6.1 AN OVERVIEW OF ENSEMBLE FORECASTING AND DATA ASSIMILATION. Thomas M. Hamill. like to find (X t t

Transcription

1 6. AN OVERVIEW OF ENSEMBLE FORECASTING AND DATA ASSIMILATION Tomas M. Hamll NOAA-CIRES Clmae Dagnoscs Cener, Boulder, Colorado. INTRODUCTION Many of e alks and posers durng s year s conference wll dscuss ow bo ensemble forecasng and amosperc daa assmlaon can work synergscally ogeer. We deal provde a bref descrpon of e underlyng eorecal bass for s researc. Te unfyng dea s a e caoc naure of e amospere can acually be pu o use o mprove daa assmlaon. Ensemble forecass provde flow-dependen esmaes of lkelood of a model-forecas sae; modern daa assmlaon eory requres jus s sor of esmae n order o deermne ow o effecvely assmlae new observaons. Tus, ensemble forecasng and daa assmlaon can be coupled no a unfed eory. I s possble a daa assmlaon sysems around e world 0 years ence wll be usng ensemble-based meodologes. I s me for s researc o emerge from a beng frnge dscplne o beng e cenral focus for ow o mprove daa assmlaon and numercal weaer forecass. 2. UNDERLYING THEORY Te leraure of numercal weaer predcon over e las 40+ years as generally focused on ow o produce a sngle bes nal condon, and from a, make e bes forecas possble. Daa assmlaon, n s conex, s usually expressed n some equaon were a frs guess, or background, s adjused oward new observaon o produce an analyss. However, consderng caoc effecs (e.g., Lorenz 993), a more reasonable goal s o predc e evoluon of probably densy, a s, o know e lkelood of any possble sae of e weaer we are neresed n. Noaonally, we wll use e convenon a a capalzed X and Y represen random vecors for e model sae and observaons, respecvely. Lowercase x and y wll represen acual samples of ose random vecors. Say we seek o esmae a dscrezed represenaon of e rue sae of e amospere X a some me, peraps represenng e sae of e weaer (wnds, emperaure, umdy, ec.) a a fne number of grd pons. We ave a pror esmae of e probably dsrbuon X, bu now we ave newly colleced observaons y o, We d lke an updaed, sarpened esmae of e probably dsrbuon. In e ermnology of Bayesan sascs, e we would Correspondng auor address: Dr. Tomas M. Hamll, NOAA-CIRES CDC, R/CDC, 325 Broadway, Boulder, CO e-mal: amll@cdc.noaa.gov lke o fnd (X = x j Y = y o ), wc we sall soren noaonally o (X jy = y o ). Ta s, we seek e poseror dsrbuon of condonal on (afer assmlang) e new daa y o. Bayes Rule (any sascs ex) ells us ow o calculae s: (X jy = yo )= (X ) (Y = y o jx R ) () () (Y = y o j)d In Bayesan ermnology, (X ) s called e pror dsrbuon. An applcaon of Bayes Rule s llusraed n Fg. for a 2-dmensonal sysem were e observaon and e background sae are assumed o measure e same quany. Te poseror s bascally a normalzed mulplcaon of e pror dsrbuon and e dsrbuon for e observaons condonal on e background (Y = y o jx ). From s nal, relavely sarp probably dsrbuon, we would lke o forecas e evoluon of probably, allowng us o esmae e lkelood of fuure saes. In probablsc erms, our goal s o esmae e probably densy (X )asncreases. If, percance, we ave a perfec model of e evoluon of e amospere,.e., dx = f(x d ), en e evoluon of s probably densy can n prncple be modeled w e Louvlle equaon, a connuy equaon for probably densy (Erendorfer ) + X (X ) = 0 (2) Fgure. Example of e applcaon of Bayes Rule n 2 dmensons. Observaon probably dsrbuon denoed by doed lne (acual observaon s eavy black do), background (pror) by e sold lne, and analyss (poseror) by dased lne. 25, 50, and 75 % of e probably le ousde e respecve conours.

