Parametrizations in Data Assimilation

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1 Parametrizatins in Data Assimilatin ECMWF raining Curse 0-4 Mar 04 Philippe Lpez Physical Aspects Sectin, Resear Department, ECMWF (Rm 00)

2 Parametrizatins in Data Assimilatin Intrductin Why are physical parametrizatins needed in data assimilatin? angent-linear and adjint cding Issues related t physical parametrizatins in assimilatin Physical parametrizatins in ECMWF s current 4D-Var system Examples f applicatins Summary and cnclusins

3 Data assimilatin Observatins with errrs a priri infrmatin frm mdel = backgrund state with errrs Data assimilatin system (e.g. 4D-Var) Analysis NWP mdel Frecast

4 mdel state 4D-Var initial time t 0 analysis time t a y All bservatins y between t a -9h and t a +3h are valid at their actual time Mdel trajectry frm crrected initial state x a x b Mdel trajectry frm first guess x b time assimilatin windw 4D-Var prduces the analysis (x a ) that minimizes the distance t a set f available bservatins (y ) under the cnstraint f sme a priri backgrund infrmatin frm the mdel (x b ) and given the respective errrs f bservatins and mdel backgrund.

5 = δx 0 B δx 0 + 4D-Var Incremental 4D-Var minimizes the fllwing cst functin: n i i 0 i i i i 0 i= ( H M δx d ) R ( H M δx d ) where: i = time index (inside 4D-Var windw, typ. h). δx 0 = x 0 x b 0 (increment). H i = tangent-linear f bservatin peratr. M i = tangent-linear f frecast mdel (t 0 t i ). d i = y i H i (M i [x b 0]) (innvatin vectr). B = backgrund errr cvariance matrix. R i = bservatin errr cvariance matrix. i x 0 = n B δx0 + Mi [ ti,t0] Hi R i i i 0 i= ( H M δx d ) i Adjint f frecast mdel with simplified linearized physics (simplified: t reduce cmputatinal cst and t avid nn-linear prcesses)

6 Why d we need physical parametrizatins in DA? Physical parametrizatins are needed in data assimilatin: ) evlve the mdel state in time during the 4D-Var assimilatin, ) cnvert the mdel state variables t bserved equivalents, s that bs mdel differences can be cmputed at bs time. Fr example: mdel initial state x 0 time t 0 qv = u v P s M time t i q u v P q q v s liq ice H bservatin equivalent = satellite cludy radiances time t i ~ y i = 3 M = frecast mdel with physics H = radiative transfer mdel

7 Why d we need physical parametrizatins in DA? angent-linear peratrs are applied t perturbatins: = = ~ ], [ i ice liq s v i s v q q P v u q P v u q t t δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δy δx H M time t i time t 0 Adjint peratrs are applied t cst functin gradient: = = s v i ice liq s v P v u q t t q q P v u q i / / / / / ], [ / / / / / / / / / / ~ x y M H time t i time t 0

8 he ice f physical parametrizatins will affect the results f 4D-Var M: input = mdel state (,q v ) utput = surface cnvective rainfall rate acbians f surface rainfall rate w.r.t. and q v M M q v M M q v frm Marécal and Mahfuf (00) Betts-Miller (adjustment seme) iedtke (ECMWF s per mass-flux seme)

9 Perturbatin scaling factr maine precisin reaed

10 he adjint test shuld be crrect at the level f maine precisin (i.e. 3 t 5 digits, typically). If nt, smething is still wrng smewhere in the cde!

11 Linearity assumptin Variatinal assimilatin is based n the strng assumptin that the analysis is perfrmed in a (quasi-)linear framewrk. Hwever, in the case f physical prcesses, strng nn-linearities can ccur in the presence f discntinuus/nn-differentiable prcesses (e.g. swites r threshlds in clud water and precipitatin frmatin). Regularizatin needs t be applied: smthing f functins, reductin f sme perturbatins. y riginal tangent in x 0 Precipitatin frmatin rate y (tangent-linear) 0 y (nn-linear) x 0 x (finite size perturbatin) x Clud water amunt

12 Linearity issue y Precipitatin frmatin rate threshld 0 x 0 y (nn-linear) tangent in x 0 x (finite size perturbatin) y (tangent-linear) = 0 x Clud water amunt

