Joint state and parameter estimation with an iterative ensemble Kalman smoother

Transcription

1 Join sae and parameer esimaion wih 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 Meeorology, Ausralia (bocque@cerea.enpc.fr) M. Bocque 9 h EnKF workshop, Bergen, Norway, June / 22

2 Conex New mehods called ensemble variaional mehods ha mix variaional and ensemble approaches (see Lorenc, 2013 for an almos perfec definiion): Hybrid mehods, 4D-Var-Ben, 4D-En-Var, Ensemble of daa assimilaion (EDA) and IEnKF/IEnKS. Lorenc A Recommended nomenclaure for EnVar daa assimilaion mehods. In Research Aciviies in Amospheric and Oceanic Modelling, WGNE. The IEnKF/IEnKS differ from he oher ones in ha hey are more naural (simple?), regardless of he numerical cos. The IEnKS has a grea poenial for parameer esimaion, as i is variaional bu avoids he derivaion of he adjoin. M. Bocque 9 h EnKF workshop, Bergen, Norway, June / 22

3 Conex The IEnKS: a he crossroad beween he EnKF and 4D-Var The IEnKS follows he scheme of he EnKF: Analysis in ensemble space Poserior ensemble generaion Ensemble forecas Excep ha The analysis in ensemble space is variaional [e.g. Zupanski, 2005] over a finie ime windows. I may require several ieraions in srongly 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 hybridizaion so o speak). M. Bocque 9 h EnKF workshop, Bergen, Norway, June / 22

4 The ieraive ensemble Kalman smooher The IEnKS: he cycling L: lengh of he daa assimilaion window, S: shif of he daa assimilaion window in beween wo updaes. L 1 L S 0 1 y L 3 L 3 y L 2 L 2 y L 1 y L L 1 L L+1 S y L+1 L+1 L+2 y L+2 L M. Bocque 9 h EnKF workshop, Bergen, Norway, June / 22

5 The ieraive ensemble Kalman smooher The IEnKS: a variaional sandpoin Analysis IEnKS cos funcion in sae space p(x 0 y L ) exp( J (x 0 )): L 1 J (x 0 ) = 2 (y k H k M k 0 (x 0 )) T β k R 1 k (y 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 anomaly 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 (y k H k M k 0 (x 0 + A 0 w)) T β k R 1 k (y k H k M k 0 (x 0 + A 0 w)) k= (N 1)wT w. (3) M. Bocque 9 h EnKF workshop, Bergen, Norway, June / 22

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 ( y 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 simply mimics he acion of he angen linear by 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 9 h EnKF workshop, Bergen, Norway, June / 22

7 The ieraive ensemble Kalman smooher The IEnKS: ensemble updae and he forecas sep Generae an updaed ensemble from he previous analysis: 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 9 h EnKF workshop, Bergen, Norway, June / 22

8 The ieraive ensemble Kalman smooher IEnKS: single vs muliple daa assimilaion L 1 L y β0 2 S y β y L 3 βl 1 L 3 y L 2 βl L 2 βl 1 y L 1 L 1 L βl y L L+1 S y β0 2 y βl 1 L+1 L+1 L+2 y 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 9 h EnKF workshop, Bergen, Norway, June / 22

9 Numerical experimens Applicaion o he Lorenz-95 model Weakly nonlinear case: Lorenz-95, M = 40, N = 20, = 0.05, R = I. Comparison of 4D-Var S = 1, EnKS S = 1, SDA IEnKS S = 1, SDA IEnKS S = L, and MDA IEnKS S = Filering analysis RMSE D-Var S=1 EnKS-N S=1 SDA IEnKS-N S=1 SDA IEnKS-N S=L MDA IEnKS-N S=1 Smoohing analysis RMSE D-Var S=1 EnKS-N S=1 SDA IEnKS-N S=1 SDA IEnKS-N S=L MDA IEnKS-N S= DAW lengh L DAW lengh L M. Bocque 9 h EnKF workshop, Bergen, Norway, June / 22

