NOTES AND CORRESPONDENCE. Ensemble Square Root Filters*

Transcription

1 JULY 2003 NOES AND CORRESPONDENCE 1485 NOES AND CORRESPONDENCE Ensemble Squre Root Filters* MICHAEL K. IPPE Interntionl Reserch Institute or Climte Prediction, Plisdes, New Yor JEFFREY L. ANDERSON GFDL, Princeton, New Jersey CRAIG H. BISHOP Nvl Reserch Lbortory, Monterey, Cliorni HOMAS M. HAMILL AND JEFFREY S. WHIAKER NOAA CIRES Climte Dignostics Center, Boulder, Colordo 31 Jnury 2002 nd 18 November 2002 ABSRAC Ensemble dt ssimiltion methods ssimilte observtions using stte-spce estimtion methods nd lowrn representtions o orecst nd nlysis error covrinces. A ey element o such methods is the trnsormtion o the orecst ensemble into n nlysis ensemble with pproprite sttistics. his trnsormtion my be perormed stochsticlly by treting observtions s rndom vribles, or deterministiclly by requiring tht the updted nlysis perturbtions stisy the Klmn ilter nlysis error covrince eqution. Deterministic nlysis ensemble updtes re implementtions o Klmn squre root ilters. he nonuniqueness o the deterministic trnsormtion used in squre root Klmn ilters provides rmewor to compre three recently proposed ensemble dt ssimiltion methods. 1. Introduction Dt ssimiltion ddresses the problem o producing useul nlyses nd orecsts given imperect dynmicl models nd observtions. he Klmn ilter is the optiml dt ssimiltion method or liner dynmics with dditive, stte-independent Gussin model nd observtion errors (Cohn 1997). An ttrctive eture o the Klmn ilter is its clcultion o orecst nd nlysis error covrinces, in ddition to the orecsts nd nlyses themselves. In this wy, the Klmn ilter produces estimtes o orecst nd nlysis uncertinty, consistent with the dynmics nd prescribed model nd observtion error sttistics. However, the error covrince cl- * Interntionl Reserch Institute or Climte Prediction Contribution Number IRI-PP/02/03. Corresponding uthor ddress: Michel K. ippett, IRI/LDEO, 223 Monell, P.O. Box 1000/61 Rt. 9W, Plisdes, NY E-mil: tippett@iri.columbi.edu cultion components o the Klmn ilter re diicult to implement in relistic systems becuse o (i) their computtionl cost, (ii) the nonlinerity o the dynmics, nd (iii) poorly chrcterized error sources. he ensemble Klmn ilter (EnKF), proposed by Evensen (1994), ddresses the irst two o these problems by using ensemble representtions or the orecst nd nlysis error covrinces. Ensemble size limits the number o degrees o reedom used to represent orecst nd nlysis errors, nd Klmn ilter error covrince clcultions re prcticl or modest-sized ensembles. he EnKF lgorithm begins with n nlysis ensemble whose men is the current stte estimte or nlysis nd whose sttistics relect the nlysis error. Applying the ull nonliner dynmics to ech nlysis ensemble member produces the orecst ensemble; tngent liner nd djoint models o the dynmics re not required. Sttistics o the orecst ensemble represent orecst errors; in its simplest orm, the EnKF only ccounts or orecst error due to uncertin initil conditions, neglecting orecst error due to model deiciencies. he orecst en Americn Meteorologicl Society

2 1486 MONHLY WEAHER REVIEW VOLUME 131 semble men nd covrince re then used to ssimilte observtions nd compute new nlysis ensemble with pproprite sttistics, nd the cycle is repeted. he new nlysis ensemble cn be ormed either stochsticlly (Houtemer nd Mitchell 1998; Burgers et l. 1998) or deterministiclly (Bishop et l. 2001; Anderson 2001; Whiter nd Hmill 2002). Deterministic methods were developed to ddress the dptive observtionl networ design problem nd to void smpling issues ssocited with the use o perturbed observtions in stochstic nlysis ensemble updte methods. he EnKF nd other ensemble dt ssimiltion methods belong to the mily o squre root ilters (SRFs), nd purpose o this pper is to demonstrte tht deterministic nlysis ensemble updtes re implementtions o Klmn SRFs (Biermn 1977; Mybec 1982; Heemin et l. 2001). An immedite beneit o this identiiction is rmewor or understnding nd compring deterministic nlysis ensemble updte schemes (Bishop et l. 2001; Anderson 2001; Whiter nd Hmill 2002). SRFs, lie ensemble representtions o covrinces, re not unique. We begin our discussion in section 2 with presenttion o the Klmn SRF; issues relted to implementtion o ensemble SRFs re presented in section 3; in section 4 we summrize our results. 