0 therwise = σ b + q q σ q b + α[ q qsat ( )] RR σ RR bs bs q Saturated backgrund q Dry

13 Illustratin f discntinuity effect n cst functin shape: Mdel backgrund = { b, q b }; Observatin = RR bs Simple parametrizatin f rain rate: RR = α {q q sat ()} if q > q sat (), 0 therwise = σ b + q q σ q b + α[ q qsat ( )] RR σ RR bs bs q Saturated backgrund q Dry backgrund { b, q b } min OK N cnvergence! Single minimum f cst functin Several lcal minima f cst functin

15 aniskvá et al. 999

Imprtance f regularizatin t prevent instabilities in tangent-linear mdel Evlutin f temperature increments (4-hur frecast) with the tangent linear

16 Imprtance f regularizatin t prevent instabilities in tangent-linear mdel Evlutin f temperature increments (4-hur frecast) with the tangent linear mdel using different appraes fr the exange cefficient K in the vertical diffusin seme. Perturbatins f K included in L Perturbatins f K set t zer in L

17 Imprtance f regularizatin t prevent instabilities in tangent-linear mdel -hur ECMWF mdel integratin (59 L60) emperature n level 48 (apprx. 850 hpa) Finite difference between tw nn-linear mdel integratins Crrespnding perturbatins evlved with tangent-linear mdel N regularizatin in cnvectin seme

18 Imprtance f regularizatin t prevent instabilities in tangent-linear mdel -hur ECMWF mdel integratin (59 L60) emperature n level 48 (apprx. 850 hpa) Finite difference between tw nn-linear mdel integratins Crrespnding perturbatins evlved with tangent-linear mdel Regularizatin in cnvectin seme (buyancy & updraught velcity reduced perturb.)

19 M Frecasts Climate runs N OK? iming: Yes N Pure cding M M(x+λδx) M(x) λm δx OK? M* <M δx,δy> = <δx,m*δy> Yes N OK? Yes Debugging and testing (incl. regularizatin) N OK? Singular Vectrs (EPS) N OK? 4D-Var (minim) NL L AD APPL

20 A shrt list f existing LP packages used in peratinal DA suyuki (996): Ku-type cnvectin and large-scale cndensatin semes (FSU 4D-Var). Mahfuf (999): full set f simplified physical parametrizatins (gravity wave drag currently used in ECMWF peratinal 4D-Var and EPS). aniskvá et al. (999): full set f simplified physical parametrizatins (Mété-France peratinal 4D-Var). aniskvá et al. (00): linearized radiatin (ECMWF 4D-Var). Lpez (00): simplified large-scale cndensatin and precipitatin seme (Mété-France). mpkins and aniskvá (004): simplified large-scale cndensatin and precipitatin seme (ECMWF). Lpez and Mreau (005): simplified mass-flux cnvectin seme (ECMWF). Mahfuf (005): simplified Ku-type cnvectin seme (Envirnment Canada).

21 ECMWF peratinal LP package (peratinal 4D-Var) Currently used in ECMWF peratinal 4D-Var minimizatins (main simplificatins with respect t the full nn-linear versins are highlighted in red): Large-scale cndensatin seme: [mpkins and aniskvá 004] - based n a unifrm PDF t describe subgrid-scale fluctuatins f ttal water, - melting f snw included, - precipitatin evapratin included, - reductin f clud fractin perturbatin and regul. in autcnversin f clud int rain. Cnvectin seme: [Lpez and Mreau 005] - mass-flux appra [iedtke 989], - deep cnvectin (CAPE clsure) and shallw cnvectin (q-cnvergence) are treated, - perturbatins f all cnvective quantities are included, - cupling with clud seme thrugh detrainment f liquid water frm updraught, - sme perturbatins (buyancy, initial updraught vertical velcity) are reduced. iatin: L and AD f lngwave and shrtwave radiatin available [aniskvá et al. 00] - shrtwave: based n Mrcrette (99), nly spectral intervals (instead f 6 in nn-linear versin), - lngwave: based n Mrcrette (989), called every hurs nly.

