Reconstruction of Surficial Rock Blocks by Means of Rock Structure Modelling of 3D TLS Point Clouds: The 2013 Long-Chang Rockfall

The Long-Chang Rockfall site is well-suited for making remote engineering surveys, as the rugged topography precludes safe personnel access. The study area was surveyed with an Optech ILRIS terrestrial laser scanner. Overlapping scans obtained from all positions were merged using shape-fitting algorithms of the software PolyWorks (InnovMetric 2018). After the merging, each laser point was referenced in the scan project Cartesian coordinate system (x, y, z) with X = E (East), Y = N (North) and positive Z = up.

Because of limited sensor range, high light absorption and occlusions during scanning, any 3-dimensional rock surface measured with TLS is non-uniformly sampled and may contain undersampled domains (Liu and Kieffer 2011, 2012). The variable sampling density affects the correctness of meshing (Fabio 2004; Berger et al. 2016) of rock surfaces. As a result, to spatially identify how discontinuities organize to form distinct removable blocks on a natural rock slope, raw point clouds are utilized rather than a derivative meshed surface.

In a point cloud, rock discontinuities and their spatial pattern can be discerned, but the data often includes obstructions such as vegetation. Because automated (algorithm-based) cloud cleaning is still a challenge (Sotoodeh 2006), and to ensure high confidence in the digital mapping results, the points falling on and defining a discontinuity surface are selected manually. The x, y and z coordinates of the selected points are then used to fit 3-dimensional discontinuities (e.g. Liu and Kieffer 2012). Another advantage of manual localization and discrimination of a geological discontinuity is that the interpreter can incorporate engineering geologic perspective and judgements that cannot be replicated by existing algorithms. The main drawbacks of manual procedures relate to efficiency and subjectivity (reproducibility). For this reason, semi-automated algorithms are considered ideal.

The key to modeling the rock structure directly with a point cloud is to estimate the direction of each point in space, which is the normal vector perpendicular to the surface on which the point is located. Consequently, the points on the same discontinuity can be extracted in space due to their identical or similar direction values. To compute normals on a point cloud, it is necessary to consider the relations of a point to its neighbors. Jaboyedoff et al. (2007) describe an approach using the eigenvalue of the covariance matrix that is defined by the nearest neighbors of the queried point. Riquelme et al. (2014) presented a semi-automatic calculation of orientations based on a neighboring point co-planarity test. Our goal is to delineate the block-forming discontinuities and identify how various discontinuities are spatially organized to form blocks. An ideal algorithm can identify planar surface regions and correctly reconstruct edge intersections. Several algorithms have been tested, including the best fitting plane approach (e.g. Hoppe et al. 1992), local 2D triangulating (e.g. Dey and Goswami 2004), quadric surface approximation (e.g. Groshong et al. 1989; Yang and Lee 1999) and the Hough transformation (Boulch and Marlet 2012, 2016). As an illustrative example, consider three discontinuities terminating as shown in Fig. 2a. The discontinuity pattern results in three sharp intersecting edges and a single corner where the three planes intersect. The discontinuity dips/dip directions are 90°/140° (F1), 70°/50° (F2), and 20°/230° (F3). The three square planes have the same side length of 1 m. Figure 2b shows a point cloud model of the three correlated planes, which were discretized using the same regular grid of 0.025 m × 0.025 m in each plane. The normal of each point is then estimated using four different algorithms and the results are depicted in Fig. 3. Each algorithm properly reconstructs the point normals within the plane area, but only the Hough transform implemented in Boulch and Marlet’s method can correctly reconstruct the normals on the intersecting edges as shown in Fig. 3d, where the normal of an edge point is consistent with one of the normals of the two intersecting planes. In Boulch and Marlet’s method, the starting point for computing is randomly selected, so the normal of a point on a sharp intersecting edge can be any one of the two intersecting planes and the normal assignment to any one of the two intersecting planes is also correct. Other estimation techniques tend to smooth the sharp features (Fig. 3a–c). As a result, the Boulch’s algorithm (Boulch 2018) is considered most suitable for computing discontinuity normals.

