Data Assimilation. Alan O Neill University of Reading, UK

Transcription

1 Data Assimilatio Ala O Neill Uiversity of Readig, UK

2 he Kalma Filter

3 Kalma Filter (expesive Use model equatios to propagate B forward i time. B B(t Aalysis step as i OI

4 Evolutio of Covariace Matrices ( ( where where ( ( ( ( ( he forecast error covariace is : : Subtract ( vectors ad the epsilos are the taget liear model, is the o - liear model, is where ( ( > =< > =< + = > =< + = = = + = = a a a m m a b b m a b b m a a b t t t M M M M ε ε ε ε ε ε ε ε ε ε ε P Q Q P M M B M x x x x M M x x x H R H B P + = a

5 x 2 he Kalma Filter t x

6 Exteded Kalma Filter Allows for the model to be o-liear ad imperfect ad for the observatio operator to oliear. Reduces to the stadard KF whe liearity holds (ad looks like it algorithmically. he EKF liearises locally i time about the oliearly evolvig state estimate. Very expesive to implemet because of the very large dimesio of the state space (~ for NWP models.

7 Esemble Kalma Filter Carry forecast error covariace matrix forward i time by usig esembles of forecasts: B + K K k ( x + k < + > ( x + k < + > Oly ~ 0 + forecasts eeded. Does ot require computatio of taget liear model ad its adjoit. Does ot require liearizatio of evolutio of forecast errors. Fits i eatly ito esemble forecastig. x x

8 he Esemble Kalma Filter same time x rajectories ot idepedet, but are coupled through K. Same weight give to each esemble member, so desity of members ear a state gives estimate of probability of state. t observatio vector perturbed slightly for each esemble member

9 Particle Filter: movig to oliear, o-gaussia methods Idea i particle filterig is to try to represet the probability desity fuctio of the state of a model by a umber of radom draws, called particles. Use Bayes s theorem to update the weight (or probability assiged to a give particle usig observatios. Has some similarity to Esemble Kalma Filter, but there are sigificat differeces.

10 Particle Filter i outlie: dealig with oliear systems ad allowig for o-gaussia statistics Ru a esemble of trajectories with a model. Give each trajectory a weight to represet the probability that the actual state correspods to the trajectory at a give time. Use observatios to update the weights by usig Bayes s theorem. I simple implemetatio, positio of trajectory ot altered by observatios, just its weight (cf. Kalma filter where positio of trajectory is altered by observatios, but weights remai equal for all trajectories.

11 Particle Filter weight or probability time t+ x weight or probabilit y time t x resample usig a proposal pdf to avoid problem of degeeracy, where oly a few particles carry most of the weight

12 Particle Filter p b t (x t w i δ(x x t i i Evolve the x i t to time t + with the model to get p b t + (x keepig the w i the same (add i some oise to represet model error. Use Bayes's theorem & observtios at time t + to update the weights p a t + (x y = t t w i p(y x + t i δ(x x + i i Expected value of x (or f(x determied by BOH desity of trajectories ear x ad by their weights (cf. Kalma filter where oly desity matters sice weights the same.

13 4d-Variatioal Assimilatio

14 4D Variatioal Data Assimilatio obs. & errors give X(t o, the forecast is determiistic t o t vary X(t o for best fit to data

15 4d-Variatioal Assimilatio J(x(t 0 = 2 N [y i h(x i ] R i [y i h(x i ] i=0 + 2 [x(t 0 x b (t 0 ] B 0 [x(t 0 x b (t 0 ] where x(t i = M 0 i (x(t 0 i.e. the model is treated as a strog costrait Miimize the cost fuctio by fidig the gradiet ( Jacobia with respect to the cotrol variables i J x( t 0 x( t 0

16 4d-VAR commets he 2 d term o the RHS of the cost fuctio measures the distace to the backgroud x b (t 0 at the begiig of the iterval. he term helps joi up the sequece of optimal trajectories foud by miimizig the cost fuctio for the observatios. he aalysis is the the optimal trajectory i state space. Forecasts ca be ru from ay poit o the trajectory, e.g. from the middle.

17 Some Matrix Algebra J = J(x(x 0 he J x 0 = x x 0 J x adjoit of the model M : x 0 x Let J have the followig form : J = 2 z (xaz(x he it ca be show that Combiig these results: J x = z x J x 0 = x x 0 Az z x Az

18 his image caot curretly be displayed. 4d-VAR for Sigle Observatio J(x(x 0 = 2 [y h(x(x 0 ] R [y h(x(x 0 ] By usig results o slide "Some Matrix Algebra": J = L x 0 t H R [y h(x(x 0 ] L 0 t d 0 where L 0 t = x x 0 L 0 t = L t t...l t t 2 L 0 t = M 0 t (x 0 x 0 obs. term oly, adjoit of taget liear model L 0 t = L 0 t L t t 2...L t t backward itegratio i time of LM

