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 Meeorology, Ausralia (bocque@cerea.enpc.fr) M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

2 Reminders: he EnKF and he EnKF-N Reminder: failure of he raw ensemble Kalman filer (EnKF) EnKF relies for is analysis on he firs and second-order empirical momens: x= 1 N N x k, P= 1 n=1 N 1 N k=1 (x k x)(x k x) T. (1) Lorenz 95 N=20 =0.05 Wihheexcepion ofgaussian and linear sysems, he EnKF fails o provide a proper esimaion on mos sysems. To properly work, i needs clever bu ad hoc fixes: localisaion and inflaion. RMSE analysis EnKF wihou inflaion EnKF wih inflaion λ=1.02 EnKF-N Analysis cycle In a perfec model conex, he finie-size EnKF (EnKF-N) avoids uning inflaion. M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

3 Reminders: he EnKF and he EnKF-N Reminder: principle of he EnKF-N The prior of EnKF and he prior of EnKF-N: { p(x x,p) exp 1 } 2 (x x)t P 1 (x x) p(x x 1,x 2,...,x N ) (x x)(x x) T + ε N (N 1)P N 2, (2) wih ε N =1 (mean-rusing varian), or ε N =1+ 1 N (original varian). Ensemble space decomposiion (ETKF version of he filers): x=x+aw. The variaional principle of he analysis (in ensemble space): J(w)= 1 2 (y H(x+Aw))T R 1 (y H(x+Aw))+ N 1 w T w 2 J(w)= 1 2 (y H(x+Aw))T R 1 (y H(x+Aw))+ N ) (ε 2 ln N +w T w. (3) M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

4 Reminders: he EnKF and he EnKF-N Reminder: he EnKF-N algorihm 1 Requires: The forecas ensemble {x n } n=1,...,n, he observaions y, and error covariance marix R 2 Compue he mean x and he anomalies A from {x k } k=1,...,n. 3 Compue Y =HA, δ =y Hx 4 Find he minimum: { ( )} w a =min (δ Yw) T R 1 (δ Yw)+Nln ε w N +w T w 5 Compue x a =x+aw a. ( ) 1 6 Compue Ω a = Y T R 1 Y+N (ε N+waw T a)i N 2w a wa T (ε N +wa Tw a) 2 7 Compue W a ={(N 1)Ω a } 1/2 U 8 Compue x a k =xa +AW a k M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

5 Conex Ieraive Kalman filers: conex The ieraive exended Kalman filer [Wishner e al., 1969; Jazwinski, 1970] IEKF The ieraive exended Kalman smooher [Bell, 1994] IEKS Much oo cosly + needs he TLM and he adjoin ensemble mehods The ieraive ensemble Kalman filer [Sakov e al., 2012; Bocque and Sakov, 2012] IEnKF The ieraive ensemble Kalman smooher [This alk...] IEnKS I s TLM and adjoin free! Don wan o be bohered by inflaion uning? The finie-size ieraive ensemble Kalman filer [Bocque and Sakov, 2012] IEnKF-N The finie-size ieraive ensemble Kalman smooher [This alk...] IEnKS-N M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

6 Conex Ieraive ensemble Kalman smooher (IEnKS): conex An exension of he ieraive ensemble Kalman filer (IEnKF), a fairly recen idea: [Gu & Oliver, 2007]: Iniial idea. [Kalnay & Yang, ]: A closely relaed idea. [Sakov, Oliver & Berino, 2012]: The pièce de résisance [Bocque & Sakov, 2012]: Bundle scheme + ensemble ransform form. Relaed bu no o be confused wih he ieraive ensemble Smooher (IEnS) in he oil reservoir modelling smoohers, where cycling is no an issue. Assumpions of he presen sudy: Perfec model. In he rank-sufficien regime. Localisaion is more challenging (bu possible) in his conex (Pavel s alk). Looking for he bes performance. Numerical cos secondary. M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

7 The ineraive ensemble Kalman smooher Theory Ieraive ensemble Kalman smooher: 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 This may or may no lead o overlapping windows. Here, we sudy he case S =1, which is close o quasi-saic condiions [Pires e al., 1996]. Le us firs focus on he single daa assimilaion (SDA) scheme. M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

