Geological rules have the potential to provide an important constraint on the classification of lithologies recovered from inversion. Such rules, like adjacency (Egenhofer and Herring, 1990), define which rock bodies can be in contact with each other and which cannot. These rules are typically expressed in geological terms as stratigraphy, where the relative age and event classification of geological units are stated. For example, a sedimentary depositional event of five separate units may define a simple subhorizontal layer cake configuration, where the oldest unit is never adjacent (or in contact) with the youngest unit. A magmatic event that follows may result in a vertical dyke that intrudes all sedimentary layers adjacent to all other rock units. Using geological rules as a constraint relies on finding those that are restrictive (such as the youngest unit never being in contact with the oldest) rather than permissive (such as the intruding dyke). Thiele et al. (2016), Pellerin et al. (2017) and Anquez et al. (2019) show how these can constrain parametric geological modelling. It is therefore important to honour geological rules if known and include them in classification schemes such as those to ensure that geological plausibility is not compromised in pursuit of an otherwise petrophysically and geophysically consistent model.

The process of post-regularization, which consists in the application of spatial–contextual filters to the classification results to eliminate geologically unrealistic features, has been shown to increase prediction accuracy in surface (2-D) geological mapping (Tarabalka et al., 2009; Stavrakoudis et al., 2014; Cracknell and Reading, 2015).

The post-regularization scheme we develop for the recovery of lithologies in 3-D relies on two hypotheses. Firstly, we assume that the presence of isolated lithologies contradicts the geological principle of continuity. Although such post-regularization has been used mostly in 2-D or shallow 3-D, there is no theoretical obstacle to the extension of this methodology to the purely 3-D classification case we present here. Secondly, we introduce the utilization of adjacency relationships between the different lithologies in post-regularization to ensure that base topological rules are respected across the entirety of the three-dimensional volume. This is particularly important for structural geological interpretation (Freeman et al., 2010; Godefroy et al., 2019). Here, we extend existing post-regularization approaches (i.e. Tarabalka et al., 2009; Stavrakoudis et al., 2014; Cracknell and Reading, 2015) by integrating geological information in the classification analysis in the form of topological relationships (see Egenhofer and Herring, 1990; Zlatanova, 2000; Thiele et al., 2016, for the different topologies) defined by geological principles.

The general formulation of post-regularization is as follows, for a given model cell: 3BMU=BMUk,wherek=argmin⁡uk-BMU|conditions, where arg min returns the argument k satisfying the conditions it precedes. Here, we set 4conditions={∃cU|c=BMUk(5)∩1TMk⋅Mf1=0nl,nl∩stratigraphy}, where 1 is nl×1 column vector of ones, and 0nl,nl is the nl×nl matrix of zeros; Mf and Mk are adjacency matrices, and Mk⋅Mf is the Hadamard product of Mk and Mf, such that Mk⋅Mfij=MkijMfij, and 0nl,nl is the zero matrix of dimension nl×nl; Mf encapsulates geological knowledge and principles about contacts between lithologies; Mk contains the adjacency relationships between the considered cell and its neighbourhood U. The derivation of Mf and Mk is detailed below.

The first part of the condition in Eq. (4) is enforced using morphological closing where isolated lithologies are replaced by the most prevalent one in their neighbourhood (see Benavent et al., 2012; Ackora-Prah et al., 2015). In such cases, the BMU is updated as follows. Isolated cells (in terms of their lithology) are identified through examination of the 26-cell 3-D cubic Moore neighbourhood U of every model cell. A cell is considered isolated if, and only if, at least 25 cells of U have a lithology that differs from it. Once such a cell is identified, its BMU is updated using the closest neuron, ensuring continuity between adjacent cells in the neighbourhood U, where the lithology to be assigned is determined by a majority vote in U subject to adjacency conditions. These conditions are determined by geological knowledge (as explained below). This process is repeated for all locations until the lithological model stops changing. The general principle of post-regularization is illustrated in Fig. 1.

