Lagrangian Data Assimilation and Manifold Detection for a Point-Vortex Model. David Darmon, AMSC Kayo Ide, AOSC, IPST, CSCAMM, ESSIC

Transcription

1 Lagrangian Data Assimilation and Manifold Detection for a Point-Vortex Model David Darmon, AMSC Kayo Ide, AOSC, IPST, CSCAMM, ESSIC

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5 Background Data Assimilation Iterative process Forecast Analysis

6 Background Data Assimilation True System Dynamics - system state - Brownian motion - deterministic evolution operator

7 Background Data Assimilation Observations - observation of system at time - Gaussian noise - observation operator

8 Background Data Assimilation Forecast and Analysis Evolve a forecasted state forward somehow Perform analysis step upon receiving observation somehow

9 Background Lagrangian Data Assimilation Consider joint state of flow (F) and drifters (D)

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11 Background Model Point-Vortex (in complex plane) - j th vortex location - k th drifter location - vorticity of j th vortex

12 Background Model Demo

13 Background Observing System Design Deploying drifters is expensive Obtaining measurements is difficult Q: Can we design a system such that we optimally place the drifters to determine the location of the vortices?

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15 Background Observing System Design Deploying drifters is expensive Obtaining measurements is difficult Q: Can we design a system such that we optimally place the drifters to determine the location of the vortices?

16 Phase I

17 Approach, Phase I Lagrangian Data Assimilation Fokker-Planck Equation, Forecast - probability density of - covariance matrix from SDE

18 Approach, Phase I Lagrangian Data Assimilation Bayes s Theorem, Analysis - posterior distribution - likelihood - prior distribution

19 Approach, Phase I Computational Simplicity Extended Kalman Filter Ensemble Kalman Filter Particle Filter Equation and Bayes Fidelity to Solution of Fokker-Planck

20 Approach, Phase I Lagrangian Data Assimilation Extended Kalman Filter (EKF) Generalization of Kalman filter to nonlinear equations Gaussian assumptions on dynamical and observational noise matrix Represent pdf by mean and covariance

21 Approach, Phase I Lagrangian Data Assimilation Extended Kalman Filter (EKF) - mean state - covariance matrix

22 Approach, Phase I Lagrangian Data Assimilation Extended Kalman Filter (EKF), Forecast - covariance matrix from SDE - Jacobian of M

23 Approach, Phase I Lagrangian Data Assimilation Extended Kalman Filter (EKF), Forecast - covariance matrix from SDE - Jacobian of M

24 Approach, Phase I Lagrangian Data Assimilation Extended Kalman Filter (EKF), Forecast - covariance matrix from SDE - Jacobian of M

25 Approach, Phase I Lagrangian Data Assimilation Extended Kalman Filter (EKF), Analysis - covariance matrix from observation - Jacobian of hk

26 Approach, Phase I Lagrangian Data Assimilation Extended Kalman Filter (EKF), Analysis - covariance matrix from observation - Jacobian of hk

27 Approach, Phase I Lagrangian Data Assimilation Extended Kalman Filter (EKF), Analysis - covariance matrix from observation - Jacobian of hk

28 Approach, Phase I Lagrangian Data Assimilation Extended Kalman Filter (EKF), Analysis - covariance matrix from observation - Jacobian of hk

29 Approach, Phase I Lagrangian Data Assimilation Extended Kalman Filter (EKF) Weakly assumes nonlinearity Assumes Gaussianity NOT valid for nonlinear systems

30 Approach, Phase I Lagrangian Data Assimilation Ensemble Kalman Filter (EnKF) Approximate pdf by an ensemble of particles: Forecast: Evolve particles forward using SDE Analysis: Perform Kalman-like analysis

31 Approach, Phase I Lagrangian Data Assimilation Ensemble Kalman Filter (EnKF) Computing (ensemble) moments

32 Approach, Phase I Lagrangian Data Assimilation Ensemble Kalman Filter (EnKF), Analysis Stochastic: Generate ensemble of observations and perform Kalman-like analysis on combined ensemble Deterministic: Square root filters

