Weight interpolation for efficient data assimilation with the Local Ensemble Transform Kalman Filter

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1 Weight interpoltion or eicient dt ssimiltion with the Locl Ensemble Trnsorm Klmn Filter Shu-Chih Yng 1,2, Eugeni Klny 1,3, Brin Hunt 1,3 nd Neill E. Bowler 4 1 Deprtment o Atmospheric nd Ocenic Science, University o Mrylnd 2 Globl Modeling nd Assimiltion Oice, NASA/GSFC 3 Institute or Physicl Science nd Technology, University o Mrylnd 4 Met Oice, FitzRoy Rod, Exeter, UK Corresponding uthor: Shu-Chih Yng Emil: scyng@tmos.umd.edu Deprtment o Atmospheric nd Ocenic Science, University o Mrylnd, MD, 20742, USA. Abstrct We investigte method to substntilly reduce the nlysis computtions within the Locl Ensemble Trnsorm Klmn Filter (LETKF) rmework. Insted o computing the LETKF nlysis t every model grid point, we compute the nlysis on very corse grid nd interpolte onto high-resolution grid by interpolting the nlysis weights o the ensemble orecst members derived rom the LETKF. Becuse the weights vry on lrger scles thn the nlysis ields or nlysis increments, there is little degrdtion in the qulity o the weight-interpolted nlyses compred to the nlyses derived with the high-resolution grid, nd the results rom the weight-interpolted nlyses re more ccurte thn the ones derived by interpolting the nlysis increments. 1

2 1. Bckground The resolution o modern numericl models nd observtion density hve gretly incresed in recent yers in order to resolve the dynmic processes in smller scles such s convective scles. This burdens the computtionl cost o the dt ssimiltion (DA) procedure, so tht or vritionl nlysis (3DVr nd 4DVr), methods to reduce the hevy computtionl cost ocus on reducing the computtion during the minimiztion process o the cost unction (Courier et l. 1994). In 4D-Vr, the so-clled inner loop is conducted by running the djoint model t low resolution nd/or with simpliied physics. For the ensemble methods s Ensemble Klmn Filter, the computtionl cost is llevited by llowing the nlysis to be computed in prllel in locl regions (Keppenne et l. 2002, Ott et l nd Hunt et l., 2007 ). However, the computtionl burden or such locl nlyses is still constrined by the ensemble size nd the number o totl locl regions in high-resolution model. It is possible to reduce the computtion urther by crrying out the ensemble nlyses on corse resolution, s done with vritionl nlyses, nd then interpolting to the iner resolution. Unortuntely, in both methods this interpoltion step degrdes the ccurcy o the nlysis, compred to ull-resolution nlysis. In this study, we investigte the esibility o reducing the computtionl cost o the ssimiltion by using the output o n ensemble o ull-resolution orecsts without scriicing the ccurcy o the nlysis. This method is developed ollowing suggestion 2

3 o Bowler (2006, see section 6.4), who computed the trnsorm mtrix derived rom the solution o Ensemble Trnsorm Klmn Filter (Bishop et l. 2001) on corse grid nd then interpolted it onto the high-resolution grid. A similr ide o weight interpoltion is used to speed up the nlysis computtion o n ocen ensemble Klmn ilter (EnKF, Keppenne et l. 2007). However, in these studies the properties o the weights nd how the weight-interpoltion ensures the nlysis ccurcy re not discussed. In this study, the weight-interpoltion method is dpted to the rmework o the Locl Ensemble Trnsorm Klmn Filter (LETKF, Ott et l., 2004 nd Hunt et l. 2007), which hs the dvntge o being conigured or locl nlyses, nd where the nlysis ensemble is expressed with weights tht linerly combine the orecst ensemble. The interpoltion is done through these weights or the bckground ensemble in order to crete n nlysis t ull resolution. As we will show, this method llows us to substntilly reduce the required number o locl nlyses or LETKF ssimiltion without scriicing nlysis ccurcy. This pper is orgnized s ollows: The implementtion o the LETKF is briely described in Section 2, including the setup o the ssimiltion experiments in Qusi- Geostrophic model. In Section 3, we explin how the sprse nlysis is done within the LETKF rmework. The results o the interpolted dt ssimiltion experiments re discussed in Section 4. A summry nd discussion is given in Section Locl Ensemble Trnsorm Klmn Filter (LETKF) 3

