Three-dimensional (3-D) geological structural modeling aims to determine geological information in a 3-D space using structural data (foliations and interfaces) and topological rules as inputs. This is necessary in any project in which the properties of the subsurface matters; they express our understanding of geometries in depth. For that reason, 3-D geological models have a wide range of practical applications including but not restricted to civil engineering, the oil and gas industry, the mining industry, and water management. These models, however, are fraught with uncertainties originating from the inherent flaws of the modeling engines (working hypotheses, interpolator's parameterization) and the inherent lack of knowledge in areas where there are no observations combined with input uncertainty (observational, conceptual and technical errors). Because 3-D geological models are often used for impactful decision-making it is critical that all 3-D geological models provide accurate estimates of uncertainty. This paper's focus is set on the effect of structural input data measurement uncertainty propagation in implicit 3-D geological modeling. This aim is achieved using Monte Carlo simulation for uncertainty estimation (MCUE), a stochastic method which samples from predefined disturbance probability distributions that represent the uncertainty of the original input data set. MCUE is used to produce hundreds to thousands of altered unique data sets. The altered data sets are used as inputs to produce a range of plausible 3-D models. The plausible models are then combined into a single probabilistic model as a means to propagate uncertainty from the input data to the final model. In this paper, several improved methods for MCUE are proposed. The methods pertain to distribution selection for input uncertainty, sample analysis and statistical consistency of the sampled distribution. Pole vector sampling is proposed as a more rigorous alternative than dip vector sampling for planar features and the use of a Bayesian approach to disturbance distribution parameterization is suggested. The influence of incorrect disturbance distributions is discussed and propositions are made and evaluated on synthetic and realistic cases to address the sighted issues. The distribution of the errors of the observed data (i.e., scedasticity) is shown to affect the quality of prior distributions for MCUE. Results demonstrate that the proposed workflows improve the reliability of uncertainty estimation and diminish the occurrence of artifacts.

Three-dimensional (3-D) geological models are important tools for decision-making in geoscience as they represent the current state of our knowledge regarding the architecture of the subsurface. As such they are used in various domains of application such as mining (Cammack, 2016; Dominy et al., 2002), oil and gas (Nordahl and Ringrose, 2008), infrastructure engineering (Aldiss et al., 2012), water supply management (Prada et al., 2016), geothermal power plants (Moeck, 2014), waste disposal (Ennis-King and Paterson, 2002), natural hazard management (Delgado Marchal et al., 2015), hydrogeology (Jairo, 2013) and archaeology (Vos et al., 2015). By definition, all models contain uncertainty, being simplifications of the natural world (Bardossy and Fodor, 2001) linked to errors about their inputs (data and working hypotheses), processing (model building) and output formatting (discretization, simplification). Reason dictates that these models should incorporate an estimate of their uncertainty as an aid to risk-aware decision-making.

Monte Carlo simulation for uncertainty propagation (MCUE) has been a widely
used uncertainty propagation method in implicit 3-D geological modeling
during the last decade (Wellmann and Regenauer-Lieb, 2012; Lindsay et al.,
2012; Jessell et al., 2014a; de la Varga and Wellmann, 2016). A similar
approach was introduced to geoscience with the generalized likelihood
uncertainty estimation (GLUE; Beven and Binley, 1992), which is a
non-predictive (Camacho et al., 2015) implementation of Bayesian Monte Carlo
(BMC). MCUE (Fig. 1) simulates input data uncertainty propagation by
producing many plausible models through perturbation of the initial input
data; the output models are then merged and/or compared to estimate
uncertainty. This can be achieved by replacing each original data input with
a probability distribution function (PDF) thought to best represent its
uncertainty called a disturbance distribution. Essentially, a disturbance
distribution quantifies the degree of confidence that one has in the input
data used for the modeling such as the location of a stratigraphic horizon
or the dip of a fault. In the context of MCUE, uncertainty in the input data
mainly arises from a number of sources of uncertainty, including but not
restricted to device basic measurement error, operator error, local
variability, simplification radius, miscalibration, rounding errors,
(re)projection issues and external perturbations. In the case of a standard
geological compass used to acquire a foliation on an outcrop,

device basic measurement error refers to error in lab and under perfect conditions (this information is typically provided by the manufacturer).

