Methods of Data Assimilation and Comparisons for Lagrangian Data

Transcription

1 Methods of Data Assimilation and Comparisons for Lagrangian Data Chris Jones, Warwick and UNC-CH Kayo Ide, UCLA Andrew Stuart, Jochen Voss, Warwick Guillaume Vernieres, UNC-CH Amarjit Budiraja, UNC-CH Elaine Spiller, SAMSI Amit Apte, TIFR Bangalore Supported by the Office of Naval Research

2 DATA: ever-improving experimental technology has led to vast amounts of accumulated data MODEL: ever-increasing computational capacity has led to greater model capability and output Scientific imperative: bring data and computations together to work in harmony to enhance prediction and estimation

3 Eddies in the Gulf

4 t = t 0 x ( t ), x ( t ) t t t = t 1 State estimate: x ( t ), x ( t ), P ( t ) a a a Model forecast: x ( t ), x ( t ), P ( t ) f f f Initial conditions: f f f x ( t ), x ( t ) t t x ( t ), x ( t ), P ( t ) Measurement: y ( t ) = x ( t ) + ε o t Gain Matrix

5 Sequential Data Assimilation Model + observations prediction t 0 model t1 t2 model t N obs update obs update interpolation interpolation

6 Ocean model: x R N f f dx = M x t dt state vector comprising all relevant dynamical variables (e.g. flow velocity, temperature, salinity, etc. at each grid point) (, ) prognos t t d = M(, t) dt + ( t) dt x x η T E[ η( t) η ( t')] = δ ( t t') Q( t) tic model actual evolution covariance of the model residual Observations: o t y i = H i[ xi] + εi H ε i y i o i observation operator observation error R L T E[ εε ], i m im i typically L N = δ R observation error covariance

7 Kalman Filter Distributions are Gaussian Model is linear-tlm (EKF) or fit to Gaussian (EnKF)

8 Extended Kalman Filter Forecast model error covariance using tangent linear model: i P = E[ ΔxΔx ]; Δx x x f T f t f dp f f T = MP + P M + Q( t) dt M M x t x f (, )/ : linearized model operator a Combine mod el and observations into a new state minimizing tr i H x = x + K d d = y H ( x ) a f o f j j j j j j j j f T f 1 a f j = j j j j j + j j = j j j ( T ) ( ) K P H H P H R P I K H P H : linearized observation function i / x x P a j

9 Ensemble Kalman Filter (EnKF) Error covariance is predicted via solution of full nonlinear system for a Monte- Carlo ensemble of states Model forecast: x ( t ), y ( t ), j = 1, N f f j 1 j 1 E 1 1 x ( t ) x ( t ), y ( t ) y ( t ) NE NE f f f f 1 = j 1 1 = j 1 NE j= 1 NE j= 1 t t x t1 y t1 ( ), ( ) t t x t0 y t0 ( ), ( ) Initial conditions: x ( t ), y ( t ), j = 1, N f f j 0 j 0 E 1 1 x ( t ) x ( t ), y ( t ) y ( t ) NE NE f f f f 0 = j 0 0 = j 0 NE j= 1 NE j= 1

10 Update step in EnKF Kalman gain matrix is computed using error covariance matrix derived from the ensemble. Ensemble members are updated with noisy observations 1 1 x x P x x x x N N E E f f f f f f f = j = j j N j= 1 1 j 1 E N = E ( )( ) T d = y + ε H x E ε ε R o f T Ensemble of observations: j j ( j) [ j j ] = Update ensemble members: x = x + Kd a f j j j ( f T ) 1 f T K= PH HPH + R

11 Lagrangian Data Floats: subsurface, pressurized to stay at certain depth Drifters: on surface, drogued to reflect behavior under turbulent layer Floats drift Drifters float

12 RAFOS Floats Floats

13 Revealing Pathways Of interest: finding pathways for salty Mediterranean water to reach North Atlantic Thanks to Amy Bower

14 Drifters

15 DART project: Marie-Poulain et al. Drifter tracks

16 Augmented system Append equations for drifters (floats) x dx dt dx dt F = x D f F f D x = = M M -- augmented state vector f F( x F, t) -- flow equations f f D( xd, xf, t) -- tracer advection equation Apply filtering to augmented system

17 Two vortices, N=2, one tracer, L=1 z = 1, z = ζ = i Γ = 2π 1,2 Δ T = 1 σ = 0.04 ρ = 0.02

18 Two vortices, N=2, one tracer, L=1 z 1 z 2 ζ z = 1, z = ζ = i Γ = 2π 1,2 ΔT = 1.5 σ = 0.04 ρ = 0.02

19 t = t 0 x ( t ), x ( t ) t t t = t 1 State estimate: x ( t ), x ( t ), P ( t ) a a a Model forecast: x ( t ), x ( t ), P ( t ) f f f Initial conditions: f f f x ( t ), x ( t ) t t x ( t ), x ( t ), P ( t ) Bayes Measurement: y ( t ) = x ( t ) + ε o t posterior obs prior P ( x y) = P ( yx) P x ( )

