Introduction

Rock joint roughness is one of the most important inputs for rock mass stability analysis in engineering design due to its significant impact on the shear strength and hydraulic properties of rock mass (Kulatilake et al. 1995; Belem et al. 2007; Liu et al. 2016; Tang et al. 2019; Jiang et al. 2020). Several indexes have been developed to represent the roughness of rock joint, such as joint roughness coefficient (JRC) (Barton 1973; Barton and Choubey 1977), fractal dimension (Xie et al. 1997; Kulatilake et al. 2006; Stigsson and Ivars 2019), θ*max/(C+1) (Grasselli and Egger 2003), and brightness area percentage (BAP) (Ge et al. 2014). Among them, as the first parameter proposed for rock joint roughness, JRC is more popular in the rock mechanics community because it is relatively straightforward (ISRM 1978). JRC can be determined through visual comparison with ten standard profiles; however, it is highly dependent on people’s subjective judgment, causing roughness estimation errors. To solve this issue, several quantitative approaches were developed to build quantitative relationships between JRC and statistical parameters, including root mean square (Z2) (Tse and Cruden 1979), standard deviation of inclination angle (SDi) (Kusumi et al. 1997), fractal dimension (Lee et al. 1990; Lee et al. 1997), and power spectral density (Ünlüsoy and Süzen 2020). Recently, artificial intelligence (AI) algorithms have been employed to predict the JRC through data training (Wang et al. 2017; Wang et al. 2020).

Several techniques were available to collect the morphology data of the rock joint surface (e.g., 2D profiles or 3D point clouds), which was treated as input parameters to determine the JRC for above-mentioned quantitative approaches. To date, data collection was mainly subdivided into two categories: contact method and non-contact method (Feng et al. 2003). A contact method was often characterized by time-consuming and low precision, especially for large-scale measurement. Therefore, there was an increasing interest in the non-contact method which mainly involves laser scanning and photogrammetry (Maerz et al. 1990; Jiang et al. 2006; Bae et al. 2011; Kim et al. 2013; Mah et al. 2013; Ge et al. 2017). Among these two non-contact techniques, laser scanning was considered as a powerful tool to collect dense point clouds of rock joint surfaces in a short time (Tang et al. 2012; Kumar and Verma 2020). In this study, two kinds of laser scanner were employed to map the rock joints: a terrestrial laser scanner for a larger scale bedding plane in the field and a portable one for small-scale rock joint specimens in the laboratory.

Numerous attempts have been made to investigate the influence of sampling scale, anisotropy, and non-stationary features on the roughness estimation of rock joint (Fardin et al. 2001; Tatone and Grasselli 2013; Kumar and Verma 2016; Ge et al. 2020). Similarly, the sampling interval (also termed sampling spacing) also has a significant impact on the data collection and roughness assessment, which was defined as the horizontal distance between two adjacent points in 2D profiles or 3D point clouds of rock joints and was directly related to the measurement resolution (Yu and Vayssade 1991). On one hand, natural rock joints always consist of two kinds of asperities: primary and secondary asperity (waviness and unevenness, non-stationary and stationary), which represents the macroscopic undulation and microscopic irregularity of the rock joint surface, respectively (Patton 1966; ISRM 1978; Yang et al. 2010). An appropriate sampling interval is needed to be selected to capture these different-scale morphological characteristics (Liu et al. 2017). On the other hand, during a laser scanning, high-resolution point clouds of a rock joint were generated under a small sampling interval, allowing acquisition of more details on the rock joint surface. However, measurement with extra high resolution tended to produce a large number of redundancy, as a result, causing overestimation on roughness and low computational efficiency. Accordingly, a high level of hardware and software capacity was required for data processing. On the other hand, if a too large sampling interval was assigned for laser scanning measurement, lots of geometrical information on the rock joint surface would be ignored, leading to an underestimation on the roughness, even for a very rough rock joint (Reeves 1985; Tatone and Grasselli 2010).

Up to now, some researches have focused on the effect of sampling interval on the estimation of rock joint roughness (Hong et al. 2008; Ge et al. 2015; Sun et al. 2019). A negative correlation between the rock joint roughness and sampling interval was observed in their findings (Ge et al. 2013; Yong et al. 2018; Han et al. 2020). Unfortunately, there was still a lack of effective methods to determine the appropriate sampling interval both considering the effect of sampling size and roughness level.