4 The LETKF scheme, described in detil in Hunt et l. (2007), is n ensemble-bsed Klmn Filter tht perorms nlysis within locl region, with the bckground error covrince in ech region described by the corresponding locl bckground ensemble. This scheme is more eicient version o the Locl Ensemble Klmn Filter proposed by Ott et l. (2004). The LETKF hs been shown to hve similr ccurcy s other sequentil ensemble-bsed Klmn Filters implemented in the sme numericl wether prediction model (Whitker et l. 2007), but its prllel implementtion, mde possible becuse the nlysis t ech grid point is independent o other grid points, becomes more eicient s the number o processors grows. We briely discuss the LETKF lgorithm, where the dt ssimiltion is perormed t the nlysis centrl grid point within locl region. The locl error sttistics re estimted bsed on the bckground sttes nd the vilble observtions within this locl region. The LETKF determines trnsorm mtrix tht converts the locl bckground ensemble perturbtions into the nlysis ensemble perturbtions. The locl nlysis error covrince cn be written s (1), where K is the ensemble size, X is the mtrix whose columns re the bckground (orecst) ensemble devitions rom the bckground ensemble men, X is the corresponding mtrix o the nlysis ensemble perturbtions, P is the nlysis error covrince nd spce. P ~, the nlysis error covrince in ensemble P 1 T ~ T = X X = X P X, (1) K 1 nd the trnsorm mtrix is computed s: 4

5 ~ T 1 1 P = [( HX ) R ( HX ) + ( K 1) I]. (2) In (2), H is observtion opertor tht converts vribles rom model spce to observtion spce nd R is the observtion error covrince. Both H nd R depend on the region becuse only locl observtions re used. The mtrix P ~ is eiciently computed within the ensemble spce. Ater P ~ is obtined, the men nlysis, x, t the centrl grid point o the locl region is computed rom the bckground ensemble men, (3). x, ccording to T = x + X P ~ ( HX ) R 1 ( yo H ( x )) = x X w x + (3) In (3), the K 1 vector o weight w, crries the inormtion bout observtionl increments. In the inl step, the nlysis ensemble perturbtions t the centrl grid point re derived by multiplying the bckground ensemble perturbtions by the symmetric squre root o ( K 1) P ~ : ~ X [( =. (4) 1 2 = X K 1) P ] X W In (4), W is multiple o the symmetric squre root o the locl nlysis error covrince in the ensemble spce. The use o symmetric squre root ensures tht the sum o the nlysis ensemble perturbtions is zero nd depends continuously on Thereore, djcent points with slightly dierent P ~ will yield similr nlysis ensemble perturbtions, necessry to ensure the smoothness o the nlysis (Hunt et l. 2007). This property o the symmetric squre root lso ensures tht the nlysis ensemble P ~. 5

6 perturbtions re consistent with the bckground ensemble perturbtion since the W is the squre root mtrix closest to the identity mtrix, given the constrint o the nlysis error covrince (Ott et l. 2004). Hrlim (2006) demonstrted tht the symmetric solution hs better perormnce thn one obtined with non-symmetric squre root, given the sme ensemble size. Eqs. (3) nd (4) show tht the nlysis ensemble t ech grid point is simply liner combintion o the bckground ensemble, with weighting coeicients given by w ( K 1 vector) or the men nlysis, nd W ( K K mtrix) or the perturbtion. So, the k th nlysis ensemble member is given by x,k = x + X,k w + W,k (5) where W is the k, k th column o W. Let v denotes vector o K ones, which is n eigenvector o the computed W t ech nlysis grid point (Hunt et l. 2007). The zero men o the bckground ensemble perturbtions, X v = 0, implies tht X v = X W v = 0, (6) which mens tht the nlysis perturbtions lso hve zero men. In this study, the LETKF is implemented on qusi-geostrophic (QG) chnnel model. There re 64 grid-points in zonl direction, 33 grid-points in meridionl direction nd totl o 7 vribles in the verticl. The model physics include processes o dvection, diusion, nd relxtion t ll levels nd Ekmn pumping t the bottom level, with 6

