Data Assimilation. Alan O Neill National Centre for Earth Observation & University of Reading

Transcription

1 Daa Assimilaion Alan O Neill Naional Cenre for Earh Observaion & Universiy of Reading

2 Conens Moivaion Univariae scalar) daa assimilaion Mulivariae vecor) daa assimilaion Opimal Inerpoleion BLUE) 3d-Variaional Mehod Kalman Filer 4d-Variaional Mehod Applicaions of daa assimilaion in earh sysem science

3 Moivaion

4 Wha is daa assimilaion? Daa assimilaion is he echnique whereby observaional daa are combined wih oupu from a numerical model o produce an opimal esimae of he evolving sae of he sysem. DARC

5 Why We Need Daa Assimilaion range of observaions range of echniques differen errors daa gaps quaniies no measured quaniies linked

6 Preliminary Conceps

7 Wha We Wan o Know ) amos. sae vecor s c ) surface flues model parameers X ) ), s ), c)

8 Wha We Also Wan o Know Errors in models Errors in observaions Wha observaions o make

9 DAA ASSIMILAION SYSEM Daa Cache A Error Saisics model observaions O DAS A Numerical Model F B

10 he Daa Assimilaion Process observaions forecass compare rejec adjus esimaes of sae & parameers errors in obs. & forecass

11 X observaion model rajecory

12 Basic Concep of Daa Assimilaion Informaion is accumulaed in ime ino he model sae and propagaed o all variables.

13 Wha are he benefis of daa assimilaion? Qualiy conrol Combinaion of daa Errors in daa and in model Filling in daa poor regions Designing observaional sysems Mainaining consisency Esimaing unobserved quaniies DARC

14 Some Uses of Daa Assimilaion sae esimaion, inverse modelling) Saellie rerievals Operaional weaher and ocean forecasing Seasonal weaher forecasing Land-surface process Surface-flu esimaion Model parameer esimaion Global climae daases Planning saellie measuremens Evaluaion of models and observaions DARC

15 ypes of Daa Assimilaion Sequenial Non-sequenial 4D-variaional) Inermien Coninuous

16 Sequenial Inermien Assimilaion obs obs obs obs obs obs analysis model analysis model analysis

17 Non-sequenial Coninuous Assimilaion obs obs obs obs obs obs analysis + model

18 Mehods of Daa Assimilaion Opimal inerpolaion or appro. o i) 3D variaional mehod 3DVar) Kalman filer wih approimaions) 4D-variaional 4DVar) DARC

19 Saisical Approach o Daa Assimilaion

20 DARC

21 Daa Assimilaion Made Simple scalar case)

22 Leas Squares Mehod Minimum Variance) he analysis should be unbiased : he observaions linear combinaion of as a Esimae he wo measuremens are uncorrelaed, ) ) + > >< < + > < > < > < > >< < + + a a a a a a # #

23 Leas Squares Mehod Coninued subjec o he consrain ) )) ) ) mean squared error : by minimizing is Esimae + + > + < > + >< < a a a a a a a a a a a # #

24 Leas Squares Mehod Coninued + a + a + a he precision of he analysis is he sum of he precisions of he measuremens. he analysis herefore has higher precision han any single measuremen if he saisics are correc).

25 Maimum Likelihood Esimae Obain or assume probabiliy disribuions for he errors he bes esimae of he sae is chosen o have he greaes probabiliy, or maimum likelihood If errors normally disribued,unbiased and uncorrelaed, hen saes esimaed by minimum variance and maimum likelihood are he same

26 Maimum Likelihood Approach Bayesian Derivaion) Assume we have already made an observaion he background forecas in daa assimilaion). hen he probabiliy disribuion of he ruh for Gaussian error saisics is :, p ) & ep$ ' % ' ) ) # he prior pdf.

27 Maimum Likelihood Coninued Bayes's formula for he poserior pdf given he observaion is : p ) p ) p ) p ) & ep$ ' % ' ) ) # & ep$ ' % ' ) ) # since p ) is normalising facor independen of. Maimize he poserior pdf or ln) o esimae he ruh. We ge he same answer as minimizing he cos funcion. Equivalence holds for muli - dimensional case for Gaussian saisics).

