Rockfall is a complex natural process that can present risks to
the effective operation of infrastructure in mountainous terrain. Remote
sensing tools and techniques are rapidly becoming the state of the practice in
the characterization, monitoring and management of these geohazards. The aim
of this study is to address the methods and implications of how the
dimensions of three-dimensional rockfall objects, derived from sequential
terrestrial laser scans (TLSs), are measured. Previous approaches are
reviewed, and two new methods are introduced in an attempt to standardize
the process. The approaches are applied to a set of synthetic rockfall
objects generated in the open-source software package Blender. Fifty
rockfall events derived from sequential TLS monitoring in the White Canyon,
British Columbia, Canada, are used to demonstrate the application of the
proposed algorithms. This study illustrates that the method used to
calculate the rockfall dimensions has a significant impact on how the shape
of a rockfall object is classified. This has implications for rockfall
modelling as the block shape is known to influence rockfall runout.
Introduction
In steep mountainous regions around the world, infrastructure such as
highways and railways may be subject to rockfall hazards. A rockfall can be
defined as discrete fragments of rock which have detached from a cliff and
subsequently fall, bounce and roll as the fragments move downslope by
gravity (Hungr et al., 2014). Rockfalls can triggered
by several factors such as freeze–thaw cycles, weathering, heavy rainfall,
root action, seismicity and others (Volkwein et al., 2011). High energy and
mobility are also characteristics of rockfalls, making them a major cause of
landslide fatalities (Guzzetti et al., 2004). Moreover,
these geohazards can result in economic losses due to service interruptions
and equipment damage.
To assist in the management of these geohazards, a rockfall hazard analysis
can be undertaken to qualitatively or quantitatively define the rockfall
hazard present along a section of linear infrastructure. A typical rockfall
hazard analysis involves the compilation of known rockfall events over a
specific spatial scale and within a set period of time (Volkwein et al., 2011). Inventories aim to
provide a better understanding about the spatio-temporal occurrence and
magnitude of events (Froude and Petley, 2018).
Ultimately, temporal trends can be identified from an inventory, which
supports a more systematic mapping of hazards in the region to help mitigate
future losses. It may also be useful to discern any long-term changes that
are projected, as extreme weather events are expected to increase in both
frequency and magnitude within a changing climate (Cloutier
et al., 2016).
Once an inventory has been assembled, power law distributions have been
suggested to characterize the frequency–magnitude relationship for rockfall
at the study slope (Hungr et al.,
1999). Using the rockfall frequency–magnitude relationship at the study
slope, characterized by specific geological and geomorphological features,
return periods for select volume ranges can be determined (Wieczorek and
Jäger, 1996; Hungr et al., 1999; Dussauge et al., 2003; Malamud et
al., 2004).
Remote sensing techniques, such as terrestrial laser scanning (TLS), have
been used to characterize and monitor rockfall hazards (Abellán
et al., 2014; Jaboyedoff et al., 2012; Telling et al., 2017). Single epoch
TLS scans can be used for structural characterization of discontinuity
orientations (Lato et al., 2009), the determination of
the size and spatial distribution of potentially unstable rock mass volumes (Sturzenegger et al., 2011), and the
back calculation of rockfall volumes based on discontinuity orientations of
identified rockfall scars (Santana et al.,
2012). Work by Lato et al. (2012),
demonstrates how TLS can be integrated into rockfall hazard assessments along
road cuts. Rockfall magnitude, block size distribution and block shape
distribution were measured using surface models derived from TLS scans. This
information can be directly integrated into rockfall modelling for rockfall
hazard evaluation.
With multi-temporal TLS datasets, change detection algorithms, such as M3C2 (Lague et al.,
2013), as an example, can be used to identify areas of loss on slopes (i.e.
rockfall) between sequential TLS scans. The location, volume and dimensions
of rockfall on the slope can be calculated and populated into a database, as
demonstrated by Rosser et
al. (2007), Guerin et al. (2014), Tonini and Abellán (2014), van Veen et al. (2017), Janeras et al. (2017) and Williams et al. (2018). In several
of these studies, smaller magnitude rockfalls have been identified, which
are generally not observed during field inspections performed from the base
of the slope. Furthermore, smaller rockfall events have been shown to bound
the area of larger deforming portions of the slope (Kromer et al., 2015).
Recent work by Williams et al. (2018) makes use of a fully automated terrestrial
laser scanning system to near-continuously monitor a section of coastal
cliff in the United Kingdom. With near-real-time processing capabilities,
they demonstrate the influence of temporal acquisition rate on the
calculated frequency–magnitude relationship for rockfall at the study slope.
They demonstrate that more frequent monitoring captures a higher proportion
of smaller magnitude rockfall events, which represents a higher frequency
magnitude scaling coefficient. However, due to the 2.5-D nature of the
volumetric analysis, smaller magnitude events resulted in a higher degree of
volumetric uncertainty, due to edge effects compounded when 3-D change maps
are converted to 2.5-D raster datasets.
With the rapid automation of TLS acquisition and change detection processing
workflows (Kromer et
al., 2017; Williams et al., 2018), practitioners are able to evaluate
potential rockfall events and their characteristics quickly and with
substantial detail (Abellán et al., 2014). TLS systems are portable and
can be deployed on a tripod as soon as the site is accessed. There is no
need to establish a baseline dataset as is the case with radar systems, for
example (Teza et al., 2008). In addition,
TLS systems can achieve high spatial resolution of measurements (Pesci et al., 2011). These strengths of TLS
systems facilitate detailed back analysis of rockfall events to assess
characteristics which can then be used for ongoing rockfall hazard analysis.
With the recent advances in rockfall modelling with rigid body physics,
models can utilize the exact shape and position of the block detachment
location (Harrap et al., 2019). These cases can be used for both calibration
of the model and development of more representative hazard mapping.