The next step is to convert the calculated mathematical normals to a perceptible 3-dimensional image of rock structures. Point cloud processing software aids an interpreter in efficient filtering and rendering the 3-dimensional rock structure. The software COLTOP-3D (Jaboyedoff and Couture 2003; Jaboyedoff et al. 2007) is perhaps the earliest rendering technique to analyze rock-slope relief using a digital elevation model (DEM) and 3-dimensional point clouds (Metzger et al. 2009). The approach is to assign a unique color to a certain spatial topographic cell quantified using the normal direction to the cell. In this way, the points on a continuous planar structure will be rendered as the same color. This automated pre-processing step greatly facilitates discrimination of discontinuities and objective interpretations of the 3-dimensional rock structure. In our approach, each point which has Hough’s normal orientation is colored using the HSV- rendering technique (Liu and Kaufmann 2015). As shown in Fig. 4a, the HSV-rendering is represented by a cylindrical color space that can be modelled as a cone. The color hue (H) and saturation (S) are defined by the bearing direction and plunge magnitude of the normal, respectively. A pure color is fully saturated (S = 1) when the normal lies at the edge of the hemisphere projection circle (i.e. vertically dipping discontinuities). Conversely, white color (S = 0) represents horizontal discontinuities that have their respective normals plotting at the center of the projection hemisphere. The value (V) of a color describes how dark the color is. In our 3-dimensional structure models, a consistent lightness of V = 0.75 has been adopted. As an example, Fig. 4a depicts six typical pure colors with full saturation (S = 1). In Fig. 4b each normal of 1° resolution, as poles in equal-angle and low-hemisphere projection, is allocated to a unique HSV-color. We use this HSV-color wheel herein for rendering 3-dimensional rock structures. Because a certain orientation is rendered only with a fixed color, the points on the same plane will appear in the same color (Fig. 4c). The points with the same color are then fitted to a discontinuity plane (Fig. 4d). The values of rendering and fitting are given in Table 1. The points on the same plane were rendered with the same color (column 2 of Table 1) and then fitted to the discontinuity plane using least-squares fitting (column 3 of Table 1). Column 4 and 5 are the positions and area size of the fitted planes.

The final step of our 3-dimensional structure extraction is to assign each fitted discontinuity a half-space code (HSC) according to Block Theory (Goodman and Shi 1985). As an example, the last column of Table 1 shows the HSC of each discontinuity in Fig. 4, in which the digit 1 and 0 represent the half-space below and above a plane, respectively. In this example, the regions of intersections of three half-spaces are defined by the joint pyramid (JP) 110, which corresponds to the block formed on the lower half-space of F1, the lower half-space of F2 and the upper half-space of F3 (Fig. 4d). Since F1 has a dip of 90°, its left side is defined here as below space. The JP identification based on HSC is decisive so that the in situ block system can be characterized using the fitted planes.

A block mold represents the negative space that remains after a block detaches from the rock outcrop. Detailed mapping and analysis of block molds provide important information regarding actual block failures that have occurred and conditions that favor the development of unstable keyblocks. Our computations concerning block molds include the location, shape, volume of the mold, followed by assessments of the corresponding block’s kinematic removability, failure mode, and limiting equilibrium conditions.

To find a removable block intersecting a slope surfaces the orientations of the slope faces must be specified to define the excavation pyramid (EP) according to Block Theory. As an example, the synthetic block mold depicted in Fig. 4d represents a block having three free planes E1, E2, and E3 (Table 2). We suggest that the surficial geometry (i.e. EP) of this synthetic rock slope is shaped by discontinuities having the same orientation of planes forming the block mold. The free (or excavation) planes are almost parallel to and 1 m from the respective discontinuities (Table 2 and Fig. 4d). The HSC of this EP has the digits 001. The closed rock block referred as the block pyramid (BP), is the intersection (∩) of the JP and EP (i.e. BP = JP∩EP = 110∩001). The complementary space region of EP is referred as space pyramid (SP), i.e. 110. According to Shi’s Theorem, removable blocks satisfy the criteria that the JP is nonempty, and the JP is completely contained within the SP (Goodman and Shi 1985). These two requirements can be readily checked via stereographic projection: (1) every JP is nonempty if it appears in the stereographic projection; and (2) the JP is removable if it is located entirely within the SP.