10 Numerical experimens Applicaion o he Lorenz-95 model Srongly nonlinear case: Lorenz-95, M = 40, N = 20, = 0.20, R = I. Comparison of 4D-Var S = 1, EnKS S = 1, SDA IEnKS S = 1, SDA IEnKS S = L, and MDA IEnKS S = 1. Filering analysis RMSE D-Var S=1 EnKS-N S=1 SDA IEnKS-N S=1 SDA IEnKS-N S=L MDA IEnKS-N S=1 Smoohing analysis RMSE D-Var S=1 EnKS-N S=1 SDA IEnKS-N S=1 SDA IEnKS-N S=L MDA IEnKS-N S= DAW lengh L DAW lengh L M. Bocque 9 h EnKF workshop, Bergen, Norway, June / 22

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 analysis of IEnKF, and comparison wih a non-scalable opimal approach Analysis 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 9 h EnKF workshop, Bergen, Norway, June / 22

12 Localisaion IEnKF/IEnKS: Localisaion Local analysis of IEnKS, and comparison wih a non-scalable opimal approach (filering performance) Analysis 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 9 h EnKF workshop, Bergen, Norway, June / 22

13 Augmened sae formalism IEnKF/IEnKS: Augmened sae formalism IEnKS reas parameers he way boh 4D-Var and EnKF rea hem. The sae space is augmened from x R M o a vecor ( ) x z = R M+P, (9) θ Technically, here is nohing more o he join sae and parameer IEnKS han in he sae IEnKS. A forward model needs o be inroduced for he parameers: For insance, he persisence model (θ k+1 = θ k ), or some jiering such as a Brownian moion (θ k+1 = θ k + ε k ). M. Bocque 9 h EnKF workshop, Bergen, Norway, June / 22

14 Augmened sae formalism Esimaion of he Lorenz-95 forcing parameer F F is saic bu unknown EnKF-N EnKS-N L=50 IEnKF-N IEnKS-N L=5 IEnKS-N L=10 IEnKS-N L=30 Analysis of parameer F Time Augmened sae vecor R 41, N = 20. The forcing of he rue model is F = 8. M. Bocque 9 h EnKF workshop, Bergen, Norway, June / 22

15 Augmened sae formalism Esimaion of he Lorenz-95 forcing parameer F Seup: Lorenz-95, M = 40, N = 20, = 0.05, R = I. Comparison of 4D-Var S = 1, EnKS S = 1, SDA IEnKS S = 1, SDA IEnKS S = L, and MDA IEnKS S = Analysis RMSE (sae variables) D-Var filering 4D-Var smoohing EnKF-N/EnKS-N filering EnKS-N smoohing MDA IEnKS-N filering MDA IEnKS-N smoohing Analysis RMSE (parameer F) D-Var filering/smoohing EnKF-N/EnKS-N filering EnKS-N smoohing MDA IEnKS-N filering/smoohing Daa assimilaion window lengh (in ) Daa assimilaion window lengh (in ) M. Bocque 9 h EnKF workshop, Bergen, Norway, June / 22

16 Augmened sae formalism Esimaion of he Lorenz-95 forcing parameer F The forcing parameer F is ime-varying. Inernalised model error (F is in he augmened sae) + unaccouned exernal model error (he rue F is ime-varying persisence assumpion). Mehod / F profile Sinusoidal Sep-wise EnKF-N EnKS-N L= D-Var L= MDA IEnKS-N L= Rerospecive analysis 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 analysis 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 9 h EnKF workshop, Bergen, Norway, June / 22

