8B.5 A HYBRID NUDGING-EnKF APPROACH TO DATA ASSIMILATION. Lili Lei* and David R. Stauffer

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1 8B.5 A HYBRID NUDGING-EnKF APPROACH TO DATA ASSIMILATION Lili Lei* and David R. Stauffer Pennsylvania State University, University Park, Pennsylvania 1. INTRODUCTION Operatinal weather analysis and numerical weather predictin (NWP) has evlved frm use f nly standard surface and upper-air bservatins twards assimilatin f many additinal data surces frm surface mesnets, satellites and radars. Data assimilatin is the prcess used t determine a mre accurate state f atmsphere frm these diverse bservatins and mdel frecast / backgrund fields. The nudging data assimilatin scheme has becme increasingly ppular fr messcale data assimilatin because f its efficiency and its use f the full set f mdel equatins. This cntinuus assimilatin methd allws the crrectins t be made gradually thereby reducing insertin shck cmmn with ther intermittent methds while allwing them t influence ther variables and future time perids thrugh the mdel equatins interactins (Stauffer and Seaman 199, Stauffer and Seaman 1994). The ensemble Kalman filter (EnKF), n the ther hand, takes advantage f ensemble frecasts, which are becming mre widely available, t get flw-dependent backgrund errr cvariance which can be used t spread the crrectins and mre efficiently seek the ptimum state between the mdel backgrund and the bservatins (Evensen 1994). Althugh the nudging scheme is cmputatinally efficient, it is ften used with ad-hc nudging cefficients based n thery and past experience but they can be better specified by adjints and/r mdel statistics (e.g., Stauffer and Ba 1993). The EnKF includes bth dynamic and statistical infrmatin but is cmputatinally expensive. Thus a hybrid nudging-enkf apprach with ptential use fr NWP is prpsed and explred here using the nn-linear Lrenz (1963) three-variable mdel system, which allws the EnKF t prvide flw-dependent / timevarying errr cvariance t cmpute the nudging cefficient matrix and t further crrect the mdel fields. * Crrespnding authr address: Lili Lei, Department f Meterlgy, 53 Walker Building, Pennsylvania State University, University Park, PA ; lzl13@psu.edu. 2. METHODOLOGY FOR HYBRID NUDGING- EnKF APPROACH T describe ur prpsed data assimilatin system, we write the equatins f the system as: d x = f ( x) + G w ( x x), (1) where x and f are the state vectr and standard frcing functin f the system respectively, x is bservatin, G is the nudging magnitude matrix fr the nudging term and w is generally the nudging spatial-tempral weighting cefficient used t map the innvatin (bservatin minus the mdel backgrund) in bservatin space and time t the mdel grid cell and time step. The prduct f G and w is defined here as the nudging cefficient. The traditinal nudging data assimilatin scheme sets the nudging cefficient with nn-zer diagnal elements and zer ff-diagnal elements s that the nudging term is relatively small cmpared t the ther physical terms f ( x). The nudging cefficient is ften specified by past experience and experimentatin (e.g., Stauffer and Seaman 1994) t emulate the errr cvariance r crrelatin in the errr at the bservatin lcatin with that in a spatial and tempral regin / windw abut the bservatin site and bservatin time. T determine flwdependent / time-dependent nudging cefficients, the hybrid nudging-enkf methd uses the gain matrix f the EnKF t cmpute the nudging magnitude matrix (Kalata 1984; Painter et al. 199; Yang et al. 26). The gain matrix f the EnKF is (Evensen 1994): T ( ) 1 T K BH HBH R = +, (2) where B is the cvariance matrix f backgrund errrs, R is cvariance matrix f bservatin errr, and H is the bservatin peratr. In the applicatin presented in this paper, the nudging magnitude matrix G is a functin f EnKF gain matrix and nudging weighting cefficient w. The nudging magnitude matrix in the hybrid nudging-enkf methd is defined as: G = t w K, (3)

