An application of nonlinear optimization method to. sensitivity analysis of numerical model *

Transcription

1 An applicain f nnlinear pimizain mehd sensiiviy analysis f numerical mdel XU Hui 1, MU Mu 1 and LUO Dehai 2 (1. LASG, Insiue f Amspheric Physics, Chinese Academy f Sciences, Beijing 129, China; 2. Deparmen f Amsphere Sciences, Ocean Universiy f China, Qingda 2663, Chin Received Sepember 2, 23; Revised Absrac A nnlinear pimizain mehd is applied sensiiviy analysis f a numerical mdel. hereical analysis and numerical experimens indicae ha his mehd can give n nly a quaniaive assessmen wheher he numerical mdel is able simulae he bservains, bu als he iniial field ha yields he pimal simulain. In paricular, when he simulain resuls are apparenly saisfacry, and smeimes bh mdel errr and iniial errr are cnsiderably large, he nnlinear pimizain mehd, under sme cndiins, can idenify he errr ha plays a dminan rle. Keywrds: sensiiviy analysis, nnlinear pimizain, adjin mehd, numerical mdel. Suppred by he Key Innvain Direcin Prjec f he Chinese Academy f Sciences (Gran N. KZCX2-28) and he Nainal Naural Science Fundain f China (Gran Ns and 4515). whm crrespndence shuld be addressed. mumu@lasg.iap.ac.cn 1

2 he accuracy f numerical weaher predicin decreases as he frecas ime increases due iniial errrs and mdel errrs. reduce hese errrs, meerlgiss have perfrmed numerus sensiiviy analysis using he numerical simulain, he adjin mehd and he linear singular vecrs (LSVS) [1~6], ec. When applied in pracice, hese mehds appear mre r less unsaisfacry. Fr example, he numerical simulain mehd cann give all pssible significan iniial fields r cmbinains f physical prcesses and parameers []. he adjin mehd used by previus auhrs and he LSVS are based n he linear hery, and nly describe he develpmen f small perurbains during he validiy perid f he angen linear mdel [5]. I is suggesed by Mu e al. [8] ha he nnlinear pimizain mehd can be applied sensiiviy analysis f a numerical mdel. Hwever, Mu e al. nly briefly described he mehd as ne f nnlinear pimizain prblems in amspheric and ceanic science, and did n verify he resuls frm he numerical experimens. his paper, based n Mu e al. [8], inrduces he nnlinear pimizain mehd fr sensiiviy analysis f numerical mdel in deail. Besides, i furher invesigaes hw idenify he errr ype ha plays a dminan rle in he predicin resuls, especially when he simulain resuls are apparenly saisfacry, and bh he mdel errr and iniial errr are nably large. As an example, wih he w-dimensinal quasi-gesrphic equain, we 2

3 perfrm a series f numerical experimens verify he hereical resuls. 1 Mehd Given he mdel = ( ), [, ], he bservain a ime and a ime M, we wan find he iniial field ha yields he pimal simulain fr. he prblem nw becmes an pimizain prblem: find he pimal such ha he cs funcin 1 J ( ) = ( M ( ) ) W ( M ( ) ) (1) 2 is minimum. W is he marix f weighing cefficiens marix which is he inverse f he cvariance marix f he bservainal errr. In rder minimize J ), we need he ( infrmain n he gradien f J ( ) wih respec. Because he number f cnrl 6 variables in a ypical meerlgical mdel is n he rder f 1, he adjin mehd is inrduced cmpue he gradien [9]. Using variainal principle fr (1), here is δj ) < W ( ( ) ), δ, (2) ( = M > where δ is he develpmen a ime f he iniial perurbain δ. Inrducing he angen linear perar M f M and subsiuing i in (2), we have δj ) = < W ( ( ) ),M ( ) δ. (3) ( M > Mrever, inrducing he adjin prpagar M f M we bain δj ) = < M ( )( W ( ( ) )), δ. (4) ( M > 3

4 hus, he gradien f J ( ) wih respec is equal J ) = M ( )( W ( ( ) )). (5) ( M hen we can minimize J and find he pimal iniial field wih he pimizain algrihm. 2 hereical analysis Le E = min J ( ). Fr a given errr bund ε, here are w cases fr E : E > ε, < E ε. When esing a mdel, E > ε means ha even if we ge he pimal iniial field, he mdel is n able simulae he bservain Namely, n maer hw we adjus, a saisfacry simulain fr prperly in he given errr bund ε. cann be bained. hen we can cnclude ha he mdel errr is cnsiderably large s ha he mdel needs be imprved. hen we cnsider he case < E ε. Nw he numerical sluin = M ) and he ( bservain have n apparen difference, which indicaes ha a saisfacry simulain can be bained by adjusing he iniial field. I shuld be pined u ha, in his case, he mdel errrs culd be large, which will be discussed laer. Wih a given nrm, defining a maximum allwable iniial errr ε, we have hree cases nw: 4

