On the simulation of continuous in scale universal multifractals, part I: Spatially continuous processes

Cascade processes generically lead to multifractal fields and have been used for simulating turbulent systems - including clouds, rain, temperature, passive scalars and the wind - as well as for solid earth fields, such as rock density, magnetization and topography. In spite of their importance, most applications use primitive discrete scale ratio processes, which singularize scales which are integer powers of integers. Realistic simulations are continuous in scale, but suffer from strong ''finite size effects'' i.e. deviations from pure power law scaling, which can take surprisingly large ranges of scale to disappear. In this two part series, we quantify and show the origin of the problem and quantify its magnitude (part I), while in part II we show how to largely overcome it and give a Mathematica code for the corresponding simulations for causal and acausal space-time simulations.

[1]  D. Schertzer,et al.  Vertical cascade structure of the atmosphere and multifractal dropsonde outages , 2009 .

[2]  Franz Durst,et al.  Turbulent Shear Flow , 1992 .

[3]  B. Mandelbrot.  Intermittent turbulence in self-similar cascades : divergence of high moments and dimension of the carrier , 2004 .

[4]  Robert F. Cahalan.  Bounded cascade clouds: albedo and effective thickness , 1994 .

[5]  D. Schertzer,et al.  Non-Linear Variability in Geophysics : Scaling and Fractals , 1990 .

[6]  D. Schertzer,et al.  Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes , 1987 .

[7]  Shaun Lovejoy,et al.  Atmospheric complexity or scale by scale simplicity? , 2009 .

[8]  Laurent E. Calvet,et al.  Multifractal Volatility: Theory, Forecasting, and Pricing , 2008 .

[9]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[10]  D. Schertzer,et al.  The morphology and texture of anisotropic multifractals using generalized scale invariance , 1997 .

[11]  Multifractal properties of visible reflectance fields from basaltic volcanoes , 1999 .

[12]  Fernando Porté-Agel,et al.  Synthetic turbulence, fractal interpolation, and large-eddy simulation. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  D. Schertzer,et al.  Horizontal cascade structure of atmospheric fields determined from aircraft data , 2010 .

[14]  Shaun Lovejoy,et al.  Nonlinear Processes in Geophysics Scaling and multifractal fields in the solid earth and topography , 2008 .

[15]  Shaun Lovejoy,et al.  NOTES AND CORRESPONDENCE Universal Multifractals Do Exist!: Comments on ''A Statistical Analysis of Mesoscale Rainfall as a Random Cascade'' , 1997 .

[16]  Shaun Lovejoy,et al.  Causal space‐time multifractal processes: Predictability and forecasting of rain fields , 1996 .

[17]  Qiuming Cheng,et al.  Multifractal modeling and spatial statistics , 1996 .

[18]  Shaun Lovejoy,et al.  The Dimension and Intermittency of Atmospheric Dynamics , 1985 .

[19]  Wojbor A. Woyczyński,et al.  Stochastic models in geosystems , 1996 .

[20]  Tapio Schneider,et al.  Comment on "Spatio-temporal filling of missing points in geophysical data sets" by D. Kondrashov and M. Ghil, Nonlin. Processes Geophys., 13, 151-159, 2006 , 2007 .

[21]  Shaun Lovejoy,et al.  Towards a new synthesis for atmospheric dynamics: Space–time cascades , 2010 .

[22]  Laurent E. Calvet,et al.  Forecasting Multifractal Volatility , 1999 .

[23]  She,et al.  Universal scaling laws in fully developed turbulence. , 1994, Physical review letters.

[24]  D. Schertzer,et al.  The simulation of universal multifractals. , 1993 .

[25]  Structures in turbulence and multifractal universality , 1995 .

[26]  Shaun Lovejoy,et al.  Generalised scale invariance in turbulent phenomena , 1985 .

[27]  E. Lutton,et al.  FRACTAL MODELING OF SPEECH SIGNALS , 1994 .

[28]  A. Kolmogorov.  The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers , 1991, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.