Variance of elastic wave scattering from randomly rough surfaces
Introduction
It is well known that surface roughness substantially affects the scattering of elastic waves, by altering the amplitude, phase and distorting the waveform. Understanding the statistics of elastic wave scattering from rough surfaces/interfaces is an important fundamental topic for a variety of problems in solid mechanics; examples include ultrasonic detection/imaging of rough defects (Zhang et al., 2011, Zhang et al., 2012), seismic wave reflection from irregular, or rough, interfaces to improve oil/gas exploration and production (Schultz and Toksoz, 1993, Makinde et al., 2005), ultra-high frequency phonon boundary reflection/transmission to understand heat transfer (Maznev, 2015, Wen et al., 2009, Ravichandran et al., 2018), biomedical ultrasound imaging (Kaleva et al., 2009), and ultrasonic sensing for corrosion (Jarvis and Cegla, 2012) and tribology (Whitehouse and Archard, 1970, Dwyer-Joyce, 2005).
The two-dimensional (2D) scattering problem is depicted in Fig. 1. A plane P wave is incident on the rough surface with an angle of , producing reflection P and mode-converted S waves. The scattered wave is accumulated from reflection wavelets from local surface points, either constructively or destructively depending on their relative phases. The specific profile of a single rough surface is rarely known, but it is often the case that the statistics of the roughness can be obtained, such that an answer by ways of statistical characterisation of the scattered waves (e.g. mean, variance and distribution of the amplitude) is very useful. For instance, for ultrasonic detection of rough cracks, knowing the statistics of the scattering amplitude helps justify the sensitivity of a proposed inspection plan (Ogilvy, 1986, Ogilvy and Culverwell, 1991). The mean field of the scattering energy is often used to calculate the ‘specularity’ of phonon boundary transport, in order to determine the thermal boundary resistance to heat flow (Ravichandran et al., 2018, Sun and Pipe, 2012, Gelda et al., 2018). The statistical features of reflection or transmission amplitude are applied for monitoring the thickness reduction of the corrosion process (Zou and Cegla, 2018), and studying tribological contacts behaviour (Dwyer-Joyce, 2005).
Most of these applications and previous research concern the mean scattering amplitude, or the intensity: where the intensity is the displacement multiplied by its conjugate value . The subscripts and refer to the coherent and the diffuse components respectively. To find the mean field, Monte Carlo simulation techniques such as the finite element (Pettit et al., 2015, Zhang et al., 2011) or the boundary element (Roberts, 2012, Jarvis and Cegla, 2012) methods are applied by executing the models with multiple surface realisations. Alternatively analytical methods such as the perturbation theory (Sun and Pipe, 2012, Ogilvy, 1991) or the Kirchhoff approximation (Ogilvy and Culverwell, 1991) have been extensively used to obtain analytical solutions of the mean intensity/amplitude. Note that such analytical expressions have been found in the acoustic (Ogilvy, 1991, Thorsos, 1988) or the optical communities (Maradudin, 2010) for a long period. Recently the counterpart expressions for elastic waves have been developed for both incident P (Shi et al., 2016, Shi et al., 2017) and S (Haslinger et al., 2020) waves, under the Kirchhoff approximation for a wide range of roughness, by taking the mode conversion occurring at the surface or the interface into consideration. The effects of material elastic properties and the choice of wave modes on the scattering are also investigated.
Apart from the mean value, perhaps an equivalently important but largely neglected statistical parameter, is the variance of the scattering amplitude/intensity. The mean value refers to the scattering amplitude most likely to be observed in a measurement. The variance quantifies to what extent the scattering amplitude deviates from its mean value for an individual surface, which is defined as: The standard deviation is the square root of the variance. The variance of elastic wave scattering from rough surfaces can be significant, or sometimes even comparable with its mean value (Zhang et al., 2012, Haslinger et al., 2020, Ogilvy, 1988). Knowing the variance of amplitudes and how it is affected by the roughness are very useful, to quantify the uncertainties of elastic wave sensing in problems such as ultrasonic detection and imaging, corrosion monitoring and seismic wave exploration. It is particularly important in the author’s research area of ultrasonic Non-Destructive Evaluation (NDE), where the probability of detecting a crack largely depends on the amplitude threshold, which is normally determined from the knowledge of the scattering from various defects. Obviously in order not to miss any possible rough defect, the amplitude threshold needs to be set as the lower bound of the scattering amplitude/intensity from a single rough defect. If the variance and the statistical distribution can be accurately predicted, the lower bound and then the detection threshold to a chosen level of confidence can be better determined.
