Abstract
Conventional evaluation of mineral resource interpolation emphasizes predictive accuracy and reporting compliance, but these criteria do not directly characterize the stability of estimation outputs with respect to method choice. This study introduces an Mutual Information and Entropy Based Interpolation Uncertainty Indicator (MUI) as a diagnostic measure of method-conditioned instability. MUI is defined from normalized entropy and normalized mutual information over controlled realization ensembles, and evaluates the coherence and stability of interpolation outputs under a prescribed realization protocol. In the case study, comparable multi-realization ensembles were constructed for inverse distance weighting (IDW) and ordinary kriging (OK) for Cu estimation in the skarn and hornfels domains of a heterogeneous porphyry-skarn deposit. The results show that OK produced a less coherent and less stable realization ensemble than IDW, with the contrast most evident in skarn. At the same time, conventional benchmarking against infill drilling data indicates that both methods remained comparatively accurate in both skarn and hornfels. These results show that realization stability and interpolation accuracy describe different aspects of estimation behavior. MUI therefore provides a useful complement to conventional validation by capturing method-dependent differences that standard accuracy-based frameworks do not directly express. Although MUI does not explicitly quantify total methodological uncertainty, it can serve as a practical diagnostic of potential method-induced bias. Interpreted together with infill-drilling benchmarking and constrained resampling, the results further provide a safety-oriented assessment of interpolation behavior in terms of accuracy, robustness, and explainability.
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This research was supported by National Natural Science Foundation of China (No.42073038).
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Appendices
Appendices
Appendix A: MUI calculation and realization generation
Appendix A.1 Multi-realization construction of target methods
Spatial sample data are defined over the estimation domain, and a set of candidate interpolation or geostatistical modeling methods is defined for evaluation. Each method is treated as a distinct estimator whose compatibility with the statistical-spatial structure of the data is to be assessed.
For each method, a realization ensemble is constructed under a fixed realization protocol. Deterministic methods generate realizations through controlled perturbations of parameters, data subsets, or initialization conditions, whereas stochastic methods produce realizations intrinsically through simulation procedures. All realizations are mapped onto a common spatial support to ensure comparability.
The choice of spatial grid (support) influences the sensitivity of entropy-based measures and is therefore treated as a fixed condition throughout the analysis.
Appendix A.2 discrete state representation and entropy
Continuous outputs are transformed into a finite set of discrete states to enable consistent estimation of information-theoretic quantities.
Let \({X}_{r}\left(u\right)\) denote the value at location \(u\) for realization \(r\), and let \(S=\left\{1,...,K\right\}\) denote the set of discrete states obtained through a predefined discretization scheme (e.g., equal-width binning). The discretized state is defined as:
where \(\varnothing (.)\) is the discretization operator applied consistently across all realizations. For each spatial location \(u\), the empirical probability distribution over state is estimated from the realization ensemble:
The local Shannon entropy at location \(u\) is defined as:
To enable comparability across different discretization choices, entropy is normalized:
The domain-averaged normalized entropy is then:
This quantity represents the overall level of state dispersion across realizations.
Appendix A.3 Mutual information across realizations
To quantify structural agreement, mutual information is computed between realization fields. Let \({X}_{r}\) and \({X}_{r{\prime}}\) denote two realizations over the domain. Their joint empirical distribution is defined as:
The marginal distributions are:
The mutual information between realizations \(r,r^{\prime}\) is:
To ensure comparability, mutual information is normalized:
where
This normalization ensures \(0\le M(r,r^{\prime})\le 1\) and provides a consistent scale for comparing structural agreement across realizations. While alternative normalization schemes exist, this formulation is adopted for stability and interpretability in discretized empirical settings.
The average normalized mutual information across all realization pairs is:
Appendix A.4 Definition of the mutual information-entropy-based interpolation uncertainty indicator (MUI)
Based on the two complementary components defined above, The Mutual Information-Entropy-Based Interpolation Uncertainty Indicator (MUI) is defined as:
Based on the definition, \(0\le MUI\le 1\).