8 The ineraive ensemble Kalman smooher Theory SDA IEnKS: a variaional sandpoin Analysis IEnKS cos funcion in sae space p(x 0 y L ) exp( J(x 0 )): J(x 0 )= 1 2 (y L H L M L 0 (x 0 ))) T R 1 L (y L H L M L 0 (x 0 ))) (x 0 x 0 )P 1 0 (x 0 x 0 ). (4) 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). (5) IEnKS cos funcion in ensemble space: J(w)= 1 2 (y L H L M L 0 (x 0 +A 0 w)) T R 1 L (y L H L M L 0 (x 0 +A 0 w)) (N 1)wT w. (6) M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

9 The ineraive ensemble Kalman smooher Theory SDA 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], ec, minimisaion schemes (no limied o quasi-newon). Gauss-Newon scheme: w (p+1) =w (p) x (p) 0 =x (0) 0 +A 0 w (p), J (p) = Y(p) T R 1 L H 1 (p) J (p) (w (p) ), H (p) =(N 1)I N +Y T (p) R 1 L Y (p), ( y L H L M L 0 (x (p) 0 ) )+(N 1)w (p), Y (p) =[H L M L 0 A 0 ] (p). (7) One alernaive o compue he sensiiviies: he bundle scheme. I simply mimics he acion of he angen linear by finie difference: Y (p) 1 ( ) ( ) ε H L M 1 0 x (p) 1 T + εa 0 I N 11T. (8) N M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

10 The ineraive ensemble Kalman smooher Theory 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. (9) Jus propagae he updaed ensemble from 0 o S : In he quasi-saic case: S =1. E S = M S 0 (E 0 ). (10) M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

11 The ineraive ensemble Kalman smooher Theory IEnKS: inroducing he MDA scheme Suppose we could assimilae he observaion vecors several imes... 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 This leads o overlapping windows. Here, we sudy he quasi-saic case S = 1. This is called muliple daa assimilaion (MDA) scheme. M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

12 The ineraive ensemble Kalman smooher Theory IEnKS: he MDA approach Two flavours of Muliple Daa Assimilaion: The spliing of observaions: Following he pariion 1= L k=1 β k, he observaion vecor y wih prior error covariance marix is spli ino L parial observaion y β k, wih prior error covariance marix βk 1 R. I is a consisen approach in he Gaussian/linear limi, and one hopes i is sill so in nonlinear condiions. The muliple assimilaion of each observaion wih is original weighs. I is correc bu he filering/smoohing pdf (essenially) becomes a power of he searched pdf! An exra sep in he analysis. MDA IEnKS does no approximae per se he filering pdf, bu a more complex pdf. To approach he correc filering/smoohing pdf, one needs an exra sep, ha we called he balancing sep which re-weighs he observaions wihin he daa assimilaion window, and perform a final analysis. M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

13 The ineraive ensemble Kalman smooher Numerical applicaions The Lorenz 95 model The oy-model [Lorenz and Emmanuel 1998]: I represens a mid-laiude zonal circle of he global amosphere. M =40 variables {x m } m=1,...,m. For m=1,...,m: dx m d where F =8, and he boundary is cyclic. =(x m+1 x m 2 )x m 1 x m +F, Chaoic dynamics, opological dimension of 13, a doubling ime of abou 0.42 ime unis, and a Kaplan-Yorke dimension of abou Seup of he experimen: Time-lag beween updae: =0.05 (abou 6 hours for a global model), fully observed, R = I M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

14 The ineraive ensemble Kalman smooher Numerical applicaions Applicaion o he Lorenz 95 model Seup: Lorenz 95, M =40, N =20, =0.05, R=I. Comparison of EnKF-N, SDA IEnKS-N, SDA Lin-IEnKS-N, EnKS-N, wih L = 20. Re-analysis rmse EnKF-N EnKS-N Lin-IEnKS-N IEnKS-N Lag (number of cycles) M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

