Disturbance distributions are probability distribution functions that are used to generate plausible datasets in MCUP. They are designed to simulate the effect of the inherent uncertainty of each observation separately. In principle, an individual disturbance distribution is associated with each observation (Fig. 1, preprocessing). Disturbance distributions are expected to be chosen and parameterized based on thorough metrological analysis of the original dataset, since disturbance distributions are expected to aggregate as many sources of input data uncertainty as possible. These sources of uncertainty relate to measurement error, rounding error, user error, local variability, miscalibration or projection issues (Bardossy and Fodor, 2001). Generally, Gaussian-like distributions make for appropriate disturbance distributions (Pakyuz-Charrier et al., 2018a). Disturbance distribution selection and parameterization is a complex topic and is outside the scope of this paper. It can be noted, however, that particular care must be taken that spherical distributions (see Appendices) should be used when handling spherical data such as orientation measurements. However, practitioners may seek guidance from recent practical metrological work on foliations (Novakova and Pavlis, 2017; Stigsson, 2016; Cawood et al., 2017) and more theoretical work on disturbance distribution selection/parameterization for MCUP (de la Varga and Wellmann, 2016; Pakyuz-Charrier et al., 2018a).

Plausible datasets are obtained by sampling from the numerous disturbance distributions that have been defined for each input observation. The sampling step is often referred to as the “perturbation” of the input data (Cherpeau et al., 2010). In this study, errors are assumed to not show any spatial dependency and the sampling is therefore performed independently. Such assumption is mostly valid when measurements can be considered to be physically independent (Pakyuz-Charrier et al., 2018a). However, spatial correlation of errors can be observed even in this case. This is especially true for cyclical datasets such as foliations in folding scenarios. Evidently, structural data that are derived from sources that naturally exhibit spatial dependency, such as seismic horizon picks, should not be perturbed in this way. The sampling step may be followed by a range of statistical checks to ensure stationarity, reject outliers or perform variographic analysis.

Plausible dataset generation is an important part of the MCUP method because it heavily predetermines its outcomes. However, plausible datasets are only as relevant as the plausible model they correspond to. MCUP is then largely dependent on the particulars of the chosen modeling engine (Fig. 1, building). Any modeling engine relies on the conceptualization of the phenomenon it is supposed to model. Conceptualization relies mainly on abstraction and simplification to make the modeling problem accessible to our minds and technology. Therefore, any workflow or method that relies on a modeling engine subsequently relies on these abstractions and simplifications which, by definition, are incomplete and uncertain. Consequently, MCUP is sensitive to this kind of “conceptual uncertainty” and care should be taken when selecting or parameterizing the modeling engine. Given that the aim of MCUP is to propagate input uncertainty through the modeling engine to the final model, several indispensable properties of the modeling engine may be identified: (i) the ability to estimate and propagate its own uncertainty, (ii) the ability to handle multiple plausible datasets without having to be reconfigured manually and (iii) the ability to function without extensive expert input. These properties are generally met by implicit modeling engines (Chilès et al., 2004; Aug et al., 2005; Calcagno et al., 2008; Chilès and Delfiner, 2009) by the virtue of them being reliant on potential field interpolation to estimate the geological surfaces from the input structural data. The interpolator is normally parameterized using variographic analysis and a geometrical rule set to solve geometrical ambiguities (Jessell, 2001). The geometrical rule set consists of a series of geometrical constraints such as the intersection priority of faults and geological units that are used to determine which interface stops on which. Conceptually, the geometrical rule set enforces the age relationships between the faults and/or geological units in the model. In this paper, the modeling engine is the GeoModeller software which uses a stochastic co-kriging interpolator and constrains surfaces using a predefined stratigraphic pile and fault relationship matrices as geometrical rule set (Guillen et al., 2008; Calcagno et al., 2008).

