Scaling Cascades in Complex Systems Non-Gaussian data assimilation via a localized hybrid ensemble transform filter Walter Acevedo, Nawinda Chutsagulprom, Maria Reinhardt and Sebastian Reich Universität Potsdam/ University of Reading Warwick, February 22nd 2016 Outline Scaling Cascades in Complex Systems Motivation UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 2 Outline Scaling Cascades in Complex Systems Motivation Hybrid Scheme Ensemble Transform Particle Filter [Reich and Cotter, 2015] Hybrid Ensemble Transform Filter [Chustagulprom et al., 2015] Example: Single 1D Assimilation Step UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 2 Outline Scaling Cascades in Complex Systems Motivation Hybrid Scheme Ensemble Transform Particle Filter [Reich and Cotter, 2015] Hybrid Ensemble Transform Filter [Chustagulprom et al., 2015] Example: Single 1D Assimilation Step Non-spatially extended systems Likelihood splitting strategies Ensemble Inflation and Particle Rejuvenation Example: Lorenz 63 model UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 2 Outline Scaling Cascades in Complex Systems Motivation Hybrid Scheme Ensemble Transform Particle Filter [Reich and Cotter, 2015] Hybrid Ensemble Transform Filter [Chustagulprom et al., 2015] Example: Single 1D Assimilation Step Non-spatially extended systems Likelihood splitting strategies Ensemble Inflation and Particle Rejuvenation Example: Lorenz 63 model Spatially extended systems Localization Example: Lorenz 96 model UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 2 Outline Scaling Cascades in Complex Systems Motivation Hybrid Scheme Ensemble Transform Particle Filter [Reich and Cotter, 2015] Hybrid Ensemble Transform Filter [Chustagulprom et al., 2015] Example: Single 1D Assimilation Step Non-spatially extended systems Likelihood splitting strategies Ensemble Inflation and Particle Rejuvenation Example: Lorenz 63 model Spatially extended systems Localization Example: Lorenz 96 model Conclusions and Prospect , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 2 Sequential Data Assimilation Strategies UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 Scaling Cascades in Complex Systems 3 Sequential Data Assimilation Strategies É Scaling Cascades in Complex Systems Ensemble Kalman Filters (Gaussian Approach) + Robust + Computationally affordable – Inconsistent for non-Gaussian PDFs UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 3 Sequential Data Assimilation Strategies Scaling Cascades in Complex Systems É Ensemble Kalman Filters (Gaussian Approach) + Robust + Computationally affordable – Inconsistent for non-Gaussian PDFs É Particle Filters (Non-parametric Approach) + consistent, suitable for non-Gaussian PDFs – Liable to the "Curse of Dimensionality” UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 3 Sequential Data Assimilation Strategies Scaling Cascades in Complex Systems É Ensemble Kalman Filters (Gaussian Approach) + Robust + Computationally affordable – Inconsistent for non-Gaussian PDFs É Particle Filters (Non-parametric Approach) + consistent, suitable for non-Gaussian PDFs – Liable to the "Curse of Dimensionality” É Hybrid schemes + trade-off between accuracy and stability , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 3 Outline Scaling Cascades in Complex Systems Motivation Hybrid Scheme Ensemble Transform Particle Filter [Reich and Cotter, 2015] Hybrid Ensemble Transform Filter [Chustagulprom et al., 2015] Example: Single 1D Assimilation Step Non-spatially extended systems Likelihood splitting strategies Ensemble Inflation and Particle Rejuvenation Example: Lorenz 63 model Spatially extended systems Localization Example: Lorenz 96 model Conclusions and Prospect , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 4 Ensemble Transform Particle Filter [Reich and Cotter, 2015] Importance weights: Scaling Cascades in Complex Systems exp − 21 (Hzif − yobs )T R−1 (Hzif − yobs ) wi = P M exp − 12 (Hzjf − yobs )T R−1 (Hzjf − yobs ) j=1 (1) , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 5 Ensemble Transform Particle Filter [Reich and Cotter, 2015] Importance weights: Scaling Cascades in Complex Systems exp − 21 (Hzif − yobs )T R−1 (Hzif − yobs ) wi = P M exp − 12 (Hzjf − yobs )T R−1 (Hzjf − yobs ) j=1 (1) Importance weigth 0.1 0.08 0.06 0.04 0.02 0 5 10 15 Particle Number 20 , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 5 Ensemble Transform Particle Filter [Reich and Cotter, 2015] Importance weights: Scaling Cascades in Complex Systems exp − 21 (Hzif − yobs )T R−1 (Hzif − yobs ) wi = P M exp − 12 (Hzjf − yobs )T R−1 (Hzjf − yobs ) j=1 0.1 0.08 0.06 ⇒ 0.04 0.02 0 5 10 15 Particle Number Optimal Coupling T∗ ⇒ 20 Importance weigth Importance weigth 0.1 (1) 0.08 0.06 0.04 0.02 0 5 10 15 Particle Number 20 , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 5 Ensemble Transform Particle Filter [Reich and Cotter, 2015] Importance weights: Scaling Cascades in Complex Systems exp − 21 (Hzif − yobs )T R−1 (Hzif − yobs ) wi = P M exp − 12 (Hzjf − yobs )T R−1 (Hzjf − yobs ) j=1 0.1 0.08 0.06 ⇒ 0.04 0.02 0 5 10 15 Particle Number Optimal Coupling T∗ ⇒ 20 T ∗ : {tij∗ ≥ 0} is a linear transformation: Importance weigth Importance weigth 0.1 (1) 0.08 0.06 0.04 0.02 0 zja = M 5 10 15 Particle Number M X zif tij∗ 20 (2) i=1 , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 5 Ensemble Transform Particle Filter [Reich and Cotter, 2015] Importance weights: Scaling Cascades in Complex Systems exp − 21 (Hzif − yobs )T R−1 (Hzif − yobs ) wi = P M exp − 12 (Hzjf − yobs )T R−1 (Hzjf − yobs ) j=1 0.1 0.08 0.06 ⇒ 0.04 0.02 0 5 10 15 Particle Number Optimal Coupling T∗ ⇒ Importance weigth Importance weigth 0.1 (1) 0.08 0.06 0.04 0.02 0 20 5 zja = M T ∗ : {tij∗ ≥ 0} is a linear transformation: 10 15 Particle Number M X zif tij∗ 20 (2) i=1 which minimizes the cost function: J({tij }) = M X tij kzif − zjf k2 (3) i,j=1 , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 5 Outline Scaling Cascades in Complex Systems Motivation Hybrid Scheme Ensemble Transform Particle Filter [Reich and Cotter, 2015] Hybrid Ensemble Transform Filter [Chustagulprom et al., 2015] Example: Single 1D Assimilation Step Non-spatially extended systems Likelihood splitting strategies Ensemble Inflation and Particle Rejuvenation Example: Lorenz 63 model Spatially extended systems Localization Example: Lorenz 96 model Conclusions and Prospect , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 6 Hybrid Ensemble Transform Filter [Chustagulprom et al., 2015] Scaling Cascades in Complex Systems Likelihood function splitting πY (yobs |z) ∝ α exp − (Hz − yobs )T R−1 (Hz − yobs ) × 2 1− α (Hz − yobs )T R−1 (Hz − yobs ) exp − 2 with bridging parameter α ∈ [0, 1] , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 7 Hybrid Ensemble Transform Filter [Chustagulprom et al., 2015] Scaling Cascades in Complex Systems Likelihood function splitting πY (yobs |z) ∝ α exp − (Hz − yobs )T R−1 (Hz − yobs ) × 2 1− α (Hz − yobs )T R−1 (Hz − yobs ) exp − 2 with bridging parameter α ∈ [0, 1] ⇒ ETPF ⇒ EnKF, ETKF, ESRF,. . . , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 7 Hybrid Ensemble Transform Filter [Chustagulprom et al., 2015] Scaling Cascades in Complex Systems Likelihood function splitting πY (yobs |z) ∝ α exp − (Hz − yobs )T R−1 (Hz − yobs ) × 2 1− α (Hz − yobs )T R−1 (Hz − yobs ) exp − 2 with bridging parameter α ∈ [0, 1] ⇒ ETPF ⇒ EnKF, ETKF, ESRF,. . . Related work Ensemble Kalman Particle Filter (EKPF) [Frei and Künsch, 2013]: bridges the EnKF with perturbed observations and a SIR particle filter. , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 7 Outline Scaling Cascades in Complex Systems Motivation Hybrid Scheme Ensemble Transform Particle Filter [Reich and Cotter, 2015] Hybrid Ensemble Transform Filter [Chustagulprom et al., 2015] Example: Single 1D Assimilation Step Non-spatially