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
Š
€
exp − 21 (Hzif − yobs )T R−1 (Hzif − yobs )
€
Š
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
Š
€
exp − 21 (Hzif − yobs )T R−1 (Hzif − yobs )
€
Š
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
Š
€
exp − 21 (Hzif − yobs )T R−1 (Hzif − yobs )
€
Š
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
Š
€
exp − 21 (Hzif − yobs )T R−1 (Hzif − yobs )
€
Š
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
Š
€
exp − 21 (Hzif − yobs )T R−1 (Hzif − yobs )
€
Š
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