Comparison of reduced-order, sequential and variational data assimilation methods in the tropical Pacific Ocean
aa r X i v : . [ phy s i c s . g e o - ph ] S e p Ocean Dynamics manuscript No. (will be inserted by the editor)
Comparison of reduced-order, sequential andvariational data assimilation methods in thetropical Pacific Ocean
C´eline Robert ⋆ , Eric Blayo , Jacques Verron LMC-IMAG UMR 5523 CNRS/INPG/UJF/INRIA, Grenoble LEGI UMR 5519 CNRS/INPG/UJF, GrenobleReceived: date / Revised version: date
November 2, 2018Abstract
This paper presents a comparison of two reduced-order, sequen-tial and variational data assimilation methods: the SEEK filter and the R-4D-Var. A hybridization of the two, combining the variational frameworkand the sequential evolution of covariance matrices, is also preliminarilyinvestigated and assessed in the same experimental conditions. The com-parison is performed using the twin-experiment approach on a model of theTropical Pacific domain. The assimilated data are simulated temperatureprofiles at the locations of the TAO/TRITON array moorings. It is shownthat, in a quasi-linear regime, both methods produce similarly good results. ⋆ Present address:
[email protected] C´eline Robert et al.
However the hybrid approach provides slightly better results and thus ap-pears as potentially fruitful. In a more non-linear regime, when TropicalInstability Waves develop, the global nature of the variational approachhelps control model dynamics better than the sequential approach of theSEEK filter. This aspect is probably enhanced by the context of the exper-iments in that there is a limited amount of assimilated data and no modelerror.
Operational oceanography is an emerging field of activity that is concernedwith real-time monitoring and prediction of the physical and biogeochemi-cal state of oceans and regional seas. Operational ocean prediction systemshave been made feasible by the concomitance of several elements: the emer-gence of relatively reliable numerical models and of appropriate computingcapabilities, the establishment of global ocean observation systems, and theprogress achieved in data assimilation techniques. It is the latter of theseadvances that is addressed in this paper. In the geophysical context, dataassimilation methods face a number of specific difficulties. In particular, dueto the very large dimensions of the systems, the computational burden andthe prescription of adequate error statistics are critical issues. In addition,there is a need to improve methods in the case of non-linear systems and/ornon-gaussian statistics. omparison of reduced-order data assimilation methods 3
Data assimilation methods are generally classified into two groups ac-cording to the approach used: the sequential approach, based on the sta-tistical estimation theory and the Kalman filter, and the variational ap-proach (4D-Var), built from the optimal control theory. It is well knownthat the 4D-Var and Kalman filter approaches provide the same solution,at the end of the assimilation period, for perfect and linear models. Theseapproaches are different however, mainly because the model is seen as astrong constraint in the 4D-Var approach and as a weak constraint in thesequential approach. In addition, the specification and time evolution of theerror statistics, the length and structure of the forecast-analysis cycles, andthe temporal use of observations may be quite different. In practice, dueespecially to non-linearity, these differences can result in significant discrep-ancies between the solutions provided by the two approaches.The full Kalman filter cannot be used in actual geophysical systems,because specifying of the error covariance matrices is difficult and also in-volves huge computational costs and impractical matrix handling. The needto circumvent these difficulties has led to the development of reduced-orderKalman filters. Here, order reduction consists in reducing the size of thebackground error covariance matrix by selecting a number of directions inthe state space along which the error variability is assumed to lie. In re-cent years, this approach has given birth in particular to the EnsembleKalman Filter (EnKF) (Evensen, 1994), the Reduced-Rank-SQuare-RooT(RRSQRT) filter (Verlaan and Heemink, 1997), the Singular Evolutive Ex-
C´eline Robert et al. tended Kalman (SEEK) filter (Pham et al. , 1998, Verron et al. , 1999) andthe ESSE method (Lermusiaux and Robinson, 1999). These four methodsbasically differ in their strategies to approximate the error covariance ma-trix and/or the way in which they propagate the state error statistics. Inthe EnKF, the error statistics are propagated using a statistically relevantensemble of states. The forecast error covariance matrix is not given ex-plicitly. The SEEK and RRSQRT filters are based on a truncation of aneigendecomposition of the error covariance matrix, and partly differ in theirinitial choice of the approximate low-rank matrix, and with respect to itstime evolution. The SEEK takes advantage of the fact that the ocean is adynamic system with an attractor, and is not intended to make correctionsin directions perpendicular to the attractor, which are naturally attenuatedby the system. In this paper, the SEEK filter is chosen.The variational 4D-Var method has long been used in meteorology ( e.g.
