Uncertainty Visualisation of a 3D Geological Geometry Model and Its Application in GIS-Based Mineral Resource Assessment: A Case Study in Huayuan District, Northwestern Hunan Province, China

The availability of three-dimensional (3D) geological models is evaluated based on the capability of a geometry to represent the input dataset and interpret geologically realistic representations of the dataset (Chen et al., 2014; Jessell et al., 2010; Li, 2008a, b; Frank et al., 2007; Maxelon and Mancktelow, 2005; Wu et al., 2005; Deutsch, 2002; Hou, 2002; Truffert et al., 2001). However, due to the inaccuracy and sparsity of geological data and imperfect geological understanding, the uncertainties associated with different aspects of the modelling process influence the quality of 3D geological models. In recent years, several papers have acknowledged uncertainty (Allmendinger et al., 2017; Carmichael and Ailleres, 2016; Jessell et al., 2014; Chilès and Delfiner, 2012; Caumon et al., 2009; Bistacchi et al., 2008; Calcagno et al., 2008; Moretti, 2008; MacEachren et al., 2005; Galera et al., 2003; Thore et al., 2002; Malengreau et al., 1999; Mann, 1993; Eisenhart, 1968) and attempted to estimate and analyse them (Pakyuz-Charrier et al., 2018; Giraud et al., 2017; Nearing et al., 2016; Thiele et al., 2016; Li et al., 2015; Lindsay et al., 2013, 2012; Wellmann and Regenauer-Lieb, 2012; Caumon et al., 2007; Glinsky et al., 2005; Snedecor and Cochran, 1989). Jessell et al. (2014) proposed that the handling of uncertainty would be of the same importance as improving modelling algorithms and integrating inversion in next-generation 3D geologic systems. Carmichael and Ailleres (2016) summarised that types of uncertainty contain (i) error, bias and imprecision; (ii) stochasticity and inherent randomness; and (iii) imprecise knowledge (Wellmann et al., 2016, 2014; Jessell et al., 2014; Wellman et al., 2010; Bárdossy and Fodor, 2001; Mann, 1993; Cox, 1982). Because all geological models are subject to several kinds of uncertainty, ranging from conceptual uncertainty and incomplete knowledge to uncertainties associated with the model construction methods and imprecision in the input data itself (e.g., Li et al., 2014; Wu et al., 2013; Dong et al., 2009; Sivia, 2006; Mann, 1993 Snedecor and, 1989), Wellmann et al. (2010) and Pakyuz-Charrier et al. (2018) proposed a comprehensive procedure with a stochastic approach based on Monte Carlo uncertainty estimation (MCUE). Essentially, MCUE relies on unsupervised models from disturbance probability distributions quantifying potential differences in the initial measurement recording (e.g., the locations and foliations of observed geological datasets). On this basis, Wellmann (2011) and Lindsay et al. (2012) proposed methods to combine stochastically plausible datasets to estimate and visualise uncertainty. Then, Thiele et al. (2016) defined a representation of topology uncertainty. All the above researches aims to quantify different kinds of geological uncertainty for upscaling the initial dataset, improving inaccurate geological understanding, and constraining multiple parameters (e.g., geology and inversion) to 3D geological modelling. Even so, there are issues with current uncertainty estimation. First, the previous integration process for uncertainty estimation (Pakyuz-Charrier et al., 2018; Lindsay et al., 2012; Wellmann, 2011) runs majority voting because it implicitly assumes that all stochastically plausible datasets are equally reliable. However, a more acceptable recognition is that the discrimination between each stochastic model and the geological reality is quite different. Second, the results in previous studies referred to how to quantify uncertainty in each cell. However, few papers present the quantitative reliability of the whole 3D geological model or input measurements. Third, a few papers referred to integrating geometric uncertainty into prospectivity mapping. Perrouty et al. (2014) showed a visualisation method, which presents lithological variability for each point of the model, to express the topography and the BV1 layer into different levels of constrained areas via colours. However, until now, there has been no paper to discuss quantitative combinations both uncertainty factors and the post-processing applications, such as model-based mineral resource assessment (Zhao, 2007).