5 << ε, ~ ε, >> ε. ( (b) (c) When he mdel errr is small, he mdel can simulae he mvemen f amsphere very well. We can esimae he bservain based n (, (b) and (c). In (, a saisfacry simulain fr he bservain can be bained frm he exising bservain direcly. We d n need rea he iniial field f he mdel paricularly, and sme rdinary inerplain is enugh. In (b), a saisfacry simulain fr frm cann be bained direcly. Bu if we imprve he iniial field f he mdel (fr example, by assimilain mehd), a saisfacry simulain can be bained. In (c), he exising bservain lacks enugh infrmain, and cann represen he real weaher and climae prcesses. If we wan bain a saisfacry simulain fr, we shuld inensify he bservainal newrk ge mre deailed bservain han he exising ne. In case ha he bservainal errr is small, he bservain is clse he real develpmen f amsphere. We can evaluae he mdel errr based n (, (b) and (c). In (, he mdel errr is small, and a saisfacry simulain fr he bservain is easily bained by adjusing he iniial field. In (b), here are cerain mdel errrs, bu a saisfacry simulain can als be bained by adjusing in he allwable errr bund. I is in fac ha an inaccurae mdel plus an inaccurae iniial field prduces a saisfacry simulain. In 5

6 (c), he difference beween and is large, is clse he real sae, s has n physical significance. We can cnclude ha he mdel errr is large, a saisfacry simulain fr is illusive, and mre wrk shuld be dne imprve he numerical mdel. In pracice, i is cmmn evaluae he mdel errr by cmparing he numerical simulain wih a relaively accurae bservain, r assess he bservain by cmparing i wih a relaively accurae numerical simulain. In hese w cases, we can bain sme significan cnclusins by he abve analyical mehd. Hwever, due sme bjecive reasns, when he mdel and bservain are bh cnsiderably inaccurae, he applicabiliy f he nnlinear pimizain mehd sensiiviy analysis is limied. In his case, when min J > ε, as menined abve, he mdel errr is cnsiderably large, and he mdel needs be imprved. When < min J ε, a saisfacry simulain can be bained by adjusing he iniial field, bu bh mdel errr and bservainal errr may be large. Le ε be he bservainal precisin ha is usually knwn. If >> ε appears, he pimal iniial field is far frm he real sae f amsphere, and has n physical significance, which implies ha he mdel errr is large. In her cases, we cann bain sme significan cnclusins by he nnlinear pimizain mehd. In summary, we can ge sme insrucive cnclusins by he nnlinear pimizain mehd fr sensiiviy analysis f a numerical mdel. In he nex secin, we will cninue discuss i 6

7 wih numerical experimens. 3 Numerical experimen he nndimensinal gverning equain is P (, ), + P = 2 P = F + f =, = + fh, (6) where P is he penial vriciy, he sream funcin, F he Planeary Frude number, f he Crilis parameer, H he characerisic verical deph f he barrpic amsphere and h he pgraphy. he Jacbian perar is (, P) = P P. x y y x Eq. (6) is slved under a duble peridical bundary cndiin. Arakawa finie difference scheme is used discreize he Jacbian perar. he empral discreizain is Adamas-Bashfrh scheme. he familiar five-pin difference scheme is emplyed discreize he Laplacian perar. We ake he space dmain [, 6.4] [, 3.2], he space sep d =.2 and he ime sep =. 18, crrespnding he dimensinal case [, 64km] [, 32 km], 2 km and 3 minues, respecively. he parameers are F =. 12, f = 1., and H = 1.. he pgraphy is h ( y) = h.112 (sin(4πy 3.2) + 1.) where he mdel errr is inrduced by changing h, and h = 1. denes he mdel is accurae. he bservain is bained by adding sme nrmal randm perurbains he rue value bained by inegraing he accurae mdel wih ru ( x, y) = sin(2π / 3.2) ru y

8 ( ru +.1 sin( 2πx 6.4) ha is, = + a where is a nrmal randm (.) perurbain, a a cefficien dening he errr and equal ru. he cs funcin is as (1) where W is diagnal and nrmalized. Euclidian nrm is emplyed. he pimizain algrihm adps he limied memry Bryden-Flecher- Gldfarb-Shanna mehd (he limied memry BFGS mehd) [1]. Numerical experimens include w pars: assess he bservain when he mdel errr is small and evaluae he mdel errr when he bservainal errr is small. 3.1 When he mdel errr is small When h =1. and here is n mdel errr,le a =. 5, 1. r 2.. hen we have (.5) bained hree differen bservains, (1.) and (2.) fr 3-day, 5-day and -day numerical experimens. he -day resuls are shwn in able 1, where ( mdel frecas f he iniial bservain. able 1. he -day resuls wihu mdel errr ( ε = 3.1, ε = 1. 6 ) ( = E min J f ( ( (%) 5 a = a = a = f is he -day ( Here, as a =. 5,here are min J < ε and << ε fr -day run. he ( iniial bservain can be used as he iniial value f he mdel direcly yield he ( saisfacry simulain fr he bservain. As a = 1., here are min J < ε and 8