operator error refers to human-related issues that affect the process of the measurement such as trembling or misinterpreting features (mistaking joints or crenulation for horizons for example).

local variability refers to the difficulty of picking up the trend of the stratigraphy appropriately because of significant variability at the scale of the outcrop (usually due to cleavage or crenulation).

simplification radius refers to the uncertainty that is introduced when several measurements made in the same area are combined into a single one.

external perturbations refer to artificial or natural phenomena that have a detrimental effect on precision and accuracy such as holding high-magnetic-mass items close to the compass (smartphone, car, metallic structures) when making a measurement or the magnetization of the outcrop itself.

Monte Carlo uncertainty propagation procedure workflow.

All these sources of uncertainty may be abstracted to individual random variables, which are all added to form a more general uncertainty variable that disturbance distributions are expected to represent. The disturbance distributions are then sampled to generate many plausible alternate models in a process called perturbation. Plausible models form a suite of 3-D geological models that are consistent with the original data set. That is, the degree of uncertainty associated with the original data set allows these models to be plausible. In layman's terms, the perturbation step is designed to simulate the effect of uncertainty by testing “what-if” scenarios. The variability in the plausible model suite is then used as a proxy for model uncertainty. Several metrics have been used to express the model uncertainty in MCUE, including information entropy (Shannon, 1948; Wellmann, 2013; Wellmann and Regenauer-Lieb, 2012), stratigraphic variability (Lindsay et al., 2012) and kriging error. The case for reliable uncertainty estimation in 3-D geological modeling has been made repeatedly and this paper aims to further improve several points of MCUE methods at the preprocessing steps (Fig. 1). More specifically, we aim to improve (i) the selection of the PDFs used to represent uncertainties related to the original data inputs and (ii) the parameterization of said PDFs. Section 2 reviews the fundamentals of MCUE methods while Sect. 3 addresses PDF selection and parameterization. Lastly, Sect. 4 expands further into the details of disturbance distribution sampling.

Recently developed MCUE-based techniques for uncertainty estimation in 3-D geological modeling require the user to define the disturbance distribution for each input datum, based on some form of prior knowledge. That is necessary because MCUE is a one-step analysis as opposed to a sequential one: all inputs are perturbed once and simultaneously to generate one of the possible models that will be merged or compared with the others. MCUE is vulnerable to erroneous assumptions about the disturbance distribution in terms of structure (what is the optimal type of disturbance distribution) and magnitude (the dispersion parameters) of the uncertainty of the input data. However, it is possible to post-process the results of an MCUE simulation to compare them to other forms of prior knowledge and update accordingly (Wellmann et al., 2014a).

The MCUE approach is usually applied to geometric modeling engines (Wellmann and Regenauer-Lieb, 2012; Lindsay et al., 2013; Jessell et al., 2010, 2014a), although it can be applied to dynamic or kinematic modeling engines (Wang et al., 2016; Wellmann et al., 2016). This choice is motivated by critical differences between the three approaches, at both the conceptual and practical level (Aug, 2004). More specifically, explicit geometric engines require full expert knowledge while implicit ones are based on observed field data, variographic analysis and topological constraints (Jessell et al., 2014a). Geometric modeling engines interpolate features from sparse structural data and topological assumptions (Aug et al., 2005; Jessell et al., 2014a); they require prior knowledge of topology and are computationally affordable (Lajaunie et al., 1997; Calcagno et al., 2008). Dynamic modeling engines require knowledge of initial geometry, physical properties and boundary conditions; the modeling process is computationally expensive. Kinematic modeling engines require knowledge of initial geometry and kinematic history (Jessell, 1981); the modeling process is computationally inexpensive. The implicit geometric approach is preferred for MCUE because knowledge of initial conditions is nearly impossible to achieve, and perfect knowledge of current conditions defeats the purpose of estimating any uncertainty.

