hal , version 1-18 Sep 2007

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1 Ocean Dynamics manuscript N. (will be inserted by the editr) Cmparisn f reduced-rder, sequential and variatinal data assimilatin methds in the trpical Pacific Ocean Céline Rbert 1, Eric Blay 1, Jacques Verrn 2 1 LMC-IMAG UMR 5523 CNRS/INPG/UJF/INRIA, Grenble 2 LEGI UMR 5519 CNRS/INPG/UJF, Grenble hal , versin 1-18 Sep 2007 Received: date / Revised versin: date September 18, 2007 Abstract This paper presents a cmparisn f tw reduced-rder, sequential and variatinal data assimilatin methds: the filter and the R- 4D-Var. A hybridizatin f the tw, cmbining the variatinal framewrk and the sequential evlutin f cvariance matrices, is als preliminarily investigated and assessed in the same eperimental cnditins. The cmparisn is perfrmed using the twin-eperiment apprach n a mdel f the Trpical Pacific dmain. The assimilated data are simulated temperature prfiles at the lcatins f the TAO/TRITON array mrings. It is shwn that, in a quasi-linear regime, bth methds prduce similarly gd results. Present address: Celine.Rbert@imag.fr

2 2 Céline Rbert et al. Hwever the hybrid apprach prvides slightly better results and thus appears as ptentially fruitful. In a mre nn-linear regime, when Trpical Instability Waves develp, the glbal nature f the variatinal apprach helps cntrl mdel dynamics better than the sequential apprach f the filter. This aspect is prbably enhanced by the cntet f the eperiments in that there is a limited amunt f assimilated data and n mdel errr. 1 Intrductin Operatinal ceangraphy is an emerging field f activity that is cncerned with real-time mnitring and predictin f the physical and bigechemical state f ceans and reginal seas. Operatinal cean predictin systems have been made feasible by the cncmitance f several elements: the emergence f relatively reliable numerical mdels and f apprpriate cmputing capabilities, the establishment f glbal cean bservatin systems, and the prgress achieved in data assimilatin techniques. It is the latter f these advances that is addressed in this paper. In the gephysical cntet, data assimilatin methds face a number f specific difficulties. In particular, due t the very large dimensins f the systems, the cmputatinal burden and the prescriptin f adequate errr statistics are critical issues. In additin, there is a need t imprve methds in the case f nn-linear systems and/r nn-gaussian statistics.

3 Cmparisn f reduced-rder data assimilatin methds 3 Data assimilatin methds are generally classified int tw grups accrding t the apprach used: the sequential apprach, based n the statistical estimatin thery and the Kalman filter, and the variatinal apprach (4D-Var), built frm the ptimal cntrl thery. It is well knwn that the 4D-Var and Kalman filter appraches prvide the same slutin, at the end f the assimilatin perid, fr perfect and linear mdels. These appraches are different hwever, mainly because the mdel is seen as a strng cnstraint in the 4D-Var apprach and as a weak cnstraint in the sequential apprach. In additin, the specificatin and time evlutin f the errr statistics, the length and structure f the frecast-analysis cycles, and the tempral use f bservatins may be quite different. In practice, due especially t nn-linearity, these differences can result in significant discrepancies between the slutins prvided by the tw appraches. The full Kalman filter cannt be used in actual gephysical systems, because specifying f the errr cvariance matrices is difficult and als invlves huge cmputatinal csts and impractical matri handling. The need t circumvent these difficulties has led t the develpment f reduced-rder Kalman filters. Here, rder reductin cnsists in reducing the size f the backgrund errr cvariance matri by selecting a number f directins in the state space alng which the errr variability is assumed t lie. In recent years, this apprach has given birth in particular t the Ensemble Kalman Filter (EnKF) (Evensen, 1994), the Reduced-Rank-SQuare-RT (RRSQRT) filter (Verlaan and Heemink, 1997), the Singular Evlutive E-

