Manuel Pulido and Takemasa Miyoshi UMI IFAECI (CNRS-CONICET-UBA) Department of Atmospheric and.

Transcription

1 Eplortory worshop DADA October 5-8 Buenos Ares Argentn Prmeter estmton usng EnKF Jun Ruz In colborton wth Mnuel Puldo nd Tems Myosh UMI IFAECI (CNRS-CONICET-UBA) Deprtment o Atmospherc nd Ocenogrphc Scences Unversty o Buenos Ares

2 Wht s prmeter estmton? Numercl models hve dynmc core nd prmeterztons o the physcs. In both components there re prmeters tht hve some mpct upon the model perormnce. Most o ths prmeters rse rom the ormulton o numercl schemes nd the smplyng ssumptons mde n prmeterztons. So these prmeters re ntrnsclly unnown nd hve to be tuned n order to get good perormnce. Some eternl orcngs my lso be treted s prmeters n the model equtons. d dt = ( tp )+ pr( tp ) + Dynmc core Prmeterztons (model physcs) Forcng

3 Why do we wnt to estmte model prmeters? Clmte chnge detecton nd ttrbuton. (??) Optmzton o clmte model or mproved representton o the current clmtology. Clmte s ndependent o ntl condtons so tht prmeters plys n mportnt role. (e.g. Annn et l. 5) To optmze model perormnce n opertonl wether orecstng. Optml prmeters should depend on the regon seson etc. (e.g. Koym nd Wtnbe Asoy et l 6) To estmte bses produced by model error. (e.g. Dee nd D Slv 5 Be et l 6) Estmte prmeter uncertnty n order to ntroduce better representton o model error ssocted wth the uncertnn prmeters n the orecst (.e. ensemble orecsts wth perturbtons n the prmeters stochstc prmeterztons etc). (e.g. Hnsen nd Penlnd 7)

4 Prmeter estmton or the erth system A stte o the rt numercl model o the tmosphere or the ocen cn hve lrge number o prmeters. The number s substntlly lrger we consder the D or 3D vrblty o the prmeters (e.g. Bocquet Kng et l. ). Trdtonl optmzton technques tht requre severl model evlutons to descrbe the model senstvty to the prmeter nd to optmze ther vlue re too epensve to optmze tme nd spce vryng prmeters. Dt ssmlton methods cn provde n ecent wy to estmte these model prmeters nd they cn te nto ccount the temporl nd sptl dependence o the prmeters.

5 Dt ssmlton nd prmeter estmton Dt ssmlton s group o methodologes tht optmlly combne normton provded by observtons wth normton comng rom numercl models (prevous observtons + system dynmcs) to obtn n ccurte representton o the stte o system. We cn consder the model prmeters s stte vrbles n dt ssmlton cycle. Ths pproch s nown s stte ugmentton nd hs been wdely used. Stte vrbles (stte vector) = n s =... n p... p M Stte vrbles + prmeters (ugmented stte vector) Observtons re now provdng normton bout the optml stte o the system nd lso bout the optml vlue o the model prmeters.

6 Dt ssmlton cycle wth prmeter estmton: Ensemble Klmn lter z y

7 Dt ssmlton bsed on the ensemble prmeter estmton b o b ) ( p m = = N j T s s s s N P ) )( ( p p = ( ) ( ) b o g b h y +K = s s ( ) T T g R + H HP H = P K g HP K I P ) = ( = = N j N P ) )( ( =+ e Klmn lter: ssmlton cycle nd T Run the model Perturb the ntl condtons nd the model prmeters. Prmeters re ssumed to be constnt durng the ntegrton tme. T )

8 = m( p p = p ) Fnl ensemble t= z y Intl ensemble t=- Integrton tme depends on the pplcton (synoptc scle dt ssmlton ~ 6 hr). Longer ntegrtons ncreses the nluence o non-lner eects. Due to chotc behvor o the system derent model trjectores usully dverge (.e. uncertnty bout the system stte grows wth tme). Ths step s computtonlly ntensve snce the tmes) model hs to be run severl tmes. (~3-5