extended systems Likelihood splitting strategies Ensemble Inflation and Particle Rejuvenation Example: Lorenz 63 model Spatially extended systems Localization Example: Lorenz 96 model Conclusions and Prospect , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 8 Example: Single 1D Assimilation Step Bayesian Inference for bimodal prior and Gaussian likelihood Scaling Cascades in Complex Systems Baysian inference 0.45 prior likelihood posterior 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -10 -5 0 5 10 , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 9 Example: Single 1D Assimilation Step Scaling Cascades in Complex Systems ETPF-ESRF performance vs ensemble size 0.4 RMS error ETPF-ESRF 0.1 101 102 103 102 103 1 bridging parameter 0.8 optimal α α = 1 : ETPF) 0.2 0 for optimally chosen (α = 0: EnKF 0.3 0.6 0.4 0.2 0 101 ensemble size , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 10 Outline Scaling Cascades in Complex Systems Motivation Hybrid Scheme Ensemble Transform Particle Filter [Reich and Cotter, 2015] Hybrid Ensemble Transform Filter [Chustagulprom et al., 2015] Example: Single 1D Assimilation Step Non-spatially extended systems Likelihood splitting strategies Ensemble Inflation and Particle Rejuvenation Example: Lorenz 63 model Spatially extended systems Localization Example: Lorenz 96 model Conclusions and Prospect , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 11 Likelihood splitting strategies É Scaling Cascades in Complex Systems Fixed: keeping bridging parameter constant , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 12 Likelihood splitting strategies É Scaling Cascades in Complex Systems Fixed: keeping bridging parameter constant É α = 0 ⇒ Pure Kalman Filter , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 12 Likelihood splitting strategies É Scaling Cascades in Complex Systems Fixed: keeping bridging parameter constant É α = 0 ⇒ Pure Kalman Filter É α = 1 ⇒ Pure Particle Filter , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 12 Likelihood splitting strategies É É Scaling Cascades in Complex Systems Fixed: keeping bridging parameter constant É α = 0 ⇒ Pure Kalman Filter É α = 1 ⇒ Pure Particle Filter Adaptive: keeping ETPF effective sample size METPF (α) := PM eff i=1 constant 1 wi (α)2 METPF (α) = θ M eff (4) , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 12 Likelihood splitting strategies É É Scaling Cascades in Complex Systems Fixed: keeping bridging parameter constant É α = 0 ⇒ Pure Kalman Filter É α = 1 ⇒ Pure Particle Filter Adaptive: keeping ETPF effective sample size METPF (α) := PM eff i=1 constant É 1 wi (α)2 METPF (α) = θ M eff (4) θ = 0 ⇒ α = 1 (pure PF) , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 12 Likelihood splitting strategies É É Scaling Cascades in Complex Systems Fixed: keeping bridging parameter constant É α = 0 ⇒ Pure Kalman Filter É α = 1 ⇒ Pure Particle Filter Adaptive: keeping ETPF effective sample size METPF (α) := PM eff i=1 constant É É 1 wi (α)2 METPF (α) = θ M eff (4) θ = 0 ⇒ α = 1 (pure PF) θ = 1 ⇒ α = 0 (pure EnkF) , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 12 Outline Scaling Cascades in Complex Systems Motivation Hybrid Scheme Ensemble Transform Particle Filter [Reich and Cotter, 2015] Hybrid Ensemble Transform Filter [Chustagulprom et al., 2015] Example: Single 1D Assimilation Step Non-spatially extended systems Likelihood splitting strategies Ensemble Inflation and Particle Rejuvenation Example: Lorenz 63 model Spatially extended systems Localization Example: Lorenz 96 model Conclusions and Prospect , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 13 Ensemble Inflation and Particle Rejuvenation Scaling Cascades in Complex Systems In order to prevent É Ensemble under-dispersion É Particle degeneracy , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 14 Ensemble Inflation and Particle Rejuvenation Scaling Cascades in Complex Systems In order to prevent