Department of Radio and Space Science / Onsala Space Observatory

H.-G. Scherneck, S. Bergstrand, M. Lidberg:

Fractals everywhere: Time series analysis and rate uncertainty

NKG Working Group for Geodynamics

Meeting in Ås, Norway, March 2006

Department of Radio and Space Science / Onsala Space Observatory

Department of Radio and Space Science / Onsala Space Observatory

Department of Radio and Space Science / Onsala Space Observatory

We use:

• Unbiased autocovariance estimator (w.r.t. missing data)

• Window: Kaiser-Bessel

• A selection scheme for fit that avoids the spectral correlation due to windowing, and still gives a heavily overdetermined case.

We find:

• Slope model fits almost always up to Nyquist

• Disadvantage: Low-frequency part is not well resolved

Department of Radio and Space Science / Onsala Space Observatory

Department of Radio and Space Science / Onsala Space Observatory

Department of Radio and Space Science / Onsala Space Observatory

Department of Radio and Space Science / Onsala Space Observatory

Department of Radio and Space Science / Onsala Space Observatory

Department of Radio and Space Science / Onsala Space Observatory

We use:

• data from beg. 1998 to end 2004

• synth. fractal noise, 1,000-10,000

• masked by the real-world time series for breaks

Then we estimate a rate where (in white noise) we’d expect none.

← shows histograms of these rates

Department of Radio and Space Science / Onsala Space Observatory

Department of Radio and Space Science / Onsala Space Observatory

To specify the rate uncertainties,

• We ought to use the fractal noise law

• Next page: assumes Gauss-Markov

Department of Radio and Space Science / Onsala Space Observatory

GaussMarkovSITE co rate sigma sigmascale--------------------------------------------ARJE.ra: 8.196 0.223 2.449BORA.ra: 3.007 0.212 1.497BRUS.ra: -1.624 0.172 1.797HASS.ra: 1.008 0.205 1.898HERS.ra: -1.286 0.087 22.761JOEN.ra: 4.704 0.159 2.485JONK.ra: 3.590 0.204 1.926KARL.ra: 5.831 0.206 1.906KEVO.ra: 5.437 0.253 4.795KIRU.ra: 8.424 0.225 3.663KIVE.ra: 8.469 0.245 2.981KOSG.ra: -1.060 0.086 1.354KUUS.ra: 12.200 0.264 5.649LEKS.ra: 8.346 0.420 2.372LOVO.ra: 5.964 0.212 1.465MADR.ra: 0.983 0.247 1.870MART.ra: 7.097 0.206 1.858MATE.ra: -0.979 0.115 1.701METS.ra: 5.242 0.234 1.355NORR.ra: 4.898 0.211 1.376NYAL.ra: 5.679 0.178 1.459OLKI.ra: 8.516 0.171 1.768

ONSA.ra: -0.179 0.360 1.135ONSW.ra: 4.077 0.483 2.274OSKA.ra: 2.231 0.211 2.449OSTE.ra: 8.331 0.209 3.290OULU.ra: 10.761 0.199 2.010OVER.ra: 9.020 0.226 1.520POTS.ra: -1.389 0.176 4.195RIGA.ra: 2.347 0.238 2.336ROMU.ra: 7.566 0.268 3.313SAAR.ra: 7.678 0.133 2.000SKEL.ra: 10.527 0.217 4.126SODA.ra: 11.143 0.217 1.692SUND.ra: 9.926 0.206 2.614SVEG.ra: 8.147 0.204 1.448TROM.ra: 3.500 0.208 1.518TUOR.ra: 6.062 0.158 1.807UMEA.ra: 10.923 0.210 2.545VAAS.ra: 10.750 0.187 1.607VANE.ra: 4.132 0.209 4.322VILH.ra: 8.508 0.210 2.508VIRO.ra: 2.681 0.171 1.774VISB.ra: 2.884 0.203 1.140WETB.ra: 0.519 0.345 1.065WETT.ra: -0.242 0.205 1.132WTZR.ra: -0.232 0.216 0.000

From Johansson et al. 2002:

Department of Radio and Space Science / Onsala Space Observatory

Department of Radio and Space Science / Onsala Space Observatory

Conclusions

• We think these considerations have a general notion; in GPS we have the advantage of long, regularly sampled time series

• We find fractal noise with a non-integer power law

= ( 0.5 to 0.9)

• Corresponding uncertainties for estimated rates must be scaled up with a factor of 4 to 15 w.r.t. white-noise results

• This is more pessimistic than Gauss-Markov