Explicit construction of operator scaling Gaussian random fields

Transcription

1 Explicit construction of operator scaling Gaussian random fields M.Clausel a, B.Vedel b a Laboratoire d Analyse et de Mathématiques Appliquées, UMR 8050 du CNRS, Université Paris Est, 61 Avenue du Général de Gaulle, Créteil Cedex, France. clausel@univ-paris12.fr Tel : b Laboratoire de Mathematiques et Applications des Mathématiques, Universite de Bretagne Sud, Université Européene de Bretagne Centre Yves Coppens, Bat. B, 1er et., Campus de Tohannic BP 573, Vannes, France. vedel@univ-ubs.fr Tel : Abstract We propose an explicit way to generate a large class of Operator scaling Gaussian random fields (OSGRF). Such fields are anisotropic generalizations of selfsimilar fields. More specifically, we are able to construct any Gaussian field belonging to this class with given Hurst index and exponent. Our construction provides - for simulations of texture as well as for detection of anisotropies in an image - a large class of models with controlled anisotropic geometries and structures. Key words: Operator scaling Gaussian random field, anisotropy, pseudo-norms, harmonizable representation MSC: 60G15, 60G18 60G60, 60G17 Preprint submitted to Elsevier October 27, 2009

2 1. Introduction Random fields are a useful tool for modelling spatial phenomenon like environmental fields, including for example, hydrology, geology, oceanography and medical images. Particularly important is the fact that in many cases these random fields have an anisotropic nature in the sense that they have different geometric characteristics along different directions (see, for example, Davies and Hall ([8]), Bonami and Estrade ([4]) and Benson, et al.([3])). Moreover, many times the chosen model has to include some statistical dependence structure that might be present across the scales. For that the usual assumption of self-similarity is made. Unfortunately, the classical notion of self-similarity (see [13]), defined for a field {X(x)} x R d on R d by {X(ax)} x R d L = {a H X(x)} x R d for some H R (called the Hurst index), is isotropic by construction and therefore has to be changed to fit anisotropic situations. For this reason, an increasing interest has been paid in defining a suitable concept for anisotropic self-similarity. Many authors have developed techniques to handle anisotropy in the scaling : We non-exhaustively refer to the works of Hudson and Mason, Schertzer and Lovejoy (see [10, 15, 16]). This motivated the introduction by Biermé, Meerschaert and Scheffler of operator scaling random fields (OSRF) in [6]. These fields satisfy the following scaling property : {X(a E L x)} x R d = {a H X(x)} x R d. (1.1) for some matrix E whose eigenvalues have a positive real part. A large class of anisotropic fields fit in this class. For example Fractional Brownian Field and Fractional Brownian Sheet are Operator Scaling Gaussian Random Fields (OS- GRF) with exponent E = Id. In [6] in then proved the existence of OSRF with stationary increments (OSSRF) in the stable case for any admissible exponent E. A special class of OSSRF is defined via a harmonizable representation. In order to recover scaling properties, these fields are required to admit a spectral density which has to satisfy specific homogeneity properties. Construction of such spectral densities are given with a non explicit integral formula in the sense that it leads to approximations for numerical computations. It seems then difficult to be of some use for practical situations. However let us recall that a simpler formula is provided in the particular case of diagonalizable matrices. In this paper, our main goal is to provide a complete description via explicit formula of the class of spectral densities of the model defined in [6]. We focus on a specific case : The Gaussian one. The motivation of this restriction is twofold. On one hand, it is a reasonable assumption in many 2

