-Var assimilation. Part I: experimentalresults with simplified physics. Q. J. R. Meteorol. Soc. , , 1143–1170.[32] Ricci, S., Weaver, A. T., Vialard, J. and Rogel, P. 2005: Incorporating state-dependent Temperature-Salinity constraints in the background error covarianceof variational ocean data assimilation. Mon. Wea. Rev. , , 317–338.[33] Schr¨oter, J., Seiler, U. and Wenzel, M., 1993: Variational assimilation of Geosatdata into an eddy-resolving model of the Gulf Stream area. J. Phys. Oceanogr. , , 925–953.[34] Spitz, Y. H., Moisan, J. R., Abbott, M. R. and Richman, J. G., 1998: Dataassimilation and a pelagic ecosystem model: parameterization using time seriesobservations. J. Mar. Syst. , , 51–68.[35] Toth, Z. and Kalnay, E., 1997: Ensemble forecasting at NCEP : the breedingmethod. Mon. Weather Rev. , , 3297–3318.[36] Veers´e, F. and Th´epaut, J.-N., 1998: Multiple-truncation incremental approachfor four-dimensional variational data assimilation. Q. J. R. Meteorol. Soc. , ,1889–1908.[37] Vialard, J., Menkes, C., Boulanger, J.-P., Delecluse, P., Guilyardi, E.,McPhaden, M. J. and Madec, G., 2001: A model study of oceanic mechanismsaffecting equatorial Pacific sea surface temperature during the 1997-98 El Nino. J. Phys. Oceanogr. , , 1649–1675.[38] Vialard, J., Vialard, A. T., Anderson, D. L. T. and Delecluse, P., 2003: Three-and four-dimensional variational assimilation with a general circulation model ofthe tropical Pacific ocean. Part II: Physical validation. Mon. Wea. Rev. , ,1379-1395.
39] Vidard, P., 2001: Vers une prise en compte des erreurs mod`ele en assimilationde donn´ees 4D variationnelle. Application `a un mod`ele d’oc´ean. PhD thesis,University of Grenoble.[40] Weaver, A. T. and Courtier, P., 2001: Correlation modelling on the sphere usinga generalized diffusion equation.
Q. J. R. Meteorol. Soc. , , 1815–1846.[41] Weaver, A. T., Vialard, J., Anderson, D. L. T. and Delecluse, P., 2003:Three- and four-dimensional variational assimilation with a general circulationmodel of the tropical Pacific ocean. Part I: Formulation, internal diagnostics andconsistency checks. Mon. Wea. Rev. , , 1360-1378.[42] Wenzel, M. and Schr¨oter, J., 1999: 4D-Var data assimilation into the LSGOGCM using integral constraints’. Pp. 141–144 in Proceedings of the Third WMOInternational Symposium on Assimilation of Observations in Meteorology andOceanography, 7-11 June 1999, Quebec City, Canada. ig. 1. First EOF. Top: surface temperature; Middle: surface salinity; Bottom: sur-face velocity. The quantities are non-dimensional. Fig. 2. Fraction of inertia conserved by the r first EOFs : P rj =1 λ j / P pj =1 λ j as afunction of r ig. 3. Locations of the TAO morrings. ig. 4. Temperature component of the optimal increment δ x for single observationexperiments. Left : horizontal structure at z = −
45 m; right : vertical section alongthe equator. Top : full-space 4D-Var; bottom : reduced-space 4D-Var. R m s Abs RMS k=2 TN Free run4D-VarReduced 4D-Var a) R m s Abs RMS k=2 SN Free run4D-VarReduced 4D-Var b) R m s Abs RMS k=2 UN Free run4D-VarReduced 4D-Var c) Fig. 5. Rms error with respect to the exact reference solution at level 2 (depth:15 m). x -axis : time (in days). y -axis : (a) T ( ◦ K), (b) S (kg.m − ), (c) u (m.s − ).The curves correspond to experiment E REF (dotted line), E F ULL (solid line) and E REDUC (dotted line). R m s Abs RMS k=20 TN Free run4D-VarReduced 4D-Var a) R m s Abs RMS k=20 SN Free run4D-VarReduced 4D-Var b) R m s Abs RMS k=20 UN Free run4D-VarReduced 4D-Var c) Fig. 6. Same as Fig. 5, but at level 20 (depth: 750 m).
20 40 60 80 100 120 1404.44.64.855.25.45.65.8 log10 Jtot J 4D−VarJ Reduced 4D−Var
Fig. 7. Cost functions vs iterations. Solid line: experiment E F ULL (22 iterationsfor each of the six one-month assimilation time-windows); Dotted line: experiment E REDUC (22 iterations for each of the six one-month assimilation time-windows)(22 iterations for each of the six one-month assimilation time-windows)