1. Assimilation of HF Radar Data into Coastal Wave Models NERC-funded PhD work also supervised by Clive W Anderson (University of Sheffield) Judith Wolf (Proudman Oceanographic Laboratory) Lee Siddons and Lucy Wyatt Department of Applied Mathematics University of Sheffield, UK

2. Overview OSCR HF radar at Holderness SWAN Wave Model Data Assimilation and Algorithms Results Future Work

4. Surface currentWind direction OSCR measurements

5. Wave height and peak direction Peak wave period and direction

6. Energy spectrum m 2 /Hz Mean direction spectrum Directional frequency spectrum m2/Hz/rad OSCR buoy OSCR wind direction

7. Wave Modelling SWAN (Simulating WAves Nearshore) The Action Balance equation: A is the action that is a function of frequency and direction i.e. C is the wave group velocity in relevant direction. S is the forcing to the system. Winds, non-linear interactions etc. See Holthuijsen L. H. et al (1999)

8. SWAN Application to Holderness Boundary Data Boundary DataBoundary Data Action 1.2km

9. Data Assimilation The Analysis - Combination of model output (Background state) and observations in an optimal way. Taking into account model and observational errors. Making assumptions about model, observation and analysis errors.

10. Data Assimilation Formulation Model Observations Initial State Since the state is a random variable, the estimate of the state is found from its probability density function (pdf).

11. The State of the Ocean The state of the ocean is often described in terms of a few wave parameters, for example: Significant Wave Height Hs Mean wave period T 1 These are the state variables used in our assimilations

12. Assimilation Algorithms There are two main approaches of assimilation. Sequential Assimilation Only considers observations from the past and to the time of the analysis. Some examples of sequential algorithms are: Optimal interpolation Kalman Filters Variational Assimilation Observations from the future can be used at the time of the analysis. Some examples of variational algorithms are: Three-dimensional variational assimilation – 3DVAR Four-dimensional variational assimilation – 4DVAR

13. The Kalman Filter The Kalman filter is derived by minimising the estimation error of the analyzed state with respect to the Kalman gain matrix

14. Variational Assimilation The aim of variational assimilation is to find the optimal estimate of the state by minimisation of a cost function. Three-dimensional variational assimilation – 3DVAR The 3dvar cost function is as follows: The solution that minimises the cost function is sought by iteratively evaluating the cost function and its gradient using a suitable descent algorithm.

15. Ensemble Kalman Filter (EnKF) EnKF introduced by Evensen - to avoid the computational load associated with Sequential method where the error statistics are predicted using Monte Carlo or ensemble integration. An ensemble of model states in integrated forward in time and statistical information is calculated from the ensemble.

16. Assimilation with Ideal Data Before assimilating radar data, the algorithms have been validated using simulated data. –SWAN is used to generate a ‘true’ state. –Model errors are assumed to be uniform over the grid and equal to background uncertainty estimated from buoy data. –Radar measurements are assumed to be available at all sites and errors also uniform.

17. Results for Simulated Case - 3DVAR

18. Results for Simulated Case - ENKF

19. Performance Error Statistics SchemeMSE HsMSE Tm No Assimilation5.26 * 10 -3 6.57 * 10 -2 3DVAR2.679 * 10 -3 2.209 * 10 -2 Ens_OI2.709 * 10 -3 2.267 * 10 -2 ENKF-161.05 * 10 -3 2.45 * 10 -2

20. Assimilation with Real Data ENS-OI

21. Assimilation with Real Data EnKF

22. Assimilation of Band Parameters Assimilation of Hs and Tm in the following frequency bands. Band 1 = 0.03Hz - 0.1Hz Band 2 = 0.1Hz - 0.2Hz Band 3 = 0.2Hz - 0.3Hz Band 4 = 0.3Hz - 0.4Hz

23. Performance Error Statistics for Test Case Assimilation of Band Parameters SchemeB1 Hs *10 -3 B1 Tm *10 -3 B2 Hs *10 -3 B2 Tm *10 -3 B3 Hs *10 -3 B3 Tm *10 -3 B4 Hs *10 -3 B4 Tm *10 -3 No Assimilation 0.160.1811.912.05.84.516.30.16 3DVAR0.160.136.86.73.62.58.20.17 Ens-OI0.160.146.76.83.72.68.70.16 ENKF-160.020.150.73.53.11.83.60.29

24. Assimilation with Real Data 3DVAR

25. Assimilation with Real Data EnKF

26. Future Work Re-estimate model and radar errors to use in the assimilation schemes. Perform a probability sensitivity analysis on the model to find model sensitivities. Extend range of assimilated parameters e.g. by using partitioned directional spectra

27. Energy spectrum m 2 /Hz Mean direction spectrum Directional frequency spectrum m2/Hz/rad OSCR buoy OSCR wind direction