Following the technical route of MCUE, the prior thinking in this paper is applied to develop an improvement method (Li et al., under review), named the global optimum MCUE method, based on the global optimum truth discovery algorithm of big-data technology (Yin et al., 2008; Li Q et al., 2014a, b, Li H et al., 2014), named geometric variance truth discovery (GVTD). The purpose of this method is not only the ability to visualise uncertainties but also to quantify each the reliability of stochastic models and to be used to mineral resources assessment. A case study was conducted to demonstrate our method in Huayuan district. First, global optimum truth discovery worked out a group of weights for stochastic-plausible geological models, which makes it possible to describe different importance levels and their sequences among stochastic models. Second, the weights were integrated to reflect 3D geological model uncertainty and visualise the constrained levels of areas. Third, the isopach map from the 3D ore-bearing formation model was constrained with the weights from the Global Optimum MCUE method for strata-bounding PbZn deposit prospect mapping in Huayuan District, China.

An uncertainty estimate procedure, named MCUE, was proposed by Wellmann and Regenauer-Lieb (2012) and Pakyuz-Charrier et al. (2018) based on the Monte Carlo simulation method (Huang et al., 2019; Evren et al., 2018; de la Varga and Wellmann, 2016; Jessell et al., 2014; Lindsay et al., 2012; Mallet, 1992; Shannon, 1948). It includes three main steps, as shown in Fig. 1: (ⅰ) work out sampling disturbance functions of perturbation; (ⅱ) simulate multi-stochastic models; and (ⅲ) integrate uncertainty in models for estimation, visualisation, and upgrading. In the first step, measurements such as orientations and positions were collected and then perturbed from several different kernel functions used, such as uniform, Gaussian, and von Mises-Fisher (vMF) distributions. Especially, the vMF distribution (Hornik and Grun, 2013; Fisher et al., 1993; Mardia et al., 1976) in Formula 1 for orientations (e.g., dip and azimuth).

Figure 1.  Flowchart of MCUE (modified from Jessell et al., 2014).

where γ^T is the transposed mean direction vector, and k is the concentration, ‖ ‖ denotes the Euclidean norm, k is analogous to the inverse of σ for the normal distribution.

The second step is to feed the stochastic datasets to implicit modelling engines to construct multiplausible models. Although there is dependence on the measured values themselves, such as foliation composed of position, dip, and dip direction, each component's error of measurement is still independent. Consequently, MCUE may sample from disturbance distributions independently from one another and respect the central limit theorem (CLT) (Sivia, 2006; Gnedenko et al., 1954). In the last step, MCUE merges the stochastic models into a probabilistic model to present the measurement uncertainty.

The MCUE method mainly depends on three kinds of samples: orientation, position and foliation (both orientation and position). Previous studies (Pakyuz-Charrier et al., 2018; Wang et al., 2016; Wellmann and Regenauer-Lieb, 2012) have shown that a high level of uncertainty occurs at interfaces among different stratigraphies. This result is reasonable because interfaces were simulated by the three kinds of data; thus, more perturbation accumulates, and the uncertainty degree is higher. Conversely, the stochastic models would converge to real natural models when geologists have obtained enough datasets. Therefore, the MCUE method could be used to evaluate the uncertainty of 3D geological models.

In this paper, we proposed a global-optimal uncertainty estimate method (Li et al., under review), which provides a useful improvement in discrimination among thousands of stochastic models via reliability per stochastic model. First, stochastic-plausible models were built by disturbance operation and they are seen as unsurprised and unknown reliability sources. Second, differences in geological information (e.g., stratigraphy, fault, intrusion) at the same unit between any two models are marked with 0 and 1, where 0 means "same" and 1 represents "different". Third, the geometric variance was calculated via global optimum truth discovery, p^* is the global-optimal uncertainty, and w_i is the weight of the i-th model. Fourth, a list of ordered stochastic models based on the weights was used to assess the uncertainty and upgrade the original measurements descriptively and quantitatively. The process proposed in this paper is shown in Fig. 2 and Process I below.

