Wrapping and unwrapping multifractal fields

TL;DR

This paper introduces a simple method to generate and analyze multifractal fields with controllable properties, and applies it to real experimental data revealing structural differences from synthetic models.

Contribution

It develops a new technique for creating and unwrapping multifractal fields with adjustable parameters, enhancing the modeling and analysis of complex spatial phenomena.

Findings

Generated synthetic multifractal fields with tunable properties

Successfully unwrapped experimental data revealing anisotropic structures

Revealed differences between synthetic models and real data, such as filamentary patterns

Abstract

We develop a powerful yet simple method that generates multifractal fields with fully controlled scaling properties. Adopting the Multifractal Random Walk (MRW) model of Bacry et al. (2001), synthetic multifractal fields are obtained from the fractional integration of non-Gaussian fluctuations, built by a non-linear transformation of log-correlated Gaussian fields. The resulting fields are parameterized by their roughness exponent $H$, intermittency $\lambda$ and multifractal range $\xi_\omega$. We retrieve all the salient features of the MRW, namely a quadratic scaling exponent spectrum $\zeta_q$, fat-tail statistics of fluctuations, and spatial correlations of local volatility. Such features can be finely tuned, allowing for the generation of ideal multifractals mimicking real multi-affine fields. The construction procedure is then used the other way around to unwrap experimental data -- here the roughness map of a fractured metallic alloy. Our analysis evidences subtle differences with synthetic fields, namely anisotropic filamental clusters reminiscent of dissipation structures found in fluid turbulence.