This paper aims to provide a method to determine the appropriate laser scanning sampling interval for calculating the JRC based on sampling size and roughness level of rock joint. Using a terrestrial and a portable laser scanner, point clouds collection was conducted both in the field and laboratory to assess the effect of sampling size and roughness level on the selection of appropriated sampling size, respectively. Furthermore, a method was proposed to determine the appropriate sampling interval from the curves of the roughness coefficient vs. sampling interval based on a power function. Eventually, a simplified workflow was provided to promptly determine the appropriate sampling interval for engineering practice.

Point clouds collection of rock joints

Point clouds collection of large-scale rock discontinuities

Point clouds of a large-scale rock discontinuity were collected in situ to investigate the effects of sampling size on the selection of the appropriate sampling interval. This rock discontinuity was exposed at an easy-access outcrop in Jiweishan landslide region in Wulong County, Chongqing City, southwest of China, which belonged to a bedding plane with an orientation of 350°∠30°. The major rock lithology in the study area is the grayish-black medium-thick limestone from Feixianguan formation in Triassic (T1f). A terrestrial laser scanner—Optech ILRIS-36D—was employed to acquire the point cloud of the discontinuity surface. The laser scanner had to be installed near the middle of the road due to the limited space. The average distance between the scanning object and the laser scanner is 11.31 m, and the sampling interval was specified as 1 mm. In this manner, the region of interest (ROI) with a dimension of 2300×2300 mm was digitized into about 5,544,113 points, and the whole scanning period lasted approximately 15 min (Fig. 1). Seventeen concentric square-shaped sampling windows were extracted from the point cloud, ranging from 60×60 to 2000×2000 mm (Fig. 2).

Fig. 1
figure 1

In situ point clouds acquisition: a configuration of a terrestrial laser scanner; b ROI locates in the red quadrilateral

Full size image
Fig. 2
figure 2figure 2

Seventeen sampling windows with different scales: a locations of various sampling windows in original point clouds, b 60×60 mm, c 80×80 mm, d 100×100 mm, e 200×200 mm, f 300×300 mm, g 400×400 mm, h 500×500 mm, i 600×600 mm, j 700×700 mm, k 800×800 mm, l 900×900 mm, m 1000×1000 mm, n 1200×1200 mm, o 1400×1400 mm, p 1600×1600 mm, q 1800×1800 mm, r 2000×2000 mm

Full size image

For each sampling window, the downsampling was performed to produce 100 new point clouds based on the original point cloud, resulting in main geometric features that were preserved but with different sampling intervals. The downsampling refers to a process of resampling point clouds with a low resolution from the original point cloud. It is time-consuming and impossible to perform 1900 scans to acquire point clouds with 100 different sampling intervals for each sampling windows using a terrestrial laser scanner in the field survey, due to the heavy workload and limited battery capacity. To reduce the testing time, the downsampling algorithm was employed to generate different resolution point clouds based on the original point cloud of the rock joint (sampling interval = 1 mm). The maximum sampling interval was equal to the half of the side of the sampling window, while the minimum one was set to the original sampling interval (1 mm).

Point clouds collection of laboratory-scale rock joints

To assess the effects of roughness degree of rock joint on the selection of appropriate sampling intervals, thirteen approximately square rock joint specimens with different roughness levels were chosen for laser scanning in the laboratory. Specimens had similar length of the side, varying from 60.35 to 71.00 mm. The rock type of specimens is quartz sandstone which belongs to Upper Jurassic Suining Formation (J3s) exposed in the Majiagou landslide region in Zigui County, Yichang City, Hubei Province, Central China. A portable laser scanner, OKIO-FreeScan X5, was utilized to capture the point clouds of these specimens. Several target points with a 3-mm diameter were placed around joint specimens, allowing to accurately merge together multiple scans into a composite point clouds through aligning the common target points to each scan. Some parameters are required to be input before scanning, and the light-emitting diode (LED) lightness level and optical intensity were appointed as 550 and 600, respectively, according to object texture, object color, and surrounding light conditions. To obtain dense point clouds of rock joint specimens, the sampling interval was set as 0.05 mm—the highest resolution the scanner can achieve. More importantly, to eliminate the influence of sampling size on the analysis, sampling windows with a fixed size of 60×60 mm were extracted from the original point cloud for each rock joint specimen (areas within the red frame in Fig. 3). In this manner, near two million points were gained for each specimen, ensuring enough resolution for performing the estimation of the effect of roughness level on the appropriate sampling interval.