2 where t w is a functin f the nudging weighting cefficient. Cmpared t the traditinal nudging apprach, the hybrid nudging-enkf apprach takes advantage f ensemble frecasts which ffer flw-dependent / time-dependent backgrund errr cvariance t prvide a flw-dependent / time-varying nudging cefficient. The hybrid nudging-enkf apprach can als extend the nudging magnitude matrix frm having nn-zer diagnal elements and zer ffdiagnal elements t a full nn-zer matrix. This may lead t mre accurate adjustment f backgrund t bservatin than the traditinal apprach. 3. MODEL DESCRIPTION AND EXPERIMENTAL DESIGN As a test bed fr data assimilatin, the nnlinear Lrenz (1963) three-variable mdel system cnsisting f three cupled and nnlinear rdinary differential equatins is applied here: dx = σ ( y x) dy rx y xz = (4) dz xy bz =, where the parameters are set as standard values: σ =1, r =28, b =8/3, and mdel errr is nt cnsidered (i.e., perfect mdel assumptin). The state vectr x and bservatin vectr x in Eq. (1) are nw clumn vectrs with dimensin 3 1, and the nudging magnitude matrix G and the EnKF gain matrix K are dimensined 3 3. We integrate the Lrenz mdel in time using a furth-rder Runge-Kutta numerical scheme and a time step Δt f.1. Equatin (4) is integrated frm a true initial value set as (1.5887, , ) t btain the true state. Fr the simulated bservatins, we add the bservatin errrs chsen randmly frm a Gaussian distributin with mean zer and variance 1. t the true values frm the true state at crrespnding times. Since the true initial value is unknwn in the real atmsphere, a central initial value, defined as the ensemble mean at the initial time step, is assumed t be an apprximate true initial value, which is the true initial value with an additive randm errr. By adding randm errrs frm a Gaussian distributin with mean zer and variance 1. t the central initial value, the initial values fr the ensemble are derived. The number f ensemble members is set t 1. T eliminate the effects f transients (Yang et al. 26), we integrate the Lrenz mdel frm the central initial value and the ensemble fr 1 time steps first and then assimilate the bservatins by either the traditinal nudging methd, the EnKF, r variatins f the hybrid nudging-enkf apprach. The integratin frm the central initial value is called as central state in the fllwing. The bservatin density is ne per 25 time steps by default, and is varied t ne per 1 and ne per 5 time steps in tw experiments. Fr the hybrid nudging-enkf apprach, the system (Eq. (4)) becmes: dx = σ + + ( y x) gxx x x) xy ) xz yx ) yy ) yz zx ) zy ) zz g y y + g z z) dy = rx y xz + g x x + g y y + g z z) dz = xy bz + g x x + g y y + g z z), (5) where each g is an element in the nudging magnitude matrix G in Eq. (1). The traditinal nudging apprach has nn-zer diagnal elements and zer ff-diagnal elements in the nudging magnitude matrix. Fr bth the traditinal nudging methd and the hybrid nudging-enkf methds applied t the Lrenz three-variable system, the nudging weighting cefficient w nly varies in time as defined by (Stauffer and Seaman 199): w = w = t 1, t t τ / 2, t + τ / 2, (6) ( τ t t )/( τ /2), t t τ, t τ /2 t + τ /2, t + τ, therwise where t is the mdel time, t is the bservatin time, and τ is the half-perid f the nudging time windw. Given τ and w, the functin t w in Eq. (3) is: t tw = 1/ w Δt. (7) i= t τ Since the nudging weighting cefficients is fixed as descriptin f Eq. (6), t w is a functin f the halfperid τ and time step Δ t as shwn in Eq. (7). In the traditinal nudging apprach (named TNGAx, where x is the value f the half-perid f the

