Modelling non-Gaussian random fields using transformations and Hermite polynomials

In the context of modelling residual roughness on nominally flat moderately polished metal surfaces, a method is proposed for solving problems related to sample function properties and/or special points such as maxima, minima, saddle points for random fields having non-Gaussian height distributions by recasting them in terms of the corresponding problems for the much more tractable Gaussian random fields by means of transformations. Special reference is made to the expansion of the transformations in series of Hermite polynomials. While the use of Hermite polynomials in connection with transformations of random fields and the useful results they yield with regard to covariance functions are well known, this paper derives the most general explicit formula for the expectation of any product of several Hermite polynomials in correlated Gaussian arguments thereby allowing their application to the higher moments of the transformed random field, in particular, to the third moment, which may be used to measure skewness.