2. he Klmn SRF Klmn SRF lgorithms, originlly developed or spce-nvigtion systems with limited computtionl word length, demonstrte superior numericl precision nd stbility compred to the stndrd Klmn ilter lgorithm (Biermn 1977; Mybec 1982). SRFs by construction void loss o positive deiniteness o the error covrince mtrices. SRFs hve been used in erth science dt ssimiltion methods where error covrinces re pproximted by truncted eigenvector expnsions (Verln nd Heemin 1997). he usul Klmn ilter covrince evolution equtions re P MP M Q, (1) P (I KH)P, (2) where P nd P re, respectively, the n n orecst nd nlysis error covrince mtrices t time t ; M is the tngent liner dynmics; H is the p n observtion opertor; R is the p p observtion error covrince; Q is the n n model error covrince mtrix nd K P (H H P H R ) is the Klmn gin; n is the dimension o the system stte; nd p is the number o observtions. he error covrince evolution depends on the stte estimtes nd observtions through the tngent liner dynmics M. he propgtion o nlysis errors by the dynmics with model error cting s orcing is described by Eq. (1). Eqution (2) shows how n optiml dt ssimiltion scheme uses observtions to produce n nlysis whose error covrince is less thn tht o the orecst. he orecst nd nlysis error covrince mtrices re symmetric positive-deinite mtrices nd cn be represented s P Z Z nd P Z Z, where the mtrices Z nd Z re mtrix squre roots o P nd P, respectively; other mtrix ctoriztions cn be used in ilters s well (Biermn 1977; Phm et l. 1998). A covrince mtrix nd its mtrix squre root hve the sme rn or number o nonzero singulr vlues. When covrince mtrix P hs rn m, there is n n m mtrix squre root Z stisying P ZZ ; in low-rn covrince representtions the rn m is much less thn the stte-spce dimension n. his representtion is not unique; P cn lso be represented s P (ZU)(ZU), where the mtrix U is ny m m orthogonl trnsormtion UU U U I. he projection x Z 2 x Px o n rbitrry n-vector x onto the mtrix squre root Z is uniquely determined, s is the subspce spnned by the columns o Z. Covrince mtrix squre roots re closely relted to ensemble representtions. he smple covrince P o n m-member nlysis ensemble is given by P SS / (m 1), where the columns o the n m mtrix S re men-zero nlysis perturbtions bout the nlysis ensemble men; the rn o P is t most (m 1). A mtrix squre root o the nlysis error covrince mtrix P is the mtrix o scled nlysis perturbtion en semble members Z (m 1) /2 S. he Klmn SRF lgorithm replces error covrince evolution equtions (1) nd (2) with equtions or the evolution o orecst nd nlysis error covrince squre roots Z nd Z in mnner tht voids orming the ull error covrince mtrices. I the model error covrince Q is neglected, (1) cn be replced by Z MZ. (3) In the ensemble context, (3) mens to pply the tngent liner dynmics to ech column o the Z, tht is, to ech scled nlysis perturbtion ensemble member. Prcticlly, (3) cn be implemented by pplying the ull nonliner dynmics to ech nlysis ensemble member. For wht ollows, we only ssume tht the orecst error covrince mtrix squre root Z is vilble nd do not ssume or restrict tht it be clculted rom (3). Section 3b discusses more sophisticted methods o generting Z tht cn include estimtes o model error nd give orecst error covrince mtrix squre root whose rn is greter thn the number o perturbtions evolved by the dynmicl model. Next, nlysis error covrince eqution (2) is replced with n eqution or the nlysis error covrince squre root Z. his eqution determines how to orm n nlysis ensemble with pproprite sttistics. Initil implementtions o the EnKF ormed the new nlysis ensemble by updting ech orecst ensemble member using the sme nlysis equtions, equivlent to pplying the liner opertor (I K H ) to the orecst perturbtion ensemble

