Introduction

Mineral resources are solid materials of economic value that occur naturally in or on the earth’s surface in which their quality and quantity support economic extraction (Stephenson et al., 2006; Joint Ore Reserves Committee, 2012). They comprise mineralized zones in varying concentrations across rock mass. Accordingly, mineral resource estimation comprises evaluating the quality, quantity, continuity, location and orientation of geological features to ensure reliable estimates. These parameters are measured directly from drillhole data or indirectly using geophysical tools. Reliable estimation of these variables and their economic viability is critical for planning and executing sustainable mining operations (Rossi & Deutsch, 2014). However, uncertainties are generally associated with mineral resources before (inherent uncertainty) and during (parametric uncertainty) the estimation process (Rossi & Deutsch, 2014; Nurmi & Rasilainen, 2015; Gandhi & Sarkar, 2016a; Haldar, 2018). These uncertainties emerge from the heterogeneity of subsurface formation, insufficient knowledge of subsurface conditions, sampling, data handling and insufficient mathematical and conceptual representations of applicable procedures (Stanley & Smee, 2007; Tartakovsky et al., 2012; Rossi & Deutsch, 2014). Uncertainty may cause financial loss due to underestimation or overestimation of grade (Li et al., 2008; Dominy & Edgar, 2012; McManus et al., 2021a). For example, a 15% discrepancy in grade estimation can result in significant loss or project failure because mining production consumes between 40 and 70% of mine revenue (Dominy & Edgar, 2012). Also, in some cases, uncertainty in the resource estimation phase has been linked to mine plan changes, operation stoppage or decrease in production rate and hence production loss (McManus et al., 2021b). Mineral resource reporting guidelines require uncertainty assessment and communication due to used data, deposit geology and estimates of mineral resources (Joint Ore Reserves Committee, 2012; McManus et al., 2021b). Quantified uncertainty can help determine drilling targets, estimate and classify mineral resources and mining planning, decide on capital allocation, and provide detailed information to investors and insurers (Dimitrakopoulos, 2005; Tajvidi et al., 2015; Witter et al., 2019).

Uncertainty in mineral resources can be encountered at both sampled and unsampled locations. The inherent uncertainty comes from the natural variability of mineral deposits while the parametric uncertainty comes from sampling procedures, processing of data and application of mathematical models to quantify uncertainty (Bárdossy & Fodor, 2001; Sagar et al., 2018). Due to the potential consequences of uncertainty, mineral resources are estimated, and the uncertainty of estimates is assessed using different methods. The method widely used to report uncertainty in mining industry is via mineral resource classification. The estimated mineral resources are classified into measured, indicated and inferred categories, and uncertainty is reported based on the level of confidence (Glacken & Snowden, 2001; Joint Ore Reserves Committee, 2012; Revuelta, 2017; Cevik et al., 2021). Other methods for assessing uncertainty can be grouped into kriging (Monteiro da Rocha & Yamamoto, 2000; Rossi & Deutsch, 2014), probabilistic (Emery, 2008; McManus et al., 2020; Riquelme & Ortiz, 2021), geostatistical simulation (Caers, 2011; Abzalov, 2016; Paithankar & Chatterjee, 2018) and machine learning (ML) (Fouedjio & Klump, 2019; Mery & Marcotte, 2022b). The development and application of these methods in resources and uncertainty assessment went through different historical phases. The stages include initial formative stage between the 1950s and 1970s (kriging methods), mature stage between the 1980s and 2000s (geostatistical simulation methods) and rapid development stage from the 2010s (ML methods) (Liu et al., 2022). These methods are scattered in the literature without systematic compilation that can serve as reference knowledge base.