3 applications. On the other hand, it is necessary to understand and classify the types of geometries, in this special case, in order to propose more pertinent models. Our main results are stated and proved in Section 3. The first one, Theorem reduces the construction of an explicit example of OSGRF for a fixed matrix E and an admissible Hurst exponent H (as defined in Section 2) to the construction of an explicit example in four particular cases related to specific geometries, 2. provides an explicit example in these four specific cases. Thus, we are able to provide an explicit example of OSGRF satisfying Equation (1.1) and then extend the already existing results. Moreover, our second result Theorem 3.2, gives a very simple relationship existing between all possible spectral densities associated to a given exponent E. This result is not formal and can also be turned into an algorithm which generates different fields - with different geometries - satisfying Equation (1.1) for the same matrix of anisotropy E. These results have important consequences. Firstly, it allows to define the studied class of OSGRF from four specific cases. Furthermore we give a complete description of the whole class of spectral densities of these fields. Finally, since the construction is explicit, it may allow to simulate the OSGRF. Thus, it provides an interesting and large class of fields for simulations of textures with new geometries. There is actually a practical motivation to be able to compare natural/real images (of clouds, bones,...) with models whose the anisotropy is controled. In the following pages, E denotes a matrix whose eigenvalues have a positive real part and we define ρ min (E) = a E is defined as min λ Sp(E) a E = exp(e log(a)) = E k log(a) k. k! k 0 As usual for any matrix E, E t denotes the transpose of E. (Re(λ)). For any a > 0 recall that 2. Presentation of the model : Operator Scaling Random Fields (OSRF) Let E be a matrix whose eigenvalues have a positive real part. We recall some preliminary facts about Operator Scaling Random Fields (OSRF) and Operator Scaling Gaussian Random Fields (OSGRF). We refer to [6] for all the material of this section. Definition 2.1. A scalar valued random field is called operator scaling if there exists a matrix E of M d (R) whose eigenvalues have positive real parts and H > 0 such that {X(a E L x)} x R d = {a H X(x)} x R d (2.1) 3

4 where (L) = denotes equality of all finite-dimensional marginal distributions. The matrix E and the real H are respectively called an exponent (of scaling) or an anisotropy and an Hurst index of the field. Remark 2.1. In general, the exponent E and the Hurst index H of an OSRF are not unique. Thus the usual notion of self-similarity is extended replacing usual scaling, (corresponding to the case where E = Id) by a linear scaling involving matrix E (see figure 2 below). It allows to define new classes of random fields with new geometry and structure E =, 0 1 λ {1,, 10} E =, 1 1 λ {1,, 10} E =, 0 1/2 λ {1,, 10} Figure 1: Action of a linear scaling x λ E x on a ellipsis. As said in the introduction, when matrix E is fixed, the class of OSSRF with exponent E may be very large. The authors of [6] prove the existence of OSSRF admitting stationary increments for any given admissible matrix E. More precisely they prove the following result that we state only in the Gaussian case : Theorem 2.1. Let ρ a continuous function with positive values such that for all x 0, ρ(x) 0. Assume that ρ is E t -homogeneous that is : a > 0, ξ R d, ρ(a Et ξ) = aρ(ξ). Then the Gaussian field X ρ (x) = (e i<x,ξ> 1)ρ(ξ) H Tr(E) 2 dŵ(ξ), R d (2.2) exists and is stochastically continous if and only if H (0, ρ min (E)). Moreover this field has the following properties : 4

5 1. Stationary increments that is for any h R d {X ρ (x + h) X ρ (h)} x R d (fd) = {X ρ (x)} x R d 2. Operator scaling : The scaling relation (2.1) is satisfied. Remark 2.2. The assumption of homogeneity on function ρ is necessary to recover linear self-similarity properties of the Gaussian field {X ρ (x)} x R d. The assumption of continuity on spectral density allows to ensure that the constructed field is stochastically continuous. The main difficulty to overcome is to define suitable spectral densities of this new class of Gaussian fields using continuous, E t -homogeneous functions with positive values. In [14] such functions are called (E t, R d ) pseudo-norms. They can be constructed using an integral formula (see theorem 2.11 of [6]) : Proposition 2.2. Function ρ defined as ρ(ξ) = (1 cos(< x, r Et θ >)) dr r 2 dµ(θ), S 0 0 is continuous with positive values and E t -homogeneous. Here S 0 denote the unit sphere of R d for a well chosen norm defined from E and µ a finite measure on S 0. Remark 2.3. This example of (E t, R d ) pseudo-norm is explicit in the mathematical sense. Nevertheless, remark that this formula does not lead to explicit numerical computations : To compute (E t, R d ) pseudo-norm using this formula need to approximate an integral. In the following, we will give simpler examples of (E t, R d ) pseudo-norms in the sense that these examples lead to exact numerical computations : We then say that we give explicit examples of (E t, R d ) pseudo-norms (in the numerical sense). Remark 2.4. We also refer to P.G.Lemarie (see [14]). Using a slightly different definition of (E t, R d ) pseudo-norms, he gives another construction leading with our definition to an integral formula. Finally, in the special case where matrix E is diagonalizable is given an explicit expression (corollary 2.12 of [6]) : Proposition 2.3. Let E a diagonalizable matrix with positive eigenvalues with associated eigenvectors 0 < λ 1 λ d, θ 1,, θ d, and C 1,, C d > 0. Then for any τ < 2ρ min (E) d ρ(x) = C j < x, θ j > τ/λj j=1 is a continuous, E t -homogeneous function with positive values. 1/τ, 5