3 JULY 2003 NOES AND CORRESPONDENCE 1487 Z. his procedure gives n nlysis ensemble whose error covrince is (I K H ) P (I K H ) nd includes nl- ysis error due to orecst error; the Klmn gin K depends on the reltive size o orecst nd observtion error, nd the ctor (I K H ) shows how much orecst errors re reduced. However, in this procedure the nlysis ensemble does not include uncertinty due to observtion error nd so underestimtes nlysis error. A stochstic solution to this problem proposed independently by Houtemer nd Mitchell (1998) nd Burgers et l. (1998) is to compute nlyses using ech orecst ensemble member nd, insted o using single reliztion o the observtions, to use n ensemble o simulted observtions whose sttistics relect the observtion error. his method is equivlent to the nlysis perturbtion ensemble updte Z (I KH)Z KW, (4) where W is p m mtrix whose m columns re identiclly distributed, men-zero, Gussin rndom vectors o length p with covrince R /m. he perturbed observtion nlysis eqution (4) gives n nlysis perturbtion ensemble with correct expected sttistics: Z (Z ) (I KH)P (I KH) KRK P. (5) However, the perturbed observtion pproch introduces n dditionl source o smpling error tht reduces nlysis error covrince ccurcy nd increses the probbility o underestimting nlysis error covrince (Whiter nd Hmill 2002). A Monte Crlo method voiding perturbed observtions is described in Phm (2001). he singulr evolutive interpolte Klmn (SEIK) ilter uses both deterministic ctoriztion nd stochstic pproches. Klmn SRFs provide deterministic lgorithm or trnsorming the orecst ensemble into n nlysis ensemble with consistent sttistics. he Potter method or the Klmn SRF nlysis updte (Biermn 1977) is obtined by rewriting (2) s P ZZ [I P H (H P H R ) H ]P Z [I Z H (H Z Z H R ) HZ ]Z Z (I VD V)Z, (6) where we deine the m p mtrix V (H Z ) nd the p p innovtion covrince mtrix D V V R. hen the nlysis perturbtion ensemble is clculted rom Z ZXU, (7) where X X (I V D V) nd U is n rbitrry m m orthogonl mtrix. As ormulted, the updted ensemble Z is liner combintion o the columns o Z nd is obtined by inverting the p p mtrix D nd computing mtrix squre root X o the m m mtrix (I V D ). V 3. Ensemble SRFs. Anlysis ensemble In mny typicl erth science dt ssimiltion pplictions the stte-dimension n nd the number o observtions p re lrge, nd the method or computing the mtrix squre root o (I V D V ) nd the updted nlysis perturbtion ensemble Z must be chosen c- cordingly. A direct pproch is to solve irst the liner system D Y H Z or the p m mtrix Y, tht is, to solve (HPH R )Y HZ, (8) s is done in the irst step o the Physicl-spce Sttisticl Anlysis System (PSAS) lgorithm (Cohn et l. 1998). hen, the m m mtrix I V D V I (H Z ) Y is ormed, nd its mtrix squre root X com- puted nd pplied to Z s in (7). Solution o (8), even when p is lrge, is prcticl when the orecst error covrince hs low-rn representtion nd the inverse o the observtion error covrince is vilble (see the ppendix). Itertive methods whose cost is on the order o the cost o pplying the innovtion covrince mtrix re pproprite when the orecst error covrince is represented by correltion model. When observtion errors re uncorrelted, observtions cn be ssimilted one t time or serilly (Houtemer nd Mitchell 2001; Bishop et l. 2001). For single observtion, p 1, V is column vector, nd the innovtion D is sclr. In this cse, mtrix squre root o (I V D V ) cn be computed in closed orm by ting the nstz, I D VV (I VV)(I VV), (9) nd solving or the sclr, which gives [D (R D ) 1/2 ]. he nlysis ensemble updte or p 1is Z Z (I VV); (10) see Andrews (1968) or generl solution involving mtrix squre roots o p p mtrices. At observtion loctions, the nlysis error ensemble is relted to the orecst error ensemble by H Z (1 V V )H Z. he sclr ctor (1 V V ) hs n bsolute vlue less thn or equl to one nd is positive when the plus sign is chosen in the deinition o. In Whiter nd Hmill (2002) the nlysis perturbtion ensemble is ound rom Z (I KH)Z, (11) where the mtrix K is solution o the nonliner eqution (I KH)P (I KH) P. (12) In the cse o single observtion, solution o (12) is 1/2 K [1 (R /D ) ] K ZV, (13) where the plus sign is chosen in the deinition o.

