Observability of flow dependent structure functions and their use in data assimilation

Transcription

1 Oservility of flow dependent structure functions nd their use in dt ssimiltion Pierre Guthier Bsed on work done in collortion with Cristin Lupu (MSc 2006, PhD thesis 2010, UQAM) nd Stéphne Lroche (Env. Cnd) Presenttion t the 2010 ESA Erth Oservtion Summer School on Erth system monitoring nd modeling Frscti, Itly, 2-13 August 2010 Deprtment of Erth nd Atmospheric Sciences Université du Quéec à Montrél

2 Outline Mesuring the impct of oservtions in dt ssimiltion systems * Impct on the nlysis (informtion content) * Impct on short-term forecsts sed on djoint methods Impct of flow-dependent structures in dt ssimiltion nd link with precursors to dynmic instility * Evlution of the oservility of structure functions (Lupu, 2010) Implictions for hyrid 4D-Vr * he erlier experiments of Fisher nd Andersson with reduced rnk Klmn filter * he hyrid 4D-Vr/EnKF (Buehner et l., 2009) Conclusions

3 Sttisticl nture of the ssimiltion * o correct short-term forecst (x, the ckground stte) with error covrince B sed on informtion contined in oservtion y with oservtion error covrince R * he resulting nlysis x hs n ccurcy mesured y its error covrince P which is less thn tht of the ckground x P x K y Hx B KHB * he weight is given y the gin mtrix K set to minimize the totl nlysis error vrince K BH * Oservtion opertor H hs een linerized round the current ckground stte R HBH 1

4 Approches to mesuring the impct of ssimilted oservtions Informtion content * sed on the reltive ccurcy of oservtions nd the ckground stte Oserving System Experiments * Dt denils * Glol view of the impct of oservtions on the qulity of the forecsts Oservtion impct on the qulity of the forecsts * Sensitivities with respect to oservtions sed on djoint methods (Bker nd Dley, 2000; Lnglnd nd Bker, 2003) * Ensemle Klmn filter methods

5 Informtion content Rtio of the nlysis error covrince to B 1 P B tri trkh N trkh tr he informtion gined from ssimilting given set of oservtions is represented y the second term, where N is the dimension of the model spce DFS = Degrees of Freedom per signl nd in oservtion spce P tr HP H B HBH 1 HP H HBH M trhk with M eing the numer of oservtions

6 Dignosing the sttisticl informtion from the results of nlysis Desroziers (2005) * use the results of the ssimiltion to estimte the oservtion, ckground nd nlysis error covrinces in oservtion spce d y Hx y Hx d H x x HKd * nd then, dd d ~ R HBH D D ~ R R D 1~ D d d d ~ HBH HP HBH ~ H HBH D D 1 ~ D ~ DD 1 1 R

7 Estimting the informtion content (or Degrees of Freedom per signl) 1 Noticing tht R R D D HBH HBH D D If the priori nd posteriori error sttistics re consistent, ~ then nd therefore, D D ~ 1~ ~ ~ R R HBH HBH Estimtion of the DFS DFS tr 1 HK tr HBH R HBH tr HBH D 1 tr tr tr DFS HK HBH D HBH D DD his gives the sme informtion content s otined from the priori error sttistics

8 Estimting the informtion content Estimte of the informtion content is sed solely on dignostics from the ssimiltion process d DFS tr y Hx d d dd y Hx x HKd d H x tr dd dd d dd Need to estimte nd invert dd which is full mtrix ecuse it contins the ckground error Alternte form ~ DFS tr R ~ HP H d d tr d Additionl ssumption: R ~ is digonl 1 1 ~ R 1 1 d d d

9 Roustness of the estimte: experiments with simple 1D-Vr 1Dvr ssimiltion of 60 oservtions with covrince model with homogeneous nd isotropic correltions Sttisticl verge over 2000 nlyses L(km) 2 2 σ o σo( t) 2 σ σ~ ) ( ~ 2 ) 2 σ ( t) ( 2 o σ,

10 Roustness of the results with the size of the smple Oservtion error Bckground error

11 Estimting the oservtion error covrince R ~ Estimte of the off-digonl terms of s function of distnce r i,j ~ R i, j i d j L = 300 km L = 500 km L = 1000 km x