É Ensemble under-dispersion É Particle degeneracy Particle rejuvenation is applied to the analysis ensemble: zja → zja + M X (zif − z̄f ) p i=1 βξij M−1 (5) , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 14 Ensemble Inflation and Particle Rejuvenation Scaling Cascades in Complex Systems In order to prevent É Ensemble under-dispersion É Particle degeneracy Particle rejuvenation is applied to the analysis ensemble: zja → zja + M X (zif − z̄f ) p i=1 βξij M−1 É β: Rejuvenation parameter É ξij ’s: i.i.d. Gaussian random variables with mean zero and variance one PM ξ = 0 so as to preserve the ensemble mean j=1 ij É (5) , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 14 Outline Scaling Cascades in Complex Systems Motivation Hybrid Scheme Ensemble Transform Particle Filter [Reich and Cotter, 2015] Hybrid Ensemble Transform Filter [Chustagulprom et al., 2015] Example: Single 1D Assimilation Step Non-spatially extended systems Likelihood splitting strategies Ensemble Inflation and Particle Rejuvenation Example: Lorenz 63 model Spatially extended systems Localization Example: Lorenz 96 model Conclusions and Prospect , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 15 Example: Lorenz 63 model Scaling Cascades in Complex Systems 50 45 40 ẋ1 = 10(x2 − x1 ) ẋ2 = x1 (28 − x3 ) − x2 35 30 25 20 15 ẋ3 = x1 x2 − 8/ 3x3 10 50 5 0 −20 0 −10 0 10 20 −50 , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 16 Example: Lorenz 63 model Scaling Cascades in Complex Systems 50 45 40 ẋ1 = 10(x2 − x1 ) ẋ2 = x1 (28 − x3 ) − x2 35 30 25 20 15 ẋ3 = x1 x2 − 8/ 3x3 10 50 5 0 −20 0 −10 0 10 20 −50 Perfect model DA experiments: É Implicit midpoint method with time step ∆t = 0.01. É x1 observed every 12 time-steps with error variance R = 8 É Particle rejuvenation β = 0.2 É 100,000 assimilation cycles É OTP solved using FastEMD algorithm , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 16 Example: Lorenz 63 model Scaling Cascades in Complex Systems hybrid ETPF-ESRF Skill dependence on M=15 M=20 M=25 M=30 M=35 3 bridging parameter 2.8 ensemble sizes using fixed RMS error for different 2.6 2.4 likelihood splitting 2.2 2 0 0.2 0.4 0.6 0.8 1 bridging parameter α , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 17 Example: Lorenz 63 model Scaling Cascades in Complex Systems hybrid ESRF-ETPF Skill dependence on M=15 M=20 M=25 M=30 M=35 3 bridging parameter ensemble sizes using fixed likelihood splitting RMS error 2.8 for different 2.6 2.4 2.2 2 0 0.1 0.2 0.3 0.4 0.5 bridging parameter α , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 18 Example: Lorenz 63 model Scaling Cascades in Complex Systems hybrid ETPF-ESRF Skill dependence on M=15 M=20 M=25 M=30 M=35 3 effective ETPF 2.8 for different ensemble sizes RMS error sample size 2.6 2.4 using adaptive 2.2 likelihood splitting 2 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 effective sample size parameter θ , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 19 Scaling Cascades in Complex Systems hybrid ETPF-LETKF Skill dependence on M=15 M=20 M=25 M=30 M=35 3 bridging parameter 2.8 ensemble sizes using fixed RMS error for different 2.6 2.4 likelihood splitting 2.2 and wrong parameter values 2 0 0.2 0.4 0.6 0.8 1 bridging parameter α (10.2, 28.2, 2.5) , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 20 Scaling Cascades in Complex Systems hybrid ETPF-LETKF Skill dependence on M=15 M=20 M=25 M=30 M=35 3.2 bridging parameter 3 ensemble sizes using fixed RMS error for different 2.8 2.6 likelihood splitting and wrong parameter values 2.4 2.2 0 0.2 0.4 0.6 bridging parameter α (10.3, 28.4, 2.9) , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 21 Outline Scaling Cascades in Complex Systems Motivation Hybrid Scheme Ensemble Transform Particle Filter [Reich and Cotter, 2015] Hybrid Ensemble Transform Filter [Chustagulprom et al., 2015] Example: Single 1D Assimilation Step Non-spatially extended systems Likelihood splitting