4 1488 MONHLY WEAHER REVIEW VOLUME 131 he corresponding nlysis perturbtion ensemble updte, Z (I KH)Z (I Z V H )Z Z (I VV), (14) is identicl to (10). Observtions with correlted errors, or exmple, rdiosonde height observtions rom the sme sounding, cn be hndled by pplying the whitening trnsormtion R to the observtions to orm /2 new observtion set with uncorrelted errors. Another method o computing the updted nlysis ensemble is to use the Shermn Morrison Woodbury identity (Golub nd Vn Lon 1996) to show tht I VD V (I Z HR HZ ). (15) he m m mtrix on the right-hnd side o (15) is prcticl to compute when the inverse observtion error covrince mtrix R is vilble. his pproch voids inverting the p p mtrix D nd is used in the ensemble trnsorm Klmn ilter (EKF) where the nlysis updte is (Bishop et l. 2001) /2 Z ZC( I) ; (16) C C is the eigenvlue decomposition o Z HR H Z. Note tht the mtrix C o orthonorml eigenvectors is not uniquely determined. 1 Comprison with (15) shows tht C ( I) C is the eigenvlue decomposition o I V D nd thus tht C ( I) /2 V is squre root o (I V D V ). In the ensemble djustment Klmn ilter (EAKF) the orm o the nlysis ensemble updte is (Anderson 2001) Z AZ ; (17) the ensemble djustment mtrix A is deined by /2 A FGC(I ) G F, (18) where P F 2 GF is the eigenvlue decomposition o P G nd the orthogonl mtrix C is chosen so tht C H F G C FHR is digonl. 2 Choosing the orthogonl mtrix C to be C G F Z C gives tht nd tht the ensemble djustment mtrix is /2 A ZC(I ) G F. (19) he EAKF nlysis updte (17) becomes /2 Z ZC(I ) G F Z. (20) he EAKF nlysis ensemble given by (20) is the sme 1 For instnce, the columns o C tht spn the (m p)-dimensionl null spce o Z HR H Z re determined only up to orthogonl trnsormtions i the number o observtions p is less thn the ensemble size m. 2 he ppernce o G in the deinition o the ensemble djust ment mtrix A seems to require the orecst error covrince P to be invertible. However, the ormultion is still correct when G is m m nd F is n m where m is the number o nonzero eigenvlues o. P ABLE 1. Summry o nlysis ensemble clcultion computtionl cost s unction o orecst ensemble size m, number o observtions p, nd stte dimension n. Anlysis method Direct Seril EKF EAKF Cost O(m 2 p m 3 m 2 n) O(mp mnp) O(m 2 p m 3 m 2 n) O(m 2 p m 3 m 2 n) s pplying the trnsormtion G F Z to the EKF nlysis ensemble. he mtrix G F Z is orthogonl nd is, in ct, the mtrix o right singulr vectors o. hereore, C (I ) /2 Z G FZ is mtrix squre root o (I V D V ). Beginning with the sme orecst error covrince, the direct, seril, EKF, nd EAKF methods produce dierent nlysis ensembles tht spn the sme sttespce subspce nd hve the sme covrince. Higherorder sttisticl moments o the dierent models will be dierent, relevnt issue or nonliner dynmics. he computtion costs o the direct, EKF, nd EAKF methods re seen in ble 1 to scle comprbly (see the ppendix or detils). here re dierences in precise computtionl cost; or instnce, the EAKF contins n dditionl singulr vlue decomposition (SVD) clcultion o the orecst with cost O(m 3 m 2 ). he computtionl cost o the seril ilter is less dependent on the rn o the orecst error covrince nd more sensitive to the number o observtions. his dierence is importnt when techniques to ccount or model error nd control ilter