Implicit geometric modeling engines use mainly three types of inputs: interfaces (3-D points), foliations (3-D vectors) and topological relationships between geological units and faults (stratigraphic column and fault age relationships). Drill holes and other structural inputs such as fold axes and fold axial planes can also be used (Maxelon and Mancktelow, 2005). Each data input is assigned to a geological unit and the model is then built according to predefined topological rules. The implicit geometric 3-D modeling package GeoModeller distributed by Intrepid Geophysics was used as a test platform for this study. The use of this specific software is motivated by its open use of co-kriging (Appendix C), which is a robust (Matheron, 1970; Isaaks and Srivastava, 1989; Lajaunie, 1990) geostatistical interpolator to generate the models (Calcagno et al., 2008; FitzGerald et al., 2009). In addition, GeoModeller allows uncertainty to be safely propagated provided that the variogram is correct (Chilès et al., 2004; Aug, 2004) as the co-kriging interpolator then quantifies the its intrinsic uncertainty. Nevertheless, MCUE is not inherently limited by the choice of the interpolator and, therefore, may be used with any implicit modeling engine. In the next section, a series of improvements are proposed to address the disturbance distribution problem.

Often, the disturbance distribution used to estimate input uncertainty is the same (same type and same parameterization) for all observations of the same nature (Wellmann et al., 2010; Wellmann and Regenauer-Lieb, 2012; Lindsay et al., 2012, 2013). Disturbance distribution parameters are defined arbitrarily (Lindsay et al., 2012; Wellmann and Regenauer-Lieb, 2012) in most cases. Additionally, uniform distributions have been regularly used as disturbance distribution and expressed as a plus minus range over the location of interfaces (Wellmann et al., 2010; Wellmann, 2013) or the dip and dip direction (Lindsay et al., 2012, 2013; Jessell et al., 2014a). Here, propositions are made about the type of disturbance distributions that should be used for MCUE, how to parameterize them and associated possible pitfalls.

The structural data collected to build the model are impacted by many random sources of uncertainty (Fig. 1) such as measurement, sampling and observation errors (Bardossy and Fodor, 2001; Nearing et al., 2016). Additionally, the uncertainty tied to each measurement is considered to be independent of the others. However, that is not to say that there is no dependence over the measured values themselves. For example, dip measurements along a fault line are expected to be spatially correlated though each measurement is an independent trial in terms of its measurement error. Consequently, MCUE may sample from disturbance distributions independently from one another. Under these conditions, the central limit theorem (CLT) holds true for these data (Sivia and Skilling, 2006; Gnedenko and Kolmogorov, 1954) if the variance of each source of uncertainty is always defined. Uncertainty would then be better represented by disturbance distributions that are consistent with the CLT, namely the normal distribution for locations (Cartesian scalar data) and the von Mises–Fisher (vMF) distribution for orientations (spherical vector data; Davis, 2003). However, MCUE does not a priori forbid the use of any kind of distribution. The normal distribution is the canonical CLT distribution (i.e., the distribution towards which the sum of random variables tends) defined as

The vMF distribution (Fig. 2) is the CLT distribution for spherical data; it
is the hyperspherical counterpart to the normal distribution (Fisher et al.,
1987) and is used under the same general assumptions for unit vectors on the

