An iterative ensemble Kalman smoother

Transcription

1 An ieraive ensemble Kalman smooher Marc Bocque 1,2, Pavel Sakov 3 1 Universié Paris-Es, CEREA, join lab École des Pons ParisTech and EdF R&D, France 2 INRIA, Paris-Rocquencour Research cener, France 3 Bureau of Meeorolog, Ausralia (bocque@cerea.enpc.fr) M. Bocque 6 h WMO Smposium on Daa Assimilaion, College Park, Marland, USA, 7-11 Ocober / 16

2 Conex Ensemble variaional mehods New mehods called ensemble variaional mehods ha mix variaional and ensemble approaches (see Lorenc, 2013 for an almos perfec definiion): Hbrid mehods, 4D-Var-Ben, 4D-En-Var, Ensemble of daa assimilaion (EDA) and IEnKF/IEnKS. The IEnKF/IEnKS differ from he oher ones in ha he are more naural (simple?), regardless of he numerical cos. Lorenc A Recommended nomenclaure for EnVar daa assimilaion mehods. In Research Aciviies in Amospheric and Oceanic Modelling, WGNE. M. Bocque 6 h WMO Smposium on Daa Assimilaion, College Park, Marland, USA, 7-11 Ocober / 16

3 Conex The IEnKS: a he crossroad beween he EnKF and 4D-Var The IEnKS follows he scheme of he EnKF: Analsis in ensemble space Poserior ensemble generaion Ensemble forecas Excep ha The analsis in ensemble space is variaional [e.g. Zupanski, 2005] over a finie ime windows. I ma require several ieraions in srongl nonlinear condiions [Gu & Oliver, 2007; Sakov e al., 2012; Bocque and Sakov, ]. The gradien of he 4D cos funcion is compued wih he ensemble [Gu & Oliver, 2007;Liu e al., 2008]: no need for he angen linear/adjoin. I generalises he ieraive exended Kalman filer/smooher [Wishner e al., 1969; Jazwinski, 1970;Bell, 1994] o ensemble mehods. I is a unified/sraighforward scheme (no hbridizaion so o speak). M. Bocque 6 h WMO Smposium on Daa Assimilaion, College Park, Marland, USA, 7-11 Ocober / 16

4 The ieraive ensemble Kalman smooher The IEnKS: he ccling L: lengh of he daa assimilaion window, S: shif of he daa assimilaion window in beween wo updaes. L 1 L S 0 1 L 3 L 3 L 2 L 2 L 1 L L 1 L L+1 S L+1 L+1 L+2 L+2 L M. Bocque 6 h WMO Smposium on Daa Assimilaion, College Park, Marland, USA, 7-11 Ocober / 16

5 The ieraive ensemble Kalman smooher The IEnKS: a variaional sandpoin Analsis IEnKS cos funcion in sae space p(x 0 L ) exp( J(x 0 )): L 1 J(x 0 )= 2 ( k H k M k 0 (x 0 )) T β k R 1 k ( k H k M k 0 (x 0 )) k= (x 0 x 0 )P 1 0 (x 0 x 0 ). (1) {β 0,β 1,...,β L } weigh he observaions impac wihin he window. Reduced scheme in ensemble space, x 0 =x 0 +A 0 w, where A 0 is he ensemble anomal marix: J(w)=J(x 0 +A 0 w). (2) IEnKS cos funcion in ensemble space [Hun e al., 2007; Bocque and Sakov, 2012]: J(w)= 1 L 2 ( k H k M k 0 (x 0 +A 0 w)) T β k R 1 k ( k H k M k 0 (x 0 +A 0 w)) k= (N 1)wT w. (3) M. Bocque 6 h WMO Smposium on Daa Assimilaion, College Park, Marland, USA, 7-11 Ocober / 16

