Terrain modelling with multifractional Brownian motion and self-regulating processes

Antoine Echelard (1) , Olivier Barrière (1) , Jacques Lévy Véhel (1, 2)

1 Regularity - Probabilistic modelling of irregularity and application to uncertainties management
2 MAS - Mathématiques Appliquées aux Systèmes - EA 4037

Approximate scale-invariance and local regularity properties of natural terrains suggest that they can be a accurately modeled with random processes which are locally fractal. Current models for terrain modeling include fractional and multifractional Brownian motion. Though these processes have proved useful, they miss an important feature of real terrains: typically, the local regularity of a mountain at a given point is strongly correlated with the height of this point. For instance, young mountains are such that high altitude regions are often more irregular than low altitude ones. We detail in this work the construction of a stochastic process called the Self-Regulated Multifractional Process, whose regularity at each point is, almost surely, a deterministic function of the amplitude. This property makes such a process a versatile and powerful model for real terrains. We demonstrate its use with numerical experiments on several types of mountains.

Mots clés: Digital elevation models; Hölderian regularity; (multifractional) Brownian motion.; (multifractional) Brownian motion

Domaines: Probabilités [math.PR]

Antoine Echelard, Olivier Barrière, Jacques Lévy Véhel. Terrain modelling with multifractional Brownian motion and self-regulating processes. ICCVG 2010, Sep 2010, Warsaw, Poland. pp.342-351. ⟨inria-00538907⟩