Estimation of State Noise for the Ensemble Kalman filter algorithm for 2D shallow water equations.

Transcription

1 Estimation of State Noise for the Ensemble Kalman filter algorithm for 2D shallow water equations. May 6, 2009

2 Motivation Constitutive Equations EnKF algorithm Some results Method Navier Stokes equations and assumptions Parameter adjustments Generation of state noise occurrences Validation method Future work

3 Motivation Motivation Constitutive Equations EnKF algorithm Some results Method Navier Stokes equations and assumptions Parameter adjustments Generation of state noise occurrences Validation method Future work

4 Sacramento Delta Motivation

5 Sacramento Delta Motivation Before the levee breach

6 Sacramento Delta Motivation After the levee breach

7 Lagrangian sensors Motivation Lagrangian sensors measure their positions as they drift along with the flow. When trying to solve the two-dimensional shallow water equations, one needs accurate initial and boundary conditions which are usually not available. Data assimilation of the Lagrangian measurements from the drifters allow to estimate the velocity field in the river even without accurate boundary conditions.

8 Constitutive Equations EnKF algorithm Some results Motivation Constitutive Equations EnKF algorithm Some results Method Navier Stokes equations and assumptions Parameter adjustments Generation of state noise occurrences Validation method Future work

9 Constitutive Equations EnKF algorithm Some results Two-dimensional shallow water equations u t + U u = g η x + F x + 1 h (hν t u) (1) v t + U v = g η y + F y + 1 h (hν t v) (2) h t + U h + h u = 0 (3) h: total water depth U = (u, v): velocity field where F x = 1 F y = 1 gn 2 cos α gn 2 cos α h 4/3 u u 2 + v 2 h 4/3 v u 2 + v 2 η: free surface elevation ν t: turbulent diffusion coefficient α = α(x, y): bottom slope n: Manning coefficient.

10 Constitutive Equations EnKF algorithm Some results Two-dimensional shallow water equations Boundary conditions: u(x, y, t) Ωland = 0, v(x, y, t) Ωland = 0, (4) (u(x, y, t), v(x, y, t)) Ωupstream = f (x, y, t), (5) η(x, y, t) Ωdownstream = g(x, y, t), (6) Initial conditions: u(x, y, 0) = u 0, v(x, y, 0) = v 0, h(x, y, 0) = h 0, (7)

11 Drifter model Constitutive Equations EnKF algorithm Some results dx Di (t) dt dy Di (t) dt = u(x Di (t), y Di (t), t), (8) = v(x Di (t), y Di (t), t), (9)

12 State Augmentation Constitutive Equations EnKF algorithm Some results We use state augmentation to simplify the observation model u(t n ) v(t n ) h(t n ) θ n = x D (t n ) y D (t n ) (10)

13 Forward model Constitutive Equations EnKF algorithm Some results θ n+1 = F n (θ n ) + w n. (11) F n : one time step in the discretized shallow water and drifter model. θ n+1 : predicted system state at time t n+1. w n : state noise (modeling error between the reality and the 2D model). Q n : covariance of the state noise.

14 Observation model Constitutive Equations EnKF algorithm Some results y n : measurements from the drifters. C n = (0 I): observation model. ɛ n : measurement noise. R n : covariance of the measurement noise. y n = C n θ n + ɛ n. (12)

15 EnKF Algorithm Constitutive Equations EnKF algorithm Some results 1. Initialization: An ensemble of N states states ξ (p) 0 indexed by p are generated to represent the uncertainty in θ Time update: ξ (p) n n 1 = Fn(ξ(p) n 1 n 1 ) + w (p) n 1 (13) θ n n 1 = 1 N states ξ (p) (14) N n n 1 states p=1 E n n 1 = [ξ (1) n n 1 θ n n 1,..., ξ (Nstates) n n 1 θ n n 1 ] (15)

16 EnKF Algorithm Constitutive Equations EnKF algorithm Some results 3. Measurement update: 1 Γ n n 1 = N states 1 E n n 1En n 1 T (16) K n = Γ n n 1 Cn T (C n Γ n n 1 Cn T + R n ) 1 (17) ξ (p) n n = ξ(i) n n 1 + K n(y n C n ξ (p) n n 1 + ɛ(p) n ) (18)

17 Twin experiment results Constitutive Equations EnKF algorithm Some results Drifters are released using a software and the EnKF algorithm runs using those measurements.