4 4 Céline Rbert et al. tended Kalman () filter (Pham et al., 1998, Verrn et al., 1999) and the ESSE methd (Lermusiau and Rbinsn, 1999). These fur methds basically differ in their strategies t apprimate the errr cvariance matri and/r the way in which they prpagate the state errr statistics. In the EnKF, the errr statistics are prpagated using a statistically relevant ensemble f states. The frecast errr cvariance matri is nt given eplicitly. The and RRSQRT filters are based n a truncatin f an eigendecmpsitin f the errr cvariance matri, and partly differ in their initial chice f the apprimate lw-rank matri, and with respect t its time evlutin. The takes advantage f the fact that the cean is a dynamic system with an attractr, and is nt intended t make crrectins in directins perpendicular t the attractr, which are naturally attenuated by the system. In this paper, the filter is chsen. The variatinal 4D-Var methd has lng been used in meterlgy (e.g. Rabier, 1998) and has been applied t several peratinal frecasting systems in its incremental frm (Curtier et al., 1994), a frm that is particularly suited t nnlinear systems. It has als been develped fr ceangraphic situatins (e.g. Greiner and Arnault, 1998a, b; Vialard et al., 2002, 2003; Weaver et al., 2003). The methd is cstly and invlves cmple sftware develpment fr the tangent linear and adjint mdels. As with sequential estimatin, lack f knwledge cncerning the errr statistics leads t the use f apprimatins and mdels fr the backgrund errr cvariance matri. T slve the prblem, the rder reductin prcedure can als

5 Cmparisn f reduced-rder data assimilatin methds 5 be used fr the 4D-Var t build the Reduced-4D-Var (Blay et al., 1998). This has been tested in a realistic cnfiguratin by Rbert et al. (2005). In the Reduced 4D-Var, the cntrl parameter (namely the initial cnditin) nw belngs t a lw-dimensin space and the backgrund errr cvariance matri can thus als be epressed using this subspace. The Incremental Reduced 4D-Var (hereafter R-4D-Var) is therefre the variatinal apprach chsen here. A majr advantage f the 4D-Var assimilatin is the simultaneus and cnsistent use f the whle bservatinal dataset ver the assimilatin time windw and the ptimisatin f the mdel trajectry is thus based n the glbal prcessing f these bservatins. A serius drawback hwever is that the backgrund errr statistics are ften cnstant ver this assimilatin time windw in actual applicatins. These characteristics are smewhat reversed with the Kalman filter: bservatins are prcessed sequentially, and are ften gruped in actual applicatins (which means that they are nt generally prcessed at the eact time f bservatin), but the state errr statistics can be prpagated frm ne assimilatin cycle t the net. T benefit frm the advantages f each f bth appraches, Veersé et al. (2000) prpsed a hybrid algrithm that cmbines the analysis perfrmed by the variatinal apprach with the state errr prpagatin f the filter. This hybrid apprach is the third type f data assimilatin algrithm that is studied here.

6 6 Céline Rbert et al. The main bjective f this study is t cmpare these three types f reduced-rder data assimilatin appraches. Nte that in the reduced-rder framewrk, the errr subspaces can be built using the same methd (the chice f the basis can be the same) and the initial subspaces will be identical. The cmparisn is cnducted using the twin-eperiment apprach, in which the data assimilated are synthetic and btained frm a free run f the mdel. Data and mdel are therefre entirely cnsistent and the assimilatin is artificially facilitated as far as the mdel errr and data errr characteristics are cncerned. Given the current levels f bservatin, this apprach was cnsidered t be the nly pssible way t cnduct a methdlgical cmparisn since the cean is nt sufficiently well bserved fr a real cmparisn eercise t be meaningful. This will clearly be the net step fllwing the present wrk. T ur knwledge, the present study represents is ne f the first attemps t cmpare these tw methds using eactly the same cnfiguratin. Since the mdel used in this study is (weakly) nn-linear and the system is f large dimensin, the results btained by bth methds shuld nt be epected t be the same. The regin chsen fr the eperiments is the trpical Pacific Ocean. The large-scale cean dynamics in this regin is weakly nn-linear ecept fr the Trpical Instability Waves (TIWs) that develp in the eastern Pacific and prpagate alng the equatr, becming increasingly intense frm mid-june/early July. The trpical Pacific Ocean was chsen because it is ne f the best-bserved regins f the wrld cean thanks t the TOGA