28 Variaional Approach for which he value of is ) ) ) # $ % & ' + J J a ) ) J )

29 Simple Sequenial Assimilaion Le b o + W ) where ) is he innovaion. a b o b o b he opimal weigh W is given by: W + ), and he analysis error variance is: b b o W ) a b

30 Commens he analysis is obained by adding firs guess o he innovaion. Opimal weigh is background error variance muliplied by inverse of oal variance. Precision of analysis is sum of precisions of background and observaion. Error variance of analysis is error variance of background reduced by - opimal weigh).

31 Simple Assimilaion Cycle Observaion used once and hen discarded. Forecas phase o updae Analysis phase o updae Obain background as b i+ ) M[ a i )] and and Obain variance of background as b b a b i b i+ b a ) ) alernaively ) i+ a a i )

32 Simple Kalman Filer Analysis sep as before. M Q i+ ) [ i )] m, < m > model no biased) hen ) M ) M ) + b, i+ b i+ a, i, i m M + where M # M a, i m # Forecas background error covariance is: $ ) ) M $ ) b, i+ < b, i+ > a, i + Q

33 Mulivariae Daa Assimilaion

34 Mulivariae Case # $ $ $ $ $ % & n ) sae vecor # $ $ $ % & y m y y ) n vecor observaio y

35 Sae Vecors sae vecor column mari) rue sae b background sae a analysis, esimae of

36 Ingrediens of Good Esimae of he Sae Vecor analysis Sar from a good firs guess forecas from previous good analysis) Allow for errors in observaions and firs guess give mos weigh o daa you rus) Analysis should be smooh Analysis should respec known physical laws

37 Some Useful Mari Properies ranspose of a produc : AB) B A Inverse of a produc : AB) B A Inverse of a ranspose : A A Posiive definieness for symmeri mari ) ) A :, he scalar A >, unless. his propery is conserved hrough inversion)

38 Observaions Observaions are gahered ino an y observaion vecor, called he observaion vecor. Usually fewer observaions han variables in he model; hey are irregularly spaced; and may be of a differen kind o hose in he model. Inroduce an observaion operaor o map from model sae space o observaion space. H )

39 Errors

40 Variance becomes Covariance Mari Errors in i are ofen correlaed spaial srucure in flow dynamical or chemical relaionships Variance for scalar case becomes Covariance Mari for vecor case COV Diagonal elemens are he variances of i Off-diagonal elemens are covariances beween i and j Observaion of i affecs esimae of j

41 he Error Covariance Mari ) # $ $ $ $ $ $ $ $ % & > < > < > < > < > < > < > < > < > < > < # $ $ $ $ $ $ $ $ % & n n n n n n n n e e e e e e e e e e e e e e e e e e e e e e e e '' ' ' P i i e i e > <

42 Background Errors hey are he esimaion errors of he background sae: b b B average bias) covariance < < b > < > ) < > ) b b >

43 Observaion Errors hey conain errors in he observaion process insrumenal error), errors in he design of H, and represenaiveness errors, i.e. discreizaion errors ha preven from being a perfec represenaion of he rue sae. y H ) < > o o R < < > ) < > ) o o o o >

44 Conrol Variables We may no be able o solve he analysis problem for all componens of he model sae e.g. cloud-relaed variables, or need o reduce resoluion) he work space is hen no he model space bu he sub-space in which we correc b, called conrol-variable space + a b

45 Innovaions and Residuals Key o daa assimilaion is he use of differences beween observaions and he sae vecor of he sysem We call y H ) b he innovaion We call y H a ) he analysis residual Give imporan informaion

46 Analysis Errors hey are he esimaion errors of he analysis sae ha we wan o minimize. a a Covariance mari A

47 Using he Error Covariance Mari Recall ha an error covariance mari for he error in has he form: C C < > y H H If where is a mari, hen y he error covariance for C HCH y is given by:

48 BLUE Esimaor he BLUE esimaor is given by: a K he analysis error covariance mari is: Noe ha: b BH + K y H HBH A I KH) B + b )) R) BH HBH + R) B + H R H) H R

49 Saisical Inerpolaion wih Leas Squares Esimaion Called Bes Linear Unbiased Esimaor BLUE). Simplified versions of his algorihm yield he mos common algorihms used oday in meeorology and oceanography.

50 Assumpions Used in BLUE Linearized observaion operaor: H ) H ) H ) b b B R and are posiive definie. Errors are unbiased: < >< y H ) > b Errors are uncorrelaed: < ) y H )) > b Linear analysis: correcions o background depend linearly on background obs.). Opimal analysis: minimum variance esimae.