Rabier, 1998) and has been applied to several operational forecasting sys-tems in its incremental form (Courtier et al. , 1994), a form that is partic-ularly suited to nonlinear systems. It has also been developed for oceano-graphic situations ( e.g.
Greiner and Arnault, 1998a, b; Vialard et al. , 2002,2003; Weaver et al. , 2003). The method is costly and involves complex soft-ware development for the tangent linear and adjoint models. As with se-quential estimation, lack of knowledge concerning the error statistics leadsto the use of approximations and models for the background error covari-ance matrix. To solve the problem, the order reduction procedure can also omparison of reduced-order data assimilation methods 5 be used for the 4D-Var to build the Reduced-4D-Var (Blayo et al. , 1998).This has been tested in a realistic configuration by Robert et al. (2005). Inthe Reduced 4D-Var, the control parameter (namely the initial condition)now belongs to a low-dimension space and the background error covariancematrix can thus also be expressed using this subspace. The IncrementalReduced 4D-Var (hereafter R-4D-Var) is therefore the variational approachchosen here.A major advantage of the 4D-Var assimilation is the simultaneous andconsistent use of the whole observational dataset over the assimilation timewindow and the optimisation of the model trajectory is thus based on theglobal processing of these observations. A serious drawback however is thatthe background error statistics are often constant over this assimilation timewindow in actual applications. These characteristics are somewhat reversedwith the Kalman filter: observations are processed sequentially, and are oftengrouped in actual applications (which means that they are not generallyprocessed at the exact time of observation), but the state error statisticscan be propagated from one assimilation cycle to the next. To benefit fromthe advantages of each of both approaches, Veers´e et al. (2000) proposed ahybrid algorithm that combines the analysis performed by the variationalapproach with the state error propagation of the SEEK filter. This hybridapproach is the third type of data assimilation algorithm that is studiedhere.
C´eline Robert et al.
The main objective of this study is to compare these three types ofreduced-order data assimilation approaches. Note that in the reduced-orderframework, the error subspaces can be built using the same method (thechoice of the basis can be the same) and the initial subspaces will be iden-tical. The comparison is conducted using the twin-experiment approach, inwhich the data assimilated are synthetic and obtained from a free run ofthe model. Data and model are therefore entirely consistent and the as-similation is artificially facilitated as far as the model error and data errorcharacteristics are concerned. Given the current levels of observation, thisapproach was considered to be the only possible way to conduct a method-ological comparison since the ocean is not sufficiently well observed for areal comparison exercise to be meaningful. This will clearly be the next stepfollowing the present work.To our knowledge, the present study represents is one of the first attempsto compare these two methods using exactly the same configuration. Sincethe model used in this study is (weakly) non-linear and the system is of largedimension, the results obtained by both methods should not be expected tobe the same. The region chosen for the experiments is the tropical PacificOcean. The large-scale ocean dynamics in this region is weakly non-linearexcept for the Tropical Instability Waves (TIWs) that develop in the easternPacific and propagate along the equator, becoming increasingly intense frommid-June/early July. The tropical Pacific Ocean was chosen because it isone of the best-observed regions of the world ocean thanks to the TOGA omparison of reduced-order data assimilation methods 7 program and the TAO mooring network in particular. In addition, manynumerical studies have been performed in this area and, as a result, direct,tangent linear and adjoint models have been reasonably well validated.The article is organized as follows: in the next section (Section 2) we de-tail the methods used, introducing in particular the hybrid approach. Then,we describe the configuration of the twin-experiment framework (Section3). Finally, we present the main results obtained in each case (Section 4),followed by conclusions and discussion (Section 5).
