Study of a bifractal process and statistical application in geology.

Summary Since the use of fractional Brownian motion for statistical applications by Mandelbrot and Van Ness in 1968, a vast literature has been built up around the estimation of self-similarity and Hölderian regularity. In the framework of the multifractal analysis of Fourier series and wavelet series (full and lacunar), random wavelet series based on a branching process are presented. Their analytical properties (bifractality, non-integer Hausdorff dimension of the graph) and the simulation of their trajectories show their ability to model intermittent processes. A method for estimating the two parameters of the model (estimation of the regularity parameter and the gap parameter) by filtering a trajectory is developed for this model. It will find a natural application for the classification, in geophysics, of stylolitic morphologies (sedimentary rocks of limestone subjected to stress dissolution processes). These series generalize the properties of fractional Brownian motion and can explain the phenomena of persistence and leptokurticity.