divergence, s described in the next section, result in n eective orecst error covrince dimension m much lrger thn the dynmicl orecst ensemble dimension. b. Forecst error sttistics In the previous section we exmined methods o orming the nlysis ensemble given mtrix squre root o the orecst error covrince. here re two undmentl problems ssocited with directly using the ensemble generted by (3). First, ensemble size is limited by the computtionl cost o pplying the orecst model to ech ensemble member. Smll ensembles hve ew degrees o reedom vilble to represent errors nd suer rom smpling error tht urther degrdes orecst error covrince representtion. Smpling error leds to loss o ccurcy nd underestimtion o error covrinces tht cn cuse ilter divergence. echniques to del with this problem re distnce-dependent covrince iltering nd covrince inltion (Whiter nd Hmill 2002). Covrince locliztion in the seril method consists o dding Schur product to the deinition o K (Whiter nd Hmill 2002). Similrly, observtions eecting nlysis grid points cn be restricted to be nerby in the EAKF (Anderson 2001). he second nd less esily resolved problem with

5 JULY 2003 NOES AND CORRESPONDENCE 1489 using the ensemble generted by (3) is the neglect o model error nd resulting underestimtion o the orecst error covrince. Since there is little theoreticl nowledge o model error sttistics in complex systems, model error prmeteriztions combined with dptive methods re liely necessry (Dee 1995). When the model error covrince Q is ten to hve lrge-scle structure, resonble representtion is n ensemble or squre root decomposition, Q ZZ d d, where Z d is n n q mtrix. hen, squre root o P is the n m mtrix: Z [MZ Z d ], (21) where m m e q nd m e is the number o dynmiclly evolved orecst perturbtions. With this model error representtion, ensemble size grows by q with ech orecst nlysis cycle. Ensemble size cn be limited by computing the singulr vlue decomposition o the ensemble nd discrding components with smll vrince (Heemin et l. 2001). A lrger ensemble with evolved nlysis error nd model error could be used in the nlysis step, nd smller ensemble used in the dynmicl orecst stge. When the model error covrince Q is pproximted s n opertor, or instnce using correltion model, Lnczos methods cn be used to compute the leding eigenmodes o M Z(M Z) Q nd orm Z (Cohn nd odling 1996). Such orecst error covrince model would resemble those used in hybrid methods (Hmill nd Snyder 2000). In this cse, the rn o Z cn be substntilly lrger thn the orecst ensemble size, ming the seril method ttrctive. Monte Crlo solutions re nother option s in Mitchell nd Houtemer (2000), where model error prmeters were estimted rom innovtions nd used to generte reliztions o model error. Perturbing model physics, s done in system simultion, explicitly ccounts or some spects o model uncertinty (Houtemer et l. 1996). 