Here

Von Mises–Fisher probability distribution function on S1 (

Regardless of which type of disturbance distribution is chosen, it is inappropriate to use the same distribution with the exact same parameters for each measurement in many cases, including but not restricted to cases in which some data inputs are actually a statistic – such as the mean – that is derived from a sample instead of an actual individual occurrence (Moffat, 1988); cases in which inputs (at the same location) are samples themselves (Kolmogorov, 1950); and cases in which the magnitude of the uncertainty of measurements may be impacted by the value of the measurement itself (Moffat, 1982). Statistics derived from samples (e.g., mean, median) or the actual sample are expected to lead to less dispersed disturbance distributions compared to single observations (Patel and Read, 1996; Bewoor and Kulkarni, 2009; Bucher, 2012; Sivia and Skilling, 2006; Davis, 2003). Therefore, making multiple measurements at each location of interest is recommended and field operators should not group, dismiss or otherwise alter these kinds of data. For example, a 15 m long limestone outcrop is expected to yield numerous structural measurements that should be fed to MCUE so that disturbance distributions are parameterized more precisely. However, because structural data inputs are sparse and often scarce, a Bayesian approach to disturbance distribution parameterization is proposed. More specifically, a prior disturbance distribution is updated by measurements over a CLT compatible likelihood function to generate a predictive posterior disturbance distribution. The following demonstration applies to both the normal and the vMF distributions.

The uncertainty about an input structural datum (location or orientation)
can be described by a distribution

When a single measurement is made at a specific location, the sample size is
1

Effect of bias over predictive posterior and posterior distributions.

For a normal distribution, the posterior predictive distribution

In Eq. (12)

In Eq. (12)

Scedasticity is defined as the distribution of the error about measured or estimated elements of a random variable of interest (Levenbach, 1973). It expresses the relationship between the measured values and their uncertainty. In the case in which uncertainty is constant across the variable space the variable is homoscedastic (Fig. 4a); such behavior is commonly assumed in gravity surveys (Middlemiss et al., 2016). When uncertainty is not constant throughout the variable space, the variable is called heteroscedastic (Fig. 4b, c). Note that heteroscedastic cases include both structured (Fig. 4b) and unstructured (Fig. 4c) relationships between the measured values and their respective errors. Structured heteroscedastic variables show a clear relationship (e.g., correlation, cyclicality) between the variable and its uncertainty while unstructured ones do not. Structured heteroscedastic behavior is observed electrical resistivity tomography (Perrone et al., 2014), magnetotellurics (Thiel et al., 2016; Rawat et al., 2014), airborne gravity and magnetics (Kamm et al., 2015), and controlled-source electromagnetic (Myer et al., 2011) surveys. It is usually possible to transform a structured heteroscedastic variable to a space where it becomes homoscedastic (commonly the log space), perform analysis and transform back to the original space. Unstructured heteroscedastic behavior is common in seismic surveys and impacts inversions (Kragh and Christie, 2002; Quirein et al., 2000; Eiken et al., 2005). The heteroscedastic case essentially allows for any level of correlation between the measured values and their uncertainty or error to be possible (Fig. 5).

Synthetic examples of different levels of scedasticity of
measurements of the same variable.

The failure to account for scedasticity often implies the assumption of
homoscedasticity as this assumption allows for a wider range of statistical
methods to be applied. With heteroscedastic data, the results of methods
that depend on the assumption of homoscedasticity, such as least-squares
methods (Fig. 4), give results of much decreased quality (Eubank and Thomas,
1993) and this may lead to the validation of incorrect hypotheses.
Scedasticity analysis from raw data without prior knowledge is challenging
(Zheng et al., 2012) and this topic of research is still being investigated
(Dosne et al., 2016). If there is no option for an appropriate transform, it
is advisable to perform an empirical analysis of scedasticity beforehand.
This is usually achieved through experimental assessment of uncertainty
under various conditions (metrological study) of measurement and over the
entire range of measured values (Allmendinger et al., 2017; Cawood et al.,
2017; Novakova and Pavlis, 2017). The results of such analysis can then be
used to define the prior dispersion (

Each data input is expected to carry its own parameterization for disturbance distribution depending on the nature of the input (single measurement, sample, central statistic). Additionally, the parameters of the disturbance distributions are better defined when scedasticity is accounted for. It is worth mentioning that both the normal distribution and the von Mises–Fisher distribution have a complete range of analytical or approximated solutions for both posterior and posterior predictive distributions (Rodrigues et al., 2000; Bagchi and Guttman, 1988; Bagchi, 1987). In the next section, disturbance distribution sampling for spherical data (orientations) is discussed.

