Influence of Roughness Digitisation Error on Predictions of Discontinuity Shear Strength

Abstract

A key component of the new stochastic approach for discontinuity shear strength (referred to as StADSS) is characterising the roughness of natural rock discontinuities at full scale to mitigate well-known scale effects on shear strength predictions. An investigation was conducted using the software Blender (v4.0) to determine the influence of camera orientation and position on the estimation of the standard deviation of gradients, a parameter used to quantify roughness and to make predictions of discontinuity shear strength. The existing literature has investigated the various distortions of images due to camera position and orientation; however, a comprehensive understanding of their unique influence on error in shear strength prediction is still missing. The investigation revealed that a ground sampling distance of less than 1.36 mm/pixel allows the standard deviation of gradients to be quantified within approximately ±10% relative error to the control data set. Based on investigations into camera orientation relative to the planar control data set, error on the roughness parameter due to perspective distortion was quantified. Recommendations were made to reduce perspective distortions and improve seed trace digitisation errors, including capturing images perpendicular to the seed trace with no rotation and getting as close as possible to the trace during capture to improve ground sampling distance. Lastly, the influence of these different trace digitisation errors on predictions of shear strength obtained using StADSS was investigated. Digitisation errors were found to have a disproportionate influence on shear strength prediction error, especially for rough discontinuities at low normal stress. The investigation highlighted the importance of accurately digitising the standard deviation of gradients to predict discontinuity shear strength.

1. Introduction

The recent growth of critical infrastructure, including open pit mines, tunnels, road cuttings, and other deep excavations, combined with increasing financial and environmental constraints, puts pressure on engineering designs to be less conservative. One source of such conservatism lies in the estimation of shear strength for large rock discontinuities, which contributes to the stability of rock masses [1]. Until recently, the prediction of discontinuity shear strength has been subject to scale effects [2,3,4,5,6,7,8,9,10,11,12,13,14], often resulting in overly conservative design. Despite decades of research into roughness and scale effects on discontinuity shear strength, there remains no consensus on how to reliably predict discontinuity shear strength at large scale [15].

There are many shear strength models that recognise different properties of discontinuities contributing to shear strength such as roughness [5,16,17,18,19,20,21], infilling [22,23,24,25,26,27,28], anisotropy [29,30,31], persistence [32,33,34], and mismatch or aperture [35,36]. However, most shear strength models are primarily based on roughness. Arguably, the current industry standard for shear strength prediction is Barton and Choubey’s empirical joint roughness coefficient (JRC) method [5]. The revised JRC model [2] accounts for scale dependent parameters, though concerns remain to its formulation and methodology [17,18,37]. Also, the scale effects on roughness were not clearly understood and only were accounted for based on an empirical chart comparing trace length and amplitude [2]. The roughness of discontinuities is difficult to quantify due to being multiscale and often non-stationary [38]. An additional challenge comes from the fact that discontinuity surfaces are usually contained within the rock mass, preventing their complete 3D roughness characterisation. Often, the only roughness datum available is the 2D profile of the discontinuity exposed on the rock face, usually referred to as a trace.

A stochastic approach for discontinuity shear strength (referred to as StADSS) was proposed by Buzzi and co-workers [20,21,38,39,40,41,42]. This novel approach predicts the peak and residual shear strength of rough rock discontinuities at full scale whilst by-passing the well-known scale effects on roughness [2,3,4,5,6,7,8,9,10,11,12,13,14]. Roughness is quantified through a multiscale statistical analysis of an exposed, daylighting seed trace at full scale, referred to as a seed trace [38]. A random field model (RFM) generates several synthetic surfaces which share the same statistical characteristics as the seed trace. Rock strength parameters and the generated synthetic surfaces are the input to a deterministic shear strength model, used to conduct a series of Monte Carlo simulations. The Newcastle discontinuity shear strength (NDSS) model [20,21] is used within StADSS to deliver a statistical distribution of peak and residual shear strengths which can be used in stability analysis. StADSS has been shown to closely predict the peak and residual shear strength of prepared specimens replicating natural joints [20,21]; more recently, it was applied to a large in situ specimen [43].

The seed trace of a discontinuity must be accurately digitised to allow for the application of StADSS. Roughness information can be obtained from contact and non-contact approaches. In the rock mechanics field, many would be familiar with the mechanical profile comb used to characterise roughness at small increments [5,30]. This method was found to be subjective and unreliable due to human error [17,18,37] and is also only applicable to small lengths (commonly 100 mm). All contact methods including the profile comb also require physical access to the rough rock surface, which is often not possible when the discontinuity is contained within the rock mass. Lastly, when applying a shear strength model, physically accessing the surface/discontinuity may not be safe or possible, making seed trace digitisation by contact an invalid approach. Non-contact approaches to digitise roughness have also been investigated, including photogrammetry [44,45,46], and laser scanning [10,47,48]. However, a seed trace is often only identifiable by its contrast in colour from the remainder of the rock mass. Discontinuities that are tight and unweathered may exhibit insufficient data to allow for digitisation by laser scanning or photogrammetry. To avoid the need to physically access a seed trace, a non-contact approach must be developed to allow application of StADSS to common discontinuities in the field where physical access is not possible or safe. The most promising non-contact approach applicable to the development of StADSS is by processing images [49,50]. Images have many advantages over other methods:

High-resolution images can be captured using hand-held digital cameras or with drones enabling remote access to potentially unsafe and difficult to reach areas.

High-quality cameras (both hand-held and mounted to drones) are now relatively affordable and cost-effective items used in industry.

Images capture a lot of data, including the colour and relative positions of objects within the image. A combination of information ideal for digitising seed traces in the field.

It is for these reasons that high-resolution images were pursued as the proposed approach to digitise seed traces for application of StADSS to natural rock discontinuities. Essentially, the digitisation methodology comprises capturing a high-resolution image of the seed trace, extracting the profile of the trace during processing, and scaling the data using survey data from the site.

Given the importance of trace roughness information for the prediction of shear strength [45,51], it is essential for the seed trace to be digitised at high resolution and with an effort to reduce errors. Capturing the profile of a natural trace in the field with high accuracy can be challenging because of limited access, size of the discontinuity and trace, weather, and lighting conditions.

