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

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)

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

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

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

Motivation – Cloud structure

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

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

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

Surrogates from airplane data

Three reconstructions

Irradiances stratocumulus

Irradiances cumulus

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

“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

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

4th order SF cumulus

Error 4th order structure function

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

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

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

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

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

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

More information  Homepage –Papers, Matlab-programs, examples  venema/themes/surrogates/  Google –surrogate clouds –multifractal surrogate time series  IAAFT in R: Tools homepage Henning Rust –  IAAFT in Fortran (multivariate): search for TISEAN (Time SEries ANalysis)