6 The ieraive ensemble Kalman smooher The IEnKS: minimisaion scheme As a variaional reduced mehod, one can use Gauss-Newon [Sakov e al., 2012], Levenberg-Marquard [Bocque and Sakov, 2012; Chen and Oliver, 2013], quasi-newon, ec., minimisaion schemes. Gauss-Newon scheme (he Hessian is approximae): w (p+1) =w (p) x (p) 0 =x (0) 0 +A 0 w (p), J L (p) = k=1 H 1 (p) J (p) (w (p) ), Y T k,(p) β kr 1 k L H (p) =(N 1)I N + Yk,(p) T β kr 1 L Y (p), k=1 ( k H k M k 0 (x (p) 0 ) )+(N 1)w (p), Y k,(p) =[H k M k 0 ] A x (p) 0. (4) 0 One soluion o compue he 4D sensiiviies: he bundle scheme. I simpl mimics he acion of he angen linear b finie difference: Y k,(p) 1 ( ) ( ) ε H k M k 0 x (p) 1 T + εa 0 I N 11T. (5) N M. Bocque 6 h WMO Smposium on Daa Assimilaion, College Park, Marland, USA, 7-11 Ocober / 16

7 The ieraive ensemble Kalman smooher The IEnKS: ensemble updae and he forecas sep Generae an updaed ensemble from he previous analsis: E 0 =x 0 1T + N 1A 0 H 1/2 U where U1=1. (6) Jus propagae he updaed ensemble from 0 o S : In he quasi-saic case: S =1. E S = M S 0 (E 0 ). (7) M. Bocque 6 h WMO Smposium on Daa Assimilaion, College Park, Marland, USA, 7-11 Ocober / 16

8 The ieraive ensemble Kalman smooher IEnKS: single vs muliple daa assimilaion L 1 L β0 2 S β L 3 βl 1 L 3 L 2 βl L 2 βl 1 L 1 L 1 L βl L L+1 S β0 2 βl 1 L+1 L+1 L+2 βl L+2 L SDA IEnKS: The observaion vecor are assimilaed once and for all. Exac scheme. MDA IEnKS: The observaion vecor are assimilaed several imes and poundered wih weighs β k wihin he window. Exac scheme in he linear/gaussian limi. More sable for long windows han he SDA scheme. M. Bocque 6 h WMO Smposium on Daa Assimilaion, College Park, Marland, USA, 7-11 Ocober / 16

9 Numerical experimens Applicaion o he Lorenz 95 model Seup: Lorenz 95, M =40, N =20, =0.05, R=I. Lin-IEnKS IEnKS wih a single ieraion (linearised IEnKS). Comparison of EnKF-N, MDA IEnKS-N, MDA Lin-IEnKS-N, EnKS-N, wih L = Re-analsis rmse EnKF-N EnKS-N Lin-IEnKS-N IEnKS-N Lag (number of ccles) M. Bocque 6 h WMO Smposium on Daa Assimilaion, College Park, Marland, USA, 7-11 Ocober / 16

10 Numerical experimens Applicaion o he Lorenz 95 model Seup: Lorenz 95, M =40, N =20, =0.20, R=I. Comparison of EnKF-N, IEnKF-N, MDA IEnKS-N, ETKS-N, wih L=10. Lin-IEnKS-N underperforms (because of he mild nonlineari) EnKF-N IEnKF-N EnKS-N Lin-IEnKS-N IEnKS-N Re-analsis rmse Lag (number of ccles) M. Bocque 6 h WMO Smposium on Daa Assimilaion, College Park, Marland, USA, 7-11 Ocober / 16

11 Localisaion IEnKF/IEnKS: Localisaion Localisaion in an EnVar conex is non-rivial because localisaion and he evoluion model do no commue: ) M k 0 (C B 0 )M T k 0 (M C k 0 B 0 M T k 0. (8) Local analsis of IEnKF, and comparison wih a non-scalable opimal approach Analsis rmse CL IEnKF (bundle, op. infl., c=10, non-scalable) N=10 IEnKF-N (bundle) N=20 LA IEnKF-N (bundle, c=10, ε N =1) N= Time inerval beween updaes M. Bocque 6 h WMO Smposium on Daa Assimilaion, College Park, Marland, USA, 7-11 Ocober / 16

12 Localisaion IEnKF/IEnKS: Localisaion Local analsis of IEnKS, and comparison wih a non-scalable opimal approach (filering performance) Analsis RMSE SDA IEnKS-N filering N=20 MDA IEnKS-N filering N=20 LA EnKS-N filering N=10 l=10 LA MDA IEnKS-N filering N=10 l=10 NSCL SDA IEnKS op.infl. filering N=10 l=10 LA SDA IEnKS-N filering N=10 l=10 EnKS-N filering N= DAW lengh L M. Bocque 6 h WMO Smposium on Daa Assimilaion, College Park, Marland, USA, 7-11 Ocober / 16