In implicit 3-D geological modeling, a model is essentially a set of spatial functions that describe the geometry of stratigraphic and intrusive interfaces and fault planes. In this form, it is difficult to apply common comparative analysis methods. Therefore, plausible models are either discretized to 3-D grids (voxets) or converted to triangulated interfaces (Fig. 1, Postprocessing). Note that in all three cases, these operations are further simplifications of the models and add more uncertainty to the final outcome. Each of these transformations allow for different comparative analyses to be run: (i) voxets are used to build probabilistic geological models and uncertainty index models such as entropy or stratigraphic variability (Wellmann and Regenauer-Lieb, 2012; Lindsay et al., 2012); (ii) the shape of triangulated surfaces may be used to estimate the variability of curvature (Lindsay et al., 2013). Furthermore, the results of these analyses can be fed to external validation systems to reduce geological uncertainty and improve understanding of the modeled volume. Examples of external validation systems include geophysical inversion (Giraud et al., 2019), concurrent geophysical forward modeling (Bijani et al., 2017; Lipari et al., 2017), 3-D restoration, fluid flow simulations or ground truthing. Lastly, the results obtained from the external validation systems may be reutilized by MCUP to further refine the models.

Topology describes the properties of special mathematical spaces that are unaltered under continuous deformation. The 3-D geological modeling mostly concerns itself with the topic of geospatial topology that focuses on spatial relationships such as adjacency, overlap or separation of geometrical objects such as points, lines, polygons and polyhedrons (Thiele et al., 2016a). Essentially, the use of topological relationships to characterize 3-D geological models allows a compact expression of a subset of their geometry (Burns, 1988). Combined with the knowledge of the intrinsic physical properties of the rock types that compose geological units, these relationships constrain the downstream predictions resulting from 3-D geological models in terms of physical processes such as fluid, heat flow and electrical flow, as well as mechanical stresses. The most common relationships between 3-D objects encountered in 3-D geological models are adjacency and separation of lithological units. In their simplest form, these relationships can be expressed using an adjacency matrix. Each element of the adjacency matrix is a Boolean, where 0 encodes separation and 1 encodes adjacency (Fig. 4). However, an adjacency matrix contains both redundant and irrelevant information. Indeed, the adjacency matrix A of a model M comprised of n geological units is symmetric and hollow. A is then of size n2, with its diagonal comprised solely of 1, while both sides are transpose of one another; it is then useful to half-vectorize A and remove unit elements from the diagonal following the triangular number sequence. For example, the 4×4 adjacency matrix, 1A=1101111101101101, is half-vectorized: 2vechAT=1101111101.

Note that vech(A) is of size n2+n2 and contains all the necessary information to fully describe the adjacency of lithological units in a 3-D geological model with n distinct lithological units. vech(A) can be also considered as a n2+n2 bit binary sequence called a basic topological signature. Although the diagonal of unit entries may seem redundant, it actually encodes the presence of a unit in the model. This is useful in MCUP because a plausible model may miss a unit as a result of the perturbation process. The total number of possible topological signatures is 2n2+n2. However, it is unlikely that all possible signatures are present in the plausible model suite given that the geometrical rule set constrains their topology. Consequently, the issue of the representativity of the plausible model suite in terms of the variability of its topological signatures comes into question. At a minimum, the variability of topological signatures should be qualitatively representative of the plausible model space to allow the clustering algorithm to delineate the right number of clusters. Cumulative observed topological signature graphs are a practical and efficient way to determine the topological representativity of the plausible model suite in real time (Thiele et al., 2016b). As the modeling engine produces new plausible models, these graphs plot the number of distinct topological signatures observed versus the number of plausible models generated so far. When the number of distinct topological signatures observed reaches a plateau, it is safe to consider that most topologies have been observed and qualitative topological stationarity may then be assumed reasonably (Fig. 5).

DBSCAN is a point-density-reliant flat data clustering algorithm (Schubert et al., 2017; Ester et al., 1996). DBSCAN is based on the notion of the reachability of border points from core points (Fig. 6). The algorithm only needs two parameters: (i) the minimum number of points Pmin that are required to form a cluster and (ii) the maximum distance ε allowed for two points to still be considered to be neighbors. On this basis, the algorithm builds a distance matrix between all points and uses that matrix to determine the neighbors of each point based on ε. Each point that has at least Pmin neighbors is a core point that forms a cluster seed to which all directly reachable points are attached. In order to build the distance matrix, DBSCAN requires each point to be characterized by a metric variable. Therefore, the variable would allow distances to be computed using regular norms such as the Euclidean distance. However, topological signatures form a series of Boolean variables that cannot provide appropriate measures for they are not additive. An alternative is to consider the whole topological signatures as a binary word and use the Hamming distance (Hamming, 1950) as the metric. The Hamming distance counts the number of individual bit switches required to match two binary words of equal lengths, effectively quantifying their disagreement. Implementation-wise, a simple XOR over two topological signatures gives the Hamming distance that separates them. As a special case, when ε=0 and Pmin=1, DBSCAN will distinguish every distinct topological signature into a separate cluster and the size of each cluster will count their occurrences.