The idea of compiling numerous methods used to quantify uncertainty in mineral resource estimation was inspired by similar studies in other subjects such as the compilation of statistics on soil properties by Phoon & Kulhawy (1999), rock properties database and ranges by Aladejare & Wang (2017), mechanical properties of high entropy alloys (RHEAs) and refractory complex concentrated alloys (RCCAs) by Couzinié et al. (2018) and comprehensive collection of empirical formulas for calculating uniaxial compressive strength (UCS) based on the mechanical and physical characteristics of rocks by Aladejare et al. (2021). Numerous methods for quantifying uncertainty are readily available in the literature, but no study has compiled them systematically for easy selection and application. To solve this problem, this paper compiles uncertainty quantification methods applied for mineral resource estimation.

Uncertainty propagates and accumulates because of interdependence among the different steps in the estimation process (Minnitt, 2007). This interdependence exists because the output of one stage is input to the next step. Figure 1 illustrates this ideal phenomenon even though there is a complex relationship among uncertainties at each step in the estimation process. This shows a necessity for uncertainty assessment in each step of estimation. Bond (2015) and McManus et al. (2021b) reviewed uncertainties in the interpretation of geological data and spatial domain models, respectively. Both reviews fall short of demonstrating stepwise uncertainty propagation and uncertainty beyond spatial domains. Therefore, to provide a global compilation of uncertainty quantification methods throughout the estimation steps, a thorough review was conducted to gather and compile these methods between 1991 and 2024. This review concentrates on probabilistic, simulation and ML methods because kriging provide error variance, which considers only the configuration of sampled data and ignores variable value effect and hence cannot be used to assess global uncertainty (Rossi & Deutsch, 2014; Ortiz, 2020). This study is crucial for the mining industry especially for mineral resource estimation experts and other personnel because it simplifies the availability, selection and application of uncertainty quantification methods and close monitoring of uncertainty propagation and accumulation in different estimation steps.

Figure 1
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Stepwise process of uncertainty propagation and accumulation in mineral resource estimation (not drawn to scale)

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Knowledge Base Development and Description

A knowledge base of different methods used to assess uncertainty in mineral resource estimation was created from a thorough literature review. In total, 87 research articles from reputable journals, textbooks or book chapters, technical reports and theses, dating from 1991 to 2024, were used to assemble information about uncertainty in mineral resource estimation. The assembled information from research articles, textbook or book chapters, technical reports and theses are summarized in Table 1. The summary shows that: 10 journal papers are from Natural Resources Research, 8 journal papers from Ore Geology Reviews, 9 textbooks or book chapters, 5 journal papers from Mathematical Geosciences, 4 journal papers from Computers and Geosciences, 3 journal papers from the Journal of Mining and Environment and so on (Table 1). The research articles are published in reputable journals in the mineral resource estimation field, peer-reviewed and considered reliable. Moreover, the references used were searched to ensure that the terms ‘mining,’ ‘minerals,’ and ‘uncertainty’ are found in their titles, abstracts or keywords. Moreover, the case studies reported in the references were from different countries worldwide, making the assembled knowledge base global (Table 1).

Table 1 Summary of journal articles, theses, textbooks and technical reports information applied to develop the knowledge base (year 1991–2024)
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Methods for Uncertainty Quantification in Mineral Resource Estimation

Mineral Resource Classification and Uncertainty Reporting

Mineral resources are classified to establish desirable orebody portions in a deposit for further exploration and investigation of the possibility of upgrading them to mineral reserves. Numerous international classification systems such as the Canadian Institute of Mining, Metallurgy and Petroleum (CIM) Guidelines, South African Code for Reporting of Mineral Resources and Mineral Reserves (SAMREC) and the Australasian Joint Ore Reserves Committee (JORC) Code have been developed and used over many years, and among them, the JORC system is the most prominent and widely adopted system (Emery et al., 2006; Joint Ore Reserves Committee, 2012). Estimates of mineral resources can be classified into measured, indicated or inferred resources, as geological knowledge and confidence increases and by consideration of other factors such as mining, processing, economic, environmental and legal (Fig. 2). In addition, Figure 2 shows a relationship between mineral resources and reserves. The resource classification is based on quantitative parameters and qualitative decisions due to the experience of technical personnel.

