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

TL;DR

This paper derives a mathematical model for wave scattering from self-affine surfaces to extract their roughness parameters, demonstrating good agreement with simulations and proposing efficient measurement configurations.

Contribution

It introduces a closed-form expression linking scattering data to self-affine surface parameters, enabling their retrieval from wave scattering measurements.

Findings

The derived expression accurately predicts scattering behavior over a wide parameter range.

Scattering data can be used to infer the Hurst exponent and topothesy of self-affine surfaces.

Proposed configurations facilitate efficient extraction of surface roughness 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 ($\alpha$-stable) L\'evy 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.