Beyond fractals: surrogate time series and fields

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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
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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)

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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

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The great tragedy of science the slaying of a
beautiful theory by an ugly fact

• Thomas Henry Huxley (18251895)

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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

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Motivation Cloud structure
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Motivation Cloud structure
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Motivation Cloud structure
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Motivation Cloud structure
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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

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The iterative IAAFT algorithmSchreiber and
Schmitz
Time series
Distribution
Flow diagram
Time series
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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

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Surrogates from airplane data
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Three reconstructions
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Irradiances stratocumulus
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Irradiances cumulus
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Validation 3D clouds

• 3D models clouds -gt 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

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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

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Structure functions

• Increment time series ?(x,l)?(tl)- ?(t)
• SF(l,q) (1/N) S ?q
• SF(l,2) is equivalent to auto-correlation
  function
• Correlated time series SF increases with l
• Higher q focuses on larger jumps

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4th order SF cumulus
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Error 4th order structure function
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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

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Comparison

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

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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

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Cloud structure is not fractal

• Scale breaks
• Waves
• Land sea mask

Satellite pictures Eumetsat
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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

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Conclusions

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

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More information

• Homepage
• Papers, Matlab-programs, examples
• http//www.meteo.uni-bonn.de/ venema/themes/surrog
  ates/
• 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)