2 If our model s mperfec, peraps our model can expressed socascally n e form dx = f(x d )+ G(x )w, were w s assumed o be nose from a Wener process (e.g., Penland 996). Ten, f ceran assumpons are made abou e socasc forcng, ere s an analogous equaon called e Fokker-Planck equaon a expresses e evoluon of uncerany (see Gardner 990). In pracce, for all bu e mos rval sysems, neer e Louvlle nor e Fokker-Planck equaons are racable, ceranly no for mllon-dmensonal sae vecors as n curren-generaon NWP models. However, we can pu a andy sascal eorem o use (e.g., Casella and Berger 990) : Le s sf e me noaon so a ; denoes an nal me, and e subsequen forecas me. Suppose x ;() = ::: nare random samples of e random varable X ; and suppose we ave some funcon (e.g., a forecas operaor) f suc a x () = f(x ;() ). Ten x () s a random sample of X. In oer words, f we randomly sample e dsrbuon of nal condons (X ) and plug ; eac no e forecas operaor, en we ge a random sample of forecass (X ) (oug, buyer beware: we ge a random sample reflecng our own NWP forecas operaor, no necessarly a random sample of e real forecas operaor, moer naure). If we are forced o work w random samples raer an explcly compung probably dsrbuons, we need o develop a meodology for dong daa assmlaon a properly ulzes ese random sample. I urns ou a Bayes Rule provdes e conex for ow o perform daa assmlaon, bu () s no drecly useful n gly dmensonal sysems. Te problem n ese g-dmensonal sysems s muc smpler f we can make Gaussan assumpons (ese assumpons may no be bad f errors are relavely small and sll growng lnearly). To see ow e Kalman fler equaons (e.g., Lorenc 986) are derved, le us assume a e pror dsrbuon s Gaussan w mean x b and varancecovarance marx P b : (X ) N(xb P b ). Ta s, (X ) / exp ; X 2 ; xb T P b; X ; xb : (3) Smlarly, le us assume a probably dsrbuon for e sae of e observaons gven e background s normally dsrbued accordng o (Y =y o jx ) / exp ; 2 TR y o ; HX ; y o ; HX : (4) Here, H s an operaor wc convers e model sae no e observaon ype. To fnd e maxmum lkelood esmaor for (X j Y = yo ), we use Bayes Rule, nong (X j Y = yo ) / (X ) (Y = y o j X ). Maxmzng s produc s equvalen o mnmzng e negave of e naural log of e produc,.e., o mnmzng e funconal J(X ) J(X )= 2 X ; xb T P b; X ; xb + TR y o ; HX ; y o ; HX (5) W muc algebra, s possble o sow a e poseror dsrbuon as e form (X j Y = yo ) N(x a P a ) were x a s defned by x a = x b + K(y o ; Hx b ) (6) and were K s e Kalman gan marx, K = P b H T (HP b H T + R) ; : (7) Te poseror varance/covarance marx s predced by e equaon P a =(I ; KH)P b : (8) I can also be sown a f one as a fauly or smplfed esmae of e background error covarance ˆP b (peraps an esmae a does no vary w e weaer of e day), e expeced analyss error covarance s ˆKHP b T P a = P b ; ˆKHP b ; (9) + ˆK HP b H T + R ˆK T were ˆK = ˆP b H T (H ˆP b H T + R) ; (Hamll and Snyder 200). Noe a predcng e rue P a va (9) requres knowledge of e rue error sascs P b, wc are lkely o be gly flow dependen. Tus, o dae, snce all operaonal ceners use ese smplfed esmaes ˆP b and don know e rue P b, ey do no aemp o predc P a, nsead focusng on esmang e mos lkely sae usng algorms akn o (6). Concepually, one could use an exended Kalman fler (e.g. Con 997, Talagrand 997), predcng e evoluon of P b usng a lnear angen and adjon of e forecas model, en usng (8) o predc P a. However, s approac s no compuaonally feasble for large-dmensonal models unless smplfyng assumpons are made (e.g., Fser 998). Bu wa mg be possble f we ave a random sample of (X ) (an ensemble of forecass) and new observaons? Frs, s possble a we can develop a gly accurae esmae of background error covarances ˆP b ' P b from e random sample. Te more accurae ese background error covarances, e more accurae e analyss, for we wll be wegng and spreadng e nuence of observaons as ecenly as possble. Furer, f we ave a qualy esmae of P b, en sould be possble o predc e covarance P a of e poseror, (Y = y o jx ). We us ave an esmae of no only e mos lkely sae, bu also some esmae of e uncerany n a sae. A random sample

