TR32 time series comparison Victor Venema

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

Wind - Measurement and synthetic

Wind - distribution – normal plot

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

Wind - Increment distribution

Daubechies wavelet family

Wind - Daubechies wavelet (db6)

Wind – Haar vs. Daubechies (db6)

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)

Wind - Increment distribution

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

Wind – Structure functions

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

Wind – power spectrum

Wind speed (Heye Bogena; 10 min.)

Wavelet - Wind speed (10 min.)

Air pressure (10 min.)

Air pressure (10 min.) - Wavelets

Water temperature

Water temperature - Wavelets

Discharge all data and zoom

Discharge Rhine - Wavelets

Discharge Rhine

Slope power spectrum vs. smoothness

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