This section provides details of the different reduced-order methods usedin our experiments. In the following, the notations proposed by Ide et al. ,(1997) are used. The superscripts a , b , f and T represent respectively theanalysis, background, forecast and mathematical transpose sign.In the full dimension space, error covariances are unknown and mustbe modeled. However, this is challenging for complex oceanic systems sincethe state vector contains several physical quantities (velocity, temperature,salinity) and is very large (it commonly reaches n = 10 components), andbecause many different spatial scales interact. One way to try to overcomethese difficulties is to consider that most of the variance can be retainedwithin a low-dimension space, spanned by a basis of a limited number ofvectors. The error covariance matrix can then be approximated by a low-rank matrix, considering only this reduced space. To make the reduction C´eline Robert et al. computationally efficient, the number of retained vectors r must be smallwith respect to the number of degrees of freedom of the system ( r ≪ n ). Tomake the data assimilation effective, however, the subspace must adequatelyrepresent the main directions of error propagation in the system. This typeof order reduction is used in both the sequential method (SEEK filter) andthe variational approach (R-4D-Var). In our experiment, an EOF basis is chosen to span the error subspace,and will be used both for the SEEK filter and R-4D-Var implementations.This means that we assume that the variability of the model state vectoris representative of the variability of the background error, which is indeedverified in the present context of twin experiments ( i.e. with no model error).Other bases can however be thought of for building this subspace, such asLyapunov, singular or breeding vectors (Durbiano, 2001), but EOFs haveproved to be efficient in the present context, probably because they takeinto account the nonlinearity of the model dynamics, and also because theircovariance matrix is relatively well known.The model solution, obtained from a previous numerical simulation, issampled and a multivariate EOF analysis of the resulting p three-dimensionalstate vectors ( x ( t ) , . . . , x ( t p )) is performed. It should be remembered thatthis analysis aims at determining the main directions of variability of themodel sample, which leads to diagonalizing the empirical covariance matrix omparison of reduced-order data assimilation methods 9 XX T , where X = ( X , . . . , X p ), with X j ( i ) = 1 σ i [ x ( t j ) − ¯ x ], ¯ x = 1 p p X j =1 x ( t j )and σ i is the empirical variance of the i -th component of the state vector: σ i = 1 p p X j =1 ( X j ( i )) . The inner product is the usual one for a state vectorcontaining several physical quantities expressed in different units: < X j , X k > = n X i =1 σ i ( x ( t j ) − ¯ x ) i ( x ( t k ) − ¯ x ) i (1)Since the size p of the sample is generally much smaller than the size n ofthe model state vector, the actual diagonalization is performed on the p × p matrix X T X rather than on the n × n matrix XX T (it is well known thatthose two matrices have the same spectrum). This diagonalization leads toa set of orthonormal eigenvectors ( L , . . . , L p ) corresponding to eigenvalues λ > . . . > λ p >
0. Since trajectories are computed with the free model,these modes represent its variability over the whole sampled period.If the background error e B is modeled as spanned by the r first EOFs: e B = r X j =1 w j L j = Lw , then its covariance matrix is modeled by B r = E ( Lww T L T ) = L E ( ww T ) L T , which is approximated by B r = LΛ r L T with Λ r = diag( λ , . . . , λ r ),since λ j is the natural estimate for the covariance of w j . The fraction ofvariability (or “inertia”) which is conserved when retaining only the r firstvectors is r X j =1 λ j / p X j =1 λ j . In the sequential approach, the SEEK filter is used following Pham et al. (1998) and Verron et al. (1999). Each error covariance matrix is decomposed in a reduced space in the form: P = SS T (2)The first estimate of the forecast error covariance matrix P f = S S T isgiven by the EOF decomposition, and can thus be written as P f = LΛ r L T .The SEEK filter algorithm is composed of successive analysis-forecastcycles. The analysis, at cycle k , is given by: x ak = x fk + K k [ y k − H k x fk ] (3) K k is the gain matrix, which minimizes the variance of the analysis errorand thus satisfies the following equation (given that P fk = S fk S fTk ): K k = S fk [ I + ( H k S fk ) T R − k ( H k S fk )] − ( H k S fk ) T R − k (4)The forecast, from cycle k to k + 1, is obtained using the model: x fk +1 = M k,k +1 [ x ak ] (5)During the analysis-forecast cycles, each error mode evolves over time. Theanalysis error