Curvelet-based Model for the Generation of Anisotropic Fractional Brownian Fields_2

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CURVELET-BASED MODEL FOR THE GENERATION OF

ANISOTROPIC FRACTIONAL BROWNIAN FIELDS

A PREPRINT

Marcos V. C. Henriques

Departamento de Ciências Exatas e Tecnologia da Informação

Universidade Federal Rural do Semi-Árido

Angicos, Brazil

viniciuscandido@ufersa.edu.br

October 6, 2022

ABSTRACT

We propose a curvelet-based model for the generation of Anisotropic Fractional Brownian Fields,

that are suited to model systems with orientation-dependent self-similar properties. The synthesis pro-

cedure consists of generating coefﬁcients in the curvelet space with zero-mean Gaussian distribution.

This approach allows the representation of natural systems having stochastic behavior in some degree

and also obeying to a given angular distribution of correlations. Examples of such systems are found

in heterogeneous geological structures, in anisotropic materials and in complex disordered media.

Keywords curvelets ·fractional brownian motion ·fractal models

1 Introduction

Systems with strong anisotropic properties are very frequent in nature, but unfortunately they present serious difﬁculties

in their characterization and mathematical description. This is the case of complex systems in geology, transport

phenomena in material science, wave propagation in disordered media and radiative transfer in systems that do not have

a spherical symmetry.

An example with technological application is related to the problem of petroleum exploration. In this problem,

the main used technique is related to the phenomenon of scattering of seismic waves in geological structures with

anisotropic inhomogeneities in all scales. To take into account the ﬂuctuations of the physical quantities of these

structures it is necessary to describe their properties in a statistical way. In these systems, the geological data are best

modeled by fractal geometries as the ones created by Mandelbrot [

Mandelbrot and Van Ness(1968)

]. For example,

it has been reported by several authors that well log data from petroleum reservoirs show long-range correlation be-

havior, which is a characteristic feature of fractal processes like the Fractional Brownian Motion [

Hewett(1986)

Sahimi and Tajer(2005)

Dashtian et al.(2011)Dashtian, Jafari, Sahimi, and Masihi

Hardy and Beier(1994)

]. It has

been also reported [

Hansen et al.(2011)Hansen, Lucena, and Da Silva

] that long range correlations in unconsolidated

sandstones and porous media can be understood as the consequence of extreme dynamics restructuring processes

occurred during the geological evolution. The onset of anisotropy in these geological systems can be related to the

inﬂuence of the gravitational ﬁeld and to macroscopic material ﬂows. Characterizing long-range correlations in these

systems, such as porous media described by the porosity logs, may prove useful for an accurate interpretation of

geological data.

When analysing layered rocks and geological ﬁelds formation due to stratiﬁcation process, we can ver-

ify that geometrical and/or transport properties may be characterized in terms of their anisotropy degree

[

Sahini and Sahimi(1994)

Dullien(2012)

Makse et al.(1995)Makse, Davies, Havlin, Ivanov, King, and Stanley

Indeed, the probability of occurrence of a given porosity (or permeability) gradient is strongly de-

pendent on the orientation of the rock properties. Consequently, these systems are characterized by

highly anisotropic correlations. Studies in other areas deal with similar phenomena. For example,

arXiv:2210.01939v1 [physics.data-an] 4 Oct 2022

Curvelet-based Model for the Generation of Anisotropic Fractional Brownian Fields A PREPRINT

Ponson et al [

Ponson et al.(2006)Ponson, Bonamy, Auradou, Mourot, Morel, Bouchaud, Guillot, and Hulin

]

have studied the self-afﬁne properties of anisotropic fracture surfaces. Jennane et al

[

Jennane et al.(2001)Jennane, Ohley, Majumdar, and Lemineur

] have observed anisotropic correlation in images of

bone X-ray tomographic microscopy projections.