Fig. 3
figure 3figure 3

Original point clouds and regions of interest (red frames) of 13 rock joint specimens in laboratory (to display clearly, the point clouds were illustrated based on a sampling interval of 0.5 m): a configuration of portable laser scanner, b JS-1, c JS-2, d JS-3, e JS-4, f JS-5, g JS-6, h JS-7, i JS-8, j JS-9, k JS-10, l JS-11, m JS-12, n JS-13

Full size image

Similar to the field survey, the downsampling algorithm was applied again to generate a series of point clouds for each specimen with 200 different sampling intervals ranging from 0.05 to 10.00 mm with a 0.05 mm step size. Figure 4 shows the 3D models of JS-5 that were reconstructed based on point clouds with intervals of 1 mm, 2 mm, and 5 mm. The sampling interval has a significant impact on the rock joint roughness. More details tended to be missed with the increase of interval, causing a smoother joint surface, while model provides more microstructural characteristics of the joint specimen with a smaller interval.

Fig. 4
figure 4

Effect of sampling interval on the 3D reconstruction of a rock joint specimen: a top view of JS-5 joint specimen, b sampling interval = 1 mm, c sampling interval = 2 mm, d sampling interval = 5 mm

Full size image

Methods for estimation of rock joint roughness and appropriate sampling interval

Although many approaches have been proposed for characterizing the roughness of rock joint, JRC is the most widely used parameter and has been adopted as a standard method, because of its easy operation and good reliability (Alameda-Hernández et al. 2014). Therefore, JRC was selected as the index to describe the roughness of rock joints in this study. For each joint specimen with a certain sampling interval, a series of profiles along the shear direction were extracted from the point cloud with a step size equaling to the sampling interval, so that there are a sufficient number of profiles to capture morphology features of rock joints in 3D level (Fig. 5). For example, if the sampling interval of a point cloud is 0.05 mm and the sampling size is 60×60 mm, 1201 profiles (60/0.05 + 1 = 1201) will be generated in total.

Fig. 5
figure 5

Extraction of 2D profiles from rock joint point clouds with different sampling intervals (just taking 4 sampling intervals for demonstration): a sampling interval = 0.5 mm, b sampling interval = 0.7 mm, c sampling interval = 1.0 mm, d sampling interval = 5.0 mm

Full size image

JRC2D for each 2D profile was computed based on two widely used parameters of Z2 and SDi, respectively, which were capable of characterizing the roughness of rock joints accurately and quickly (Eqs. 1, 2, 3, 4, and 5), due to the strong correlation with JRC, straightforward calculation, and high computational efficiency (Tse and Cruden 1979; Kusumi et al. 1997; Yang et al. 2001; Jang et al. 2014).

$$ {Z}_2=\sqrt{\frac{1}{L}\sum \limits_{j=1}^{N-1}\frac{{\left({z}_{j+1}-{z}_j\right)}^2}{y_{j+1}-{y}_j}} $$
(1)
$$ {JRC}^{2D}=32.2+32.47\cdot {\log}_{10}{Z}_2 $$
(2)
$$ {SD}_i={\tan}^{-1}\sqrt{\frac{1}{L}\sum \limits_{j=1}^{N-1}{\left(\frac{z_{j+1}-{z}_j}{y_{j+1}-{y}_j}-\tan {i}_{ave}\right)}^2\cdot \left({y}_{j+1}-{y}_j\right)} $$
(3)
$$ {i}_{ave}={\tan}^{-1}\left(\frac{1}{L}\sum \limits_{j=1}^{N-1}\left|{z}_{j+1}-{z}_j\right|\right) $$
(4)
$$ {JRC}^{2D}=1.14\cdot {SD}_i-3.88 $$
(5)

where zj+1 and zj are the Z coordinates of (j+1)th and jth points in the given profile; yj+1 and yj are the Y coordinates of (j+1)th and jth points; N is the total points number of the profile, L is the profile length; and iave is the average inclination angle.

