Extended ensemble Kalman filters for high-dimensional hierarchical state-space models

Transcription

1 Exended ensemble Kalman filers for high-dimensional hierarchical sae-space models Jonahan Sroud McDonough School of Business Georgeown Universiy Join work wih Mahias Kazfuss (Texas A&M) and Chris Wikle (Missouri) Jonahan Sroud (Georgeown) Exended EnKFs 1 / 31

2 Ouline Ouline 1 SSMs and Exising DA Mehods 2 Exended EnKFs 3 Numerical examples 4 Conclusions Jonahan Sroud (Georgeown) Exended EnKFs 2 / 31

3 SSMs and Exising DA Mehods Ouline 1 SSMs and Exising DA Mehods 2 Exended EnKFs 3 Numerical examples 4 Conclusions Jonahan Sroud (Georgeown) Exended EnKFs 3 / 31

4 SSMs and Exising DA Mehods The sae-space model (SSM) From a saisical perspecive: DA is filering in a SSM SSM wih addiive Gaussian error in discree ime = 1, 2,...: y = H x + v, v N m (0, R ), x = M (x 1 ) + w, w N n (0, Q ), where y is m 1 observaion vecor. x is n 1 sae vecor. H is m n observaion marix. M ( ) is evoluion operaor. v, w are independen errors. iniial sae x 0 N n (a 0, P 0 ) assume parameers known (for now) Jonahan Sroud (Georgeown) Exended EnKFs 4 / 31

5 SSMs and Exising DA Mehods Sequenial Daa Assimilaion Mehods Wan filering disribuion p(x y 1:) for = 1, 2,..., T. Kalman Filer (KF) Ensemble Kalman Filer (EnKF) Paricle Filer (PF) (Kalman, 1960) (Evensen, 1994) (Gordon e al., 1993) Poserior N (a 1, P 1 ) x (i) 1 p(x 1 y 1: 1 ) a ime 1 x (i) 1 p(x 1 y 1: 1 ) Prior N (a f, Pf ) a ime a f = M a 1 x f (i) p(x x (i) 1 ) P f = M P 1M + Q (i) xf p(x x (i) 1 ) Poserior N (a, P ) Resample x f (i) a ime a = (I K H )a f + K y x(i) = (I ˆK H )x f (i) + ˆK y (i) where w (i) P = (I K H )P f wih weighs w (i) p(y x f (i) ) y (i) N (y, R ) o obain poserior samples x (i) Jonahan Sroud (Georgeown) Exended EnKFs 5 / 31

6 SSMs and Exising DA Mehods Illusraion of he updae sep Illusraion of differen updaing schemes a a single ime poin poserior likelihood prior Exac EnKF Paricle Filer Paricle weighs degenerae for large n (Snyder e al., 2008; Slivinski and Snyder, 2016) EnKF requires regularizaion of sample forecas covariance/gain marix (Houekamer and Michell, 1998, 2001; O e al., 2004; Hamill e al., 2001; Anderson and Anderson, 1999; Whiaker and Hamill, 2002; Zhang e al., 2004,...) Jonahan Sroud (Georgeown) Exended EnKFs 6 / 31

7 Exended EnKFs Ouline 1 SSMs and Exising DA Mehods 2 Exended EnKFs 3 Numerical examples 4 Conclusions Jonahan Sroud (Georgeown) Exended EnKFs 7 / 31

8 Exended EnKFs Hierarchical sae-space model (HSSM) Saring wih x 0 N n (a 0, P 0 ), we assume for = 1, 2,...: z y, θ g(y ; θ ) y = H (θ ) x + v, v N m ( 0, R (θ ) ) x = M (x 1 ; θ ) + w, w N n ( 0, Q (θ ) ) θ θ 1 p(θ θ 1 ) where z R m are he acual measuremens, y R m is a laen variable, θ are model parameers, and we have added a ransformaion layer and a parameer layer o he addiive Gaussian SSM (in black). Wan filering disribuion p(x, θ z 1: ) for = 1, 2,.... Jonahan Sroud (Georgeown) Exended EnKFs 8 / 31