Once the plausible model suite has been segregated into clusters based on their topology, a range of statistical methods may be applied to the results to (i) evaluate the quality and relevance of the clusters, (ii) determine data leverage in relation to the clusters, (iii) perform standard MCUP comparative analysis on the clusters and (iv) feed the clusters to an external rejection system. Cluster quality may be evaluated by computing the internal binary information entropy matrix E for each cluster: 3Eij=-∑k=1cAkijclog⁡∑k=1cAkijc-1-∑k=1cAkijclog⁡1-∑k=1cAkijc, where Ak is the kth adjacency matrix of the cluster, c is the cardinality of the cluster, and ij are standard matrix indices. For a given cluster, E informs the user about the internal variability of the binary topological relationships between each lithological couple. Note that writing ∑k=1cAkij implies, for convenience, that each matrix entry is considered like a real number instead of a bit. Most entries are expected to be null, thus indicating no variations. Non-null entries indicate topological “switches” inside the cluster itself. That is, E highlights topological changes that the clustering algorithm considered not to be significant enough to warrant a split in the cluster. Importantly, this is directly translatable into geological insights: “these two models are different because in only one of them is the sandstone unit found adjacent to the shale unit”. Naturally, Eq. (3) may be applied to the whole suite of adjacency matrices as a practical reference to compare the internal information entropy matrices of each cluster to a global information entropy matrix. Standard MCUP comparative analysis tools may be applied to the individual clusters concurrently to, for example, obtain per-cluster/scenario uncertainty indices and sub-probabilistic geological models. Given that common MCUP uncertainty index models are sums of matching positive elements, per-cluster uncertainty index model voxets are guaranteed to yield equal or lower values compared to their global equivalent. Moreover, per-cluster uncertainty index models are expected to be better structured as a common effect of all clustering algorithms is to reduces intra-class variance. Clustered plausible models may be traced back to their plausible input datasets (structural measurements) to conduct cluster leverage analysis. The aim of cluster leverage analysis is to determine which parts of the datasets are responsible for the topological switches that induce the formation of new clusters. A straightforward way to achieve this aim would be to compute a central statistic such as the mean or the median for every individual datum input in every cluster. 4u‾=d‾l=1…d‾t, where u‾ is the vector of central statistics, d‾l is the central statistic for the plausible input observation l, and t is the cardinality of the input data. The next step is to compare every matching individual input datum central statistic between all cluster pairs: 5Δu‾a,b=u‾a-u‾b∘u‾a-u‾b, where a,b identifies a cluster pair and “∘” stands for the Hadamard product. The results of this procedure should be ranked to find the highest leverage plausible input data differences between clusters.

The CarloTopo 3-D geological model features eight lithological units distributed into five series and two faults (Fig. 7). All of the 25 foliations and 46 interface (Table 1) points for all units and faults are placed onto a single N–S vertical median cross section. This design decision was made to ensure that the cross sections discussed in the subsequent sections are representative of the models. CarloTopo simulates a normally faulted basin (cyan and green) placed on top of a mafic formation (blue) that sits on an erosional surface. Below the erosional surface is a metamorphic folded series (pink) comprised of three individual formations. The metamorphic series rests onto the basement and both are intruded by a pluton (red). The mafic and metamorphic units were both interpolated with an assumption of strong anisotropy over the x axis while other units were left to be isotropic. This design decision was made to prevent excessive variations within the plausible model suite and ease interpretation. The geometries for each unit were designed to manifest as many common geological features as possible without compromising its relevance for practical issues such as mining/oil and gas exploration. More specifically, several potential traps for sedimentary-hosted deposits were included in the original model along with a network of faults that serve as theoretical channels or barriers (Fig. 8). The case study was split into two separate MCUP experiments with different disturbance distribution parameterization with over a thousand perturbations each. The first run aims to simulate a high-input data confidence scenario applicable to well-surveyed areas. Conversely, the second run simulates a low-confidence scenario applicable to legacy data or early stages of exploration. Disturbance distributions in the high-input data confidence scenario were chosen to be of the Gaussian type with relatively low dispersion, whereas uniform-type distribution parameterized with large ranges were used for the low-input data confidence scenario (Table 2).

