1 Introduction

Rock masses usually contain various discontinuities, such as joints, faults, bedding planes, fractures, and other mechanical defects, which make rock masses have significant characteristics of heterogeneity, anisotropy, and discontinuity. Structural surfaces play an important role in the instability and deformation of rock masses, and the structural surface roughness has a great impact on rock masses mechanical and hydrological properties, such as strength and permeability. The shear strength and deformation characteristics of the structural face of rock masses are controlled by the joint controls of structural face roughness and structural face strength [1]. In order to assess the stability and safety of rock mass, it is crucial to investigate structural surface properties.

Since Barton et al. [2] proposed the concept of rock surface roughness coefficient, domestic and foreign scholars have proposed various calculation methods for structural surface JRC estimation of structural surface contour curves, and Chen SJ et al. [3] summarized the main description methods for quantitative characterization of roughness of two-dimensional structural surface contour lines. However, it is difficult to accurately and comprehensively reflect the overall characteristics of structural surface roughness by a single contour curve. With advances in technology, non-contact measurement methods provide the possibility to accurately and comprehensively refine the 3D structural surface morphology features; the digital extraction of structural surface morphology develops from two-dimensional contour lines to three-dimensional point cloud data; and the research of structural surface roughness gradually shifts from two-dimensional structural surface to three-dimensional one. In the study of three-dimensional structural surface roughness, based on the geometric characteristics of the structural surface, Belem et al. [4] described the three-dimensional morphological characteristics of the structural surface through five parameters, such as three-dimensional average inclination θs, apparent anisotropy degree Ka, joint surface average gradient Z2s, surface distortion parameter Ts, etc. Grasselli [5] proposed to use the parameters \({\theta }^{*}_{\text{max}}/c\) (\({\theta }^{*}_{\text{max}}\) is the maximum inclination angle along the shear direction, c is the joint surface roughness parameter) to describe the anisotropy of the structural surface; Tatone et al. [6] proposed to use \({\theta }^{*}_{\text{max}}/c+1\) to show the anisotropic characteristics of the structural surface, which is considered to be more physically meaningful. Yin HM et al. [7] used three-dimensional fractal theory to estimate the structural surface roughness, and established the relationship between structural surface roughness and the fractal dimension. Ge YF et al. [8, 9] proposed a new method for characterizing the three-dimensional structural surface roughness, based on the principle of light and shadow by using image segmentation technology, and carried out a research on anisotropy, size effect, and interval effect of the structural surface roughness. Based on the rotation sampling method, Sun Qi et al. [10] used UAV close photogrammetry technology to carry out the change law of structural surface roughness characterization value at different sizes and different intervals, and made a new attempt for the study of structural surface roughness. The use of close-up photogrammetry could also evaluate the three-dimensional structural surface roughness [11], which was convenient and less expensive.

There are still certain issues that require ongoing improvement despite the ongoing optimization and development of the present three-dimensional roughness characterization methods: (1) The complexity of three-dimensional morphology is ignored by the method of averaging based on the two-dimensional contour calculation results, and there is a disconnect in the quantitative characterization of 2D and 3D roughness. (2) The sampling intervals will have a significant impact on the roughness evaluation of the structural surface, but there are few studies on the evaluation index of the sampling intervals. (3) In the study of the three-dimensional roughness quantification method that combines the undulation degree and undulation angle and considers the shear direction, the discretization of the undulation degree and undulation angle of the structural surface is not taken into consideration.

In this paper, the point cloud data of rock structural surface was obtained by using 3D laser scanning equipment, the evaluation index of sampling interval of 3D point cloud data of rock structure surface was proposed, the representative volume element of the sampling interval was determined, the 3D roughness quantification characterization method of rock structure surface based on undulation degree and undulation angle was established, and the research of structural roughness under different sampling intervals and different shear directions was carried out by using this method. On the basis of the above study, the influence of sampling interval on the roughness of the structural surface was further clarified, and a three dimensional roughness measurement characterization method based on undulation degree and undulation angle was established.

2 Determination of sampling interval

The morphological characteristics of the structural surface are the key to study the roughness, and the roughness of the structural surface obtained under different sampling interval conditions is different. Therefore, this paper firstly completes the determination of the sampling interval characterization unit from the perspective of size effect, which not only can establish the quantitative relationship between roughness and sampling interval, but also lays the foundation for the subsequent study of the mechanical parameters of the structural surface.