covariance is evaluated directly at each analysis step as fol-lows: P ak = S fk [ I + ( H k S fk ) T R − k ( H k S fk )] − S fTk (6)In this formula, the diagnostic of the forecast error modes depends on theformulation. With the “fixed basis” SEEK filter, the forecast error modesare equal to the analysis error modes at the previous cycle k : S fk +1 = S ak omparison of reduced-order data assimilation methods 11 With the “evolutive basis” SEEK filter, the forecast error modes evolve withthe fully non-linear model:[ S fk +1 ] j = M [ x ak + [ S ak ] j ] − M [ x ak ] j = 1 , . . . , r This procedure makes it possible to follow the time evolution of the modelvariability, but increases the computational cost by a factor of r . In thepresent study, the implementation with a fixed basis is chosen. As mentioned earlier, for the variational approach a reduced-order approxi-mation of the Incremental 4D-Var algorithm (Courtier et al. , 1994) has beenused. In this algorithm, we assimilate data available at different times t ,..., t N , and the initial condition at time t is controlled through an increment δx . The following cost function must then be minimized: J ( δ x ) = 12 ( δ x ) T B − δ x +12 N X i =1 ( H i M i δ x − d i ) T R − ( H i M i δ x − d i ) (7)where δ x = x ( t ) − x b is the increment, and x b the first guess (or “back-ground” value) for the model state at the initial time t . M i is the tangentlinear model between time t and time t i , H i is the linearized observationoperator at time t i and d i the innovation vector d i = y oi − H i M i x b ( y oi isthe observation vector at time t i ).Because of its size, the covariance matrix is never explicitly calculated inthe full 4D-Var method. The B matrix is built as an operator composition in order to represent error covariances, generally as gaussian-like functions(Weaver et al. , 2001).In the reduced-order approach, as proposed in section 2.1, the increment δ x is looked for in a low-dimension space spanned by the r first EOFs: δ x = r X j =1 w j L j = Lw , which results in the use of the low-rank covariancematrix B r = LΛ r L T . Formally, the same cost function (Eq. 7) must beminimized (only the expression of B and δ x change), but the minimizationphase is performed on a very limited number of coefficients w , . . . , w r .When this reduced-order approach is compared to the full 4D-Var algorithmusing the same twin-experiment framework as in the present paper, it isfound that only 10-15 iterations are needed to reach the minimum of thecost function while almost 40 (and often more) are necessary with the full4D-Var (Robert et al. , 2005). For the assimilation of real data, however,designing a relevant reduced basis becomes a challenge, because the modelis no longer perfect. These two reduced-order methods, SEEK filter and R-4D-Var, present sev-eral similarities. In particular, the choice of the initial error subspace can beexactly the same). However, intrinsic differences remain. For example, forthe SEEK filter, the observations are unrealistically co-located in time ac-cording to the analysis window (they are typically gathered every 10 days),unlike the 4D-Var in which the observations are correctly distributed over omparison of reduced-order data assimilation methods 13 time throughout the assimilation window (typically one month here). A sec-ond fundamental difference is that, in the SEEK filter, the error subspaceevolves in time at every analysis step, which makes it possible to follow theevolution of the error. In the R-4D-Var approach, the initial subspace gen-erally remains constant during the assimilation period, even if this periodis divided into successive time windows (e.g. one month) for the validity ofthe tangent linear approximation.In an attempt to combine the best features of both these methods, Veers´e et al. (2000) proposed a hybrid algorithm using the 4D-Var and the SEEKsmoother. This method, developed only from a theoretical point of view,has never been implemented in a real numerical configuration.Although the theoretical context is slightly different here, since Veers´e et al. (2000) used a SEEK smoother instead of a SEEK filter, we retainedthe idea of making the covariance matrix B of the R-4D-Var evolve in timethanks to the SEEK filter. The following hybrid algorithm can thus beproposed: – Initialize B = P f using the r first vectors provided by an EOF analysis – Perform R-4D-Var and SEEK filter assimilations on successive time win-dows. In the present implementation, these windows are one month long.R-4D-Var processes this window in one go. For the SEEK filter, sincethe observations are artificially gathered every ten days, three analysissteps are performed during each window. – At the end of each window, B is updated in the R-4D-Var by the newvalue of P f provided