6 In this paper, our main goal is to extend these results and describe for any given admissible matrix E and Hurst index H, all the spectral density of this model of OSGRF with stationary increments in an explicit way. 3. Construction of explicit spectral densities As it has already been said in Section 2, the main difficulty to overcome in order to construct explicitly the class of operator scaling Gaussian random field defined in [6] is to define appropriate spectral densities. To this end, we remark that the class of the spectral density used in [6] is intimately related to the class of the so-called pseudo-norms defined in [14]. We then explicit the link between two (R d, E) pseudo-norms when matrix E is fixed. Thereafter using a Jordan reduction, for each matrix E whose eigenvalues have positive real parts we give an explicit example of a suitable spectral density of the studied model. Combining these two results, we entirely describe in a explicit way the class of spectral densities of the Gaussian fields considered in [6] More about pseudo-norms Let us first recall some well known facts about pseudo-norms which can be found with more details in [14]. This concept is a fundamental one when defining anisotropic functional spaces since using pseudo-norms allows to introduce anisotropic topology on R d. Thus, even if the introduction of this concept is not necessary to the construction of spectral densities, it is of great importance to relate the notion of anisotropic spectral densities to the concept of pseudonorms. This give us indeed all the tools of Anisotropic functional analysis to study, for eg., the regularity of sample paths of the fields in anisotropic spaces (see [7]) and to better understand the inherent topology of these spaces. Definition 3.1. A function ρ defined on R d is a (R d, E) pseudo-norm if it satisfies the three following properties : 1. ρ is continuous on R d, 2. ρ is E-homogeneous, i.e. ρ(a E x) = aρ(x) x R d, a > 0, 3. ρ is strictly positive on R d \ {0}. Remark 3.1. Our definition of (R d, E) pseudo-norm is a slightly modified version of the concept of pseudo-norm on (R d, A) defined by P.G.Lemarié in [14]. In [14], A denotes a matrix whose eigenvalues have a modulus greater than one. A pseudo-norm on (R d, A) is a function satisfying properties 1 and 3 of the previous definition and the following property : ρ(ax) = det(a) ρ(x), for any x in R d. Further for any matrix A whose eigenvalues have a modulus greater than one, an example of pseudo-norm on (R d, A) is provided by ρ(x) = j Z det(a) j φ(a j x), 6