In 1991, Yu et al. [12] found that if the sampling interval of the joint contour line was different, the regression model was also different. In the summer of 1996, the concept of sampling point interval was proposed [13], and Abolfazli M. [14] used three different sampling intervals sizes to calculate the parameters related to the structural surface roughness. The study of two-dimensional and three-dimensional roughness eigenvalues [15,16,17,18] showed that the sampling interval is correlated with the structural surface roughness value, but the current evaluation index of the sampling interval effect was unclear. In this paper, the structural surface volume and the structural surface equivalent height index were used as the evaluation basis of the sampling interval.

At present, the sampling interval is mostly determined by scholars according to their personal research needs, and the sampling interval is mainly 0.01–1.00 mm. The smaller the sampling interval, the higher the requirements for 3D laser scanning equipment, and more structural surface information can be obtained, and the roughness calculation accuracy is higher, but the calculation amount is larger and more time-consuming. The larger the sampling interval, the less structural surface information is obtained, and the roughness calculation accuracy is lower, but the calculation amount is smaller, and less time-consuming. In this paper, the sampling interval was set to n = 0.5–1, 0.50, 0.51, 0.52, 0.53, 0.54 and 0.55.

2.1 The determination index of size effect of the sampling interval on the structural surface

In this paper, a typical flat and smooth structural surface of sandstone with an area size of 1.5 × 0.6 m2 was selected (Fig. 1). The 3D laser scanning equipment was used to obtain the point cloud data of the structure surface according to the set sampling intervals. In this research, roughness calculation was carried out by intercepting point cloud data based on the similarity in the surface morphology of the chosen structural surfaces.

Fig. 1
figure 1

The typical photo of the structural surface

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The 3D laser scanning equipment was used to obtain the point cloud data of the natural rock structural surface and carried out 3D reconstruction. The point cloud model of the structural surface is shown in Fig. 2.

Fig. 2
figure 2

Three-dimensional point cloud model of the natural rock structural surface

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In the i-th triangle after triangulation of the structural surface, the coordinates of the three corner points defined are A(xi1,yi1,zi1), B(xi2,yi2,zi2), C(xi3,yi3,zi3) respectively (Fig. 3); n is the exterior normal vector \(l_{{{\text{A}} - {\text{B}}}}\), \(l_{{{\text{B}} - {\text{C}}}}\), \(l_{{{\text{A}} - {\text{C}}}}\) can be obtained based on the Euclidean distance [19].

$$l_{{{\text{A}} - {\text{B}}}} = ((x_{i1} - x_{i2} )^{2} + (y_{i1} - y_{i2} )^{2} + (z_{i1} - z_{i2} )^{2} )^{0.5}$$
(1)
$$l_{{{\text{B}} - {\text{C}}}} = ((x_{i2} - x_{i3} )^{2} + (y_{i2} - y_{i3} )^{2} + (z_{i2} - z_{i3} )^{2} )^{0.5}$$
(2)
$$l_{A - C} = ((x_{i1} - x_{i2} )^{2} + (y_{i1} - y_{i2} )^{2} + (z_{i1} - z_{i2} )^{2} )^{0.5}$$
(3)
Fig. 3
figure 3

The i-th triangular surface

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The formula for the area of the i-th triangular surface is Si = pi(pi−ai)(pi−bi)(pi−ci), where pi = 0.5 × (ai + bi + ci).

The length of projection of each side of the triangular plane in the XOY plane is ai' = ((xi1−xi2)2 + (yi1−yi2)2)0.5, bi' = ((xi2−xi3)2 + (yi2−yi3)2)0.5, ci' = ((xi1−xi3)2 + (yi1−yi3)2)0.5, and its area is S0i' = pi'(pi'−ai')(pi'−bi')(pi'−ci'), where pi' = 0.5(ai' + bi' + ci'), the oblique truncated prism volume is \(V_i=\frac13(z_{i1}+z_{i2}+z_{i3})S_{i1}^{'}\).

According to the volume of the structural surface and the projection area, the equivalent height of the structural surface can be obtained, and the equivalent height can be used as the evaluation index of the spaced effect of the structural surface. The specific formula of the equivalent height is as follows:

$$h = \frac{V}{{S_{0} }} = \frac{{V_{{{\text{ABCXOY}}}} }}{{S_{{{\text{XOY}}}} }}$$
(4)

in which h is the equivalent height, which can reflect undulation degree of the structural surface; the greater undulation degree is, the higher the equivalent height will be; V is the structural surface volume; S0 is the projected area of the structural surface.