9 Dt ssmlton bsed on the ensemble prmeter estmton b o b ) ( p m = = N j T s s s s N P ) )( ( p p = ( ) ( ) b o g b h y +K = s s ( ) T T g R + H HP H = P K g HP K I P ) = ( = = N j N P ) )( ( =+ e Klmn lter: ssmlton cycle nd T Estmte the PDF o the orecst rom the ensemble. Compute the rst nd the second moments rom the ensemble. T )

10 P N N j= ( s s )( s s ) T Ensemble orecst t tme t= t= - z y Intl ensemble Estmte the rst nd second moments o the PDF o the orecst. P contns n ths cse covrnces between the stte vrbles nd the prmeters. P evolves n tme ccordng to the non-lner dynmcs o the system. The PDF o the orecst t tme s ssumed to be Gussn. When non-lner eects re mportnt the ctul PDF my be r rom Gussnty.

11 Dt ssmlton bsed on the ensemble prmeter estmton b o b ) ( p m = = N j T s s s s N P ) )( ( p p = ( ) ( ) b o g b h y +K = s s ( ) T T g R + H HP H = P K g HP K I P ) = ( = = N j N P ) )( ( =+ e Klmn lter: ssmlton cycle nd T T ) Compute the nlyss nd the optml prmeters Get the nlyss nd ts uncertnty.

12 s K g = s = P b +K H T g ( ( ) o y h b ( T HP H + R ) Forecst ensemble t= z y observtons Intl ensemble Get the observtons or tme. Errors ssocted to the observtons re ssumed to be Gussn. Errors n the observtons re oten ssumed to be uncorrelted mong ech other. Errors n the orecst nd n the observtons re ssumed to be unbsed. Compre the ensemble men gnst the observtons. (note tht prmeters re not drectly observed) The nlyss men s obtned or the stte vrbles nd the prmeters. Prmeters re estmted bsed on the covrnces between errors n the stte vrbles nd n the prmeters.

13 s = s b +K g ( ( ) o y h b Forecst ensemble P = ( I K ) HP g t= z y observtons Intl ensemble Usng the Klmn lter equtons obtn the nlyss PDF o model sttes t tme. Obtn the PDF o the nlyss. Under the ore mentoned ssumptons the shpe o the nlyss PDF s lso Gussn. The vrnce o the stte vrbles nd prmeters s reduced due to the normton provded by the observtons. Ths s consequence o ssumng tht the dstrbuton o the errors n the model nd n the observtons s Gussn.

14 Dt ssmlton bsed on the ensemble prmeter estmton b o b ) ( p m = = N j T s s s s N P ) )( ( p p = ( ) ( ) b o g b h y +K = s s ( ) T T g R + H HP H = P K g HP K I P ) = ( = = N j N P ) )( ( =+ e Klmn lter: ssmlton cycle nd T T ) Obtn the new ensemble members Smple them rom the nlyss PDF.

15 P = N N j= ( )( ) T New nlyss t= z y Intl ensemble Generte new ensemble members consstentt wth the nlyss PDF. Ths new ensemble members or the model vrbles nd or the prmeters wll be used to produce new short rnge ensemble orecst nd the cycle s closed!

16 A smple model emple: Prmeter estmton n the Lorenz s three vrble model. (Lorenz 963). d dt dy dt dz dt = ( y ) = b y = y cz z y z re model vrbles b c re model prmeters strnge. yet ttrctve.

17 Twn eperments Twn eperment nd perect model The model s ntegrted wthout ssmltng ny normton or tme unts. Ths run s ssumed to be the true system evoluton. Prmeters re constnt durng the ntegrton. Synthetc observtons re generted every tme unt ddng rndom Gussn nose to the true stte. A 3 members ensemble s used to estmte the PDF o the model sttes. (n ths smple eperment the ensemble sze s lrger thn the dmenson o the ugmented stte). P = y z b c y yy zy y by cy z yz zz z bz cz y z b c b yb zb b bb cb c yc zc c bc cc h = H =

18 Assmlton eperment : constnt prmeters PDF men A dt ssmlton cycle s strted t the begnnng the model prmeters nd the model stte re mperectly nown. True X evoluton estmted evoluton nd observtons X Tme In ths emple we cn see how the dt ssmlton scheme s ble to reconstruct the true evoluton usng the normton provdedd by the observtons nd by the model.