Figure 2.  Flowchart of Global Optimum MCUE.

In addition, pre-processing was conducted before geometric variance truth discovery started. First, we called the 3D model constructed by the measurement dataset the measured model. Second, discrimination of geological information (e.g., stratigraphy, fault, intrusion) at the same units between measurement model and any stochastic model was marked from the short enumeration set {-1, 0, 1}, where 0 means "same age", 1 represents "different and younger", and -1 represents "different and order". Third, each stochastic model was compared with the measured model and then represented as a vector composed of {-1, 0, 1}, named the compared model. Fourth, GVTD started to run for uncertainty estimation, named the uncertainty model, as Process I shows. It was coded by Python, Protocol Buffer and business package's secondary development interface for the methodology.

Step 1. Randomly take N models from the compared model set with size N=$ \frac{4k}{\alpha {\beta }^{2}}log\frac{16{k}^{2}}{{\beta }^{2}} $ into subsets $ K(\alpha =\frac{{ϵ}^{3}\mathrm{log}n}{2{\left(\mathrm{log}\mathrm{log}(nΔ/ϵ)\right)}^{2}\mathrm{log}\left(\frac{nΔ}{ϵ}\right)} $, and $ =\frac{{ϵ}^{2}}{\mathrm{log}\mathrm{log}\frac{nΔ}{ϵ}} $).

Step 2. Sequentially enumerate all the subsets K, finding vertexes {q₁, q₂, q₃, …, q_k} of the simplex by the centre clustering analysis method, $ k=\frac{1}{{ϵ}^{2}}{\left(loglog\frac{nΔ}{ϵ}\right)}^{2} $.

Step 3. Mesh the simplex, and then interpolate each point in the simplex as a potential p^* set.

Step 4. Heuristically infer $ {p}^{*} $ by trying all the grid points by the objective function, $ min{\sum }_{i=1}^{n}{w}_{i}{‖{p}_{i}-{p}^{*}‖}^{2}, s.t.{\sum }_{i=1}^{n}{e}^{-{w}_{i}}=1 $, and output the one with the smallest value seen as $ {p}^{*} $ shown that is, i.e., uncertainty model.

In this paper, we build the geometry of measurement and modification in Step 1, Step 2, and Step 4 of the flowchart. Thousands of stochastic models were simulated via the above program. On the other hand, geometric variance truth discovery in Step 3 was coded by Python, such as the packages of Numpy, Pandas, and Scikit-learn, which calculate the weights of all models and expect their uncertainty. In addition, visualisation of uncertainty appeared in MinESoft package.

The study collected geological maps and geological survey reports, as shown in Table 1. It is important, according to aeromagnetic anomalies, large concealed rocks haven't been detected in northwestern Hunan area.

The Huayuan district contains a significant concentration of Pb-Zn mineralisation. This area is located in the Bamianshan orogen (fold belt), which is related to the Ordovician to Devonian convergence of the Yangtze Block's southeastern margin and the western part of the Jiangnan terrane (Fig. 3). The Huayuan area has a long geological history and contains sedimentary units located in the northwestern part of an arc-shaped structure in northwestern Hunan Province. The geological units in the area include a stable platform comprising an ~10-km-thick sedimentary Neoproterozoic succession, which is unconformably overlain by an ~4-km-thick succession of nonmetamorphosed early Paleozoic marine sedimentary units. The marine succession includes an ~1.8-km-thick carbonaceous succession that hosts significant Pb-Zn ore deposits. The spatially zoned Pb-Zn deposits in the area are hosted by tectonic structures in the Early Cambrian Qingxudong Formation (Cheng et al., 2011). The host rocks include algal reef limestone, dolomitic limestone, and calcarenite, containing bioclastics, oolitic limestone, oncolite, and carbonaceous rudite (Duan, 2014; Fu, 2011; Yang and Lao, 2007).

Figure 3.  Geotectonic classification in the Yangtze continental block.