2.2 Size effect of sampling interval of structural surface

The structural surface in Fig. 1 was scanned according to the set sampling interval, and the calculation area was intercepted. Taking Fig. 1 as an example, point cloud data at different sampling intervals was triangulated, and the numbers of triangular polygons were calculated as 2068, 8550, 34762, 139412, 557648, 2233656, and 8934624, respectively. The area, projection area, volume, and equivalent height of the structural surface at different sampling intervals were statistically counted. The statistical results are shown in Table 1.

Table 1 Statistics of characteristic parameters under different interval sizes of structural surfaces
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The mesh size and area to projected area ratio (S/S0) were taken as logarithms to obtain the fractal dimension of the structural surface. The relationship curve is shown in Fig. 4.

Fig. 4
figure 4

Fractal dimension of structural surface area

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It can be obtained from Fig. 2 that the correlation coefficient is 0.98 and the fitting effect is good, indicating that the structural surface has typical fractal characteristics and that the structural surface reconstruction model is better at different intervals. The slope of the fitted straight line is −0.00645, and the fractal dimension value is 2−(−0.00645) = 2.00645. Wang JA et al. [20] showed that fractal dimensions can describe the degree of irregularity indicated by roughness, and the larger the fractal dimension value, the coarser the surface, so the roughness is positively correlated with the fractal dimension. The smaller the sampling interval, the richer the details of the structural surface, the larger the calculated area, and the larger the area ratio (S/S0) value, so the sampling interval has a negative correlation with the area ratio. With the increase in value, there is gradually a transition from plane to body.

The volume and interval size of the oblique truncated prism were used to establish a single logarithmic relationship curve, as shown in Fig. 5.

Fig. 5
figure 5

Size effect of the sampling interval of the structural surface

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As can be seen from Table 1 and Fig. 5, the volume and equivalent height values of the structural surface gradually stabilize after the sampling interval is less than 0.125 mm, which can be used as a representative elementary volume for the volume of the structural surface to select the interval size, and the interval setting should be less than or equal to 0.125 mm.

3 Structural surface undulation and undulation angle distribution characteristics

The original data obtained by 3D laser scanning technology is the point cloud data, which is the basis of all subsequent calculation. The errors will inevitably occur in data acquisition, and the errors can be mainly divided into several kinds, such as instrument error, error related to the reflective surface of the target, and error caused by external conditions, and the error of the scanning equipment itself is the most frequent. In this paper, the instrument system was calibrated by the self-calibration method before collecting data. At the same time, the instrument errors were systematic, and this paper avoided the influence caused by the instrument error by establishing a relative coordinate system.

The two key indicators affecting the morphological characteristics of the structural surface are undulation degree and undulation angle. Therefore, the distribution characteristics of undulation degree and undulation angle of the structural surface were investigated. The average value of the height of each corner point is Hi, Hi = (zi1 + zi2 + zi3)/3, and the angle between the outer normal and the Z axis is θi (Fig. 3), and the distribution characteristics of H value and θ value are studied at different sampling intervals.

3.1 Characteristics of the undulation distribution of triangular surfaces

3.1.1 H distribution law of undulation degree of triangular surface

The distribution of the triangular surface height values H at different sampling intervals of the triangular surface is fitted, as shown in Fig. 6.

Fig. 6
figure 6

Distribution characteristics of the H value at different sampling intervals

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Figure 4 presents the H values following the normal distribution at different sampling intervals, and the expected μ and standard deviation of the probability density function of the normal distribution at different sampling intervals are σ shown in Statistical Table 2. The adjusted R-square of the fitting curve is all greater than 0.95, and the fitting effect is good. At the same time, the skewness value and kurtosis value are close to 0, which can judge that the data obey normal distribution.

Table 2 Statistical table of normal distribution characteristics of the H value at different sampling intervals
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In Table 2, with the decrease of the sampling interval, the mean and standard deviation gradually increases, and the increase rate changes fast at first and then slows down, and gradually tends to be smooth, and all of them tend to be stable after being less than 0.125 mm, indicating that the size effect exists in the sampling interval.

3.2 Distribution law of the angle θ between the triangular outer normal and the Z-axis

The distribution fit of the angle θ between the triangle out-of-plane normal and the Z axis at different sampling intervals of the triangular surface is shown in Fig. 7.