Figure 2
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Simplistic relationship between exploration results, mineral resources and reserves (source: JORC, 2012)

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Different methods and criteria are used for classifying mineral resources and reporting their uncertainty based on the mining industry's accepted guidelines and regulations (Glacken & Snowden, 2001; Joint Ore Reserves Committee, 2012; Rossi & Deutsch, 2014; Revuelta, 2017; Cevik et al., 2021). These methods include the 15% rule, whereby estimated grade, metal concentrate and tonnage will have at most 15% error at a confidence interval (CI) of 90%. For example, Tajvidi et al. (2015) applied this method to classify a multi-element deposit with Cu, Mo and Ag in combination with coefficient of variation. Another method used to classify resources is based on quantified uncertainty whereby if a block is known within an error of ± 15% at a 90% CI, it is classified as measured resource, between ± 15 and  ± 30% as indicated resource, and above ± 30% as an inferred resource (Battalgazy & Madani, 2019; Verly & Parker, 2021).

Exploration drillhole spacing can be used for classification as well. In this method, when small blocks are estimated and used to classify mineral resources, a block with at least three drillholes with no extrapolation can be classified as measured mineral resources, a block with two to three drillholes and minimum extrapolation can be classified as indicated mineral resources and one drillhole with reasonable extrapolation is classified as an inferred mineral resource. Also, resources can be classified using ML methods. Cevik et al. (2021) classified mineral resources using random forest and support vector machine methods. Other criteria/methods for resource classification include search neighborhood, kriging variance, projection pursuit multivariate transform, conditional variance, relative kriging variance and relative conditional variance (Emery et al., 2006; Silva & Boisvert, 2014; Battalgazy & Madani, 2019). Therefore, mineral resource classification is based on the confidence of estimates whereby high confidence implies less uncertainty and low confidence shows high uncertainty (Tajvidi et al., 2015; Cevik et al., 2021; Verly & Parker, 2021). However, uncertainty reported due to resource classification often lacks the inclusion of other sources of uncertainty such as geological domaining and grade spatial variability (Verly & Parker, 2021).

Geostatistical Techniques

Geostatistics deals with the spatial distribution of data, i.e., each data value is linked with a location in space in which there are connections between the data value and location. The unknown samples at different locations are interpolated/extrapolated or simulated using known data points (Chiles & Delfiner, 2012; Daya, 2015b; Oliver & Webster, 2015). Geostatistics describes quantitatively natural variable distribution over time or in space. The variables may be, among others, grade of mineral deposits, geological layer thickness and depth, porosity and permeability in porous media (Caers, 2011; Chiles & Delfiner, 2012). Estimation using geostatistics is possible because of the spatial correlation among samples taken nearby that are dependent on each other. According to Daya (2015a) and Abzalov (2016), geostatistical techniques are used for characterizing spatial variation, spatial data modeling, simulation, spatial interpolation, optimizing sampling and characterizing uncertainty. Hence, geostatistical techniques provide the mechanism to quantify uncertainty in mineral resource estimation (Chiles & Delfiner, 2012; Rossi & Deutsch, 2014).

Geostatistical methods can be classified into three categories, namely linear kriging approaches, nonlinear kriging (probabilistic) methods and simulation methods (Chiles & Delfiner, 2012; Rossi & Deutsch, 2014). Linear kriging is a statistical method used for interpolation/extrapolation of known samples to predict values at unknown locations during mineral resource estimation. The common types of linear kriging methods include simple kriging (SK), ordinary kriging (OK), universal kriging (UK), kriging with external drift (KED) and factorial kriging (Krige, 1985; Cressie, 1989; Goovaerts, 1997; Abzalov, 2016). Linear kriging methods are famous for quantifying optimal weights based on minimizing expected error variance. Linear kriging expresses uncertainty using an error variance based on data configuration, hence providing a less practical assessment of uncertainty.