19 4d-VAR Procedure x, x b Choose 0 0 for example. Itegrate full (o-liear model forward i time ad calculate d for each observatio. Map d back to t=0 by backward itegratio of LM, ad sum for all observatios to give the gradiet of the cost fuctio. Move dow the gradiet to obtai a better iitial state (ew trajectory hits observatios more closely Repeat util some SOP criterio is met. ote: ot the most efficiet algorithm

20 Commets 4d-VAR ca also be formulated by the method of Lagrage multipliers to treat the model equatios as a costrait. he adjoit equatios that arise i this approach are the same equatios we have derived by usig the chai rule of partial differetial equatios. If model is perfect ad B 0 is correct, 4d-VAR at fial time gives same result as exteded Kalma filter (but the covariace of the aalysis is ot available i 4d- VAR. 4d-VAR aalysis therefore optimal over its time widow, but less expesive tha Kalma filter.

21 Icremetal Form of 4d-VAR he 4d-VAR algorithm preseted earlier is expesive to implemet. It requires repeated forward itegratios with the o-liear (forecast model ad backward itegratios with the LM. Whe the iitial backgroud (first-guess state ad resultig trajectory are accurate, a icremetal method ca be made much cheaper to ru o a computer.

22 Icremetal Form of 4d-VAR he icremetal form of the cost fuctio is defied by J(δx 0 = 2 (δx 0 B 0 (δx 0 where δx 0 = x(t 0 x b (t 0 N + [y 2 i H(x f (t i H i L(t 0,t i δx 0 ] R i [y i H(x f (t i H i L(t 0,t i δx 0 ] i=0 x b (t 0 δx 0 aylor series expasio about firstguess trajectory startig from x b (t 0 x f ( t i Miimizatio ca be doe i lower dimesioal space

23 4D Variatioal Data Assimilatio Advatages cosistet with the goverig eqs. implicit liks betwee variables Disadvatages very expesive model is strog costrait

24 Summary of basic priciples DA is cocered with estimatig the state of a system give: observatios (direct [e.g. i-situ] ad idirect [e.g. remotely sesed], forecast models (to provide a-priori data, give too-few obs, observatio operators (to coect model state with obs. All data have ucertaities, which must be quatified. DA estimates are sesitive to ucertaity characteristics, which are ofte poorly kow. May observatios ad model have systematic as well as radom errors. Should take ito accout all sources of error i the system. DA theory is suited mostly to errors that are Gaussia distributed. Most errors are o-gaussia ad o-liearity is syoymous with o-gaussiaity. DA problems are computatioally expesive ad require itesive developmet effort.

25

26 Leadig methods of solvig the DA problem Method Descriptio Pros Cos A. Data isertio B. Variatioal data assimilatio C. Kalma filterig D. Esemble Kalma filterig Set grid poits to observatio values Miimize a cost fuctio May flavours: 3D, 4D, weak/strog costrait Evaluate KF equatios Approximate KF equatios with esemble of N model rus May flavours. Easy to do. No respect of ucertaity 2. What about observatio voids? 3. Ca t deal with idirect observatios. Respect of data ucertaity 2. Direct ad idirect observatios 3. P f gives smooth ad balaced fields 4. Efficiet 5. Ca deal with (weakly o-liear h. As B., B.2, B.3 2. P f adapts with the state. As B.,B.2, B.4, B.5, C.2 2. h: do ot eed H ad H 3. Have measure of aalysis spread E. Hybrid Cross betwee C/D. As B., B.2, B.3, B.4, B.5, C.2. As D.2. P f is difficult to kow, ofte static ad suboptimal 2. High developmet costs 3. h: eed taget liear, H ad adjoit, H 4. Gaussia pdf. As B.3, B.4 2. Difficult to use with o-liear h 3. Prohibitively expesive for large. As B.4 2. Serious samplig issues whe N << 3. Need esemble iflatio ad localizatio schemes to overcome D.2 F. Particle filter Assig weights to esemble members to represet ay pdf. As. B., B.2 2. Ca deal with o-liear h 3. Ca deal with o-gaussia pdf 4. Have measure of aalysis spread. As D.2 2. Iefficiet members ofte become redudat 3. Need special techiques to overcome F.2

27 Some Useful Refereces Atmospheric Data Aalysis by R. Daley, Cambridge Uiversity Press. Atmospheric Modellig, Data Assimilatio ad Predictability by E. Kalay, C.U.P. he Ocea Iverse Problem by C. Wusch, C.U.P. Iverse Problem heory by A. aratola, Elsevier. Iverse Problems i Atmospheric Costituet rasport by I.G. Etig, C.U.P. Dyamic Data Assimilatio, Lewis et al. C.U.P Data Assimilatio: the esemble KF, G. Evese, Spriger Quatitative Remote Sesig of Lad Surfaces, S Liag, Wiley ECMWF lecture otes:

28 END