3 of s dsrbuon s jus wa we re afer n ensemble forecasng. Le s ake s a sep furer: can we develop a eory so a f we are gven a random sample from (X ), n performng e daa assmlaon we auomacally generae a random sample from e poseror (Y = y o jx )? Muc of e researc n ensemble daa assmlaon s based on s premse. 3. ENSEMBLE KALMAN FILTER DATA ASSIMILATION SYSTEM In e orse race for e frs operaonal ensemblebased daa assmlaon meod, e ensemble Kalman fler, or EnKF as e early lead (Evensen 994, Houekamer and Mcell 998, 200, Hamll and Waker 200, and references eren). Many of e alks and posers durng e conference wll dscuss varans on e EnKF. We provde a quck revew ere. Sar w an ensemble of n analyses a some me 0 w suffcen spread amongs members. Ten repeaedly follow a ree-sep process for eac daa assmlaon cycle: () Make n forecass o e nex analyss me. Tese forecass wll be used as background felds for n parallel analyses. (2) Gven e already mperfec observaons a s nex analyss me (ereafer called e conrol observaons), generae = ::: nndependen ses of perurbed observaons y o by addng random nose o e conrol observaons y o. Te nose s drawn from e same dsrbuon R as e observaon errors, and e nose s consruced o ensure a e mean of e perurbed observaons s equal o e conrol observaon. (3) Perform n objecve analyses, updang eac of e n background forecass usng e assocaed se of perurbed observaons. Te analyss equaon for e member s x a = xb + ˆP b H T ; H ˆP b H T + R y o ; Hxb : (0) x b s e m-dmensonal model sae vecor for e member background forecas of an n-member ensemble, and x a s e subsequenly analyzed sae for e member. ˆP b s now an approxmaon of e background error covarances generaed from e collecon of background forecass. In s mos smple form, ˆP b s approxmaed by ˆP b = n ; = T x b ; xb x b ; xb () P were x b = n n = xb s e ensemble mean. I can be sown (Burgers e al. 998) a under ceran assumpons, e ensemble mean from e EnKF s opmal and e poseror covarance calculaed from e ensemble of analyses maces e P a predced by (9). Addonal complexy s ofen nroduced o e sandard EnKF desgn o deal w e dermenal process known as fler dvergence (e.g., Houekamer and Mcell 998, van Leeuwen 999). In s process, e ensemble progressvely gnores observaonal daa more and more n successve cycles, leadng o a useless ensemble. One approac s o modfy backround error covarances by applyng a Scur produc w a correlaon funcon, as dscussed n Houekamer and Mcell (200) and Hamll e al. (200). Anoer approac wc can amelorae e endency oward fler dvergence s o use a double EnKF (Houekamer and Mcell 998), wereby ensemble members are kep n wo separae baces; e covarance model from one bac s used n e assmlaon of e oer. Ts can elp preven e feedback cycle oward smaller and smaller background error covarances. Hamll and Snyder (2000) suggesed a ybrd EnKF, wereby covarances are modeled as a combnaon of covarances from e ensemble and from a saonary model lke 3D-Var. Anderson and Anderson (999) suggesed ncreasng background error covarances somewa by nflang e devaon of background members w respec o er mean by a small amoun. Toug covarances are generally modeled n e EnKF as n (), drec applcaon of () s compuaonally probve. Tus, for compuaonal effcency, e marx operaons ˆP b H T and H ˆP b H T n (9) are compued ogeer usng daa from e ensemble of background saes. Defne Hx b = n = Hx b wc represens e mean of e esmae of e observaon generaed from e background forecass. Ten and ˆP b H T = H ˆP b H T = n ; n ; = x Hx b ; xb b ; Hxb T (2) Hx b ; HxB Hx b ; HxB T = 4. ENSEMBLE DATA ASSIMILATION: SIMPLE EXAMPLES : (3) We now dscuss some smple example o llusrae ow e EnKF works. We ake e same daa as was used o generae Fg., a 2-D sysem. However, n acualy we do no know e full pror backgrounderror covarance as assumed o generae Fg.. Le us assume a we ave an ensemble, a random sample from s dsrbuon (Fg. 2(a), small dos), and a conrol