20 Forecast step: p( x, t ) p( x, t ) p ( Mp) 1 ( Qp ij ) + i = t x 2 x x i i j Bayes step (update/analysis): p x t p x t y o (, 1) (, 1 ) p( y x) p( x, t ) o o 1 1 y o 1 p( x, t ) = p( y z) p( z, t ) dz But: computationally prohibitive, state ~ 6 10

21 Particle Filter Elaine Spiller, Amarjit Budiraja, Kayo Ide, CJ Initialize particle cloud and evolve under noisy dynamics: N 500 Reinitialize by choosing best particles using weight forecast tracer location observed tracer location Horizontal: best particles. Vertical: number reinitialized

22 Correcting particle filter Issue: particle cloud pulls away from obs Exploit different time-scales of Eulerian and Lagrangian dynamics Backtrack two time steps and reinitialize: i. Cloud expansion using Gaussian ii. Redistribute along line toward and away from obs (directed doubling)

23 Results

24 Gulf of Mexico 3 active layer, reduced gravity Modeling of the loop current in the GoM Limited area model km Lagrangian vs Eulerian data

25 Twin Experiment: Control

26 Twin Experiment: Control

27 Synthetic observation: Fixed stations(u,v) Surface floats (x,y) Isopycnal floats (x,y,z)

28 Synthetic observation: 4 Fixed stations (u,v)

29 Synthetic observation: 4 Surface floats (x,y)

30 Synthetic observation: 4 Isopycnal floats (x,y,z)

31 Results: ssh field Isopycnal floats Surface drifters Fixed stations

32 Results: rms(truth-analysis) of interface s depths Interface between layer 1 and 2 EuDA (u,v) LaDA (x,y,z) LaDA (x,y) Interface between layer 2 and 3

33 EuDA LaDA (x,y,h) LaDA (x,y)

34 Lagrangian DA and State Estimation Augmented model : dx dx F = M F F D D F dt dt, D ( x, t) ; = M ( x, x t) with initial conditions x ( x x ) 0 F(0), D(0) Observations : y i = x D ( t i ) + ξ i ; ξ i ~ N (0, R 2 ) at times (t 1,t 2, t m ) Goal : Estimate initial conditions x(0) using the observations Idea : Use Bayesian formulation and Langevin sampling Note: This is really hindcasting/nowcasting, not forecasting

35 Bayesian formulation prior Prior distribution: P ( x) from initial initial condition Observational likelihood: P obs ( y x) from (Lagrangian) data For example: obs 1 P ( y x) exp y Hx 2 2σ ( ) 2 Bayes rule: P posterior obs prior ( x y) P ( y x) P ( x) State estimation problem : Calculate the statistics (mean / variance) of the posterior distribution of initial conditions (assume a perfect model)

36 State Estimation Model runs + observations state estimate x 0 0 t t 1 t2 model model model t 3 + OBS t N ( ) xt N y( t0) y( t1) y( t2) y( t3) Bayes: posterior obs prior P ( x y) = P ( y x) P x ( ) y t N ( ) sample posterior } Model runs

37 Lagrangian DA and State Estimation State estimation is particularly appropriate for float data Natural to use an augmented state-space approach Obs are in a clearly defined low-dimensional subspace but encode key aspects of full dynamics, for instance large scale features

38 Model Problem for Lagrangian DA Linearized shallow water model: 2 mode approximation: u 0 Geostrophic mode with amplitude Inertial gravity mode that is time periodic k = l = m= 1

39 Augmented System obs at: tk = kδ k = 1,..., N with Gaussian errors uncorrelated and independent of each other

40 Experiments and Methods 1. Short trajectory 2. Long trajectory staying in cell 3. Trajectory crossing cell boundaries A. Langevin Stochastic DE B. Metropolis Adjusted Langevin Algorithm C. Random Walk Metropolis Hastings D. EnKF

41 Short Trajectory

42 Long Trajectory in Cell 3 observations sets, # of obs: Obs set 1: 100 Obs set 2: 20 Obs set 3: 6

43 Comparison MALA improves with increased number of observations (frequency kept same) but EnKF does not. Obs set 4, has same fequency as 3, but extends trajectory and makes 20 obs

44 Scatter Plots EnKF is handicapped by trying to effectively approximate by a Gaussian and thus not accounting for nonlinear effects

45 Trajectory crossing cell boundary

46 Scatter Plots

47 Advantages of Langevin sampling Smoothing technique Works for nonlinear evolution Both the prior and observational distributions can be non-gaussian

48 Conclusions LaDa works and reveals saddle problem, but saddles are crucial due to directed launch strategy Effective approaches to saddle problem (EnKF, Langevin sampling) are emerging Statistical sampling has edge in cases it can be applied, but not developed for high-dimensions. Nonlinearity in statistical filtering approach is well addressed, but saddle issue is not resolved Filtering vs. smoothing is a significant debate because of chaotic effects