9 Exended EnKFs Exising mehods for parameer esimaion Sae augmenaion (Anderson, 2001). Mos popular mehod, bu does no work well if he saes and parameers are weakly correlaed (DelSole and Yang, 2010) Sequenial maximum likelihood/bayes (Dee and da Silva, 1999; Michell and Houekamer, 2000; Sroud and Bengsson, 2007; Frei and Künsch, 2012; Sroud e al., 2018) for specific parameers Ieraive EnKF/EnKS (Gu and Oliver, 2007; Chen and Oliver, 2012; Bocque and Sakov, 2013) sae augmenaion wih opimizaion. Also much work on choosing uning parameers (e.g., Anderson, 2007a,b, 2009; Li e al., 2009; Miyoshi, 2011,...). Jonahan Sroud (Georgeown) Exended EnKFs 9 / 31

10 Exended EnKFs Non-Gaussian Daa Assimilaion Paricle Filers (e.g., van Leeuwen, 2010,...). Local Paricle Filers (Poerjoy, 2016) Hybrid PF/EnKF (Frei and Künsch, 2012, 2013; Slivinski e al., 2015) Robus EnKF (Roh e al., 2013) Rank Hisogram Filers (Anderson, 2010) Anamorphism (Lien e al., 2013; Amezcua and van Leeuwen, 2014) GIGG-EnKF (Bishop, 2016) Momen Maching EnKF (Lei and Bickel, 2011) Reviews/Comparisons (Bocque e al., 2010; Lei e al., 2010)... Jonahan Sroud (Georgeown) Exended EnKFs 10 / 31

11 Exended EnKFs Basic idea of exended EnKFs Condiional on y and θ, he HSSM reduces o he sandard SSM, for which he EnKF is applicable Take exising echniques for Bayesian inference (e.g., Gibbs sampler, paricle filer), bu replace he par requiring inegraing ou or sampling from x by he EnKF Examples: Gibbs-EnKF Paricle-EnKF Gibbs-EnKS... Jonahan Sroud (Georgeown) Exended EnKFs 11 / 31

12 Exended EnKFs Basic idea of exended EnKFs Condiional on y and θ, he HSSM reduces o he sandard SSM, for which he EnKF is applicable Take exising echniques for Bayesian inference (e.g., Gibbs sampler, paricle filer), bu replace he par requiring inegraing ou or sampling from x by he EnKF Examples: Gibbs-EnKF Paricle-EnKF Gibbs-EnKS... Jonahan Sroud (Georgeown) Exended EnKFs 11 / 31

13 Exended EnKFs Algorihm 1: Gibbs EnKF (GEnKF) Assume: forecass of sae and parameers are independen, M (x 1 ) does no depend on θ, and θ are independen over ime. Then: For = 0: Draw (x (i) For 1: 0, θ(i) 0 1. Forecas sep: x (i) 2. Iniialize y (i) ) from Nn(a0, P0)p(θ0), i = 1,..., M. = M (x (i) 1 ), i = 1,..., M. and θ (i), for i = 1,..., M. 3. For i = 1,..., M, ierae beween he following seps unil convergence: (a) Sample x (i) (b) Sample y (i) (c) Sample θ (i) from ˆp(x y (i) from p(y x (i) from p(θ y (i), θ (i), x (1:M) ) using EnKF updae., θ (i), z )., x (i), z ). Then, each (x (i), θ (i) ) is a join sample from p(x, θ z 1: ). Jonahan Sroud (Georgeown) Exended EnKFs 12 / 31