The geological formations in the Huayuan District (Fig. 4) are the Early Cambrian Shipai and Qingxudong formations, the Middle Cambrian Gaotai Formation, and the Middle–Late Cambrian Loushanguan Formation (Xue et al., 2017). The Qingxudong Formation (Є₂q) is mainly exposed in the Limei, Tudiping, and Laohuchong ore deposits. Both the hanging wall sequences (striated fine dolomite) and footwall sequences (argillaceous limestone) have lower porosity and permeability than the ore-bearing strata, facilitating the formation of a fluid reservoir. Meanwhile, stratigraphic control of the mineralisation is also manifested in the following two aspects, i.e., the Banxi Group to the Cambrian strata provided metallogenic materials. In contrast, the wall rocks provided organic matter for thermochemical sulfate reduction and space for ore accumulation. Other strata are widely distributed across Bamaozhai, Taiyangshan, Changdengpo, and Yangjiazhai.

Figure 4.  Geological map of Huayuan district.

Three major folds, namely, the NE-trending Limei-Bamozhai anticline, the NNE-trending Taiyangshan syncline and the sub-EW-trending Yutang anticline, were delineated in the study area based on contours within the ceiling wall of the ore-bearing unit (Є₂q₂). The Limei-Bamaozhai anticline is located within the northern region of the Huayuan district. The fold trends 40° NE with a length of 6–7 km and a width of 3–4 km. The average dip in the anticline is 2° to 3°, and the dip in the two edges of the anticline is approximately 6° to 8°. The emergence stratum at the core of the anticline is the Early Cambrian Qingxudong Formation (Є₂q), and the edges of the anticline are composed of the Middle Cambrian Gaotai Formation (Є₂g) and the Middle–Late Cambrian Loushanguan Formation (Є_2–3ls).

Regional faults mainly trending ENE to NE include the Huayaun-Zhanjiajie fault, the northern segment of the Songtao-Zhangjiajie tectonic zone across the Huayuan area. Geological data and lithofacies paleogeographic studies show that the ore-bearing formation named Qingxudong algal limestone extends across the Huayuan-Zhangjiajie fault. Therefore, the fault is the key to providing the fluid migration channel. A series of NE-trending fractures subjected to regional faults directly control the occurrence of the orebodies at Huayuan.

According to geological survey reports (Zhang et al., 2013; Huang et al., 2011), the sedimentary strata in the Huayuan area exhibit weak to no susceptibility, which implies that there are no large concealed rocks at depth in this area.

According to study (Wei et al., 2020) of modeling-based mineral system approach to prospectivity mapping, the following sub-sections constructed 3D geometry in the districts containing structures and formations at the district scale in the Huayuan region (24×18 km²). Then the global optimum MCUE was run to predict, improve, and visualise the uncertainty of 3D geological models in Hunan Province, central-south China.

As discussed in section 3.1, the geologically modelled favourable features in Huayuan in this study case mainly contain formations and major structures, including the Huayaun-Zhanjiajie fault and Limei-Bamaozhai anticline. In this section, we describe how to evaluate uncertainty of model via global optimum MCUE.

First, the measurement was fed to an implicit modelling tool (Li et al., 2019) to restructure the 3D geological model in Huayuan. Second, this paper used vMF distribution perturbing orientations on the surface and inferred cross-sections to build stochastic models. Emphatically, it does not mean the paper's method has a priori forbidden the use of any other distribution. The vMF can be replaced with another distribution or work with other distributions of different kinds of data, such as normal distribution for locations. Third, parameters of distribution functions are given by geologists depending on their empirical values. To be more specific, perturbing azimuths of formations and faults were assigned to k=100 (approximately ±10°), and dips of formations and faults were assigned to k=120 (approximately ±5°) and k=65 (approximately ±20°), respectively. Fourth, these perturbations were used to construct stochastic models, as shown in Fig. 5.

Figure 5.  Part of stochastic models.

