In deep learning-based MPM, ore-controlling features sometimes should be transformed into evidential layers according to mineral deposit models, which are then integrated to predict mineral potential by learning their relationships with known mineralization. However, this process inherently introduces uncertainty due to incomplete understanding of mineralization processes, insufficient spatial resolution for finer modeling grid, and training dataset construction (Zuo et al., 2021). In this study, data uncertainty is represented by stochastic simulation of evidential layers, including geological, geochemical, and geophysical features.

CNNs rely on both positive and negative samples to learn discriminative features that distinguish mineralization from non-mineralized backgrounds. In practice, known mineral deposit locations are used as positive samples, while negative samples are typically selected randomly from non-mineralized areas (Carranza et al., 2008). However, the definition of non-mineralized regions is inherently uncertain, as geological processes are continuous and ore-forming environments may extend beyond the boundaries of known mineral deposits. In this study, the uncertainty associated with negative samples selection is modeled by randomly sampling points from regions located at least 10 km from known mineral deposits, ensuring minimize potential mineralization influence. The 10 km exclusion distance was determined empirically based on the geological characteristics of the study area and is consistent with the maximum mineralization control distance adopted in this study. The effects of negative sample selection on model performance can then be quantified, thereby providing a measure of data uncertainty arising from training sample preparation (Zuo and Wang, 2020).

For uncertainty quantification, variance and entropy are two of the most widely used statistical measures. Given T stochastic predictions ptt=1T, where pt is the probability predicted by the Bayesian CNN, the variance can be expressed as: 6Varp=1T∑t=1T(pt-p‾)2, where p‾ denotes the mean prediction across all T forward passes at each location (i,j). Variance reflects the dispersion of predictions around the mean, indicating the degree of disagreement among stochastic realizations.

Entropy, introduced by Shannon (1948) in information theory, measures the expected amount of information contained in a probability distribution. It has been extensively applied in geoscientific modeling to characterize uncertainty in spatial predictions (Wellmann and Regenauer-Lieb, 2012). Mathematically, entropy is defined as (Shannon, 1948): 7Hp=-∑t=1Tptlog⁡pt.

In this study, entropy quantifies the expected uncertainty of the predicted probability distribution for mineral occurrence. Thus, a higher entropy value indicates a more diffuse probability distribution, reflecting greater uncertainty and a lower level of confidence in the predicted prospectivity map.

The case study area is situated in western Henan Province of China (Fig. 4). This region is recognized as one of the country's most important gold-producing areas, hosting more than 81 large to super-large size of gold deposits (Deng and Wang, 2016; Fan et al., 2016; Zhang et al., 2020). Such findings highlight substantial potential for further gold exploration (Mao et al., 2002a; Mao, 2005; Tang et al., 2013). These gold deposits are predominantly magmatic-hydrothermal in origin, formed mainly during the Early Cretaceous under an extensional tectonic regime of the Yanshanian period (Mao et al., 2002b). According to the established mineral deposit models (Wu et al., 2012), mineralization and its spatial distribution are primarily controlled by regional faults, Yanshanian intrusions, and the strata of the Xiong'er and Taihua rock groups. Specifically, ore-forming materials are sourced from the deep mantle, while intrusions of intermediate to acidic magmatic rocks supplied the thermal energy for fluid migration. Faults served as major conduits for fluid transport, ultimately leading to gold mineralization within the Taihua and Xiong'er Group formations (Lu et al., 2002; Pang et al., 2020).

Geochemical and aeromagnetic data provide indirect indicators or proxy evidence of regional geophysical and elemental anomalies linked to gold mineralization. The geochemical exploration dataset was sourced from the Chinese National Geochemical Mapping Project (Xie et al., 1997), comprising of 1131 stream sediment samples collected at a 1:200 000 scale. The original stream sediment geochemical data were conducted at an average density of one sample per 1 km², after which four adjacent samples were composited to represent a 4 km² area grid cell (Xie et al., 1997). Each sample contains concentrations of 39 major and trace elements, determined using analytical techniques such as inductively coupled plasma atomic fluorescence spectrometry, X-ray fluorescence, and inductively coupled plasma atomic emission spectrometry (Xie et al., 1997). Geochemical exploration data are typical compositional data, which can induce spurious spatial and statistical correlations among elements (Zuo et al., 2013). Accordingly, geochemical exploration data were preprocessed using the centered-log ratio transformations to eliminate the influence of closure effects (Aitchison, 1982). The geophysical data were acquired from the China Geological Survey, comprising aeromagnetic gridded raster at the same scale. To reduce distortions associated with skewed magnetization, the vertical first-order derivatives were applied to enhance shallow magnetic anomaly signatures. The resulting raster was subsequently used as the geophysical input for simulation. Consequently, 44 mineralization-related evidential features, including faults, Yanshanian intrusions, Xiong'er and Taihua rock groups strata, geophysical aeromagnetic data, and geochemical data, are extracted to construct the evidential layers for MPM (Fan et al., 2023; Yang and Zuo, 2024; Yang et al., 2024b).