14 Exended EnKFs Algorihm 2: Paricle EnKF (for low-dim parameers) Iniialize he algorihm wih an ensemble of ensembles: (θ (i) 0, x(i,j) 0 ) wih w (i) 0 = 1/M, i = 1,..., M; j = 1,..., N. Then, for each ime 1: 1. For i = 1,..., M: (a) Sample a paricle θ (i) from p(θ θ (i) (b) Propagae he ensemble: x (i,j) (c) Calculae paricle weigh: w (i) (d) Generae x (i,j) from ˆp(x z, θ (i) 1 ) 2. Filering disribuion: p(θ, x z 1:) M = M (x (i,j) 1, θ(i) ), j = 1,..., N. w (i) 1 LZ (z θ (i), x (i,1:n) )., x (i,1:n) ), j = 1,..., N, using EnKF. i=1 w (i) 1 N N j=1 δ (θ (i),x (i,j) ) (θ, x) 3. If desired, resample he paricles (θ (i), x (i,1:n) ) wih weighs w (i) o obain an unweighed sample Likelihood L Z (z θ, x (i,1:n) ) is approximaed using EnKF In he case of forecas independence of saes and parameers, only need a single ensemble (cf. Frei and Künsch, 2012) Jonahan Sroud (Georgeown) Exended EnKFs 13 / 31

15 Exended EnKFs Algorihm 3: Gibbs-EnKS Wan smoohing disribuion p(x 1:T, θ 1:T z 1:T ) for fixed T. 1. Iniialize y 1:T and θ 1:T. 2. Ierae beween following seps unil convergence: (a) Draw a sample x 1:T from p(x 1:T θ 1:T, y 1:T ) using he EnKS*. (b) Draw a sample y 1:T from p(y 1:T z 1:T, x 1:T, θ 1:T ). (c) Draw a sample θ 1:T from p(θ 1:T x 1:T, y 1:T, z 1:T ). *Ensemble Kalman Smooher (Evensen and van Leeuwen, 2000). Works even for high-dimensional parameers, if full condiional disribuion is available in closed form Jonahan Sroud (Georgeown) Exended EnKFs 14 / 31

16 Exended EnKFs Properies of he exended EnKFs For linear Gaussian SSMs, algorihms converge o rue poserior as N and M or he number of MCMC ieraions increase For small N, algorihms will end o perform well for HSSMs for which EnKF/EnKS works well for he embedded SSM. Compuaional complexiy: EnKF: Have o apply M o N ensemble members; updae is O(nN 2 ) for mos EnKF varians (e.g. Tippe e al., 2003) Exended EnKFs: In general, have o carry ou EnKF several imes. Bu: Ofen only a small number of ieraions or paricles is necessary, and only he updae has o be repeaed Thus, increased compuaional cos of exended EnKFs is minor in some applicaions Jonahan Sroud (Georgeown) Exended EnKFs 15 / 31

17 Numerical examples Ouline 1 SSMs and Exising DA Mehods 2 Exended EnKFs 3 Numerical examples 4 Conclusions Jonahan Sroud (Georgeown) Exended EnKFs 16 / 31

18 Numerical examples Daa wih ouliers Heavy-ailed noise disribuion: v l ν (0, σ 2 ), ν small Special case of our HSSM: z = y and R(θ) = σ 2 diag(θ 1,..., θ m ), where θ l ind. IG(ν/2, ν/2) GEnKF (Robus EnKF): Updae sep: For j = 1,..., N, repea G imes (unil convergence): (a) EnKF updae of x (j) from x f (j) based on y, θ (j), x f (1:N). (b) Sample θ (j) ind. l IG(ν/ , ν/2 + ( y l (Hx (j) ) l ) 2/2), l = 1,..., m σ Jonahan Sroud (Georgeown) Exended EnKFs 17 / 31