On the basis of the procedure in section 2.3, the uncertainty of the 3D geological model in Huayuan was estimated by the GVTD method. First, a block (cell) model was used to estimate uncertainty as a geometric basement; second, formations, lithologies, and faults were marked sequence integers in the disturbance process. For example, we supposed that the Early Cambrian Formations were equal to 1, the Middle Cambrian Formations were equal to 2, and the Middle–Late Cambrian Formations were equal to 3. However, discrimination (or diversity) metrics between the measured model and any stochastic model cannot be simply represented by Euclidean distances among comparison of sequence numbers of different age structural data. It is common two things are concerned with uncertainty when we compared any two geological data, i.e., same or different on time-space. Thus, as section 2 mentioned, this paper designed pre-processing with ages rather than distance, which defined the compared model and uncertainty model.

Finally, the uncertainty of the 3D geological model was estimated, as shown below in Fig. 6. Global optimum provides a particular advantage in that stochastic models can be ordered depending on sequential set $ W=\left\{{w}_{i}\right|{w}_{1}, {w}_{2}, \dots, {w}_{n}\} $ from larger ones to smaller ones in Table 2, and $ {w}_{i} $ can be seen as reliable metrics. The reason is that, according to the definitions and lemmas of GVTD, a larger $ w $ is closer to discovering the truth that was integrated by not only majority voting but also wisdom minority voting. This means that the global optimum MCUE method can remove unreliable models or provide a statistical standard to ask geologists that some models are more reasonable to be involved in the estimation process than others.

Figure 6.  Uncertainty of 3D geological model in Huayuan. (a) Uncertainty of model; (b) uncertainty of measured cross-section in Table 1 with fault; (c) uncertainty with younger age changing; (d) uncertainty with older age changing.

On the other hand, GVTD was compared with traditional majority voting methods such as entropy and diversity in this section. The same main points among the methods are the uncertainty that can be represented as a possibility, and higher uncertainty often happens at nearby structural contacts, i.e., faults and different formations. Meanwhile, GVTD brings two special characteristics here. One point is that the uncertainty from GVTD presents different details compared to the majority voting method. An important difference is that $ W=\left\{{w}_{i}|{w}_{1}, {w}_{2}, \dots, {w}_{n}\right\} $ can be seen as a process of dimensionality reduction to thousands of 3D models so that it finds a way to dig deeper via statistical methods; the other is that the uncertainty model can present a visualisation of uncertainty with changing age i.e., uncertainty of the younger or older shown in Figs. 6c, 6d. The interesting point in here provides insight into how uncertainty changes with time. Comparing previous methods, the GVTD asks geologists how strong uncertainty is at a position and makes different partitions to clarify younger or order uncertainty.

To verify the uncertainty of the global optimal MCUE in Huayuan district, we designed the simulation experiment as follows. First, the observations were perturbed, and 50 stochastic models were constructed, named Init50. Second, we hypothesised that geologists would have obtained enough deep data on the Qingxudong Formation so that the program would not disturb the orientation, position, and foliation related to the Qingxudong Formation and simulated 50 models, named Middle50. Third, we created 50 models in which the program would not perturb the Huayaun-Zhanjiajie fault and respected the rule in the second step, named Final50.

The experiment is shown in Fig. 7. Comparing the three results obeying a normal distribution shown in Fig. 7e, and they showed that the standard deviation of Final50 is lowest in Fig. 7c, and Init50 is largest in Figs. 7a, 7d. The performance matches the description in section 2 and verifies the uncertainty in section 3.2.

Figure 7.  Histograms of perturbation with constraints.

In the Huayuan district, algal limestone is mainly composed of algal agglomerate limestone and pebbly/sandy algal limestone, which are rich in algal decay pores and intergranular pores. These rocks are brittle, chemically active, and prone to metasomatism. Such features provide the conditions for migration and deposition of ore-forming fluids. Abundant borehole data indicate that orebodies generally occur in algal limestone that is over 150 m thick, revealing that the ore-bearing strata are only a few metres to tens of metres thick and that the mineralisation potential is low (Wei et al., 2020). Hence, we consider that the possibility of mineralisation is positively correlated with the thickness of algal limestone, and the isopach map is seen as a major favourable feature. Unfortunately, as Table 1 shows, geologists have not collected deep samples of ore-bearing formations because of technology issues. Isopach mapping based on drill holes or 3D models has many uncertainties. In this study, we integrated the weights of Init50 in Table 2 into an isopach map, as shown in Fig. 8.