Nonlinear kriging methods (also referred to as probabilistic methods) use the nonlinear function of data to estimate in situ and recoverable resources (Vann & Guibal, 1998; Triantafilis et al., 2004; Rahimi et al., 2018a, 2018b). They are similar to linear kriging methods but use nonlinear transformation of real data (Rahimi et al., 2018a, 2018b). These methods have become popular due to their ability to estimate unknown attribute values that exceed a particular threshold value, reduce smoothing effect (Emery, 2008; Rahimi et al., 2018a, 2018b; Liu et al., 2022) and assess uncertainty based on probability distribution (Manchuk et al., 2015). Methods belonging to this category include multi-Gaussian kriging, disjunctive kriging (DK), probability kriging, multiple indicator kriging (MIK), lognormal kriging, uniform conditioning and residual indicator kriging (Vann & Guibal, 1998; Emery, 2008; Rahimi et al., 2018a, 2018b; Liu et al., 2022).

Geostatistical simulation has been adopted as a mineral resource estimation method because kriging techniques are heavily affected by smoothing effect and produce deterministic solutions (Goovaerts, 1997; Ortiz, 2020). Geostatistical simulation methods produce a set of realizations for each geological domain, depending on the parameter being estimated (Asghari et al., 2009; Paravarzar et al., 2015; Rahimi 2018a, 2018b; Ortiz, 2020). The generated realizations depict the uncertainty of regionalized variables at each location (Mery & Marcotte, 2022b), in a block or domain (Emery et al., 2006; Pyrcz et al., 2006; Ortiz, 2020). Simulation, unlike kriging, reproduces the histogram or the trend of a model and data variability, and hence, it provides a quantification of both local and global uncertainty (Harrison et al., 2009; Liu et al., 2022). Because of this capability of simulation methods, their application has increased significantly. However, a model of uncertainty obtained using simulation methods relies on conditioning data characteristics, the random function model utilized and the deposit type (Ortiz, 2020). The most adopted and applied simulation technique is sequential Gaussian simulation (SGS) (Ortiz, 2020; Liu et al., 2022). It is based on an assumed multi-Gaussian random function model and Gaussian conditional distribution. SGS combines simple kriging (which determines the local distribution of uncertainty) and Monte Carlo simulation (MCS) (which draws a simulated value randomly from distribution). A high number of realizations is suitable for uncertainty assessment; nonetheless, 20–50 realizations are generally adequate to characterize the probable range of simulated values (Harrison et al., 2009; Rossi & Deutsch, 2014).

Other common geostatistical simulation techniques include: (1) turning bands simulation (TBS), in which data are transformed into Gaussian distribution first, can reproduce histogram and variogram from transformed data, it can be used to model multi-element deposits, it preserves the realizations variability of unconditional simulation, and it honors the conditioning data (Paravarzar et al., 2015; Abzalov, 2016); (2) matrix decomposition method, which is applied when the amount of realizations needed is high and the sum of conditioning data and the quantity of nodes to be simulated is low (at most a few thousands); (3) truncated Gaussian simulation; (4) sequential indicator simulation (SIS); (5) sequential indicator co-simulation (SIcS); and (6) plurigaussian simulation (Bastante et al., 2008; Ortiz, 2020; Liu et al., 2022). Methods numbered (1) and (2) are alternatives to SGS and can produce multi-Gaussian random functions (Davis, 1987; Paravarzar et al., 2015). Methods numbered (3) to (6) are often used to simulate categorical variables codifying geological domains (Rossi & Deutsch, 2014). Figure 3 illustrates the percentage of each class of methods application to assess uncertainty among the 87 articles in the knowledge base. So far, probabilistic, ML and kriging methods have been applied moderately. The most applied methods to assess uncertainty are simulation techniques due to their ability to provide multiple solutions compared to other methods (Fig. 3).