19 Numerical examples Daa wih ouliers Heavy-ailed noise disribuion: v l ν (0, σ 2 ), ν small Special case of our HSSM: z = y and R(θ) = σ 2 diag(θ 1,..., θ m ), where θ l ind. IG(ν/2, ν/2) GEnKF (Robus EnKF): Updae sep: For j = 1,..., N, repea G imes (unil convergence): (a) EnKF updae of x (j) from x f (j) based on y, θ (j), x f (1:N). (b) Sample θ (j) ind. l IG(ν/ , ν/2 + ( y l (Hx (j) ) l ) 2/2), l = 1,..., m σ Jonahan Sroud (Georgeown) Exended EnKFs 17 / 31

20 Numerical examples Daa wih ouliers Heavy-ailed noise disribuion: v l ν (0, σ 2 ), ν small Special case of our HSSM: z = y and R(θ) = σ 2 diag(θ 1,..., θ m ), where θ l ind. IG(ν/2, ν/2) GEnKF (Robus EnKF): Updae sep: For j = 1,..., N, repea G imes (unil convergence): (a) EnKF updae of x (j) from x f (j) based on y, θ (j), x f (1:N). (b) Sample θ (j) ind. l IG(ν/ , ν/2 + ( y l (Hx (j) ) l ) 2/2), l = 1,..., m σ Jonahan Sroud (Georgeown) Exended EnKFs 17 / 31

21 Numerical examples Example: Heavy-ailed daa Simulaed heavy-ailed daa (wih v l /σ 2 ) daa rue sae exac pos. GEnKF EnKF PF Huber Jonahan Sroud (Georgeown) Exended EnKFs 18 / 31

22 Numerical examples Example: Threshold Models Challenge: Observaion disribuions wih poin masses (e.g., binary) Use ransformaion equaions involving indicaor funcions Example: Rainfall amouns: { yl κ, y l > 0 z l = g(y l ; θ) = 0, y l 0 for some κ > 1. Assume R = σ 2 I m. Gibbs-EnKF for rainfall daa: Sep 3(c): Independenly: { = z 1/κ y l z l, x, κ l, z l > 0 N ( (Hx) l, σ 2), z l = 0 Sep 3(b): If κ is unknown, sample from p(κ x (j), z) Jonahan Sroud (Georgeown) Exended EnKFs 19 / 31

23 Numerical examples Example: Threshold Models Challenge: Observaion disribuions wih poin masses (e.g., binary) Use ransformaion equaions involving indicaor funcions Example: Rainfall amouns: { yl κ, y l > 0 z l = g(y l ; θ) = 0, y l 0 for some κ > 1. Assume R = σ 2 I m. Gibbs-EnKF for rainfall daa: Sep 3(c): Independenly: { = z 1/κ y l z l, x, κ l, z l > 0 N ( (Hx) l, σ 2), z l = 0 Sep 3(b): If κ is unknown, sample from p(κ x (j), z) Jonahan Sroud (Georgeown) Exended EnKFs 19 / 31

24 Numerical examples Example: Threshold Models Challenge: Observaion disribuions wih poin masses (e.g., binary) Use ransformaion equaions involving indicaor funcions Example: Rainfall amouns: { yl κ, y l > 0 z l = g(y l ; θ) = 0, y l 0 for some κ > 1. Assume R = σ 2 I m. Gibbs-EnKF for rainfall daa: Sep 3(c): Independenly: { = z 1/κ y l z l, x, κ l, z l > 0 N ( (Hx) l, σ 2), z l = 0 Sep 3(b): If κ is unknown, sample from p(κ x (j), z) Jonahan Sroud (Georgeown) Exended EnKFs 19 / 31

25 Numerical examples Example: Threshold models Simulaed rainfall daa (wih κ = 3) daa rue rain exac pos. GEnKF EnKF PF Jonahan Sroud (Georgeown) Exended EnKFs 20 / 31