Preliminary studies of the rockfall runout with respect to rockfall shape
have been conducted by Glover (2015) and Sala (2018). Both authors found
that the shape of the rockfall object has a pronounced effect on runout
behaviour. This behaviour has direct implications for rockfall hazard
zoning. Therefore, the ability to characterize the shape of rockfall events
is a key component which needs to be considered in generating rockfall
databases from sequential TLS scans. A variety of different formulations
have been proposed to measure the shape of rockfalls from point cloud
datasets, which record the before and after failure geometry. The authors
highlight several methods which have been used in other studies and present
two new methodologies to determine the dimensions of a rockfall object.
Rockfall shape and dimensions
The quantification of the shape of a rockfall scar can provide insight into
the kinematics of failure and potential runout of detached material
fragments. The use of remote sensing techniques and 3-D change detection
algorithms permits extraction of true rockfall shape, yet limited work has
been completed to quantify shape, despite its pronounced effect on runout
behaviour (Glover, 2015; Sala, 2018). Shape, as
noted by Blott and Pye (2008), is a function of four primary characteristics
which include form, roughness, irregularity and sphericity. Readers are
referred to Blott and Pye (2008) for further details on these
characteristics.
Overview of the Sneed and Folk ternary diagram adapted from Blott
and Pye (2008). (a) Visual representation of the different shape forms as
defined by Sneed and Folk (1958). Inset diagrams display the divisions for
each shape class. (b) Overview of the rounded synthetic blocks generated
using Blender. (c) Overview of the angular blocks generated using Blender.
In 1958, Sneed and Folk (1958) introduced a ternary diagram (Fig. 1a) to
describe the shape of pebbles based on relations between the long (A),
intermediate (B) and short orthogonal axes (C). The three ratios are listed
below.
1CA2A-BA-C3BA
Based on the three relations described above, particles can be classified
into 10 different shape classes. The end members of the ternary diagram
are compact (cubic), platy (tabular), and elongated (rod shaped).
In addition to granular particle shape classification, the Sneed and Folk
ternary diagram has been used in rockfall studies to characterize rockfall
dimensions and shape (Benjamin, 2018; van
Veen et al., 2017; Williams, 2017). In the aforementioned studies, the
rockfall dimensions were computed using a bounding box approach. A bounding
box defines the minimum extents of a cuboid which fully encloses the set of
points defining the object. In this study and the aforementioned studies,
the bounding box is oriented such that the edges of the calculated box are
parallel to Cartesian coordinate axes.
Currently there is no standardized method to determine the dimensions of a
rockfall object extracted from remote sensing data, such that the rockfall
shape can be classified. There is uncertainty in evaluating both the
distance and orientation of the axis lengths. This uncertainty is compounded by the fact
that there is ambiguity about whether the axis measurements are to be
mutually orthogonal or not. In this work, the authors address these uncertainties and
propose standardized methods to evaluate the dimensions of a rockfall
object.
Objectives
In this work, the authors address the process of extracting information regarding
rockfall dimensions from remotely sensed datasets. The primary objectives of
this work are summarized below.
Review current approaches used to determine the dimensions of 3-D rockfall
objects.
Present two new approaches for extracting the dimensions from 3-D rockfall
objects represented by point clouds.
Apply all of the approaches to a dataset of synthetic 3-D rockfall objects.
Implement the proposed approaches on a rockfall database derived from
terrestrial laser scanning (TLS) at the White Canyon in the Thompson River
valley in Interior British Columbia, Canada (Fig. 2).
Determine which method(s) provide(s) the most accurate measurements of the
objects' three mutually orthogonal principal axes.
A rockfall object in this context is defined as a three-dimensional (3-D) point
cloud or mesh that approximates the geometry and volume of rock that
detached from the slope.
Location map of the White Canyon. (a) October 2015 orthophoto of
the White Canyon. The White Canyon is delineated by the red dashed line. (b) July 2016 panoramic photograph from track level looking northeast at the
complex morphology of the study slope. (c) July 2016 photograph from track
level of the Mt. Lytton batholith. (d) April 2017 photograph displaying the
TLS system setup looking at the study slope from across the Thompson River.
(e) February 2018 photograph from track level looking at one of the rock sheds
on the eastern portion of the canyon. The rock shed is 20 m in width.
Methodology
The methodology involves applying six different approaches to measure the
geometry of irregularly shaped blocks and evaluating the output using the
range of shapes described in the Sneed and Folk ternary diagram (Sect. 2.1). Both synthetically generated (Sect. 2.1) and real rock shapes were
assessed. The methods by which data were collected and processed for the
real rock shapes using both terrestrial laser scanning (TLS) and
structure-from-motion multi-view-stereo (SfM-MVS) photogrammetry are
discussed in Sect. 2.2. Section 2.3 presents the methodology used to
extract rockfall information from 3-D change detection. Section 2.4 describes
the six approaches used to extract dimension information from 3-D point
clouds of rockfall shapes.
Synthetic block dataset
A synthetic block dataset was generated in the open-source software package
Blender (Blender, 2018) using the process described by Sala (2018) to generate synthetic blocks for rockfall
simulation. In general, the process involves the sculpting of cubic meshes
that encompass 1 m3 of volume. Mesh sculpting in Blender allows for
the displacement of mesh geometries into a variety of different shapes,
taking into consideration block form characteristics, such as angularity.
Once a shape has been created, its mesh is subdivided, increasing the number
of vertices on the shape's surface to better match the point density which
can be achieved from the TLS data described in the following sections. The
mesh vertices are then exported, creating a synthetic rockfall block point
cloud. Blocks corresponding to each major class in the Sneed and Folk
ternary diagram were created. For each class, (i.e. platy, elongate, cubic,
etc.) a rounded and an angular version, as defined by Powers (1953), of the block was generated. Figure 1b displays examples of the blocks used in this study.
