A Model for Information Support of Environmental Management by Bill Davey, Clive Mathews

By Bill Davey, Clive Mathews

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178) The local curvature of a surface corresponds to the variations in the unit surface normal n with position on the surface. 179) It can be shown that b is symmetric; thus it can be diagonalized in an orthogonal coordinate system. 180) These quantities are the basic quantities which can be used to characterize the shape of the surface. The study of random surfaces has been stimulated by various systems such as microemulsions, sheet polymers and foams. For instance, physicists introduced the study of membranes as two-dimensional generalizations of linear polymer chains; an overview of this topic can be found in Nelson et al.

116) For the sake of notational simplicity, in most of this section, (j is assumed to be equal to 1. 110) are taken into account, roughness (j is chosen as the unit length l. It is crucial to realize that Z (x) cannot have independent increments except for H = 1/2. 120) Hence, increments are not independent except for the standard Brownian motion (H = 1/2). 2. 109) and their expressions for the particular case of a FBM are given. 122) or in terms of b.. 124) Some general properties can be derived from this formula i.

47) Now let's consider two Gaussian variables {Xl! X 2 }. Because of the previous properties, this couple is a random variable just as any linear combination is a Gaussian. Thus, this couple is an example of the multivariate Gaussian variables (Adler, 1981), which can be defined for reference purposes. An Rd-valued random variable X = {Xl! , X d } is called multivariate Gaussian if any nontrivial linear combination of its components is Gaussian. 50) ill the specific case of two independent variables (Xl, X 2), the matrix V is diagonal.

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