Abstract. 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.Key words: Digital elevation models, Hölderian regularity, (multifractional) Brownian motion. Motivation and backgroundA digital elevation model (DEM) is a digital representation of ground surface topography or terrain. DEM are widely used for geographic information systems and for obtaining relief maps. In most places, the surface of the earth is rough and rapidly varying. 2D random processes are thus often used as models for DEM. In addition, most terrains possess a form of approximate self-similarity, i.e. the same features are statistically observed at various resolutions. For these reasons, fractal stochastic processes are popular models for DEM.The most widely used set of models is based on fractional brownian motion (fBm) and its extensions. fBm has been used for mountain synthesis as well as in the fine description of the sea floor [PP98]. One of the reasons for the success of fBm is that it shares an important property of many natural grounds: statistically, a natural ground is the same at several resolutions. fBm allows to model this scale-invariance property as well as to control the general appearance of the ground via a parameter H taking values in (0, 1): H close to 0 translates into an irregular terrain, while H close to 1 yields smooth surfaces. More precisely, one can show that, almost surely, the local regularity of the paths of fBm, as measured by the Hölder exponent (see definition 22), is at any point equal to H. A large body of works has been devoted to the synthesis and estimation of fBm from numerical data.A major drawback of fBm is that its regularity is the same at every point. This does not fit into reality: for example, erosion phenomena will smooth parts of a mountain more than others. Multifractional Brownian motion (mBm) goes beyond fBm by allowing H to vary in space. Multifractional Brownian motion simply replaces the real parameter H of fBm with a function, still ranging in (0, 1), with the property that, at each point (x, y), the Hölder exponent 2 Ground modelling using extensions of the fractional Brownian motion. of a re...
Self‐regulating processes are stochastic processes whose local regularity, as measured by the pointwise Hölder exponent, is a function of amplitude. They seem to provide relevant models for various signals arising for example in geophysics or biomedicine. We propose in this work an estimator of the self‐regulating function (that is, the function relating amplitude and Hölder regularity) of the self‐regulating midpoint displacement process and study some of its properties. We prove that it is almost surely convergent and obtain a central limit theorem. Numerical simulations show that the estimator behaves well in practice.
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