The orientation of a camera relative to an object (in this case, a seed trace) is important to ensure the objects captured within the image are least affected by perspective distortion. Ideally, the camera should be oriented perpendicular in all directions to the object dimensions desired to be proportionally correct within the image. This orientation reduces the effects of pose, distance, and foreshortening [52]. Whilst corrective models for perspective distortion do exist [52,53,54,55], these will not be investigated within this process for simplicity and because they become complex when applied to irregular 3D scenes such as rock faces [56].

This paper presents the results of a study on the quantification of errors on a discontinuity roughness descriptor resulting from camera rotations around two axes and photograph resolution. Poor information capture will result in errors on the estimation of roughness, which, in turn, will result in error in shear strength predictions. Although the significance of ground sampling distance and image distortion have been extensively studied in other applications, there is no information in the literature on those effects on the quantification of roughness for large in situ discontinuities. As far as the authors know, there are no published data on identifying the influence of camera position (distance, and orientation) on the relative error of standard deviation of gradients, a roughness indicator, and its effect on predictions of shear strength for rough rock discontinuities. Hence, the novelty of this investigation lies in determining the magnitude of the influence of different image distortion errors and systematic errors in the trace extraction methodology on predictions of shear strength using a recently developed shear strength prediction approach named StADSS [45].

2. Methodology

The seed trace digitisation methodology comprises a sequence of steps beginning with the capture of the images on site followed by post-processing:

• Capture a high-resolution image containing the entire seed trace and take survey measurements (tape measure, ruler, or theodolite survey) on site for scale.
• Trace the profile of the seed trace in the image at single pixel increments manually or using a customised Python script. The rock mass and seed trace should be coloured white and black, respectively, to increase the contrast between intact rock and discontinuity trace.
• Extract the coordinates of the seed trace profile within the image at single-pixel increments. Depending on the trace thickness, a top and bottom trace can be extracted at each pixel position on the horizontal axis. In this investigation, only the top trace was extracted.
• Scale the seed trace data using the physical measurements taken on site. This converts the trace coordinates from pixels to millimetres using linear interpolation.

Examples of steps 1, 2, and 3 are illustrated in Figure 1 [57].

The coordinates of the trace profile are extracted using a top–down approach. A Python script starts the identification in the top left pixel of the image; if this pixel is coloured as rock mass, it then moves down a single pixel and repeats this process. If the next pixel is coloured as seed trace, the script identifies this as the coordinate of the top trace for that column of pixels. Extracting the coordinates of the bottom trace is not applicable within this paper. The script then progresses to the next column of pixels and repeats the process. This is demonstrated visually in Figure 2.

The main sources of errors when digitising the trace are expected to come from steps 1 and 2. This paper aims to quantify the errors encountered primarily during step 1 of the digitisation methodology by investigating the influence of the camera’s distance from the trace and the orientations of the camera relative to the trace. Some errors are also expected to occur during post-processing due to image types, data compression, and geometric anomalies caused by digitising roughness at pixel increments. This paper does not address errors due to survey accuracy on site used to scale the trace data, edge detection of the trace in photographs of real rock surfaces, or errors due to methods used to digitise very long traces not containable within a single image. These issues will be addressed in further research. To quantify the desired errors, the following steps shown in Figure 3 will be followed, and the following sections provide specific information on each step of the methodology.

2.1. Random Field Model and Control Trace Generation

Control traces with known roughness statistics were generated using a multiscale Gaussian random field model. The control traces were generated to simulate real in situ traces because the roughness statistics of real in situ traces are unknown. The simulation of the traces in a 3D software is explained in Section 2.2. Three control traces of increasing roughness were generated. They were 5 m in length and characterised by their standard deviation of gradients (sd_i), which was 0.31 m/m, 0.71 m/m, and 1.12 m/m, respectively. More information on the input traces is found in

Table 1. Summary of roughness characteristics (from left to right, including standard deviation of gradients, mean of heights, standard deviation of heights, and mean of gradients) of the input multiscale seed traces generated by the random field model.

sdi [m/m]	${\mathit{\mu }}_{\mathit{z}}$ [mm]	${\mathit{s}\mathit{d}}_{\mathit{z}}$ [mm]	${\mathit{\mu }}_{\mathit{i}}$ [m/m]
1.12	58.7	11.3	0.00
0.71	61.5	12.1	0.00
0.31	54.1	9.9	0.00

. The trace length of 5 m was nominally adopted to cover situations the authors believed most likely to encounter in the field where StADSS or CFR would be required to gain predictions of shear strength of a natural rock discontinuity.

The random field model was written using components of the GSTools (v1.5.0) random field modelling package in Python [58], with a Gaussian covariance model for spatial correlation. The GSTools (v1.5.0) package provides the framework to write unique random field models for specific tasks; in this case, a model was written to generate 2D seed trace data with known output statistics and inputs. The inputs to the random field model are the mean and variance of asperity height, the dimension of the field created (5001 points for a trace 5000 mm long), and the correlation length. These values were based off decoupled trace data [21] but were altered through trial and error to achieve the desired statistics of each trace.

Each trace was obtained by superposing three daughter traces with different roughness profiles at small, intermediate, and large scales of roughness [38]. The large, intermediate, and small scales were generated separately with the random field model and then superimposed to form a single multiscale seed trace as demonstrated in Figure 4.

The z values of each trace were integer values between 0 and 100. The random field model generated values with decimal places; however, each height value (z) was rounded to the nearest whole number to prescribe a pixel coordinate within the output images (i.e., pixels form an array with positions 0, 1, 2, …, 3, etc.). The resulting roughness characteristics of each trace are summarised in Table 1 and they are shown in Figure 5.

The control traces were then plotted onto an image (format png); with the trace plotted black, and the remainer of the image plotted white. The images were 5000 pixels wide and 200 pixels tall. The thickness of the trace on the image was arbitrarily set to 7 pixels to increase visibility of the trace on the photographs to be taken. Such a thickness is consistent with photographs of real discontinuity traces. At a given x-coordinate, the z-coordinate of the trace was in the middle pixel of the seven within the thickness of the trace.