Remote sensing data acquisition
The following subsections (Sect. 2.2.1 and 2.2.2) outline the remote sensing
techniques that are used in this study to collect point cloud data to define
the geometry of real rockfalls.
Terrestrial laser scanning (TLS)
Terrestrial laser scans were taken with an Optech Ilris 3D-ER terrestrial
laser scanner (Fig. 2d). The Optech Ilris has a manufacturer-specified
accuracy of 7 mm in range and 8 mm in vertical and horizontal directions for
data collected from a distance of 100 m (Optech, 2014). The
maximum range for the Optech Ilris is approximately 800 m with 20 % target
reflectivity (Pesci et al., 2011).
Due to the complex geometry of the rock slopes in the White Canyon, several
overlapping scans from different vantage points were captured to minimize
occlusions and to decrease the lateral incidence angle in the scans of the
slope. Point spacing for each scan varied between 7 and 10 cm. The scan site
locations are displayed in Fig. 3, along with a timeline of the scans used
in the study. Scans were taken approximately every 2–3 months starting in
November 2014. The last set of TLS scans used in the analysis were taken in
December 2017.
Overview of the scan site locations from across the Thompson
River. The timeline across the bottom of the figure indicates the times when
TLS scans were captured (green dots).
To process the TLS scans, the scans were first parsed using Optech Parsing
software. Once parsed, vegetation, mesh, and railway infrastructure
components such as slide detector fences were manually removed from the raw
point cloud using PolyWorks PIFEdit. After the point clouds were cleaned,
they were aligned using PolyWorks ImAlign to a common baseline (November
2014). The alignment process consisted of a coarse alignment using point
picking and then a fine alignment using an iterative closest point (ICP)
algorithm (Besl and McKay, 1992). Areas of known
change on the slope were excluded from the alignment process to help improve
the alignment between sequential scans (Lato et al.,
2015).
Structure-from-motion multi-view-stereo (SfM-MVS) photogrammetry models were
generated of both White Canyon East (WCE) and White Canyon West (WCW) (Fig. 4). The Agisoft PhotoScan Professional V1.3.2 software package
(Agisoft LLC, 2018) was used to create the models. The
models were generated following a typical SfM-MVS photogrammetry processing
workflow (Smith
et al., 2016; Westoby et al., 2012).
Overview of the SfM-MVS photogrammetry models. (a) Model of White Canyon West (WCW)
taken on 30 January 2017. (b) Model of White Canyon East (WCE) taken on 4 April
2017. (c) Classified model of WCE. The model was remotely mapped in PhotoScan
using a combination of the RGB point cloud and visual inspection of the
panoramic photography.
A Nikon D750 DSLR camera with a Nikkor 50 mm f/1.8 prime lens was used for
all image acquisitions. An external global positioning system (GPS) was
attached to the camera to geotag each photograph. The 282 images used to
generate the White Canyon West (WCW) model were captured on 30 January 2018. The
452 images used to generate the White Canyon East (WCE) model were captured
on 7 April 2018. Images were captured with approximately 50 % to 60 % overlap.
Each of the SfM-MVS photogrammetry models were mapped in PhotoScan to
delineate the boundaries of bedrock outcrops and channels to create masks
(Fig. 4c). This process is described in detail by Jolivet
et al. (2015). The photogrammetry models and masks were exported and
aligned to the TLS datasets in CloudCompare for further analysis. The masks
are used in the semi-automated rockfall extraction process that is described
below.
Rockfall extraction process
In this study, a similar process as utilized by Tonini
and Abellán (2014), Carrea et al. (2015), Janeras et al. (2017) and van
Veen et al. (2017) to semi-automatically identify rockfall locations and
extract information related to the dimensions of each rockfall event is
implemented. A generalized rockfall extraction process is illustrated in the
flow chart in Fig. 5.
Structured flow chart of the semi-automated process of extracting
rockfall from sequential TLS scans.
The process can be summarized as follows: once the TLS scans are cleaned and
aligned, the process involves computing the change between sequential scans
taken at times A and B. Distances are computed from A to B and then B to A. This
process determines the front and back of each rockfall event in each
respective scan. A minimum change threshold is then applied; this threshold
is typically based on the calculated limit of detection. The point clouds of
the fronts and backs of all rockfall events are then merged to generate
rockfall objects. Variants of DBSCAN (Ester et al.,
1996) are then implemented to cluster individual rockfall events which have
occurred between time A and B. The dimensions, volume and other parameters of
each individual rockfall event can be calculated and then populated into a
database for further analysis.
In this study, to compute the change between sequential TLS point clouds,
the process outlined by Kromer et al. (2015)
is utilized. The distance calculation is very similar to M3C2 (Lague et al.,
2013), where distances are calculated along normal vectors defined by slope
geometry within a specified radius from the point. The change is then
filtered based on the limit of detection. The limit of detection (LOD) can
be defined based on the registration error (Abellán et al., 2014). In this study,
the authors take 2 times the standard deviation (95 % confidence interval) of the
registration error to define the limit of detection. The LOD equates to
approximately 5 cm in the summer months and 7 to 10 cm in the winter months
(i.e. October to February). The higher limit of detection in the winter
months corresponds to a higher standard deviation in the registration error
(alignment). Higher standard deviations correspond to the winter scans,
where there is generally more humidity in the air and possibly water on the
slope surface, which have been found to influence the alignment process
(Abellán et al., 2014).
Detectable change was then filtered based on the LOD, to resolve clusters of
points that represent the scars (backs) of rockfall events. This process was
repeated, conducting the change detection in the opposite direction to
resolve the fronts of the rockfall objects. DBSCAN (Ester et al., 1996) was then used to cluster areas
of change. The same parameters as van Veen et al. (2017) are used for the
DBSCAN clustering (i.e. search radius of 30 cm and a minimum of 12 points to
define a cluster).
To resolve rockfall events as opposed to debris movements, we utilized the
masks mapped on the SfM-MVS photogrammetry models. The geometric centroids
of each cluster are used to search and find the 10 nearest neighbours within
the mask point cloud. Based on the classification of the 10 nearest
neighbours within the mask point cloud, a vote is conducted to classify the
centroid as either a debris movement or rockfall depending on the mask
classification (i.e. bedrock vs. channel).
Model fitting
The following subsections present the background for each of the models used
to determine the dimensions of the rockfall objects. Each of the approaches
were implemented in MATLAB (Mathworks, 2018).
Bounding box
A bounding box or enclosing box defines the minimum extents of a box within
which all points are contained. In this study, the bounding box is oriented
with the edges of the calculated box parallel to the Cartesian coordinate
axes (Fig. 6a).
Visual representation of each of the model fitting methods used in
the study. (a) Bounding box approach (e.g. van Veen et al., 2017; Benjamin, 2018; Williams, 2017). (b) Adjusted bounding box approach. (c) Least-squares
ellipsoid fit. (d) Minimum bounding sphere fit. (e) RFSHAPZ approach. (f) RFCYLIN approach.
Adjusted bounding box
The adjusted bounding box approach differs from the bounding box approach in
that the orientation of the box is not subjected to any constraints. In this
study, singular value decomposition (SVD) (Golub and Loan,
1996) is used to determine the orientation of the object relative to the
principal axes in Cartesian space. SVD is used because this process can
handle any m×n matrix, whereas eigenvalue decomposition can only be applied to
certain classes of square matrices (Golub and Loan, 1996). The
direction of most variance using SVD is determined and the point cloud is
rotated to align with the direction of maximum variance with the x axis in
Cartesian space. This results in the x axis of the box being aligned with
the longest dimension of the object. A bounding box can then be calculated
for the point cloud (Fig. 6b).
Least-squares ellipsoid
An ellipsoid can be defined as a closed quadric surface that is the analogue
of an ellipse. To fit an ellipsoid to the point cloud defining a rockfall
object, an algebraic form linear-least-squares ellipsoid fit
(Schneider and Eberly, 2003) is implemented. An algebraic
fitting model was selected as opposed to an orthogonal fitting ellipsoid to
reduce computing time and to benefit from the advantages of solving linear-least-squares problems (Li and Griffiths, 2004). The
algorithm generates a least-squares ellipsoid fit of the input point cloud
(Fig. 6c). Further details on the algorithm and derivation can be found in
Schneider and Eberly (2003).
Minimum bounding sphere
To fit a minimum bounding sphere to the point cloud, Welzl's
1991 algorithm is implemented. The algorithm computes the smallest sphere
enclosing a set of points in 3-D space in linear time (Fig. 6d). For further
details on the algorithm, readers are referred to Welzl (1991).
RFSHAPZ
In this study, the RFSHAPZ (rockfall shape) approach is introduced. The
approach can be broken into four main steps: (1) point cloud preparation,
(2) voxelation, (3) distance calculations, and (4) curve fitting. Figure 7
outlines a flowchart for the process used to determine the dimensions of
each rockfall object.
Structured flow chart of the RFSHAPZ algorithm.
Point cloud preparation involves translating each rockfall object so that
the object's geometric centroid is centered at the origin of a locally
defined Cartesian coordinate system. Once the object is centered at the
origin, SVD is used to rotate the object so that the longest dimension is
parallel with the x axis in Cartesian space.
The next step involves generating a voxel grid of the point cloud. A voxel
is a 3-D volume element that represents a numerical value. For this study,
the default voxel cube size is defined as a function of the point spacing.
We calculate the average point spacing of the surfaces that make up the
rockfall object and then double the value to determine the voxel size. The
size of the voxel is therefore a function of the point spacing and can be
adjusted depending on the rockfall object. The voxel grid is used to provide
a spatial context for the rockfall object and allows all points within each
voxel to be stored for further analysis. Once the voxel grid is established,
for each voxel grid line in the XY and XZ planes, we calculate the maximum
Euclidean distance between points within populated voxels (Fig. 6e). The
calculated distances are plotted along each grid line. Curves are then fit
to each of the distributions, utilizing a Fourier series fit, a Gaussian fit
and a sum of sines fit. An overview of each of the fitting methods is
provided below.
The Fourier series is a sum of sine and cosine functions that describes a
periodic signal. In this study, we use the trigonometric form of the series
which can be expressed as
y=a0+∑i=1naicos(iwx)+bisin(iwx),
where a0 is a constant term and is associated with the i=0 in the
cosine term. w represents the fundamental frequency of the signal, and n is the
number of terms in the series. For this study, n is fixed at a constant value
of one.
The Gaussian model fits peaks in a data series and is given by
y=∑i=1naie-x-bici2,
where a is the amplitude, b is the centroid (location), c is related to the peak width and n is the number of peaks to fit. For this study, n is fixed at a
value of one.
The last curve fitting function used is the sum of sines model. This model
fits periodic functions and is given by
y=∑i=1naisinbix+ci,
where a is the amplitude, b is the frequency and c is the phase constant for
each sine wave term. n defines the number of terms in the series. This
equation is closely related to the Fourier series described in Eq. (4). The main difference is that the sum of sines equation includes the
phase constant and does not include a constant (intercept) term. For this
study, n is fixed at a value of one.
RFCYLIN
The last approach introduced and implemented in this study, RFCYLIN
(rockfall cylinders), draws inspiration from the M3C2 methodology (Lague et al.,
2013). The point cloud preparation is the same as is described for the
RFSHAPZ approach discussed in Sect. 2.4.5.
For all points in the cloud defining the rockfall object, we calculate the
vector and Euclidean distance from each point to the geometric centroid. The
vector is oriented towards the calculated centroid. A cylinder is then
projected from each point through the geometric centroid of the rockfall
object. The length of the cylinder is set to be greater than the distance
calculated between each point and the geometric centroid. After the cylinder
has been projected, points are found to be within the cylinder. These points
are projected on the vector line, and the maximum distance between all points
through the centroid is determined. This process results in determining the
maximum (longest) dimension of the rockfall object.
Once the maximum distance and vector orientation has been calculated,
orthogonal vectors to the vector of maximum distance are then calculated
through SVD. To do this step, a plane is projected perpendicular to the
vector defining the maximum dimension. Points defining the rockfall object
are projected onto the plane. SVD is then used to determine the direction of
maximum and minimum variance. These define the vector orientations of the
other axes. Once the orientations of the orthogonal vectors have been
determined, cylinders are projected along each vector to find points which
lie within the cylinder. If no points are found to be within the cylinder,
we incrementally increase the diameter of the cylinder until points are
found to be within the cylinder. These points are then projected onto the
vector line defining the centerline of the cylinder. The distances between
points along each of the orthogonal vectors are calculated and define the
intermediate and shortest dimensions of the rockfall object (Fig. 6f). A
flowchart outlining this algorithm is displayed in Fig. 8.
Structured flow chart of the RFCYLIN algorithm.
Results
The calculated dimensions of the rockfall objects, using each of the
techniques described in Sect. 2, are tabulated for analysis. Section 3.1
presents the results from the analysis of the synthetic block dataset.
Section 3.2 presents the results of the analysis on the rockfall objects
extracted from the TLS monitoring in the White Canyon.
Synthetic block dataset
The dimensions of the 20 synthetic blocks described in Sect. 2.1 were
measured using the six methods outlined in Sect. 2.4. In addition,
independent sets of measurements were made manually by two different members
of the research team.
The calculated dimensions were plotted on Sneed and Folk ternary diagrams
in order to examine the geometric results, as shown in Fig. 9. The data for
the smooth (rounded) and angular synthetic objects are shown on separate
diagrams to highlight differences in the distribution of these datasets. The
observations made of these datasets include the following.
The angular synthetic block dataset displayed the largest spread in the
geometry represented by the calculated dimensions, when compared to the
smooth rockfall objects.
The measured dimensions of the very bladed and very elongate blocks, at the
bottom left and right corners of the ternary diagram respectively, were
closely aligned for all methods and manual measurements.
The angular compact series (i.e. compact platy, compact bladed and
compact elongate) showed the greatest divergence between the manual
measurements and the automated methods. A number of the methods, including
the manual measurements, classified the angular compact-platy block as
platy. The methods which did correctly categorize this shape include the
bounding box, the adjusted bounding box, the RFSHAPZ Gaussian fit and the
manual measurements. These measurements, however, are not closely aligned
and display significant spread between the data points.
With increasing compactness of the synthetic shapes, there are challenges
with assessing what is the shortest axis with manual measurements. This
effect is compounded with increasing angularity of the rockfall object.
For the angular compact-elongate block, the two manual measurements
incorrectly classify the block as compact bladed while all of the calculated
dimensions classify the block as compact elongate.
The results of the rounded synthetic block dataset displayed significantly
less spread in the calculated and measured block dimensions relative to
their angular counterparts. Only the rounded compact-elongate block had
classification issues based on the measured or calculated dimensions. The
RFCYLIN approach, RFSHAPZ and adjusted bounding box all classified the block
as compact bladed.
Sneed and Folk ternary diagrams separated to highlight shape
classification results. (a) The results of each of the nine fits for
each of the rounded synthetic blocks. (b) The results of each of
the fits for the angular synthetic blocks. BB: bounding box;
BB_ADJ: adjusted bounding box; EL: least-squares ellipsoidal
fit; RFSHAPZ_FOR: RFSHAPZ Fourier fit; RFSHAPZ_GAU: RFSHPZ Gaussian fit; RFSHAPZ_SINS: RFSHAPZ sum of sines
fit; RFCYLIN: RFCYLIN fit.
Error in dimension measurement for each fit compared to a set of
manual measurements for the rounded synthetic blocks. EL: least-squares
ellipsoidal fit; FOUR: RFSHAPZ Fourier fit; GAUSS: RFSHPZ Gaussian fit;
SINES: RFSHAPZ sum of sines fit; CYLIN: RFCYLIN fit.
In order to analyze the results, the manual measurements were selected as a
basis of comparison with the synthetic blocks. Figure 10 displays the
results for the rounded synthetic blocks, and Fig. 11 presents the results
for the angular set. The bounding box and adjusted bounding box approaches
were excluded from this analysis since they are a component of the process
of how the synthetic blocks were generated within Blender (Sala,
2018).
Error in dimension measurement for each fit compared to a set of
manual measurements for the angular synthetic blocks. SPH: minimum-bounding
sphere fit; EL: least-squares ellipsoidal fit; FOUR: RFSHAPZ Fourier fit;
GAUSS: RFSHPZ Gaussian fit; SINES: RFSHAPZ sum of sines fit; CYLIN: RFCYLIN
fit.
Overall, the errors associated with the angular dataset are an order of
magnitude higher than the rounded dataset (A axis). In addition, none of the
calculated fits underestimated the A-axis dimension for both the angular
and rounded datasets. Relative to the rest of the shapes, the platy series
(i.e. compact platy, platy, very platy) showed the highest deviations from
the manual measurement. Within the angular data series, errors on the order
of 20 cm (∼20 %) were reported for the A-axis measurement.
White Canyon rockfall dataset
The White Canyon (50.266261∘, -121.538943∘), located
in the Thompson Rail Corridor in Interior British Columbia, Canada, is an
operationally challenging rock slope (Fig. 2). Rockfall and the movement of
debris originating from the steep slopes present hazards to the safe
operation of the Canadian National (CN) rail line, which runs at the base of
the slope adjacent to the Thompson River (Bonneau and
Hutchinson, 2017; Kromer et al., 2015; van Veen et al., 2017).
The morphology of the White Canyon is highly complex; differential erosion
has formed a morphology which varies across the canyon and consists of
vertical spires and deeply incised channels. The active portion of the
canyon reaches up to 500 m in height above the railway track. The canyon
spans approximately 2.2 km between mile 093.1 and 094.6 of the CN Ashcroft
subdivision. A series of short tunnels mark the entrances to the canyon; a
tunnel can be found on either side of the canyon. A third short tunnel is
located in the middle of the canyon through a ridge which separates the
eastern and western portions of the site.
Two dominant geological units comprise the geology of the White Canyon. The
primary unit is the Lytton Gneiss. The Lytton Gneiss is a quartzofeldspathic
gneiss with amphibolite bands, containing massive quartzite, amphibolite and
gabbroic intrusions (Monger, 1985). In the most western extent
of the canyon towards the west tunnel portal is the other dominant unit,
the Mt. Lytton batholith. The Mt. Lytton batholith is a distinctly red
stained unit which is composed of granodiorite with local diorite and
gabbro. The red staining of the rock mass is thought to be a direct result
of fluids originating from the weathering of hematite in overlying
mid-Cretaceous continental clastic rocks. Two sets of dykes have intruded
the Lytton Gneiss within the White Canyon. The first dyke set consists of
tonalitic intrusions which are believed to be related to the emplacement of
the Mt. Lytton batholith (Brown, 1981). The second dyke set is a
series of dioritic intrusions which cross-cut the Lytton Gneiss and
tonalitic dykes. These dioritic intrusions are believed to be part of the
Kingsvale andesites (Brown, 1981).
The White Canyon rockfall database. The centroid of each rockfall
event is displayed as a red dot on the photogrammetry model. The blue dots
correspond to the 50 rockfall events analyzed in detail. The light green
dots correspond to the events analyzed in Fig. 14. (a) White Canyon West
results. (b) White Canyon East results.
Analysis of the TLS data collected at the White Canyon study slope between
November 2014 and December 2017 using the semi-automated rockfall
extraction process resulted in a database of 4960 rockfall events: 2566
events in WCW and 2394 events in WCE. The centroid of each of the detected
rockfalls is displayed in Fig. 12. The data plotted in this figure display
that the spatial distribution of rockfall is quite varied across the entire
canyon. Rockfalls were documented to occur in all lithologies present in the
slope.
A subset of rockfall events were identified and selected from the overall
database for further analysis. The volumes of the selected rockfalls ranged
from 1 up to 130 m3, which was the largest event recorded
during this period. As a first pass, only events larger than 1 m3 were
selected from the full database. This selection was based on a criterion in
CN's Rockfall Hazard Rating System (RHRS: Abbott et al., 1998) which focuses
on the rockfall events that are greater than 1 m3. The resulting 160
rockfall events were considered large enough that a reasonable estimate of
their shape could be made from the point cloud where the data points are
spaced at approximately 7 cm apart. A total of 110 rockfall events were removed from
the subset due to their shapes. It is probable that numerous smaller
failures have occurred from that same location (van Veen et al., 2017;
Williams et al., 2018) during the 3 to 4 months elapsed time between
scanning campaigns. The change detection, therefore, will generate the
geometry of an apparent single rockfall which is in fact likely to be the
result of several coalesced smaller events, resulting in complex,
multi-lobed shapes. With more frequent scanning intervals, the authors would have more confidence that these rockfalls are single events or multiple coalescing events. The remaining 50 events were
large enough to be of potential impact on the railway infrastructure and
were interpreted to be the result of discrete individual events, based on
fairly well constrained shape, relative to rock mass structure present at the
rockfall source location. Using the six methods outlined in Sect. 2.4, the
dimensions of the 50 blocks were measured. In addition to the automated
measurements, two sets of independent manual measurements were also made.
Sneed and Folk ternary diagrams for each of the model fits for
the 50 rockfall events that occurred in the White Canyon. The bar chart at the
bottom highlights the percentage of classes for each of the fits. BB:
bounding box; BB_ADJ: adjusted bounding box; EL:
least-squares ellipsoidal fit; FOUR: RFSHAPZ Fourier fit; GAUSS: RFSHPZ
Gaussian fit; SINES: RFSHAPZ sum of sines fit; RFCYLIN: RFCYLIN fit.
Figure 13 displays the Sneed and Folk ternary diagram for each model fit
applied to the 50 rockfall cases in the White Canyon. The bounding box
approach resulted in a distribution on the Sneed and Folk ternary diagram
that is quite scattered; however, the overall trend is towards a more cubic
shape for all of the measured rockfall objects. All possible shapes in the
Sneed and Folk classification (i.e. compact, very elongate) are represented
by rockfall object shapes assessed using this method.
The results of the other fitting methods and the manual measurements are in
stark contrast to the results of the bounding box approach. None of the
other fitting methods or manual measurements classify any of the 50
rockfall events as cubic or in the compact series (i.e. compact platy,
etc.). All the other fitting methods and manual measurements trend towards
very bladed to very elongate shape classifications and are distributed
across the lower portion of the diagram.
Overview of the two rockfall events analyzed in more detail, with
manual measurements made by five different people. (a) The spatial
location and shape of the rockfall event in White Canyon West. The red
points correspond to the front of the object while the blue points
correspond to the back of the object. (b) The spatial location and
shape of the rockfall event in White Canyon East. The red points correspond
to the front of the object while the blue points correspond to the back of
the object. (c) The results of the different fitting methods for
the rockfall event shown in panel (a). (d) The results of the different
fitting methods for the rockfall event shown in panel (b). BB: bounding box;
BB_ADJ: adjusted bounding box; EL: least-squares ellipsoidal
fit.
Two rockfall events were isolated from the 50 events to illustrate the
complexities inherent in working with real rockfall shapes, as well as the
variations in the calculated dimensions (Sneed and Folk shape
classification) using each of the methods implemented in the study (Fig. 14). A notch in one of the rockfalls (Fig. 14a) gives one of the objects more
geometric complexity than its counterpart (Fig. 14b). For the geometrically
complex object, the results of the rockfall event dimension measurements are
displayed in Fig. 6. The rockfall occurred in the western portion of the
White Canyon between June 2015 and August 2015. The rockfall fell from a
height of approximately 20 m above track level, and a number of impact points
along the rockfall trajectory were documented from the change detection
analysis. The volume of the rockfall event was estimated to be approximately
1.7 m3, and the shape, although complex, is considered to be well
enough constrained that this could be the result of a single event that
occurred during that 3-month period between scans. The other rockfall
event analyzed (Fig. 14b) occurred in the eastern portion of the White
Canyon between October 2015 and February 2016. This 1 m3 rockfall
occurred above a debris channel in the quartzofeldspathic gneiss host unit.
Based on the orientation and spacing of the discontinuities in the
surrounding rock mass, it is thought that this event is likely the result of
a single event that occurred between the scan dates.
Five different independent manual measurements of the dimensions were
conducted for both of these rockfall objects. For the first object (Fig. 14a), all of the manual measurements indicated that the rockfall object is
classified as very bladed. The adjusted bounding box, least-squares
ellipsoid, RFSHAPZ fits and the RFCYLIN approach all resulted in the
rockfall object being classified as very elongate. The bounding box
classified the rockfall object as either compact platy or platy. The
spherical fit, as always, classified the rockfall object as compact. This is
a direct result of the fact that all calculated dimensions are equal when
using the spherical fit.
In comparison, the less complex object was classified as very elongate with
all manual measurements and the adjusted bounding box, least-squares
ellipsoid, RFSHAPZ fits and the RFCYLIN approach. The bounding box approach
resulted in the object being classified as compact bladed, and the spherical
fit classified the rockfall object as compact.
In general, when comparing the shape classifications of the dataset of the
50 rockfall objects to the manual shape classifications, there were
significant differences. When the automated shape classifications were
compared to the Manual 1 and Manual 2 shape classifications, the bounding
boxing agreed with Manual 1 and Manual 2 in 3 of the 50 cases. The adjusted
bounding box, ellipsoid fit, each of the RFSHAPZ fits (Fourier, Gaussian and
sum of sines) and RFCYLIN agreed with the Manual 1/2 shape classifications
in 29/25, 25/29, 29/30, 29/29, 28/30 and 24/29 of the cases, respectively.
Interestingly, there were only 34 cases where the shape classifications of
the two manual measurements agreed with one another. Furthermore, there were
only 8 cases of the 50 analyzed where both manual measurements matched all
of the automated approaches, excluding the bounding box and spherical fits.
Neither of the two manual classification datasets aligned with any of the
spherical fits.
Discussion
The shape or form of an object can be classified by assessing aspect ratios
of major axes. However, it has been noted that there is no standardized
method used to determine axis length or if the axis measurements should be
orthogonal or not. Additionally, there is an inherent ambiguity in selecting
the geometric axes of a particle (Blott and Pye, 2008). The ambiguity arises
with increasing compactness of particles, where all axes lengths are almost
equal. In these situations, it is very difficult as well as subjective as to
how these dimensions are measured since it is difficult to manually define
an orthogonal frame. In this study, the authors have presented and compared
six different methods for assessing a rockfall object's dimensions and
resulting shape. All of the algorithms have been presented, allowing these
approaches to be replicated for future works. In addition, the authors have
created a synthetic dataset of rockfall objects that can be used to assess
new algorithms aimed at determining a rockfall object's dimensions. This
synthetic dataset represents a step forward in standardizing methodologies
for best practice in generating remotely sensed rockfall inventories.
The results of this study confirm that the method in which dimensions are
measured results in significantly different shape classifications. The
authors have demonstrated that a bounding box approach (e.g. van Veen et
al., 2017; Benjamin, 2018) can potentially bias the dimension
measurements toward a more cubic form, if the orientation of the longest
axis of the rockfall object is not parallel with one of the major Cartesian
axes. If opting for a bounding box approach to determine the object's
dimensions, the adjusted approach should be used instead.
A minimum bounding sphere was shown to be highly inappropriate for dimension
extraction in the cases analyzed in this work. The approach results in all
dimensions of the object being equal and every single object being
classified as compact or equant (i.e. A=B=C). This may be valid for
rockfall objects in some narrowly defined geological settings where equant
blocks are released by the slope rock mass. For example, further work is
required to assess the applicability of the minimum bounding sphere approach
to determine the dimensions of detached and rounded cobbles and boulders
from select horizons of postglacial river terraces (Bonneau and Hutchinson, 2018, 2019).
In comparison to the other methods, the RFCYLIN approach introduced in the
study is the most computationally demanding algorithm. The method tries to
standardize an approach to measure dimensions, where each axis is measured
orthogonally to one another after the longest dimension has been defined.
However, occlusions and edge effects in change detection analysis can result
in inaccurate distance calculations that can compound when trying to
quantify the object's dimensions. Therefore, while this method most closely
aligns with the definition of measuring three mutually orthogonal axes, it
is sensitive to data occlusions. In comparison to the RFCYLIN, the adjusted
bounding box approach guarantees that the maximum dimensions will be
measured. Therefore, this method is sensitive to outlier points, whereby the
maximum extent of complex geometry is defined using this approach. The
RFSHAPZ approach attempts to bypass these complications by utilizing curve
fitting approaches to assess the object dimensions when there is non-uniform
distribution of point density. This approach will not result in the maximum
dimensions being selected but rather representative dimensions based on the
input point cloud. In comparison to the other methods, this method is the
second most computationally demanding algorithm. The bounding box and
adjusted bounding boxes are the least computationally demanding of the
presented approaches. The adjusted bounding box is a robust approach that
would work well in all environments when the input point clouds do not
contain outlier points. When the input point clouds contain a uniform
distribution of points, at the cost of some increased computational time,
the RFCYLIN approach remains the closest to the definition of three mutually
orthogonal axes based on the input point cloud.
Automated approaches have several advantages over manually classifying the
dimensions of rockfall objects for a shape analysis. An automated approach
removes the subjectivity that could potentially be induced by manually
measuring an object. Increasing angularity or complex geometric features in
the shapes can make it increasingly difficult to define orthogonal
measurements for both manual and automated approaches. In comparison to
manual measurements, however, an automated approach is repeatable because
there is inherent subjectivity in the manual measurements depending on the
skill and experience of the person doing the work. In this work, the two
independent sets of manual measurements for the White Canyon dataset
differed depending on the complexity of the object. The classification
resulting from the manual methods agreed in only 34 of the 50 analyzed
cases.
In the interpretation of the results, it should be noted that there are hard
cut-offs between the different classes in the Sneed–Folk diagram. This can
lead to circumstances where the object plots directly on the line between
two classes yet is assigned a single shape class.
Almost all the automated fits attempt to find the maximum distance in order
to define one of the dimensions, except the RFSHAPZ approach. In comparison
to the manual measurements, the measured dimension could be reflective of
the overall dimension as opposed to the maximum. As illustrated with the
case study of the synthetic objects, with increasing angularity and
compactness, there is greater difficulty in defining the shortest axis of
the objects. Underestimation of the shortest axis will result in the objects
being classified as flatter shapes, while overestimation leads to
classification in a more compact class.
Assessing the overall dataset of the 50 rockfall objects from the White
Canyon, the shape classes trend toward very elongate to very bladed.
This is the direct result of the orientations of the joint sets and
foliation within the rock mass which promotes the generation of these shapes.
This result is in contrast to work previously done on this slope by van Veen
et al. (2017), where the rockfall shapes were mostly cubic: this is a direct
result of the bounding box approach implemented in their study.
The differences in shape classification have direct implications for rockfall
modelling. The size and shape of rock blocks are vital components when
considering and assessing potential runout trajectories. Shape has been
noted to affect the degree to which rolling can be sustained for blocks
(Kobayashi et al., 1990). Furthermore, the degree of angularity of a block
also has implications for transitions between translational and rotational
motion (Pfeiffer and Bowen, 1989).
Industry standard rockfall modelling software packages such as RockyFor3D
(Dorren, 2016) still use relatively simple geometric shapes
(rectangles, ellipsoids, spheres). Therefore, if we consider the simulation
of a cuboid or rectangular prism, where the volume can be defined as a
product of the three axes, the measured dimensions directly influence the
volume of the rockfall being simulated. The volume then defines the mass of
the object and, as a result, the moment of inertia.
Conclusions
In this study, the authors have demonstrated that the method used to measure the
dimensions of rockfall objects matters. Depending on the method used, the
object's shape may be misclassified into non-representative geometric
categories. The classification of shape has implications for the assessment
of rockfall hazards when using shape classifications as input to rockfall
modelling. Therefore, it is imperative to select a robust method that can
accurately and efficiently determine the dimensions of a rockfall object.
As illustrated with the analysis of synthetic blocks, increasing compactness
and angularity results in the most difficulties in measuring the dimensions
of a rockfall object. All automated methods and manual measurements
displayed less scatter for the rounded dataset in comparison to their
angular counterparts. Furthermore, there is a decrease in differences
between the calculated dimensions as the object becomes less compact. This
is best illustrated with the synthetic very bladed and very elongate blocks.
The dimensions of both the angular and rounded version resulted in minimal
scatter in the calculated dimensions. The differences between the lengths of
the long, intermediate and short axes for these blocks are quite apparent.
Therefore, both the manual and automated methods can converge on a dimension
length and are not subject to the uncertainties created when there are
similarities in the length of two or three of the axes.
The shapes of real rockfall objects are quite complex, as displayed with the
White Canyon dataset, where the results of the different axis measuring
methods were quite variable. Angularity, non-uniform point spacing and
occlusion in the rockfall object point clouds results in complications for
extracting dimensions with both manual and automated methods. From the
analysis of 50 rockfall events in the White Canyon, it appears that the
RFSHAPZ method most closely aligns with the manual measurements (Fig. 13).
However, a comparison between the manual measurements shows that these
measurements are different. The manual method relies on the user
consistently determining a representative length, as opposed to automated
methods which attempt to find the length of the maximum dimension.
The adjusted bounding box is the most robust approach presented in this
study; however, the results can be sensitive to outlier points, leading to a
potentially significant over estimation of the block volume, particularly
for complex objects. The bounding box approach should not be used in any
future studies. The two methods introduced in this study, RFCYLIN and
RFSHAPZ, were designed to standardize the way in which the dimensions are
measured and work around challenges such as non-uniform point density and
increasing angularity, while staying most closely aligned with the
definition of measuring three mutually orthogonal axes. These two methods
are, however, the most computationally demanding of all the presented
approaches.
Data availability
The underlying research data for this project can not be made publicly available as they are commercially sensitive and permission must be granted by our industry partners to access the data.
Author contributions
DAB coordinated the study, generated all algorithms and code,
and drafted the manuscript. DAB and DJH
co-analyzed the data. MC and PMD ran the code. ZS generated the synthetic block dataset. All co-authors reviewed and
edited the manuscript.
Competing interests
The authors declare that they have no conflict of
interest.
Acknowledgements
Past and present Queen's RGHRP team members are thanked for their help with data collection. The comments and suggestions by two anonymous reviewers are gratefully acknowledged.
Financial support
This research has been supported by the Natural Sciences and Engineering Research Council of Canada Discovery Grant (grant no. 05668), Collaborative Research and Development (CRD; grant no. 470162), and the NSERC Graduate Scholarship Program.
Review statement
This paper was edited by Andreas Günther and reviewed by two anonymous referees.
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