Figure 8.  (a) Isopach map of measured model. (b) Isopach map of the true-value model; (c) the uncertainty of the true-value model.

The thickness of ore-bearing strata varies significantly on the southeastern side of the fault. The ore-bearing strata in middle and southern Huayuan are thicker, corresponding to the Limei and Danaopo deposits, respectively, gradually thinning to the east and abruptly to the north. Combining the Huayuan-Zhangjiajie fault movement analysis withthe ore-bearing stratoisohypse mapping, three levels of prospecting targets are delineated (shown in Fig. 9a modified by Wei et al., 2020). Meanwhile, in this paper, we integrated uncertainty of ore-bearing stratoisohypse maps into Fig. 9c so that targets were improved. Finally, this study provides additional insights for further mineral exploration in the northern part of the ore field in the future.

Figure 9.  (a) Target map in 2D; (b) isopach map of ore-bearing strata, target map in 3D (after Wei et al., 2020); (c) isopach map of true-value model, target map in 3D.

Another important advantage of set W from globally optimal value calculation is to assess the confidence of the measurement and the model constructed. In statistics, a confidence interval (CI) is a type of estimate computed from the observed data's statistics. The interval has an associated confidence level (γ) that the true parameter is in the proposed range. In this paper, if we find the stochastic models as a sample of size "n" that are taken from a population having a normal distribution, then there is a well-known application that allows a test to be made of whether the variance of the population has a predetermined value. The test statistic "T" could be set to be the sum of squares about the mean of $ \left\{{w}_{i}|{w}_{1}, {w}_{2}, \dots, {w}_{n}\right\} $ (samples), divided by the nominal value for the variance. Finally, acceptance regions for "T" for several significance levels of 5% and 1% are calculated, and then w_measurement is distinguished for acceptance or rejection.

In this paper, we use the chi-square (x²) test to evaluate confidence for inputs such as orientations and positions and outputs, i.e., model. Carmichael and Ailleres (2016) discussed inherent uncertainty due to data density and used k-means clustering and self-organising maps (SOMs) to optimise the initial dataset's information. This paper's method uses GVTD to upscale the initial dataset and assess its confidence level. In this case, the input dataset mainly points to orientations of formations and faults. Both azimuth and dip are consecutive ranges of variables to be directly fed to the GVTD program. Then, the initial dataset's confidence level is evaluated, and a group of possible datasets is inferred. On the output side, as described in section 3.2, the compared models are involved in GVTD. w_m represents the reliability of the compared model converted from the measured model. Then, the confidence level of the measured model is calculated, as shown in Fig. 10.

Figure 10.  Reliability estimation, μ is the mean value and σ2 is the variance; (a), (b) and (c) represent three groups respectively; (d) is the comparation.

As shown in Fig. 10, the mean values of the three groups from section 3.3 respected a normal distribution and implied that the more inaccurate group was, the larger the variance was and the smaller the mean value was. Specifically, by comparison, the variance of the first group is the largest, and the range of the normal distribution curve is relatively dispersed. The mean value of Middle50 is larger than that of the first group, and the variance is relatively smaller. The third group's range of distribution, Final50, was relatively concentrated and showed the largest mean value and the smallest variance value.

There are three points in Figs. 10a, 10b, 10c, and blue represents the measured model. In the first group, the measured model is located to the curve's left side, near the mean value. In the second group, the model is located to the curve's right side, near the mean value. In the third group, the value appeared at a position that was quite close to the 2-fold variance, which implied that the confidence was more than 95%.

Figure 10 shows that the reliability of the measured model in the 1st group was 38.8%, that in the 2nd group was 58.0%, and that in the 3rd group was 97.0%. Interestingly, based on 50 perturbations, the test without any deep constraint exhibited more geological diversity and greater uncertainty. This means that if 3D geological model only depended on 2D geological map data, the reliability of the 3D model would be less than 50%, as shown as 38.8% in Fig. 10a.

