About
The multifractal fields above were generated using the Fractionally Integrated Flux (FIF) model, a mathematical method proposed in 1987 by Daniel Schertzer and Shaun Lovejoy for simulating scale invariant random fields. It was originally developed to model geophysical fields shaped by turbulence, like rain and cloud fields, which display scale invariance, i.e. structure at every modeled scale.
Remarkably, the full range of possible scale invariant fields can
be described with just three parameters:
- H (Hurst Exponent): Controls the "smoothness" of the field by determining how much stronger or weaker larger-scale fluctuations are compared to smaller-scale ones. Higher values create smoother fields.
- α (Lévy Index): Determines how likely extreme jumps are. Lower values (α≈1.2) create occasional giant spikes, while higher values (α≈2) produce fewer extreme events.
- C₁ (Intermittency Parameter): Controls whether extreme spikes are clustered together. When large, the field becomes "intermittent"—all the "activity" concentrates in small parts of the field rather than being distributed evenly.
This implementation includes finite-size corrections following Lovejoy & Schertzer (2010) to reduce
grid artifacts. However, the method still produces less small-scale variance than required by scale invariance. To account for this, all the fields above were created initially at higher resolution before being manually coarsened, effectively increasing grid-scale variance.
All fields generated using the Python package scaleinvariance.