The generation of geological evidence layers relies on buffer analysis combined with weighting functions. This procedure involves counting the number of mineral deposits that fall within different buffer zones surrounding geological features, thereby quantifying their spatial correlation with mineralization. The resulting spatial correlation patterns are subsequently used to derive weighting functions for corresponding evidence layers. Specifically, singularity indexes α are calculated from log–log plots of paired datasets [ρ,d], yielding values of 1.324, 1.74, 1.093, and 1.37 for faults (Fig. 5a), Yanshanian intrusions (Fig. 5b), and the Xiong'er (Fig. 5c) and Taihua (Fig. 5d) rock groups strata, respectively. Fractal analysis suggests that the Xiong'er group strata and faults exert a stronger spatial influence on gold mineralization compared with other geological features. This finding is consistent with established mineral deposit models and existing geological knowledge (Wu et al., 2012), thereby supporting the reliability of the geological evidence layer construction.

The uncertainty of geological evidential features can be modeled by randomly sampling the parameter dm within a predefined interval. In this study, the uncertain dm, representing the maximum controlling distance of a geological feature, is empirically assigned as a random value within the estimated interval [5, 10]. Accordingly, Monte Carlo sampling is performed by randomly selecting dm values from the defined intervals, generating 100 weighting functions (Fig. 6) and corresponding weighted evidential layers for each geological feature (Fig. 7). These multiple realizations not only capture the variability arising from weighting function but also provide a probabilistic representation of the spatial relationships between geological features and mineralization, thereby offering a more robust characterization of data uncertainty.

To model the spatial distribution patterns of geochemical elements, the DS algorithm is employed to transform discrete geochemical measurements into spatially continuous evidential layers on a 500 m grid from the original 2 km scale. This enhancement in spatial resolution inevitably introduces data uncertainty associated with the estimation of unsampled locations. In this study, geochemical layer serves as both conditioning data and the TI for the simulation process. Two parameters, the similarity threshold t and the maximum fraction f of the TI to be scanned, are empirically set to 0.1 and 0.7, respectively. In this regard, a total of 100 realizations of geochemical maps are generated with a spatial resolution of 500 × 500 m, each representing an independently simulated distribution pattern of element concentrations. These multiple simulations infill unsampled locations while preserving the spatial structures of the original datasets. The spatial distribution of Au geochemical samples and three randomly selected simulated evidential layers are shown in Fig. 8a. Geophysical evidence layers are produced following the same simulation procedure. Figure 8b presents three representative simulated aeromagnetic evidence layers.

Furthermore, to evaluate the reliability of the simulated realizations, cumulative frequency distribution and experimental variogram curves are employed to compare the similarity between observed and simulated patterns. Using Au as an example, the cumulative frequency distribution curve (Fig. 9a) clearly demonstrate that the DS reproduces the statistical distribution pattern of geochemical features in the study area. Experimental variograms, which characterize the spatial continuity and autocorrelation of geoscientific data, are used to assess the preservation of spatial structure. The close agreement between the variogram curves of the observed and simulated realizations (Fig. 9b) further indicates that the DS approach successfully reproduces the major spatial structures of the original geochemical layer. These results demonstrate that the simulations preserve not only the statistical distribution of the evidential layers but also their spatial dependence, enabling the generation of realistic geochemical evidence layers while simultaneously accounting for uncertainty associated with unsampled regions.

It is worth noting that the application of DS to geochemical and geophysical data also has limitations. These variables are generally continuous and may exhibit anisotropy and heterogeneous spatial characteristics, which may not be fully represented by a single TI. Consequently, the quality and representativeness of the TI can strongly influence the simulated realizations. Moreover, when DS is applied to pre-interpolated geophysical data, the resulting simulations may inherit part of the smoothing effects and spatial assumptions introduced during the initial interpolation, thereby potentially propagate interpolation-related bias into the simulated realizations.