A.5 Limiting Cases
If realizations are identical: \({\overline{H}}^{*}=0, \bar{M}=1 \Rightarrow MUI=0\);
If realizations are highly dispersed but structurally consistent:
If realizations are both dispersed and structurally inconsistent:
MUI is conditional on discretization, spatial support, and realization design.
A higher MUI value indicates weaker compatibility between method assumptions and data characteristics under the prescribed realization protocol. It should therefore be interpreted as a diagnostic signal of potential method-induced bias, rather than as a direct measurement of methodological bias. Conversely, a lower MUI reflects stronger convergence among realizations and higher method-data compatibility.
Appendix B Method-Conditioned MUI Behavior (Heuristic Approximate)
This appendix provides method-conditioned, first-order approximations of MUI behavior under simplified assumptions. These expressions are not closed-form derivations but serve as interpretive tools grounded in the formal definition:
All method-specific approximations in this appendix are derived by examining how a given method class tends to affect the normalized state-dispersion and the normalized structural-consistency. The use of MUI under this assumption helps identify when a method is well-suited to the data spatial structure.
Appendix B.1 Geometry-based methods
Geometry-based methods (e.g., nearest-neighbor, triangulation and polygon partitioning) are assumed to have uncertainty mainly arising from geometric interpretation.
However, the methods do not generate realizations intrinsically. Realization ensembles are constructed through perturbations of sample configuration or partition boundaries.
Under stable partitioning, state assignments remain invariant (Simplified conditions of spatially uniform sampling and stable partition geometry):
Under irregular sampling or unstable boundaries (clustered sampling pattern, irregular boundaries, or local support is poorly constrained):
Also, under these conditions, a first-order heuristic approximation can be interpreted as:
MUI scales with geometric sensitivity to data configuration and partition stability.
Appendix B.2 Variogram-based and kriging methods
Variogram-based methods (e.g., simple kriging, ordinary kriging, indicator kriging) impose an explicit covariance structure on the data.
Realizations are constructed through perturbations of variogram parameters, search neighborhoods, or data subsets.
Let \({\delta}_{\gamma}\) denote the mismatch between the covariance model and the spatial organization of the data.
Under well-specified conditions small \({\delta}_{\gamma}\):
\({\overline{H}}_{0}^{*}\) is the baseline normalized entropy under well-aligned conditions.
For small mismatch, first-order approximations can be written as:
where \({c}_{1}\), \({c}_{2}>0\) are method and dataset-dependent proportionality constants:
In the near-ideal regime \({{\updelta}}_{{\upgamma}}\ll 1\).
This indicates that MUI increases linearly with covariance or anisotropy mismatch in the near-ideal regime.
Stronger mismatch from stationarity, anisotropy, or variogram stability will produce nonlinear propagation in both entropy and structural inconsistency.
Appendix B.3 Simulation-based methods
Simulation-based methods (e.g., sequential Gaussian simulation and conditional smoothing) generate realization ensembles intrinsically to represent conditional spatial variability. Since the realizations are generated from a Gaussian process, the variability in method outputs is generally constrained within a certain range.
Let \({\rho}_{cond}\in \left[\text{0,1}\right]\) denote the effective conditioning imposed by data and model constraints, the realizations are expected to exhibit high entropy due to the inherent stochastic nature of the method, but the structural agreement (mutual information) remains stable unless conditioning becomes weak.
Under well-conditioned simulation:
Under weak conditioning (sparse data or unstable covariance):
Thus, for simulation-based methods, MUI is controlled by the extent to which dispersion is accompanied by loss of structural coherence, rather than dispersion alone.
Appendix B.4 Bayesian methods
Bayesian approaches generate realizations through posterior sampling governed by a prior–likelihood structure.
Let \({{\updelta}}_{{\uppi}}\) denote the effective mismatch between prior assumptions and data-supported structure.