22 ECMWF peratinal LP package (peratinal 4D-Var) Vertical diffusin: - mixing in the surface and planetary bundary layers, - based n K-thery and Blackadar mixing length, - exange cefficients based n Luis et al. [98], near surface, - Mnin-Obukhv higher up, - mixed layer parametrizatin and PBL tp entrainment recently added. - Perturbatins f exange cefficients are smthed (esp. near the surface). Orgraphic gravity wave drag: [Mahfuf 999] - subgrid-scale rgraphic effects [Ltt and Miller 997], - nly lw-level blcking part is used. Nn-rgraphic gravity wave drag: [Or et al. 00] - istrpic spectrum f nn-rgraphic gravity waves [Scincca 003], - Perturbatins f utput wind tendencies belw 00 hpa reset t zer. ROV is emplyed t simulate radiances at individual frequencies (infrared, lngwave and micrwave, with clud and precipitatin effects included) t cmpute mdel satellite departures in bservatin space.

23 Impact f linearized physics n tangent-linear apprximatin Cmparisn: Finite difference f tw NL integratins L evlutin f initial perturbatins Examinatin f the accuracy f the linearizatin fr typical analysis increments: M ) M ( x ) M ( x ( xan bg an xbg typical size f 4D-Var analysis increments ) Diagnstics: mean abslute errrs: ε = [ M ( x ) M ( x )] M ( x x ) an bg an bg ε EXP REF relative errr ange: 00% (imprvement if < 0) ε ε REF here: REF = adiabatic run (i.e. n physical parametrizatins in tangent-linear)

24 emperature Impact f peratinal vertical diffusin seme ε EXP - ε REF REF = ADIAB relative imprvement [%] X EXP

25 Impact f dry + mist physical prcesses ε EXP - ε REF relative imprvement [%] EXP X

26 Impact f all physical prcesses (including new mist physics & radiatin) ε EXP - ε REF relative imprvement [%] X X X X EXP

27 Applicatins

28 D-Var with radar reflectivity prfiles Backgrund x b =( b,q b, ) b = ( x x ) B ( x x ) b b x b = B ( x x ) b Initialize x=x b x=(,q, ) n x 0? MINIM x x Mist Physics Reflectivity Mdel yes Analysis x a =x Mist Physics (AD) Reflectivity Mdel (AD) Z md = K k = Z Z k σ k md bs k bs Z = Z k md Z k bs k ( σ bs ) k =, K Reflectivity bservatins Z bs with errrs σ bs K = number f mdel vertical levels

29 D-Var with RMM/Precipitatin ar data rpical Cyclne Ze (6 December UC; Suthwest Pacific) AQUA MODIS image RMM Precipitatin ar Crss-sectin RMM-PR swath

30 D-Var with RMM/Precipitatin ar data A5 Rain Backgrund Rain D-Var Analysed Rain A5 Reflectivity Backgrund Reflect. D-Var Analysed Reflect. rpical Cyclne Ze (6 December UC) Vertical crss-sectin f rain rates (tp, mm h - ) and reflectivities (bttm, dbz): bserved (left), backgrund (middle), and analyzed (right). Black islines n right panels = D-Var specific humidity increments.

31 Impact f ECMWF linearized physics n frecast scres Cmparisn f tw 5 L9 4D-Var 3-mnth experiments with & withut full linearized physics: Relative ange in frecast anmaly crrelatin. > 0 =

32 Physics parameter ptimizatin (new resear) Idea: It might be feasible t ptimize the values f parameters used in the physical semes with the variatinal data assimilatin appra. his wuld require t include the parameter(s) in the cntrl vectr f the data assimilatin system (4D-Var, fr instance): ( ) ( ) b b b b H H y p x R y p x p p B p p x x B x x p + + = ), ( ), ( ) ( ) ( ) ( ) ( Limitatins: Only parameters that are present in bth the frecast mdel and the linearized simplified physics (L & AD) can be treated in this way. Discrepancies between the full nn-linear physics and the L & AD physics (used in the minimizatin f ) might lead t sub-ptimal results.

33 Influence f time and reslutin n linearity assumptin in physics Results frm ensemble runs with the MC mdel (3 km reslutin) ver the Alps, frm Walser et al. (004). Cmparisn f a pair f ppsite twin experiments. L Linearity Crrelatin between ppsite twins 3-km scale 9-km scale Frecast time (hurs) he validity f the linear assumptin fr precipitatin quickly drps in the first hurs f the frecast, especially fr smaller scales.