4 observaon y o, denoed by e eavy black do. We know e observaonal error covarance R and generae perurbed observaons (damonds) conssen w ese error sascs. A background error covarance, or sample pror s esmaed from our ensemble; noe a snce s s esmaed from a random sample, s no e same dsrbuon as e rue pror (one dased conour llusraed for comparson n Fg. 2(a)). Now, parallel assmlaon cycles are conduced usng eq. (0) updang one background o one perurbed observaon. Te resulng analyses (dos n Fg 2(b)) are generally conssen w e rue poseror dsrbuon (dased lnes n Fg. 2(b), reproduced from Fg. ). As llusraed n Fg. 2, ere s a samplng error n esmang covarances from a random sample of background saes. As long as e rue dsrbuon s Gaussan, s esmae wll mprove as more members are added. We noe, also, a ere s a samplng error assocaed w e use of perurbed observaons. As dscussed n Burgers e al. (998), e perurbed observaons were orgnally added o ensure a e expeced value of P a from e EnKF maced a predced by eq. (8). However, especally for small ensemble szes, e perurbed observaons can be spurously correlaed w e background. Fg. 3 (a)- (b) llusraes ow, by reassocang ceran perurbed observaons w ceran background members, very dfferen ses of analyses may resul. Oer presenaons durng s conference by Anderson and Waker wll sugges alernave meods o e EnKF wc can overcome s parcular samplng problem. 5. CONCLUSIONS Ensemble-based daa assmlaon scemes are a very promsng new approac wc may dramacally mprove daa assmlaon. By provdng flow-dependen nformaon on e errors n e background sae, e nfluence of new observaons can be used muc more effecvely. Daa assmlaon and ensemble forecasng, now consdered dsncly dfferen pars of e forecas process, may be unfed, mprovng e qualy of eac. Researc no ensembles and daa assmlaon s growng, bu ere are sll a os of ssues a need o be explored and resolved before e ensemble-based daa assmlaon ecnques wll be ready for operaonal use. Among e expeced problems a wll need o be addressed are: How do we mnmze e compuaonal expense of ensemble-based daa assmlaon scemes? Generally, e ensemble scemes are more accurae w more members, bu e scemes scale n cos w e sze of e ensemble. I may be possble o reduce coss usng dynamcally consraned perurbaons (e.g., Heemnk e al. 200), and clever meods of explong parallelsm may reduce e compuaonal demands (e.g., Houekamer and Mcell 200). Fgure 2. (a) Example of EnKF algorm. Perurbed observaons (damonds) creaed by addng nose o conrol observaon (eavy do). Background saes (dos) ere assumed o be randomly sampled from rue pror dsrbuon n Fg.. Sample pror dsrbuon from pror overploed n sold, and observaonal error dsrbuon w dos. (b) Poseror (analyzed) saes from EnKF (dos), w rue poseror dsrbuon from Fg. ploed over op (dased). How do we deal w non-gaussan sascs? To wa exen are e dsrbuons really non-gaussan? Ts s an acve bu relavely new area of researc. Wll ere be complcaons (suc as balance ssues) wen e ecnque s appled o full prmveequaon models w complex pyscal parameerzaons? How do we deal w model errors? Do we ave models cas n a socasc framework (e.g., Buzza e al. 999, Penland 996) or add sysem nose durng e daa assmlaon (e.g., Mcell and Houekamer 2000)?