3 nudging time windw, 8, 4, 2 and 1), the diagnal elements f G are set t 1. (Yang et al. 26). After the initial integratin f 1 time steps, we cntinue t integrate the central state frward in time by assimilating the bservatins using traditinal nudging fr anther 3 time steps. This and the fllwing experiments are als integrated fr a secnd 3-step perid frm 4 t 7 time steps. The secnd experiment, which integrates the ensemble frward and assimilates bservatins by the EnKF, is named as EKFA. The HNKFx (HNKDx) experiment uses the full nudging magnitude matrix (diagnal elements nly) and assimilates the bservatins by the hybrid nudging-enkf apprach in the central state while the nudging magnitude matrix G is determined by creating an ensemble by adding randm errrs t the central value at ne bservatin time step and integrating the ensemble frm this bservatin time t the next bservatin time. Then we calculate the results f the average rt mean square () errrs f the central state (TNGAx, HNKFx and HNKDx) and the ensemble mean (EKFA) cmpared t the truth fr all three variables tgether every time step during the 3 time steps. The r cntrl experiment is the result f integrating the mdel frward fr 3 time steps withut assimilating any bservatins. Fr experiments HNKFx and HNKDx, we test the hypthesis that a temprally averaged nudging cefficient may be cmputed fr the first 3-step perid and then frzen fr use in these experiments (FHNKFx and FHNKDx) during the first 3-step perid and again fr a future 3-step perid. 4. PRELIMINARY RESULTS During the first 3-step perid, the results f, EKFA, TNGAx, HNKFx and HNKDx are shwn in Table 1. All data assimilatin schemes prduce much smaller analysis errrs than the cntrl withut data assimilatin, The EKFA shws the smallest analysis errr (1.2). Bth hybrid nudging-enkf appraches (ne with the full nudging magnitude matrix and the ther with diagnal elements nly) generally have smaller errrs than the traditinal nudging, since these tw hybrid appraches use the ensemble spread t estimate the backgrund errr infrmatin and use this in the nudging cefficients. The hybrid methds can prduce errrs that are nly slightly larger than the EnKF. Als, the errrs using the full nudging magnitude matrix are smaller than thse using diagnal elements nly. The hybrid nudging-enkf with nly diagnal elements f the nudging magnitude matrix (see Eq. (5)) still imprves the statistical scres cmpared t the traditinal nudging methd by using the EnKF t prvide time varying nudging cefficients that may be larger r smaller than the traditinal nudging cefficient f 1. Furthermre, the hybrid nudging TNGA8 2.1 HNKF8/HNKD8 1.8/2.2 FHNKF8/FHNKD8 2.1/2.8 EKFA 1.2 TNGA4 2.5 HNKF4/HNKD4 1.3/1.9 FHNKF4/FHNKD4 1.8/2.6 TNGA2 7.2 HNKF2/HNKD2 1.3/2.1 FHNKF2/FHNKD2 1.9/2.9 TNGA HNKF1/HNKD1 1.5/1.8 FHNKF1/FHNKD1 1.6/2.3 Table 1. The average errr in the first 3-step perid f the n data assimilatin cntrl experiment, EnKF data assimilatin experiment EKFA, traditinal nudging data assimilatin experiments TNGAx, hybrid nudging-enkf data assimilatin experiments HNKFx and HNKDx, and frzen hybrid nudging-enkf data assimilatin experiments FHNKFx and FHNKDx, respectively. 18. TNGA8 2.7 FHNKF8/FHNKD8 2.4/3.3 EKFA 1.4 TNGA4 3.5 FHNKF4/FHNKD4 2. 6/3.7 TNGA2 1.9 FHNKF2/FHNKD2 2.8/4. TNGA1 1.8 FHNKF1/FHNKD1 2.2/3.4 Table 2. The average errr in the secnd 3-step perid f the n data assimilatin cntrl experiment, EnKF data assimilatin experiment EKFA, traditinal nudging data assimilatin experiments TNGAx, and frzen hybrid nudging-enkf data assimilatin experiments FHNKFx and FHNKDx, respectively.