4. Summry nd discussion Ensemble orecst/ssimiltion methods use low-rn ensemble representtions o orecst nd nlysis error covrince mtrices. hese ensembles re scled mtrix squre roots o the error covrince mtrices, nd so ensemble dt ssimiltion methods cn be viewed s squre root ilters (SRFs; Biermn 1977). Ater ssimiltion o observtions, the nlysis ensemble cn be constructed stochsticlly or deterministiclly. Deterministic construction o the nlysis ensemble elimintes one source o smpling error nd leds to deterministic SRFs being more ccurte thn stochstic SRFs in some exmples (Whiter nd Hmill 2002; Anderson 2001). SRFs re not unique since dierent ensembles cn hve the sme covrince. his lc o uniqueness is illustrted in three recently proposed ensemble dt ssimiltion methods tht use the Klmn SRF method to updte the nlysis ensemble (Bishop et l. 2001; Anderson 2001; Whiter nd Hmill 2002). Identiying the methods s SRFs llows clerer discussion nd comprison o their dierent nlysis ensemble updtes. Accounting or smll ensemble size nd model deiciencies remins signiicnt issue in ensemble dt ssimiltion systems. Schur products cn be used to ilter ensemble covrinces nd eectively increse covrince rn (Houtemer nd Mitchell 1998, 2001; Hmill et l. 2001; Whiter nd Hmill 2002). Covrince inltion is one simple wy o ccounting or model error nd stbilizing the ilter (Hmill et l. 2001; Anderson 2001; Whiter nd Hmill 2002). Hybrid methods represent orecst error covrinces with combintion o ensemble nd prmeterized correltion models (Hmill nd Snyder 2000). Here we hve shown deterministic methods o including model error into squre root or ensemble dt ssimiltion system when the model error hs lrge-scle representtion nd when the model error is represented by correltion model. However, the primry diiculty remins obtining estimtes o model error. he nonuniqueness o SRFs hs been exploited in estimtion theory to design ilters with desirble computtionl nd numericl properties. An open question is whether there re ensemble properties tht would me prticulr SRF implementtion better thn nother, or i the only issue is computtionl cost. For instnce, it my be possible to choose n nlysis updte scheme tht preserves higher-order, non-gussin sttistics o the orecst ensemble. his question cn only be nswered by detiled comprisons o dierent methods in relistic setting where other detils o the ssimiltion system such s modeling o systemtic errors or dt qulity control my prove to be s importnt. Acnowledgments. Comments nd suggestions o two nonymous reviews improved the presenttion o this wor. he uthors thn Ricrdo odling (NASA DAO) or his comments, suggestions, nd corrections. IRI is supported by its sponsors nd NOAA Oice o Globl Progrms Grnt NA07GP0213. Crig Bishop received support under ONR Project Element N, Project Number BE , nd lso ONR Grnt N APPENDIX Computtionl Costs Here we detil the computtionl cost sclings summrized in ble 1. All the methods require pplying the observtion opertor to the ensemble members to orm H Z, nd we do not include its cost. his cost is importnt when compring ensemble nd nonensemble methods, prticulrly or complex observtion opertors. he cost o computing H Z is ormlly O(mnp), but my be signiicntly less when H is sprse or cn

