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.