7 zonlly periodic solution (Rotunno nd Bo 1996). The model vribles re potentil vorticity, in the ive internl levels, nd potentil tempertures t the bottom nd top levels. They re the nlysis vribles in the DA experiments. Observtions re generted by dding rndom Gussin errors to the true proiles o zonl nd meridionl winds nd potentil tempertures. Yng et l. (2007) compred the perormnce o LETKF with other DA schemes in this model. They showed tht with perect model conigurtion (no model errors), the nlysis derived rom the LETKF ws more ccurte thn the nlyses rom the 3D-Vr or rom the 4D-Vr with short ssimiltion window (12-hour), but with 24-hour windows 4D-Vr ws more ccurte. A similr DA setup s in Yng et l. (2007) is used here to test the method o interpolted LETKF nlyses. There re 128 rwinsonde observtions uniormly distributed within the model domin. The observtion error is 0.8 ms -1 or the zonl wind, 0.5 ms -1 or the meridionl wind nd 0.8 C or potentil temperture. In this study, we llow the model to be imperect by chnging the verticl mixing prmeter used or the Ekmn pumping rom 5 or the truth run to 4.75 or the dt ssimiltion cycle. For the LETKF nlysis, 20 ensemble members re used in this study nd the locl ptch is chosen to be 7 7 grid-points in the horizontl nd to include the whole column (7 levels). A Gussin unction with decorreltion length o 5 grid points is pplied to the observtion error covrince to reduce the impct o more distnt observtions on the 7

8 nlysis (Miyoshi, 2005). In ddition to multiplictive covrince inltion o 8%, n dditive perturbtion is used to optimize the LETKF perormnce by dding very smll mount o rndom perturbtions onto the nlysis ensemble perturbtions (Corzz et l., 2007). 3. Interpoltion o the nlysis weights The locl nlysis error covrince in the LETKF is estimted by combining the contributions rom ech vilble observtion in ech locl region. Such contributions re represented in the locl ensemble spce by the weighting coeicients ( w, W in Eqs. (3-4)). The sme inormtion rom observtions nd bckground sttes is used over severl regions due to the overlpping between locl regions. This ensures tht the weights vry slowly nd llows us to perorm locl nlysis on limited number o grid points ( corse-resolution grid), nd to spred out the inormtion o the error sttistics to the higher-resolution grid through the interpoltion o the weights. Figure 1 illustrtes how the sprse nlysis is done within the LETKF conigurtion. The bckground ensemble is vilble t ll the grid points o high resolution denoted s dots nd crosses in Figure 1. The dots, rrnged on corse grid, denote the grid points where the LETKF nlyses nd the weighting coeicients ( w vector nd W mtrix) re ctully computed. In this exmple, only one nlysis is computed or every 3 3 grid box so tht the coverge o the grid (i.e., the number o nlysis grid points divided by the 8

9 totl number o grid points, except or modiictions t the northern nd southern boundries) is 11%. Ater the weight coeicients re collected on the corse grids, we interpolte w nd W onto the high-resolution grid points where no nlysis hs been computed, in order to generte ields o weights. Corresponding to ech orecst ensemble member, we will hve one mp o weight coeicients ssocited with the men nlysis increment, nd K mps o weights ssocited with the nlysis ensemble perturbtions. As discussed in Section 2, the irst mp o weights represents the observtionl impct in correcting the men stte, nd the ltter K mps pply errors o the dy structures bsed on the dynmic evolution o the ensemble perturbtions to generte the nlysis perturbtions. Yng et l. (2007) showed tht both these two sources o inormtion re importnt in improving the ccurcy o the LETKF nlysis. The unstble spce o growing perturbtions domintes the orecst errors, nd the ensemble perturbtions provide inormtion tht mkes the nlysis increments project well onto this unstble spce. For the interpoltion o the weights, we pply smooth bi-vrite interpoltion scheme (Akim, 1978) bsed on loclly itting quintic polynomils s the unction o the zonl nd meridionl positions o the nlysis grid. The chosen interpoltion scheme is liner in the interpolted vlues. This ensures tht the vector v o K ones is still n eigenvector o the interpolted W, so tht the nlysis ensemble perturbtion clculted rom the interpolted weights will mintin the property o zero men s in Eq (6). We lso conducted ssimiltion experiments with even corser nlysis grids thn the one shown in Figure 1, so tht there is one LETKF nlysis computed in every 3 3, 5 5, or 7 7 horizontl grid-box. The corresponding nlysis coverge is 11%, 4% nd 2% 9

