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Lagrangian Data Assimilation and Manifold Detection for a Point-Vortex Model
David Darmon, AMSC
Kayo Ide, AOSC, IPST, CSCAMM, ESSIC
Mid-year Progress Report
Today’s Presentation
Background
Database Generation
The EKF, Validation and Testing
The EnKF, Validation and Testing
Moving Forward
Background
Iterative processData Assimilation
Forecast
Analysis
True State
Observation
Database Generation
−2 −1 0 1 2−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
x
y
Database Generation
True System Dynamics
�
- system state
� - Brownian motion
- deterministic evolution operator
Database Generation
Point-VorticesDeterministic Dynamics
�
�
Database Generation
DriftersDeterministic Dynamics
�
�
Database Generation
Stochastic Differential EquationScalar vs. Vector Case
�
Database Generation
Runge-KuttaNumerical Solution
� �
Database Generation
Runge-KuttaNumerical Solution
Database Generation
Runge-KuttaNumerical Solution
Database Generation
ValidationSimple Scalar Case
�� � �
1 1.2 1.4 1.6
5
10
15
t
X(t)
True solutionRK solution
Database Generation
Stochastic Differential EquationTrue Dynamics
�
−1 −0.5 0 0.5 1 1.5 2−1.5
−1
−0.5
0
0.5
1
1.5
x
y
Vortex 1Vortex 2Drifter
−1 −0.5 0 0.5 1 1.5 2−1.5
−1
−0.5
0
0.5
1
1.5
x
y
Vortex 1Vortex 2Drifter
σ = 0.02
−1 −0.5 0 0.5 1 1.5 2−1.5
−1
−0.5
0
0.5
1
1.5
x
y
Vortex 1Vortex 2Drifter
σ = 0.02ρ = 0.02
Data Assimilation
Forecast
Analysis
True State
Observation
Data Assimilation
Approach 1:EKF
Approach 1
Extended Kalman Filter (EKF), Forecast
- covariance matrix from SDE�� - Jacobian of M
Approach 1
Extended Kalman Filter (EKF)Validation - Jacobian via Tangent Linear Model
- Reference Trajectory- Perturbed Trajectory
Approach 1
Extended Kalman Filter (EKF)Validation - Jacobian via Tangent Linear Model
- Reference Trajectory- Perturbed Trajectory
Approach 1
Extended Kalman Filter (EKF)Validation - Jacobian via Tangent Linear Model
- Reference Trajectory- Perturbed Trajectory
Tangent Vector (small)
Approach 1
Extended Kalman Filter (EKF)Validation - Jacobian via Tangent Linear Model
����
Approach 1
Extended Kalman Filter (EKF)Validation - Jacobian via Tangent Linear Model
Evolve reference and perturbed trajectories forward using deterministic ODE
Evolve tangent vector forward using Tangent Linear Model
Compare and
0 0.5 1 1.5 27.5
8
8.5
9
9.5
10 x 10−11
t
y 1
Tangent LinearRK
Approach 1
Extended Kalman Filter (EKF), Analysis
- covariance matrix from observation
- Jacobian of hk
��
Approach 1
Extended Kalman Filter (EKF)Validation - No Noise
Approach 1
Extended Kalman Filter (EKF)Validation - Full Observation, Imperfect Data
Observe vortices and drifter under observational noise
−1 −0.5 0 0.5 1 1.5 2−1.5
−1
−0.5
0
0.5
1
1.5
x
y
0 5 10 15 20−2
−1
0
1
t
Drifter
ξ1
0 5 10 15 20 25
−1
−0.5
0
0.5
1
t
Vortex 1
x1
0 20 40 600
0.5
1
1.5
t
||Xtru
e − X
filte
r|| 2
Approach 1
Extended Kalman Filter (EKF)Results - Partial Observation, Imperfect Data
Observe only drifter under observational noise
−1 −0.5 0 0.5 1 1.5 2−1.5
−1
−0.5
0
0.5
1
1.5
x
y
0 10 20 30
−2
−1
0
1
t
Drifter
ξ1
0 20 40 60−4
−2
0
2
4
6
8
t
Vortex 1
x1
0 20 40 600
5
10
15
t
||Xtru
e − X
filte
r|| 2
Data Assimilation
Approach 1:EnKF
Approach 2
Ensemble Kalman Filter (EnKF)
Approximate pdf by an ensemble of particles:
Evolve particles forward using SDE
Perform Kalman-like analysis
Forecast:
Analysis:
Approach 2
Ensemble Kalman Filter (EnKF)
Approximate pdf by an ensemble of particles:
Evolve particles forward using SDE ODE
Perform Kalman-like analysis
Forecast:
Analysis:
Approach 2
Ensemble Kalman Filter (EnKF)
Approximate pdf by an ensemble of particles:
Evolve particles forward using ODE
Perform Kalman-like analysis
Forecast:
Analysis:
Approach 2a
EnKF with Observational Perturbations
�
�
Approach 2a
EnKF with Observational Perturbations
�
Forecast ensembleObservation ensemble
�
Approach 2a
EnKF with Observational PerturbationsKalman-like update
Approach 2a
