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

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Database Generation

True System Dynamics

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- system state

� - Brownian motion

- deterministic evolution operator

Database Generation

Point-VorticesDeterministic Dynamics

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Database Generation

DriftersDeterministic Dynamics

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Database Generation

Stochastic Differential EquationScalar vs. Vector Case

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Database Generation

Runge-KuttaNumerical Solution

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Database Generation

Runge-KuttaNumerical Solution

Database Generation

Runge-KuttaNumerical Solution

Database Generation

ValidationSimple Scalar Case

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True solutionRK solution

Database Generation

Stochastic Differential EquationTrue Dynamics

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Vortex 1Vortex 2Drifter

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Vortex 1Vortex 2Drifter

σ = 0.02

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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

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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

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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

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Drifter

ξ1

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Vortex 1

x1

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filte

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Approach 1

Extended Kalman Filter (EKF)Results - Partial Observation, Imperfect Data

Observe only drifter under observational noise

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Drifter

ξ1

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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

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Approach 2a

EnKF with Observational Perturbations

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Forecast ensembleObservation ensemble

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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

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Approach 2

Ensemble Kalman Filter (EnKF)Validation - Full Observation, Imperfect Data

Observe vortices and drifter under observational noise

N = 6 ensemble members

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Drifter

ξ1

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Vortex 1

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Approach 2

Ensemble Kalman Filter (EnKF)Results - Partial Observation, Imperfect Data

Observe only drifter under observational noise

N = 6 ensemble members

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Drifter

ξ1

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x1

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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

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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

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Testing, Phase I

Failure statistic: time to failure

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Time to Failure

Frac

tion

Faile

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EKF

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Time to Failure

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tion

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EnKF,PerturbedObservation

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tion

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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

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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???