51 Opimal Inerpolaion observaion observaion operaor a b + K y H )) b analysis lineariy H H background forecas) mari inverse K BH HBH + R) - limied area

52 y H ) a obs. poin b daa void b

53 Spreading of Informaion from Single Pressure Obs. p q

54 Ozone a hpa, Z 3rd Sep Analysis MIPAS observaions 6 day model forecas

55 3D variaional daa assimilaion - ozone a hpa b

56 3D variaional daa assimilaion - ozone a hpa y h ) b b

57 3D variaional daa assimilaion - ozone a hpa y h ) b b K y h b))

58 he daa assimilaion cycle: ozone a hpa y h ) b b + K y h )) K y h )) b b b

59 Esimaing Error Saisics Error variances reflec our uncerainy in he observaions or background. Ofen assume hey are saionary in ime and uniform over a region of space. Can esimae by observaional mehod or as forecas differences NMC mehod). More advanced, flow dependen errors esimaed by Kalman filer.

60 Esimaing Covariance Mari for Observaions, O O usually quie simple: diagonal or for nadir-sounding saellies, non-zero values beween poins in verical only Calibraion agains independen measuremens

61 Esimaing he Error Covariance Mari B Model B wih simple funcions based on comparisons of forecass wih observaions: B # # ep d L) horiz. fn ver. fn ij i j ij Error growh in shor-range forecass verifying a he same ime NMC mehod) B < [ 48h) 4h)][ 48h) 4h)] f f f f > sae vecor a ime from forecas 48h or 4 h earlier

62 3d-Variaional Daa Assimilaion

63 Variaional Daa Assimilaion J ) vary o minimise J ) a

64 Equivalen Variaional Opimizaion Problem BLUE analysis can be obained by minimizing a cos penaly, performance) funcion: J ) ) B ) + y H )) R y J ) a J min b ) + J b J o ) b H )) a he analysis is opimal closes in leassquares sense o ). If he background a and observaion errors are Gaussian, hen is also he maimum likelihood esimaor.

65 Remarks on 3d-VAR Can add consrains o he cos funcion, e.g. o help mainain balance Can work wih non-linear observaion operaor H. Can assimilae radiances direcly simpler observaional errors). Can perform global analysis insead of OI approach of radius of influence.

66 Variaional Daa Assimilaion J ) b ) B b ) + y H )) R y H )) nonlinear operaor assimilae y direcly global

67 Maimum Probabiliy or Likelihood For Gaussian errors he background, observaion and analysis pdfs are: P B b b b o ) b ep[ ) )] a b o P ) o ep[ y H )) R y H ))] P ) a P ) P ) where b, o, and a are normalizing facors. Maimum probabiliy esimae minimizes J ) ln P ) a

68 Commens Biases occur in background and observaions. Remove hem if known, oherwise analysis is sub-opimal. Monior O-B), bu is he bias in he model or in observaions? B and O errors usually uncorrelaed, bu could be correlaions in saellie rerievals. Error in he linearizaion of H should be much smaller han observaional errors for all values of me in he analysis procedure. b

69 Choice of Sae Variables and Precondiioning Free o choose which variables o use o define sae vecor, ) We d like o make B diagonal may no know covariances very well wan o make he minimizaion of J more efficien by precondiioning : ransforming variables o make surfaces of consan J nearly spherical in sae space

70 Cos Funcion for Correlaed Errors

71 Cos Funcion for Uncorrelaed Errors

72 Cos Funcion for Uncorrelaed Errors Scaled Variables

73 he Kalman Filer

74 Kalman Filer epensive) Use model equaions o propagae B forward in ime. B B) Analysis sep as in OI

75 Evoluion of Covariance Marices adjoin. is is he angen linear model; is where ) ) ) ) ) ) he forecas error covariance is : : Subrac ) he non - linear model is where ) M M Q Q P M M P B M > < + > < m m n n a n n b n b n f m n a n b m n n n a n b M M M H R H B P + a

76 he Kalman Filer

77 Remarks In OI and 3d-VAR) isolaed observaion given more weigh han observaions close ogeher forecas errors have large correlaions a nearby observaion poins). When several observaions are close ogeher calculaion of weighs may be illposed. herefore combine ino a super observaion.

