Pandora Box of Multifractals: Barely Open?

Abstract

Three decades ago, multifractals were a major breakthrough in nonlinear geophysics by providing a general framework to understand, analyze, and simulate fields that are extremely inhomogeneous over a wide range of space-time scales. They have remained on the forefront of nonlinear methodologies, but they are still far from being used or even developed to their full extent. Indeed, they have been too often limited to scalar-valued fields, whereas the relevant geophysical fields are vector fields. This chapter therefore gives new insights on current developments to overcome this limitation. This is done in an inductive manner. For instance, it takes hold on simple considerations on “spherical” and “hyperbolic” rotations to introduce step by step the Clifford algebra of Lévy stable generators of multifractal vectors that have both universal statistical and robust algebraic properties.