1. TR32 time series comparison Victor Venema

2. Content  Jan Schween –Wind game: measurement and synthetic –Temporal resolution of 0.1 seconds  Heye Bogena –Wind, air pressure, water temperature –Temporal resolution of 10 minutes –Rollesbroich  Global Runoff Data Centre –Runoff Rhine Cologne –Daily, years: 1817 to 2001

3. Wind - Measurement and synthetic

4. Wind - distribution – normal plot

5. Increment distribution  Measurement:  (t)  Increment time series for lag l:  (x,l) =  (t+l) -  (t)  Distribution jumps sizes  Width of the distribution is the mean variance at scale l

6. Wind - Increment distribution

7. Daubechies wavelet family

8. Wind - Daubechies wavelet (db6)

9. Wind – Haar vs. Daubechies (db6)

10. Intermittency / Intermittence  On-off intermittency –Rain, eddy in laminar flow  Operationalisation: variance of variance (at a certain scale)  Intermittence is typically strongest at small scales  Time series modelling: Autoregressive conditional heteroskedasticity (ARCH, GARCH)  Multi-fractal models (not all)

11. Wind - Increment distribution

12. Structure functions  Increment time series:  (x,l)=  (t+l)-  (t)  SF(l,q) = (1/N) Σ |  | q  SF(l,2) is equivalent to auto-correlation function  Correlated time series SF increases with l  Higher q focuses on larger jumps  For large l, SF equivalent to the moments

13. Wind – Structure functions

14. Fourier decomposition  Decompose a time domain signal in sinuses of varying wavelength  Wavelength -> scale  Fourier coefficients -> variance as function of scale

15. Wind – power spectrum

16. Wind speed (Heye Bogena; 10 min.)

17. Wavelet - Wind speed (10 min.)

18. Air pressure (10 min.)

19. Air pressure (10 min.) - Wavelets

20. Water temperature

21. Water temperature - Wavelets

22. Discharge all data and zoom

23. Discharge Rhine - Wavelets

24. Discharge Rhine

26. Slope power spectrum vs. smoothness

27. Conclusions  Some signals showed annual, diurnal cycle  Except for this no frequency was special –Variability on all scales –Large scales:  white noise or even correlated  variance is never gone  All signals showed intermittence –Typical for complex systems