EnKF with Observational PerturbationsKalman-like update
�Sample Covariance Matrix
Approach 2
Ensemble Kalman Filter (EnKF)
Approximate pdf by an ensemble of particles:
Evolve particles forward using SDE ODE
Perform Kalman-like analysis
Forecast:
Analysis:
Approach 2b
Ensemble Transform Kalman Filter (ETKF)
Approach 2b
Ensemble Transform Kalman Filter (ETKF)
Use sample estimates:
Approach 2b
Ensemble Transform Kalman Filter (ETKF)
� �Define
Approach 2b
Ensemble Transform Kalman Filter (ETKF)
� �Define
Approach 2b
Ensemble Transform Kalman Filter (ETKF)
� �Define
Approach 2b
Ensemble Transform Kalman Filter (ETKF)
Define
is the matrix square root of
Choose
Approach 2b
Ensemble Transform Kalman Filter (ETKF)
Analysis Ensemble
−1 −0.5 0 0.5 1 1.5 2−1.5
−1
−0.5
0
0.5
1
1.5
x
y
Approach 2
Ensemble Kalman Filter (EnKF)Validation - Full Observation, Imperfect Data
Observe vortices and drifter under observational noise
N = 6 ensemble members
5 10 15 20 25 30
−3
−2
−1
0
1
t
Drifter
ξ1
0 20 40 60−1.5
−1
−0.5
0
0.5
1
1.5
t
Vortex 1
x1
0 20 40 600
0.5
1
1.5
2
2.5
Time (s)
||Xtru
e − X
filte
r|| 2
Approach 2
Ensemble Kalman Filter (EnKF)Results - Partial Observation, Imperfect Data
Observe only drifter under observational noise
N = 6 ensemble members
−1 −0.5 0 0.5 1 1.5 2−1.5
−1
−0.5
0
0.5
1
1.5
x
y
5 10 15 20 25−2
−1
0
1
2
t
Drifter
ξ1
0 20 40 60−2
−1
0
1
2
t
Vortex 1
x1
0 20 40 600
1
2
3
4
5
6
Time (s)
||Xtru
e − X
filte
r|| 2
Data Assimilation
Testing
Testing, Phase I
Compare EKF, perturbed observations EnKF and ETKF
Failure statistic: time to failure
Testing, Phase I
Generate M = 500 instances of the SDE at each of L = 4 drifter locations
Start vortex 1 at (0, 1)Start vortex 2 at (0, -1)
Add observational noise according to the model
Failure statistic: time to failure
−2 0 2
−2
−1
0
1
2
x
y
Testing, Phase I
Record when the distance between the analyzed state and the true state for either vortex is greater than 1
Failure statistic: time to failure
−2 0 2
−2
−1
0
1
2
x
y
Testing, Phase I
Failure statistic: time to failure
0 20 40 600
0.05
0.1
0.15
0.2
Time to Failure
Frac
tion
Faile
d
EKF
0 20 40 600
0.05
0.1
0.15
0.2
Time to Failure
Frac
tion
Faile
d
EnKF,PerturbedObservation
0 20 40 600
0.05
0.1
0.15
0.2
Time to Failure
Frac
tion
Faile
d
ETKF
Moving Forward
Improvements to EnKF
Covariance Inflation
Localization Local Ensemble Transform Kalman Filter (LETKF)
Moving Forward
Implement Particle Filter
Approximate pdf by an ensemble of weighted particles:
Evolve particles forward using SDE
Perform full Bayesian update at analysis
Forecast:
Analysis:
Moving Forward
Phase II - Manifold Detection for Observing System Design
−2 −1 0 1 2−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
x
y
Timeline
– Produce database: now through mid-October– Develop extended Kalman Filter: now through mid-October– Develop ensemble Kalman Filter: mid-October through mid-November– Develop particle filter: mid-November through end of January– Validation and testing of three filters (serial): Beginning in mid-October, complete by February
Phase I
Timeline
– Produce database: now through mid-October– Develop extended Kalman Filter: now through mid-October– Develop ensemble Kalman Filter: mid-October through mid-November– Develop particle filter: mid-November through end of January– Validation and testing of three filters (serial): Beginning in mid-October, complete by February
Phase I
(Some) References
D.J. Higham. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM review, pages 525–546, 2001.
A.H. Jazwinski. Stochastic processes and filtering theory. Dover Publications, 2007.
E. Kalnay. Atmospheric modeling, data assimilation, and predictability. Cambridge University Press, 2003.
G. Evensen. Data assimilation: the ensemble Kalman filter. Springer Verlag, 2009.
T. DelSole and M. Tippett. The ensemble square root filter. Accessed: http://mason.gmu.edu/%7Etdelsole/AdvStat/
Questions???
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