strategies Ensemble Inflation and Particle Rejuvenation Example: Lorenz 63 model Spatially extended systems Localization Example: Lorenz 96 model Conclusions and Prospect , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 22 R-Localization Scaling Cascades in Complex Systems Localized measurement error covariance elements: LOC rqq (xk ) = rqq ρ kxk −xq k Rloc , with ρ a compactly supported tempering function, e.g., Gaspari-Cohn function. , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 23 R-Localization Scaling Cascades in Complex Systems Localized measurement error covariance elements: LOC rqq (xk ) = rqq ρ kxk −xq k Rloc , with ρ a compactly supported tempering function, e.g., Gaspari-Cohn function. , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 23 R-Localization Scaling Cascades in Complex Systems Localized measurement error covariance elements: LOC rqq (xk ) = rqq ρ kxk −xq k Rloc , with ρ a compactly supported tempering function, e.g., Gaspari-Cohn function. É Local updates for each gridpoint É Directly applicable to Kalman Filters É Directly applicable to ETPF [Cheng and Reich, 2015] (local weights and transport cost) , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 23 Outline Scaling Cascades in Complex Systems Motivation Hybrid Scheme Ensemble Transform Particle Filter [Reich and Cotter, 2015] Hybrid Ensemble Transform Filter [Chustagulprom et al., 2015] Example: Single 1D Assimilation Step Non-spatially extended systems Likelihood splitting strategies Ensemble Inflation and Particle Rejuvenation Example: Lorenz 63 model Spatially extended systems Localization Example: Lorenz 96 model Conclusions and Prospect , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 24 Example: Lorenz 96 model Scaling Cascades in Complex Systems ẋj = (xj+1 − xj−2 )xj−1 − xj + F, xj = xj+N where F = 8 and N = 40. , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 25 Example: Lorenz 96 model Scaling Cascades in Complex Systems ẋj = (xj+1 − xj−2 )xj−1 − xj + F, xj = xj+N where F = 8 and N = 40. Perfect model DA experiments: É Implicit midpoint method with time step ∆t = 0.005. É Odd variables observed every 22 time-steps É Particle rejuvenation β = 0.2 É Localisation radius is Rloc = 4 É 50,000 assimilation cycles É OTP solved using FastEMD algorithm , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 25 Example: Lorenz 96 model Scaling Cascades in Complex Systems hybrid ETPF-LETKF 1.75 Skill dependence on bridging parameter ensemble sizes using fixed likelihood splitting RMS error for different 1.7 1.65 1.6 M=20 M=25 M=30 1.55 0 0.1 0.2 0.3 0.4 0.5 0.6 bridging parameter α , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 26 Example: Lorenz 96 model M=20 M=25 M=30 effective ETPF ensemble sizes using adaptive 1.7 RMS error for different hybrid ETPF-LETKF 1.75 Skill dependence on sample size Scaling Cascades in Complex Systems 1.65 1.6 likelihood splitting 1.55 0.75 0.8 0.85 0.9 0.95 1 effective sample size parameter θ , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 27 Conclusions UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 Scaling Cascades in Complex Systems 28 Conclusions É Scaling Cascades in Complex Systems Hybrid approach outperforms both ETKF/ESRF and ETPF for a suitable bridging parameter UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 28 Conclusions É Scaling Cascades in Complex Systems Hybrid approach outperforms both ETKF/ESRF and ETPF for a suitable bridging parameter É Applying ETPF before the ensemble Kalman filter appeared more profitable, especially for Lorenz 63 model. Arguably due to its strongly nonlinear dynamics. UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 28 Conclusions É Scaling Cascades in Complex Systems Hybrid approach outperforms both ETKF/ESRF and ETPF for a suitable bridging parameter É Applying ETPF before the ensemble Kalman filter appeared more profitable, especially for Lorenz 63 model. Arguably due