Each seed trace image was then densified in Photoshop using the bilinear interpolation method. The final image was 30,000 pixels wide and 1200 pixels tall. The purpose of densifying each image at this stage was to allow for high-resolution input of data into Blender to best simulate a real discontinuity [57]. This was performed now rather than during the generation step due to limitations in the RFM; data had to be generated at 1 mm increments (or single-pixel increments) to maintain control of the input statistics. Please note the bilinear interpolation method was deemed most suitable by trial and error.

A summary of inputs to the random field model can be found in Appendix A.

2.2. Trace Photography and Digitisation

The seed trace digitisation methodology was applied to the generated seed trace data in a variety of cameras distance and orientations, relative the rock face, to quantify the associated errors on roughness characterisation. The control traces printed onto high resolution images were input into Blender, for photographs to be virtually captured, replicating real scenarios in a virtual environment. Blender (v4.0) is an opensource 3D creation software that supports 3D models, lighting, camera positioning and setup, simulations, and rendering, amongst other features [59]. Blender has been used in the scientific community for the presentation of data, medical scans, and animations [60,61]. A key capability of Blender is being able to capture images of 3D scenes with cameras of different technical specifications and orientations and in various lighting conditions. The images captured within Blender closely represent those that would be captured in the same scenarios in the ‘real world’. The virtual camera properties used for the investigations presented in this paper within Blender are summarised in

Table 2. Properties of the virtual camera used in Blender.

Property	Value
Resolution	3840 × 2160 pixels (W × H)
Focal length	35 mm

.

For all Blender scenarios considered, a vertical rock face was used, and sunlight was applied to the trace at 45° to simulate ideal lighting conditions (i.e., uniform, and consistent daylight on the rock face). Lens distortion was minimised by adopting a prime lens (i.e., fixed focal length) for all images captured in Blender. The boundary of the output images was checked for square within Photoshop, no obvious curved distortions were noticed at each camera position, so no corrections for lens distortion were required.

A specific programme of investigations was developed to study the effect of several relevant parameters, one at a time, on the possible error made on the roughness quantification.

Part 1: The error due to developing the input seed trace using the random field model was evaluated by applying those data into png image format and densifying the input image for use in Blender. Errors may arise from possible image compression during the formation of the png image or during densification. For this study, Blender was not used. The three control traces were developed and then put straight into the Python trace extraction script, bypassing any photography to determine the magnitude of any systematic errors.

Part 2: The effect of ground sampling distance (GSD) was investigated. GSD is a parameter commonly used in scientific photography and photogrammetry. It describes the real length in the scene which a pixel covers in mm per pixel. The influence of GSD on roughness data was investigated to determine the optimal distance to position a camera. For this part, the camera is ideally orientated perpendicular in all directions to the rock face the trace is exposed on.

Part 3: The error arising from camera rotation around the z-axis was investigated (see Figure 6 for axis definition). The idea is to simulate human error in positioning the camera perpendicular from the rock face the trace is exposed on with the camera oriented toward the centre. Each camera position relative to the seed trace is shown in Figure 7. Angles of 90°, 85°, 80°, and 75° were considered.

Part 4: The effect of camera rotation around the x-axis was investigated (see Figure 6 for axis definition). Here, two angles are relevant, as illustrated in see Figure 8c: the angle between the rock face and the camera orientation (denoted by $\delta$) and the angle between the rock face and the average discontinuity plan (denoted by ${\alpha }_r$). The latter angle’s relevance lies in the fact that the asperity heights should be measured perpendicular to the average discontinuity plane. In any other position than a discontinuity plane perpendicular to the rock face, the asperity heights are seen larger than they are (i.e., exaggerated by perspective), and a geometric correction should be applied. The correction is expressed in Equation (1):

$$z=z_c\times {\displaystyle \frac{\sin {\alpha }_r}{\sin \delta }}$$

(1)

where z is the corrected height value representative of the trace amplitude perpendicular to the discontinuity plane, $z_c$ is the distorted amplitude of the seed trace captured in the original image of the seed trace (see Figure 8a), $\delta$ is the angle between to average plane of the rock face where the seed trace is exposed upon and the orientation vector of the camera (see Figure 8c), and ${\alpha }_r$ is the angle between the average discontinuity plane and the average plane of the rock face where the seed trace is exposed upon (see Figure 8c). It should be noted that the camera is positioned perpendicular to the rock face around the z-axis (see Figure 8b) and level. Unfortunately, it is not possible to change angle ${\alpha }_r$ using Blender, so a constant value of 90° was adopted. Angle $\delta$ was varied from 90° to 45° in 5° increments. The data were then corrected at each data point during post-processing using Equation (1). An example of a planar rock face containing a trace oriented 15° from perpendicular is shown in Figure 9.

Part 5 combines incremental rotations around x and z axes to assess the possible compounding digitising errors due to rotation around two axes. This was conducted to simulate the potential for poor put results when the operator capturing the images is not careful or diligent. Like part 3 above, the camera was positioned with rotation around the z-axis at 5° from perpendicular; however, the trace was also tilted by 5° at the same time rather than staying perpendicular. The trace and camera were oriented at the same respective orientations from 90° to 75° at 5° increments.

A summary of inputs for each part of the investigation can be found in

Table 3. Summary of all parameters used during the investigations into the trace digitisation approach. Contained within parts 2, 3, and 5 are subsections (labelled a, b, c, …, etc.) which explain the variations tested. In part 2, for example, the 5 m trace with input image size of 30,000 × 12,000 pixels was captured at 12 m, 9 m, and 7 m to result in GSDs of 2.2, 1.6, and 1.3 mm/pixel.