15 The ineraive ensemble Kalman smooher Numerical applicaions Applicaion o he Lorenz 95 model Beyond L>25, he performance of he SDA IEnKS slowly degrades. Seup: Lorenz 95, M =40, N =20, =0.05, R=I. Comparison of EnKF-N, MDA IEnKS-N, MDA Lin-IEnKS-N, EnKS-N, wih L = Re-analysis rmse EnKF-N EnKS-N Lin-IEnKS-N IEnKS-N Lag (number of cycles) M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

16 The ineraive ensemble Kalman smooher Numerical applicaions 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 has (undersandably) diverged EnKF-N IEnKF-N EnKS-N Lin-IEnKS-N IEnKS-N Re-analysis rmse Lag (number of cycles) M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

17 The ineraive ensemble Kalman smooher Numerical applicaions Applicaion o he Lorenz 95 model Seup: Lorenz 95, M =40, =0.05, R=I. Filering performance of he EnKF-N, IEnKF-N, MDA IEnKS-N for an increasing L. Rmse analysis EnKF-N L=0 IEnKS-N L=1 IEnKS-N L=2 IEnKS-N L=3 IEnKS-N L=4 IENKS-N L=5 IEnKS-N L=10 IEnKS-N L=20 IEnKS-N L= Ensemble size M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

18 Voriciy q, =251 The ineraive ensemble Kalman smooher Voriciy q, =252 Numerical applicaions Voriciy q, = Forced 2D urbulence model Forced 2D urbulence model q +J(q,ψ)= ξq+ ν 2 q+f, q = ψ, (11) where J(q,ψ)= x q y ψ y q x ψ, q is he voriciy 2D field, ψ is he curren funcion 2D field, F is he forcing, ξ ampliude of he fricion, ν ampliude of he biharmonic diffusion, grid: small enough o be in he sufficien-rank regime. Seup of he experimen: Time-lag beween updae: =2, decorellaion of 0.82, fully observed, R = 0.09I M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

19 The ineraive ensemble Kalman smooher Numerical applicaions Applicaion o 2D urbulence Seup: 2D urbulence, 64 64, N =40, =2, R=0.09I. Comparison of EnKF-N, MDA Lin-IEnKS-N, MDA IEnKS-N, EnKS-N, wih L = 20, wih balancing. Voriciy re-analysis rmse (s -1 ) EnKF-N EnKS-N Lin-IEnKS-N IEnKS-N Lag (number of cycles) M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

20 Conclusions Conclusions The ieraive ensemble Kalman smooher (IEnKS) is a way o eleganly combine he advanages of variaional and ensemble Kalman filering, and avoids some of heir drawbacks. The IEnKS is a generalisaion of he ieraive ensemble Kalman filer (IEnKF). I is an En-Var mehod. I is angen linear and adjoin free. I is, by consrucion, flow-dependen. Though based on Gaussian assumpions, i can offer (much) beer rerospecive analysis han sandard Kalman smoohers in mildly nonlinear condiions. When affordable, i beas oher Kalman filer/smoohers in srongly non-linear condiions. (Properly defined) muliple assimilaion of observaions can sabilise he smooher over very large daa assimilaion window (20 days of Lorenz 95). More generally he IEnKF/IEnKS have he poenial o bea boh he EnKF and he 4D-Var (IEnKS already does so wih oy-models). Localisaion remains a fundamenal issue in his conex (a glimpse ono i in Pavel s alk). M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

21 References References 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, Bocque, M., Ensemble Kalman filering wihou he inrinsic need for inflaion. Nonlin. Processes Geophys. 18, Sakov, P., Oliver, D., Berino, L., An ieraive EnKF for srongly nonlinear sysems. Mon. Wea. Rev. 140, Bocque, M., Sakov, P., Combining inflaion-free and ieraive ensemble Kalman filers for srongly nonlinear sysems. Nonlin. Processes Geophys. 19, M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