4 EnKF with the full nudging magnitude matrix (Eq. (5)) extends the nudging magnitude matrix frm nly nn-zer diagnal elements and zer ff-diagnal elements matrix t a full nn-zer matrix. Figure 1 shws the average value f the nudging cefficients in the first 3-step perid f HNKF4. The diagnal nudging cefficients are larger than the ffdiagnal nes in the y and z equatins, and the ffdiagnal nudging cefficients relating t variable z are much smaller than thse ff-diagnal elements f x and y. These findings are cnsistent with the results f Yang et al. (26). The use f the EnKF gain matrix t prvide infrmatin t the nudging cefficients has been prven. The diagnal nudging cefficients are always psitive since they are cmputed frm the variance f the backgrund and they determine the e-flding time fr reducing the errrs f the system. Hwever, the ff-diagnal nudging cefficients may be negative, like the nes f variable z (Fig. 1), because they are based n the cvariance f the backgrund which may be negative. The ff-diagnal nudging cefficients have n direct effect n the e-flding time f the system. Frsen Hybrid Nudging EnKF Cefficients x eqn y eqn z eqn Figure 1. The average (frzen) hybrid nudging-enkf cefficients in the first 3-step perid in each equatin in HNKF4. The dashed line dentes the nudging cefficient magnitude f the traditinal nudging apprach applied here. Because the hybrid nudging-enkf apprach with the full nudging magnitude matrix btains better results than that with the diagnal elements nly, the ff-diagnal nudging cefficients play an imprtant rle in further reducing the errr althugh they are small. The Lrenz system with three variables can be seen as a trajectry f a particle in a three-dimensinal space: we can see that the effects f the ff-diagnal nudging cefficients ffer a mre accurate directin and magnitude f frcing n the particle frm the backgrund twards the bservatin, which leads t faster and mre accurate x y z interactin between the backgrund and the bservatin and smaller errrs. This may be the reasn that using the full nudging magnitude matrix prduces better results than using the diagnal elements nly. Since running ensembles and applying the EnKF can be quite expensive in real-time NWP applicatins, we wanted t determine prf f cncept that histrical nudging cefficients frm hybrid nudging-enkf schemes can be effective in future data assimilatins. We have rerun the previus HNKFx and HNKDx experiments with time invariant (frzen) nudging cefficients based n the temprally averaged values f the previus experiments t create FHNKFx and FHNKDx, in the first and secnd 3- step perids. The frzen nudging cefficients f FHNKF4 are temprally averaged frm HNKF4 and are shwn in Fig. 1. The results f FHNKFx and FHNKDx during the first 3-step perid are als shwn in Table 1. The average analysis errrs f FHNKFx and FHNKDx are smewhat larger than HNKFx and HNKDx respectively, but are generally similar r smaller t thse frm TNGAx. When the half-perid f the nudging time windw decreases (t x=2 r x=1), the FHNKFx and FHNKDx get much better results than the TNGAx. This is likely because the nudging cefficients have becme much larger in magnitude, pssibly larger than the physical frcing terms, thereby vilating the assumptin that the artificial nudging terms shuld nt be a dminant frcing in the mdel equatins. Table 2 shws the results f using these average (frzen) hybrid nudging-enkf cefficients in the secnd 3-step perid. The average analysis errrs f FHNKFx and FHNKDx are smaller than thse f TNGAx (except fr FHNKD8 and FHNKD4). S the ff-diagnal terms helped t prduce errrs that were still smaller than thse using traditinal nudging, but these errrs were still larger than thse using the EnKF thrugh this perid, as expected. Thus the histrical hybrid nudging- EnKF cefficients in the previus 3 time steps may have sme value in the next 3 time steps. This suggests that after integrating the ensemble frecast and calculating the EnKF gain matrix nce, the average hybrid nudging-enkf cefficients, especially the full matrix, can be determined and applied in the future. This is the same general idea used in sme calibratins f ensemble frecasts that apply crrectins frm the recent past t future frecasts (Raftery et al. 25). analysis errr in the traditinal nudging scheme increases with decreasing the half-perid f the nudging time windfrm x = 8 t x = 1). Results f TNGA1 with the same nudging cefficients as TNGA2, TNGA4, and TNGA8 had

5 minimal impact as expected because decreasing the nudging time windw reduces the time that the system is gradually frced by the bservatins, which in turn reduces the impact f the bservatins. Hwever, there is nt a simple linear relatinship fr the average analysis errr with the half-perid f nudging time windw in the tw hybrid nudging- EnKF schemes. As shwn in Table 1, the errr first decreases with decreasing nudging time windw and then increases in HNKFx experiments, and the errr in HNKDx experiments decreases first then increases, and decreases again. Thus, the nudging cefficient prvided by the EnKF gain matrix results in a nnlinear respnse f the average errr t the nudging time windw. This may be due t the highly nnlinear Lrenz system and the calculatin f EnKF gain matrix, and the pssibility that large nudging cefficients decuple the mdel equatins. All f the previus experiments used an bservatin density f ne per 25 time steps, and we cmputed the errrs against the truth state every time step. Nw we investigate the effects f different bservatin densities (ne per 1 time steps and ne per 5 time steps) fr traditinal nudging and bth hybrid methds. Similarly the average analysis errr in the traditinal nudging scheme (TNGA4) increases while the bservatin density in time is reduced, as shwn in Fig. 2a. Figures 2b and 2c shw that the tw hybrid nudging-enkf schemes (HNKF4 and HNKD4) have similar relatinships t the bservatin density with the traditinal nudging scheme, but the difference between the results using an bservatin density f ne per 25 time steps and ne per 1 time steps in the tw hybrid nudging- EnKF schemes is much smaller than that in the traditinal nudging scheme. It appears that the (a) (b) (c) bs.5 bs.25 bs.1 bs.5 bs.25 bs.1 bs.5 bs.25 bs Figure 2. The errr during the first 3-step perid f (a) TNGA4, (b) HNKF4 and (c) HNKD4 with different bservatin densities. The slid green line dentes an bservatin density f ne per 5 time steps (bs.5), the dashed blue line indicates an bservatin density f ne per 25 time steps (bs.25), and the ted red line represents an bservatin density f ne per 1 time steps (bs.1). The hrizntal lines and values n the right side f the figure dente the average errr during the first 3-step perid.

