1. Statistical characteristics of surrogate data based on geophysical measurements Victor Venema 1, Henning W. Rust 2, Susanne Bachner 1, and Clemens Simmer 1 1 Meteorological Institute University of Bonn 2 PIK, Potsdam Institute for Climate Impact Research

2. 2 Content  Surrogate data: time series generated based on statistical properties of measurements –Distribution and/or power spectrum  7 Geophysical time series  Generated surrogates with 7 different algorithms from their statistics  Compared the measurements to their surrogates –Increment distribution –Structure functions

3. 3 Motivation  Need time series with a known structure –Statistical reconstruction –Bootstrap confidence intervals –Studying non-local process –…  FARIMA & Fourier methods vs. Multifractals  Multifractals vs. surrogates

4. 4 Satellite pictures: Eumetsat Motivation - generator  Empirical studies –Exact measured distribution –Measured power spectrum  Scale breaks  Waves  Deviations large scales  …

5. 5 7 Generators for surrogate data  D: distribution –PDF surrogates  S: spectrum –Fourier surrogates –FARIMA surrogates –Seasonal cycle and logarithm if needed  DS: distribution + spectrum –AAFT, IAAFT, SIAAFT surrogates –FARIMA + IAAFT surrogates –seasonal cycle and log. if needed

6. 6 Measurements

7. 7 Surrogate types DS S D S DS

8. 8 Increment distribution  Measurement:  (t)  Increment time series for lag l:  (x,l) =  (t+l) -  (t)  Distribution jumps sizes  Next plots: l is 1 day

9. 9 Increment distribution temperature

10. 10 Increment distribution Rhine

11. 11 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  Higher q focuses on larger jumps

12. 12 Structure function Salzach

13. 13 Structure function stratocumulus

14. 14 RMSE 4th order structure functions  Best surrogate in bold  Multifractal means: power law fit

15. 15 Extension IAAFT algorithm  2D and 3D fields with PDF(z)  PDF(t), i.e. distribution varies as function of –Season, time of day –Break point  Multivariate statistics, cross correlations  Increment distribution at small scales –More accurate increment distribution –Asymmetric increment distribution (runoff)  Downscaling –Extrapolate spectrum –Iterate the original coarse mean values

16. 16 Conclusions  DS-Surrogates of geophysical reproduce measurements accurately –spectrum –increments –structure functions  IAAFT algorithm –Flexibly –Efficiently –Many useful extensions are possible  Surrogates for empirical work  Multifractals for theoretical work (use IAAFT)

17. 17 More information  Homepage –Papers, Matlab-programs, examples  http://www.meteo.uni-bonn.de/ venema/themes/surrogates/  Google –surrogate clouds –multifractal surrogate time series  IAAFT in R: Tools homepage Henning Rust –http://www.pik-potsdam.de/~hrust/tools.html  IAAFT in Fortran (multivariate): search for TISEAN (Time SEries ANalysis)