# Roughness Evolution of Granite Flat Fracture Surfaces during Sliding Processes
## 2 Roughness Measurement Indicators  
Roughness is measured in a variety of indicators, which are usually categorized into three types [22]: amplitude parameter, spacing parameter, and hybrid parameter, of which amplitude parameter is the most common. The amplitude parameter is mainly based on the vertical coordinate value of the surface. For two-dimensional profile lines, there are the following types of amplitude parameters. The arithmetic mean deviation \(Ra\) of the profile, defined as Equation (1), indicates the average of the absolute value of the vertical coordinate value \(Z(x)\) within the sample length, which can most directly reflect the degree of the friction surface being worn out, and \(l\) is the sample length of the profile line. \[Ra=\frac{1}{l}\int_{0}^{l}|Z(x)|dx \tag{1}\]  
The second amplitude parameter is, the root mean square (RMS) deviation of the profile, \(Rq\), defined as in Equation (2), which represents the root mean square of the vertical coordinate, \(Z(x)\), within the sample length, and reflects the extent to which the profile line deviates from the mean value. \[Rq=\sqrt{\frac{1}{l}\int_{0}^{l}Z(x)^{2}dx} \tag{2}\]  
The \(Rsk\) skewness parameter is used to reflect the tendency of the assessed profile to be concave or convex; a negative value of \(Rsk\) indicates that the profile lies more below the mean plane (valley), but since skewness is affected by a number of factors in the profile, it is discarded here. In addition, JRC (joint roughness coefficient) proposed by [PERSON] and [PERSON] [24] is also a common indicator of surface roughness. Many scholars have also carried out a large number of related studies around JRC, including those on the method of determining JRC [25; 26], and those exploring the new, adapted to different conditions of the relationship between rock fracture surface shear strength and JRC [27; 28]. However, JRC is controversial in its own quantification, is influenced by the subjective factors of the measurer. And when it is verified by back-calculation using the strength results obtained from the test, it tends to have a large discrepancy with the measurement results, and is currently only adapted to the 2D plane. Therefore, this indicator is not used in this paper for the time being. In the above representation of roughness, the amplitude parameters such as \(Ra\), \(Rq\) only reflect the statistics of the longitudinal coordinate data of the rough surface, and miss the information of the horizontal direction of the rough profile. For example, for the two sinusoidal profiles shown in Figure 1, the amplitude is the same, but the curve frequency is different. If roughness indicators such as \(Ra\), \(Rq\) are used for calculation, the roughness obtained will be identical. Although there are also indicators that represent the spacing parameters in the horizontal direction, but they are all more one sided. Some recent studies [23; 29; 30; 31] have shown that the roughness profile can be treated as a special signal, and the power spectral density obtained by discrete Fourier transform has a stable power-law relationship with the wavelength (or spatial frequency), such as Equations (3) and (4), or there is a linear relationship between the logarithms of the two, as written in Equations (5) and (6). \[p=C\cdot\lambda^{-\beta} \tag{3}\]  
\[p=C\cdot f^{\beta} \tag{4}\]  
\[\lg p=\lg C-\beta\lg\lambda \tag{5}\]  
\[\lg p=\lg C+\beta\lg f \tag{6}\]  
where \(\lambda\) is the wavelength, \(f\) is the spatial frequency, \(f=1/\lambda\). \(C\) and \(\beta\) are constants. \(p\) is the power spectral density (PSD), which is expressed as in Equation (7) for discrete data.