12 Estimtion of the informtion content ~ ( 1) ~ ~ 1 DFS APOS tr HBH D 2) ~ DFS APOS tr R ~ ( 1 L (km) ~ HP H DFSHEOR D FS ~ : only the digonl terms of the second DIAG method re used DFS ~ (1) GIRARD DFS APOS ~ D (2) FS APOS D FS ~ DIAG Esiest to compute DFS GIRARD DFS HEOR : estimtion otined from pertured nlysis : estimtion otined from the true vlues

13 Informtion content in 3D-Vr nd 4D-Vr nlyses from Environment Cnd s system Results from the ssimiltion experiments of Lroche nd Srrzin (2010,) over the period Decemer 21, 2006 to Ferury 28, 2007 * Exclude the first 11 dys (spin-up of the ssimiltion cycle) Oservtions include * Rdiosondes, ircrft, surfce nd ship dt, wind profilers * Atmospheric motion vectors from geosttionry stellites * Rdinces from polr-oriting stellites (AMSU-,) nd geosttionry stellites (GOES-Est nd West) Dignostic of sttisticl consistency: 2 /M ~ 1 * Both in 3D-Vr nd 4D-Vr it ws found tht 2 /M = 0.56 * Error sttistics used in the system re overestimted * Desroziers nd Ivnov (2001) nd Chpnik et l. (2004) use this informtion to reclirte the sttistics * his ws not the oject of this work

14 otl DFS estimted over different regions for 3D-Vr nd 4D-Vr (Jnury-Ferury 2007)

15 Computtion of DFS for ech type of oservtions in MSC s 3D-Vr nd 4D-Vr systems DFS Region Os _ type (%) DFS 100 DFS Region Os _ type Gloe otl _ os Region : Gloe Os_types : AI, GO, PR, SF, SW, AMSU-A, AMSU-B, RAOB Lupu et l. (2010)

16 Oservtion impct per oservtion in ech region IC(%) DFS 100 p Region k k Lupu et l. (2010)

17 Oserving System Experiments (OSEs) Experiments reported in Lroche nd Srrzin (2010-,) Evlution of the impct of oservtions through dt denils * ke n nlysis using ll oservtions s reference nd then remove one oservtion type nd mesure the degrdtion * Modifiction of the oservtion environment lters the reltive importnce of the oservtions Comprison of the informtion content for these experiments gives detiled view of the interctions etween oservtions

18 OSEs experiments: 3D-Vr nd 4D-Vr, North Americ DFS p k NA k DFS vlues per ostype normlized y the numer of oservtions. NO_RAOB: DFS per single oservtion notly increses, especilly for AMSU-A nd GO; NO_AIRCRAF: DFS per single oservtion notly increses, especilly for RAOB nd PR; For other oservtions (GO, SW nd AMSU-B) DFS per os lso increses slightly.

19 Summry Informtion content cn e evluted y dignosing the results of n ssimiltion Provides detiled view of the impct of the oservtions within the originl oservtion environment Appliction to the results from OSEs show how the impct of oservtions on nlyses depend on the oservtion environment OSEs on the other hnd mesure the impct of oservtions on the susequent forecsts

20 Oservtion Impct Methodology (Lnglnd nd Bker, 2004) e e OBSERVAIONS ASSIMILAED e 30 e 24 00UC + 24h Oservtions move the model stte from the ckground trjectory to the new nlysis trjectory he difference in forecst error norms, 24 30, is due to the comined impct of ll oservtions ssimilted t 00UC e e

21 Oservility of flow dependent structure functions Forecst Verifying nlysis Anlysis error (e 24 ) Anlysis X x Forecst error (e 30 ) Bckground X 0-h 24-h 26/01 12 UC 28/01 12 UC e e 30 e 24 x L J x L J x J J K y Hx L L x x

22 Evlution of the impct of oservtions At initil time 0 t t J J e x x K H x y where : oservtion deprture from the ckground stte y H x

23 Evlution of the impct of oservtions At initil time J J e x L x L K H x y where : oservtion deprture from the ckground stte Computtion of the Oservtion Impct: One cn otin w y slightly dpting the ssimiltion to solve y H x t t J J F x x w Hw R Hw w B w w Hw R x x K 1 0 t t J J t t 0 J J x x