13 Localisaion IEnKF/IEnKS: Localisaion Localisaion of he IEnKF works fine on a forced 2D-urbulence model ( ) Forced 2D urbulence model 256x256 Local IEnKF, N=16, pariall observed grid (1/16) Analsis roo mean square error # ieraions IEnKF filering rmse IEnKF smoohing rmse Localisaion lengh True vorici q, =25 0 Analsed vorici q, =25 0 Vorici error q, = M. Bocque 6 h WMO Smposium on Daa Assimilaion, College Park, Marland, USA, 7-11 Ocober / 16

14 Parameer esimaion IEnKS: parameer esimaion The augmened sae formalism is convenien for he IEnKS, and offers an eas implemenaion of echnicall challenging daa assimilaion problems. Lorenz 95 wih join esimaion of he forcing parameer F (41 variables): RMSEs. Mehod / F profile Sinusoidal Sep-wise EnKF-N EnKS-N L= D-Var L= MDA IEnKS-N L= Rerospecive analsis of parameer F 8,8 8,6 8,4 8,2 8 7,8 7,6 EnKF-N EnKS-N L=50 4D-Var L=50 MDA IEnKS-N L=50 Truh Rerospecive analsis of parameer F 9 8,5 8 7,5 EnKF-N L=50 EnKS-N L=50 4D-Var L=50 MDA IEnKS-N L=50 Truh 7, Time Time M. Bocque 6 h WMO Smposium on Daa Assimilaion, College Park, Marland, USA, 7-11 Ocober / 16

15 Conclusions Conclusions The ieraive ensemble Kalman smooher (IEnKS) is a wa o eleganl combine he advanages of variaional and ensemble Kalman filering. The IEnKS is a generalisaion of he ieraive ensemble Kalman filer (IEnKF). I is an EnVar mehod. I is flow-dependen, angen linear and adjoin free. Though based on Gaussian assumpions, i can offer (much) beer rerospecive analsis han sandard Kalman smoohers in weak and mildl nonlinear condiions. Much more performing han oher Kalman filer/smoohers in srongl non-linear condiions. Properl defined muliple assimilaion of observaions can sabilise he smooher over ver large daa assimilaion window (20 das of Lorenz 95). More generall he IEnKF/IEnKS have he poenial o ouperform boh he EnKF and he 4D-Var (IEnKS alread does so wih o-models). Localisaion remains a fundamenal issue in his conex (work in progress). Weak-consrain formalism no reall explored e (work in progress). M. Bocque 6 h WMO Smposium on Daa Assimilaion, College Park, Marland, USA, 7-11 Ocober / 16

16 References References Bocque, M., Sakov, P., Combining inflaion-free and ieraive ensemble Kalman filers for srongl nonlinear ssems. Nonlin. Processes Geophs. 19, Bocque, M., Sakov, P., An ieraive ensemble Kalman smooher. Q. J. R. Meeorol. Soc., in press, doi: /qj Bocque, M., Sakov, P., Join sae and parameer esimaion wih an ieraive ensemble Kalman smooher. Nonlin. Processes Geophs., in press. Gu, Y., Oliver, D. S., An ieraive ensemble Kalman filer for muliphase fluid flow daa assimilaion. SPE Journal 12, Hun, B. R., Koselich, E. J., Szunogh, I., Efficien daa assimilaion for spaioemporal chaos: A local ensemble ransform Kalman filer. Phsica D 230, Sakov, P., Oliver, D., Berino, L., An ieraive EnKF for srongl nonlinear ssems. Mon. Wea. Rev. 140, Yang, S.-C., E. Kalna, and B. Hun, Handling nonlineari and non-gaussiani in he Ensemble Kalman Filer: Experimens wih he hree-variable Lorenz model. Mon. Wea. Rev., 140, M. Bocque 6 h WMO Smposium on Daa Assimilaion, College Park, Marland, USA, 7-11 Ocober / 16