Another geological understanding from the 3D model is that mineralisation triggers are generally interpreted to be related to faults (Leach et al., 2010). Extensional faults and associated fractures are the most important structural controls of MVT deposits or ore fields, and they have close genetic links with compressive events in the nearby orogeny (Bradley and Leach, 2003; Leach et al., 2001; Lajaunie et al., 1997). Regional geological data and lithofacies paleogeographic studies show that the Qingxudong Formation algal limestone is widely distributed (in the NE-NNE direction) and extends across the Huayuan-Zhangjiajie fault. Therefore, the key to prospecting prediction is to find the fluid migration channel. A series of NE-trending fractures directly control the orebodies' occurrence at Huayuan, and these fractures are subjected to regional faults. The Limei-Bamaozhai-Chengdengpo-Tudiping-Laohuchong ore belt strikes NE 30° and is approximately 15 km long and 3–5 km wide. There is a 1.2 km wide gap between the Limei and Danaopo deposits, which is the interruption between two en echelon ore belts. In the future, the uncertainty of faults should be involved in prospectivity mapping in this district (Gümplová et al., 2018; Xiao et al., 2015).

This study also discussed a possible novel in the future that could be used to evaluate the reliability of observations. Essentially, stochastic models should be seen as unsurprising and unknown reliability sources. Meanwhile, we consider that the reliability of stochastic models or the measured model is not equal, i.e., some models are always closer to reality than others. As one of the important outputs of the GVTD, compared models were ordered depending on weights. The bigger w_i is, the more reliable model_i is. On the basis of set$ W=\left\{{w}_{i}\right|{w}_{1}, {w}_{2}, \dots, {w}_{n}\} $, the concept of significance level and the chi-square (x²) test were used to evaluate the reliability of the model under a confidence level of 0.95. Comparing the previous studies on uncertainty, this is the first paper attempting to quantify the reliability of geological models. Furthermore, the capability of GVTD is suitable not only for modelling but also for observation.

This paper used a big-data-based method to assess a 3D geological model's uncertainty and put the quantitative uncertainty into GIS-based prospectivity mapping for deep prospecting in Huayuan District. It is a novel way to share the extent of uncertainty via the optimal global solution and to constrain geometric uncertainty into quantitative prospectivity mapping.

One hand, this study demonstrated a quantitative process of 3D geological model's uncertainty via the global optimal value method concerning both wisdom minority and majority voting, named GVTD. Before using GVTD, three types of models were defined: measured model, compared model, and uncertainty model. The measured model, also named the initial model in other topic papers, is the geometry of input measurement modelling; the compared model is composed of the difference between the measured model and any other stochastic model, and the discrimination model represents {-1, 0, 1} (i.e., {-1, 0, 1}), the older, the same, and the younger); the uncertainty model is the smallest objective value, p^*, of process I. By the GVTD, hundreds of compared models were integrated into the uncertainty model, and the uncertainty was calculated based on a group of weights, $ W=\left\{{w}_{i}|{w}_{1}, {w}_{2}, \dots, {w}_{n}\right\} $, considering both majority voting and wisdom minority. Furthermore, the uncertainty of each cell, u_i, or the entire model, M_measured, can be assessed by this paper. In addition, another progress from the novel is to upscale the uncertainty visualisation. The variation tendency of uncertainty can be presented based on age changing. If u_i < 0, then the uncertainty in cell_i becomes old. If u_i > 0, then the uncertainty becomes younger, comparing to the measured model. If the absolute value is larger, then the uncertainty is higher. It is useful to intuitively distinguish uncertainty caused by different tectonic phenomena.

On the other hand, the method condensed a group of 3D models, including thousands of cells, into a group of weights via a dimensionality reduction algorithm. Thus, the models were replaced with the weights so that geologists could use the weights to analyse and obtain deep information about the source, level, and improvement of uncertainty of the observed datasets. This characteristic can improve the integration between geometry uncertainty and favourable parameters of GIS-based prospectivity mapping. On this basis, we integrated isopach mapping and its uncertainty and targeted in Huayuan district for strata-bounding Pb-Zn deposit.