Data Assimilation. Alan O Neill Data Assimilation Research Centre University of Reading

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1 Dt Assimiltion Aln O Neill Dt Assimiltion Reserch Centre University of Reding

2 Contents Motivtion Univrite sclr dt ssimiltion Multivrite vector dt ssimiltion Optiml Interpoltion BLUE 3d-Vritionl Method Klmn Filter 4d-Vritionl Method Applictions of dt ssimiltion in erth system science

3 Motivtion

4 DARC Wht is dt ssimiltion? Dt ssimiltion is the technique whereby observtionl dt re combined with output from numericl model to produce n optiml estimte of the evolving stte of the system

5 Why We Need Dt Assimiltion rnge of observtions rnge of techniques different errors dt gps quntities not mesured quntities linked

6 DARC

7 Some Uses of Dt Assimiltion Opertionl wether nd ocen forecsting Sesonl wether forecsting Lnd-surfce process Globl climte dtsets Plnning stellite mesurements Evlution of models nd observtions DARC

8 Preliminry Concepts

9 Wht We Wnt o Know x t tmos stte vector s t c surfce fluxes model prmeters X t x t, s t, c

10 Wht We Also Wnt o Know Errors in models Errors in observtions Wht observtions to mke

11 DAA ASSIMILAION SYSEM Dt Cche A Error Sttistics model observtions O DAS A Numericl Model F B

12 he Dt Assimiltion Process observtions forecsts compre reject djust estimtes of stte & prmeters

13 X observtion model trjectory t

14 DARC Dt Assimiltion: n nlogy Driving with your eyes closed: open eyes every 0 seconds nd correct trjectory

15 Bsic Concept of Dt Assimiltion Informtion is ccumulted in time into the model stte nd propgted to ll vribles

16 DARC Wht re the benefits of dt ssimiltion? Qulity control Combintion of dt Errors in dt nd in model Filling in dt poor regions Designing observtionl systems Mintining consistency Estimting unobserved quntities

17 DARC Methods of Dt Assimiltion Optiml interpoltion or pprox to it 3D vritionl method 3DVr 4D vritionl method 4DVr Klmn filter with pproximtions

18 ypes of Dt Assimiltion Sequentil Non-sequentil Intermittent Continous

19 Sequentil Intermittent Assimiltion obs obs obs obs obs obs nlysis model nlysis model nlysis

20 Sequentil Continuous Assimiltion obs obs obs obs obs

21 Non-sequentil Intermittent Assimiltion obs obs obs obs obs obs nlysis + model nlysis + model nlysis + model

22 Non-sequentil Continuous Assimiltion obs obs obs obs obs obs nlysis + model

23 Sttisticl Approch to Dt Assimiltion

24 DARC

25 Dt Assimiltion Mde Simple sclr cse

26 Lest Squres Method Minimum Vrince re two mesurements the 0, 0 > < > < > < > >< < + + ε ε σ ε σ ε ε ε ε ε t t

27 Estimte of the observtio ns : t s liner combintio n + he nlysis should be unbised : < > t +

28 Lest Squres Method Continued subject to the constrint : men squred error by minimizing its Estimte + + > + < > + >< < t t t σ σ ε ε σ

29 Lest Squres Method Continued σ σ σ + σ σ σ + σ σ σ + he precision of the nlysis is the sum of the precisions of the mesurements he nlysis therefore hs higher precision thn ny single mesurement if the sttistics re correct

30 Vritionl Approch 0 for which the vlue of is + J J σ σ J

31 Mximum Likelihood Estimte Obtin or ssume probbility distributions for the errors he best estimte of the stte is chosen to hve the gretest probbility, or mximum likelihood If errors normlly distributed,unbised nd uncorrelted, then sttes estimted by minimum vrince nd mximum likelihood re the sme

32 Mximum Likelihood Approch Bysin Derivtion A s s u m e w e hve lredy mde n observtio n the bckground forecst in dt ssimiltion hen the probbilit y distributi on of the truth for Gussin error sttistics is :, p exp σ the prior pdf

33 Mximum Likelihood Continued Byes' s formul for the posterior p d f given the observtio n i s : p p p p exp σ exp σ since p Mximize We get the i s the normlisin g fctor independen t sme posterior nswer p d f minimizing the cost Equivlenc e holds for multi - dimension l cse s o r ln to estimte o f the function for truth Gussin sttistics

34 Simple Sequentil Assimiltion error vrince i s : nlysis the nd, by : given i s weight optiml he the " innovtion " i s where Let b o b b b o b o b o b W W W W σ σ σ σ σ + +