7 Cmparisn f reduced-rder data assimilatin methds 7 prgram and the TAO mring netwrk in particular. In additin, many numerical studies have been perfrmed in this area and, as a result, direct, tangent linear and adjint mdels have been reasnably well validated. The article is rganized as fllws: in the net sectin (Sectin 2) we detail the methds used, intrducing in particular the hybrid apprach. Then, we describe the cnfiguratin f the twin-eperiment framewrk (Sectin 3). Finally, we present the main results btained in each case (Sectin 4), fllwed by cnclusins and discussin (Sectin 5). 2 The reduced-rder apprach This sectin prvides details f the different reduced-rder methds used in ur eperiments. In the fllwing, the ntatins prpsed by Ide et al., (1997) are used. The superscripts a, b, f and T represent respectively the analysis, backgrund, frecast and mathematical transpse sign. In the full dimensin space, errr cvariances are unknwn and must be mdeled. Hwever, this is challenging fr cmple ceanic systems since the state vectr cntains several physical quantities (velcity, temperature, salinity) and is very large (it cmmnly reaches n = 10 6 cmpnents), and because many different spatial scales interact. One way t try t vercme these difficulties is t cnsider that mst f the variance can be retained within a lw-dimensin space, spanned by a basis f a limited number f vectrs. The errr cvariance matri can then be apprimated by a lwrank matri, cnsidering nly this reduced space. T make the reductin

8 8 Céline Rbert et al. cmputatinally efficient, the number f retained vectrs r must be small with respect t the number f degrees f freedm f the system (r n). T make the data assimilatin effective, hwever, the subspace must adequately represent the main directins f errr prpagatin in the system. This type f rder reductin is used in bth the sequential methd ( filter) and the variatinal apprach (R-4D-Var). 2.1 Definitin f the subspace In ur eperiment, an EOF basis is chsen t span the errr subspace, and will be used bth fr the filter and R-4D-Var implementatins. This means that we assume that the variability f the mdel state vectr is representative f the variability f the backgrund errr, which is indeed verified in the present cntet f twin eperiments (i.e. with n mdel errr). Other bases can hwever be thught f fr building this subspace, such as Lyapunv, singular r breeding vectrs (Durbian, 2001), but EOFs have prved t be efficient in the present cntet, prbably because they take int accunt the nnlinearity f the mdel dynamics, and als because their cvariance matri is relatively well knwn. The mdel slutin, btained frm a previus numerical simulatin, is sampled and a multivariate EOF analysis f the resulting p three-dimensinal state vectrs ((t 1 ),...,(t p )) is perfrmed. It shuld be remembered that this analysis aims at determining the main directins f variability f the mdel sample, which leads t diagnalizing the empirical cvariance matri