19 Assmlton eperment : constnt prmeters PDF covrnce Z vrnce nd error n the estmton s uncton o tme Erro or / ensemble vr rnce Tme P y z = b c y yy zy y by cy z yz zz z bz cz y z b c b yb zb b bb cb c yc zc c bc cc The method cptures lso chnges n the PDF o the model vrbles. Usully lrge error n the estmted vlue o vrble s ssocted wth lrger vrnce n the estmted PDF. These chnges n the PDF re cused by the system dynmcs. The PDF o the orecst errors s stte dependent

20 Assmlton eperment : constnt prmeters Estmted prmeters Tme evoluton o the estmted prmeter Estmted prmeter True prmeter Tme At the begnnng o the eperment vlue o 3 s ssumed or. Ater severl dt ssmlton cycles the vlue o the prmeter ensemble men (red lne) converges towrds the true vlue (blc lne). Observtons provde enough normton to nd the optml vlue or the prmeters (smlr results re obtned or the other two prmeters).

21 Assmlton eperment : constnt prmeters Covrnce Covrnce between nd nd between nd z s uncton o tme P y z = b c y yy zy y by cy z yz zz z bz cz y z b c b yb zb b bb cb c yc zc c bc cc Tme Covrnce between the model vrbles nd the prmeters re lso stte dependent. In ths cse the sgn o the covrnce cn chnge dependng on the stte vrbles. The EnKF estmte these covrnces rom the ensemble.

22 Assmlton eperment : constnt prmeters Anlyss BIAS Anlyss RMSE Dt ssmlton Dt ssmlton & prmeter estmton The optmzton o model prmeters mproves the qulty n the estmton o the model vrbles (y nd z n ths cse).

23 Assmlton eperment : tme-vryng prmeters A new true evoluton s generted ssumng tht the model prmeters chnges slowly wth tme. Estmted prmeter True prmeter Tme Estmted prmeters cn cpture the tme chnges n the optml prmeters. In ths cse results re more nosy. We re stll ssumng tht the prmeter s constnt wth tme durng the model ntegrton but ths s not good ssumpton or ths cse. The uncertnty n the estmted prmeter ncreses n ths cse. The dptve nlton o Myosh () s used n ths cse to estmte the uncertnty o the prmeters. In most pplcton o prmeter estmton spce nd tme. the prmeters beng estmted chnges n

24 Assmlton eperment 3: Imperect model Only prmeter s estmted. b nd c re ed nd they hve errors. Estmted prmeter True prmeter Tme In ths cse the estmted prmeter shows lrge osclltons round the true prmeter. The estmted prmeter s compenstng errors ssocted wth the other two prmeters tht re not beng estmted. It s not possble or the method (s mplemented n ths cse) to dstngush between errors ssocted wth the estmted prmeters nd other sources o model error. The estmton o the model vrbles re stll mproved by the estmton o even when t does not converge to ts true vlue.

25 Prmeter estmton bsed on the model clmtology

26 Clmte model optmzton (Annn nd Hrgreves 4) Insted o ssmltng observtons every tme unt buld n ensemble o long runs. Study the senstvty o the clmtology o the model to the model prmeters. Use the clmtology o the observtons nsted o observtons t prtculr tmes. Long ensemble run (-3 yers) Estmte the optml model prmeters usng EnKF Use the updted prmeter ensemble to ntlze the long run ensemble Severl ssmlton cycles re perormed but n ech cycle the sme observtons re used. Ths my help to del wth non-lnertes n the system? Smple eperments usng the Lorenz model re perormed.