Part	Primary Targeted Source of Error	Trace Length(s) [mm]	Input Image Properties [Pixels]	Distance(s) to Rock Face [m]	GSD(s) [mm/Pixel]	Trace Orientation(s) Around x-Axis	Camera Orientation(s) Around z-Axis
1	Image densification, generation of png image	5000	30,000 × 1200	N/A	N/A	N/A	N/A
2	Ground sampling distance/resolution	(a) 5000 (b) 1000	(a) 30,000 × 1200 (b) 6000 × 1200	(a) 12, 9, 7 (b) 6, 3, 1.5	(a) 2.2, 1.6, 1.3 (b) 1.1, 0.6, 0.3	90°	90°
3	Perspective distortion	5000	30,000 × 1200	7.5	1.4 *	90°	(a) 90° (b) 85° (c) 80° (d) 75°
4	Perspective distortion (geometrically corrected)	5000	30,000 × 1200	7.5	1.4 *	90° to 45° at 5° increments	90°
5	Perspective distortion	5000	30,000 × 1200	7.5	1.4 *	(a) 90° (b) 85° (c) 80° (d) 75°	(a) 90° (b) 85° (c) 80° (d) 75°

* GSD of the image relative to the seed trace is calculated at the centre of the image, but perspective distorts the GSD in different areas of the image due to rotations of the camera and trace. Variations within each step are denoted by parts a, b, and c, and d in some cases.

. The output images captured in Blender were all densified during post-processing prior to extracting the coordinates of the trace using the nearest neighbour approach in Photoshop [57]. The orientation of each is always the same for each of the investigations referred to above. The z-axis is always vertical, the x-axis is parallel to the rock face plane and is level, the y-axis is perpendicular to the rock face plane and level, and the origin of axes is set at the centre of the seed trace. This is visually demonstrated in Figure 6.

2.3. Calculation of Errors

The investigations into the digitisation approach were evaluated by calculating the relative error on sd_i and the standard deviation of absolute errors for each output trace in each different scenario. The relative error on sd_i (%) was calculated by Equation (2) and the standard deviation of absolute errors was calculated by Equation (3).

$${RE}_{{sd}_i}={\displaystyle \frac{{sd}_{i-outputtrace}-{sd}_{i-original}}{{sd}_{i-original}}}\times 100$$

(2)

where ${RE}_{{sd}_i}$ is the relative error percentage of sd_i, ${sd}_{i-outputtrace}$ is the sd_i of the output trace of a particular scenario, and ${sd}_{i-original}$ is the sd_i of the original input trace data.

$${sd}_{absolute-error}=\sqrt{{\displaystyle \frac{{(\left(z_{i-output}-z_{i-original}\right)-{\mu }_{z-differences})}^2}{N}}}$$

(3)

where ${sd}_{absolute-error}$ is the standard deviation of absolute errors, $z_{i-output}$ is each height value of the output trace at x position i, $z_{i-original}$ is each height value of the original trace at x position i, ${\mu }_{z-differences}$ is the average of height differences between the output and original traces at 1 mm increments on the x-axis, and N is the number of points on the trace. In most cases, N was equal to 5001; however, some investigations used smaller traces, in which case N was equal to 1001.

Quantification of the standard deviation of absolute errors was conducted to provide readers (and the authors) with additional insight into how the digitisation method behaved. Just because the quantification of sd_i may be accurate in the tested scenarios does not necessarily indicate the method is accurate when the absolute errors are high.

Like quantifying the errors on trace digitisation, errors on the predictions of peak and residual shear strength were also calculated relative to the real values using Equation (4) and Equation (5), respectively.

$${RE}_{{\tau }_p}={\displaystyle \frac{{\tau }_{p-error}-{\tau }_{p-actual}}{{\tau }_{p-actual}}}\times 100$$

(4)

$${RE}_{{\tau }_r}={\displaystyle \frac{{\tau }_{r-error}-{\tau }_{r-actual}}{{\tau }_{r-actual}}}\times 100$$

(5)

where ${RE}_{{\tau }_p}$ and ${RE}_{{\tau }_r}$ are the relative error of the peak and residual shear strength predictions, ${\tau }_{p-error}$ and ${\tau }_{r-error}$ and the peak and residual shear strength predictions made with known error input of sd_i, and ${\tau }_{p-actual}$ and ${\tau }_{r-actual}$ are the peak and residual shear strength predictions made with no error input of sd_i.

2.4. Influence of Digitisation Error on Shear Strength Predictions

The digitisation of a seed trace yields the roughness data required to make predictions of peak and residual shear strength using the Newcastle discontinuity shear strength model (NDSS) and the continued fraction regression (CFR) model [51]. To assess the significance of the error made during data capturing on shear strength prediction, a parametric study was conducted using the CFR model, which is more computationally efficient than the NDSS.

The CFR model was developed by applying a continued fraction mathematical regression model to a set of 14,000 shear strength predictions produced by the Newcastle discontinuity shear strength model. The primary inputs to the model are the sd_i (in m/m) computed in the shear direction, the material strength parameters of the rock (${\sigma }_{ci}$, mi, and ${\varphi }_b$), and effective normal stress (${{\sigma }^{\prime }}_n$). The model predicts the number of sheared facets (N_CFo) of a 3D surface with a 4 m² projected area (A_tot-o) at a 1 mm spatial increment (Res_o). The model is defined mostly by Equation (6) (further broken down into Equations (7)–(9)):

$$\ln \left(N_{CFo}\right)=g_o\left({\mathit{x}}^{\prime }\right)+{\displaystyle \frac{h_o\left({\mathit{x}}^{\prime }\right)}{g_1\left({\mathit{x}}^{\prime }\right)}}$$

(6)

where

$$g_o\left({\mathit{x}}^{\prime }\right)=0.497568\times {sd}_i^{{\displaystyle \frac{1}{2}}}+0.051503\times {{{\sigma }^{\prime }}_n}^{{\displaystyle \frac{1}{3}}}-0.604231\times {{\sigma }_{ci}}^{{\displaystyle \frac{1}{3}}}-0.125605$$

(7)

$$h_o\left({\mathit{x}}^{\prime }\right)=0.001748\times {\tan }^2\left({\varphi }_b\right)+0.017096\times {{{\sigma }^{\prime }}_n}^{{\displaystyle \frac{1}{3}}}+0.000020\times {{\sigma }_{ci}}^{{\displaystyle \frac{1}{3}}}+0.002485$$

(8)

$$g_1\left({\mathit{x}}^{\prime }\right)=-0.000307\times {sd}_i^{{\displaystyle \frac{1}{2}}}+0.000973\times {{{\sigma }^{\prime }}_n}^{{\displaystyle \frac{1}{3}}}+0.000544$$

(9)

The sd_i may be computed from a 3D point cloud of a surface or from a trace exposed on a rock face and should ideally be computed in the shear direction whenever possible to avoid effects of possible anisotropy.