We point out that in contrast to Tarabalka et al. (2009) and Stavrakoudis et al. (2014), who used the first and second Chamfer neighbourhoods in 2-D around the considered model cell, we do not follow the same approach in 3-D. Our implementation of the extension of their approach to 3-D showed that, in our application case study, the adjustments of the recovered lithological model it imposes are detrimental to the consistency of the classification with geophysical measurements. That is, the perturbation of the corresponding geophysical response of the model it generates exceeds noise level and compromises the geophysical validity of the recovered model (see geophysical validation subsection below for more details). The same remark applies to the utilization of a mode filter with a 3×3×3 kernel.

The conditions relating to adjacency relationships forces the model to honour adjacency relationships extracted from surface geology (Burns, 1988; Thiele et al., 2016) in the recovered lithological model.

We determine lithological topology by identifying the contacts between adjacent model cells and represent the topological signature of lithological models using the adjacency representation of Godsil and Royle (2001). Let the adjacency matrix M of a given cell be defined as 5Mi>j,j=nij,Mi≤j,j=0, where nij is the number of contacts between lithologies i and j. Similarly, geological laws and knowledge allow the derivation of a matrix Mf defined as follows: 6Mi>j,jf=1if contact betweeniandjcontradicts geology.Mi>j,jf=0otherwise.

From there, it is straightforward to identify occurrences of forbidden contacts by calculating Mk⋅Mf. Therefore, 1TMk⋅Mf1=0 (Eq. 4) indicates that no contact violating the condition imposed by Mf is observed. The last condition in Eq. (4) can be used to prevent the local stratigraphy (in the Moore neighbourhood U of the considered cell) from violating geological rules such as “lithology B must be in conformable sequence between lithologies A and C”.

After identification of configurations forbidden by the conditions set in Eq. (4), its BMU is updated using the closest neuron honouring the set of conditions (Fig. 2, box 4a).

The next stage of the methodology we introduce is the calculation of the apportionment of each neuron in terms of the lithologies of the testing data vectors (from the synthetic survey) they predict (Fig. 2, box 5a). For instance, in a two-lithology scenario, a given node may be found to predict lithology A using the validation dataset correctly 80 % of the time (80 % accuracy) and lithology B correctly 20 % of the time (20 % accuracy). This process is described below.

The methodology is summarized in Fig. 2.

The prediction accuracy τi of lithology iϵ[1,…,nl] (with nl the total number of lithologies) is the ratio of correct predictions to the total number of predictions. It is obtained from the “matching matrix” of the recovered lithologies Mc. We remind that the “matching matrix” (or “confusion matrix” in supervised learning) is a matrical representation of the number of occurrences of true/false positives/negatives.

We use the prediction accuracy τi as a metric measuring the capability of the node i of the trained SOM to recover lithologies. Let τi be expressed as 7τi=MiicMiic+∑j≠ij=1nlMijc, which is a particular case of the overall accuracy τ: 8τ=∑i=1nlMiic∑i=1nl∑j=1nlMijc.

From Eq. (7), it appears that τi is equivalent to the frequentist probability of the ith lithology over the entire SOM. When considering a specific node j, it becomes the relative frequentist probability of the lithology i for cells classified as having the jth node as their BMU, noted τij.

The consistency of the classification performed using SOM after application of post-regularization with field geophysical measurements might be altered by both the classification itself and by post-regularization. It is therefore necessary to ensure that the approximation of the dataset by values from the units of SOM is consistent with geophysical measurements. To this end, we verify that the geophysical response of the density contrast model mSOM corresponding to the BMUs of each cell in the studied area fits the field measurement d within a certain tolerance assumed to approximate noise level. Consequently, we ensure that the difference between minv and mSOM, Δm=minv-mSOM, satisfies the following condition: 9Wdd-gminv-d-gmSOM22=WdGΔm22≤tol, where tol is the threshold depending on noise levels in the data above which minv and mSOM are not considered geophysically equivalent.