Distribution of errors for the cases described in Fig. 3. Homoscedastic case shows constant uncertainty and no relationship of uncertainty to the data. The structured heteroscedastic case has a linear relationship of uncertainty to the data. The unstructured heteroscedastic case demonstrates no obvious relationship of uncertainty to the data and is not constant.

In the geoscience, the orientation of planar features such as faults and bedding is described by foliations. These foliations can be recorded in the form of dip vectors using the dip and dip-direction system. This system is equivalent to a reversed right-hand rule spherical coordinates system. The following covers sampling strategies for such spherical data and demonstrates their impact on MCUE results.

Recent research using MCUE (Lindsay et al., 2012, 2013; Wellmann and Regenauer-Lieb, 2012; de la Varga and
Wellmann, 2016) uses dip and dip-direction values independently (as two
scalars) from one another. The dip and dip-direction system is a practical
standard for field operators to record and make sense of orientation data.
However, it is highly inappropriate for statistics. Geoscientists generally
perform statistical analysis on stereographic projections of the dip vectors
to the planes. Because stereographic projection involves the transform of
dip vectors to pole vectors (normal vector to the plane), it gives a sound
representation of the underlying prior uncertainty distribution. The pole
transform step is essential to avoid variance distortion (Fisher et al.,
1987) as shown in Fig. 5. The distortion will increase as the dip of the
plane diverges from

Distortion of the maximum likelihood estimation (MLE) of
concentration or spherical variance of 100 spherical unit vector samples
of a size of 1000 individuals drawn from a von Mises–Fisher
distribution with

Effect of sampling over dip vectors or pole vectors on bounded
uniform spherical distribution at a range

The use of distributions in MCUE makes it very sensitive to scedasticity
over inputs. The uncertainty of a dip vector which is quantified by any
dispersion parameter similar to

The impact of pole versus dip vector sampling on the results of MCUE is evaluated on a simple synthetic model and on a realistic synthetic model. The simple model is a standard symmetric graben with four horizontal units; it has been chosen for its simplicity and is commonly used as a test case (Wellmann et al., 2014b; de la Varga and Wellmann, 2016; Chilès et al., 2004) in MCUE for proof of concepts. The realistic model is a modification of a real demonstration case that is part of the GeoModeller package based on a location near Mansfield, Victoria, Australia. It features a Carboniferous sedimentary basin oriented NW–SE that is in a faulted contact (Mansfield Fault) on its SW edge to a Silurian–Devonian set of older, folded basins. Outcropping units are almost all of the siliceous detritic type ranging from mildly deformed sandstones to siltstones and shales; the basement is made of Ordovician–Cambrian serpentinized sandstone. The original data for the Mansfield model were not altered in any way, instead data based on the Mansfield geological map (Cayley et al., 2006) geophysical map (Haydon et al., 2006) and airborne geophysical survey (Wynne and Bacchin, 2009; Richardson, 2003) were added to refine it.

The graben model is built using orientations and interfaces only, with three
interfaces and three foliations per unit and one interface and one foliation per
fault (Figs. 8, 9c). The Mansfield model is built with 281 interface
points and 176 foliations over six units and three faults (Figs. 10, 11c). For
both models, perturbation is performed as described in Sect. 3. For the
graben model, units' interfaces are isotropically perturbed over a normal
distribution with the mean centered on the original data point and a standard
deviation of 25 m. The orientations of the faults are perturbed over a von
Mises–Fisher distribution with the original data as the mean vector and
concentration of 100 (

Structural data for the graben model and modeled surfaces for units and faults. Spheres represent interfaces and cones represent pole vectors.

