Wave scattering from two-dimensional self-affine Dirichlet and Neumann surfaces and its application to the retrieval of self-affine parameters

Abstract

Wave scattering from two-dimensional self-affine Dirichlet and Neumann surfaces is studied for the purpose of using the intensity scattered from them to obtain the Hurst exponent and topothesy that characterize the self-affine roughness. By the use of the Kirchhoff approximation a closed form mathematical expression for the angular dependence of the mean differential reflection coefficient is derived under the assumption that the surface is illuminated by a plane incident wave. It is shown that this quantity can be expressed in terms of the isotropic, bivariate (α-stable) Lévy distribution of a stability parameter that is two times the Hurst exponent of the underlying surface. Features of the expression for the mean differential reflection coefficient are discussed, and its predictions compare favorably over large regions of parameter space to results obtained from rigorous computer simulations based on equations of scattering theory. It is demonstrated how the Hurst exponent and the topothesy of the self-affine surface can be inferred from scattering data it produces. Finally several possible scattering configurations are discussed that allow for an efficient extraction of these self-affine parameters.

Non-parametric reconstruction of the statistical properties of penetrable, isotropic randomly rough surfaces from in-plane, co-polarized light scattering data: Application to computer generated and experimental scattering data

“…Figure 7 compares results for the glass scattering system for s-and p-polarization with a polar angle of incidence of θ 0 = 26°, when the reconstruction is performed on the basis of either the iterative or non-iterative reconstruction approaches, that is, on the basis of the expressions in Eqs. (15) or (16). From the results presented in this figure, several interesting observations should be made.…”

“…It should be remarked that for the input scattering data presented in Fig. 5, only marginal changes are found in the results of the reconstruction by basing it on the iterative approach (15), as compared to using the non-iterative approach (16) to produce the results presented in this figure. It should also be mentioned that for the numerous reconstruction tests that we have performed based on scattering data produced by computer simulations, we have found that the accuracy in the reconstruction of the correlation functions typically is the same, or better, than the accuracy in the reconstruction of the surface roughness.…”

“…Our reconstruction approach is non-parametric and based on the expressions in Eqs. (16) or (15). To be performed, it requires the numerical evaluation of a one-dimensional integral and a few standard mathematical functions for each value of x for which one wants to reconstruct the correlation function W (x ).…”

“…In passing we mention that a set of inverse wave scattering approaches recently were presented for the determination of the Hurst exponent and the topothesy of self-affine surfaces [15,16]; such surfaces are examples of scale-free surfaces (if cut-offs can be neglected), and they display fractal properties at sufficiently small length scales.…”

Statistically representative estimators of multi-scale surface topography: example of aluminum blasted rough samples

“…An interesting limit is obtained in the case of power-law correlations which can also be considered as a continuum of characteristic length scales. This case of long-ranged height correlations corresponds to selfaffine surfaces [39,[69][70][71]. Such surfaces have the particular property to remain statistically invariant by the family of anisotropic transformations: r → λr, z → λ H z where λ is a positive real scaling factor and H an exponent usually in the range [0, 1].…”