Ultimately, the JRC of a rock joint was determined by averaging JRC2D for all profiles (Eq. 6).

$$ JRC=\frac{1}{M}\sum \limits_{k=1}^M{JRC}_k^{2D} $$
(6)

where M is the total profile number of the given rock joint specimen and k is the kth profile. Note that roughness calculations were carried out after removing the non-stationary features for the laboratory-scale rock joints, which represents large-scale undulation of the rock joint surface and can be eliminated through coordinate transformation. For more details about the removal of non-stationary features, please refer to Ge et al. (2020).

The inflection point method was referred to determine the appropriate sampling interval in this study, which was originally proposed to separate the large-scale undulation and small-scale unevenness based on the plots of grid size vs. surface area (Fig. 6a) (Sun et al. 2019; Li et al. 2018). Figure 6 b and c illustrate typical plots of the roughness coefficient (i.e., Z2 and SDi) vs. sampling interval. The roughness coefficients dropped sharply when small sampling intervals were specified; then roughness coefficients experienced a gradual decrease with the increase of sampling interval as the sampling interval continued beyond a certain point (appropriate sampling interval); eventually, the roughness coefficients fluctuated along with a constant. That is, the roughness coefficients were more sensitive to small sampling interval than the large one. When the sampling interval was specified less than the appropriate sampling interval, although more details on the morphology of rock joint surfaces could be captured, the roughness was overestimated, making the evaluations unreliable. Nevertheless, if the sampling interval was set larger than appropriate sampling interval, the change in roughness coefficients tended to be stable; however, many surface details of rock joint would be ignored, resulting in underestimation of rock joint roughness. Therefore, errors caused by the effect of sampling interval can be eliminated when collecting the point clouds with an appropriate sampling interval.

Fig. 6
figure 6

Schematic diagram of searching the appropriate sampling interval using: a dual-linear function, b power function for Z2, and c power function for SDi

Full size image

Rock joint roughness always presents a strong nonlinear characteristic so that the linear fitting (Fig. 6a) is no longer suitable for this issue. Additionally, previous work did not provide a method to exactly locate the infection point. In this study, the above-mentioned approach was modified to detect the infection point from the curve of the sampling interval vs. roughness coefficient as the appropriate sampling interval. To describe the nonlinear characteristics of the rock joint roughness, a power function was used to create the fit to the data instead of the dual-linear function (Eq. 7).

$$ RC=a\times {SI}^b $$
(7)

where RC is the roughness coefficient, such as Z2 and SDi; SI denotes the sampling interval; and a and b are the fitting parameters that need to be determined. The length of the curvature vector (k) of each point in the power fitting line can be defined as,

$$ k=\frac{\left|R{C}^{{\prime\prime}}\right|}{{\left( 1+R{C^{\prime}}^2\right)}^{\frac{3}{2}}}=\frac{a\times b\times \left(b\hbox{-} 1\right){SI}^{b\hbox{-} 2}}{{\left[ 1+{\left(a\times b\times {SI}^{b\hbox{-} 1}\right)}^2\right]}^{\frac{3}{2}}} $$
(8)

Then, the point with the maximum length of the curvature (kmax) was considered as the appropriated sampling interval in the curve (cyan dots in Fig. 6 b and c). To facilitate the utility of Eqs. (7) and (8) to find the appropriate sampling interval in a simple way, the main steps were given: ① obtain the plots of roughness coefficients (Z2 and SDi) vs. sampling interval; ② determine the fitting parameters a and b through fitting power function curves (Eq. 7) to the plots of roughness coefficient (Z2 and SDi) vs. sampling interval in step (1); ③ calculate the length of the curvature vector k for each sampling interval, which is a function of a, b, and sampling interval (Eq. 8); and ④ find the maximum k values (kmax), and the sampling interval with the kmax is regarded as the appropriate sampling interval.

Influence of rock joint size on the appropriate sampling interval

For the large-scale rock discontinuity in situ situations, roughness coefficients (Z2 and SDi) of 17 rock joint specimens with different sizes were calculated under different sampling intervals. Figure 7 shows that all roughness coefficients (Z2 and SDi) decreased with the increase of sampling interval. For a small sampling interval, a sharp decrease of roughness coefficients was observed, while the roughness coefficients experienced a gradual reduction when the sampling interval became large.

