The validity of Kirchhoff theory for scattering of elastic waves from rough surfaces

Abstract

The Kirchhoff approximation (KA) for elastic wave scattering from two-dimensional (2D) and three-dimensional (3D) rough surfaces is critically examined using finite-element (FE) simulations capable of extracting highly accurate data while retaining a fine-scale rough surface. The FE approach efficiently couples a time domain FE solver with a boundary integration method to compute the scattered signals from specific realizations of rough surfaces. Multiple random rough surfaces whose profiles have Gaussian statistics are studied by both Kirchhoff and FE models and the results are compared; Monte Carlo simulations are used to assess the comparison statistically. The comparison focuses on the averaged peak amplitude of the scattered signals, as it is an important characteristic measured in experiments. Comparisons, in both two dimensions and three dimensions, determine the accuracy of Kirchhoff theory in terms of an empirically estimated parameter σ 2 /λ 0 ( σ is the RMS value, and λ 0 is the correlation length, of the roughness), being considered accurate when this is less than some upper bound c , ( σ 2 /λ 0 < c ). The incidence and scattering angles also play important roles in the validity of the Kirchhoff theory and it is found that for modest incidence angles of less than 30°, the accuracy of the KA is improved even when σ 2 /λ 0 > c . In addition, the evaluation results are compared using 3D isotropic rough surfaces and 2D surfaces with the same surface parameters.

Diffusely scattered and transmitted elastic waves by random rough solid-solid interfaces using an elastodynamic Kirchhoff approximation

“…The theoretical results again match very well with those computed from numerical simulations for both P-P and P-S modes, over a wide range of scattering angles. At grazing angles (|θ s | ≥ 70 o ), the Kirchhoff approximation is not reliable according to previous 245 studies Thorsos (1988); Shi et al (2015); here the errors at these angles are not clearly seen in Fig. 5(c) and (d), due to the very small values of < σ sc >.…”

“…(27) vanish, which clearly leads to the linear relationship between λ 0 and σ d sc . It should be noted that the simple linear dependence is found within 355 the valid region of the Kirchhoff approximation Shi et al (2015). For weakly correlated surfaces (λ 0 is very small) the linearity might break down, and it is argued to be caused by the energy converted from the surface waves Maznev (2015).…”

“…The KA has been found to be accurate in modelling the scattering from surfaces with roughness up to σ = λ/3, when the correlation length λ 0 is comparable with or larger than one wavelength Thorsos (1988); Shi et al (2015). Despite the early application of KA in acoustics Thorsos (1988), it is only recently that the elastodynamic KA has been developed to model theoretically the 85 diffuse field from an elastic rough surface including mode conversions Shi et al (2016); there are details in elasticity, the mode conversion at surfaces, two types of bulk waves, that prevent a direct generalization.…”

“…For each realization of the surface a purely numerical method, the finite element boundary integral (FEBI) approach Shi et al (2015Shi et al ( , 2014, is performed to compute σ sc . The FEBI method is highly efficient as it computes the very local scattering field on the 190 interface using an explicit time domain FE scheme inside a small domain, and then performs a boundary integral to globally calculate the scattered waves.…”

A time-domain finite element boundary integral approach for elastic wave scattering

“…Classical approaches such as separation of variables (SEP) [43] provide exact mathematical formulae to calculate scattered waves but only from scatterers with regular and simple geometries, such as a spherical voids, and cannot handle complex branched cracks or irregular shapes. In terms of frequency one can employ methods such as the Kirchhoff approximation (KA) which is based on the assumption of an infinite tangential plane can predict scattering signals from defects with irregular shapes, but these approaches can generate unacceptable errors when the roughness is large or with grazing incidence/scattering angles [36,37].…”