10 respectively. We will show in the next section tht even though the number o totl locl nlyses is substntilly reduced, the nlyses derived by interpolting the weights to ull-resolution show little degrdtion. Bowler (2006) indicted tht the sptil consistency o the weighting coeicients is wht mkes dvntgeous the interpoltion o weight to perorm nlyses on corse grid. With the use o the symmetric squre root mtrix, the weighting coeicients derived t djcent points re typiclly close to ech other, s illustrted in Figure 2()-(d) showing the weight coeicients or the observtionl correction o the bckground ensemble men. We choose to show s n exmple the contribution rom the 4 th ensemble member (n element o the w vector o mps) to the nlysis men. Figure 2() is the weight mp obtined rom the ull nlysis t n rbitrry time. The empty spots in Figure 2() re res where there re no observtions vilble in the corresponding locl ptch nd the bckground ensemble is thereore not updted in these regions. Figure 2(b)-(d) re the interpolted mps o this weighting coeicient with corser nlysis grids. Becuse the weights tend to be consistent t djcent points, the interpolted weight structures cn represent the originl structures resonbly well. We now exmine the weights pplied to orm the nlysis ensemble perturbtions, bsed on digonl element o W nd n o-digonl element. Figure 2(e)-(h) show the mps o the weight coeicients o the irst element in the irst column o the mtrix W, representing the contribution rom the irst bckground ensemble perturbtion to the irst 10

11 nlysis ensemble perturbtion. The vlues derived rom the ull nlysis coverge re shown in Figure 2(e). Figure 2()-(h) re the sme weights ter interpoltion on corser nlysis grids. As could be expected rom (2) nd (4), the irst ensemble orecst perturbtion hs the most inluence in determining the irst nlysis perturbtion, with weights rnging rom 0.7 to 1.0. Regions with lower vlues in these mps indicte tht the contributions rom other ensemble become lso importnt. The min etures o the weight mp derived rom ull nlysis coverge re well recovered in the interpolted mps, except or the 2% cse, in which the ptterns re smoothed out. As n exmple o the o-digonl elements, Figure 2(i)-(l) show the mps o the weight coeicients o the ourth element in the irst column o the mtrix W, representing the contribution rom the ourth bckground ensemble perturbtion to the irst nlysis ensemble perturbtion. The lrge vlues in Figure 2(i)-(l) correspond to smll vlues ppering in Figure 2(e)-(h). This gin indictes tht t this loction, the other ensemble perturbtions contribute more. Overll, their mplitude is much smller compred to Figure 2(e)-(h), due to the property tht W is close to the identity mtrix. The min etures shown in Figure 2(i) re well recovered in Figure 2(j) (l) ter interpoltion. Besides reducing the nlysis computtion, the interpolted weights lso provide n dditionl beneit in obtining resonble weights or those locl ptches without vilble observtions (the empty points in the interior domin o Figure 2 (), (e) nd (i)). For these points, w is simply vector o zeros nd W is equl to the identity mtrix in the originl nlysis setting without weight interpoltion, nd these vlues my not be optiml. This interpoltion provides esible wy to updte the bckground ensemble in 11

12 the regions tht hve not been observed, or tht re under-observed compred to neighboring regions. 4. Anlysis Results In this section, we compre the LETKF sprse nlyses constructed rom weight interpoltion with the nlysis derived t ull resolution. For comprison, we lso conduct sprse nlyses using interpolted nlysis increments, ter running the ull-resolution LETKF or 20 dys. The increment interpoltion is trditionl method used to convert corse resolution nlysis into ull-resolution nlysis. The time series o the root men squre o nlysis error in terms o the potentil vorticity is shown in Figure 3. The LETKF nlysis derived rom ull resolution or weights interpoltion rom sprse gridpoint outperorm the 3D-Vr nlysis derived t the high (ull) resolution. Results show tht the sprse nlyses with the interpolted weights t dierent nlysis coverge (blue lines) hve qulity similr or even slightly better to the one obtined t ull resolution. Such result suggests tht the interpolted weights re very useul in retining the qulity o the nlysis. The nlysis ccurcy rom the interpolted weights becomes poorer only when the nlysis grid is so corse (2% or less) tht the locl ptches o ech nlysis do not overlp. In ddition, the sprse nlyses with the interpolted increments (shown s red lines) re more sensitive to the sprseness o the nlysis grid-point nd hve much lower nlysis ccurcy thn the ones derived rom the interpolted weights. By contrst, the nlysis ccurcy rom the interpolted increments remins stisctory only with high nlysis coverge o 50%, where the nlysis grid-point is rrnged s 12