26 Numerical examples Simulaion sudy: Non-Gaussian obs. a a single ime poin Simulaed heavy-ailed and rainfall daa. Compared GEnKF updae o (marix-free) EnKF and PF (imporance sampler): Heavy-ailed Rainfall Rain (κ unkn.) MSPE CRPS MSPE CRPS MSPE CRPS exac GEnKF EnKF >100 >100 PF Huber (Roh e al., 2013) Deails: Simulaed 100 rue saes of size n = 100; m = 75 randomly chosen observaions; H is subse of ideniy marix True sae disribuion: mean 0.2, powered exponenial covariance wih power 1.8 and scale 10. σ 0.2 EnKF and PF: N = 100. GEnKF: N = 30; 1 or 3 ieraions Wendland aper wih range 20 Jonahan Sroud (Georgeown) Exended EnKFs 21 / 31

27 Numerical examples Smoohing for Lorenz-96 n = 40 equally-spaced locaions on a circle Observaions a all locaions Observaions every δ = 0.2, T = 10 observaion imes Gaussian observaions (z = y ) M (x 1 ) = θ Lorenz 8,0.2 (x 1 ) H = R = I n, Q = 0.2Σ L Prior: θ N (0.8, ). Goal: Find smoohing disribuion p(θ, x 1:T y 1:T ) Compared hree mehods on 100 simulaed daases: 1. EnKS wih sae augmenaion wih N = 1000, θ N (θ 1, ) 2. Gibbs-EnKS 3. Paricle Gibbs sampler (Andrieu e al., 2010) For Gibbs-EnKS and paricle-gibbs: N = 50; 100 Gibbs ieraions; p(θ x 1:T, y 1:T ) is in closed form Jonahan Sroud (Georgeown) Exended EnKFs 22 / 31

28 Numerical examples Lorenz-96 resuls for one simulaed daase sae daa rue x GEnKS PG θ rue GEnKS PG (a) Sae a loc. 1 over ime (x 1:T,1 ) Ieraion (b) Trace plos for θ Figure: (a) True sae and poserior means and 80% inervals for sae variable x 1. (b) True value and race plos for parameer θ. Jonahan Sroud (Georgeown) Exended EnKFs 23 / 31

29 Numerical examples Lorenz-96 resuls averaged over 100 daases Parameer θ Sae x 1:T MSPE CRPS MSPE CRPS Gibbs-EnKS EnKS+SA >100 > Paricle Gibbs Prior EnKS wih sae augmenaion diverges Paricle Gibbs sampler produces worse inference on θ han simply using he prior disribuion (i.e., compleely ignoring he daa) Jonahan Sroud (Georgeown) Exended EnKFs 24 / 31

30 Conclusions Ouline 1 SSMs and Exising DA Mehods 2 Exended EnKFs 3 Numerical examples 4 Conclusions Jonahan Sroud (Georgeown) Exended EnKFs 25 / 31

31 Conclusions Summary EnKF handles high-dimensional, nonlinear SSMs, bu is less appropriae under non-gaussianiy or for unknown parameers Exended EnKFs handle more general, hierarchical SSMs, including unknown parameers and non-gaussian observaions In some cases, compuaional effor is similar o EnKF Resuls are approximae, bu asympoically correc for linear models In general, exended EnKFs can only work well if he embedded EnKF for known parameers works well for he problem a hand Jonahan Sroud (Georgeown) Exended EnKFs 26 / 31

32 Conclusions References This alk is largely based on 2 papers: Kazfuss, M., Sroud, J.R., and Wikle, C.K Undersanding he ensemble Kalman filer. The American Saisician, 70(4), Kazfuss, M., Sroud, J.R., and Wikle, C.K Exended ensemble Kalman filers for high-dimensional hierarchical sae-space models. arxiv: Funding Kazfuss: NSF DMS and DMS Wikle: NSF SES Jonahan Sroud (Georgeown) Exended EnKFs 27 / 31