35 Comments he nlysis is obtined by dding first guess to the innovtion Optiml weight is bckground error vrince multiplied by inverse of totl vrince Precision of nlysis is sum of precisions of bckground nd observtion Error vrince of nlysis is error vrince of bckground reduced by - optiml weight

36 Simple Assimiltion Cycle Observtion used once nd then discrded Forecst phse to updte Anlysis phse to updte Obtin bckground s b t i+ M [ t i ] nd nd Obtin vrince of bckground s b b b i b i+ σ b σ σ t σ t lterntively σ t i+ σ t i

37 Simple Klmn Filter,,,,,,, M covrince i s : e r r o r bckground Forecst where M M hen bised! not model, ] [ before step s Anlysis Q M M M Q t M t i i b i b m i m i t i i t b i b m m i t i t + > < + + > < σ ε σ ε ε ε ε ε ε

38 Multivrite Dt Assimiltion

39 Multivrite Cse x n x x t x stte vector y m y y t n vector observtio y

40 Stte Vectors x stte vector column mtrix x t true stte x b bckground stte x nlysis, estimte of x t

41 Ingredients of Good Estimte of the Stte Vector nlysis Strt from good first guess forecst from previous good nlysis Allow for errors in observtions nd first guess give most weight to dt you trust Anlysis should be smooth Anlysis should respect known physicl lws

42 Some Useful Mtrix Properties rnspose of product : A B B A Inverse of product : A B B A Inverse of trnspose : A A Positive definitene s s for symmetrix mtrix A : x, the sclr x A x > 0, unless x 0 this property is conserved through inversion

43 Observtions Observtions re gthered into n y observtion vector, clled the observtion vector Usully fewer observtions thn vribles in the model; they re irregulrly spced; nd my be of different kind to those in the model Introduce n observtion opertor to mp from model stte spce to observtion spce x H x

44 Errors

45 Vrince becomes Covrince Mtrix Errors in x i re often correlted sptil structure in flow dynmicl or chemicl reltionships Vrince for sclr cse becomes Covrince Mtrix for vector cse COV Digonl elements re the vrinces of x i Off-digonl elements re covrinces between x i nd x j Observtion of x i ffects estimte of x j

46 he Error Covrince Mtrix > < > < > < > < > < > < > < > < > < > < 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 εε ε ε P i i e i e σ > <

47 Bckground Errors hey re the estimtion errors of the bckground stte: ε b x b x t B verge bis covrince < < ε b > ε < ε > ε < ε > b b >

48 Observtion Errors hey contin errors in the observtion process instrumentl error, errors in the design of H, nd representtiveness errors, ie discretizton errors tht prevent x t from being perfect representtion of the true stte ε y H x < ε > o t o R < ε < ε > ε < ε > o o o o >

49 Control Vribles We my not be ble to solve the nlysis problem for ll components of the model stte eg cloud-relted vribles, or need to reduce resolution he work spce is then not the model spce but the sub-spce in which we correct x b, clled control-vrible spce x x b + δx

50 Innovtions nd Residuls Key to dt ssimiltion is the use of differences between observtions nd the stte vector of the system We cll y H x b the innovtion We cll y H x the nlysis residul Give importnt informtion

51 Anlysis Errors hey re the estimtion errors of the nlysis stte tht we wnt to minimize ε x x t Covrince mtrix A

52 Using the Error Covrince Mtrix Recll tht n error covrince mtrix x for the error in hs the form: C C < εε > y H x H If where is mtrix, then the error covrince for y C HCH y is given by:

53 BLUE Estimtor he BLUE estimtor is given by: he nlysis error covrince mtrix is: Note tht: b b + + R H B H B H K x y K x x H K H B I A + + R H H R H B R H B H B H

54 Sttisticl Interpoltion with Lest Squres Estimtion Clled Best Liner Unbised Estimtor BLUE Simplified versions of this lgorithm yield the most common lgorithms used tody in meteorology nd ocenogrphy

55 Assumptions Used in BLUE Linerized observtion opertor: H x H x b H x x b B R nd re positive definite Errors re unbised: < x x >< y H x > b t t Errors re uncorrelted: < x x y H x > b t t Liner nlysis: corrections to bckground depend linerly on bckground obs Optiml nlysis: minimum vrince estimte 0 0

56 Optiml Interpoltion observtion observtion opertor x x b + K y H x b nlysis linerity H H bckground forecst mtrix inverse K B H H B H + R limited re

57 y H x b t obs point dt void x b

58 END

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