Effect of pole

The influence of dip vector (Figs. 9b, 11b) versus pole vector (Figs. 9a,
11a) sampling of orientations is very noticeable over the output
information entropy uncertainty models. Information entropy is a concept
derived from Boltzmann equations (Shannon, 1948) that is used to measure
chaos in categorical systems. Because of this, it is possible to use
information entropy as an index of uncertainty in categorical systems. Dip
vector sampling appears to add a layer of artificial “noise” on top of the
uncertainty models. The noise prevents expected structures of the
starting model (Figs. 9c, 11c) to be easily distinguishable. In cases
in which the orientation data are more vulnerable to improper sampling error
(away from 45

Structural data for the Mansfield model and modeled surfaces for units and faults. Spheres are interfaces and cones are orientations.

Effect of pole

Generally, CLT distributions are valid choices as prior uncertainty
distributions (and disturbance distributions) because they describe the
behavior of uncertainty well. However, there may be scenarios in which
alternatives can offer a better solution. More specifically, the uniform or
the Laplace distribution may better describe location uncertainty than the
normal distribution. The uniform distribution indicates a lack of
constraints as to the prior uncertainty distribution; it is a valid choice
when there is little knowledge about data dispersion. The Laplace
distribution is suitable if the measured data abide by the first law of
errors instead of the second (Wilson, 1923). For example, to model the
uncertainty on the thickness of a geological unit along a drill core, one
might observe that the uncertainty of the location of the top and bottom
interfaces of the unit is best represented by an exponential distribution. In
this instance, the Laplace distribution would be a suitable option to model
the thickness' uncertainty. Under similar circumstances, a spherical
exponential distribution could be swapped with the vMF distribution. The
Kent

The Kent distribution is the spherical analogue to a
bivariate normal distribution; it takes an additional concentration
parameter along with a covariance matrix. Together, these two parameters
allow for any level of elliptic anisotropy on

In this paper, it is explicitly assumed that the dispersion of prior uncertainty distributions is a deterministic function. Note that this does not necessarily make this function a constant and it might depend on the observed data. The dispersion function of field measurements (using a compass) of structural data would be expected to be nearly constant. Conversely, the dispersion function of interpreted measurements (using geophysics) would be expected to be dependent on the sensitivity of the intermediary method. Additionally, dispersion functions may be probabilistic as well as deterministic (Bucher, 2012). Determinism is a strong assumption when no metrological study was conducted beforehand to assess its plausibility. Such metrological studies involve experimental testing of devices and procedures in order to estimate precision, accuracy, bias, scedasticity or drift about measured data. These estimates can then be compiled into a dispersion function that can be used as an input parameter for other purposes, including prior uncertainty distributions for MCUE. Probabilistic dispersion functions apply non-negligible uncertainty to the dispersion function for prior uncertainty distributions. Uncertainty about dispersion makes the proposed workflow for disturbance distribution parameterization inadequate. Indeed, Eq. (5) may not be simplified into Eq. (6) anymore and the following statements (Eqs. 7 to 14) would then ignore the probabilistic nature of the dispersion function. Both the normal and the vMF distributions have analytical solutions or good approximations for such cases; the authors recommend the readers refer to relevant works (Gelman et al., 2014; Bagchi and Guttman, 1988) if required. Note that there is significant metrological work about borehole data (Nelson et al., 1987; Stigsson, 2016) as opposed to usual structural data such as foliations, fold planes, fold axes or interfaces.

Although the authors make the case for scedasticity analysis in MCUE, it is left open in this paper. Scedasticity is essentially an untouched subject in geological 3-D modeling and it was pointed out to make the geological 3-D modeling community aware of this fact and its potentially nefarious influence on MCUE outputs. However, standard metrological studies can determine scedasticity and include it in a dispersion function to be a parameter of the prior uncertainty distributions (Bewoor and Kulkarni, 2009; Bucher, 2012).