33 Conclusions References I Amezcua, J. and van Leeuwen, P. J. (2014). Gaussian anamorphosis in he analysis sep of he EnKF: A join sae-variable/observaion approach. Tellus A, 66: Anderson, J. L. (2001). An ensemble adjusmen Kalman filer for daa assimilaion. Monhly Weaher Review, 129: Anderson, J. L. (2007a). An adapive covariance inflaion error correcion algorihm for ensemble filers. Tellus A, 59: Anderson, J. L. (2007b). Exploring he need for localizaion in ensemble daa assimilaion using an hierarchical ensemble filer. Physica D, 230: Anderson, J. L. (2009). Spaially and emporally varying adapive covariance inflaion for ensemble filers. Tellus, 61: Anderson, J. L. (2010). An non-gaussian ensemble filer updae for daa assimilaion. Monhly Weaher Review, 138: Anderson, J. L. and Anderson, S. L. (1999). A Mone Carlo implemenaion of he nonlinear filering problem o produce ensemble assimilaions and forecass. Monhly Weaher Review, 127: Andrieu, C., Douce, A., and Holensein, R. (2010). Paricle Markov chain Mone Carlo. Journal of he Royal Saisical Sociey, Series B, 72:1 33. Bishop, C. H. (2016). The GIGG-EnKF: Ensemble Kalman filering for highly skewed non-negaive uncerainy disribuions. Quarerly Journal of he Royal Meeorological Sociey, 141: Bocque, M., Pires, C. A., and Wu, L. (2010). Beyond Gaussian saisical modeling in geophysical daa assimilaion. Monhly Weaher Review, 138(8): Bocque, M. and Sakov, P. (2013). Join sae and parameer esimaion wih an ieraive ensemble Kalman smooher. Nonlinear Processes in Geophysics, 20(5): Chen, Y. and Oliver, D. S. (2012). Ensemble randomized maximum likelihood mehod as an ieraive ensemble smooher. Mahemaical Geosciences, 44(1):1 26. Dee, D. P. and da Silva, A. M. (1999). Maximum likelihood esimaion of forecas and observaion error covariance parameers. Par I: Mehodology. Monhly Weaher Review, 127: DelSole, T. and Yang, X. (2010). Sae and parameer esimaion in sochasic dynamical models. Physica D, 239(18): Jonahan Sroud (Georgeown) Exended EnKFs 28 / 31

34 Conclusions References II Evensen, G. (1994). Sequenial daa assimilaion wih a nonlinear quasi-geosrophic model using Mone Carlo mehods o forecas error saisics. Journal of Geophysical Research, 99: Evensen, G. and van Leeuwen, P. J. (2000). An ensemble Kalman smooher for nonlinear dynamics. Monhly Weaher Review, 128: Frei, M. and Künsch, H. R. (2012). Sequenial sae and observaion noise covariance esimaion using combined ensemble Kalman and paricle filers. Monhly Weaher Review, 140: Frei, M. and Künsch, H. R. (2013). Bridging he ensemble Kalman and paricle filers. Biomerika, 100: Gordon, N. J., Salmond, D. J., and Smih, A. F. M. (1993). Novel approach o nonlinear/non-gaussian Bayesian sae esimaion. In IEE Proceedings, volume F-140, pages IEE. Gu, Y. and Oliver, D. S. (2007). An ieraive ensemble Kalman filer for muliphase fluid flow daa assimilaion. SPE Journal, 12(4): Hamill, T. M., Whiaker, J., and Snyder, C. (2001). Disance-dependen filering of background error covariance esimaes in an ensemble Kalman filer. Monhly Weaher Review, 129: Houekamer, P. L. and Michell, H. L. (1998). Daa assimilaion using an ensemble Kalman filer echnique. Monhly Weaher Review, 126: Houekamer, P. L. and Michell, H. L. (2001). A sequenial ensemble Kalman filer for amospheric daa assimilaion. Monhly Weaher Review, 129: Kalman, R. (1960). A new approach o linear filering and predicion problems. Journal of Basic Engineering, 82(1): Kazfuss, M., Sroud, J. R., and Wikle, C. K. (2016). Undersanding he ensemble Kalman filer. The American Saisician, 70(4): Kazfuss, M., Sroud, J. R., and Wikle, C. K. (2017). Exended ensemble Kalman filers for high-dimensional hierarchical sae-space models. Technical repor, ArXiv: Lei, J. and Bickel, P. (2011). A momen maching ensemble filer for nonlinear non-gaussian daa assimilaion. Monhly Weaher Review, 139: Jonahan Sroud (Georgeown) Exended EnKFs 29 / 31