For this run, a global information entropy uncertainty index model voxet was produced to serve as a reference against matching topology-based estimates. Three vertical N–S cross sections were extracted from the voxet at 250, 500 and 750 m easting (Fig. 9). The 250 and 750 m information entropy cross sections are almost identical because the original model is symmetrical about the N–S median cross section where all structural data are located. Both sections display low-to-medium levels of entropy (0.20 to 0.40) distributed around the original interfaces' trace and forming entropy halos of about 70 m apparent thickness for non-triple-point areas. Conversely, triple points and areas of potential geometrical ambiguities display medium-to-high levels of entropy (0.50 to 0.70) and thicker halos (∼100 m). The 500 m information entropy cross section exhibits lower levels of entropy and much thinner halos (∼20 m) because of its extreme proximity to the structural data inputs.

To verify topological stationarity, each plausible model was exported to a voxet that was used to build its corresponding adjacency matrix (Fig. 5). Every “new” topology was placed into a standard topological stationarity graph (Fig. 10). The number of distinct topologies observed over the process of generating plausible models appears to follow a logarithmic pattern. That is, the greater part of possible topologies is “discovered” quickly and further plausible model generation yields diminishing returns. In this instance, a third of topologies are discovered in a mere 3 % of the total number of perturbations and the next third is completed in under 25 % of said number. The total number of observed distinct topologies represents about 5 % of the total number of plausible models. Note that these finds are in accordance with previous work on topological stationarity in 3-D geological modeling (Thiele et al., 2016b). Based on these observations, it is safe to assume topological stationarity for this run. Several parameter sets for DBSCAN were tested and it appeared that the only working set for this case is ε=0 and Pmin=1. Otherwise, DBSCAN returns a single cluster along with a small number of unclustered topological signature. That is, each distinct topological signature has to be considered as a cluster in itself in order to obtain more than one cluster. Such behavior is not entirely unexpected because of the low dispersion parameters set for the disturbance distributions. Indeed, low dispersion of disturbance distributions is partially and non-linearly correlated to low plausible model topological variability. This is confirmed by the low number (nine) of non-null elements in the global internal information entropy matrix (Table 3), which indicates that few topological relationships were affected by the perturbation process. With the aforementioned settings, DBSCAN returned 55 clusters that correspond to the 55 distinct topological signatures present in the plausible model suite. For practical purposes, a significance threshold of 60 occurrences was applied (Fig. 11) to retain only the six most significant topological signatures and make subsequent steps more manageable, and such operation is only justified on the basis that topological stationarity is adequately met.

A representative plausible model was selected from each significant topological signature cluster and three vertical N–S cross sections were taken (Fig. 12) to obtain a qualitative view of the topological and geometrical differences between them. The 500 m easting median cross section is mostly invariant throughout the cluster as pointed out by the low value observed on the global information entropy uncertainty index model voxet (Fig. 9). The 250 and 750 m easting cross sections appear to be significantly more variable throughout the clusters in terms of distinct topological features and geometrical variations. Evident differences in section view include (i) the basin lower unit (Fig. 12, green) gaining or losing contact with the metamorphic folded series (Fig. 12, pink shades) with the mafic cover separating the two series (Fig. 12, blue), (ii) the basement (Fig. 12, brown) coming into contact with the mafic cover and (iii) the upper metamorphic folded unit (Fig. 12, light pink) being in direct contact with the lower metamorphic unit (Fig. 12, dark pink). Additionally, the potential traps highlighted in the original model are seen to change size and shape, to close and open throughout the clusters. These results indicate that topological signatures may help differentiate favorable scenarios in ore reservoir or oil and gas modeling applications.