Under prior–data consistency:
For small mismatch:
For weak mismatch, \({\delta}_{\pi}\ll 1\), this reduces to.
This shows that MUI increases when posterior realizations lose structural coherence due to prior–data inconsistency or insufficient data support.
Appendix B.5 Data-Driven or Machine-Learning Methods
In data-driven methods, spatial structure is learned implicitly from data rather than imposed through explicit covariance models.
Let \({\delta}_{fit}\) denote instability arising from training sensitivity, feature dependence, or overfitting.
Under stable training:
Under instability:
Thus, MUI scales with training instability and sensitivity to perturbations in data representation.
Appendix B.6 Scope of the approximations
These method-conditioned heuristic approximations derived from the unified MUI formulation provide a structured interpretation of how estimation mechanisms influence entropy and mutual information. Conversely, the use of MUI under methods assumptions helps identify whether a method is well-suited to the data spatial structure.
Their role is interpretive and diagnostic: to guide understanding of MUI behavior across method classes and to support practical evaluation in technical and commercial settings.
Appendix C: Preprocessing and MUI interaction
Preprocessing alters the statistical and spatial structure of input data and therefore influences MUI through both entropy and mutual information components. Changes in MUI under preprocessing should be interpreted as the combined effect of improved method–data alignment and potential modification of underlying variability.
Appendix C.1 Clustering and denoising of geological data
Let \(X=\left\{{x}_{1}, {x}_{2},...{x}_{n}\right\}\) be the set of samples with grade \(g\left(x\right)\).
Clustering reduces intra-group variance and compresses the range of discretized states.
For a realization ensemble\(\left\{{s}_{r}\right(u\left)\right\}\), the empirical distribution \({\widehat{p}}_{u}\left(k\right)\) is:
This leads to a reduction in entropy and consequently in MUI.
However, excessive clustering or outlier removal may suppress genuine geological heterogeneity, artificially stabilizing the realization ensemble. In such cases, reduced MUI reflects information loss rather than improved structural consistency.
Appendix C.2 Spatial discretization and mesh segmentation
Spatial discretization defines the support over which probability distributions are constructed. Let \(r\) denote grid resolution. Changes in \(r\) affect empirical state distributions:
Finer grids → reducing the variance within each cell → lowering entropy.
Coarser grids → increased aggregation → higher entropy.
Thus:
This dependence reflects scale conditioning rather than methodological deficiency.
Excessively fine discretization under limited sampling density may produce artificially low entropy by isolating sparse observations, while overly coarse discretization may inflate entropy by aggregating heterogeneous regions. Both cases distort the interpretation of method-output stability.
Appendix C.3 Covariance and model calibration
Adjusting covariance structures in kriging-based methods can alter the model’s response to spatial data variations. Overfitting covariance parameters to minimize MUI can result in smoother estimates with reduced variability, which misrepresents geological heterogeneity.
The mutual information \(I({X}_{r};{X}_{r{\prime}})\) between realizations r and r’ can be impacted by the optimization of covariance parameters:
Overfitting covariance parameters may reduce marginal entropy but also reduce joint consistency, potentially lowering \(\bar{M}\) in a non-intuitive manner.
Thus, minimizing MUI through parameter tuning does not necessarily imply improved model validity; it may reflect over-smoothing or reduced structural variability.
Appendix C.4 Interpretation
Changes in MUI induced by preprocessing reflect two possibilities, improved compatibility between method assumptions and data structure, or suppression or distortion of inherent variability. Therefore, reductions in MUI should not be interpreted as direct evidence of reduced uncertainty. Instead, MUI should be evaluated jointly with preprocessing choices to distinguish genuine stabilization from artificially induced consistency.
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He, X., Huang, Y., Wu, B. et al. Diagnosing method conditioned bias in mineral resource estimation using a mutual information and entropy uncertainty indicator. Sci Rep (2026). https://doi.org/10.1038/s41598-026-53627-9
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DOI: https://doi.org/10.1038/s41598-026-53627-9