Figure 3
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Frequency of methods used to assess uncertainty associated with mineral resource estimates

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ML Techniques

To transform and automate the mining industry, different tasks have been undertaken using ML methods. For the past two to three decades, the mining industry adopted ML in mineral resource estimation and prospecting (Dutta et al., 2010; Abuntori et al., 2021; Jung & Choi, 2021; Chudasama et al., 2022; Liu et al., 2022). ML techniques are divided into supervised and unsupervised learning whereby each class tend to train, test and apply models. The solution is known during training for supervised learning and unknown for unsupervised learning (Samson, 2020; Dumakor-Dupey & Arya, 2021; Zhou, 2021). Mineral resources have been estimated using different ML methods (Rodriguez-Galiano et al., 2015; Hyder et al., 2018; Dumakor-Dupey & Arya, 2021). ML applications tend to reduce human interference and hence reduce bias (Samson, 2020). Compared to kriging methods to produce accurate estimates, ML methods consider few assumptions and can take a relatively small amount of data when optimized (Li et al., 2013; Soltani-Mohammadi et al., 2022). ML was adopted due to weaknesses in the geostatistical methods, which require an experimental variogram and fitted model, assumptions, skills and long processing time (Soltani-Mohammadi et al., 2022). Even though ML methods do not assume stationarity in estimation, they are data-driven and are form of regression; hence, they are unable to reproduce data (Samson, 2020). Dumakor-Dupey & Arya (2021) studied ML techniques in mineral resource estimations and found that artificial neural networks (ANN), random forest (RF), support vector machine (SVM) and neuro-fuzzy are the most applied methods. ML can also be applied to assess uncertainty associated with resource estimation. This is conducted by quantifying errors between predicted and observed data (real-world process). ML approaches show mineral resource uncertainty using CI of estimates (Solomatine & Shrestha, 2009; Mery & Marcotte, 2022a, 2022b). Mery & Marcotte (2022a) studied uncertainty in recoverable mineral resources using ANN and multiple linear regression (MLR) trained by geostatistical simulations.

The ML models used to assess uncertainty are affected by noise and inference errors. Data used to develop them can be erroneous, noisy and mismatched between training and testing data for model development and cause uncertainty (Abdar et al., 2021). Also, uncertainty associated with an estimated variable can be caused by errors in different learning and approximation steps along the estimation process (Hüllermeier & Waegeman, 2021). Despite the broad adoption of ML in estimating mineral resources, few ML methods have been used to assess associated uncertainty (Mery & Marcotte, 2022a).

Advantages, Limitations and Applicability of Uncertainty Quantification Methods

The applicability, advantages and limitations of different methods for uncertainty quantification depend on deposit type (geological context), data available and expertise of technical personnel. Kriging methods, which are simple and straightforward to implement, can provide the best guess of a value being estimated (deterministic solution) at an unsampled location but they do not provide alternative values (Caers, 2011; Chiquini & Deutsch, 2020). Kriging methods yield a smoothed reality representation, and this underestimates real data variability (Bastante et al., 2008; Rossi & Deutsch, 2014). They are variogram-based, often assume stationarity and cannot capture the spatial complexity of a deposit (Yamamoto, 1999, 2000; Oliver & Webster, 2015; Paithankar & Chatterjee, 2018); in particular, they cannot model uncertainty across domain boundaries.

Probabilistic methods may use experience (prior probability) at the beginning of exploration to support deposit study and analysis. They can be used to model uncertainty in numerous types of deposit including new deposits provided they have similar geological formation and characteristics as other studied and known deposits. In the early stages of exploration and resource estimation, very few data and knowledge related to deposit are available. This complicates the assessment of resources and uncertainty quantification. Probabilistic methods such as Bayesian approximation can be used even with few data due to their ability to consider opinion of an expert. However, they cannot be applied independently but in conjunction with other methods (McManus et al., 2020).