In total, 44 evidence layers are constructed as model inputs, including geological features (faults, intrusions, strata), aeromagnetic data, and geochemical data comprising 39 elemental concentration maps. Each evidential layer is simulated 100 times, providing sufficient realizations to quantify data-related uncertainty in MPM.

The Bayesian CNN architecture (Table 1) comprises three 3×3 convolutional layers, three normalization layers, two 2×2 max-pooling layers, and two fully connected layers. Each convolutional layer is followed by a dropout layer to facilitate uncertainty estimation. The final output layer employs a sigmoid activation function to produce probability predictions for MPM. Key hyperparameters, including the learning rate, batch size, number of epochs, activation function, and optimizer, are listed in Table 2. These parameters are adjusted through grid search (Liashchynskyi and Liashchynskyi, 2019). Model performance is evaluated using binary cross-entropy loss and classification accuracy. It should be noted that the sensitivity analysis of these parameters is beyond the scope of the present study.

A 9 × 9 sliding window is used to draw training samples from the evidential layers. This approach ensures that each training sample encompasses not only the central pixel but also the contextual information from its surrounding 9 × 9 neighborhood. Within this window, each pixel is characterized by 44 evidential features derived from geological, geochemical, and geophysical datasets, forming a comprehensive feature space that enables CNN to learn spatial-attribute relationships associated with mineralization. A total of 483 positive samples with dimensions of 9 × 9 × 44 are extracted from 51 known gold mineral deposits and their surrounding neighborhoods within a 3 × 3 km radius. This data augmentation strategy has been demonstrated effectively to improve the generalization ability of deep learning algorithms in MPM (Li et al., 2022; Zuo, 2025). To ensure dataset balance, an equal number of negative samples are also randomly generated from areas distant from known mineralization during each training stage. Of the total samples, 80 % are used for model training, while the remaining 20 % are reserved for validation.

A deterministic CNN is first trained using the original evidence layers to validate the effectiveness of the proposed model's architecture and parameters settings. The training convergence curves of loss and accuracy over epochs (Fig. 10) indicate that the deterministic model achieved stable and satisfactory training performance. The resulting prospectivity map of gold deposits (Fig. 11) reveals that high-potential areas closely align with the spatial distribution of known mineral deposits. The outcomes provide additional evidence supporting the reliability and efficacy of the proposed CNN architecture for MPM.

Data uncertainty is quantified by generating multiple simulated evidence layers with randomly selected negative samples, thereby capturing variability arising from incomplete geological knowledge, relatively coarse spatial resolution, and randomness of training labels. A deterministic CNN model is trained on simulated evidence layers to produce 100 prospectivity predictions. Figure 12 presents the mean, variance, and entropy maps across these predictions. The mean map (Fig. 12a) represents the most probable spatial distribution of mineralized potential by averaging the ensemble predictions. High mean values indicate areas of consistently high prospectivity, which are primarily distributed along major faults and within the Xiong'er rock groups strata. These regions closely coincide with known gold mineral deposits, confirming that CNN model effectively captures the key mineralization patterns. Most gold deposits in the study area are quartz-vein and altered-rock types (Mao et al., 2002b), mainly controlled by faults which act as pathways for mineralizing fluids, while the surrounding strata provide favorable host environments, jointly contributing to the high prospectivity in these regions.

The variance and entropy maps reveal different aspects of data uncertainty induced by the stochastic simulation of evidence layers. The variance map (Fig. 12b) quantifies the degree of variability among the 100 prospectivity predictions under varying evidence inputs. Regions with low variance indicate stable predictions and high confidence. Conversely, areas with high variance demonstrate strong sensitivity to data perturbations, suggesting that small variations in evidence layers can substantially affect the predicted probabilities. Although both metrics display broadly similar spatial patterns, the entropy map (Fig. 12c) offers additional insight beyond variance by characterizing the uncertainty in the probability distribution of predictions. In other words, entropy captures the degree of ambiguity in predictions, even when the magnitude of variation is moderate. Overall, areas exhibiting high mean and low entropy values represent the most reliable exploration targets, while those characterized by high prospectivity but high uncertainty deserve further geological investigation.