6 accuracy f the simulated hybrid nudging state using the full matrix is less sensitive t the bservatin density than the ther tw nudging experiments. Figure 2 als shws that the errr in the tw hybrid nudging-enkf appraches are smaller than that in the traditinal nudging apprach given the same bservatin density, and the errr in the hybrid nudging-enkf with the full nudging magnitude matrix is the smallest f all three experiments. Since the hybrid nudging-enkf appraches use the EnKF gain matrix t prvide nudging cefficients that prduce smaller errrs in the mdel, the underlying assumptin fr nudging stated by Stauffer and Seaman (199; 1994) that the nudging terms shuld be cnstrained t be smaller than the mdel s physical terms in rder t retain the physical prperties and dynamic balance / intervariable cnsistency f the system may be vilated. Figure 3 shws the average ratis f the nudging terms ver the sum f the physical terms in each equatin in the first 3-step perid f experiment HNKF4. The ratis f the y nudging term in the x-equatin and the x and y nudging terms in the y-equatin are less than.5 but larger than.1 (an rder f magnitude difference in the magnitudes f the nudging and physical terms). The ther nudging terms have ratis smaller than.1. In ther wrds, the time scale f the nudging term (1/G) shuld be larger than that f the smallest physical terms in the mdel, n average. Otherwise, the artificial nudging frcing will disrupt the natural balance f frces within the mdel, which diminishes the rle f the mdel equatins in the data assimilatin. Rati x eqn y eqn z eqn x term y term z term Figure 3. The ratis f the nudging terms cmpared t the physical terms averaged ver the first 3-step perid in each equatin in HNKF4. The dashed line dentes a rati f.1, where the nudging term is an rder f magnitude smaller than the sum f the physical frcing terms. In cntrast t the synchrnizatin f the master (truth run) with the slave (mdel with cupling/nudging terms) in Yang et al. (26) and the use f sme intermittent assimilatin strategies, atmspheric data assimilatin schemes that make small crrectins t the mdel backgrund t crrect the mdel trajectry minimize the insertin nise and maximize the natural dynamic balance within the nudged mdel state. Thus cmplete success with any nudging apprach in the atmspheric sciences shuld require that the magnitude f the nudging terms be cnstrained relative t the physical frcing terms, s that dynamic balance and cnsistency are retained in additin t reducing the errrs cmpared t the truth state. A nudging perid which is t shrt may cause the nudging cefficients / nudging strength t becme t large relative t the ther frcing terms, and create numerical nise and data rejectin. Anther measure f success fr a data assimilatin scheme in NWP is t imprve the frecast. Fr example, it is imprtant t knw if the hybrid nudging-enkf apprach with the full nudging magnitude matrix is als effective in prviding a better frecast. Frm abve, the hybrid nudging- EnKF apprach with nly the diagnal elements f the nudging magnitude matrix has similar r smaller errrs than traditinal nudging but larger errrs than the hybrid nudging-enkf apprach with the full nudging magnitude matrix. S here we discuss the hybrid nudging-enkf apprach with the full nudging magnitude matrix applied t the dynamic initializatin prblem. Figure 4 shws the results f dynamic initializatin n EKFA, TNGA4 and HNKF4, respectively. The scale f the nudging cefficients is 1 as shwn in Fig. 1, s the e-flding time is.1. Since the minimum time needed fr effective nudging is generally 2 r 3 e-flding times, assuming that this perid is less than the errr crrelatin time scale, the pre-frecast dynamic-initializatin perid is chsen