9 Cmparisn f reduced-rder data assimilatin methds 9 XX T, where X = (X 1,...,X p ), with X j (i) = 1 σ i [(t j ) ], = 1 p p (t j ) and σi 2 is the empirical variance f the i-th cmpnent f the state vectr: σi 2 = 1 p (X j (i)) 2. The inner prduct is the usual ne fr a state vectr p j=1 cntaining several physical quantities epressed in different units: σ 2 i=1 i j=1 n 1 < X j,x k >= ((t j ) ) i ((t k ) ) i (1) Since the size p f the sample is generally much smaller than the size n f the mdel state vectr, the actual diagnalizatin is perfrmed n the p p matri X T X rather than n the n n matri XX T (it is well knwn that thse tw matrices have the same spectrum). This diagnalizatin leads t a set f rthnrmal eigenvectrs (L 1,...,L p ) crrespnding t eigenvalues λ 1 >... > λ p > 0. Since trajectries are cmputed with the free mdel, these mdes represent its variability ver the whle sampled perid. If the backgrund errr e B is mdeled as spanned by the r first EOFs: r e B = w j L j = Lw, then its cvariance matri is mdeled by B r = E(Lww T L T ) = j=1 LE(ww T )L T, which is apprimated by B r = LΛ r L T with Λ r = diag(λ 1,..., λ r ), since λ j is the natural estimate fr the cvariance f w j. The fractin f variability (r inertia ) which is cnserved when retaining nly the r first r p vectrs is λ j / λ j. j=1 j=1 2.2 The sequential apprach In the sequential apprach, the filter is used fllwing Pham et al. (1998) and Verrn et al. (1999). Each errr cvariance matri is decmpsed

10 10 Céline Rbert et al. in a reduced space in the frm: P = SS T (2) The first estimate f the frecast errr cvariance matri P f 0 = S 0S T 0 is given by the EOF decmpsitin, and can thus be written as P f 0 = LΛ rl T. The filter algrithm is cmpsed f successive analysis-frecast cycles. The analysis, at cycle k, is given by: a k = f k + K k[y k H k f k ] (3) K k is the gain matri, which minimizes the variance f the analysis errr and thus satisfies the fllwing equatin (given that P f k = Sf k SfT k ): K k = S f k [I + (H ks f k )T R 1 k (H ks f k )] 1 (H k S f k )T R 1 k (4) The frecast, frm cycle k t k + 1, is btained using the mdel: f k+1 = M k,k+1[ a k] (5) During the analysis-frecast cycles, each errr mde evlves ver time. The analysis errr cvariance is evaluated directly at each analysis step as fllws: P a k = S f k [I + (H ks f k )T R 1 k (H ks f k )] 1 S ft k (6) In this frmula, the diagnstic f the frecast errr mdes depends n the frmulatin. With the fied basis filter, the frecast errr mdes are equal t the analysis errr mdes at the previus cycle k: S f k+1 = Sa k

11 Cmparisn f reduced-rder data assimilatin methds 11 With the evlutive basis filter, the frecast errr mdes evlve with the fully nn-linear mdel: [S f k+1 ] j = M[a k + [S a k] j ] M[ a k] j = 1,...,r This prcedure makes it pssible t fllw the time evlutin f the mdel variability, but increases the cmputatinal cst by a factr f r. In the present study, the implementatin with a fied basis is chsen. 2.3 The variatinal apprach As mentined earlier, fr the variatinal apprach a reduced-rder apprimatin f the Incremental 4D-Var algrithm (Curtier et al., 1994) has been used. In this algrithm, we assimilate data available at different times t 1,...,t N, and the initial cnditin at time t 0 is cntrlled thrugh an increment δ. The fllwing cst functin must then be minimized: J(δ) = 1 2 (δ)t B 1 δ N i=1 (H i M i δ d i ) T R 1 i (H i M i δ d i ) (7) where δ = (t 0 ) b is the increment, and b the first guess (r backgrund value) fr the mdel state at the initial time t 0. M i is the tangent linear mdel between time t 0 and time t i, H i is the linearized bservatin peratr at time t i and d i the innvatin vectr d i = y i H im i b ( y i is the bservatin vectr at time t i ). Because f its size, the cvariance matri is never eplicitly calculated in the full 4D-Var methd. The B matri is built as an peratr cmpsitin