Since the early work of Mandelbrot [

Mandelbrot and Van Ness(1968)

], the Fractional Brownian Motion model (FBM)

is largely being used to treat a wide variety of natural phenomena having self-similarity and long-range correla-

tions. In particular, the fractal analysis study applied to bidimensional ﬁelds has proved to be very useful to de-

scribe some physical properties such as roughness and porosity. However, the standard FBM model is only suitable

for describing materials and media that show isotropic symmetry. In that context, generalizations of the FBM to

anisotropic models have been proposed in last years. Kamont [

Kamont(1995)

] studied the Fractional Anisotropic

Wiener Field. Bonami and Estrade [

Bonami and Estrade(2003)

] analysed the properties of various models of anisotropic

Gaussian ﬁelds and proposed a new procedure to characterize the anisotropy of such ﬁelds. Tavares and Lucena

[

Tavares and Lucena(2003)

] and Heneghan [

Heneghan et al.(1996)Heneghan, Lowen, and Teich

] have implemented

wavelet-based models for anisotropic hypersurfaces. Biermé and Richard [

Biermé and Richard(2008)

] have applyied

Radon Transform to estimate the anisotropy of Fractional Brownian Fields. Xiao [

Xiao(2009)

] has studied sample path

properties of such anisotropic ﬁelds. It will be desirable to generalize these concepts and ideas in simple and automatic

ways that can allow sparse representations of highly complex anisotropic ﬁelds.

In this paper, we introduce a new method for generating 2-D Anisotropic Fractional Brownian Fields (AFBF)

based on the Curvelet Transform. This is a new multiscale transform with strong directional character that

provides an optimal representation of objects that have discontinuities along edges [

Candes and Donoho(2000)

Starck et al.(2002)Starck, Candès, and Donoho

Candès and Donoho(2004)

]. The curvelets are localized not only in

the spatial domain (location) and frequency domain (scale), but also in angular orientation, which is a step ahead

compared to Wavelet Transform [

Stephane(1999)

]. This directional feature is the one we use to achieve the anisotropy.

2 The Fractional Brownian Motion

We start with the model of Fractional Brownian Motion proposed by Mandelbrot [

Mandelbrot and Van Ness(1968)

An isotropic two-dimensional Fractional Brownian ﬁeld (FBF)

BH(u)

u∈R2

, with Hurst index

taking values in

(0,1), is deﬁned by the correlation function

E[BH(u)BH(v)] = CH|u|2H+|v|2H+|u−v|2H,∀u, v ∈R2,(1)

where

E[·]

is the expectation operator and

is a constant depending on

BH(u)

is not a stationary process, but its

increments form a stationary, zero-mean gaussian process, with variance depending only on the distance ∆u:

Eh|BH(u+ ∆u)−BH(u)|2i∝∆uH.

It follows from (1) that BHis a self-similar ﬁeld:

BH(λu)d

=λHBH(u),

where

means equal in distribution, and

λ > 0

is a constant. It can be proved that

has an average spectral density

of the form:

S(ξ)∝|ξ|−2H−2(2)

in which

ξ= (ξ1, ξ2)

are the frequency coordinates in the Fourier domain. As proposed in [

Bonami and Estrade(2003)

anisotropic ﬁelds with stationary increments can be deﬁned by taking orientation-dependent spectral densities of the

form:

S(ξ)∝|ξ|−2Hθ−2

where the Hurst index

Hθ

now depends on the direction of

, labeled by the index

. We consider

a parameter related

to the roughness and regularity of the surface. So, in the case of anisotropy, we would expect that

Hθ

would be related

to the self-similarity properties of the ﬁeld on a given direction. A 1-D process obtained by selecting any straight line of

an isotropic FBF

is a FBM with Hurst index

. However, contrary to what one would expect, the estimation of the

orientation-dependent Hurst index

in an anisotropic Gaussian ﬁeld cannot be performed simply by analysing sample

lines of the ﬁeld, as in the isotropic case, since the regularity along a line do not appear to be dependent on the direction.

In order to measure the anisotropy dependence of Hurst exponent

Hθ

, Bonami and Strade [

Bonami and Estrade(2003)

]

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CURVELET-BASEDMODELFORTHEGENERATIONOFANISOTROPICFRACTIONALBROWNIANFIELDSAPREPRINTMarcosV.C.HenriquesDepartamentodeCiênciasExataseTecnologiadaInformaçãoUniversidadeFederalRuraldoSemi-ÁridoAngicos,Brazilviniciuscandido@ufersa.edu.brOctober6,2022ABSTRACTWeproposeacurvelet-basedmodelforthegenerationofAn...

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Curvelet-based Model for the Generation of Anisotropic Fractional Brownian Fields_2

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