## 1 Introduction
\begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Formula** & **Ref.** \\ \hline \(Z_{2}=\left[\frac{1}{L}\sum\limits_{i=1}^{n-1}\frac{(y_{i+1}-y_{i})^{2}}{x_{i+ 1}-x_{i}}\right]^{1/2}\) & \(JRC=32.2+32.47\log Z_{2}\) & [18] \\  & \(JRC=130.87(Z_{2})^{2}-2.73\) & [20] \\  & \(JRC=32.69+32.98\lg Z_{2}\) & [21] \\  & \(JRC=51.85Z_{2}^{0.6}-10.37\) & [22] \\  & \(JRC=51.16Z_{2}^{0.531}-11.44\) & [23] \\  & \(JRC=55.7376Z_{2}-4.1166\) & [24] \\  & \(JRC=32.172\log Z_{2}+32.202\) & [25] \\  & \(JRC=57.82[1-(bZ_{2})]^{-0.26}\sqrt{a(Z_{2})}-13.5\) & [26] \\ \hline \(JRC=37.28+16.58\log SF\) & [18] \\  & \(JRC=16.67\log(SF)+37.726\) & [20] \\  & \(JRC=-3.28+121.13\sqrt{SF}\) & [21] \\  & \(JRC=37.63+16.5\log(SF)\) & [23] \\  & \(JRC=73.95(SF)^{0.266}-11.38\) & [25] \\ \hline \(JRC=77.06(R_{P}-1)^{0.172}-29.27\) & [21] \\  & \(JRC=47.62+93.74(R_{P}-1)^{0.1235}\) & [22] \\  & \(JRC=63.69\sqrt{R_{P}-1}-2.31\) & [23] \\  & \(JRC=\left(0.036+\frac{0.00127}{\ln(R_{P})}\right)^{-1}\) & [25] \\  & \(JRC=411(R_{P}-1)\) & [27] \\  & \(JRC=65.9(R_{P}-1)^{0.302}-9.65\) & [28] \\ \hline \(\theta_{p^{+}}=\tan^{-1}\Biggl{\{}\frac{1}{M_{+,+}}\sum\limits_{i=1}^{M_{+,+}} \left[\left(\frac{y_{i+1}-y_{i}}{x_{i+1}-x_{i}}\right)_{+}\right]_{i}\Biggr{\}} \)[29] & \(JRC=-106.67(\theta_{p^{+}})^{-0.36}+55.34\) & [30] \\ \hline \(Z_{2}^{\prime}=\left[\frac{1}{M(x)^{2}}\sum\limits_{i=1}^{M}(\max(0,y_{i+1}-y_ {i}))^{2}\right]^{1/2}\)[31] & \(JRC=-9.43(Z_{2}^{\prime})^{-0.687}+35.11\) & [30] \\ \hline \(\tan\theta^{*}=-\tan\theta\cos\alpha\) & \(A_{\theta^{*}}=A_{0}\left(\frac{\theta_{\max}-\theta^{*}}{\theta_{\max}^{2}} \right)^{C}\)[32] & \(JRC=3.95(\theta_{\max}^{*}/[C+1])_{2\mathrm{D}}^{0.7}-7.98\) & [22] \\ \hline \(r_{i}=\frac{A_{i}}{\sum\limits_{i=1}^{M}A_{i}},R_{i}=\theta_{i}^{*}r_{i},WPA= \sum\limits_{i=1}^{i=m}R_{i}^{+}\)[33] & \(JRC=14.602\ln(WPA)-12.536\) & [33] \\ \hline \end{tabular}
\end{table} 
Table 1: The statistical parameter and characterization formula by single parameter. the contributions of different orders of asperity to the shear strength are different [44; 45; 46; 47]. The joint roughness would be underestimated if it were characterized by a single statistical parameter [48]. In order to solve this problem, multiple joint roughness formulas characterized by two parameters have been proposed, as shown in Table 2. However, the parameters proposed by previous researchers ([9; 49; 50; 51; 52; 53]) do not quantify waviness and unevenness. [PERSON] et al. [54; 55] characterized joint roughness with two statistical parameters by decomposing the joint surface into waviness and unevenness. However, this formula considers only the morphology contribution, and not the full mechanical contribution. At present, several methods, such as the Fourier series method [56; 57; 58; 59; 60], different sampling interval methods [61; 62; 63; 64], and the wavelet analysis method [65], are used for the separation of waviness and unevenness from the joint surface. However, these methods are limited by the characteristics of themselves [66]. The Fourier series method [67] and wavelet analysis both need a priori function, and this function will affect the final result. At the same time, a variety of methods have been proposed to identify waviness and unevenness. [PERSON] [56] proposed a method to distinguish waviness and unevenness by visual observation. [PERSON] [61] proposed distinguishing waviness and unevenness through the change law of joint surface area at different sampling intervals. [PERSON] [65] proposed a law to identify the cut-off level by the variance law of the approximate component. However, a method with clear mathematical definition to judge the critical decomposition level has not yet been proposed. In summary, the morphology of the joint specimen is multi-scale. The joint roughness will be underestimated if it is characterized by a single statistical parameter. At same time, the mechanical contribution and the morphology contribution of each asperity order are different.