Figure 5a shows the projection of the six boundaries of the BP defined in Table 2, using whole sphere upper-focal-point stereographic projection. The dotted line represents the reference circle, and solid circles represent the three discontinuity planes (F1, F2, and F3). Dashed circles are the free or excavation planes (E1, E2, and E3). The lower and upper half spaces of a plane lie inside (expressed as digit 1) and outside (expressed as digit 0) its corresponding circle, respectively. For instance, JP 110 is the shaded region on the left side of Fig. 5a, which is simultaneously inside the circle of F1, inside the circle of F2 and outside the circle of F3. Because JP 110 is completely within the SP, it is kinematically removable. The digit “3” in parentheses below the JP digits 110 of Fig. 5a indicates the failure mode, in this case planar sliding along F3. Other possible modes in Block Theory include “0” for a block lifting or falling without having contact to any bounding plane, a single digit for sliding along the respective joint plane, e.g. 3 for sliding along F3, and double digits for wedge sliding along the intersection of the respective two joint planes, e.g. 13 along the intersection of F1 and F3.

The stability of JP 110 depends on the effective friction angle on the actual sliding plane F3. The friction angle required for limiting equilibrium of JP 110 can be readily assessed in 3-dimension via stereographic projection (Goodman and Shi 1985; Goodman 1989). In Fig. 5b, the location of the gravitational force R reveals that the required friction angle for JP 110 is 20°. Additional block computations are performed to determine block volumes and surface areas.

The Kaili limestone bedrock exhibits varying grades of weathering. Selected representative rock samples were subjected to uniaxial compressive testing and direct shear testing. The goal was to assign representative strength parameters according to weathering grade, and scanning electron microscopy was utilized to facilitate correlations.

Figure 6a shows boxplots of the unconfined compressive strength (σ_c). The slightly weathered limestone has a mean σ_c of 74 Mpa and for moderately weathered limestones the mean σ_c is 33 Mpa. Here the class “slightly weathered” means that the limestones are just discolored along micro fractures, some of which may have been opened slightly (Fig. 6b). The slightly weathered samples were taken in the interior of the failed blocks. Moderate weathering, on the other hand, is the collectively weathered or altered part in the vicinity of highly weathered joints surfaces, on which the rocks have brownish-red colors (Fig. 1c, d) and a significant portion of calcites has been removed by dissolution (Fig. 6c). The brownish-red colored surfaces (Fig. 1b, d) evidence the water circulation passing through open joint system. Figure 7 shows the peak shear strength parameters of the intact limestones having different weathering, which help us to appropriately find post-peak residual shear strength on discontinuities.

The area size of the main failure planes of the Long-Chang rockfalls (Fig. 1b) varies between 5000 and 9000 m². The mean dips of the failure planes are all steeper than 80°. The total volume of the first collapse body is approximately 200,000 m³ and the second about 40,000 m³ (Dong et al. 2015). The dry unit weight of Kaili limestones changes from 2.50 to 2.73 g/cm³. Due to the identifiable solution features (Fig. 1b–d) we take the low limit of 2.5 g/cm³ to represent the weathered blocks. Thus, the mean normal stress (σ_n) of the failure planes would be less than 0.15 MPa. Barton’s ratio (Barton 1973, 1976) of unconfined compressive strength σ_c to the normal stress σ_n, i.e. σ_c/σ_n, would be much higher than 100, which indicates that the shear strength of discontinuities under low normal stress is frictional and consisted of dilation angle (d_n) due to joint roughness and the residual frictional angle (ϕ_r) of weathered limestone. Compared to the peak shear strength of 35.05° (Fig. 7), a residual friction angle of 30° was assumed to represent karstified discontinuities of Kaili limestone. As enumerated in the results section of this paper, block kinematics, failure modes and stability are controlled by very steeply dipping discontinuities and are thus rather insensitive to the selection of residual shear strength.