78 Eended Kalman Filer Assumes he model is non-linear and imperfec. he angen linear model depends on he sae and on ime. Could be a gold sandard for daa assimilaion, bu very epensive o implemen because of he very large dimension of he sae space ~ 6 7 for NWP models).

79 Ensemble Kalman Filer Carry forecas error covariance mari forward in ime by using ensembles of forecass: P f K # K k $ f k < > ) < > ) Only ~ + forecass needed. Does no require compuaion of angen linear model and is adjoin. Does no require linearizaion of evoluion of forecas errors. Fis in nealy ino ensemble forecasing. f f k f

80 4d-Variaional Assimilaion

81 4D Variaional Daa Assimilaion obs. & errors given X o ), he forecas is deerminisic o vary X o ) for bes fi o daa

82 4d-VAR For Single Observaion a ime J, )) ~ where H ~ ) y

83 4d-Variaional Assimilaion as a srong consrain reaed is he model i.e. )) ) where )] ) [ )] ) [ )] [ )] [ )) M H H J i i b b i i i i N i i B y R y + # Minimize he cos funcion by finding he gradien Jacobian ) wih respec o he conrol variables in ) J )

84 4d-VAR Coninued he nd erm on he RHS of he cos funcion measures he disance o he background a he beginning of he inerval. he erm helps join up he sequence of opimal rajecories found by minimizing he cos funcion for he observaions. he analysis is hen he opimal rajecory in sae space. Forecass can be run from any poin on he rajecory, e.g. from he middle.

85 a M : Some Mari Algebra Az z Az z Az z Combining hese resuls : hen i can be shown ha ) ) have he following form : Le hen )) # $ % & ' ' # $ $ % & ' ' ' ' # $ % & ' ' ' ' ' ' # $ $ % & ' ' ' ' J J J J J J J J adjoin of he model

86 4d-VAR for Single Observaion LM ime of backward inegraion in linear model angen adjoin of, ) where ))] [ By using resuls on slide Some Mari Algebra: ))] [ ))] [ )) # # $ $ % & ' ' ) # # + * * * # # * *,,,,,,,,,,, *, * * * n n M H J H H J L L L L L L L L L d L y R H L y R y obs. erm only

87 4d-VAR Procedure Choose, b for eample. Inegrae full non-linear) model forward in ime and calculae d for each observaion. Map d back o by backward inegraion of LM, and sum for all observaions o give he gradien of he cos funcion. Move down he gradien o obain a beer iniial sae new rajecory his observaions more closely) Repea unil some SOP crierion is me. noe: no he mos efficien algorihm

88 Commens 4d-VAR can also be formulaed by he mehod of Lagrange mulipliers o rea he model equaions as a consrain. he adjoin equaions ha arise in his approach are he same equaions we have derived by using he chain rule of parial differenial equaions. If model is perfec and B is correc, 4d-VAR a final ime gives same resul as eended Kalman filer bu he covariance of he analysis is no available in 4d-VAR). 4d-VAR analysis herefore opimal over is ime window, bu less epensive han Kalman filer.

89 Incremenal Form of 4d-VAR he 4d-VAR algorihm presened earlier is epensive o implemen. I requires repeaed forward inegraions wih he non-linear forecas) model and backward inegraions wih he LM. When he iniial background firs-guess) sae and resuling rajecory are accurae, an incremenal mehod can be made much cheaper o run on a compuer.

90 Incremenal Form of 4d-VAR ] ), )) [ ] ), )) [ ) ) ) he cos funcion is defined by form of he incremenal H L y H L y B i i i f i i N i i i f i H H J + # aylor series epansion abou firs-guess rajecory ) i f Minimizaion can be done in lower dimensional space ) ) where b

91 4D Variaional Daa Assimilaion Advanages consisen wih he governing eqs. implici links beween variables Disadvanages very epensive model is srong consrain

92 Some Useful References Amospheric Daa Analysis by R. Daley, Cambridge Universiy Press. Amospheric Modelling, Daa Assimilaion and Predicabiliy by E. Kalnay, C.U.P. he Ocean Inverse Problem by C. Wunsch, C.U.P. Inverse Problem heory by A. aranola, Elsevier. Inverse Problems in Amospheric Consiuen ranspor by I.G. Ening, C.U.P. ECMWF Lecure Noes a

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