7 where φ is a compactly supported smooth function. Remark that if ρ is a (R d 1, E) pseudo-norm then ρ( ) Tr(E) is a pseudo-norm on (R d, A) in the sense of [14] with A = a E for any given a > 0. The properties satisfied by (R d, E) pseudo-norms are very similar to those satisfied by pseudonorms on (R d, A) proved in [14]. Moreover the example of pseudo-norm on (R d, A) given in [14] can be adapted to our case. Indeed for any compactly supported smooth function ρ(x) = + 0 φ(a E x)da, is a (R d, E) pseudo-norm. This example also leads to numerical approximations and thus is a non explicit one. The term of pseudo-norm is justified by the following Proposition : Proposition 3.1. Let ρ a (R d, E) pseudo-norm. There exists a constant C > 0 such that ρ(x + y) C(ρ(x) + ρ(y)), x, y R d Relationship between two given pseudo-norms The main result of this section is the description of all the (R d, E) pseudonorms for a given matrix E : Theorem 3.2. Let ρ 1 be a (R d, E) pseudo-norm. Then, ρ 2 is a (R d, E) pseudonorm if and only if there exists a continuous and positive function g defined on R d \ {0} such that ρ 2 (ξ) = g(ρ 1 (ξ) E ξ)ρ 1 (ξ). Remark 3.2. Note that, ρ 2 = ρ 1 g = 1 on the unit sphere S0 E(ρ 1). Conversely, if g is not identically equal to 1 on S0 E (ρ 1 ) and if X 1 (x) = (e i<x,ξ> T r(e) (H+ 1)ρ 1 (ξ) 2 ) dŵξ, X 2 (x) = (e i<x,ξ> T r(e) (H+ 1)ρ 2 (ξ) 2 ) dŵξ, then X 1 and X 2 have not the same spectral densities and therefore are no equal in law. Thus, taking g non-trivial on S0 E (ρ) allows us to define different OSGRF of same exponent E. Proof. Let ρ 1 and ρ 2 be two (R d, E) pseudo-norms. Then the function g = ρ2 ρ 1 is continuous, positive on R d \ {0} and satisfies for all a > 0, g(a E ξ) = g(ξ). In particular, for a fixed ξ and a = ρ 1 (ξ) 1, it follows that ρ 2 (ξ) = g(ξ)ρ 1 (ξ) = g(ρ 1 (ξ) E ξ)ρ 1 (ξ). 7

8 The converse is straightforward. Consider the special case E = Id. Since ξ ξ is a (Id, R d ) pseudo-norm, we then proved that any self-similar Gaussian field with stationary increments and Hurst index H admitting a continuous spectral density is of the form X(x) = e i<x,ξ> 1 R ξ S H+d/2 d ξ dξ ξ with S = 1/g H+d/2. Recall that we recover well known results since Dobrushin in [9] has given a complete description of self-similar generalized Gaussian fields with stationary rth increments and then in particular of self similar Gaussian fields with stationary increments admitting a continuous spectral density. This model of anisotropic Gaussian field has been widely studied (see [5, 4]). Recently in [12] J.Istas has proposed an estimation of the anisotropy function S using shifted generalized quadratic variations. Let us emphasis that if an anisotropy E may be known, using Theorem 3.2 one can probably deduce in a similar way in our case an estimate of function S and thus of the spectral density of the considered Gaussian field. We now need to define explicit examples of pseudo-norms associated with a given matrix E. This is the aim of next Section Explicit construction of pseudo-norms associated with a given matrix E The result of this section is based on the Jordan decomposition of any matrix E. Proposition 3.3. Let E be a matrix of M d (R) whose eigenvalues have a positive real parts. The matrix E can be written, using the Jordan reduction in M d (R) as E = P E E m1+m 2 (where (m 1, m 2 ) N 0 N 0 \ {(0, 0)}) with P 1, 1. For all l 1 {1,, m 1 }, λ l E l1 = λ l1 Id or E l1 = 1, 0 λ l1 8