here as 5, 1 and 2 e-flding times. This is the same as assimilating 3, 5 and 9 bservatins with the bservatin density f ne per 25 time steps. The results f the hybrid nudging-enkf (Fig. 3g-3i) are similar with thse f EnKF (Fig. 3a-3c) wing t the use f EnKF gain matrix. Cmparing Fig. 3a, 3d, 3g using 2 e-flding times t the thers, assimilating mre bservatins during a lnger pre-frecast perid des nt lead t better frecasts necessarily. Assimilating 5 bservatins (1 e-flding times) in each data assimilatin scheme btains the best frecast (Fig. 3b, 3e and 3h). The dynamic initializatin experiments appear t be mst effective at prducing a better frecast than the fr a time perid f 2 t 3 at best. In this time perid, HNKF4 and EKFA prduce a better frecast than

7 (a) EKFA 6 5 (b) EKFA 6 5 (c) EKFA (d) TNGA4 6 5 (e) TNGA4 6 5 (f) TNGA (g) HNKF4 6 5 (h) HNKF4 6 5 (i) HNKF Figure 4. The errr as a functin f time during the dynamic-initializatin perid and subsequent frecast with different data assimilatin schemes and varius lengths fr the dynamic-initializatin perid in terms f e-flding times (1/G). EKFA: (a) 2 e-flding times, (b) 1 e- flding times, (c) 5 e-flding times; TNGA4: (d) 2 e-flding times, (e) 1 e-flding times, (f) 5 e-flding times; HNKF4: (g) 2 e-flding times, (h) 1 e-flding times, (i) 5 e-flding times. The green line is the errr f (n assimilatin f bservatins), which is the same in every figure, the blue line is the errr f each experiment using data assimilatin during a pre-frecast dynamic-initializatin perid fllwed by a free frecast, and the circles dente the times f bservatins used in the pre-frecast assimilatin. TNGA4 except fr the lngest pre-frecast perid assimilating 9 bservatins. Thus the hybrid nudging-enkf is als mre effective at imprving the frecast accuracy in this case using dynamic initializatin than the traditinal nudging methd ver the shrter pre-frecast perids. Traditinal nudging utperfrmed the hybrid methd fr the lngest pre-frecast perid. Althugh these results may change with different time windws, initial cnditins, and bservatin densities, the hybrid nudging-enkf apprach als appears t be effective fr mdel initializatin and imprved shrt-term frecasting. These results suggest that the length f the pre-frecast perid must be chsen very carefully in highly nnlinear cases. 5. SUMMARY A hybrid nudging-enkf apprach with ptential use fr NWP was explred here, which allws the EnKF t prvide flw-dependent / timevarying errr cvariance t cmpute the nudging cefficients rather than using ad-hc nudging cefficients derived frm thery and experience. Traditinal multiscale nudging methds (Stauffer and Seaman 1994) have been prven t be very effective and efficient fr messcale NWP nwcasting (Stauffer et al. 27a) and frecasting (Stauffer et al. 27b) applicatins. In the frmer, multiscale nudging is used in a mdel frecast fr current

8 cnditins (called a nwcast ), available just ahead f the clck and cntinuusly assimilating standard and special bservatins as the mdel integrates frward in time t prvide high quality meterlgical infrmatin fr Army and Marines Crps use. In the latter, a running start dynamic initializatin using multiscale nudging f standard and special bservatins at 36-km, 12-km, 4-km and 1.3-km hrizntal mdel reslutins ver the cmplex terrain f the Italian Alps and Trin plains fr the 26 Winter Olympics, prduced imprved initial cnditins and subsequent 24-h frecasts fr the Department f Defense fr hazard predictin and cnsequence assessment. The mdel clud, precipitatin and lcal circulatins were spunup at the initial time, which imprved the shrt-term frecasts. The nudging magnitude and weighting functins were based n thery and past experience including terrain cnsideratins. This