by the SEEK filter, and the state vector x f of theSEEK filter is reinitialized using the final state provided by the R-4D-Var at the end of the window.The cost of this algorithm is the sum of the costs of both methods. As mentioned previously, we compared the different assimilation methodsin a unique configuration in the tropical Pacific Ocean. The general oceancirculation in this area is weakly non-linear, which was seen as a convenientproperty for conducting a first comparison of the methods.The numerical model used in the experiments is the OPA model (Madec et al. , 1998), in the so called OPA-TDH configuration (Vialard et al. , 2003).The extent of the domain is shown in Fig. 1. The horizontal grid of themodel is 1 ◦ in longitude and 0 . ◦ in latitude at the equator, stretched toreach 2 ◦ at the northern and southern limits of the domain. The verticalgrid is composed of 25 levels, spaced at intervals ranging from 5 m at thesurface to 1000 m for the deepest levels.Following Weaver et al. (2003) and Vialard et al. (2003), the year 1993was chosen as our simulation period. During this year, the circulation of thetropical Pacific Ocean was marked by the weak influence of the last El Ni˜no omparison of reduced-order data assimilation methods 15 event (the last big one had occurred in 1982-1983, the ones in 1986-1987 and1991-1992 had been weaker and the next big one would begin only in 1997).The year 1993 can therefore be seen as a “normal” year from a dynamicalpoint of view and was thus suitable for conducting a comparison of the twodifferent approaches. All experiments last one year, beginning on January1, 1993. The winds used to force the numerical model were based on bothsatellite ERS measurements and in-situ TAO winds (Menkes et al. , 1998).The atmospheric heat fluxes came from ECMWF data files (ERA 40). All the experiments discussed here are conducted using the twin experimentframework. The initial true state is obtained from a previous simulation overyear 1992 and is used to generate a reference one-year free run consideredas the truth. This solution is then sampled to generate simulated temper-ature data. The distribution of these simulated data is chosen as close aspossible to the distribution of the real TAO/TRITON array (Fig. 1) andXBT profiles. Temperature is sampled from the surface down to a depthof 500 meters every six hours. A gaussian noise is added to the simulatedobservations with a standard error set at σ T = 0 . ◦ C .For the computation of the EOFs, the model state x consists of 4 vari-ables: temperature, salinity and the two horizontal components of velocity.A free run experiment trajectory is sampled over one year, using a 2-dayperiodicity to build the covariance matrix. A large part of the total variance is represented by a few EOFs: 80% for the first 13 EOFs, 92% for the first30 EOFs.Since the initial state used in the assimilation experiments is not thecorrect one, we want to control an error on the initial condition throughdata assimilation. This error is more or less corrected naturally by the freerun in roughly six months, thanks to the forcings. After six months, the errorto be controlled is no longer an error on the initial condition but mainlyconcerns the non-linear dynamics of the model.In the R-4D-Var experiment, the error on the initial condition is in-troduced via the background. For the initial background state, we use thesolution of the reference simulation on April 1 ( i.e. omparison of reduced-order data assimilation methods 17 the solution and acting more specifically on the dynamics. Moreover, itshould be noted that the limitation of the R-4D-Var method arising fromthe fact that the model is a strong constraint does not play a role here sincethe model is supposed to be perfect in these twin experiments.For the variational approach we used the OPAVAR package, developedand validated by Weaver et al. (2003) and Vialard et al. (2003), and for theSEEK filter experiments, the SESAM package (Testut et al. , 2001). The results shown below are presented for two different periods, from Jan-uary to June 1993 and from July to December 1993. There are two mainreasons for this: (i) in such a quasi-linear model and twin-experiment frame-work, the model is naturally restored from the erroneous initial conditionover a time scale of some months, (ii) physically, June is also the time ofthe onset of the non-linear Tropical Instability Waves (TIWs) in the easterntropical Pacific Ocean. Schematically, the first six-month period concernsthe control of the error on the initial condition, while the next six-monthperiod concerns the control of the non-linear dynamics in the system. In thelatter period, the intensity of TIWs begins to increase, a development thatappears to have considerable influence on data assimilation.