Fig. 7
figure 7

Curves of sampling interval vs. roughness coefficient for seventeen larger-scale rock joint specimens: a Z2 and b SDi

Full size image

Table 1 summarizes the fitting parameters a and b for 17 different-scale rock discontinuities, which were determined through fitting power function curves (Eq. 7) to the plots of Z2 vs. sampling interval and SDi vs. sampling interval, respectively. The length of the curvature vector k is a function of a, b, and sampling interval SI (Eq. 8). k values for each sampling interval were calculated, and the sampling interval with a maximum k (kmax) was regarded as the appropriate sampling interval (cyan dots in Fig. 7).

Table 1 Fitting parameters and appropriate sampling intervals for 17 large-scale rock discontinuities
Full size table

It was found that sampling sizes had a significant impact on the appropriate sampling interval (Fig. 8a), and the appropriate sampling interval increased with the increase of sampling size. The relationship between them can be written as,

$$ \left\{\begin{array}{l}{SI}_a= 0.0238\times SS+ 5.8479\kern0.5em 60 mm\mathit{\pounds} SS\mathit{\pounds} 2000 mm{r}^2= 0.9147\begin{array}{c}\end{array}\left({Z}_2\right)\\ {}{SI}_a= 0.0363\times SS+ 9.2085\begin{array}{cc}& 60 mm\mathit{\pounds} SS\mathit{\pounds} 2000 mm\end{array}\begin{array}{c}\end{array}{r}^2= 0.9582\begin{array}{c}\end{array}\left({SD}_i\right)\end{array}\right. $$
(9)

where SIa indicates the appropriate sampling interval with a unit of mm, SS is the sampling size with a unit of mm, and r2 is the correlation coefficient.

Fig. 8
figure 8

Roughness estimation for large-scale rock discontinuities: a relationship between sampling size and appropriate sampling interval and b comparison of roughness estimation between Grasselli’s method and our methods using Z2 and SDi

Full size image

Furthermore, a comparison of rock joint roughness—obtained using Grasselli (2001) and Z2 and SDi methods presented in this study, respectively—was performed with the same appropriate sampling interval. Table 2 shows the values of JRC, C, \( {\theta}_{\mathrm{max}}^{\ast } \), and \( {\theta}_{\mathrm{max}}^{\ast }/\left(C+1\right) \) for 17 rock discontinuities with different sizes, which were computed based on the appropriate sampling interval mentioned in Table 1. Figure 8 b illustrates the comparisons of estimation of rock joint roughness based on Z2 SDi and Grasselli’s methods, and good agreements were observed among these methods for the large-scale rock discontinuities with R2 = 0.8844 and 0.8471, verifying the robustness of the proposed method.

Table 2 Summary of roughness estimation for 17 large-scale rock discontinuities using three methods and appropriate sampling intervals
Full size table

Influence of rock joint roughness level on the appropriate sampling interval

Roughness coefficients (Z2 and SDi) were calculated for thirteen small-scale rock joint specimens under different sampling intervals in the laboratory. Similar trends were observed from the plots of sampling interval vs. roughness coefficients, and the roughness coefficients decreased as the sampling interval increased (Fig. 9). Afterwards, power functions were adopted to fit the data, and appropriate interval sampling was determined by calculating the curvature of each point. Table 3 lists the fitting parameters a and b of 13 rock joint specimens that were determined based on Eq. (7) for Z2 and SDi methods, respectively. In addition, the length of curvature k for each sampling interval was calculated according to Eq. (8), and the sampling interval that had a maximum length of curvature kmax was considered as the appropriate sampling interval (cyan points in Fig. 9).