24 Comined Use of ADJ nd OSEs (Gelro et l., 2008) ADJ pplied to vrious OSE memers to exmine how the mix of oservtions influences their impcts Removl of AMSUA results in lrge increse in AIRS (nd other) impcts Removl of AIRS results in significnt increse in AMSUA impct Removl of ros results in significnt increse in AMSUA, ircrft nd other impcts (ut not AIRS)

25 Frction of Oservtions tht Improve the Forecst GEOS-5 July z (Gelro, 2008) AIRS Control No AMSU-A AMSU-A Control No AIRS only smll mjority of the oservtions improve the forecst

26 Key nlysis errors lgorithm configurtion (Lroche et l., 2002) rue Stte of the Atmosphere GEM Reference nlysis Key nlysis error Sensitivity nlysis Initil nlysis Forecst error (e 24 ) 0 hr 24 hr Key nlysis error GEM x 0 x ( ngent liner ) 24 Minimiztion lgorithm 3 itertions J x 0 GEM (Adjoint) J x 24 J J=Energy of ( x 24 e 24 )

27 Modelling ckground-error covrinces using sensitivities he dpted 3D-Vr Structure functions defined with respect to posteriori sensitivities; Flow dependent structure functions were introduced in the 3D-Vr; ~ ~ 2~ ~ B I B I ff ξ 1 Error vrince long f: σ1 Does flow-dependent ckground error formultion improve the nlysis nd susequent forecst? (Lupu 2006)

28 Cse study of Jnury 27, 2003 Forecst verifiction, 12 UC Jnury 28, 2003 CMC nlysis Glol-GEM 24hr opertionl forecst Se Level Pressure (4 hp)

29 Cse study Glol sensitivity function Initil temperture corrections for the 12 UC Jnury 27, 2003 nlysis 700hP Corrections responsile for the forecst improvement of the Cndin Mritimes system nd cross section of initil temperture correction mde long the rrow.

30 Impct of the dpted 3D-Vr in the nlysis 700hP Difference etween the temperture nlysis increments for 12 UC Jnury 27, 2003 nlysis 3D dpted -3D stndrd nd cross section.

31 Cse study Forecst improvement Energy (totl) of the forecst error verge over Northern Hemisphere Extr-tropics (25N - 90N) Glol-GEM opertionl forecst Energy (J/Kg) Glol-GEM dpted forecst Glol-GEM sensitivity forecst Forecst hour

32 Fit to the oservtionl Dt Do the corrections decrese or increse the deprture etween the nlysis nd the oservtions? Δ J o J( x ) J( x ) o 1,2 3DVr o 3DVr J( o x ) > 0 = increse < 0 = decrese Difference reltive en Jo (%) 1- Sensitivity nlysis Difference reltive en Jo (%) 2- Adpted 3D-Vr nlysis RAOB AIREP SURFC AOV SAWIND OAL RAOB AIREP SURFC AOV SAWIND OAL

33 Fit to the oservtionl Dt Positive vlues men tht the sensitivity nlysis is further wy from the os. thn the initil nlysis (sme conclusions from ECMWF, Isksen et l., 2004); Negtive vlues men tht the dpted 3D-Vr nlysis is closer to the os. (due to the increse ckground-error vrince); Difference reltive en Jo (%) 1- Sensitivity nlysis Difference reltive en Jo (%) 2- Adpted 3D-Vr nlysis RAOB AIREP SURFC AOV SAWIND OAL RAOB AIREP SURFC AOV SAWIND OAL

34 Oservility of flow-dependent structures Adpted 3D-Vr for which the structure functions where defined y normlizing the posteriori sensitivity function 2 Consider the cse where B vv nd the nlysis increment is then with σ δx 2 K y Hx Kd αv 1 ( Hv) R ( Hv) R 1 d ( Hv) σ 1 σ 2 C1 2 C 2 nd 1 C 1 ( Hv) R d C2 ( Hv) R ( Hv) 1

35 Associted informtion content nd oservility Evlution of the DFS in this cse 2 C2 DFS trhk 2 C lim DFS Correltion etween the innovtions nd structure function 1 ( Hv) R d C1 ρ 1 1/ 2 1 1/ 2 1/ ( Hv) R ( Hv) d R d (2C2J o(0)) 2 his defines the oservility of structure functions * Cn the oservtions detect given structure function