Fig. 7
figure 7

Distribution characteristics of θ at different sampling intervals

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It can be seen from Fig. 7 that the angle θ between the triangle out-of-plane normal and the Z axis obeys the Weibull distribution at different sampling intervals. Distribution characteristics can be represented by the Weibull distribution [21] as follows:

$${{F}}(x,\theta ,\eta ) = 1 - \exp ( - (\frac{x}{\theta })^{\eta } ),x > 0,\theta > 0,\eta > 0$$
(5)
$$f(x,\theta ,\eta ) = \frac{\eta }{{\theta^{\eta } }}x^{\eta - 1} \exp ( - (\frac{x}{\theta })^{\eta } ),x > 0,\theta > 0,\eta > 0$$
(6)

where \(\theta\) is the scale parameter; \(\eta\) is the shape parameter, noted as Weib (\(\theta\), \(\eta\)).

The scale parameter plays the role of enlarging or shrinking the curve, but does not affect the shape of the distribution, while the shape parameter is the most cucial one of the characteristic parameters of the Weibull distribution and it determines the basic shape of the distribution density curve.

The adjusted R-square of the fitting curve is all greater than 0.96, and the fitting effect is good. The Weibull distribution characteristic parameters and statistics of the angle θ between the triangle out-of-plane normal and the Z-axis at different sampling intervals are shown in Table 3. In Table 3, with the decrease of the sampling interval, the scale parameters show an increasing trend; the increase rate has a trend of fast at first and then slow down, and the inflection point is 0.125 mm; the shape parameters gradually decrease, and the rate is basically the same. The variation in scale parameters indicates that there is the size effect in the sampling interval.

Table 3 Statistical table of Weibull distribution characteristics of the H value at different sampling intervals
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4 Quantitative characterization of structural surface roughness

4.1 Structural surface undulation and discretization of the outer normal

The undulation degree H of the triangular plane, the angle between the outer normal and the Z axis θ were statistically analyzed, and the statistical indicators were minimum, maximum, median, coefficient of variation, skewness coefficient, and kurtosis coefficient. The minimum, maximum, and median reflect the overall change in the data. Variation data is a measure of the fluctuation degree of the overall value, which has the evaluation advantage of being dimensionless. Both the skewness coefficient and the kurtosis coefficient are used to describe the shape characteristics of the distribution, with the former depicting the symmetry of the distribution and the latter describing the steepness of the distribution. The example of the sampling interval of 0.5 mm and 2 mm can be seen in Figs. 8 and 9, and the statistical results are shown in Tables 4 and 5.

Fig. 8
figure 8

Triangular height value at a sampling interval of 2 mm

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Fig. 9
figure 9

Triangular height value at a sampling interval of 0.5 mm

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Table 4 Statistical table of the H value at different sampling intervals
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Table 5 Statistical table of θ value at different sampling intervals
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From Figs. 8 and 9, the undulation H value of the triangular surface and the θ value of the triangular surface at different sampling intervals are consistent, but there are still tiny differences.

For the structural surface statistical parameters in this paper, the H value obeys the normal distribution, while the θ value obeys the Weibull distribution. Although the value does not change much, less than 7 mm, it still has discrete characteristics. Individual sample of structural surface topography feature parameters has randomness and uncertainty, but with the superposition of sample size, the average result of random phenomenon will be stable.

It can be seen from Table 4 that the minimum value of undulation H tends to decrease with the decrease of the sampling interval, and the decrease is fast at first and then slow; the maximum value tends to decrease and then increase; the median first increases sharply and then tends to stabilize; the coefficients of variation are all less than 1, indicating that the degree of data dispersion is small and that the trend is first decreasing sharply and then leveling off with the decrease of the sampling interval. The skewness value is greater than 0 and less than 0.5, indicating that the H value is a right-biased normal distribution and slightly right-biased, that is, the mean value is slightly right-biased to the peak, and the skewness value changes steadily after a sharp increase at first with the decrease of the sampling interval. The kurtosis coefficient is greater than 0; the distribution is steep, and it first decreases sharply and then tends to stabilize with the decrease of the sampling interval.