Fig. 9
figure 9

Curves of sampling interval vs. roughness coefficient for thirteen rock joint specimens: a Z2 and b SDi

Full size image
Table 3 Fitting parameters and appropriate sampling intervals for 13 small-scale rock joint specimens with different roughness level
Full size table

Figure 10 a illustrates that negative correlations between JRC (calculated based on Z2 and SDi methods, respectively) and the appropriate sampling interval were observed, following the linear distribution functions. It can be seen that a smaller interval was required to estimate the roughness of a rock joint with a rougher surface, and vice versa. For an absolutely smooth rock joint in an extreme situation, the point cloud obtained in a large sampling interval still can accurately represent morphology features of the rock joint; while for a rough rock joint, a fine measurement with a smaller sampling interval is required to avoid ignoring any details. Their relationships created based on Z2 and SDi could be described as,

$$ \left\{\begin{array}{l}{SI}_a=\hbox{-} 0.0481\times JRC+ 0.8045\kern0.5em {r}^2= 0.8500\begin{array}{cc}& \end{array}\left({Z}_2\right)\\ {}{SI}_a=\hbox{-} 0.0955\times JRC+ 2.5228\begin{array}{cc}& \end{array}{r}^2= 0.9327\begin{array}{cc}& \end{array}\left({SD}_i\right)\end{array}\right. $$
(10)
Fig. 10
figure 10

Roughness estimation for small-scale rock joint specimens: a relationship between roughness level and appropriate sampling interval and b comparison of roughness estimation between Grasselli’s method and our methods using Z2 and SDi

Full size image

Furthermore, the JRC values calculated using Z2 and SDi methods in this study matched the roughness estimation obtained using Grasselli’s method. Both Table 4 and Fig. 10b provide more details about the comparison, and strong correlations among three methods were observed for the small-scale rock joint specimens through linear fitting analysis, with an R2 of 0.9529 and 0.9047, respectively, proving the robustness of the proposed method again.

Table 4 Summary of roughness estimation for 13 small-scale rock joint specimens calculated using three methods and appropriate sampling intervals
Full size table

Discussions

Nevertheless, the linear fitting lines in Eq. (9) and Fig. 8a did not pass through the origin, causing unreasonable determination on the appropriate sampling interval when roughness estimation was performed for small size rock joints. For example, if the size of a rock joint was 60 mm, the appropriate sampling size would be specified as 7.2759 mm or 11.3865 mm based on Eq. (9), which were too large to collect the detailed information on the rock joint surface. The main cause of the intercept of Eq. (9) lied in there was an obvious size effect on JRC estimation for the 17 larger scale rock discontinuities (Fig. 11): JRC sharply decreased with sampling sizes ranging from 60 to 400 mm, followed by an increase on JRC when sampling size ranged from 500 to 800 mm, and then JRC slightly fluctuated between 10 and 14 with the increase of the sampling size. As a result, the JRC values of rock discontinuities varied from each other. Theoretically, to better understand the effect of sampling size on the appropriate sampling intervals, the roughness for all size rock joints should keep constant. However, it is impossible that natural joints with different scales have the same roughness level. Therefore, some calculation errors were introduced during investigating the effect of sampling size on the appropriate sampling interval mentioned in the “Influence of rock joint size on the appropriate sampling interval” section, due to the roughness difference.

Fig. 11
figure 11

Variance of JRC with the sampling size ranging from 60 to 2000 mm

Full size image

Alternatively, to produce point clouds of rock joints with different sizes but without changing on roughness, the original point cloud of a small size joint specimen was proportionally resized in XYZ coordinates. In this manner, eighteen artificial point clouds of joint specimens with different sizes (from 60 to 2000 mm) were generated. Meanwhile, 75 sampling intervals were set for each new joint, and the maximum and minimum interval was set as 37.5% and 0.5% of the side length, respectively. Then, roughness coefficients (Z2 and SDi) were computed based on Eqs. (1), (3), and (4). Figure 12 shows the variance of roughness coefficients with the sampling interval for 18 artificial point clouds. Similar changes on curves of sampling interval vs. roughness coefficient were observed. Also, a power function fitting analysis and curvature calculations were performed to identify appropriate sampling intervals from these curves. Two linear fitting lines with an intercept closing to 0 were obtained to describe the relationship between the sampling size and appropriate sampling interval (Fig. 13), which can be written as Eq. (11). Both formulas suggested that a larger sampling interval was required for a larger size rock joint so that the key geometrical information was not ignored and more accurate estimation was also achieved at the same time.

