Beyond multifractional Brownian motion: new stochastic models for geophysical modelling

J. Lévy Véhel

doi:10.5194/npg-20-643-2013

Abstract

Abstract. Multifractional Brownian motion (mBm) has proved to be a useful tool in various areas of geophysical modelling. Although a versatile model, mBm is of course not always an adequate one. We present in this work several other stochastic processes which could potentially be useful in geophysics. The first alternative type is that of self-regulating processes: these are models where the local regularity is a function of the amplitude, in contrast to mBm where it is tuned exogenously. We demonstrate the relevance of such models for digital elevation maps and for temperature records. We also briefly describe two other types of alternative processes, which are the counterparts of mBm and of self-regulating processes when the intensity of local jumps is considered in lieu of local regularity: multistable processes allow one to prescribe the local intensity of jumps in space/time, while this intensity is governed by the amplitude for self-stabilizing processes.

Highlights

Background on fBm and mBmLet us start by recalling some basic facts about fractional Brownian motion (Kolmogorov, 1940; Mandelbrot and Van Ness, 1968)
This is a centred Gaussian process that essentially depends on a single parameter, usually denoted by H and called the Hurst exponent, which belongs to (0,1)
Due to the shape of g, the synthetic terrain automatically adjusts its regularity to be smoother at lower altitudes and rougher at higher ones

Summary

Background on fBm and mBm

Let us start by recalling some basic facts about fractional Brownian motion (fBm) (Kolmogorov, 1940; Mandelbrot and Van Ness, 1968) This is a centred Gaussian process that essentially depends on a single parameter, usually denoted by H and called the Hurst exponent, which belongs to (0,1). Fractional Brownian motion exhibits features that make it a useful model in various fields such as geophysics, financial and teletraffic modelling, image analysis and synthesis, and more These features include self-similarity, long-range dependence (when H > 1/2), and the ability to match any prescribed constant local regularity. Due to the shape of g, the synthetic terrain automatically adjusts its regularity to be smoother at lower altitudes and rougher at higher ones

Midpoint-displacement self-regulating process

Application to the characterization of natural terrains

Findings

Temperature records