13 stggered grid vilble every other grid-point. Once the coverge decreses, the ccurcy degrdes quickly. With the 25% nlysis coverge, the nlysis increments re smoothed out nd stretch isotropiclly nd thus the nlysis hs ccurcy similr to the 3D-Vr nlysis. With lower nlysis coverge o 11%, the LETKF with the interpolted increments diverges (the dshed red line in Figure 3). These results show tht interpolting the nlysis increments or the ull ields (not shown) leds to serious degrdtion o the nlysis, wheres by interpolting the weights, we retin the nlysis ccurcy nd the dvntges o the LETKF nd the eiciency o low-resolution nlysis. The results shown in Figure 3 cn be understood by the structures o the nlysis increments (the dierences between nlysis nd bckground) rom the sprse nlyses constructed with interpolted weights. The nlysis increments, s shown or potentil vorticity t the mid-level, in Figure 4, represent orecst errors stretched by the low, nd they hve elongted structures nd scles similr to tht o bred vectors (Corzz et l. 2002). We ound tht the nlysis increments obtined rom interpolted weights t dierent nlysis coverge (11%, 4% nd 2%) re very similr, s shown in n exmple in Figure 4()-(d). It is cler tht interpolting the weights succeeds in recovering quite well the ull nlysis increments, nd becuse o the lrge scles o vritions in the weighting mps, the obtined nlysis increments re insensitive to the nlysis coverge. By contrst, i the interpoltion is done on the nlysis increments, like Figure 4(e)-(h), the nlysis increment structures re quickly smoothed out s the nlysis grid becomes corser. Only 13

14 with high nlysis-coverge o 50%, cn the locl chrcteristics in the nlysis increment t ull resolution be retined nd thus mintin n dvntge over 3D-Vr. In Figure 4(h), the pttern o interpolted nlysis increment with 2% nlysis coverge hs n unrelistic lrge-scle eture. As could be expected, this will impose lse corrections to the bckground stte nd lose the dvntge o using the time-dependent error sttistics in the LETKF. 5. Summry In this study, we investigted n eicient method to reduce the nlysis computtionl cost within the rmework o the LETKF ollowing suggestion by Bowler (2006). The LETKF nlyses re computed on corse grids, but the weights used to updte the bckground ensemble re interpolted onto the high-resolution grids. Insted o repetedly using the observtions nd the bckground ensemble to perorm the LETKF, the nlysis t the high-resolution is now derived through estimting the interpolted weights. In LETKF, the weights o the nlysis ensemble represents two sources o inormtion: one is ssocited with the observtions contribution to the men bckground stte nd the other is ssocited with the dynmiclly evolving error structures obtined rom bckground ensemble perturbtions. Interpoltions re done seprtely on these two components o weights by tking the dvntge o the symmetric squre root solution o the trnsorm mtrix used in LETKF. 14

15 We showed tht the weights derived rom LETKF re smooth nd consistent mong nerby points so the interpolted weight mps or corse nlysis-grid cn represent evolving etures very well. Furthermore, interpoltion results in some smoothing o the weights, which my provide dditionl blnce in the nlysis increments. Becuse the weights vry on lrger scles thn the nlysis or nlysis increments, there is little degrdtion in the qulity o the weight-interpolted nlyses compred to the nlyses derived with the high-resolution grid. Thereore, the corresponding nlysis ccurcy is still high even when we reduce the nlysis grid to just 2% o the grid points o the ull resolution. In ddition, the results re insensitive to the nlysis coverge in this study (but they re sensitive to the size o the locl ptch nd the ensemble s discussed in Yng et l. 2007). The results lso show tht interpolting the weights gives n nlysis tht is much more ccurte thn the nlysis constructed rom interpolting the nlysis increments (or the ull ields, not shown). The qulity o interpoltion o the nlysis increments is much more sensitive to the sprseness o the nlysis-grid due to their chrcteristic stretched dynmicl scles. Besides the purpose o perorming sprse nlyses, the weight-interpoltion cn lso be used to provide nlysis weights or regions without locl observtions, so tht insted o returning the bckground ensemble vlues without updting them, s in the originl LETKF procedure, the nlysis ensemble cn still be computed or these under-observed regions. This dvntge o smoothing the weights in hndling under-observed regions my explin why the interpolted weight results re not just comprble but even slightly better thn the ull resolution nlysis, so tht smoothing the weights my be 15