9 2. For all l 2 {1,, m 2 }, A A l2 0 l2 I 2 0 E m1+l 2 = or E m1+l 2 = I2, 0 A l2 0 A l2 αl2 β with A l2 = l2 1 0, I β l2 α 2 =. l2 0 1 As a consequence of Jordan decomposition, we state the following proposition : Proposition 3.4. The notations are as above. Let us assume that for each 1 l m 1 + m 2, there exist a pseudo-norm τ l such that, for all ξ l R d l with d l is the size of E l, τ l (a E l ξl ) = aτ l (ξ l ). Then, the function ρ defined by ρ( ) = ϕ(p ) with ϕ defined, for ξ = (ξ 1,, ξ p,, ξ m1+m 2 ) with ξ p R dp, by ϕ(ξ) = (τ 1 (ξ 1 ) τ 2 m 1+m 2 (ξ m1+m 2 )) 1/2, is a (R d, E ) pseudo-norm and φ = ρ (2H+Tr(E)) is a suitable spectral density of an operator scaling Gaussian random field with stationary increments. Proof. Let F = P 1 EP. Then : It follows that ϕ(a F t ζ) = ( τ 2 1 (aet 1 ζ1 ) + + τ 2 m 1+m 2 (a Et m 1 +m 2 ζ m1+m 2 ) = ( a 2 τ 2 1 (ζ 1) + + a 2 τ 2 m 1+m 2 (ζ m1+m 2 ) ) 1 2 = aϕ(ζ) ρ(a Et ξ) = ϕ(p t a Et ξ) = ϕ(a F t P t ξ) = aϕ(p t ξ) = aρ(ξ) The conclusion is then straightforward. Let us illustrate Proposition 3.4 through an example : ) 1 2 Example 3.5. Let us consider E = 2 1, 0 1 which is a diagonalizable matrix since it admits two different eigenvalues. One has E = PDP 1 with D =, P =

10 Remark that a (R d, D) pseudo-norm can be defined as ρ D (ξ) = ξ 1 1/2 + ξ 2. Hence Proposition 3.4 allows to give an explicit expression of a (R d, E t ) pseudonorm : ρ E (ξ) = ρ D0 (P t ξ) = ξ 1 1/2 + ξ 1 + ξ 2 Remark that in this case using corollary 2.12 of [6] would have led exactly to the same result since the authors of [6] have given an explicit example of (E t, R d ) pseudo-norms in the special where matrix E is diagonalizable. Thus, it is sufficient to construct an explicit pseudo-norm for the four following matrices. λ 0 1. E = λ λ E = λ A 0 3. E =... α β with A =. β α 0 A A I E = α β... I2 with A =. β α 0 A We emphasize that Proposition 3.4 above has two important consequences : The first consequence will be that, combining Theorem 3.6 in which we define in each generic case an example pseudo-norm, Proposition 3.4 and Theorem 3.2 we can give a complete description of the spectral densities used to define the class of Gaussian field defined in [6]. Moreover, it implies that all the Gaussian fields belonging to the studied class can be generated from four generic cases corresponding to four specific geometries. In the following Proposition, we define a (R d, E) pseudo-norm in each generic case above : Theorem 3.6. One has : 10

11 1. The function, defined for ξ R d by ρ(ξ) = ξ 1/λ is a (R d, E 1 ) pseudo-norm. 2. For 1 i d and ξ = (ξ 1,, ξ d ) R d, let us define the functions τ i and ρ i by If i = 1 τ 1 (ξ) = ξ 1 ρ 1 (ξ) = ξ 1 { ξi if ξ 1 = ξ 2 = = ξ i 1 = 0 if i 2 τ i (ξ) = ρ i 1 (ξ) ( ρ i 1 (ξ) E λ ξ) i else ρ i (ξ) = ρ i 1 (ξ) + τ i (ξ). Then, the function ρ defined for ξ R d by is a (R d, E 2 ) pseudo-norm. ρ(ξ) = ρ d (ξ) 1 λ 3. The function ρ defined for ξ R 2 by ρ(ξ) = ξ 1/α is a (R 2, E 3 ) pseudo-norm. 4. For 1 i d and ξ = (ξ 1,, ξ n ) R d, let us define the functions τ i and ρ i by If i = 1 τ 1 (ξ) = ( ξ ξ 2 2 ) 1 2 ρ 1 (ξ) = ( ξ ξ 2 2 ) 1 2 { ( ξ2i ξ 2i 2 ) 1 2 if ξ 1 = ξ 2 = = ξ 2i 2 = 0 If i 2 τ i (ξ) = ρ i (ξ) ρ i 1 (ξ) (( ρ i 1 (ξ) E α ξ) 2 2i 1 + ( ρ i 1(ξ) E α ξ) 2 2i )1 2 = ρ i 1 (ξ) + τ i (ξ). else Then, the function ρ defined for ξ R d by is a (R d, E 4 ) pseudo-norm. ρ(ξ) = Φ d (ξ) 1 α Remark 3.3. In the third case, we have given a pseudo-norm which is isotropic. Up to a multiplicative constant, it is the unique isotropic one. Indeed, let ρ be an other (R d, E ) pseudo-norm such that ρ is isotropic, in the sense that ρ(a γid ξ) = aρ(ξ) ( a > 0) 11