In contrast to kriging methods, geostatistical simulation methods provide multiple solutions (Caers, 2011). However, conditional simulation (CS) methods are heavily affected by stationarity and multi-Gaussian distribution (Mery & Marcotte, 2022b) and require skills to apply. Numerous geostatistical methods have been used to assess uncertainty in different complex deposits. For example, multiple-point statistics (MPS) is mostly applied in a structurally complex mineral deposit because it can model finely intercalated folded lithologies, which are challenging with classical methods. It can provide good estimates even with sparse data (Hong & Oh, 2021; van der Grijp et al., 2021). MPS uses a geological conceptual model (training image) to assess uncertainty in the geology and mineralization envelope of a deposit with complex nonlinear features (Jones et al., 2013). This method is useful because it can overcome the limitations of conventional two-point spatial models (Paithankar & Chatterjee, 2018). In addition, plurigaussian and multi-Gaussian have been used to model uncertainty in multi-element grades of interest in numerous rock types within a deposit (Mery et al., 2017) to overcome the weakness of single-variable simulation methods (Tajvidi et al., 2015).

ML methods have been used to assess uncertainty in different geological contexts. They are advantageous and applicable because of the availability of powerful computers for their execution. They consider few assumptions and can model uncertainty in any deposit, even deposits with multiple variables, e.g., the variable of interest and auxiliary variables (Fouedjio & Klump, 2019). ML methods are safe from the influence of stationarity and multi-Gaussian distribution (Mery & Marcotte, 2022b). The application of ML in uncertainty quantification is still trailing compared to other tasks in mining industry.

Due to the potential impact that uncertainty can have on estimates, researchers have realized the need to identify uncertainty sources, quantify them and examine their propagation, and look for reduction and control mechanisms. Understanding where most of the uncertainties come from and their contribution to total uncertainty is necessary for directing more efforts toward their management (Solomatine & Shrestha, 2009). Therefore, as discussed in the following sections, two major sources of uncertainties related to mineral resources are classified as (1) uncertainties at sampled locations and (2) uncertainties at unsampled locations.

Uncertainty at Sampled Locations

Sampling is undertaken to collect data necessary to describe the material of interest such as mineral deposits. The collected samples are analyzed to establish the deposit constituents through physical investigation in site, crushing and laboratory testing (Dominy & Petersen, 2005; Stanley & Smee, 2007; Dominy, 2016). The collected information comprises geophysical and geochemical data that can be divided into two groups, i.e., continuous and categorical data (Emery & Silva, 2009). On the one hand, categorical data include information such as lithology, mineralogy, alteration, rock textures, description of mineralization and rock structures. These data are used in creating geological models, domains of mineralized bodies and block models. On the other hand, continuous data such as geochemical values (e.g., metal grades), petrophysical information (e.g., density) and coordinates in XYZ space are used to estimate resource grade and tonnage and help to define deposit domains (Emery & Silva, 2009; Caers, 2011). These data may display varying properties depending on the sampling rate, accuracy and uncertainty (Karpatne et al., 2018).

A large amount of data is often required to estimate resources (Narciso et al., 2019). This is because a mineral resource model is close to truth when large amounts of data are used (Wilde, 2010). However, collecting data in abundance is difficult and expensive due to budget and technical constraints. This forces data for resource estimation to be collected at spaced locations (drillholes) with the objective that the collected sample population represent the entire mineral deposit as well as possible (Li et al., 2008; Fouedjio & Klump, 2019). The sparse data often obtained are not adequate to acquire enough knowledge about the entire deposit properties (Rossi & Deutsch, 2014; Sadeghi & Cohen, 2023). The raw collected data have different distribution characteristics due to outliers and skewness. This compels data transformation before their application in estimation of variables at unsampled locations. Data distribution for estimating mineral resources varies depending on the method of estimation and purpose for estimation. Some methods such as SGS require a Gaussian distribution of data before their application. The compositing and transformation of data may in turn contribute to the uncertainty of a mineral resource (Abzalov, 2016).