22 References More references I Bishop, C. H., Eheron, B. J., Majumdar, S. J., Adapive sampling wih he ensemble ransform Kalman filer. Par I: Theoreical aspecs. Mon. Wea. Rev. 129, Bocque, M., Ensemble Kalman filering wihou he inrinsic need for inflaion. Nonlin. Processes Geophys. 18, 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. Roy. Meeor. Soc. 0, 0 0, submied. Burgers, G., van Leeuwen, P. J., Evensen, G., Analysis scheme in he ensemble Kalman filer. Mon. Wea. Rev. 126, Chen, Y., Oliver, D. S., Ensemble randomized maximum likelihood mehod as an ieraive ensemble smooher. Mah. Geosci. 44, Chen, Y., Oliver, D. S., Levenberg-marquard forms of he ieraive ensemble smooher for efficien hisory maching and uncerainy quanificaion. Compu. Geosci. 0, 0 0, in press. Cohn, S. E., Sivakumaran, N. S., Todling, R., A fixed-lag kalman smooher for rerospecive daa assimilaion. Mon. Wea. Rev. 122, Cosme, E., Brankar, J.-M., Verron, J., Brasseur, P., Krysa, M., Implemenaion of a reduced-rank, square-roo smooher for ocean daa assimilaion. Mon. Wea. Rev. 33, M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

23 References More references II Cosme, E., Verron, J., Brasseur, P., Blum, J., Auroux, D., Smoohing problems in a bayesian framework and heir linear gaussian soluions. Mon. Wea. Rev. 140, Emerick, A. A., Reynolds, A. C., Ensemble smooher wih muliple daa assimilaion. Compuers & Geosciences 0, 0 0, in press. Evensen, G., Sequenial daa assimilaion wih a nonlinear quasi-geosrophic model using Mone Carlo mehods o forecas error saisics. J. Geophys. Res. 99 (C5), 10,143 10,162. Evensen, G., The ensemble Kalman filer: Theoreical formulaion and pracical implemenaion. Ocean Dynamics 53, Evensen, G., Daa Assimilaion: The Ensemble Kalman Filer, 2nd Ediion. Springer-Verlag. Evensen, G., van Leeuwen, P. J., An ensemble Kalman smooher for nonlinear dynamics. Mon. Wea. Rev. 128, Ferig, E. J., Harlim, J., Hun, B. R., A comparaive sudy of 4D-VAR and a 4D ensemble Kalman filer: perfec model simulaions wih Lorenz-96. Tellus A 59, 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, M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

24 References More references III Pham, D. T., Verron, J., Roubaud, M. C., A singular evoluive exended Kalman filer for daa assimilaion in oceanography. J. Marine Sysems 16, Sakov, P., Evensen, G., Berino, L., Asynchronous daa assimilaion wih he EnKF. Tellus A 62, Sakov, P., Oliver, D., Berino, L., An ieraive EnKF for srongly nonlinear sysems. Mon. Wea. Rev. 140, Wang, X., Hamill, T. M., Bishop, C. H., 2007a. A comparison of hybrid ensemble ransform Kalman-opimum inerpolaion and ensemble square roo filer analysis schemes. Mon. Wea. Rev. 135, Wang, X., Snyder, C., Hamill, T. M., 2007b. On he heoreical equivalence of differenly proposed ensemble-3dvar hybrid analysis schemes. Mon. Wea. Rev. 135, Yang, S.-C., Kalnay, E., Hun, B., Handling nonlineariy in an ensemble Kalman filer: Experimens wih he hree-variable Lorenz model. Mon. Wea. Rev. 140, M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

25 Addiional slides 0.47 (a) 0.29 (b) Filering RMSE Smoohing RMSE MDA IEnKS-N Time inerval Time inerval Filering RMSE (c) (d) Smoohing RMSE 0.16 MDA Lin-IEnKS-N Time inerval Time inerval EnKS-N Time inerval Filering RMSE (e) 0.20 Time inerval (f) Smoohing RMSE DAW lengh L DAW lengh L M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26

26 Addiional slides Applicaion o 2D urbulence Seup: 2D urbulence, 64 64, N =40, =2, R=0.1I. Comparison of EnKF-N, MDA Lin-IEnKS-N, MDA IEnKS-N, EnKS-N, wih L = 50, wihou balancing. Voriciy re-analysis rmse (s -1 ) EnKF-N EnKS-N Lin-IEnKS-N IEnKS-N Lag (number of cycles) M. Bocque 8 h EnKF workshop, Bergen, Norway, May / 26