Fig. 12
figure 12

Curves of sampling interval vs. sampling size for eighteen artificial rock joints (60–2000 mm): a Z2 and b SDi

Full size image
Fig. 13
figure 13

The relationship between sampling size and appropriate sampling interval for artificial point clouds of joints

Full size image
$$ \left\{\begin{array}{l}{SI}_a= 0.0150\times SS+ 0.3832\kern0.5em 60 mm\mathit{\pounds} SS\mathit{\pounds} 2000 mm\begin{array}{c}\end{array}{r}^2= 0.9908\begin{array}{c}\end{array}\left({Z}_2\right)\\ {}{SI}_a= 0.0298\times SS+ 0.2057\begin{array}{cc}& 60 mm\mathit{\pounds} SS\mathit{\pounds} 2000 mm\end{array}\begin{array}{c}\end{array}{r}^2= 0.9846\begin{array}{c}\end{array}\left({SD}_i\right)\end{array}\right. $$
(11)

Several empirical equations—Eqs. (9), (10), and (11)—were proposed to estimate the appropriate sampling interval according to the sampling size and roughness level of rock joints. During performing roughness estimation of a given rock joint in practical engineering, the size of the interested rock joint is easy to be measured by using common devices, such as a ruler and a laser meter, while the JRC of the rock joint is unknown, which is difficult to be calculated directly in the field. However, a general subjective understanding of the roughness level could be achieved through visual inspection, for example, ten standard profiles (Barton and Choubey 1977). Therefore, for a particular rock joint, an appropriate sampling interval is suggested to be determined mainly based on the sampling size. Because Eq. (11) has almost no intercept, to facilitate the application of the findings presented in this study in a simple way, the appropriate sampling interval is suggested to be set as 0.015×sampling size (a small value) for a rock joint with a high-level roughness, due to the negative correlation between the roughness level and appropriate sampling interval (Eq. 10). For a low-level roughness rock joint, 0.03×sampling size (a large value) is recommended to be considered as the appropriate sampling interval. A flow chart was provided for a quick determination of the appropriate sampling interval in Fig. 14, and the main procedures are as follows:

  1. (1)

    Preliminary study on the roughness level of a given rock joint, which was coarsely determined by comparing with ten standard profiles.

  2. (2)

    Measurement of the sampling size of the rock joint using common devices.

  3. (3)

    Determination of the appropriate sampling interval. For a rock joint with a high-level roughness, the appropriate sampling interval was specified as 0.015×sampling size, whereas 0.030×sampling size for a rock joint with a low-level roughness.

Fig. 14
figure 14

Flow chart for quick determination of the appropriate sampling interval in practice

Full size image

Conclusions

The sampling interval is an important parameter for roughness estimation of rock joint using laser scanning technique. To describe rock joint roughness accurately and effectively, it is necessary to choose an appropriate sampling interval to collect point clouds and perform roughness investigation. Based on above-mentioned analyses, the following conclusions of this study were drawn:

  1. (1)

    The effect of the sampling size on the selection of the appropriate sampling interval was investigated based on 17 rock discontinuities with different sampling size ranging from 60×60 to 2000×2000 mm in the field survey. Fitting analysis showed that there was a positive correlation between the sampling size and the appropriate sampling interval. Furthermore, a similar trend was observed for 18 artificial point clouds of joint specimens with the same roughness level but different sampling sizes.

  2. (2)

    Based on the point clouds collected by a portable laser scanner in the laboratory, 13 small-scale rock joint specimens with different roughness levels were used to assess the effect of roughness level on the selection of the appropriate sampling interval. The results demonstrated that the appropriate sampling interval decreased with the increase of the roughness level of rock joints.

  3. (3)

    A method was developed to detect the appropriate sampling interval from curves of the roughness coefficient vs. sampling interval that obeyed the power distribution. The inflection point with the maximum curvature on the curves was considered as the appropriate sampling interval. The roughness coefficients were more sensitive to the sampling intervals less than the inflection point. Under the condition of the appropriate sampling interval, the roughness estimation obtained using Z2 and SDi methods corresponded to the results of a 3D method.

  4. (4)

    To facilitate practical engineering applications, a simplified approach was proposed to determine the appropriate sampling interval: for a rock joint with a high-level roughness, it was suggested to specify the appropriate sampling interval as 0.015×sampling size, while 0.03×sampling size was recommended to be considered as the appropriate sampling interval for a low-level roughness rock joint.