In this first time period, the error on the initial condition is quite large,since the forcings of the model have not had time to correct it, so that mostof the work done by data assimilation is to control the error on the initialcondition. Moreover, the dynamics is quite stable and the EOFs representmodel variability perfectly. In this case, and as would be expected from thepurely linear and optimal context, it can be observed in Fig. 2 that theassimilation methods are almost equivalent. We can see that the correctionis substantial and that all methods provide roughly the same solution. Thethree algorithms work well, not only in terms of temperature misfit in thearea of observations but also at greater depths and for the other variables,thanks to the multivariate nature of the EOFs. This can be seen for examplein Fig. 3.Concerning the hybrid algorithm, a slightly lower level of error is ob-tained as shown in Fig. 2, for example for ( u, v ) variables. The diagnostic ofthe analysis and forecast errors performed by the SEEK filter is correct withregard to the dynamics. The solution is thus very good. Finally, this hybridmethod succeeds in combining two intrinsic aspects of the reduced-ordermethods, which leads to slightly improved results. The fact that the im-provement is not more significant is because both the SEEK and R-4D-Varmethods already obtain excellent results in decreasing the error. omparison of reduced-order data assimilation methods 19
The second period starts in July, when the intensity of TIWs increasessignificantly. The dynamics of these waves is non-linear. Since the first TAOpoint is quite distant from the eastern coast, the TIWs rise before the firstdata point can see the change. In this case, it can take a significant time(even more than 10 days, which is the duration of the SEEK cycle) beforethe easternmost points of the TAO array register these changes.In addition, the error due to the initial condition is very weak, since thedynamics of the model has already naturally corrected this error. Thus, themajor part of the remaining error is driven by the non-linear dynamics ofthe ocean. As we can see in Fig. 4, the difference in temperature betweenthe reference simulation and the SEEK filter simulation is mainly locatedin the eastern basin, near the equator.An important difference between the algorithms is their sequential ver-sus global processing of the observations. Since the observations availableduring the whole assimilation window are taken into account in the R-4D-Var method, this approach can anticipate the propagation of a physicalphenomenon at this time scale. For example, the Tropical Instability Waves(TIWs) rise in the eastern part of the basin and propagate along the equa-tor in roughly one month. When they become more intense in early July,the variational system takes into account observations of these waves in theanalysis conducted before their actual occurrence. This is not the case inthe SEEK filter because the analysis at the same time takes into account only past observations. Consequently, there is a time lag with this approachbetween the occurrence of the physical phenomenon (the intensification)and its integration into the analysis. This explains the differences observedin Fig. 5 which concerns the eastern part of the domain. The comparisonis almost the same in the western part (Fig. 6), but with a lower rms errorlevel, probably because more observations are available.In that second period, the hybrid method continues to draw advantagesfrom the quality of the analysis of the R-4D-Var. However, since the evolu-tion of its covariance matrix provided by the SEEK filter is less accurate,the hybrid method do not succeed in that case in providing better resultsthan the R-4D-Var (see Fig 6 and 5).