The implication of Eq. (9) is that the difference Δm belongs to the null space of the inverse problem considered. The null space is characteristic of geophysical inversion's non-uniqueness. It is defined as the ensemble of models that reproduce geophysical data with a comparable misfit. The models honouring Eq. (9) can therefore be considered equivalent from a geophysical data point of view (Muñoz and Rath, 2006; Chen et al., 2007; Deal and Nolet, 1996). For qualitative assessment of geophysical consistency, we complement the utilization of Eq. (9) with the visual comparison of data misfit maps corresponding to the model corresponding to minv and mSOM.

The Paleoproterozoic Yerrida Basin is located in the southern part of the Capricorn Orogen (WA) and covers approximately 150 km N–S and 180 km E–W (Pirajno and Adamides, 2000) (Fig. 3a). The structures of interest in this work are Archean greenstone belts (Fig. 3), as they are prospective for Au and Ni and underlie the younger basin rocks. The basement to the Yerrida Basin is considered to be Archean granite–gneiss or greenstone rocks of the Yilgarn Craton. Lithospheric extension initiated the formation of the Yerrida Basin at approximately 2200 to 1990 Ma with deposition of the Windplain Group (Occhipinti et al., 2017; Pirajno and Adamides, 2000). The Goodin Inlier remains exposed in the central part of the basin and is in unconformable contact with the Windplain Group. A hiatus ensued, followed by deposition of the younger Mooloogool Group, which was then overlain in the east by the Tooloo Group of the Earaheedy Basin.

The density contrast of the lithologies observed in the area ranges between 0 and 330 kg m-3, making it appropriate for gravity modelling and inversion (Giraud et al., 2019a; Lindsay et al., 2018). While basin rocks exhibit some density contrast, the greenstone is conspicuous in gravity data with a density contrast expected to lie between 190 and 270 kg m-3, making it an attractive subject for gravity inversion. Field geological measurements (orientation data in the form of interfaces and foliations) and petrophysical data were used to build the reference geological model. Airborne geophysical data, Landsat and Aster 8 satellite data were also used to support the interpretation of geological measurements.

The gravity anomaly dataset we consider (Fig. 3b) is comprised of a total of 4882 measurement points. The model is discretized into 100×100×42 cells of dimensions 2.335 km × 1.875 km × 1.0475 km down to approximately 44 km depth. Weights and parameters used for the inversion of synthetic data follow the settings of Giraud et al. (2019a) on field data.

This subsection introduces the semi-synthetic survey we performed for the training of SOM.

The volumes of most probable lithology, WS and the starting model are shown in Fig. 4. Volumes shown in Fig. 4a, b and c are used for the training and validation dataset for SOM training as explained in Sect. 2.2.

We use the modelling results shown in Fig. 4 to calculate a synthetic geophysical dataset (Fig. 5d). The model used to generate the synthetic geophysical measurements is shown in Fig. 5a. The corresponding inverted model and its gradients are shown in Fig. 5b and c. Volumes shown in Fig. 5b and c are used for training and validation.

The density contrast model obtained from inversion of field geophysical data and its gradients are shown in Fig. 6a and b. Visual comparison of inverted models shown in Figs. 6a and 5b reveals that mesoscale structures are similar with the exception of large structures presenting low-density contrasts at depth (darker shades of blue in Fig. 6a). This is reflected in Fig. 6b, which exhibits low gradient values in these areas.

The classification of lithologies using SOM is performed applying the trained network to volumes shown in Figs. 4b, c and 6a, b. The next subsections describe the geological laws used for post-regularization (Sect. 3.1.4) and the classification process (Sect. 3.2.2).

As mentioned above, for clarity in this demonstration, only adjacency relationships are considered. The utilization of simple relationships such as adjacency is also chosen because in areas of sparse data, a full description of geological rules (fault relationships with fault and stratigraphy) is often not known. Given the complexity of the Yerrida Basin and its magmatic and deformation history, several base geological rules can be derived to assess the plausibility of recovered lithological models. Using fundamental geological principles (such as uniformitarianism, superposition, Walther's law, cross-cutting relationships and original horizontality), the two most likely restrictive adjacency rules are as follows. We assume that the mafic greenstone bodies cannot be in contact with the Killara Formation (in the Mooloogool Group) since our field data suggest that the Killara Formation is a volcanic unit that is restricted to the Yerrida Basin and thus not in contact with the mafic greenstone. In addition, we assume that the mafic greenstone cannot be in contact with the Goodin Inlier and background (or basement), as the mafic greenstone is modelled to be enveloped by the felsic component of the greenstone.