12 12 Céline Rbert et al. in rder t represent errr cvariances, generally as gaussian-like functins (Weaver et al., 2001). In the reduced-rder apprach, as prpsed in sectin 2.1, the increment δ is lked fr in a lw-dimensin space spanned by the r first EOFs: r δ = w j L j = Lw, which results in the use f the lw-rank cvariance j=1 matri B r = LΛ r L T. Frmally, the same cst functin (Eq. 7) must be minimized (nly the epressin f B and δ change), but the minimizatin phase is perfrmed n a very limited number f cefficients w 1,..., w r. When this reduced-rder apprach is cmpared t the full 4D-Var algrithm using the same twin-eperiment framewrk as in the present paper, it is fund that nly iteratins are needed t reach the minimum f the cst functin while almst 40 (and ften mre) are necessary with the full 4D-Var (Rbert et al., 2005). Fr the assimilatin f real data, hwever, designing a relevant reduced basis becmes a challenge, because the mdel is n lnger perfect. 2.4 The hybrid methd These tw reduced-rder methds, filter and R-4D-Var, present several similarities. In particular, the chice f the initial errr subspace can be eactly the same). Hwever, intrinsic differences remain. Fr eample, fr the filter, the bservatins are unrealistically c-lcated in time accrding t the analysis windw (they are typically gathered every 10 days), unlike the 4D-Var in which the bservatins are crrectly distributed ver

13 Cmparisn f reduced-rder data assimilatin methds 13 time thrughut the assimilatin windw (typically ne mnth here). A secnd fundamental difference is that, in the filter, the errr subspace evlves in time at every analysis step, which makes it pssible t fllw the evlutin f the errr. In the R-4D-Var apprach, the initial subspace generally remains cnstant during the assimilatin perid, even if this perid is divided int successive time windws (e.g. ne mnth) fr the validity f the tangent linear apprimatin. In an attempt t cmbine the best features f bth these methds, Veersé et al. (2000) prpsed a hybrid algrithm using the 4D-Var and the smther. This methd, develped nly frm a theretical pint f view, has never been implemented in a real numerical cnfiguratin. Althugh the theretical cntet is slightly different here, since Veersé et al. (2000) used a smther instead f a filter, we retained the idea f making the cvariance matri B f the R-4D-Var evlve in time thanks t the filter. The fllwing hybrid algrithm can thus be prpsed: Initialize B = P f 0 using the r first vectrs prvided by an EOF analysis Perfrm R-4D-Var and filter assimilatins n successive time windws. In the present implementatin, these windws are ne mnth lng. R-4D-Var prcesses this windw in ne g. Fr the filter, since the bservatins are artificially gathered every ten days, three analysis steps are perfrmed during each windw.

14 14 Céline Rbert et al. At the end f each windw, B is updated in the R-4D-Var by the new value f P f prvided by the filter, and the state vectr f f the filter is reinitialized using the final state prvided by the R-4D- Var at the end f the windw. The cst f this algrithm is the sum f the csts f bth methds. 3 Eperiments 3.1 Cnfiguratin As mentined previusly, we cmpared the different assimilatin methds in a unique cnfiguratin in the trpical Pacific Ocean. The general cean circulatin in this area is weakly nn-linear, which was seen as a cnvenient prperty fr cnducting a first cmparisn f the methds. The numerical mdel used in the eperiments is the OPA mdel (Madec et al., 1998), in the s called OPA-TDH cnfiguratin (Vialard et al., 2003). The etent f the dmain is shwn in Fig. 1. The hrizntal grid f the mdel is 1 in lngitude and 0.5 in latitude at the equatr, stretched t reach 2 at the nrthern and suthern limits f the dmain. The vertical grid is cmpsed f 25 levels, spaced at intervals ranging frm 5 m at the surface t 1000 m fr the deepest levels. Fllwing Weaver et al. (2003) and Vialard et al. (2003), the year 1993 was chsen as ur simulatin perid. During this year, the circulatin f the trpical Pacific Ocean was marked by the weak influence f the last El Niñ