35 Conclusions References III Lei, J., Bickel, P., and Snyder, C. (2010). Comparison of ensemble Kalman filers under non-gaussianiy. Monhly Weaher Review, 138: Li, H., Kalnay, E., and Miyoshi, T. (2009). Simulaneous esimaion of covariance inflaion and observaion errors wihin an ensemble Kalman filer. Quarerly Journal of he Royal Meerological Sociey, 135: Lien, G.-Y., Kalnay, E., and Miyoshi, T. (2013). Effecive assimilaion of global precipiaion: Simulaion experimens. Tellus A, 65: Michell, H. L. and Houekamer, P. L. (2000). An adapive ensemble Kalman filer. Monhly Weaher Review, 128: Miyoshi, T. (2011). The Gaussian approach o adapive covariance inflaion and is implemenaion wih he local ensemble ransform Kalman filer. Monhly Weaher Review, 139: O, E., Hun, B. R., Szunyogh, I., Zimin, A. V., Koselich, E. J., Corazza, M., Kalnay, E., Pail, D. J., and Yorke, J. A. (2004). A local ensemble Kalman filer for amospheric daa assimilaion. Tellus, 56A: Poerjoy, J. (2016). A localized paricle filer for high-dimensional nonlinear sysems. Monhly Weaher Review, 144: Roh, S., Genon, M. G., Jun, M., Szunyogh, I., and Hoei, I. (2013). Observaion qualiy conrol wih a robus ensemble Kalman filer. Monhly Weaher Review, 141(12): Slivinski, L. and Snyder, C. (2016). Exploring pracical esimaes of he ensemble size necessary for paricle filers. Monhly Weaher Review, 144: Slivinski, L., Spiller, E., Ape, A., and Sandsede, B. (2015). A hybrid paricle-ensemble Kalman filer for Lagrangian daa assimilaion. Monhly Weaher Review, 143: Snyder, C., Bengsson, T., Bickel, P., and Anderson, J. L. (2008). Obsacles o high-dimensional paricle filering. Monhly Weaher Review, 136: Sroud, J. R. and Bengsson, T. (2007). Sequenial sae and variance esimaion wihin he ensemble Kalman filer. Monhly Weaher Review, 135: Sroud, J. R., Kazfuss, M., and Wikle, C. K. (2018). A Bayesian adapive ensemble Kalman filer for sequenial sae and parameer esimaion. Monhly Weaher Review, 146(1): Jonahan Sroud (Georgeown) Exended EnKFs 30 / 31

36 Conclusions References IV Tippe, M. K., Anderson, J. L., Bishop, C. H., Hamill, T. M., and Whiaker, J. S. (2003). Ensemble square-roo filers. Monhly Weaher Review, 131: van Leeuwen, P. J. (2010). Nonlinear daa assimilaion in Geosciences: An exremely efficien paricle filer. Quarerly Journal of he Royal Meeorological Sociey, 136: Whiaker, J. S. and Hamill, T. M. (2002). Ensemble daa assimilaion wihou perurbed observaions. Monhly Weaher Review, 130: Zhang, F., Snyder, C., and Sun, J. (2004). Impacs of iniial esimae and observaion availabiliy on convecive scale daa assimilaion wih an ensemble Kalman filer. Monhly Weaher Review, 132: Jonahan Sroud (Georgeown) Exended EnKFs 31 / 31