Analysis of Vector-Field Multifractal Cascades

Multifractals provide a powerful framework to describe systems that exhibit variability over a wide range of scales together with strong intermittency. By encoding scale-dependent fluctuations through multiplicative cascades, multifractal models capture non-Gaussian statistics, heavy tails, and scale invariance in a compact and predictive manner. These properties have made multifractals particularly successful in the analysis of a wide variety of geophysical phenomena.

 

From the outset, multifractal fields have been formulated on domains of arbitrary dimension, allowing to represent space, space–time, or higher-dimensional parameter spaces. In contrast, the codomain of multifractal constructions has most often been restricted to scalar-valued fields. Although simpler for modeling and inference, the scalar setting omits directional information, anisotropy, and cross-component couplings that are essential in vector observations. Recent works, such as (Schertzer and Tchiguirinskaia 2020), have explored the use of Clifford algebras for constructing cascade generators, offering a natural algebraic framework to represent vector-valued multifractals while preserving their multiscale and symmetry properties.

 

In this work, we consider and simulate Clifford multifractal cascades as an extension of scalar models, capable of capturing directional variability and the internal geometry of multiscale fields. Rather than relying on a scalar stability exponent, we work in a framework where the stability can be encoded by algebra-valued or operator-like parameters, enabling anisotropic scaling and nontrivial coupling between different components of the Clifford field across scales.

 

To characterize the resulting operator-scaling structure, we extended the scalar analysis methods and developed inference methods that enable the direct estimation of multifractal parameters. Numerical experiments on synthetic cascades demonstrate that the proposed approach reliably recovers these parameters. The results demonstrate that extending multifractal analysis to vector-valued fields is both feasible and essential for the characterization of complex multiscale phenomena.