15 Cmparisn f reduced-rder data assimilatin methds 15 event (the last big ne had ccurred in , the nes in and had been weaker and the net big ne wuld begin nly in 1997). The year 1993 can therefre be seen as a nrmal year frm a dynamical pint f view and was thus suitable fr cnducting a cmparisn f the tw different appraches. All eperiments last ne year, beginning n January 1, The winds used t frce the numerical mdel were based n bth satellite ERS measurements and in-situ TAO winds (Menkes et al., 1998). The atmspheric heat flues came frm ECMWF data files (ERA 40). 3.2 Assimilatin Eperiments All the eperiments discussed here are cnducted using the twin eperiment framewrk. The initial true state is btained frm a previus simulatin ver year 1992 and is used t generate a reference ne-year free run cnsidered as the truth. This slutin is then sampled t generate simulated temperature data. The distributin f these simulated data is chsen as clse as pssible t the distributin f the real TAO/TRITON array (Fig. 1) and XBT prfiles. Temperature is sampled frm the surface dwn t a depth f 500 meters every si hurs. A gaussian nise is added t the simulated bservatins with a standard errr set at σ T = 0.5 C. Fr the cmputatin f the EOFs, the mdel state cnsists f 4 variables: temperature, salinity and the tw hrizntal cmpnents f velcity. A free run eperiment trajectry is sampled ver ne year, using a 2-day peridicity t build the cvariance matri. A large part f the ttal variance

16 16 Céline Rbert et al. is represented by a few EOFs: 80% fr the first 13 EOFs, 92% fr the first 30 EOFs. Since the initial state used in the assimilatin eperiments is nt the crrect ne, we want t cntrl an errr n the initial cnditin thrugh data assimilatin. This errr is mre r less crrected naturally by the free run in rughly si mnths, thanks t the frcings. After si mnths, the errr t be cntrlled is n lnger an errr n the initial cnditin but mainly cncerns the nn-linear dynamics f the mdel. In the R-4D-Var eperiment, the errr n the initial cnditin is intrduced via the backgrund. Fr the initial backgrund state, we use the slutin f the reference simulatin n April 1 (i.e. 3 mnths later than the true state). Fr the filter, accrding t the thery, the initial state is a mean state calculated frm the free run fr the year In bth cases, the initial errr between the assimilatin eperiment and the reference run is large enugh t make the crrectin, prvided by the data assimilatin methd, significant. A third assimilatin eperiment using the hybrid methd presented previusly was als perfrmed. Finally, an additinal furth simulatin, withut data assimilatin and using the same false initial cnditin as in the R-4D-Var eperiment was als perfrmed in rder t help quantify the efficiency f the assimilatin. Nte that assimilating nly temperature data at TAO pints represents a departure frm mst previus filter analysis eperiments. Surface fields like SSH are als usually assimilated, cnsiderably helping t cnstrain

17 Cmparisn f reduced-rder data assimilatin methds 17 the slutin and acting mre specifically n the dynamics. Mrever, it shuld be nted that the limitatin f the R-4D-Var methd arising frm the fact that the mdel is a strng cnstraint des nt play a rle here since the mdel is suppsed t be perfect in these twin eperiments. Fr the variatinal apprach we used the OPAVAR package, develped and validated by Weaver et al. (2003) and Vialard et al. (2003), and fr the filter eperiments, the SESAM package (Testut et al., 2001). 4 Results The results shwn belw are presented fr tw different perids, frm January t June 1993 and frm July t December There are tw main reasns fr this: (i) in such a quasi-linear mdel and twin-eperiment framewrk, the mdel is naturally restred frm the errneus initial cnditin ver a time scale f sme mnths, (ii) physically, June is als the time f the nset f the nn-linear Trpical Instability Waves (TIWs) in the eastern trpical Pacific Ocean. Schematically, the first si-mnth perid cncerns the cntrl f the errr n the initial cnditin, while the net si-mnth perid cncerns the cntrl f the nn-linear dynamics in the system. In the latter perid, the intensity f TIWs begins t increase, a develpment that appears t have cnsiderable influence n data assimilatin.