33 Approach, Phase I Lagrangian Data Assimilation Ensemble Kalman Filter (EnKF) Better captures nonlinearity (in forecast) (up to ensemble approximation) Still assumes Gaussianity (in analysis) NOT valid for nonlinear systems

34 Approach, Phase I Lagrangian Data Assimilation Particle Filter Approximate pdf by an ensemble of weighted particles: Forecast: Evolve particles forward using SDE Analysis: Perform full Bayesian update at analysis

35 Approach, Phase I Lagrangian Data Assimilation Particle Filter, Analysis - observation of system at time - Gaussian noise - observation operator

36 Approach, Phase I Lagrangian Data Assimilation Particle Filter, Analysis Likelihood of Observation:

37 Approach, Phase I Lagrangian Data Assimilation Particle Filter, Analysis Likelihood of Observation:

38 Approach, Phase I Lagrangian Data Assimilation Particle Filter, Analysis Weight of particle i at analysis step k: Bayesian Update:

39 Approach, Phase I Lagrangian Data Assimilation Particle Filter Better captures nonlinearity (up to ensemble approximation) Does not assume Gaussianity on posterior or prior

40 Approach, Phase I Lagrangian Data Assimilation Particle Filter BUT Requires a large number of particles For large, nonlinear systems, requires frequent resampling

41 Phase II y x

42 Approach, Phase II Manifold Detection for Observing System Design Stream functions Consider the trajectories in phase space Trajectories will lie along the level curves of the stream function for steady flows Streamlines Trajectories may not cross streamlines Uniqueness

43 Stream function in corotating frame y x

44 Approach, Phase II Manifold Detection for Observing System Design Mendoza and Mancho s Lagrangian Descriptor M

45 Approach, Phase II Manifold Detection for Observing System Design Mendoza and Mancho s Lagrangian Descriptor M C. Mendoza and A.M. Mancho. Hidden geometry of ocean flows. Physical review letters, 105(3):38501, 2010.

46 Approach, Phase II Manifold Detection for Observing System Design Mendoza and Mancho s Lagrangian Descriptor M Improve Phase I by using knowledge of dynamics

47 Implementation Develop serial code for EKF, EnKF, and Particle Filter in MATLAB on MacBook Pro Parallelize the EnKF and particle filters Parallelize computation of M Use MATLAB Parallel Computing Toolbox

48 Databases Phase I Archived numerical solutions to point-vortex model Generate using stochastic 4 th -order Runge-Kutta method Phase II No databases: completely model dependent

49 Validation, Phase I Validate filter by varying noise Three stages Stage 1: Vortices positions known, low noise Stage 2: Vortices positions unknown, low noise Stage 3: Vortices positions unknown, realistic noise

50 Validation, Phase I Validate filter by varying noise Validation Metric

51 Validation, Phase I Validate filter by comparison Compare to published studies with Nv = 2 Nd = 1

52 Validation, Phase II Validate M by comparison Compare M to analytically known stream function Compare M to finite-time Lyapunov exponents

53 Testing, Phase I Failure statistic Compare EKF, EnKF, and particle filter across databases using

54 Testing, Phase II Failure statistic Compare EKF, EnKF, and particle filter, including decision of initial drifter position

55 Project Schedule and Milestones

56 Project Schedule and Milestones

57 Deliverables Database of sample trajectories used for validation and testing Suite of parallelized software for performing EKF, EnKF, and particle filtering for point-vortex model Software to compute M for point-vortex model

58 References (See Proposal for More) A.H. Jazwinski. Stochastic processes and filtering theory. Dover Publications, G. Evensen. Data assimilation: the ensemble Kalman filter. Springer Verlag, A. Doucet, N. De Freitas, and N. Gordon. Sequential Monte Carlo methods in practice. Springer Verlag, C. Mendoza and A.M. Mancho. Hidden geometry of ocean flows. Physical review letters, 105(3):38501, 2010.

59 Questions???