Here, the sd_i was taken from the three generated seed traces used throughout the paper equal to 1.12 m/m, 0.71 m/m, and 0.31 m/m, respectively. Based on the prediction of N_CFo, the CFR model may predict peak and residual shear strength using Equation (10) and Equation (11), respectively.

$${\tau }_p={\displaystyle \frac{Re{s}_o^2{\times N}_{CFo}}{2\times A_{tot-o}}}\times c(N_{CFo})+{{\sigma }^{\prime }}_n\tan \left(\varphi \left(N_{CFo}\right)\right)$$

(10)

$${\tau }_r={{\sigma }^{\prime }}_n\times \tan \left(\varphi \left(N_{CFo}\right)\right)$$

(11)

The Hoek brown criterion is used to derive both $\varphi$ and $c$ for a given stress state [62] by Equations (12) and (13).

$$c=\left({\displaystyle \frac{1}{\sin \left(\varphi \right)}}-1\right){\displaystyle \frac{{\cos \left(\varphi \right)m}_i{\sigma }_{ci}}{8}}-{\sigma }_{local-i}\tan \left(\varphi \right)$$

(12)

where ${\sigma }_{local-i}$ is the local normal stress acting on contributing asperities, m_i and ${\sigma }_{ci}$ are the material parameter and intact unconfined compressive strength of the Hoek Brown criterion obtained by fitting experimental triaxial data, and $\varphi$ is defined as

$$\varphi =\mathrm{atan}\left({\displaystyle \frac{1}{\sqrt{4h\times {\cos }^2\left(\theta \right)-1}}}\right)$$

(13)

where $h=1+{\displaystyle \frac{16{(m}_i{\sigma }_{local-i}+{\sigma }_{ci})}{3{\sigma }_{ci}{m}_i^2}}$ and $\theta ={\displaystyle \frac{1}{3}}\left(90+\mathrm{atan}\left({\displaystyle \frac{1}{\sqrt{h^3-1}}}\right)\right)$.

The material strength inputs for the parametric study are presented in

Table 4. Parameters used as input to the CFR model for the investigation into peak shear strength prediction error induced by error on sd_i. The strength parameters correspond to limestone previously tested [43].

sdi of Input Trace [m/m]	${\mathit{\sigma }}_{\mathit{c}\mathit{i}}$ [MPa]	${\mathit{m}}_{\mathit{i}}$ [−]	${\mathit{\varnothing }}_{\mathit{b}}$ [°]
0.31, 0.71, and 1.12	97	8.8	37.4

.

The three input traces with standard deviation of gradients (sd_i) of 0.31 m/m, 0.71 m/m, and 1.12 m/m were also used as a base for the inputs. The sd_i was input with relative error ranging from −20% to +20% at 5% increments at a variety of normal stresses. The normal stress scenarios were 0.01 MPa, 0.1 MPa, 1 MPa, 10 MPa, and 100 MPa. This process was conducted for each seed trace resulting in plots of input sd_i error and output relative errors of peak and residual shear strength.

3. Results

3.1. Error Due to Data Extraction Algorithm and Image Densification

This section reports the results of the investigation into systematic error produced by generating png-format images of the control traces that underwent pixel densification by the bilinear interpolation method in Photoshop. Each trace image was processed using a trace coordinate extraction script written in Python. Here, Blender was not used, and photographs of the traces were not taken. Rather, the objective was to assess the error resulting from the image creation, densification, and data extraction. The error relative to the original trace data both for sd_i and absolute error is shown in Figure 10a and Figure 10b, respectively. See Section 2.3 for definitions and the calculations of errors.

The results show that the process of converting the generated seed trace data to a png image, its densification, and then the extraction of the trace using a python script does result in some error. Interestingly, the rougher the surface, the higher the absolute error on heights (Figure 10b) but the lower the relative error (absolute) on gradients (Figure 10a). The fact that the relative error on gradients is lower for rougher traces can be attributed to simple math; a higher sd_i on the denominator of the relative error calculation results in a smaller relative error percentage. The increasing relationship between the standard deviation of absolute error and roughness can be attributed to the formation of the png image type, where hard edges are slightly softened during development using deflate compression [63]. The relative error on sd_i is negative for all three traces, implying that the process slightly underestimates sd_i. This underestimation can again be attributed to the formation of the image type, where some extremity asperity peaks are softened, resulting in a slightly smoother trace profile.

The standard deviations of absolute error for each trace are maximum ~1 mm for all traces. Overall, the error due to the image processing and extraction Python script falls within the range −7.5% to −2%, which is quite low.

3.2. Error Due to Ground Sampling Distance (GSD)

In this section, the image of each control trace was imported to Blender, photographs were taken with the virtual camera (orientated ideally, i.e., perpendicularly, to the rock face), the photographs were densified, and the roughness profile was extracted. The distance between the rock face and the camera was increased from 6 m to 12 m. This range of distances results in a GSD ranging from 1.1 mm/pixel to 2.2 mm/pixel given the resolution of the camera (refer to Table 2). This process was applied to the three control traces of Section 2.1; however, eventually the entire trace was not contained within the image frame as the camera was positioned closer. To continue investigating at GSDs less than 1.2 mm/pixel, each trace was cropped to the first 1 m of length, enabling the entire trace to be contained within the image frame. The relative errors for these cases were calculated relative to the trace statistics representative of the trace length captured within the image frame over 1 m of length and the results depict this in Figure 11.

Figure 11 shows the evolution of relative error on sd_i and standard deviation of absolute error on heights as the GSD increases from 1.1 mm/pixel to 2.2 mm/pixel. When the camera was positioned too close to the rock face (in our case, closer than 7 m), it was not possible to capture the full 5 m of trace length. In that case, only 1 m of trace was photographed, analysed, and compared to the same 1 m of control trace.