18 18 Céline Rbert et al. 4.1 January-June 1993 In this first time perid, the errr n the initial cnditin is quite large, since the frcings f the mdel have nt had time t crrect it, s that mst f the wrk dne by data assimilatin is t cntrl the errr n the initial cnditin. Mrever, the dynamics is quite stable and the EOFs represent mdel variability perfectly. In this case, and as wuld be epected frm the purely linear and ptimal cntet, it can be bserved in Fig. 2 that the assimilatin methds are almst equivalent. We can see that the crrectin is substantial and that all methds prvide rughly the same slutin. The three algrithms wrk well, nt nly in terms f temperature misfit in the area f bservatins but als at greater depths and fr the ther variables, thanks t the multivariate nature f the EOFs. This can be seen fr eample in Fig. 3. Cncerning the hybrid algrithm, a slightly lwer level f errr is btained as shwn in Fig. 2, fr eample fr (u, v) variables. The diagnstic f the analysis and frecast errrs perfrmed by the filter is crrect with regard t the dynamics. The slutin is thus very gd. Finally, this hybrid methd succeeds in cmbining tw intrinsic aspects f the reduced-rder methds, which leads t slightly imprved results. The fact that the imprvement is nt mre significant is because bth the and R-4D-Var methds already btain ecellent results in decreasing the errr.

19 Cmparisn f reduced-rder data assimilatin methds July-December 1993 The secnd perid starts in July, when the intensity f TIWs increases significantly. The dynamics f these waves is nn-linear. Since the first TAO pint is quite distant frm the eastern cast, the TIWs rise befre the first data pint can see the change. In this case, it can take a significant time (even mre than 10 days, which is the duratin f the cycle) befre the easternmst pints f the TAO array register these changes. In additin, the errr due t the initial cnditin is very weak, since the dynamics f the mdel has already naturally crrected this errr. Thus, the majr part f the remaining errr is driven by the nn-linear dynamics f the cean. As we can see in Fig. 4, the difference in temperature between the reference simulatin and the filter simulatin is mainly lcated in the eastern basin, near the equatr. An imprtant difference between the algrithms is their sequential versus glbal prcessing f the bservatins. Since the bservatins available during the whle assimilatin windw are taken int accunt in the R-4D- Var methd, this apprach can anticipate the prpagatin f a physical phenmenn at this time scale. Fr eample, the Trpical Instability Waves (TIWs) rise in the eastern part f the basin and prpagate alng the equatr in rughly ne mnth. When they becme mre intense in early July, the variatinal system takes int accunt bservatins f these waves in the analysis cnducted befre their actual ccurrence. This is nt the case in the filter because the analysis at the same time takes int accunt

20 20 Céline Rbert et al. nly past bservatins. Cnsequently, there is a time lag with this apprach between the ccurrence f the physical phenmenn (the intensificatin) and its integratin int the analysis. This eplains the differences bserved in Fig. 5 which cncerns the eastern part f the dmain. The cmparisn is almst the same in the western part (Fig. 6), but with a lwer rms errr level, prbably because mre bservatins are available. In that secnd perid, the hybrid methd cntinues t draw advantages frm the quality f the analysis f the R-4D-Var. Hwever, since the evlutin f its cvariance matri prvided by the filter is less accurate, the hybrid methd d nt succeed in that case in prviding better results than the R-4D-Var (see Fig 6 and 5). 5 Cnclusins This paper presents the results f a first cmparisn f three reduced-rder data assimilatin methds implemented in a mdel f the trpical Pacific Ocean. The first tw methds, the filter and the reduced-rder 4D- VAR, are respectively derived frm the sequential and the variatinal appraches. The third methd, cmbining features f the filter and the R-4D-VAR, is a hybrid versin f the first tw methds. T ur knwledge, the present study is prbably ne f the first side-by-side implementatins and cmparisns f these techniques ever made in a realistic cntet. Investigatins are eplratry in nature due t the cmpleity f the methdlgy and mre especially because simulatins have been carried ut in a twin-