to its strongly nonlinear dynamics. É Adaptive likelihood splitting was beneficial for Lorenz-96 system and detrimental for Lorenz-63 model. Other adaptation criteria must be tested UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 28 Conclusions É Scaling Cascades in Complex Systems Hybrid approach outperforms both ETKF/ESRF and ETPF for a suitable bridging parameter É Applying ETPF before the ensemble Kalman filter appeared more profitable, especially for Lorenz 63 model. Arguably due to its strongly nonlinear dynamics. É Adaptive likelihood splitting was beneficial for Lorenz-96 system and detrimental for Lorenz-63 model. Other adaptation criteria must be tested É Proposed hybrid filter can be combined with R-localization. UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 28 Ongoing Research and Prospect É Scaling Cascades in Complex Systems Hybrid scheme being currently implemented into the DA system of the Deutsche Wetterdienst , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 29 Ongoing Research and Prospect É Scaling Cascades in Complex Systems Hybrid scheme being currently implemented into the DA system of the Deutsche Wetterdienst É Spatial regularity of analysis fields after localisation , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 29 Ongoing Research and Prospect É Scaling Cascades in Complex Systems Hybrid scheme being currently implemented into the DA system of the Deutsche Wetterdienst É Spatial regularity of analysis fields after localisation É Impact of systematic model errors on hybrid scheme , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 29 Ongoing Research and Prospect É Scaling Cascades in Complex Systems Hybrid scheme being currently implemented into the DA system of the Deutsche Wetterdienst É Spatial regularity of analysis fields after localisation É Impact of systematic model errors on hybrid scheme É Hybrid ensemble transform smoother , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 29 References I Scaling Cascades in Complex Systems Cheng, Y. and Reich, S. (2015). Nonlinear Data Assimilation, chapter Assimilating data into scientific models: An optimal coupling perspective, pages 75–118. Springer-Verlag. Chustagulprom, N., Reich, S., and Reinhardt, M. (2015). A hybrid ensemble transform filter for nonlinear and spatially extended dynamical systems. submitted, available from ArXiv. Frei, M. and Künsch, H. R. (2013). Bridging the ensemble Kalman and particle filters. Biometrika. Reich, S. and Cotter, C. (2015). Probabilistic Forecasting and Bayesian Data Assimilation. Cambridge University Press. , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 30 Scaling Cascades in Complex Systems Thanks! , UofP/ UofR, Hybrid ensemble transform filter, University of Warwick, February 22nd 2016 31 Scaling Cascades in Complex Systems Mathematisches Forschungsinstitut Oberwolfach Oberwolfach Seminars 2016 Data Assimilation: The Mathematics of Connecting Dynamical Systems to Data Organizers: Jana de Wiljes, Potsdam Sebastian Reich, Potsdam and Reading Andrew Stuart, Warwick Date (ID): 15 – 21 May 2016 (1620a) Deadline: 13 March 2016 UofP/ UofR, Hybrid Recent Advances on the Global Nonlinear Stability of Einstein Spacetimes Organizers: Mihalis Dafermos, Princeton Philippe LeFloch, Paris Qian Wang, Oxford Date (ID): 15 – 21 May 2016 (1620b) ensemble Deadline: transform University of Warwick, 13 filter, March 2016 Mathematical Theory of Evolutionary Fluid-Flow Structure Interactions Organizers: Barbara Kaltenbacher, Klagenfurt Igor Kukavica, Los Angeles Irena Lasiecka, Memphis Roberto Triggiani, Memphis Date (ID): 20 – 26 November 2016 (1647b) Deadline: 18 September 2016 About Oberwolfach Seminars The Oberwolfach Seminars are organized by leading experts in the field, and address to postdocs and Ph.D. students from all over the world. The aim is to introduce the participants to a particular interesting development. February 22nd 2016 , 32