The overall trend was that increasing the GSD resulted in increased error, both relative and absolute. The GSD correlates to the resolution of the photograph: the higher the GSD, the more physical area a pixel covers, and hence, the lower the resolution. Roughness information was lost as GSD increased, which explains the increased error. These results indicate that a GSD equal to 1 mm/pixel delivered the best results overall, because the relative error on standard deviation of gradients trended towards zero for all traces. Whilst the results presented in Figure 11 indicate that output data generally improve as the camera gets closer to the seed trace, the access to and size of the trace may limit the ability to place the camera very close to the rock face. Indeed, in this study, the trace was required to be captured in a single photograph. A GSD of 1.36 mm/pixel was practical as it corresponded to 7.5 m from the rock face for this camera and it allowed for the capture of the entire 5 m trace.

3.3. Errors Due to Camera Rotation Around the z-Axis

In this section, the images of the three control traces were imported into Blender and photographs were taken with the camera positioned at various positions to the rock face such that the camera was not perpendicular to the trace plane around the z-axis. The errors were computed for each camera orientation to reveal the effect of camera rotation around the z-axis (refer to Figure 7). The relative error on sd_i is shown in Figure 12a, and the standard deviation of absolute errors is shown in Figure 12b.

As seen in Figure 12 any deviation around the z-axis from perpendicular results in a decay in data quality from both a gradient perspective and an absolute error perspective. The range of absolute errors increases (represented by an increase in standard deviation) as the camera is rotated further from perpendicular for each trace. The roughest trace (sd_i = 1.12 m/m) experienced the greatest increase in absolute error as the camera increased its rotation from perpendicular to the trace. As the camera rotated around the z-axis, roughness data were lost, with all traces exhibiting a standard deviation of absolute error greater than 3 mm after only 5° rotation. The data loss exhibited by the output traces occurred due to parts of the rough profile falling within the resolution of the image, disabling the ability to characterise the profile by pixel location.

These results highlight the importance of ensuring a perpendicular orientation of the camera relative to the rock face. This can be achieved with relatively high accuracy using basic measuring techniques and survey data on site.

3.4. Errors Due to Camera Rotation Around the x-Axis

In this section, the three input traces were imported into Blender and incrementally rotated around the x-axis at 5° increments to determine the error where the camera cannot capture images of the trace whilst perpendicular to the trace plane around the x-axis. The errors were computed for each trace orientation to reveal the effect of a camera rotation around the x-axis (refer to Figure 9). Prior to evaluating the error on the output traces, each output was geometrically corrected using Equation (1). The relative error on sd_i is shown in Figure 13a and the standard deviation of absolute errors is shown in Figure 13b.

The results presented in Figure 13 show that, as the camera progressively rotates from its initial position (angle of 90°), the initially positive relative error reduces to 0% for an angle that depends on the roughness of the trace, and then its magnitude increases again, this time in the negative range. For the roughest trace, it takes a rotation of 40° to reach a relative error of 15% in magnitude. In contrast, for the smoothest trace, the maximum magnitude of relative error measured in 12%. Figure 13b shows a consistent increase in absolute error on heights as the camera orientation increases. The rougher the trace (or surface), the higher the absolute error.

Considering these results, one would suggest that rotations over 35° from vertical give poor results that may not be reliable for accurate shear strength prediction. However, Figure 13b indicates that the data suffered a notable decay in quality from inclinations over 15° from vertical, demonstrated by the standard deviation of absolute errors exceeding 2 mm for the smoothest trace in Figure 13b. Hence, whilst the relative errors on sd_i remain within ±10% for rotations up to 35° from perpendicular, the actual output traces are not necessarily representative of the real trace because the absolute errors increase with any rotation.

Data quality decayed as the seed trace plane was rotated around the x-axis because some asperities’ amplitude fell within the resolution of the image. Hence, some asperities were lost due to the camera’s inability to prescribe them a location within a single pixel. This is demonstrated in Figure 14b after Figure 14a is rotated by 15° around the horizontal axis. The shaded pixel boxes show the locations of pixels extracted to digitise the seed trace. The shaded pixel profile for the image captured perpendicular to the seed trace (Figure 14a) is clearly rougher and captures more roughness data than the image captured at 15° to the seed trace (Figure 14b).

Based on the results of this investigation presented in Figure 13 and Figure 14, it is recommended that the angle between the camera and the seed trace plane does not exceed 15° from perpendicular.

As mentioned in Section 2.2, it was not possible to change angle ${\alpha }_r$ using Blender, so a constant value of 90° was adopted, and only angle $\delta$ was varied (refer to Figure 8). Whilst ${\alpha }_r$ was not independently varied, the simulations presented above cover its effects on results because the geometric influence of ${\alpha }_r$ is the same as $\delta$. This investigation resulted in a recommendation for $\delta$ not to exceed 15° from perpendicular; this can be extrapolated to a basic geometric relations ship with ${\alpha }_r$. Geometrically, no distortion of the trace will occur when the sum of ${\alpha }_r$ and $\delta$ is 180° (i.e., the camera direction is parallel to the discontinuity plane). Hence, the sum of ${\alpha }_r$ and $\delta$ should not be less than 165° in accordance with the findings in this section; this allows for a combination of these angles where required.

3.5. Effect of Combined Rotations

In this section, the three input traces were imported into Blender and images were captured whilst incrementally moving the camera to rotate its perspective around the z-axis and tilting the trace around the x-axis simultaneously at 5° increments. The purpose of the investigation was to determine whether the output trace errors were exaggerated by a combination on rotations when an operator capturing the images is not careful or diligent on site.

The errors were computed for each trace orientation to reveal the effect of combined rotations (refer to Figure 7 and Figure 9). Prior to evaluating the error on the output traces, each output was geometrically corrected using Equation (1) to correct for the x-axis rotation. The relative error on sd_i is shown in Figure 15a, and the standard deviation of absolute errors is shown in Figure 15b.

