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

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

3. 1D Linear level model level Dynamics (M) space time Measurements3 2 n
n-1
n-2 i
Dynamics (M)
level
time
space
Measurements
(exact solution)
average
amplitude
wave period
analysis gain
meas. operator
Kalman Filter

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

5. Sudden change Amplitude: 1m  0.5m
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. Optimal Interpolation
1D Hydrodynamic model
Shallow water equations √ H h  u
Kalman Filter methods
Optimal Interpolation

7. EnKF Wf Pf Wf first guess P0 Predictor Corrector f State ensemble
...
Wf1
WfM f
M >= 100 Wo
Predictor
time R f
Wf: Ensemble mean o
... Pf
...
Corrector 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. =
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 = (levels or velocities)

10. First results – levels DA

11. First results – velocities DA

12. First results - statistics
True   Wrong (time equivalent to 300 assimilations):
Levels: RMSE= ; CORR=
Velocities: RMSE= ; CORR=
True   Wrong assim. levels (300 assimilations):
Levels: RMSE= ; CORR=
Velocities: RMSE= ; CORR=
True   Wrong assim. velocities (300 assimilations):
Levels: RMSE= ; CORR=
Velocities: RMSE= ; CORR=
Better to assimilate velocities?
Seems not advantageous to assimilate...
More tuning of DA parameters needed!

13. Future work EnKF: Implement other DA methods
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
RRSQRT 
Eigen values decomposition
value p. 1
value p. 2
...
value p. m
value p. r
dominant 
Predictor
(Linearized model)
Redução
(r < m)
Corrector wo 

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