5 Fgure 3. Two more examples of samples of e EnKF analyses (dos) for dfferen assocaons of e perurbed observaons w e background saes. REFERENCES Anderson, J. L., and S. L. Anderson, 999: A Mone Carlo mplemenaon of e nonlnear flerng problem o produce ensemble assmlaons and forecass. Mon. Wea. Rev., 27, Buzza, R., M. Mller, and T. N. Palmer, 999: Socasc smulaon of model uncerany n e ECMWF ensemble predcon sysem. Quar. J. Roy. Meeor. Soc., 25, Casella, G., and R. L. Berger, 990: Sascal Inference. Duxbury Press, 650 pp. Con, S. E., 997: An nroducon o esmaon eory. J. Meeor. Soc. Jap., 75(B), Erendorfer, M., 994: Te Louvlle equaon and s poenal usefulness for e predcon of forecas skll. par I: eory. Mon. Wea. Rev., 22, Evensen, G., 994: Sequenal daa assmlaon w a nonlnear quasgeosropc model usng Mone Carlo meods o forecas error sascs. J. Geopys. Res., 99 (C5), , and P. J. van Leeuwen, 996: Assmlaon of Geosa almeer daa for e Agulas curren usng e ensemble Kalman fler w a quasgeosropc model. Mon. Wea. Rev., 24, Fser, M., 998: Developmen of a smplfed Kalman fler ECMWF Researc Deparmen Teccal Memorandum 260. European Cenre for Medum- Range Weaer Forecass. 6 pp. Avalable from Lbrary, ECMWF, Snfeld Park, Readng, Berksre, RG2 9AX, England. Gardner, C. W., 990: Handbook of socasc meods. Sprnger-Verlag, 442 pp. Hamll, T. M., and C. Snyder, A ybrd ensemble Kalman fler - 3D varaonal analyss sceme. Mon. Wea. Rev., 28, , and, 200: Usng mproved background error covarance esmaes from an ensemble Kalman fler for adapve observaons. Mon. Wea. Rev., submed. Avalable from amll@cdc.noaa.gov., J. S. Waker, and, 200: Dsancedependen flerng of background error covarance esmaes n an ensemble Kalman fler. Mon. Wea. Rev., acceped. Heemnk, A. W., M. Verlaan, and A. J. Segers, 2000: Varance-reduced ensemble Kalman flerng. Mon. Wea. Rev., n press. Houekamer, P. L., and H. L. Mcell, 998: Daa assmlaon usng an ensemble Kalman fler ecnque. Mon. Wea. Rev., 26, , and, 200: A sequenal ensemble Kalman fler for amosperc daa assmlaon. Mon. Wea. Rev., 29, Lorenc, A. C., 986: Analyss meods for numercal weaer predcon. Quar. J. Roy. Meeor. Soc., 2, Lorenz, E. N., 993: Te essence of caos. Unversy of Wasngon Press. Mcell, H. L., and P. L. Houekamer, 2000: An adapve ensemble Kalman fler. Mon. Wea. Rev., 28, Penland, C., 996: A socasc model of IndoPacfc sea surface emperaure anomales. Pysca D 98, Talagrand, O., 997: Assmlaon of observaons: an nroducon. J. Meeor. Soc. Jap., 75 (B), van Leeuwen, P. J., 999: Commen on "Daa assmlaon usng an ensemble Kalman fler ecnque." Mon. Wea. Rev., 27,