27 Clmte model optmzton (Annn nd Hrgreves 4) Estmted prmeters s uncton o the number o ssmlton cycles. Ech ssmlton cycle requres obtnng n ensemble o model clmtologes wth the updted vlues o the prmeter ensemble. c The convergence rte s uncton o the ntl vrnce o the prmeter ensemble. (red vs blue lnes)

28 Clmte model optmzton (Annn nd Hrgreves 4) Non-lner senstvty o the model to the prmeters. Jont cost uncton o b nd c prmeters. c In ths cse the Gussn ppromton or the shpe o the PDF s not so bd. But s cn be seen n the gure the PDF o the uncertn prmeters s not Gussn.

29 DADA nd eperment wth the Lorenz model (Annn 5) d dt dy dt dz dt = ( y ) = b y z + = y cz + Forcng terms were dded to the Lorenz equtons. C wrmng volcnoes erosols solr orcng Annul men or X

30 DADA nd eperment wth the Lorenz model Annn 5 Tme dependence o the orcng s ssumed to be nown = α + α + α + α solr solr ghg ghg volcnoes volcnoes erosols erosols In ths cse the prmeters α re estmted usng the observtons but the shpe o the orcng s ssumed to be nown pror. X evoluton beore the ssmlton X evoluton ter the ssmlton The totl orcng s ccurtely reconstructed here. In ths smple model the senstvty o the clmtology to the prmeters s lmost lner. The ssue o model error s not ten nto ccount n ths smple eperments.

31 Eperments wth smple GCM SPEEDY hs 4897 grd nd smple prmeterztons (convecton pbl sol model rdton lrge scle condenston) The EnKF method (LETKF Hunt et l. 7) s used or the smultneous estmton o model vrbles nd prmeters n smple GCM (SPEEDY). Three prmeters ssocted wth the convectve scheme re estmted (P P nd P3). (Ruz et l. )

32 Twn eperments wth the SPEEDY model Ruz et l Kng et l 9 Fertg et l 7 Myosh et l 5. Nture run. A three month run wth the SPEEDY model usng the orgnl set o prmeters s used s the nture run. Observtons re generted tng the nture run vlues t every other grd pont t ll vertcl levels nd every s hours. Dt ssmlton Observtons re ssmlted every s hours. The model used n the ssmlton eperment my only der rom the true model n the vlue o the convectve prmeter scheme.

33 Eperments wth smple GCM Ruz et l. The success o prmeter estmton depends on the senstvty o the model to the estmted prmeters. I the model s not senstve to chnges n these prmeters then prmeter estmton wll l. 6 hr Forecst ERROR ERROR P P The lrger the response the smller the uncertnty n the estmton o the prmeter. P3

34 Vertcl levels Eperments wth smple GCM Ruz et l. RMSE U t hp P Zonlly verged RMSE (temp) P Zonlly verged RMSE (V) P Lttude Lttude

35 Eperments wth smple GCM Ruz et l. Covrnce between the temperture t md levels nd P (contours) nd orecsted precptton (shded). Lower P mens weer convecton whch produces less wrmng nd lower tempertures t md levels. Covrnce structure s strongly low dependent becuse convecton s ntermtent n spce nd tme.

36 Eperments wth smple GCM Ruz et l. Estmted prmeters s uncton o tme nd prmeter uncertnty (gry shded) P P3 P Convectve schemes prmeter re ccurtely estmted nd the spn-up tme s round 5 dys (ncludng the spn-up o the ntl condtons).

37 Eperments wth smple GCM Ruz et l. ( ) ( t ) T t RMSE= A Imperect prmeters (no estmton) Anlyss s RMSE Prmeter estmton Perect model Tme (cycles) Prmeter estmton produces strong mpct upon nlyss error. In ths twn eperment the prmeter estmton eperment s s good s the perect model.

38 Eperments wth smple GCM Ruz et l. RMSE o short rnge precptton orecst (shded) nd ts BIAS (contours) Imperect prmeters Estmted prmeters Estmted prmeters produce postve mpct upon the precptton orecst. Most orecsted precptton n the tropcs s produced by the convectve scheme. The ncrese n the ccurcy o the estmted model vrbles (ntl condtons or the orecst) lso eplns the ncrese n the sll o the precptton orecst.