21 Cmparisn f reduced-rder data assimilatin methds 21 eperiment framewrk where n mdel errr is present and data are simulated. In additin, nly ne year f cmparative simulatins is perfrmed. Hwever, we believe that the first results prvide useful insights: In a quasi-linear regime, as epected frm linear thery, the three methds prvide rather similar results in reducing the initial cnditin system errr. The hybrid methd prvides slightly better results, which wuld mean, as epected, that cmbining the evlutin f errr cvariance matrices and the variatinal analysis is, at the very least, feasible and ptentially fruitful (as sn as further tuning is dne). In a regime where strng nnlinearities develp at reginal scales (crrespnding t the nset f Trpical Instability Waves within the eastern trpical Pacific), the 4D-VAR succeeds in keeping the errr t a lw level whereas the filter, due t its sequential nature, fails t fully cntrl the increasing instability using the data available. The hybrid methd fllws the divergent nature f the filter in the first stages but, after several weeks, resumes a mre cnvergent path. The twin-eperiment set-up entails bvius limits and may influence the cnclusins. The n mdel errr assumptin may favr the variatinal slutin since the twin eperiments are eactly within the strng cnstraint variatinal framewrk. It als favrs the perfrmances f all reduced-rder methds since the reduced basis can be built frm a perfect reference simulatin. With statistical lw-cst methds like the filter, the amunt and the nature f assimilated data are a key factr. The data used in the

22 22 Céline Rbert et al. present study mimic the real data that are acquired frm the TAO array in the Pacific, but this array prly samples the eastern Pacific Ocean. It is likely that with the additin f sme higher frequency cmplementary data (such as altimetric data), the filter wuld behave mre satisfactrily (see, fr eample, Castrucci et al., 2006). Preliminary eperiments were als perfrmed with real TAO data, thus including mdel errrs. In this case, first results seem t indicate a much mre balanced behavir between the filter and the reduced 4D-VAR. Acknwledgments J.-M. Brankart and A. Weaver are gratefully acknwledged fr supplying respectively the SESAM and OPAVAR packages and fr prviding supprt in using these tls. This wrk has been supprted by CNES and the MER- CATOR prject. Idpt is a jint CNRS-INPG-INRIA-UJF research prject. Fig. 1 TAO/TRITON array (

23 Cmparisn f reduced-rder data assimilatin methds 23 FREE RUN FREE RUN (a) Temperature (b) Salinity FREE RUN FREE RUN (c) Velcity u (d) Velcity v Fig. 2 Abslute Rms errr btained by each methd during the first 6 mnths f simulatin, at 15m. depth. Slid line with +: free run, slid line: R-4D-Var, dashed line with : filter and dashed-dtted line: hybrid methd.

24 24 Céline Rbert et al. FREE RUN FREE RUN (a) Velcity u (b) Velcity v Fig. 3 Rms errr btained by each methd during the first 6 mnths f simulatin, belw the bservatins, at 750 m. depth. Slid line with +: free run, slid line: R-4D-Var, dashed line with : filter and dashed-dtted line: hybrid methd.

25 Cmparisn f reduced-rder data assimilatin methds 25 (a) End f June (b) Beginning f July Fig. 4 Difference between temperature field at 15 m. depth f the reference simulatin and f the filter simulatin.

26 26 Céline Rbert et al. (a) Temperature (b) Salinity (c) Velcity u (d) Velcity v Fig. 5 Rms errr the last 6 mnths f simulatin, at 15 m. depth, in the eastern part f the basin. Slid line: R-4D-Var, dashed line with : hybrid methd and dashed-dtted line with : filter.

27 Cmparisn f reduced-rder data assimilatin methds 27 FREE RUN FREE RUN (a) Temperature (b) Salinity FREE RUN FREE RUN (c) Velcity u (d) Velcity v Fig. 6 Rms errr the last 6 mnths f simulatin, at 15 m. depth, in the western part f the basin. Slid line with +: free run, slid line: R-4D-Var, dashed line with : filter and dashed-dtted line: hybrid methd.