16 dvntgeous even when using the ull resolution nlysis grid. We lso point out tht ny conserved quntity tht is liner unction o the model stte will be eqully conserved in the originl LETKF nlysis nd in the nlysis perormed with the interpolted weights. With the method o weights interpoltion, the nlysis remins in the subspce o the orecst ensemble, so tht properties such s conservtion o totl mss nd blnce, stisied by ech ensemble member, re lso stisied in the nlysis. Although the gol o this study ws to test whether sprse nlyses with interpolted weights is suitble method or opertionl use, the results obtined in this study my be over-optimistic since the qusi-geostrophic model used in the experiments ws only slightly imperect. 6. Reerences Akim, H. (1978). A Method o Bivrite Interpoltion nd Smooth Surce Fitting or Irregulrly Distributed Dt Points. ACM Trnsctions on Mthemticl Sotwre, 4, Bishop, C. H., B. J. Etherton, nd S. J. Mjumdr, 2001: Adptive smpling with the ensemble trnsorm Klmn ilter. Prt I: Theoreticl spects. Mon. We. Rev., 129, Bowler, N. 2006: Comprison o error breeding, singulr vectors, rndom perturbtions nd ensemble Klmn ilter perturbtion strtegies on simple model. Tellus A, 58,

17 Courtier, P., J. N. Théput, nd A. Hollingsworth, 1994: A strtegy or opertionl implementtion o 4DVAR, using n incrementl pproch. Qurt. J. Roy. Meteor. Soc., 120, Corzz, M., E. Klny, D. J. Ptil, E. Ott, J. Yorke, I. Szunyogh, M. Ci, 2002: Use o the breeding technique in the estimtion o the bckground error covrince mtrix or qusigeostrophic model. AMS Symposium on Observtions, Dt Assimiltion nd Probbilistic Prediction, Orlnd, Florid, , E. Klny, nd S.-C. Yng, 2007: An implementtion o the Locl Ensemble Klmn Filter or simple qusi-geostrophic model: Results nd comprison with 3D-Vr dt ssimiltion system. Nonliner Processes in Geophysics, 14, Hunt, B., E. Kostelich, I. Szunyogh, 2007: Eicient dt ssimiltion or sptiotemporl chos: Locl Ensemble Trnsorm Klmn Filter. Physic D, 230, Hrlim, H., 2006: Errors in the initil conditions or numericl wether prediction: A study o error growth ptterns nd error reduction with ensemble iltering. PhD disserttion, University o Mrylnd, 86 pges. Keppenne, C.L., nd M.M. Rienecker, 2002: Development nd initil testing o prllel Ensemble Klmn ilter or the Poseidon isopycnl ocen generl circultion model. Mon. We. Rev., 130, , M. M. Rienecker, J. P. Jcob nd R. Kovch, 2007: Error covrince modeling in the GMAO ocen ensemble Klmn ilter. Mon. We. Rev., (in press). Miyoshi, T., 2005: Ensemble Klmn ilter experiments with Primitive-Eqution globl model. Doctorl disserttion, University o Mrylnd, College Prk, 197pp. Avilble t 17

18 Morss, R. E. 1998: Adptive observtions: Idelized smpling strtegies or improving numericl wether prediction. PhD thesis, Msschusetts Institute o Technology. 225 pp. Rotunno, R. nd Bo, J. W., 1996: A cse study o cyclogenesis using model hierrchy. Mon. We. Rev., 124, Ott, E., B. R. Hunt, I. Szunyogh, A. V. Zimin, E. J. Kostelich, M. Corzz, E. Klny, D. J. Ptil, nd J. A. Yorke, 2004: A locl ensemble Klmn ilter or tmospheric dt ssimiltion. Tellus A, 56, Wng, X., nd S. J. Julier, 2004: Which is better, n ensemble o positive-negtive pirs or centered sphericl simplex ensemble? Mon. We. Rev., 132, Whitker, J. S. nd T. M. Hmill, 2002: Ensemble Dt Assimiltion without Perturbed Observtions. Mon. We. Rev., 130, , T. M. Hmill, X. Wei, Y. Song nd Z. Toth, 2007: Ensemble Dt Assimiltion with the NCEP Globl Forecst System. Mon. We. Rev., (in press). Yng, S-C., M. Corzz, A. Crrssi, E. Klny nd T. Miyoshi, 2007: Comprisons o ensemble-bsed nd vritionl-bsed dt ssimiltion schemes in Qusi- Geostrophic model. Mon. We. Rev. (under revision). 7. Acknowledgements We re grteul or helpul interctions with Je Whitker, Istvn Szunyogh, Eric Kostelich, nd Milij Zupnski. Rebecc Morss nd Mtteo Corzz provided the originl 3D-Vr code. S.-C. Yng ws supported by NASA grnts NNG004GK78A nd NNG06GB77G. 18