The output traces for the combined rotations exhibited features of the results shown in Figure 12 and Figure 13 where the rotations were individually investigated. The relative error on sd_i exhibited nearly the same profile and magnitude for all traces as the errors for rotation around the z-axis (i.e., Figure 12a). The smoothest trace (sd_i = 0.31 m/m) error began to reduce from 80° to 75°; possibly, this was the influence of rotation around the x-axis where the relative error decreases from positive to negative with rotation (refer to Figure 13a). The standard deviation on absolute errors exhibited little difference to the errors where only the z-axis was rotated (refer to Figure 12b).

Based on these observations, a combination of rotations does not necessarily exaggerate the errors on output traces using the digitisation methodology. However, the results reinforce the importance of aligning the camera perpendicular to the rock face ideally in all directions. If the camera cannot be orientated perpendicular around the x-axis, a geometric correction may be made. Unfortunately, no real solution allows for rotation around the z-axis due to complex perspective distortions. Hence, alignment around the z-axis remains critical to check to ensure quality trace output results.

3.6. Effect of Roughness Error on Shear Strength Predictions

Figure 16 presents the relative error on peak and residual shear strength prediction resulting from a relative error input on the standard deviation of gradients sd_i, as per Section 2.4. All other input parameters for the CFR model are given in Section 2.4. Each sub-figure corresponds to a trace with a given sd_i, on which the error is made. The results are plotted for a variety of normal stresses denoted in the legend at the base of the figure.

The relative errors on strength and on standard deviation are positively correlated: an overestimation of roughness yields an overestimation of strength, which is consistent with results from the literature. The most important observation is the disproportionate influence of input error of sd_i on predictions of peak shear strength. This strong effect is exacerbated under low normal stress and high roughness: for a stress of 0.01 MPa and a rough trace (sd_i = 1.12 m/m), an error of +20% on sd_i yields in a peak shear strength prediction error of +300%. However, results improved with increases in normal stress, with predictions of peak shear strength falling within ±10% relative error at 100 MPa normal stress. As seen in Figure 16b,c, the sensitivity to error input of sd_i reduces with roughness, with the maximum errors decreasing to 125% and 38%, respectively. These findings highlight the need to take extra care when assessing the roughness of rougher discontinuities subjected to low normal stress.

Residual shear strength is dependent on the number of contributing facets as defined by Equation (11). Despite this, the influence of sd_i error on predictions of residual shear strength is far less significant than on predictions of peak shear strength as demonstrated in Figure 17. Similarly, here, results are plotted for a variety of normal stresses denoted in the legend at the base of the figure.

The results shown in Figure 16 and Figure 17 demonstrate the sensitivity of shear strength predictions to different input errors of sd_i. These results are exacerbated by roughness and normal stress to varying degrees for both peak and residual shear strength. Unlike peak shear strength, the relative errors on predictions of residual shear strength are contained between ±7%. Predictions of residual shear strength are fundamentally less sensitive to roughness compared to peak shear strength due to their mathematical formulation [51]. The equations used to calculate peak and residual shear strength may assist the reader understand this phenomenon; refer to Equation (10) and Equation (11), respectively.

4. Discussion

4.1. Remarks on the Digitisation Methodology

The new seed trace digitisation methodology was shown to be effective in several scenarios expected to encounter in the field. The influence of camera rotation and GSD relative to the seed trace plane were investigated. The key observations taken from each investigation include the following:

The systematic relative error on sd_i induced by generating images of seed trace data ranged from −1.7% to −7.5%.

The ideal conditions simulating a camera perpendicular to the seed trace plane produced a relative error on sd_i from 4.1% to 11.2%, with the smoothest trace experiencing the highest relative error. For all traces, this process exhibited a maximum standard deviation of absolute errors of approximately 0.6 mm.

Rotation of the camera up to 5° from perpendicular around the z-axis resulted in suitably low relative error on sd_i. Based on the results of this investigation, the camera should ideally be placed perpendicular to the rock face around the z-axis in all scenarios. Failure to do so may results in unreliable results.

Rotation of the camera around the x-axis relative to the seed trace plane was shown to be correctable for rotations up to 15° from perpendicular. Any rotations exceeding that limit began to exhibit high levels of error, especially for rough traces.

A captured ground sampling distance (GSD) equal to or better (i.e., smaller) than 1.4 mm/pixel was found best to characterise seed trace roughness. All traces exhibited greater than 10% relative error for sd_i where the GSD exceeded 1.4 mm/pixel.

A combination of rotation around the z and x axes resulted in similar results to only rotating the camera around the z-axis. The combination of rotations did not exaggerate error, and this investigation reinforced the importance of orienting the camera perpendicular around the z-axis.

4.2. Ramifications of Errors on Predictions of Peak Shear Strength

The investigation into shear strength prediction errors due to input error of sd_i enabled several key observations to be made:

Seed trace errors resulting in an overestimation or underestimation of sd_i disproportionately influence predictions of peak shear strength but have relatively little influence on predictions of residual shear strength. This can be explained by the mathematical formulation of the CFR model, a model that is based on theoretical shear mechanics and experimentally validated.

Rougher seed traces experience greater errors in predictions of shear strength due to input errors of the standard deviation of gradients calculated at 1 mm increments (sd_i).

Predictions of peak shear strength are most disproportionately influenced by input errors of sd_i at normal stresses less than 1 MPa.

Peak shear strength predictions must be correctly interpreted based on the level of roughness and magnitude of normal stress to ensure predictions are made within acceptable bounds.

These observations give valuable insights into how site investigations must be conducted in the future and how predictions of shear strength made after digitising an in situ seed trace must be interpreted. Following this reasoning, the following recommendations can be made:

During a site investigation, firstly, an accurate estimation of normal stress must be gained. For discontinuities experiencing low normal stress (i.e., <1 MPa), there is less flexibility in data capture methods, limiting the following steps of the investigation. Where normal stress is low, ideal conditions (i.e., ground sampling distance (GSD) $\le$1.4 mm/pixel, camera perpendicular to seed trace, seed trace perpendicular to rock face) must be adhered to as closely as possible to make accurate predictions of peak shear strength.