Model uncertainty is quantified using the Bayesian CNN trained on the original evidence layers, reflecting the inherent randomness in network architecture and parameters estimation. The resulting mean, variance, and entropy maps (Fig. 13) provide a comprehensive view of both predictive outcomes and the associated uncertainty in deep learning-based MPM. High-probability regions in the mean map (Fig. 13a) appear primarily distributed within the Xiong'er rock groups, implying favorable conditions for gold mineralization. The strong spatial correspondence between high-probability regions and known gold mineral deposits confirms that the Bayesian CNN model effectively captures the principal ore-controlling geological features of gold mineralization. The variance (Fig. 13b) and entropy maps (Fig. 13c) exhibit similar spatial patterns. Areas with low variance and low entropy generally coincide with high mean probabilities and known gold mineral deposits, reflecting strong model confidence and stable predictive behavior. Conversely, regions with high variance and entropy indicate greater model uncertainty, implying limited model knowledge or insufficient feature representation for inferring mineralization potential in those regions.

Notable differences are observed between the data and model uncertainty maps. Data uncertainty primarily arises from the heterogeneity and stochasticity of evidential layers, whereas model uncertainty reflects the limitations of the network in learning stable and generalizable relationships from the data. Overall, data uncertainty appears broader and more pronounced than model uncertainty, implying that variability in the evidence layers exerts a greater influence than model's architecture or parameter settings. This finding further suggests that, in geologically complex regions, CNN model faces challenges in accurately interpreting gold prospectivity when the input data themselves are highly uncertain.

The total predictive uncertainty represents the combined effects of both data and model uncertainties, derived through stochastic simulation of evidence layers with a Monte Carlo dropout-based Bayesian CNN. Specifically, 100 stochastic realizations of the evidence layers are generated to characterize data uncertainty. For each realization, 100 Monte Carlo Dropout stochastic forward passes were performed to sample model uncertainty. In this way, 10 000 predictions are generated, enabling a comprehensive characterization of the total uncertainty. The mean, variance, and entropy maps are shown in Fig. 14. Compared with the individual uncertainty quantification, the joint consideration of data and model uncertainties yields smoother and more spatially coherent patterns that align better with the distribution of known gold mineral deposits. The mean prospectivity map (Fig. 14a) preserves the main high-prospectivity areas identified in the data and model uncertainties, highlighting the stabilizing effect achieved by combining model stochasticity with variations in evidential inputs. The variance map (Fig. 14b) indicates that the total predictive uncertainty effectively captures variability arising from both geological features and CNN model, thereby enhancing predictive robustness. The spatial distribution of predictive entropy (Fig. 14c) provides direct implications for exploration strategy and model refinement. High entropy regions suggest either input data or model limitations, whereas low entropy areas represent priority targets because the model provides a low-risk predictive indication. Importantly, multiple predictions allow the model to account for variations in both evidence layers and network architecture, resulting in highly consistent variance and entropy patterns across the study area.

To quantify the relative contribution of data and model uncertainties to the total predictive uncertainty in MPM, a dominant uncertainty map is presented using the variance decomposition (Grömping, 2007). This statistical approach enables the partitioning of total predictive variance into components attributable to data variability and model stochasticity, thereby providing a clearer understanding of uncertainty sources and supporting more informed exploration decisions. The data uncertainty (Vardata) exhibits geologically meaningful patterns, reflecting the influence of heterogeneous simulated evidence layers. In contrast, the model uncertainty (Varmodel) primarily captures limitations of the network architecture and parameters, highlighting areas where CNN struggles to generalize. Then, the relative contribution of each component can be calculated as their ratio to the total predictive uncertainty (Vartotal).

Compared with the uncertainty maps in Figs. 12 and 13, the uncertainty decomposition maps (Fig. 15) identify the dominant source of predictive uncertainty rather than its magnitude at each location. Obviously, high data uncertainty contribution (Fig. 15a) exhibits broader spatial influence, whereas high model uncertainty contribution (Fig. 15b) is more localized with lower magnitude. This finding suggests that variability in the evidence layers is the primary source of total uncertainty. Moreover, the clustering of data uncertainty around known mineral deposits further reflects its dominant contribution to total uncertainty. Such decomposition provides additional diagnostic information beyond individual uncertainty maps and helps identify whether future efforts should focus on improving input data quality or model development. Consequently, enhancing the quality and representativeness of input evidence layers is likely to be more effective in reducing total MPM uncertainty than solely optimizing CNN architecture or parameters.