19 Figure Cptions Figure 1 The grid rrngement or n 11% corse nlysis. The dots re the grids where the LETKF nlysis is ctully perormed. The crosses indicte high-resolution grid points whose nlysis will be derived by weight-interpoltion or increment interpoltion (see the explntion in the text). Figure 2 () The weighting coeicients corresponding to the ourth element o the LETKF men weighting vector or updting the bckground ensemble men, derived with 100% nlysis grid-point coverge (ull resolution); (b)-(d) the sme s (), except tht the weight is interpolted t the 11%, 4% nd 2% nlysis grid coverge. (e)-(h) re the irst element o the irst column o the LETKF weighting mtrix or constructing the irst nlysis ensemble perturbtion derived t the sme resolutions used in ()-(d). (i)-(l) re the ourth element o the irst column o the LETKF weighting mtrix or constructing the irst nlysis ensemble perturbtion derived t the sme resolutions used in ()-(d). The empty spots denote the locl regions with no vilble observtions. Their weight vlues re thereore zero or () nd (i) nd one or (e). Figure 3 The time series o the RMS nlysis error in terms o the potentil vorticity rom dierent DA experiments. The LETKF nlysis rom the ull-resolution is denoted s the blck line nd the 3D-Vr derived t the sme resolution is denoted s the grey line. The LETKF nlyses derived rom weight-interpoltion with dierent nlysis coverge re indicted with blue lines. The LETKF nlyses derived ter the irst 20 dys rom increment-interpoltion with dierent nlysis coverge re indicted with the red lines. 19

20 Figure 4 The nlysis increment or potentil vorticity t the 3rd level t n rbitrry time rom the LETKF with () ull resolution, (b) obtined through interpolted weights with 11% nlysis coverge, (c) the sme s (b) except or 4% nlysis coverge, (d) the sme s (b), except or 2% nlysis-grid coverge. (e)-(h) re the interpolted nlysis increments computed by tking 50%, 11%, 4% nd 2% nlysis coverge o nlysis increment obtined t the ull-resolution nd interpolting bck to the ull-resolution. 20

21 Figure 1 The grid rrngement or n 11% corse nlysis. The dots re the grids where the LETKF nlysis is ctully perormed. The crosses indicte high-resolution grid points whose nlysis will be derived by weight-interpoltion or increment interpoltion (see the explntion in the text). 21

22 Figure 2 () The weighting coeicients corresponding to the ourth element o the LETKF men weighting vector or updting the bckground ensemble men, derived with 100% nlysis grid-point coverge (ull resolution); (b)-(d) the sme s (), except tht the weight is interpolted t the 11%, 4% nd 2% nlysis grid coverge. (e)-(h) re the irst element o the irst column o the LETKF weighting mtrix or constructing the irst nlysis ensemble perturbtion derived t the sme resolutions used in ()-(d). (i)-(l) re the ourth element o the irst column o the LETKF weighting mtrix or constructing the irst nlysis ensemble perturbtion derived t the sme resolutions used in ()-(d). The empty spots denote the locl regions with no vilble observtions. Their weight vlues re thereore zero or () nd (i) nd one or (e). 22

23 Figure 3 The time series o the RMS nlysis error in terms o the potentil vorticity rom dierent DA experiments. The LETKF nlysis rom the ull-resolution is denoted s the blck line nd the 3D-Vr derived t the sme resolution is denoted s the grey line. The LETKF nlyses derived rom weightinterpoltion with dierent nlysis coverge re indicted with blue lines. The LETKF nlyses derived ter the irst 20 dys rom increment-interpoltion with dierent nlysis coverge re indicted with the red lines. 23

24 Figure 4 The nlysis increment or potentil vorticity t the 3 rd level t n rbitrry time rom the LETKF with () ull resolution, (b) obtined through interpolted weights with 11% nlysis coverge, (c) the sme s (b) except or 4% nlysis coverge, (d) the sme s (b), except or 2% nlysis-grid coverge. (e)-(h) re the interpolted nlysis increments computed by tking 50%, 11%, 4% nd 2% nlysis coverge o nlysis increment obtined t the ull-resolution nd interpolting bck to the ullresolution. 24