17 Towards using he IEnKF in a chemisry and ranspor model Exending he Lorenz-95 model An online racer model: Lorenz-95 (wind field) + racer x m 1 c m 1 2 x m c m+ 1 2 x m+1 Φ m 1 E m 1 2 Φ m E m+ 1 2 Φ m+1 The racer is adveced by he wind field of he Lorenz-95 model. We have chosen o use he simple Godunov/upwind scheme which is posiive and conservaive. dx m d = (x m+1 x m 2 )x m 1 x m + F, (10) dc m+ 1 2 = Φ m Φ m+1 λc d m+ 1 + E 2 m+ 1, 2 (11) where Φ m = x m c m 1 2 if x m 0, (12) = x m c m+ 1 2 if x m < 0. (13) Uniform emission of he racer wih he flux E m+ 1. Deposied on he whole domain, 2 using a simple scavenging scheme paramerised by a scavenging raio λ. M. Bocque 9 h EnKF workshop, Bergen, Norway, June / 22

18 Towards using he IEnKF in a chemisry and ranspor model Exending he Lorenz-95 model Time evoluion of he wind (op) and concenraion (boom) fields of he coupled Lorenz-95 - racer model. M. Bocque 9 h EnKF workshop, Bergen, Norway, June / 22

19 Towards using he IEnKF in a chemisry and ranspor model Exending he Lorenz-95 model Analysis RMSE (wind) D-Var filering 4D-Var smoohing EnKS-N filering EnKS-N smoohing SDA IEnKS-N filering SDA IEnKS-N smoohing MDA IEnKS-N filering MDA IEnKS-N smoohing Analsyis RMSE (concenraion) D-Var filering 4D-Var smoohing EnKS-N filering EnKS-N smoohing SDA IEnKS-N filering SDA IEnKS-N smoohing MDA IEnKS-N filering MDA IEnKS-N smoohing Daa assimilaion window lengh (in ) Daa assimilaion window lengh (in ) Mean filering and smoohing analysis rmses of he wind variables (lef) and concenraion variables (righ) of he online racer model, as a funcion of he daw lengh for he IEnKS (finie-size varian), he EnKF/EnKS, and 4D-Var (wih opimal inflaion of he prior). M. Bocque 9 h EnKF workshop, Bergen, Norway, June / 22

20 Towards using he IEnKF in a chemisry and ranspor model Exending he Lorenz-95 model wind-wind concenraion-concenraion wind-concenraion Correlaion coefficien Disance Srucure funcions of he mean correlaion of he errors of he iniial condiion from he IEnKS applied o he online racer model. The full error covariance marix obained from he IEnKS can be used o help 4D-Var by building beer background saisics. I does help 4D-Var in he esimaion of parameers, bu does lile o he esimaion of he sae variables whose error covariance marix is quie dynamical. M. Bocque 9 h EnKF workshop, Bergen, Norway, June / 22

21 Conclusions Conclusions The ieraive ensemble Kalman smooher (IEnKS) is a mehod o seamlessly 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. The IEnKF/IEnKS have he poenial o (significanly) ouperform boh he EnKF and he 4D-Var in all regimes. IEnKS already does so wih oy-models. IEnKS is very well suied for parameer (or join sae/parameer) esimaion, and does so in a very simple way via he augmened sae formalism. More complex reacive air qualiy oy-model under developmen in order o es he IEnKS on challenging amospheric chemisry problems. M. Bocque 9 h EnKF workshop, Bergen, Norway, June / 22

22 References References Bocque, M., Sakov, P., Combining inflaion-free and ieraive ensemble Kalman filers for srongly nonlinear sysems. Nonlin. Processes Geophys. 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 Geophys., 20, 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., Szunyogh, I., Efficien daa assimilaion for spaioemporal chaos: A local ensemble ransform Kalman filer. Physica D 230, Sakov, P., Oliver, D., Berino, L., An ieraive EnKF for srongly nonlinear sysems. Mon. Wea. Rev. 140, Yang, S.-C., E. Kalnay, and B. Hun, Handling nonlineariy and non-gaussianiy in he Ensemble Kalman Filer: Experimens wih he hree-variable Lorenz model. Mon. Wea. Rev., 140, M. Bocque 9 h EnKF workshop, Bergen, Norway, June / 22