39 Eperments wth smple GCM Ruz et l. See lso Kng et l. () Koym nd Wtnbe (). Tme dependent prmeters re ncluded n the nture run. P3 P P Tme dependent prmeters re well estmted. However smll lg s present n the estmted prmeters. Prmeter evoluton s not nown pror. Ths lg s due to the persstnce model used n the prmeter estmton.

40 Prmeter estmton n the presence o other sources o model error. In the eperments presented so r the model ws (lmost) perect. All model error s due to the uncertnty n the convectve scheme prmeters. Wht hppens when those re not the only source o errors nd when the model error cnnot be completely corrected by tunng some model prmeters (s n rel pplctons)? How cn prmeter estmton be combned wth other methods tht nclude representton o model error n the ensemble Klmn lter?

41 Imperect model eperments: Other sources o model mperecton re ntroduced. To smulte other sources o model mperecton the vlue o some prmeters (numercl duson surce echnge coecents) re moded. Convectve scheme prmeter re the only ones beng estmted n the prmeter estmton eperments. All other settngs re s n the perect model eperment.

42 Prmeter estmton n the presence o other model error. P P3 P In the presence o other sources o model error the estmted prmeters do not converge to the true prmeter vlue. Estmted prmeters shows n ncresed vrblty n tme.

43 Conclusons. Prmeter estmton bsed on dt ssmlton s very promsng tool or objectvely tunng the model. It cn prtlly correct model error. Implementton o prmeter estmton n most stte-o-the-rt dt ssmlton systems s strghtorwrd nd produces lmost no etr computtonl cost. Some ssues: How to estmte the prmeter uncertnty? Ths s very mportnt to hve good results n prmeter estmton. How to del wth prmeters tht re not constnt n tme. Non-lner senstvty o the model to the prmeters. KF ssumes Gussn PDF multmodl or symmetrc PDF tht results rom non-lner senstvtes re not well represented n ths rmewor. (Posselt nd Bshop ). Prmeter estmton usully ncreses the non-lnerty o the model. Physcl nterpretton o the estmted prmeters n the presence o other sources o model error. Cn ths eect be qunted? Some pplctons wll suer rom computtonl ssues. Clmte model tunng my be too epensve. (Tune the clmte model s short rnge numercl wether predcton model?).

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45 Impct upon the nlyss Imperect model wth perect convectve scheme prmeters Anlyss RMSE Perect model Imperect model wth prmeter estmton Estmted prmeters produced lower nlyss RMSE thn the true convectve scheme prmeters. Prmeter estmton cn mprove the nlyss even n the presence o other sources o model error.

46 Combne prmeter estmton wth dptve nlton nd ddtve nlton. Addtve nlton (ppled to the stte vrbles only): Rndom smples o true model error re used s perturbtons or the ddtve nlton pproch. Ths cn compenste or low ensemble spred ssocted wth model error nd cn mody the structure o the ntl condtons perturbtons. (L et l 9) Adptve multplctve nlton (ppled to the stte vrbles only): Adptve multplctve nlton uses the observtons to nd the optml nlton level (L et l 9 Myosh ). Ths cn compenste low ensemble spred due to error sources not consdered n the ensemble ormulton. Kng 9 showed tht multplctve nlton ppled to the stte vrbles hs posstve mpct upon the estmton o the prmeters.

47 Impct upon the nlyss Imperect model wth perect convectve scheme prmeters Anlyss RMSE Addtve + dptve nlton (no prmeter estmton) Perect model Addtve + dptve nlton (wth prmeter estmton) Addtve nd dptve nlton produces lrger mpct thn prmeter estmton lone. Combnng prmeter estmton wth these technques produces n smll mprovement o the nlyss.

48 Impct upon the short to medum rnge orecst: Imperect model. Imperect model Estmted prmeters Addtve + dptve nlton Addtve + dptve nlton + prmeter estmton Eventhough the estmted prmeters do not converge to the true prmeters they produce n mprovement o the orecst sll.