Table 5 shows that the lowest value of undulation degree H first declines quickly, then slowly increases, and then decreases continuously as the sample interval is reduced. The highest value displayed a pattern of first increasing gradually and then climbing steadily. The coefficients of variation were less than 1, suggesting that there was little data dispersion, and the trend indicated linear development with the reduction of sampling intervals. The median middle segment rises dramatically in the front segment, while the rear segment tends to stay steady. The skewness value is greater than 0 and less than 1, indicating that the H value is moderately skewed and right-skewed, and that the skewness value increases with the decrease of the sampling interval, and first increasing sharply and then changing steadily. The kurtosis coefficient is greater than 0; the distribution is steep, and the anterior section shows an increasing trend with the decrease of the sampling interval, but the sampling interval decreases first and then increases after 0.25 mm.

4.2 Structural surface roughness quantification model

The area projected by the triangular plane into the xoy plane is Si, and the discrete coefficients correspond to the undulation degree Hi of each triangular surface, the angle θi of the surface normal vector is ni and the z-axis are cv1 and cv2, and the dispersion coefficient formula is \(c_{v} = \frac{\sigma }{\mu }\), where \(\sigma\) is the standard deviation, and \(\mu\) is mean.

Using the structural surface undulation degree, and the angle between the outer normal of the triangular surface and the Z-axis, the correction coefficient is defined as:

$$K_{1} = (1 + c_{v1} )^{0.5} = (1 + \frac{{\sigma_{1} }}{{\mu_{1} }})^{0.5}$$
(7)
$$K_{2} = (1 + c_{v2} )^{0.5} = (1 + \frac{{\sigma_{2} }}{{\mu_{2} }})^{0.5}$$
(8)

The clip** direction moves from x to y in a 5° direction, and the angle between the triangular outer hair line and the shear direction is defined as:

$$A_{{\text{i}}} = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {A_{i} \le pi/2} \hfill \\ {A_{i} ,} \hfill & {A_{i} > 0} \hfill \\ \end{array} } \right.$$
(9)

The three-dimensional structural surface roughness quantitative characterization parameter is W, and the equation is:

$$W = K_{1} K_{2} \sum\limits_{1}^{n} {A_{i} H_{i} V_{i} /V}$$
(10)

where Vi is the equivalent volume of the structural surface, defined as \(V_{i} = H_{i} S_{i}\).

4.3 Quantitative analysis of structural surface roughness characterization values

The characteristic values of roughness obtained from different shear directions at the condition of a sampling interval of 2 mm — 0.0625 mm are shown in Fig. 10. It can be seen from Fig. 8 that the data obtained with different shear directions at intervals of 2 mm and 1 mm are less effective, and the effect tends to be better at intervals less than 0.25 mm. According to the roughness characterization parameters obtained in different shear directions, the structural surface roughness is quitely large in the shear direction of 270° — 290°, while it is relatively small at 70° — 200°. Therefore, the structural surface roughness should be fully considered when calculating it, and the roughness calculation and related research should be carried out according to the anisotropic characteristics on the basis of the size effect.

Fig. 10
figure 10

Three-dimensional characterization under different sampling intervals and different shear directions. a The sampling interval is 2mm. b The sampling interval is 1mm. c The sampling interval is 0.5mm. d The sampling interval is 0.25mm. e The sampling interval is 0.125mm. f The sampling is 0.0625

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5 Conclusions

  1. (1)

    The volume and equivalent height values of the structural surface are employed as the judgment indicators of the characteristic parameters of the sampling interval, and the sampling interval is associated with the roughness value of the structural surface. It can also be utilized as a new indicator for figuring out the sampling interval and offer technical assistance to scholars looking into the quantification and characterization of three-dimensional rough.

  2. (2)

    That the undulation degree obeys the normal distribution and the undulation angle obeys the Weibull distribution is showed in the research on the distribution laws of the two key indicators of undulation degree and undulation angle affecting the roughness measurement characterization value of the structural surface. The structure of the surface’s undulation degree and undulation angle are discrete, the degree of dispersion is minimal, and they all increase linearly as sampling intervals are shrunk. The distribution of both is steep, right-biased, and moderately skewed.

  3. (3)

    According to the 3D roughness refinement characterization method of the rock structural surface developed by the integrated characteristics of undulation degree and undulation angle, for the studied structural surface, the change law of structural roughness characterization value under various sampling intervals and different shear directions shows that the effect gradually improves after the interval is less than 0.25 mm, and the sampling interval has a size effect; the structural surface roughness in the shear direction is quite large at 270° to 290°, and the roughness is relatively small at 70° to 200°; different shear directions have a great influence on the roughness characterization value. Therefore, in further research, the shear direction and interval effect should be comprehensively considered and used as important indicators to study the structural surface roughness.