Information entropy cross sections were extracted from the uncertainty index model voxets (Fig. 9) that were generated for each significant topological signature. Although the information entropy values look similar throughout the clusters, there are noticeable differences in terms of sharpness and triple-point differentiation. Predictably, the 500 m easting section shows very little extra-cluster variability and is very similar to its global counterpart. This is most likely because of its relative proximity to the original structural data inputs. In contrast, the 250 and 750 m easting sections display significant extra-cluster variability in terms of entropy halos thickness (from 150 to 50 m), triple-point differentiation (right ellipses) and sequence repetition in the metamorphic folded series (middle and left ellipses). As expected, cluster-based information entropy cross sections are all sharper than their non-clustered counterpart. This constitutes a strong indication that topological clusters are geometrically consistent and supports the thesis that topology is an efficient determinant for geological coherence. Additionally, sharper information entropy cross sections imply sharper probabilistic geological models which allows for an increased external applicability of MCUP results. In general, these results underline the plausible model-discriminating efficiency of topological signatures even when they are considered individually.

As with the previous run, a global information entropy uncertainty index model voxet was produced to serve as a reference against matching topology-based estimates. Equivalent cross sections were taken (Fig. 13) and exhibit very similar features to the high-input data confidence run. However, attention is brought to the increased fuzziness of the information entropy halos. These patterns can be explained by the disturbance distribution selection and parameter selection for this run. The uniform distributions that were selected in this instance always have a higher innate entropy compared to Gaussian distributions. Furthermore, the ranges selected largely exceed those of the previous run. Although at a lesser degree, the topological stationarity graph (Fig. 14) expresses the same diminishing return effect as the high-input data confidence run. More specifically, a third of topologies were in the first 13 % of plausible models, another third in the next 20 % of plausible models and the final third in the last 70 % of plausible models. In this instance, DBSCAN was parameterized with ε=2 and Pmin=2 and returned two topological signature clusters of size 953 and 39, respectively, along with eight outliers. Lower or higher values for ε and Pmin returned either a single cluster of size 1000 or a thousand clusters of size 1.

Cross sections extracted from representative models of both clusters (Fig. 15) display stark differences at the geometrical and topological levels. Significant topological changes between the two clusters include the disappearance of the middle the metamorphic folded unit (purple) from cluster 2, the emergence of the lower metamorphic folded unit (dark pink) against the lower basin unit (green) and the contact of the intrusion unit (red) with the upper metamorphic folded unit (light pink) in cluster 2. This is not surprising given the high number of non-null elements in the global internal information entropy matrix (Table 4). Indeed, a total of 20 topological relationships were affected by the perturbation process to varying degrees. Moreover, per-cluster internal information entropy matrices result in a significant number of non-null elements (Table 4) which can be used to determine the main “breaking” topological relationships when compared against each other and against the global matrix. Most topological shifts between the two clusters relate to internal topological relationships of the metamorphic folded unit and the basement. These shifts are consistent with the representative cross sections and indicate that per-cluster internal information entropy matrices may be used to draw geological inferences from their topological differences. When the internal entropy matrices of the clusters are compared against the global one, small differences become visible because of the inclusion of the unclustered plausible models. Notably, the intermediate metamorphic folded unit entries are non-null against all other units and themselves, which suggests that the unit may be absent from some of the unclustered plausible models.

The information entropy uncertainty index model cross section for cluster 1 shows little variation to its global counterpart (Fig. 13). This is mainly due to the large size of cluster 1 compared to the number of plausible models. About 95 % of plausible models carry a topological signature that links them to cluster 1. Given the convex nature of information entropy, large clusters are likely to be near undiscernible with the global population. Overall, cluster 2 displays sharper entropy halos than cluster 1 or the global cross sections. It also features strong aliasing because of its relatively small size (39). Information entropy peaks about the metamorphic folded series appear to be shifted by a half of a fold wavelength between the two clusters (ellipses), while other features remain mostly constant. The relative similarity between the information entropy cross sections for both clusters (Fig. 13) despite their strong geological, structural and topological disagreement suggests that topological clustering holds potential as a differentiation tool in MCUP comparative analysis. Topological clustering would then be a way to mitigate the weaknesses of global information entropy uncertainty index models in regard to structural relevance.