To investigate the relationship between predictive probability and uncertainty, the mean predictive probability and uncertainty were jointly analyzed using a pixel-wise scatter plot. The mean-uncertainty scatter plots (Fig. 16) show the relationship between the mean predictive probability and predictive uncertainty in the study area, providing a direct visualization of how data uncertainty and model uncertainty vary with the mean predictive values. In this inverted U-shaped distribution plot, each scatter point represents one pixel of the study area, with the x-axis indicating the mean predictive probability obtained from multiple stochastic forward passes and the y-axis representing the corresponding entropy value. Data uncertainty exhibits a generally greater magnitude than model uncertainty, indicating that variability in the geological evidence layers contributes more to the predictive uncertainty than instability in the model's architecture and parameters. The area under the receiver operating characteristic curve (AUC) (Hanley and McNeil, 1982) is another robust metric widely adopted to quantitatively measure model performance, particularly in the presence of class imbalance. Figure 17 presents the frequency distribution of AUC values derived from 100 realizations for data uncertainty and model uncertainty. Both distributions are centered around AUC ≈ 0.95, indicating that CNN model maintains consistently high predictive accuracy across multiple simulations. However, the wider spread of the data uncertainty curve reflects greater variability in predictive performance due to fluctuations in the evidence layers. In contrast, the narrower spread of the model uncertainty curve demonstrates that the models' architecture and parameters remain relatively stable during stochastic forward passes. Such results further emphasize the importance of optimizing mineral deposit models and weighting strategy by identifying the most influential geological features, as they exert more influence on predictive uncertainty.

To further facilitate the exploration target prioritization, an exploration reliability map (Fig. 18) is delineated by integrating both the mean predictive probability with predictive uncertainty: 8Rp=p‾*-H(p)*, where p‾* and H(p)* denote the normalized mean predictive prospectivity and normalized predictive entropy, respectively. This normalization places both variables on the same scale, allowing predictive potential and reliability to be considered simultaneously. Areas with high values represent locations correspond to both high prospectivity and low uncertainty, and are therefore regarded as the most reliable exploration targets. Compared with the mean prospectivity map alone, this map suppresses regions with high uncertainty while preserving areas with consistently high prospectivity, thereby providing a more reliable basis for exploration decision-making.

Uncertainty quantification provides essential insights into the reliability of MPM. On the one hand, it delineates regions characterized by low uncertainty and high prospective probability, thereby guiding exploration efforts toward areas with the greatest potential of new mineral discoveries. On the other hand, it facilitates the interpretation of spatial relationships between mineral prospectivity and major geological features. By decomposing total uncertainty into data uncertainty and model uncertainty, the proposed framework establishes a quantitative basis for evaluating the credibility of predictive outcomes. Such differentiation enables the identification of whether uncertainty stems from inappropriate evidence layers or model limitations. Specifically, regions with high prospectivity and low uncertainty represent high-confidence exploration targets, whereas high uncertainty areas correspond to high-risk but potential opportunities. Consequently, uncertainty quantification not only enhances the interpretability of MPM but also supports informed decision-making in mineral exploration.

Future advances in uncertainty quantification for MPM are expected to focus on improvements in data quality, modeling strategies, interpretability, and decision optimization to reduce predictive uncertainty. From a data perspective, enhancing the geological representativeness of evidence layers and developing more reliable strategies for negative sample selection are critical for reducing data uncertainty and more accurately capturing the spatial variability of mineralization controls. Regarding the uncertainty modeling, progress should extend beyond conventional Monte Carlo dropout toward Bayesian inference and ensemble learning capable of jointly characterizing both data and model uncertainties. In terms of the uncertainty interpretability and visualization, embedding geological knowledge directly into deep learning architectures and loss functions offers a promising way to constrain model training, reduce high-uncertainty output, and enhance the transparency of predictions (Zuo et al., 2023, 2024). In addition, feature attribution is also valuable for identifying which evidential layers drive most to high prospectivity or high uncertainty, thereby strengthening geological interpretability (Dwivedi et al., 2023). Finally, integrating uncertainty quantification with risk-return analysis will support the selection of exploration targets that balance mineralization potential against geological risk, translating probabilistic predictions into more actionable decision support (Wang et al., 2020).