34 General cnclusins Physical parametrizatins have becme imprtant cmpnents in variatinal data assimilatin systems because they allw: - the cmparisn f the mdel state with bservatins (frward mdel), - the minimizatin f the cst functin (tangent-linear and adjint mdels). Hwever, their linearized versins (tangent-linear and adjint) require sme special attentin (regularizatins/simplificatins) in rder t eliminate r smth pssible discntinuities and nn-differentiability f the physical prcesses they represent. his is particularly true fr the assimilatin f bservatins related t precipitatin, cluds and sil misture. Develping new simplified parametrizatins fr data assimilatin requires: Cmprmise between realism, linearity and cmputatinal cst, Validatin f simplified frward cde against bservatins, Cmparisn t full nn-linear cde used in frecast mde (4D-Var trajectry), Numerical tests f tangent-linear and adjint cdes fr small perturbatins, Validity f the linear hypthesis fr perturbatins with larger size (typical f analysis increments) n spurius perturbatin grwth. Successful cnvergence f 4D-Var minimizatins.

35 hank yu!

36 = = 3 n n q q H y x Example f bservatin peratr H (radiative transfer mdel): = n n n n n n q q q q q q H and its tangent-linear peratr H:

37 REFERENCES Variatinal data assimilatin: Lrenc, A., 986, Quarterly urnal f the Ryal Meterlgical Sciety,, Curtier, P. et al., 994, Quarterly urnal f the Ryal Meterlgical Sciety, 0, Buttier, F. and P. Curtier, 999: Data assimilatin cncepts and methds, ECMWF lecture ntes, available at Rabier, F. et al., 000, Quarterly urnal f the Ryal Meterlgical Sciety, 6, he adjint tenique: Erric, R.M., 997, Bulletin f the American Meterlgical Sciety, 78, angent-linear apprximatin: Erric, R.M. et al., 993, ellus, 45A, Erric, R.M., and K. Reader, 999, Quarterly urnal f the Ryal Meterlgical Sciety, 5, aniskvá, M. et al., 999, Mnthly Weather Review, 7, Physical parameterizatins fr data assimilatin: Mahfuf,.-F., 999, ellus, 5A, aniskvá, M. et al., 00, Quarterly urnal f the Ryal Meterlgical Sciety, 8, mpkins, A. M., and M. aniskvá, 004, Quarterly urnal f the Ryal Meterlgical Sciety, 30, Lpez, P., and E. Mreau, 005, Quarterly urnal f the Ryal Meterlgical Sciety, 3, aniskvá, M., and P. Lpez, 0: Linearized physics fr data assimilatin at ECMWF, enical Memrandum 666, available frm ECMWF, Reading, UK.

38 REFERENCES Micrwave iative ransfer Mdel fr data assimilatin: Bauer, P., May 00, Satellite Applicatin Facility fr Numerical Weather Predictin NWPSAF-EC- R-005, 7 pages. Assimilatin f bservatins affected by precipitatin and/r cluds: Lpez, P. 007: urnal Atmspheric Sciences, Special Issue n the Assimilatin f Satellite Clud and Precipitatin Observatins in NWP Mdels, 64, (Review paper n the use f mist physical parameterizatins in data assimilatin). Bauer, P., P. Lpez, A. Benedetti, D. Salmnd, and E. Mreau, 006a, Quarterly urnal f the Ryal Meterlgical Sciety, 3, Bauer, P., P. Lpez, A. Benedetti, D. Salmnd, S. Saarinen, and M. Bnazzla, 006b, Quarterly urnal f the Ryal Meterlgical Sciety, 3, Fillin, L., and R.M. Erric, 997, Mnthly Weather Review, 5, Ducrcq, V., et al., 000, Quarterly urnal f the Ryal Meterlgical Sciety, 6, Hu, A. et al., 000, Mnthly Weather Review, 8, Marécal, V., and..-f. Mahfuf, 00, Mnthly Weather Review, 30, Marécal, V., and..-f. Mahfuf, 003, Quarterly urnal f the Ryal Meterlgical Sciety, 9,

The challenges of Linearized Physics (LP) in Data Assimilation (DA)

The challenges of Linearized Physics (LP) in Data Assimilation (DA) The challenges f Linearized Physics (LP) in Data Assimilatin (DA) Philippe Lpez, ECMWF with thanks t Marta aniskvá (ECMWF). Outline Brief reminder f what DA is abut. Physics in (4D-Var) DA: Why, where