hybrid nudging-enkf apprach explred here cntributed t a mre rapid assimilatin f the data and a better fit f an analysis t the data (decreased errrs cmpared t the truth), and imprved frecasts, than the traditinal nudging data assimilatin scheme. This is prmising because the hybrid methd als accunts fr mdel backgrund errr based n ensemble spread, and bservatin errrs. Hwever, the magnitude f the hybrid nudging cefficients shuld be cnstrained s that the nudging terms are smaller than the physical terms in the mdel equatins n average, s that the time scale fr the nudging term is lnger than that f the smallest physical terms in the mdel (e.g., Stauffer and Seaman 199). A set f temprally averaged hybrid nudging- EnKF cefficients was btained during a histrical perid using the hybrid nudging-enkf apprach. These temprally averaged hybrid nudging-enkf cefficients were fund t have added value in a future perid fr analysis. Therefre, archived hybrid nudging-enkf cefficients may prvide sme useful statistical infrmatin fr defining the nudging cefficients during the dynamic initializatin perid fr future frecast applicatins since the EnKF can be quite expensive in real-time NWP applicatins. This wrk with a ty mdel representing a highly nnlinear system such as the atmsphere serves as a test bed fr further experimentatin in mre sphisticated mdels such as a shallw water mdel and three-dimensinal messcale mdel. These results ffer a prf f cncept that will be further analyzed and used t begin develpment and testing f a hybrid nudging-enkf in the Weather Research and Frecast (WRF) mdel. 6. ACKNOWLEDGMENTS This research is supprted by DTRA cntract n. HDTRA1-7-C-76. The authrs wuld like t thank Sue Ellen Haupt and Gerge S. Yung fr helpful discussins and cmments. 7. REFERENCES Evensen, G., 1994: Sequential data assimilatin with a nnlinear quasi-gestrphic mdel using Mnte Carl methds t frecast errr statistics. J. Gephys. Res., 99(C5), Kalata, P. R., 1984: The tracking index: A generalized parameter fr α-β and α-β-γ target trackers. IEEE transactins n aerspace and electrnic systems, AES-2, Lrenz, E., 1963: Deterministic nn-peridic flw. J. Atms. Sci., 2, Painter, J. H., D. Kerstetter, S. Jwers, 199: Recnciling steady-state Kalman and Alpha-Beta filter design. IEEE transactins n aerspace and electrnic systems, 26, Stauffer, D. R., and N. L. Seaman, 199: Use f furdimensinal data assimilatin in a limited-area messcale mdel. Part Ι: Experiments with synptic data. Mn. Wea. Rev., 118, Stauffer, D. R., and J.-W. Ba, 1993: Optimal determinatin f nudging cefficients using the adjint equatins. Tellus, 45A, Stauffer, D. R., and N. L. Seaman, 1994: Multiscale fur-dimensinal data assimilatin. J. Appl. Meter., 33, Stauffer, D.R., A. Deng, G.K. Hunter, A.M. Gibbs, J.R. Zielnka, K. Tinklepaugh, and J. Dbek, 27a: NWP ges t war, 22 nd Cnference n Weather Analysis and Frecasting/18 th Cnference n Numerical Weather Predictin, June 25-29, Park City, UT. Stauffer, D.R., G.K. Hunter, A. Deng, J.R. Zielnka, K. Tinklepaugh, P. Hayes, and C. Kiley, 27b: On the rle f atmspheric data assimilatin and mdel reslutin n mdel frecast accuracy fr the Trin Winter Olympics, 22 nd Cnference n Weather Analysis and Frecasting/18 th Cnference n Numerical Weather Predictin, June 25-29, Park City, UT. Raftery, A. E., T. Gneiting, F. Balabdaui, and M. Plakwski, 25: Using Bayesian mdel averaging t calibrate frecast ensembles. Mn. Wea. Rev., 133, Yang, S., D. Baker, H. Li, K. Crdes, M. Huff, G. Nagpal, E. Okereke, J. Villafane, E. Kalnay, and G. Duane, 26: Data Assimilatin as synchrnizatin f truth and mdel: experiments with the three variable Lrenz system. J. Atms. Sci., 63,

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