18 Limitations Constitutive Equations EnKF algorithm Some results The state noise used here is realistic for twin experiments, since we know the discrepancy between the reality and the model, as they are both from the same software. We need more realistic state noise in order to have the algorithm run with the real data from November experiment.

19 Method Navier Stokes equations and assumptions Parameter adjustments Generation of state noise occurrences Motivation Constitutive Equations EnKF algorithm Some results Method Navier Stokes equations and assumptions Parameter adjustments Generation of state noise occurrences Validation method Future work

20 Method Navier Stokes equations and assumptions Parameter adjustments Generation of state noise occurrences Eulerian sensors - Boundary Conditions Mike 3 θ n Interpolation Mike 3 TELEMAC 2D θ n+1 F (θ n) State noise estimation w n = θ n+1 F (θ n) Statistical study Covariance of the state noise Q n

21 Navier-Stokes equations Method Navier Stokes equations and assumptions Parameter adjustments Generation of state noise occurrences 1 ρc 2 s P t + u j x j = SS (19) u i t + (u iu j ) x j + 2Ω ij u j = 1 P + g i ρ x i + x j (ν T { u i x i + u j x j } 2 3 δ ijk) + u i SS (20)

22 Testing the hydrostatic assumption Method Navier Stokes equations and assumptions Parameter adjustments Generation of state noise occurrences Hydrostatic assumption: pressure in the river can be estimated using the hydrostatic pressure. Valid for L H >> 1, where L is the horizontal characteristic length of the river.

23 Method Navier Stokes equations and assumptions Parameter adjustments Generation of state noise occurrences Navier-Stokes equations using hydrostatic assumption u t + u2 x + vu y + wu z 1 ρ 0 h ( sxx x + s xy y v t + uv x + v 2 y + wv z 1 ρ 0 h ( syx x + s xy y u x + v y + w z = 0 (21) η = g ) + F u + z η = g ) + F v + z x 1 p a ρ 0 x g ρ ( ) 0 u ν t z y 1 p a ρ 0 y g ρ ( ) 0 v ν t z η z η z ρ x dz ρ y dz (22) (23)

24 Method Navier Stokes equations and assumptions Parameter adjustments Generation of state noise occurrences Telemac2D Mike 3 Vertical Eddy Viscosity ν t Eddy viscosity ν t Horizontal Eddy Viscosity A Manning s n Roughness height k s Figure: Equivalence of parameters between the two dimensional and the three dimensional models. A relationship between the roughness height and the Manning s n can be found as: n = k 1 6 s 25.4 (24)

25 Three dimensional simulations Method Navier Stokes equations and assumptions Parameter adjustments Generation of state noise occurrences

26 Boundary conditions Method Navier Stokes equations and assumptions Parameter adjustments Generation of state noise occurrences

27 Method Navier Stokes equations and assumptions Parameter adjustments Generation of state noise occurrences

28 Result of the 3D simulation Method Navier Stokes equations and assumptions Parameter adjustments Generation of state noise occurrences

29 State noise characteristics Method Navier Stokes equations and assumptions Parameter adjustments Generation of state noise occurrences

30 Validation method Future work Motivation Constitutive Equations EnKF algorithm Some results Method Navier Stokes equations and assumptions Parameter adjustments Generation of state noise occurrences Validation method Future work

31 Real data Validation method Future work

32 Validation method Future work Cross validation principle No data are available to validate. Remove 1 or 2 drifters from the data set. Run the EnKF on the remaining drifters. Compute the trajectory of the extracted drifters using the estimated velocity field. Compare it with the real trajectory.

33 Validation method Future work Future work Run the EnKF using the real data set and the generated state noise. Generate a state noise model for the drifters.

34 Validation method Future work Questions?