Kalman Filter in the Real Time URBIS model

1. Kalman Filter in the Real Time URBIS model Richard Kranenburg Master scriptie June 11, 2010

2. Kalman Filter in the Real Time URBIS model • Introduction • Real Time URBIS model • Problems and Goals • Method • Kalman filter equations • Results • Extensions on the Kalman Filter • Conclusions

3. Introduction • Company: TNO • Business Unit: ‘BenO’ • Department: ‘ULO’ • Accompanists: • Michiel Roemer (TNO) • Jan Duyzer (TNO) • Arjo Segers (TNO) • Kees Vuik (TUDelft)

4. Real Time URBIS model

5. Real Time URBIS Model • URBIS Model, standard concentration fields • 11 sources, 4 wind directions, 2 wind speeds

6. Real Time URBIS model • Every hour interpolation between standard concentration fields • Correction for background and traffic fields • μ is the weight function dependent of wind direction (φ), wind speed (v), temperature (T), hour (h), day (d), month (m) • : standard concentration fields

7. Real Time URBIS model

8. Real Time URBIS model • Linear correction used by DCMR • Average concentration of three stations • Schiedam • Hoogvliet • Maassluis

9. Real Time URBIS model

10. Problems and Goals • The model simulation can become negative • No information about the uncertainty of the simulation • Goal: Find an uncertainty interval for the concentration NO , which does not contain negative concentrations

11. Method • Kalman filter connects the model simulations with a series of measurements • Kalman filter corrects the model in two steps • Forecast step • Analysis step • Result is a (multivariate) Gaussian distribution of the unknown • Mean • Covariance matrix

12. Kalman filter equations • Correction factor ( ) for each standard concentration field • Kalman filter calculates a Gaussian distribution for the unknown variable ( ) • The concentration NO can be found in a log-normal distribution

13. Kalman filter equations • Second equation not linear in ( ), thus a linearization around • H: projection matrix, A: correlation matrix • represents the uncertainty of the measurements on time k

14. Kalman filter equations • The linearization results in: • with:

15. Kalman filter equations • Forecast step • represents the uncertainty of the model • is the covariance matrix after the forecast step • The temporal correlation matrix A is determined with information from the measurements

16. Kalman filter equations • Analysis step • Minimum variance gain

17. Kalman filter equations • Start of the iteration process: • Screening process: • Before the analysis step is executed, the measurements are screened. • If difference between simulation and observation is too large, that observation will be thrown away. • In this application the criterion is twice the standard deviation

18. Results

19. Results • For the whole domain on each hour an uncertainty interval for the concentration NO can be calculated • Annual mean of the widths of these uncertainty intervals • Population density on the whole domain • Connection between population and uncertainty

20. Results

21. Results

22. Results • Connection between uncertainty and population • Kalman filter reduced the uncertainty • Absolute uncertainty: 14.5 % • Relative uncertainty: 16.1 %

23. Extensions of the Kalman filter • Goal: Minimize the uncertainty connected with the population • Methods: • Add extra monitoring stations to the system • Apply Kalman filter with different time scale and add stations with other time scales • Analyse the values of the correction factors