The evidence brought at the theoretical and practical levels allows us to
strongly advocate for the use of pole vectors over dip vectors. In fact, dip
vector sampling shows poor performance away from 45

Good prior knowledge about input uncertainty is critical to the propagation
of uncertainty in general. This, in turn, makes metrological work mandatory
to any form of modeling that relies on actual measured data. Note that it is
acceptable to use preexisting metrological studies to define the priors
(Allmendinger et al., 2017; Cawood et al., 2017; Novakova and Pavlis, 2017)
provided that the measurement device and procedure used are similar to those
of the studies. To gather multiple observations per site is strongly
recommended as this practice sharply increases the quality of the
disturbance distributions. From a practical point of view this would require
field operators to perform several measurements on the same outcrop. If
that is not possible one may group measurements of clustered outcrops
together provided that the scale of the modeled area compared to that of the
cluster allows it. The authors recommend not grouping clusters that are
spread out more than 3 orders of magnitude below the model size (e.g.,
for a 10 km

Propagation of uncertainty is the process through which different kinds and sources of uncertainties about the same phenomenon are combined into a single final estimate. MCUE methods seek to achieve propagation of uncertainty using Monte Carlo-based systems in which input uncertainty is simulated through the sampling of probability distributions called a disturbance distribution. Disturbance distributions are the distributions that normally best represent the uncertainty about the input data in the context of uncertainty propagation in geological 3-D modeling.

This paper discusses the importance of disturbance distribution selection and proposes a simple procedure for better disturbance distribution parameterization and a pole-vector-based sampling routine for spherical data (orientations) used to represent the geometry of planar features. Pole-vector-based sampling for spherical data and Bayesian disturbance distribution parameterization are proved – either through demonstration or through experiment – to be valid and practical choices for MCUE applied to implicit 3-D geological modeling. Namely, the normal and the vMF distributions are shown to be the best candidates for disturbance distributions for location and orientation, respectively. A Bayesian approach to disturbance distribution parameterization is shown to avoid underestimation of input data dispersion, which is important as such underestimation artificially decreases the output uncertainty of the 3-D geological models. Such underestimation may give a false sense of confidence and lead to poor decision-making. Pole vector sampling is evidenced to be the best alternative because it is guaranteed not to distort the disturbance distribution shape or generate artifacts in the output uncertainty models the way dip vector sampling does.

The proposed framework and methods are compatible with previous MCUE work on 3-D geological modeling and can be easily added to existing implementations to improve their accuracy. As MCUE is applicable to all fields in which 3-D geological models are needed, so is the proposed framework. The primary domains of application are the mining and oil and gas industries at the exploration, development and production steps. In addition, numerous secondary domains of potential application are available to this work, such as civil engineering and fundamental research.

Both the Mansfield and graben GeoModeller models
(including the perturbed data sets and series of plausible models) showcased
in the present study are available online openly at

Von Mises–Fisher sampling on the usual sphere is not new (Wood, 1994) and
this appendix serves as a reminder for the reader. To generate a von
Mises–Fisher distributed pseudo-random spherical 3-D unit vector

The maximum likelihood estimation

The co-kriging algorithm used in GeoModeller interpolates a 3-D vector field and converts it into a potential (scalar) field (Calcagno et al., 2008; Guillen et al., 2008) that is then contoured to draw interface surfaces. The space between surfaces is defined as belonging to a specific unit based on topological rules. The topological rules are set by (i) the stratigraphic column for units versus unit topological rules, (ii) the fault network matrix for faults versus fault topological rules and (iii) the fault affectation matrix for faults versus unit topological rules.

The potential field co-kriging interpolator is

The authors declare that they have no conflict of interest.

The authors would like to thank the Geological Survey of Western Australia, the Western Australian Fellowship Program and the Australian Research Council for their financial support. In addition, the authors make special distinction to Intrepid Geophysics for their outstanding technical support. In addition, the authors express their gratitude to Ylona van Dinther, Guillaume Caumon and Eric de Kemp, for their insightful comments and dedication to help us improve this paper. Edited by: Ylona van Dinther Reviewed by: Eric de Kemp and Guillaume Caumon