This paper presents the results of a first comparison of three reduced-orderdata assimilation methods implemented in a model of the tropical PacificOcean. The first two methods, the SEEK filter and the reduced-order 4D-VAR, are respectively derived from the sequential and the variational ap-proaches. The third method, combining features of the SEEK filter and theR-4D-VAR, is a hybrid version of the first two methods. To our knowledge,the present study is probably one of the first side-by-side implementationsand comparisons of these techniques ever made in a realistic context. Investi-gations are exploratory in nature due to the complexity of the methodologyand more especially because simulations have been carried out in a twin- omparison of reduced-order data assimilation methods 21 experiment framework where no model error is present and data are simu-lated. In addition, only one year of comparative simulations is performed.However, we believe that the first results provide useful insights: – In a quasi-linear regime, as expected from linear theory, the three meth-ods provide rather similar results in reducing the initial condition systemerror. The hybrid method provides slightly better results, which wouldmean, as expected, that combining the evolution of error covariancematrices and the variational analysis is, at the very least, feasible andpotentially fruitful (as soon as further tuning is done). – In a regime where strong nonlinearities develop at regional scales (cor-responding to the onset of Tropical Instability Waves within the easterntropical Pacific), the 4D-VAR succeeds in keeping the error to a low levelwhereas the SEEK filter, due to its sequential nature, fails to fully controlthe increasing instability using the data available. The hybrid methodfollows the divergent nature of the SEEK filter in the first stages but,after several weeks, resumes a more convergent path.The twin-experiment set-up entails obvious limits and may influence theconclusions. The“no model error” assumption may favor the variational so-lution since the twin experiments are exactly within the “strong constraint”variational framework. It also favors the performances of all reduced-ordermethods since the reduced basis can be built from a perfect reference simu-lation. With statistical low-cost methods like the SEEK filter, the amountand the nature of assimilated data are a key factor. The data used in the present study mimic the real data that are acquired from the TAO array inthe Pacific, but this array poorly samples the eastern Pacific Ocean. It islikely that with the addition of some higher frequency complementary data(such as altimetric data), the SEEK filter would behave more satisfactorily(see, for example, Castruccio et al. , 2006). Preliminary experiments werealso performed with real TAO data, thus including model errors. In thiscase, first results seem to indicate a much more balanced behavior betweenthe SEEK filter and the reduced 4D-VAR.
Acknowledgments
J.-M. Brankart and A. Weaver are gratefully acknowledged for supplyingrespectively the SESAM and OPAVAR packages and for providing supportin using these tools. This work has been supported by CNES and the MER-CATOR project. Idopt is a joint CNRS-INPG-INRIA-UJF research project.
Fig. 1 oo o o o o
FREE RUN o R−4DVSEEKHYBRID (a) Temperature o o o o o o
FREE RUN o R−4DVSEEKHYBRID (b) Salinity o o o o o o
FREE RUN o R−4DVHYBRIDSEEK (c) Velocity u o o o o oo FREE RUN o R−4DVSEEKHYBRID (d) Velocity vFig. 2
Absolute Rms error obtained by each method during the first 6 monthsof simulation, at 15m. depth. Solid line with +: free run, solid line: R-4D-Var,dashed line with o: SEEK filter and dashed-dotted line: hybrid method.4 C´eline Robert et al. o o o o o o
FREE RUN o R−4DVSEEKHYBRID (a) Velocity u o o o o o o FREE RUN o R−4DVSEEKHYBRID (b) Velocity vFig. 3
Rms error obtained by each method during the first 6 months of simula-tion, below the observations, at 750 m. depth. Solid line with +: free run, solidline: R-4D-Var, dashed line with o: SEEK filter and dashed-dotted line: hybridmethod.omparison of reduced-order data assimilation methods 25(a) End of June(b) Beginning of July
Fig. 4
Difference between temperature field at 15 m. depth of the reference sim-ulation and of the SEEK filter simulation.6 C´eline Robert et al.
HYBRIDSEEKR−4DV o xx x o o o o o o o o x x x x x x x x x x x x x x (a) Temperature HYBRIDSEEKR−4DV o x o o o o o o o o xxxxxxxxxxxxxxxx (b) Salinity HYBRIDSEEKR−4DV o xx x x x x x x x x x x x x x x x o o o o o o o o (c) Velocity u HYBRIDSEEKR−4DV o xx x x x o o o o o o x x o o x x x x x x x x x x (d) Velocity vFig. 5 Rms error the last 6 months of simulation, at 15 m. depth, in the easternpart of the basin. Solid line: R-4D-Var, dashed line with o: hybrid method anddashed-dotted line with x: SEEK filter.omparison of reduced-order data assimilation methods 27 o o o o o o
FREE RUN o R−4DVSEEKHYBRID (a) Temperature o o o o o o
FREE RUN o R−4DVSEEKHYBRID (b) Salinity o o o o o o
FREE RUN o R−4DVSEEKHYBRID (c) Velocity u o o o o o o FREE RUN o R−4DVSEEKHYBRID (d) Velocity vFig. 6vFig. 6