12 for a γ > 0. For a > 0 such that lna = 2π/β, one has a E ξ = a α ξ and aρ(ξ) = ρ(a α ξ) = ρ(a γ ξ) Writing a = b γ with b > 0, it comes b γ = b α and γ = α since a 1. Now, consider the function g defined for ξ R d \ {0} by g = ρ/ ξ α. As in the proof of Theorem 3.2, since α is a (R n, E ) and a (R d, αid) pseudo-norm, we have, for all ξ R d and a > 0, g(a E ξ) = g(ξ) = g(a α ξ). The second equality shows that g is isotropic. But, the other equality leads to g(ξ) = g(a E ξ) = g(a α R θ (ξ)) = g(r θ ξ) ( a > 0) where R θ is the rotation of angle β log(a). Thus g is radial and so constant on R d \ {0}. Remark 3.4. Using Theorem 3.2, with g non trivial (i.e. non constant on the isotropic unit ball) and ρ 1 = 1/α, we are able to construct non-isotropic (R d, E ) pseudo-norms with a property of spiral anisotropy. Proof. Cases 1 and 3 are straightforward. For the second case, it is clear that ρ d 0 and ρ d (ξ) = 0 if and only if ξ = 0. Therefore, the function ρ is well-defined and positive on R d \ {0}. To show that ρ is continuous, the only point to check is that for all 1 i d, τ i is continuous. For that, we observe that ρ i 1 (ξ) ( ρ i 1 (ξ) E /λ ξ) i = ρ i 1 (ξ) Id E /λ ξ) i i 1 = ξ i (ξ ξ k )c(i, k)(lnρ i 1 (ξ)) k and prove by recurrence on 1 i d, that, for all 1 k i, and for all 1 p d, lim (ξ 1,,ξ i) (0,,0) (ξ ξ k )(ln ρ i (ξ)) p = 0. The last point is to check that ρ satisfies the homogeneity condition. It can be done by recurrence on i, showing that, for all 1 i d, one has k=1 τ i (a E ξ) = a λ τ i (ξ) and ρ i (a E ξ) = a λ ρ i (ξ) Indeed, if the result is true for i 1, for a > 0, and for ξ such that ξ 1,, ξ i are not all equal to 0 (the other case being trivial), one has τ i (a E ξ) = ρ i 1 (a E ξ) ( ρ i 1 (a E ξ) E λ a E ξ) i = a λ ρ i 1 (ξ) ((a λ ρ i 1 (ξ) ) E λ a E ξ) i = a λ τ i (ξ). 12