9 (. he saisfacry simulain fr cann be bained direcly frm ( ~ ε. Bu if we imprve he iniial field f he mdel by an assimilain mehd, a ( saisfacry -day simulain can be bained. As a = 2., here are min J < ε and >> ( ε. Because he exising bservain lacks enugh infrmain, we shuld ( imprve he exising bservainal daa bain a saisfacry simulain fr. When h =. 99 and he mdel has sme small errrs, he numerical resuls are he same as when h =1. and he mdel has n mdel errr. ables are negleced. 3.2 When he bservainal errr is small In he case wihu he bservainal errr, namely, when a =. and he bservain is, we inrduce hree mdel errrs h =. 99, h = 1. 3 and h = 1. 8 respecively fr (.) 3-day, 5-day and -day numerical experimens. he -day resuls are shwn in able 2. able 2. he -day resuls wihu bservainal errr ( ε = 3.1, ε =. 9 ) (.) = E min J f (.) (.) (%) 2 h =.99 4 h = h = Here, when h =. 99, here are min J < ε and (.) << ε. he mdel errr is very ( small, and he saisfacry simulain fr he bservain can be easily bained by adjusing he iniial field. When h =1. 3, here are min J < ε and (.) ~ ε. 9

10 here exis cerain mdel errrs, bu he saisfacry simulain can als be bained by adjusing he iniial field in he allwable errr bund. When h = 1. 8, here are min J < ε ( and >> ε. he pimal iniial field has n physical significance, he mdel errr is large and he saisfacry simulain is illusive. (.5) When he bservain has small errr (fr he bservain ), he numerical resuls (.) are he same as when he bservain has n errr (fr he bservain ). ables are negleced. 4 Cnclusin and discussin Wih he w-dimensinal quasi-gesrphic mdel, we explre he applicain f he nnlinear pimizain mehd sensiiviy analysis f a numerical mdel. he resuls sugges ha by his mehd he abiliy f a mdel simulaing he bservain can be assessed quaniaively, and even he iniial field ha yields he pimal simulain can als be fund u, which herefre becme he main advanages f he nnlinear pimizain mehd cmpared her sensiiviy mehds. Besides, fr he siuain ha he mdeling is apparenly saisfacry, we als explre he mdel errr and iniial errr f he numerical mdel. I is shwn ha, in his case, he mdel errr and iniial errr may be large. Hwever, here are evidences ha nnlinear pimizain mehd, under sme cndiins, can idenify he errr ha has a dminan cnribuin he uncerainies f he simulain resuls. Alhugh he w-dimensinal quasi-gesrphic mdel is simple, he hereical analysis f 1

11 he numerical resuls is bained lgically. herefre, i is reasnable reckn ha fr he cmplex mdels, nnlinear pimizain mehd is f imprance in he invesigain f sensiiviy analysis f a numerical mdel. References 1 Hall, M. C. G. e al. Sensiiviy analysis f a radiaive-cnvecive mdel by he adjin mehd. J. Ams. Sci., 1982, 39: Hall, M. C. G. Applicain f adjin sensiiviy hery an amspheric general circulain mdel. J. Ams. Sci., 1986, 43: Erric, R. M. e al. Sensiiviy analysis using an adjn f he PSU- NCAR messcale mdel. Mn. Wea. Rev., l992, 12: l Rabier, F. e al. An applicain f adjin mdels sensiiviy analysis. Beir. Phys. Amsph., 1992, 65: 1. 5 Rabier, F. e al. Sensiiviy f frecas errrs iniial cndiins. Quar. J. Ry. Meer. Sc., 1996, 122: Gelar, R. e al. Sensiiviy analysis f frecas errrs and he cnsrucin f pimal perurbains using singular vecrs. J. Ams. Sci., 1998, 15: 112. Zu, X. e al. Incmplee bservains and cnrl f graviy waves in variainal daa assimilain. ellus, 1992, 44A: Mu, M. e al. Nnlinear pimizain prblems in amspheric and ceanic sciences, Eas-Wes Jurnal f Mahemaics, 22, Special Vlume: alagrand, O. e al. Variainal assimilain f meerlgical bservain wih he adjin 11

12 vriciy equain, I: hery. Quar. J. Ry. Meer. Sc., 198, 113: Dng, C. L. e al. On he limied memry BFGS mehd fr large scale pimizain. Mahemaical Prgramming, 1989, 45: