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Accepted Manuscript https://doi.org/10.1016/j.jsg.2019.03.001

Extraction of high-resolution structural orientations from

digital data: A Bayesian approach

Samuel T. Thiele1, Lachlan Grose1, Tiangang Cui2, Alexander R. Cruden1, Steven

Micklethwaite1

1School of Earth, Atmosphere and Environment, Monash University, Melbourne, 3800,

Australia

2School of Mathematical Sciences, Monash University, Melbourne, 3800, Australia

Correspondence to: Samuel T. Thiele (sam.thiele@monash.edu)

Keywords: Digital outcrop geology, plane-fitting, orientation measurement, structure normal

estimate, uncertainty

Abstract

Measurement of structure orientations is a key part of structural geology. Digital outcrop

methods provide a unique opportunity to collect such measurements in unprecedented

numbers, and are becoming widely applied. However, orientation estimates produced by plane

fitting can be highly uncertain, especially when observed data are approximately collinear or

the structures of interest comprise differently oriented segments. Here we present a Bayesian

approach to plane fitting that can use data extracted from digital outcrop models to constrain

the orientation of structures and their associated uncertainty. We also describe a moving-

window search algorithm that exploits this Bayesian formulation to estimate local structure

orientations for segmented structures. These methods are validated on synthetic datasets for

which both the structure orientation and associated uncertainty is known. Finally, we

implement the method in the point cloud analysis package CloudCompare and use it to estimate

the orientation and thickness of dykes exposed in cliffs on the island of La Palma (Spain). The

results highlight the potential of this method to generate structural data at unprecedented spatial

resolution, while simultaneously characterising the associated uncertainties.

Accepted Manuscript https://doi.org/10.1016/j.jsg.2019.03.001

1. Introduction

Understanding and quantifying the orientation of geological structures is a key component of

many geological studies, from resource exploration, tectonic reconstructions and three-

dimensional (3D) modelling to geotechnical analyses. One challenge is quantifying the

uncertainties associated with primary structural observations, which is often further

complicated by limited access to outcrops and the multiscale nature of geological observations.

It is now practical to create accurate, sub-cm resolution 3D digital reconstructions of outcrops,

over areas as large as 1 km2, with the advent of Structure-from-Motion (SfM) photogrammetry

and terrestrial laser scanners (McCaffrey et al., 2005; Vollgger and Cruden, 2016; Dering et

al., 2019a). These digital outcrops form the basis for an objective, reproducible and quantitative

approach for collecting basic field information and orientation measurements at unprecedented

spatial intensity.

Analyses of these high resolution datasets allow detailed structural and stratigraphic mapping

(Pringle et al., 2006; Jones et al., 2009; Nesbit et al., 2018), and have provided new insight into

geological processes such as faulting (Bemis et al., 2014; Bond et al., 2017; Corradetti et al.,

2017; Kirsch et al., 2018), folding (Schober and Exner, 2011; Vollgger and Cruden, 2016;

Menegoni et al., 2018), dyke intrusion (Healy et al., 2018; Magee et al., 2018; Dering et al.,

2019b) and vein formation (Thiele et al., 2015). Descriptions of fracture populations derived

from digital outcrops are also widely used in the engineering community for rock stability

analyses (Haneberg, 2008; Ferrero et al., 2009; Salvini et al., 2013; Bonilla-Sierra et al., 2015;

Mancini et al., 2017; Thoeni et al., 2018). Regardless of the application, the orientations of

specific structures (e.g., bedding, veins and fractures) are of interest, and are typically

represented locally as a plane described by dip and dip-direction or strike angles.

Accepted Manuscript https://doi.org/10.1016/j.jsg.2019.03.001

Structures that exert a control on outcrop morphology, such as joints, can be measured directly

where they form a sub-planar region on the outcrop surface. This is achieved by fitting planes

to regions of co-planar points (“facets”) on the reconstructed outcrop surface, which is typically

combined with some form of clustering algorithm to allow automatic detection (e.g., García-

Sellés et al., 2011; Lato and Vöge, 2012; Dewez et al., 2016). Assuming an accurate digital

outcrop model, the results of these estimates tend to be robust because the area of exposed

structure will be far from co-linear, and uncertainty around the orientation estimate, although

rarely quantified, will typically be quite small (Gallo et al., 2018).

In contrast, many structures intersect the outcrop surface at a large angle to form an intersection

trace (Fig. 1). These traces can also be used to estimate the orientation of a structure using

plane-fitting routines; however, the accuracy of the result is highly dependent on the relief of

the outcrop. As Fernández (2005) and Jones et al. (2016) show, the best-fit plane through a

trace becomes progressively less constrained as the co-linearity of the trace increases; an

infinite number of equally valid planes will fit a co-linear trace. Both Fernández (2005) and

Jones et al. (2016) suggests that traces above a conservative linearity threshold should be

excluded from orientation analyses because they will not produce robust orientation estimates.

Seers and Hodgetts (2016b) build on this work, pointing out that even perfectly co-linear traces

constrain the orientation of a structure to a single axis of rotation, and they quantify the

relationship between co-linearity and uncertainty by stochastically generating synthetic traces

and characterising the spread of the resulting orientation estimates. Their method, however,

cannot be applied in practice because it relies on the stochastic sampling of many hundreds of

intersection traces for each structure, while in reality only a single trace is observed.

Accepted Manuscript https://doi.org/10.1016/j.jsg.2019.03.001

Figure 1. Synthetic examples of intersections between a simple planar structure (a) and more

complex composite structure (b) with a randomly generated fractal outcrop surface. The

intersection trace (heavy black line) between the structure and the outcrop surface is commonly

observed and, for non-planar outcrops, can be used to constrain the geometry of the structure.

The structure normal n, represented in spherical coordinates by θ→ϕ, and outcrop normal u is

also shown, as well as the angle γ between the structure and the outcrop.

Plane-fitting algorithms applied to digital outcrop models produce the best results if they can

average the orientation of a structure over a large area, as doing so exploits often subtle

topographic variation to reduce the co-linearity of the trace. This is advantageous for structures

that are truly sub-planar at the large-scale, as it removes local variation that would add error to

compass-clinometer measurements. However, many real structures have more complex

geometries. For example, faults often contain jogs, step-overs and restraining bends at larger

scales while dykes and veins often change orientation in broken bridges or near fracture tips.

These variations in structure geometry, and errors incurred during trace digitisation, will result

in variance along the trace that is co-planar with the outcrop surface. This variance, especially

if combined with large-scale non-planarity of the structure, will bias structure orientation

Accepted Manuscript https://doi.org/10.1016/j.jsg.2019.03.001

estimates towards parallelism with the outcrop surface (e.g., Fig. 7 in Thiele et al., 2015).

Overcoming this “outcrop-normal bias” is essential for producing reasonable estimates of the

orientation of a structure.

Bayesian statistics provides a rigorous framework for integrating expert knowledge with

observed data to provide either single (maximum a postiori) or multiple estimates of

parameters of interest, by deriving and then sampling from a posterior distribution (Tarantola,

2006). Bayesian methods are commonly applied to fit multivariate linear models (planes in 3D)

to point data (Tiao and Zellner, 1964) and reconstruct surfaces from point-clouds (e.g., Torr,

2002; Erdogan et al., 2012). Here we develop a Bayesian method for the specific purpose of

estimating a geological structure’s orientation from its intersection with the surface in a way

that: (1) probabilistically describes the increase in uncertainty as traces become co-linear; (2)

accounts for outcrop-normal bias, and; (3) provides orientation estimates at each point, and so

can be applied to structures that are non-planar at the large scale. We validate this approach,

which we term “structure-normal estimation”, by applying it to a series of synthetic datasets

and present a case study in which we estimate the orientation and thickness of dykes exposed

in cliffs on the island of La Palma (Canary Islands, Spain).

2. Bayesian plane fitting

The orientation of a planar structure can be described in spherical coordinates by the

geographical bearing or trend ϕ and inclination or plunge θ of its normal vector n, also known

as the pole to the plane and written here as θ→ϕ (Fig. 1a). It is also useful to express n in

Cartesian coordinates by computing its direction cosines:

Eq. 1.

Accepted Manuscript https://doi.org/10.1016/j.jsg.2019.03.001

A set of N position vectors (u0, u1 … uN) can readily be sampled from the intersection trace

using, for example, digital outcrop methods (e.g., Seers and Hodgetts, 2016a; Thiele et al.,

2017; Guo et al., 2018), providing the raw data with which we wish to constrain ϕ and θ. The

spatial distribution of these points is summarised by their covariance, or specifically for our

purposes, by constructing an unscaled covariance matrix X, hereafter referred to as a scale

matrix (Eq. 2).

Eq. 2.

The average normal vector to the outcrop in which a trace is observed, o, is also easily measured

using field or digital outcrop methods. It is well established that traces formed by structures

with orientations similar to the outcrop (n ≈ o) are less likely to be observed (Terzaghi, 1965),

and hence we can define a priori that an observed trace is more likely to originate from a

structure intersecting the outcrop at a higher angle.

Bayes theorem (Eq. 3) provides a robust framework for combining this prior knowledge, or

P(ϕ, θ; o), with the observed data X by using a likelihood function P(X | ϕ, θ) to derive a

posterior distribution P(ϕ, θ | X) that constrains the orientation of the planar structure:

Eq. 3.

In practice, Equation 3 can be simplified because P(X) is constant for an observed trace,

allowing the calculation of an unnormalised posterior distribution (Eq. 4). Note that numerical

techniques such as the Metropolis-Hastings Monte-Carlo Markov chain method (Metropolis et

al., 1953; Hastings, 1970) can be used to sample directly from the unnormalised posterior

distribution.

Accepted Manuscript https://doi.org/10.1016/j.jsg.2019.03.001

Eq. 4.

In the following sections we derive the prior distribution P(ϕ, θ; o) based on the commonly

used Terzaghi correction factor (Terzaghi, 1965), and likelihood function P(X | ϕ, θ) that uses

the Wishart distribution (Wishart, 1928) to quantify how well an observed scale matrix X fits

structures of different orientations.

2.1. Terzaghi Prior

The Terzaghi correction provides a scaling factor that accounts for biases caused by the low

probability of observing structures with a similar orientation to the outcrop or scan-line on

which data are collected (Terzaghi, 1965). We adapt this correction factor and use it to describe

the prior-probability of observing a structure given its angle of intersection with the outcrop in

which it is exposed. Assuming an arbitrary set of planar structures with true-spacing S0, and a

single planar outcrop surface, the projected spacing (S) of intersection traces observed on the

outcrop surfaces is a function of the acute angle γ between the structure and outcrop normals n

and o (Eq. 5, 6):

Eq. 5

Eq. 6.

Structures from this set will be observed with a frequency f in traverses along the outcrop that

are perpendicular to the projected structure orientation, where:

Eq. 7.

Therefore, for a structure belonging to a hypothetical set of similarly oriented structures, the a

priori probability of observing a structure trace intersecting a sub-planar outcrop of known

Accepted Manuscript https://doi.org/10.1016/j.jsg.2019.03.001

orientation and limited extent will be proportional to sin(γ). Hence, noting that n can be

explicitly calculated from ϕ and θ as per Equation 1, the prior probability distribution P(ϕ, θ;

o) becomes:

Eq. 8.

2.2. Wishart likelihood function

The Wishart distribution (Eq. 9; Wishart, 1928) expresses the probability of observing scale

matrix X from d + 1 independent observations drawn from a multivariate normal distribution

with an underlying population covariance matrix V, expressed here for 3D data as:

Eq. 9,

where Γ3 is the 3D gamma function, and tr and det are the trace and determinant functions

respectively. An arbitrary postulated covariance matrix V can be decomposed into a matrix of

eigenvectors ε1, ε2 and ε3 and eigenvalues λ1, λ2 and λ3 such that:

Eq. 10.

If we assume fixed eigenvalues, estimated from the observed covariance of the structures

intersection-trace, then V can be derived from three angles representing the rotation of the

eigenvectors: ϕ, θ and a third angle, α, which we introduce below. The third eigenvector (ε3)

of a 3D covariance matrix gives the direction of lowest variance, and hence the normal to the

best-fit plane, and so will be equal to the direction cosines of a structure with orientation θ→ϕ:

Accepted Manuscript https://doi.org/10.1016/j.jsg.2019.03.001

Eq. 11.

By defining the third rotation, α, as the angle between ε2 and the projection of ε3 onto the

horizontal plane (assuming ε3 is not vertical), we can describe ε2 as:

rotated by angle α around ε3, which simplifies to:

rotated by angle α around ε3.

This can be expanded using Rodrigues’ rotation formula (Rodrigues, 1840; Koks, 2006) to

give:

Eq. 12.

Finally, ε1 can be calculated from ε2 and ε3 using the vector product:

Eq. 13.

Hence, a postulated population covariance matrix Vϕ,θ,α can be constructed for any

hypothesised structure orientation (ϕ, θ) and additional rotational term α using Equation 10.

Substituting Vϕ,θ,α into the Wishart distribution, it follows that the likelihood an observed trace

Accepted Manuscript https://doi.org/10.1016/j.jsg.2019.03.001

and associated scale matrix X result from a structure oriented at θ→ ϕ (and with an eigensystem

rotated by angle α to give a proposed population covariance matrix Vϕ,θ,α) is:

Eq. 14.

Using the Terzaghi prior described in Section 2.1, and a uniform prior P(α) for α, the

unnormalised posterior distribution P(ϕ, θ, α | X ) becomes:

Eq. 15.

If necessary, α can be marginalised by integration to give a posterior distribution for ϕ and θ

only:

Eq. 16.

Examples of the Terzaghi prior, Wishart likelihood and resulting posterior distributions are

shown in Fig. 2, visualised on equal area, lower hemisphere stereographic projections. These

highlight the anisotropic distribution of uncertainty that results from the often high degree of

co-linearity in observed structure trace data (Seers and Hodgetts, 2016b).

It is important to note at this point that the confidence given to the observed scale matrix X,

and hence the tightness of the resulting posterior (Fig. 3), depends on the number of

independent observations used to estimate it (i.e., the degrees of freedom d in Eq. 14). In real-

world applications, and the synthetic examples we present in Section 5, points defining the

trace of a structure will show local autocorrelation and systematically distributed errors (i.e.,

not be statistically independent), meaning that d needs to be chosen based on the quality of the

data.

Accepted Manuscript https://doi.org/10.1016/j.jsg.2019.03.001

Figure 2. Lower hemisphere stereographic projections of the unnormalised prior (a), likelihood

(b) and posterior (c) distributions for a poorly constrained (largely co-linear) structural trace

observed on an outcrop with a normal vector (o) oriented 25˚→330˚. Areas of high probability

density are shown in red.

Figure 3. Wishart distributions showing the reduction in variance as the degrees of freedom d

increases.

3. Structure-normal estimation

The Bayesian plane-fitting approach assumes that structure traces result from truly planar

structures. In reality this is often not the case as structures, especially dykes, faults and veins,

Accepted Manuscript https://doi.org/10.1016/j.jsg.2019.03.001

often comprise differently oriented segments (Fig. 4a). In such instances, a single orientation

estimate is inappropriate. Instead, we treat the problem as a surface-normal estimation problem

where we create “structure normal estimates” (SNEs) at every point along the trace.

The creation of SNEs is achieved by the application of a moving-window search algorithm,

which, for windows defined by start point ui and end point uj in the ordered list of points (u0,

u1 … uN):

1. Calculates and stores the scale (X) matrices of points (ui, ui+1, … uj) inside each window

(Fig. 4b).

2. Evaluates the unnormalised posterior probability density P(ϕ, θ, α | X) for each window

at the maximum likelihood estimate for ϕ, θ and α. These are easily calculated from the

eigenvectors of X, and will in almost all cases represent the maxima of P(ϕ, θ, α | X),

the only exception being where the plane falls almost sub-parallel to the outcrop

orientation and the posterior becomes bimodal. Once computed, P(ϕ, θ, α | X) is stored

in a symmetric two-dimensional (2D) matrix such that Mij contains the posterior density

of a window starting at point i and ending at point j (Fig. 4c). Higher values of Mij result

from windows with tighter posterior distributions, and hence represent better

constrained orientation estimates.

3. Finds the window with the largest unnormalised posterior probability density at each

point (u0, u1 … uN), and hence the best-constrained orientation estimate at that location

(i.e. finds the maximum value in Mij where i < p < j for each point up; Fig. 4d).

4. Draws orientation samples (ϕ, θ) from the posterior distribution defined by the best-

constrained window for each point, as identified in the previous step, using the

Metropolis-Hastings Monte-Carlo Markov-Chain method.

Accepted Manuscript https://doi.org/10.1016/j.jsg.2019.03.001

Figure 4. Example of a dyke comprising sub-planar segments with different orientations

(a). A moving window is applied (b) to create a matrix where each element Mij contains the

unscaled posterior probability density of the best-fit plane through data within a window

starting at i and ending at j (c). This matrix is searched to find the most constrained best-fit

plane for each point p (d) and hence derive the structure normal estimates (SNEs).

The results of this are distinct orientation estimates for every point, allowing different sections

of the trace to have different orientations. The size of acceptable search windows is also

constrained to define the scale at which the orientation estimates are expected to be averaged

over and avoid spurious small or large search windows.

Accepted Manuscript https://doi.org/10.1016/j.jsg.2019.03.001

4. Implementation.

The methods described above have been implemented in the Compass plugin (Thiele et al.,

2017), which comes bundled with the standard version of CloudCompare

(www.cloudcompare.org). Source code for the entire program, including the Compass plugin,

is available at https://github.com/CloudCompare. A python library that implements the SNE

algorithm is also available at https://github.com/samthiele/pycompass. This python library also

includes a variety of utility functions for loading and visualising xml files exported from the

Compass plugin. Code for the following synthetic tests of the SNE method can also be found

as part of this library (https://github.com/samthiele/pycompass/tree/master/examples).

5. Results and validation

5.1. Synthetic examples

In the following section, we generate synthetic structural traces by intersecting a fractal surface

representing topography with a second, differently oriented fractal surface representing a sub-

planar structure. Monte Carlo sampling of the best-fit plane through these intersection traces

then provides an independent estimate of uncertainty, as described by Seers and Hodgetts

(2016b), that can be compared with results from our Bayesian model.

First, we generate a two-dimensional elevation grid representing the surface of a structure

dipping 55˚ to the south and add a small amount (1% of the length) of fractal variation using

the spectral synthesis method (Fisher et al., 2012). This synthetic structure is then intersected

with 1000 randomly generated, sub-horizontal outcrop topographies (also generated with the

spectral synthesis method) to create a set of synthetic intersection traces (Fig. 1a). The 3rd

eigenvector of each of these traces are then computed, resulting in a sample of 1000 plausible

orientation estimates as per Seers and Hodgetts (2016b). The amplitude of topographic relief

Accepted Manuscript https://doi.org/10.1016/j.jsg.2019.03.001

is varied in different experimental groups to explore the associated change in orientation

uncertainty.

Our Bayesian method is then applied to the last-generated (1000th) structural trace, and the

resulting posterior distribution explicitly evaluated for comparison with the Monte-Carlo

sampled population. As our synthetic trace has a long axis parallel to the strike of the structure,

orientation uncertainty will almost entirely cause variation of the estimated dip, and so for easy

comparison we only compare variations in dip. These results suggest a close fit (overlaps of

73-88%; Fig. 5) between the “true-uncertainty” sampled using the Monte Carlo method and

our Bayesian posterior distribution.

Figure 5. Kernel density estimates of structure orientations observed by stochastically

generating random topographies (blue line) and the posterior distribution generated by applying

our Bayesian method to a single structural trace (red line). Traces were generated by

intersecting a sub-planar fractal surface with fractal topographies with amplitudes of 4% (a),

8% (b) and 20% (c) of the trace length, and a degrees of freedom d = 10 was used when

evaluating the posterior distribution. The dashed black line shows the actual orientation of the

structure, and the green area the overlap between the two kernel density estimates.

Accepted Manuscript https://doi.org/10.1016/j.jsg.2019.03.001

We also tested a more complex example where the synthetic structure changes in orientation

along strike (Fig. 1b). Similar to the example above, the best-fit plane to each differently

oriented subsection of the Monte-Carlo sampled traces were computed to give an independent

estimate of orientation uncertainty. Unlike the previous example, we do not explicitly evaluate

the posterior, as doing so on every point in the trace (each of which has a potentially different

orientation estimate) is computationally expensive. Instead, a Metropolis-Hastings sampler

was used to generate 100 samples from the posterior of each point. These were then aggregated

to provide a sample describing the plausible variation of structure orientations that could be

derived from the trace. The sampler was initialised at the maximum-likelihood position

because, in our case, this generally coincides with the maxima of the posterior distribution,

avoiding the need for an initialisation period (burn-in; Kass et al., 1998). Proposed samples

were generated by sampling perturbations from a normal distribution with a standard deviation

of 0.075 radians and accepted or rejected using the Metropolis-Hastings method.

As with the previous example, the Bayesian and Monte-Carlo uncertainty estimates are similar,

although the Bayesian results clearly show a larger spread, suggesting higher uncertainty (Fig.

6). This is reasonable given the Bayesian results are estimated from a single trace, and suggest

that the method is capable of identifying subdomains within complex, multi-planar structures

and quantifying the range of credible orientations that could be attributed to them. As in the

previous example, a d ≈ 10 degrees of freedom was found to produce the best match.

5.2. Application to the Taburiente dyke swarm

We now present an application to real field data from the island of La Palma (Canary Islands,

Spain). Dykes within the Taburiente volcano are spectacularly exposed here along a series of

cliffs that resulted from erosion of a collapse-scarp formed by catastrophic volcano edifice

failure at ca. 550 Ma (Carracedo, 1994). The cliffs are accessible at a location known locally

Accepted Manuscript https://doi.org/10.1016/j.jsg.2019.03.001

as Hoyo Verde (Fig. 7a), where a swarm of ~5–50 m spaced, ~0.5–4 m thick mafic dykes and

sills can be observed (Fig. 7b). These intrusions exhibit geometries ranging from tabular dykes

to complex, anastomosing dyke networks and small saucer-shaped sills. They are also exposed

across much of their section as they intersect the cliff face, which dips broadly at ~50-60° but

locally contains significant topographic variation, allowing the estimation of the dyke

orientations from their intersection traces.

An Unmanned Aerial Vehicle (UAV) survey was completed using a DJI Phantom 4 Pro and

integrated camera (20-megapixel CMOS sensor). Flight lines were manually controlled and

flown horizontally at a distance of ~35 m from the cliff. The images were acquired frequently

enough to maintain ~80% overlap both laterally and vertically. The camera was alternated

between pointing forwards and angled at ~30˚ downwards to (1) ensure sufficient vertical

overlap between images and (2) avoid the systematic distortions that can occur when camera

orientation is fixed (James and Robson, 2014). This resulted in an approximate ground-

sampling distance of ~1 cm/pixel over the surveyed area. Significant topography resulting in

poor GPS reception precluded the deployment of accurately surveyed ground control points.

We also measured dyke orientations and thicknesses along a traverse bisecting the model using

a compass and tape measure, providing a reference with which to validate orientation estimates.

A Structure-from-Motion Multi-View-Stereo workflow, as implemented in Agisoft Photoscan

1.4.3, was then used to reconstruct the geometry of the outcrop in 3D. This reconstruction

comprised ~100 million points covering an area of ~38,000 m2 (~2600 points/m2) and was

georeferenced using approximate camera locations stored by the UAV using its on-board GPS.

This georeferencing was then refined by comparison with the publicly available 2014 Spanish

LiDAR survey series (which has a resolution of ~0.75 points/m2 at this location) and optimised

using the iterative closest point algorithm in CloudCompare.

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Fig. 6. Lower hemisphere stereographic projection of orientation estimates of normals to a

planar structure consisting of two differently oriented planar segments. As with Figure 5,

Monte Carlo sampled orientation estimates (a) indicate the amount of uncertainty present in

the orientation estimates for different degrees of topographic relief. Orientation estimates

produced from a single structural trace by Monte Carlo Markov Chain sampling the posterior

distributions of each structure-normal (b) estimated using the moving window method

described in section 4 match these closely, although the moving-window method results in a

noticeably wider spread of SNEs.

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Figure 7. Dyke swarms from Hoyo Verde on the island of La Palma, Spain (a) recorded using

a UAV to capture 203 images in a regular survey pattern, an example of which is shown in (b).

Dykes mapped using a digital outcrop model constructed from these images (c) have been used

to test the structure normal estimation method and measure dyke thickness (d), visualised here

using the colour table. Field measurements of dyke orientation and thickness have been

included on (c) for reference.

Accepted Manuscript https://doi.org/10.1016/j.jsg.2019.03.001

The upper and lower contacts of dykes visible in the model were then extracted using the semi-

automatic “trace tool” of the Compass plugin in CloudCompare (Thiele et al., 2017), which

uses the colour (RGB) gradient and a least-cost-path algorithm to “follow” contacts between

user-defined waypoints (Fig. 7c). We then used the CloudCompare implementation of the

method presented in this study to estimate the orientation and thickness of the dykes at each

point along these traces.

The resulting SNEs were then carefully censored using the segment tool in CloudCompare.

This was necessary as the method described in Section 3 assumes a locally planar structure,

while the dykes sometimes have a curved otherwise non-planar geometry. This complexity can

result in well-constrained but incorrect best-fit planes, which are generally sub-parallel to the

outcrop surface. The large angle between the outcrop surface and incorrect SNEs, and typically

anomalous orientations compared to other SNEs from the same structure meant that, in most

cases, spurious results could be manually identified and removed without significant

subjectivity.

After vetting, SNEs remained for 68% of the original ~5.2 km of digitised dyke margins, spaced

every ~7 cm (Fig. 7d). The orientation of these remaining SNEs (Fig. 8a) hint at several

differently oriented dyke sets, one shallow dipping and striking roughly west, and two (possibly

three) steeper sets striking north and north-north-west.

In many studies, the true-thicknesses of sedimentary units, dykes or other structures is of

interest (e.g., Krumbholz et al. 2014; Nesbit et al., 2018; Dering et al., 2019a). True-thickness,

as opposed to the apparent thickness observed on the outcrop, requires knowledge of the

structures orientation (see Fig. 9 in Dering et al., 2019a), providing a relevant application of

the SNE algorithm. We use SNEs on opposite sides of dykes extracted from the Hoyo Verde

model to calculate the true thickness of each intrusion and the associated uncertainty (Fig. 8b).

Accepted Manuscript https://doi.org/10.1016/j.jsg.2019.03.001

These measurements suggest that the dykes have a median true thickness of 50 cm and

generally follow a lognormal distribution, similar to thickness data for other dykes in Caldera

Taburiente (Krumbholz et al., 2014).

Hand-measured dyke orientation and thickness measurements were also acquired at 26 sites

and subsequently located on the digital outcrop model using recorded GPS location and field-

notes. Of these, 15 corresponded to dyke sections with SNEs and so have been directly

compared (Fig. 9). These comparisons show broadly consistent results, but also highlight some

differences in orientation (>30°) and thickness (>20 cm). Of the 15 comparisons (Fig. 9b), 10

are within 15°, which we suggest is reasonable given some of the dykes contain a significant

amount of magnetite, meaning the compass measurements are probably only accurate to ± 15°.

Figure 8. Equal-area lower hemisphere stereographic projection (a) overlying hand-measured

orientations (red and black squares) on the contoured log-density of SNE samples and

histogram (b) of associated thickness estimates. Orientation measurements with corresponding

SNEs (and hence included in Fig. 9) are shown in red, while the black triangle shows the

average orientation of the normal to the outcrop. Several statistical distributions fitted to the

thickness data are also provided for reference.

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Figure 9. Differences between the SNEs and directly measured field data. Angular differences

are shown both in aggregate (a) and at an individual measurement level (b). Differences

between thicknesses estimated using the SNEs and corresponding hand-measured values are

shown (c), excluding multiple-intrusions.

6. Discussion

The Hoyo Verde case study highlights the ability of our method to provide quantitative

measurements of structure orientations and thicknesses from intersection-traces captured in a

digital outcrop model. The unprecedented number of these orientation and thickness

measurements, and associated characterisation of orientation uncertainty, provide a far more

robust description of dyke geometry than the spot orientations or thickness measurements

traditionally employed in the field. Most measurements extracted from the Hoyo Verde model

are also within entirely inaccessible terrain, illustrating the potential of digital outcrop methods

for analyses of inaccessible areas and features.

The degrees of freedom used in the Wishart distribution (d) change the confidence placed in

the observed structural trace, and hence the uncertainty surrounding the SNEs. It is likely that

an analysis of autocorrelation within the structural traces could provide guidance as to a

Accepted Manuscript https://doi.org/10.1016/j.jsg.2019.03.001

reasonable value for d, although this is beyond the scope of this contribution. In the absence of

further information, and in light of the synthetic tests presented in Section 4, we suggest using

a lower value (~10). As planes become well constrained, the importance of d decreases,

meaning the choice of value will mostly affect SNEs with high uncertainty.

Despite both careful vetting and the use of a low value for d, direct comparisons between

structures measured in the field and corresponding SNEs revealed some discrepancies of >20-

40°. Visualisations of the digital outcrop model suggests that the SNEs are a more reasonable

estimation of the dyke orientation than the compass measurements in these cases. We suggest

that it is likely they occur because point-orientation estimates acquired with a compass are not

representative of large-scale orientation, especially because dyke contacts at Hoyo Verde are

irregular at the 1-5 m scale. This highlights the importance of averaging structure orientation

over large areas, at least for these dykes, which is difficult to achieve using traditional methods.

The ability outlined here to represent orientation estimates as posterior probability density

functions (rather than individual measurements with unknown uncertainty) also provide

opportunities for probabilistic discrimination between structure sets and therefore estimation

of the probability that an observed structure belongs to a predefined set. As such, the method

may allow a distinction between intra-set variance and measurement uncertainty, improving

our ability to characterise structure sets during, for example, fracture-network modelling or

geotechnical analysis. The development of such methods would further improve the rigour of

classic and widely applied structural geology techniques.

Finally, in terms of future work, our case study demonstrates the importance of careful vetting

to remove misleading orientation estimates caused by local geometric complexities. It is

possible that this vetting process could be automated or semi-automated by identifying and

removing statistical outliers and, in the case of structures such as dykes or sedimentary units,

Accepted Manuscript https://doi.org/10.1016/j.jsg.2019.03.001

checking the consistency of independent SNEs from the upper and lower surfaces. Regardless,

we recommend that all SNEs are carefully checked before use, especially since this is

straightforward in an interactive 3D environment such as CloudCompare.

Acknowledgements

The authors would like to gratefully acknowledge the staff at Parque Nacional Caldera de

Taburiente for their generous support and hospitality during collection of the field data

presented in this study. ST was supported by a Westpac Future Leaders Scholarship and

Australian Postgraduate Award. LG was supported by ARC grant LP170100985. Finally, we

would like to acknowledge Clare Bond and Florian Wellmann for their thoughtful and

constructive reviews.

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