1. Beyond fractals: surrogate time series and fields Victor Venema and Clemens Simmer Meteorologisches Institut, Universität Bonn, Germany Cloud measurements: Susanne Crewell, Ulrich Löhnert, Sebastian Schmidt Climate data & analysis: Susanne Bachner, Alice Kapala, Henning Rust Radiative transfer & analysis: Sebastián Gimeno García, Anke Kniffka, Steffen Meyer, Sebastian Schmidt 3D cloud modelling: Andreas Chlond, Frederick Chosson, Siegfried Raasch, Michael Schroeter

2. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line Benoit B. Mandelbrot in The Fractal Geometry of Nature (1983)

3. Fractals  Implied: nature is fractal  Fractal, self-similar –Zoom in, looks the same –Structure measure is a power law of scale –Linear on a double logarithmic plot  Beginning of complex system sciences?  Structure on all scales  My experience: good approximation for turbulence and stratiform clouds, but often see different signals

4. The great tragedy of science — the slaying of a beautiful theory by an ugly fact Thomas Henry Huxley (1825–1895)

5. Content  Motivation – What I do –Radiative transfer through clouds –Basic algorithm  Case study – 3D clouds  Validation - 3D clouds  Structure functions of surrogates  Motivation – compare multifractals  Conclusions  More information

6. Motivation – Cloud structure

10. Motivation  Can not measure a full 3D cloud field  Need 3D field for radiative transfer calculations  Can measure many (statistical) cloud properties  Generate cloud field based on statistics measurements  Nonlinear processes –Precise distribution  Non-local processes –E.g. power spectrum (autocorrelation function)  In geophysics you generally do not have full fields, but can estimate these two statistics

11. Time series The iterative IAAFT algorithm Schreiber and Schmitz DistributionFlow diagramTime series

12. Case study  Two flights: Stratocumulus, Cumulus  Airplane measurements –Liquid water content –Drop sizes  Triangle horizontal leg (horizontal structure)  A few ramps, for vertical profile  Three cloud generators  Irradiance modelling and measurement

13. Surrogates from airplane data

14. Three reconstructions

15. Irradiances stratocumulus

16. Irradiances cumulus

17. Validation – 3D clouds  3D models clouds -> 3D surrogates  Full information, perfect statistics  Test if the statistics are good enough  The root-mean-square (RMS) differences are less than 1 percent (not significant)  Significant differences –Fourier surrogates: distribution is important –PDF surrogates: correlations are important  Trivial problem, but just numerical result

18. “Validation” time series  1D climate time series and clouds  4th order structure function –Surrogates more accurate (as multifractal)  Full information, perfect statistics  Numerical test how good the statistics are

20. 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

21. 4th order SF cumulus

22. Error 4th order structure function

23. Generators  Iterative amplitude adjusted Fourier transform algorithm –Schreiber and Schmitz (1996, 2000) –Masters and Gurley (2003) –...  Search algorithm –Simulated annealing (Schreiber, 1998) –Genetic algorithm (Venema, 2003)  Geostatistics: stochastic simulation –Search algorithms –Gaussian distribution

24. Comparison  FARIMA modelling, Fourier methods –Gaussian distribution  AR modelling & Multifractals –Idealised structure  Linear statistics –Kriging –Assimilation –Optimal estimation –Kernel smoothing –....

25. Surrogates vs. multifractals  Measured power spectrum  Perfect distribution  Indirect over distribution  One specific measured field  Empirical studies  Power law fit  Indirect control distribution  Direct intermittence  Ensemble of fields  Theoretical studies

26. Cloud structure is not fractal  Scale breaks  Waves  Land sea mask  … Satellite pictures: Eumetsat

27. Land surface is not fractal  15 Reasons the surface is not uni-fractal (Steward and McClean, 1985): –Fractal landscape have the same number of tops and pits –Glacial cirques has a narrow size range and size dependent shape

28. Conclusions  IAAFT algorithm can generate structures –Accurately –Flexibly –Efficiently  Many useful extensions are possible –Local values –Increment distributions –Downscaling

29. 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)