Regardless of normal stress, the seed trace digitisation methodology may be an iterative approach to ensure the process remains economical where an initial application is used to determine the approximate magnitude of sd_i. Where sd_i is approximately greater than 0.3 m/m, perhaps an additional application of the method is necessary where extra expense and effort is spent to improve the camera quality, camera position, or the capture GSD. This is due to rough traces producing the highest shear strength prediction errors due to input error of sd_i.

The predictions of peak shear strength must be correctly interpreted based on the quality of the investigation, the magnitude of normal stress, and the magnitude of sd_i in conjunction with the results shown in Figure 16. Predictions of peak shear strength for smoother discontinuities (sd_i < 0.3 m/m) experiencing higher normal stress (>1 MPa) can be treated with greater confidence, especially where the trace was digitised under ideal conditions (i.e., GSD $\le$ 1.4 mm/pixel, camera perpendicular to rock face, and discontinuity perpendicular to rock face).

4.3. Comments on Practical Application

The work presented in this paper gave valuable insight into further developing and applying the digitisation methodology for shear strength predictions. Based on the discussion in Section 4.1 and Section 4.2, some key tips on applying the methodology have been compiled and are summarised in

Table 5. A summary of practical tips on applying the digitisation methodology in the field.

Keyword	Comment
Camera type	A high-quality and -resolution camera will produce better results. A minimum 20-megapixel sensor should be used, and contrast should be adjusted on site to best define the trace profile from the rock mass.
Camera lens	The operator should select a suitable lens for the camera that minimises any lens distortion. Whilst corrective models exist for this issue, typically lenses with longer focal lengths than fisheye lenses will produce more accurate photos. A prime lens (i.e., no zoom) is also recommended to improve image quality.
Ground sampling distance	Careful consideration must be made during the planning of image capture on site. The camera should be placed perpendicular from the rock face at a distance that achieves a suitable GSD (i.e., $\le$1.4 mm/pixel).
Camera position	The camera should be oriented as close to perpendicular as possible to the rock face to minimise any perspective distortion. Where the rock face is safe and accessible, this can be achieved with basic measurement tools (i.e., tape measure) or by coordinates with accurate survey tools (i.e., theodolite, differential GPS).
Drone camera	Inaccessible discontinuities may require image capture using a drone. Here, additional planning must be conducted to ensure the drone camera is oriented perpendicular to the rock face. For discontinuities of high importance, the operator may choose to develop a 3D model of the site and map a 3D coordinate to position the drone when capturing images of the seed trace to ensure the camera is perpendicular.
Lighting	Lighting may critically influence image quality. Ideally, the rock face should be well lit with ambient light but not in direct sunlight. Direct sunlight can be too intense and may degrade the quality of image capture. Good lighting can be achieved (depending on the location of the specimen) at certain times of the day or artificially with instrumentation. Shadows cast around the region of the trace should be avoided to maximise the contrast between the (usually) dark discontinuity profile and lighter intact rock. Harsh lighting may produce very dark shadows and obscure areas of the trace due to the geometry of the rock face. In this scenario, the operator may need to choose a better time of day to capture the images or employ the use of artificial lighting the simulate more ideal conditions.
Trace preparation	The trace may be obscured by vegetation or other debris obscuring parts of the trace. Any debris should be removed (whenever possible) to allow digitisation of the continuous trace at full scale. Where the trace cannot be fully exposed and is not visible to a camera, another method must be applied.

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4.4. Recommendations for Further Research

The work presented in this paper provides a robust foundation for digitising the seed trace data of planar seed traces in the field. However, limitations to the approach must be addressed, with further research focusing on the following issues:

Developing an approach to robustly extract the profile of a seed trace from images of real, natural rock discontinuities. The research presented in this paper focusing on simplified conditions (i.e., black and white images); however, images of natural discontinuities contain different colours which complicates the identification of the profile of the seed trace.

Lighting conditions must be considered in the future. Standard photography practices are well known and can be employed to improve the image quality (i.e., non-direct sunlight, consistent lighting across rock face). However, the influence of exposure settings and contrast is not yet understood.

Discontinuities may contain infilling, have openings, are weathered, or be mismatched. These issues are known to influence predictions of shear strength [23,24,26,35]. Additional work must be undertaken to determine how to robustly digitise a seed trace in these circumstances and also how to incorporate these properties into StADSS.

The question of how to extend the approach to discontinuities longer than 5 m needs to be addressed. A single image may not provide a suitable GSD; hence, multiple images may be required to digitise the length of a seed trace accurately. Possible avenues include panorama images and stitching trace segments after processing.

The influence of curved rock faces on seed trace data collection. Rock faces are not always planar as considered within this paper. Additional research is required to determine how much curve or undulation is acceptable before seed trace data becomes unusable with the standard approach.

5. Conclusions

A new method to digitise the in situ traces of natural rock discontinuities at full scale using high resolution images was investigated from an error’s perspective. The purpose of the research was to determine how sensitive the method is to scenarios likely to encounter in the field and how the resulting errors on trace digitisation influence predictions of shear strength made using the stochastic approach for discontinuity shear strength (StADSS) and the continued fraction regression (CFR).

Investigations into rotations of the camera and trace plane around the z and x axes, respectively, confirmed that orienting the camera perpendicular to the trace plane produces the most accurate results. Rotations around the x-axis were found to be correctable to within 15° of perpendicular, and output traces from rotations around the z-axis were only suitably accurate within 5° of perpendicular. The output traces were found to be highly sensitive to the capture ground sampling distance (GSD), and a recommendation of a GSD equal to or better than 1.4 mm/pixel was made. Combinations or rotations around the z and x axes at the same time were found not to exaggerate the output trace errors more than when the rotations occurred independently.

Lastly, predictions of peak shear strength were found to be highly sensitive to overestimations of the standard deviation of gradients calculated at 1 mm increments (sd_i), especially at low normal stress. Generally, the digitisation method always slightly overestimated the sd_i of the trace, which highlighted the importance of interpreting predictions of peak shear strength based on the magnitude of sd_i and normal stress. Residual shear strength predictions were found to be far less sensitive to input errors of sd_i. Based on the recommendations made within this paper to achieve acceptable results, predictions of peak and residual shear strength made using trace data from the digitisation methodology can be treated with confidence.