Most of the work regarding the variance of elastic wave scattering from randomly rough surfaces are based on computer simulations. Ogilvy (1988) has investigated numerically the standard deviation of the ultrasonic signal amplitude scattered by Gaussian random rough defects, via Monte Carlo simulations with the Kirchhoff model. Numerical results suggest that for monochromatic waves, the amplitude is approaching the Rayleigh distribution for very rough surfaces or in the backscattering direction; this is consistent with the findings by Berry (1973) for wave echoes diffracted from rough surfaces. Recently Monte Carlo FE simulations (Pettit et al., 2015, Zhang et al., 2011) have been run to find the variance and the lower bound of the scattering amplitude, and the results are verified via experiments. The mean and the variance of both incident compressional and shear waves have been obtained from numerical simulations (Haslinger et al., 2020). The distributed point source method (Jarvis and Cegla, 2012) has been applied to compute the ultrasonic reflection from corroded surfaces, and the variance of thickness gauging is analysed. However, numerical simulations are often computational demanding, and also it hardly allows for the general trend to be drawn. So far in the elastic wave community, a theoretical approach is lacking for rapid predictions of the variance of the scattering amplitude and intensity. Such theoretical solutions exist for optical and acoustic waves, but not for elastic waves in solids. The main contribution in this study is to provide analytical solutions for the variance of elastic wave scattering. The second contribution is to use high-fidelity Monte Carlo FE simulations to carefully evaluate the accuracy of the theoretical approach.
This article is organised as follows: Section 2 presents the theoretical formulae to predict the variance of the scattering intensity and the amplitude. The theoretical results are verified by numerical Monte Carlo FE simulations in Section 3, and the significance for ultrasonic applications is explained in Section 4.
Section snippets
Rough surface
In this article, the surface is assumed to obey Gaussian statistics because previous research has suggested that it is suitable for characterising morphologies of some real surfaces (Zhang et al., 2011, Ogilvy, 1991, de Billy et al., 1980, Whitehouse and Archard, 1970). The pdf for a Gaussian surface is expressed as: where is the root-mean-square (rms) of the height data. The normalised correlation function between two surface points and is:
Validation with finite element Monte Carlo simulations
High-fidelity finite element simulations capture all the scattering mechanisms, and are used in this section as the benchmark to carefully evaluate the accuracy of the elastic wave theory. Similar with the Monte Carlo simulations using the Kirchhoff model in the previous section, for finite element models 500 Gaussian surfaces are generated for each roughness. The surface is assumed to have a finite length of 7 mm. The correlation length is half wavelength and the rms height ranges from
Significance for ultrasonic applications
In this section, the significance of the variance of the scattering amplitude will be illustrated using one important application example taken from the field of ultrasonic NDE. The reliability of ultrasonic defect detection largely relies on a predefined amplitude threshold, which is set based on the knowledge of elastic wave scattering from various types of regular defects (e.g. a flat crack or a side-drilled-hole). However, regular cracks are rarely seen, and the morphologies of defects
Conclusions
In this article, the variance of elastic wave scattering from randomly rough surfaces is investigated. Theoretical solutions are developed to predict the standard deviation of the scattering amplitude and the intensity, apart from the previously established means of calculating the mean value. The variance of the scattering intensity is obtained directly from the Kirchhoff integral, by taking the fourth order statistics into consideration. A high-frequency asymptotic solution is derived, and
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
The author is grateful to Prof. Mike Lowe for the encouragement of this work, discussions and comments on the manuscript.
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