Data assimilation applied to simple hydrodynamic cases in MATLAB

1. Data assimilation applied to simple hydrodynamic cases in MATLAB Ângela Canas MARETEC

2. Measurements Analysis First Guess Known dynamics Past measurements Past measurements Future measurements Data assimilation generalities DA methods: Sequential: Kalman Filter (KF, EKF, EnKF, RRSQT, SEIK, SEEK), Optimal Interpolation Statistical Interpolation Uncertainty Variational: 3D-Var, 4D-Var

3. n 2 3 i n-1 1 n-2 level Dynamics (M) space amplitude time Measurements (exact solution) wave period Kalman Filter average analysis gain meas. operator 1D Linear level model

4. no assimilation with assimilation exact solution Measurements Wf0 Wrong model Cr = 0.5 (k = 0.5) Kalman Filter time step DA twin test True model Cr = 1 (k = 1) Cr = (k.c)/(x)  N. Courant Assimilation every 5 time steps

5. Sudden change 25 inst. after Amplitude: 1m  0.5m Introduced at time 150 instantes Introduced at time 25 instants: 25 inst. after Later introduction prejudicates convergence 40 inst. after

6. Shallow water equations Kalman Filter methods Optimal Interpolation √  H h u 1D Hydrodynamic model

7. Wf first guess Wf P0 f Pf ... State ensemble Wf1 WfM M >= 100 Corrector EnKF Wo Predictor time R Wf: Ensemble mean f ... o Pf ... a Wa: Ensemble mean ... Pa

8. Implementation details • Model: • Velocity and water level discretization: upwind, implicit (except when H in equations - explicit) • Levels at cells centers, velocities at cells faces • Level solved first then velocities calculated • Boundaries: • level first cell - imposed sine function (solution linear model) • level last cell – radiative • velocity first cell – 0 (not needed for calculation) • EnKF (based on Evensen, 2003, Ocean Dynamics): • State: levels and velocities in each cell • Initial state: null levels, null velocities; • Initial ensemble: • random perturbations based on covariance matrix; • run in model without error for proper correlations to develop (1 wave T) • Measurement error: randomly generated (time, members) assuming a variance (R) equal for each measure • Model error: randomly generated (time, members) independently for each variable assuming variance (Qlevel, Qveloc)

9. First test case • Twin test • Constant h = 5m • Test rational: different spatial discretization: • True model: deltax=1m, 100 cells • Wrong model: deltax=5m, 20 cells • Deltat = 1s • Bottom stress coef. = 0.0025 • Assimilation every 3s • Initial state: • Only levels perturbed (variance = 1) • Correlation length (exp. model) = 6 cells • Number members (ensemble) = 100 • Model error: Qlevel = 0.003; Qveloc = 0.03 • Measurements taken cells 28 and 73 of True; 6 and 15 of Wrong • Measurement error: R = 0.002 (levels or velocities)

10. First results – levels DA

11. First results – velocities DA

12. Better to assimilate velocities? Seems not advantageous to assimilate... More tuning of DA parameters needed! First results - statistics • True   Wrong (time equivalent to 300 assimilations): • Levels: RMSE=0.000317; CORR=0.955961 • Velocities: RMSE=0.004322 ; CORR=0.594234 • True   Wrong assim. levels (300 assimilations): • Levels: RMSE=0.021871; CORR=0.947670 • Velocities: RMSE=0.587055; CORR=0.848449 • True   Wrong assim. velocities (300 assimilations): • Levels: RMSE=0.021112; CORR=0.952238 • Velocities: RMSE=0.355114; CORR=0.791911

13. Future work • EnKF: • Sensibility analysis to filter parameters (Q, R, initial condition) • Consider other tests: • Non constant h • Bottom stress • ... • Implement other DA methods • Compare methods performance for same case • Implement DA methods in MOHID

14. Eigen values decomposition value p. 1 value p. 2 ... value p. m  value p. 1 ... value p. r dominant Predictor  (Linearized model) Corrector wo RRSQRT  Redução (r < m) 

15. a ... Wa1 War SEIK (LULT) EOF analysis Wa Pa • Lower computational cost than EnKF (r < m) Predictor mean Wf Pf Wo Corrector R SEEK = SEIK without ensemble and linearized model