The matrix Mf defining the contacts forbidden by geology as described above is given in Fig. 7.

In this work, we use approximately 500 neurons (units) for the training of SOM. This number is inferred from the analysis of the elbow curve we calculated using the validation datasets (see Fig. 8) and approximately matches the proposed value of 5N (with N the total number of observation) proposed by Vesanto and Alhoniemi (2000) and commonly used since (Shalaginov and Franke, 2015). The chosen number of units is corroborated by the lithology prediction accuracy (Fig. 8) as approximating the point of diminishing returns, i.e. the number of nodes beyond which additional nodes are becoming nearly redundant.

The trained map presents a mean quantization Q equal to 0.075, which indicates relatively good approximation of the datasets by the trained SOM. This is illustrated by Fig. 8, where all lithologies are recovered with a prediction accuracy superior to 90 %.

After classification of the recovered model, we performed post-regularization to remove geologically unrealistic features from the classified lithological volume. Figure 9 shows the classification results before and after post-regularization, along with the associated adjacency matrix.

As can be inferred from the adjacency matrices plotted in Fig. 9, the number of contacts between units with indices 1 and 2, respectively, is reduced by the application of post-regularization. One reason for this is the presence of a number of inclusions of lithology 1 in lithology 2, and vice versa; a total of 2561 such inclusions was identified. Overall, the number of contacts between lithologies 5 and 2 increased slightly due to post-regularization because a large percentage of contacts between lithologies 5 and 1 has been reassigned as contacts between lithologies 5 and 2. The elimination of contacts between lithologies 5 and 6 is also visible in Fig. 9.

For each node of the SOM, we calculate the prediction accuracy τi (Eq. 3) for the different lithologies observed in the area using the cross-validation dataset. After application of the trained SOM for the classification of inversion results obtained from the inversion of field geophysical data, we obtain a frequentist probability volume for each lithology. Figure 10 shows the resulting frequentist probability volumes for the six lithologies present in the Yerrida Basin.

Figure 10 exhibits probabilities between 0.3 and 0.6 in the area marked by the two arrows. This suggests that these zones are the least well constrained. Note that from Fig. 10, we can interpret the presence of mafic greenstone with confidence, as it shows high frequentist probability nearly everywhere classification suggests its presence. For completeness, assessment of the prediction accuracy of the different lithologies is shown by the corresponding box plot in Appendix B.

Applying Eq. (9), we obtain WdGΔm22=6×10-4, indicating that the model can be considered geophysically equivalent overall. This is illustrated by the map of the corresponding data misfits (see Fig. B1 in Appendix B), which indicates that we can consider the recovered lithological model after application of post-regularization as reflective of both geophysical and geological information. Focusing on the mafic greenstone belts of interest in the area (Fig. 11), the classification results allow us to propose the following geological interpretations.

Figure 11 shows that the northern portion of the greenstone belts A and B recovered by geophysical inversion and SOM classification is thinner in their northern part than was proposed by the initial geological model. Likewise, greenstone belt C seems to be much thinner near its centre than expected. Given the data density and lack of understanding we have about the depth of this greenstone belt, this observation is plausible. It confirms and refines considerably the crude, preliminary lithology differentiation of Giraud et al. (2019a) that was based only on density contrast value. The cause of the thinning of mafic greenstone belt C could be attributed to faulting, folding or the topography of the palaeoenvironment where the protoliths to the belt were formed, which are not captured directly by surface geology. The portion of the southern Merrie Greenstone Belt (mafic greenstone C) is shown to be thinner than expected, prompting a review of the structure of existing models. A plausible reason is the presence of structure that has not been identified from the initial interpretation of geophysical data. In addition, the potential presence of deep-penetrating faults or shear zones, as shown in Fig. 11b by the arrows around greenstone C, hints at a possible false assumption that the Merrie Greenstone Belt is a single and coherent geological body.