Visualization of Roughness Indices in Amplitude-Frequency Space from Wavy Simulated Profiles
Authors: Gray R. , Hudyma N. W. , Chittoori B. C. S. , McLaughlin M. M. 
Published: August 2024

ABSTRACT:
Rock surface roughness influences strength, stiffness, and fluid flow dynamics. Roughness algorithms, classified as either Euclidian or fractal, have been used to quantify roughness through computing roughness indices. Because of the complexity of the microtopography of the rock surface, it is difficult to put into context the meaning of a roughness index. The purpose of this study is to visualize roughness indices of simple simulated rough surfaces in amplitude-frequency space. We used a 1000 mm sine wave with varying amplitudes and frequencies to simulate the rough surface. Five Euclidian roughness algorithms were used to quantify the roughness. To visualize these methods, simulated wavy profiles were created using sine waves with specific bounds to capture both the waviness and unevenness of a profile. The findings highlight the superior performance of the Z2 and Mean Absolute Angle algorithms while indicating that care should be taken when using the Sinuosity to assess low frequency (&lt;2 Hz) and low amplitude (&lt;2 mm) profiles, and Root Mean Square, and Energy algorithms when assessing high frequency (&gt;1 Hz) profiles.

1. INTRODUCTION
Roughness is a parameter that is ubiquitous across many science and engineering fields. Surface roughness is a controlling factor in the strength, stiffness, and fluid flow properties and an assessment tool for weathering.
The separate works of Patton and Barton are fundamental to understanding and quantifying rock roughness. Patton (1966) introduced the concept of two length-based roughness scales of rock joints. Waviness or large-scale roughness occurs on a decimeter-to-meter scale, and unevenness or small-scale roughness occurs on a millimeter-to-centimeter scale. Waviness is low-frequency and high amplitude, whereas unevenness is high-frequency and low amplitude.
Barton (1973) and Barton and Choubey (1977) introduced the joint roughness coefficient (JRC) parameter. The later reference provided the first visualization of a rock profile linked to a roughness index (JRC), which prompted new research efforts to quantify roughness. Contact and non-contact methods were developed to measure surface microtopography. Existing roughness algorithms were applied to quantify the JRC profiles and natural rock surfaces. More importantly, new roughness algorithms were and are continuously being developed.
Methods to quantify rock surface roughness are categorized into Euclidean geometry methods, fractal geometry methods, and empirical methods. The Euclidean geometry methods, 2D and 3D, are further classified as statistical, amplitude-based, slope-based, and amplitude-and-slope-based methods (Kulatilake and Ankah, 2023). These authors state Euclidian geometry methods are ill-suited to accurately represent the highly erratic rock joint surfaces. These methods are still extensively used because they are less complex than fractal geometry. Artificial intelligence, specifically deep learning, has recently been used to assess rock joint roughness (Zhang et al., 2023).