Using the high-resolution point cloud data, the roughness-related dilation angle of a sliding plane is determined. All points of the undulating surface are first selected and the orientations of their Hough’s normals are plotted in equal area stereographic projection. The peak dilation angle is estimated as the angle between the cone axis (mean join plane) and the surface formed by the maximum scatter of the surface normals. As an example the grey discontinuity surface of Fig. 8a is discretized by the point cloud, on which the Hough’s normal of each point is represented as a short black line. This surface has a mean attitude of 3°/289°, shown as a great circle (global mean plane) in Fig. 8b and the maximum scatter around the mean pole (87°/109°) indicates a peak dilation angle of 14°.

Field observations (e.g. Goodman and Kieffer 2000) of rock slope failures indicate that structurally controlled failures can occur when existing discontinuities have limited persistence, but this requires progressive rupturing of intact rock bridges. Identifying and documenting the location of rock bridges is thus important for understanding the conditions promoting the initiation and progression of failure.

The TLS cloud of the Long Chang cliff includes 17,7333,810 points (Fig. 9a). The portion of the point cloud corresponding to the 2013 rockfall is marked with a white frame in Fig. 9a. The varying brightness of the point model is due to the location-dependent reflectivity of the outcrop surface. The intact limestone is generally brighter than the scree deposits and vegetation. To elucidate the 3-dimensional rock structure the normal of each point was computed by Hough’s method, and the two normal orientations of a point were then used for visualization, i.e. the outerward pointing normal is rendered as white and the inward pointing normal as black. As shown in Fig. 9b, such rendering greatly facilitates the differentiation of discontinuities and their spatial organization. Location-dependent rock structure is quantified by assigning a specific HSV color based on the normal orientation of the respective point. Figure 9c shows the HSV colored rock structure of the cliff. The HSV rendering shows that the surface relief of the cliff is dominated by systematic structural elements, including joint surfaces forming the overhanging (green) and non-overhanging (violet) areas of the main slope face, pervasive bedding planes which are the flat-lying boundaries between green and violet areas, and the two additional sets of persistent joints (blue and yellow) intersecting the cliff surface. In the following, we use the part of 2013 rockfall (Fig. 10a) to show the HSV-color based extraction of in situ rock structure.

The distribution of HSV colors in Fig. 10a indicates the existence of seven discontinuity sets: the pink and violet areas are clustered into one set; the green is the second set; the blue is the third; yellow the fourth; aqua the fifth; and red the sixth. These six sets are all joints. The seventh set, i.e. the very flat-lying bedding surfaces, is traceable by means of the undulation of the main slope faces (green and violet). Figure 10b–h shows each of the individual discontinuity sets that were extracted from Fig. 10a.

To reduce sampling bias due to slope orientation, approximately 35 small areas are randomly selected in each of the filtered HSV-colored sets (Fig. 10b–h) and the points of each selected area are fitted to a plane to obtain the average set orientation over the domain sampled. The orientations of 248 sampled areas are plotted as poles in Fig. 11. The cliff has a mean slope orientation of 84°/106°. Table 3 shows the results of kinematic analysis. With the assumed residual friction angle of 30°, discontinuity sets J₁₁, J₂₁ and J₃₂ (dipping out of the cliff) tend to fail in-plane and wedge sliding modes, while the sets J₁₂, J₂₂ and J₃₁ (dipping into the cliff) are prone to toppling failure. To assess how failure mode tendencies change according to the frictional resistance of discontinuities, the effective friction angle was systematically increased from the residual value up to 65°. As shown in Table 3, the failure modes remain constant over this range of friction angle.

The failure scenario analysis included: (1) global stability analysis of the entire failure area assuming a single large block was involved; and (2) separate analysis of block molds to reconstruct the first and the secondary collapse events.