36 Exmple from 1D-Vr experiments Consider the following cses * Oservtions re generted from the sme structure function s tht used in the ssimiltion * Oservtions re generted from different structure function (phse shift) * Signl hs n mplitude lower thn the level of oservtion error

37 Oservility s function of oservtion error y' 2( Hv) N os. C 1 C 2 ρ 10 os os os y' 2( Hv) 2 o =1 ε o 10 os os os y' 2( Hv) 2 o =4 ε o 10 os os os

38 Experiment with the sme function

39 Experiment with shifted function

40 Experiments with n dpted 3D-Vr A posteriori sensitivities depend on * rget re * Norm used to mesure the forecst error * Initil norm * Definition of the tngent-liner nd djoint model Experiments with n dpted 3D-Vr sed on EC s 3D-Vr ssimiltion * Dry energy norm * Four cses documented in Cron et l. (2007): Jnury 19, 2002, 00UC, Feurry 6, 2002, 00UC Jnury 6, UC; Jnury 27, UC * rget re: glol, hemispheric (25-90N) nd locl (re on the Est Cost of North Americ) * Imposition of nonliner lnce constrint (Cron et l., 2007)

41 Preliminry test: does it work? Normlized nlysis increment of 3D-Vr s structure function * Limiting cse where B = 2 vv * Does the dpted 3D-Vr recover the right mplitude * his prticulr choice insures tht we hve structure tht cn fit the oservtions.

42 Oservility for the test cse Os. type Jnury 27, 2003 Correltion coefficient Jnury 06, 2003 Ferury 06, 2002 Jnury 19, 2002 RAOB AIREP AMV SURFC AOVS OAL

43 Oservility of different structure functions sed on key nlyses Structure functions Os. type Jnury 27, 2003, correltion coefficient Jnury 06, 2003 Ferury 06, 2002 Jnury 19, 2002 GLOBAL RAOB AIREP AOVS OAL LOCAL RAOB AIREP AOVS OAL HEMISPHERIC RAOB AIREP AOVS OAL PV-BAL RAOB AIREP AOVS OAL

44 Oservility of pseudo-inverse otined from finite numer of singulr vectors (Mhidji et l., 2007) Leding singulr vectors re the structures tht will grow the most rpidly over finite period of time * Leding 60 SVs were computed sed on totl dry energy norm t led time of 48-h * he forecst error is projected onto those SVs t the finl time which llows to express the error t initil time tht explins tht forecst error (pseudo-inverse) Experiments * 18 cses were considered in Decemer 2007 * Are those structures oservle from ville oservtions? * Oservility of SV 1, the leding singulr vectors * Oservility of the pseudo-inverse

45 Oservility of the leding singulr vector nd pseudoinverse Dte Os. type SV no. 1 Initil time Correltion coefficient SV no. 1 Finl time Pseudo-inverse OAL OAL OAL OAL OAL OAL OAL OAL OAL OAL OAL OAL OAL OAL OAL OAL OAL OAL

46 Summry nd conclusions Evlution of the informtion content of oservtions cn e otined from simple dignostics using informtion generted y ny ssimiltion system * Impct of oservtions depends on the oserving environment * Offer mesure of the consistency etween the sttistics used in the ssimiltion nd those dignosed through comprison to oservtions Impct of oservtions on forecsts cn e quntified s well, sed on method proposed y Lnglnd nd Bker (2004) * Mesurement sed on ckwrd integrtion of the djoint model * Sme ingredients tht re used to compute key nlyses to pinpoint the source of the forecst error * Oservtion impct is defined with respect to their correltion with respect to tht prticulr structure * Our results my explin in prt why only hlf the oservtions hve positive impcts

47 Conclusion (cont d) Oservility of structure functions hs een defined in oservtion spce s correltion etween innovtions nd the structure function Even though those structures do correspond to structure tht will grow the most or grow to correct the forecst error t given led time * A posteriori sensitivities re not well correlted with oservtions his hs een tested for different wys to compute the sensitivities * Singulr vectors were not found to e oservle either Reduced rnk Klmn filters do not seem to e pproprite to represent the ckground error covrinces in n ssimiltion system Evolved covrinces s estimted with n Ensemle Klmn filter would e more pproprite for n hyrid 4D-Vr ssimiltion