13 The fourth case can be treated in a similar way by noticing that if, for all 1 i d, one puts r i = ( ξ 2i ξ 2i 2 ) 1 2, it leads to the same definition of the pseudo-norm that for the second case (with r i instead of ξ i ) Examples in dimension 2 We now consider the specific case of dimension 2. Up to a change of basis, E is a matrix of the form : λ E 1 = with (λ 0 λ 1, λ 2 ) (R 2 +) 2. λ 0 2. E 2 = with λ R 1 λ +. α β 3. E 3 = with (α, β) R β α 2. Let us remark that in dimension 2 there is not four generic cases but three since A I matrix E cannot be equivalent to α β... I2 with A =. β α 0 A We now give an explicit example in each of the case above using the results of Theorem 3.6 : 1. If E = E 1, the the function ρ 1 (ξ 1, ξ 2 ) = ( ξ 1 2/λ1 + ξ 2 2/λ2 ) 1/2 is a (E t 1, R2 ) pseudo-norm. 2. If E = E 2, the function ρ 2 (ξ 1, ξ 2 ) = ( ξ 1 + ξ 2 ξ1 λ ln ξ 1 ) 1/λ is a (E t 2, R2 ) pseudo-norm. 3. If E = E 3 the function ρ(ξ 1, ξ 2 ) = ξ 1/α is a (E 3, R 2 ) pseudo-norm. More interestingly, the function with is also a (E t 3, R2 ) pseudo-norm. ρ 3 = ξ 1 cos(β/α ln(r)) ξ 2 sin(β/α ln(r) r 2/α r = ( ξ ξ 2 2 ) 1 2 Combining with Proposition 3.4, it leads us to an explicit example of (E t, R 2 ) pseudo-norms for any matrix E of M 2 (R) whose eigenvalues have positive real parts (see 3.4 below). Thereafter Theorem 3.2 brings us a complete description of the whole class of (E t, R 2 ) pseudo-norms for any matrix E (E t, R 2 ) pseudonorms and thus for spectral densities of the class of OSGRF defined by H.Biermé, M.Meerschaert and H.P.Scheffler. 13

14 E = E = E = /2 E = Figure 2: Four pseudo-norms corresponding to generic cases in dimension two. References [1] Arneodo, A., Decoster, N. and Roux, S.G (2000). A wavelet-based method for multifractal image analysis. I. Methodology and test applications on isotropic and anisotropic random rough surfaces. European Physical Journal B 15, [2] Ayache, A., Léger, S., Pontier and Pontier, M. (2002). Drap Brownien Fractionnaire. Pot. Anal. 17, [3] Benson, D., Meerschaert, M.M., Baumer, B., and Sheffler, H.P. (2006). Aquifer Operator- Scaling and the effect on solute mixing and dispersion. Water Resour.Res. 42 W01415,1 18. [4] Bonami, A. and Estrade, A. (2003). Anisotropic analysis of some Gaussian models. The Journal of Fourier Analysis and Applications 9, [5] A.Benassi,S.Jaffard,D.Roux (1997) Elliptic Gaussian random processes Revista Matematica Iberoamericana 13(1) [6] Biermé, H., Meerschaert, M.M. and Scheffler, H.P. (2007). Operator Scaling Stable Random Fields. Stoch. Proc. Appl. 117(3), Bownik, M.: Anisotropic Hardy spaces andwavelets. Mem.Am. Math. Soc. 164(781), 122pp (2003) [7] Clausel, M., Vedel, B.(2009)Two optimality results about sample paths properties of Operator Scaling Gaussian Random Fields Submitted [8] Davies, S. and Hall, P. (1999). Fractal analysis of surface roughness by using spatial data (with discussion). J. Roy. Statist. Soc. Ser. B 61, [9] Dobrushin, R.L. (1979). Gaussian and their subordinated self-similar random fields. Ann. Proba. 7, [10] Hudson, W., and Mason,J.D. (1982). Operator-self-similar processes in a finitedimensional space, Trans. Am. Math. Soc., 273, [11] Kamont, A. (1996). On the Fractional Anisotropic Wiener Field. Prob. and Math. Stat. 16(1) [12] Istas, J. (2007). Identifying the anisotropical function of a d-dimensional Gaussian selfsimilar process with stationary increments Stat.Inf.Stoch.Proc. 10(1) [13] Lamperti, J.W. (1962). Semi-stable stochastic processes Trans.Math.Amer.Soc

15 [14] Lemarié-Rieusset, P.G. (1994). Projecteurs invariants, matrices de dialatation, ondelettes et analyses multi-résolutions. Revista Matematica Iberoamericana Vol 10, [15] Schertzer, D., Lovejoy, S. (1985), Generalised scale invariance in turbulent phenomena, Phys. Chem. Hydrodyn. J., 6, [16] Schertzer, D., Lovejoy, S. (1987), Physically based rain and cloud modeling by anisotropic, multiplicative turbulent cascades, J. Geophys. Res., 92(D8),