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

400 600 800 1000 1200 1400 1600 18000

5

10

Time (pixel)

Va

lue p-model

1894 1894.5 1895 1895.5 1896 1896.5 1897 1897.50

20

Time (year)

Ra

in (

mm

/d)

daily rain sums

1828 1828.5 1829 1829.5 1830 1830.5 1831 1831.5

500100015002000

Time (year)

Ru

no

ff (m

3/s

)

runoff Burghausen

1818 1818.5 1819 1819.5 1820 1820.5 1821 1821.5

2000400060008000

Time (year)

Ru

no

ff (m

3/s

)

runoff Cologne

0 200 400 600 800 1000 1200 1400 16000

0.2

0.4

Time (s)

LW

C (

g m

-3)

cumulus

0 200 400 600 800 1000

0.4

0.6

Time (s)

LW

C (

g m

-3)

stratocumulus

1894 1894.5 1895 1895.5 1896 1896.5 1897 1897.5-10

01020

Time (year)

Te

mp

. (°C

)

temperature

Discharge Rhine - Wavelets

Discharge Rhine

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