6 1490 MONHLY WEAHER REVIEW VOLUME 131 be pplied eiciently. We lso ssume tht the inverse observtion error covrince R is inexpensive to pply.. Direct method 1) Solve (H P H R )Y H Z or Y.IR is vilble, the solution cn be obtined using the Shermn Morrison Woodbury identity (Golub nd Vn Lon 1996), (HP H R ) R R HZ [I (HZ ) R (HZ )] (H Z ) R, nd only inverting m m mtrices. Cost: O(m 3 m 2 p). 2) Form I (H Z ) Y. Cost: O(pm 2 ). 3) Compute mtrix squre root o the m m mtrix I (H Z ) Y. Cost: O(m 3 ). 4) Apply mtrix squre root to Z. Cost: O(m 2 n). otl cost: O(m 3 m 2 p m 2 n). b. Seril method For ech observtion: 1) Form D. Cost: O(m). 2) Form I VV nd pply to. Cost: O(nm). otl cost: O(mp mnp). c. EKF 1) Form Z H R HZ. Assume R inexpensive to pply. Cost: O(m 2 p). 2) Compute eigenvlue decomposition o m m mtrix. Cost: O(m 3 ). 3) Apply to Z. Cost: O(m 2 n). otl cost: O(m 2 p m 3 m 2 n). d. EAKF Cost in ddition to EKF: 1) Eigenvlue decomposition o P (low rn). Cost: O(m 2 n m 3 ). 2) Form F Z. Cost: O(m 2 p). otl cost: O(m 2 p m 3 m 2 n). Z REFERENCES Anderson, J. L., 2001: An ensemble djustment ilter or dt ssimiltion. Mon. We. Rev., 129, Andrews, A., 1968: A squre root ormultion o the Klmn covrince equtions. AIAA J., 6, Biermn, G. J., 1977: Fctoriztion Methods or Discrete Sequentil Estimtion. Vol. 128, Mthemtics in Science nd Engineering, Acdemic Press, 241 pp. Bishop, C. H., B. Etherton, nd S. J. Mjumdr, 2001: Adptive smpling with the ensemble trnsorm Klmn ilter. Prt I: heoreticl spects. Mon. We. Rev., 129, Burgers, G., P. J. vn Leeuwen, nd G. Evensen, 1998: On the nlysis scheme in the ensemble Klmn ilter. Mon. We. Rev., 126, Cohn, S. E., 1997: An introduction to estimtion theory. J. Meteor. Soc. Jpn, 75, , nd R. odling, 1996: Approximte dt ssimiltion schemes or stble nd unstble dynmics. J. Meteor. Soc. Jpn, 74, , A. M. d Silv, J. Guo, M. Sieniewicz, nd D. Lmich, 1998: Assessing the eect o dt selection with the DAO Physiclspce Sttisticl Anlysis System. Mon. We. Rev., 126, Dee, D. P., 1995: On-line estimtion o error covrince prmeters or tmospheric dt ssimiltion. Mon. We. Rev., 123, Evensen, G., 1994: Sequentil dt ssimiltion with nonliner qusi-geostrophic model using Monte Crlo methods to orecst error sttistics. J. Geophys. Res., 99, Golub, G. H., nd C. F. Vn Lon, 1996: Mtrix Computtions. 3d ed. he Johns Hopins University Press, 694 pp. Hmill,. M., nd C. Snyder, 2000: A hybrid ensemble Klmn ilter 3D vritionl nlysis scheme. Mon. We. Rev., 128, , J. S. Whiter, nd C. Snyder, 2001: Distnce-dependent iltering o bcground error covrince estimtes in n ensemble Klmn ilter. Mon. We. Rev., 129, Heemin, A. W., M. Verln, nd A. J. Segers, 2001: Vrince reduced ensemble Klmn iltering. Mon. We. Rev., 129, Houtemer, P. L., nd H. L. Mitchell, 1998: Dt ssimiltion using n ensemble Klmn ilter technique. Mon. We. Rev., 126, , nd, 2001: A sequentil ensemble Klmn ilter or tmospheric dt ssimiltion. Mon. We. Rev., 129, , L. Leivre, J. Derome, H. Ritchie, nd H. L. Mitchel, 1996: A system simultion pproch to ensemble prediction. Mon. We. Rev., 124, Mybec, P. S., 1982: Stochstic Models, Estimtion, nd Control. Vol. 1. Acdemic Press, 423 pp. Mitchell, H. L., nd P. L. Houtemer, 2000: An dptive ensemble Klmn ilter. Mon. We. Rev., 128, Phm, D., 2001: Stochstic methods or sequentil dt ssimiltion in strongly nonliner systems. Mon. We. Rev., 129, , J. Verron, nd M. Roubud, 1998: A singulr evolutive extended Klmn ilter or dt ssimiltion in ocenogrphy. J. Mr. Syst., 16, Verln, M., nd A. W. Heemin, 1997: idl low orecsting using reduced rn squre ilters. Stochstic Hydrol. Hydrul., 11, Whiter, J., nd. M. Hmill, 2002: Ensemble dt ssimiltion without perturbed observtions. Mon. We. Rev., 130,