Therefore, the uncertainty of data is a result of inaccurate and imprecise measurements arising from, among others, sample collection, preparation and the whole data handling process, inherent randomness, simplification requirement and inadequate amount of data (Wellmann et al., 2010; Schweizer et al., 2017; Soares et al., 2017; Fouedjio & Klump, 2019). These uncertainties require an appropriate assessment to eliminate or reduce their adverse impact on mineral resource estimates (Sketchley, 1998; Stanley & Smee, 2007; Caers, 2011) and a mining project's potential profit (Gursuren et al., 2018). As explained in the following subsections, uncertainties at sampled locations originate from three sources, namely geological logging, geochemical and geophysical data, and drillhole positioning.

Uncertainty Due to Geological Logging

Geological logging is performed to establish different rock types or ore types and their boundaries along with other geological parameters that control and characterize the rocks including potential mineralized bodies during exploration drilling (Gazley et al., 2014; Revuelta, 2017). Logging results are used to create geological models and domains. The results drawn from logging may become inconsistent due to the subjective nature of reporting by different geologists’ judgment, experience, drilling methods and varying logging conditions (Hill et al., 2021; McManus et al., 2021a). These factors introduce uncertainty to geological logging outputs necessary for geological modeling (Revuelta, 2017; Hill et al., 2021). Uncertainties associated with geological logging can be assessed using QA/QC (quality assurance/quality control). The samples sent to laboratory for analysis are compared with those logged from core samples and other previously known results. The lower the discrepancy the lower the uncertainty, and vice versa. Bayesian approximation can be used to quantify interpretation uncertainty through simulation and sampling of the posterior probability distribution derived from logging results and classify drillhole intercepts (McManus et al., 2020). The lower the probability range, the lower the uncertainty. Other sources and approaches for uncertainty assessment in connection with geological logging are shown in Table 2.

Table 2 Techniques for uncertainty assessment at sampled locations
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Uncertainty Due to Sample Collection and Analysis

The quality of collected data during sampling can be compromised by the drilling equipment and method used (e.g., reverse circulation, diamond core drilling, trenches or pits) (Sketchley, 1998; Rossi & Deutsch, 2014; Witter et al., 2019). In addition, sampled data accuracy can be affected by conditions of sampling such as the presence of groundwater or fractures, drillhole inclination relative to orebody dip, contamination of samples in the drillhole or during handling and processing (Dominy et al., 2002; Abzalov, 2011). Hence, the process of sampling to obtain representative data is susceptible to a variety of errors as explained in detail by Dominy & Petersen (2005) and Dominy (2016). These sampling issues can make data uncertain and consequently affect the quality of a resource estimated using this information (Dominy, 2007).

Collected samples are processed and analyzed to determine their minerals constituents and enable their application in predicting values at unsampled locations but are affected by numerous factors. According to Revuelta (2017), the challenges faced when processing and analyzing geochemical data that contribute to uncertainty include some elements being below-used instrument detection limit and reported as a value below the limit or zero, missing element value in a sample of a multi-element deposit, and uncertainty due to experimental methods and equipment. Also, the time interval in data collection durations due to technological change and biases in sampling programs contributes to uncertainty in geochemical data (Harrison et al., 2009).

There are good quality QA/QC procedures in place, such as insertion of duplicates and blanks into sample batches sent for geochemical analysis (Sketchley, 1998). These procedures are used to test the laboratory analyzed results and assess uncertainty (Smee et al., 2024). Discrepancies between laboratory and already known results of formally tested samples, blanks and duplicates indicate the presence of uncertainty, expressed as fundamental sampling error (Dominy & Petersen, 2005). However, lack of precision can still affect the quality of data due to the randomness of the sampling process and collection of unrepresentative samples (Stanley & Smee, 2007; Smee et al., 2024). Other methods for assessing uncertainty include the probability distribution of measurement orientation and fault contact points (Table 2).

Uncertainty Due to Drillhole Positioning

Drillhole positioning refers to the location, dip and spacing between drillholes in an area being studied (Wilde, 2010). Areas rich with minerals are often heavily sampled compared to other places, and this can add bias and hence uncertainty to the outcomes of the process (Caers, 2011). Heavily sampled areas tend to have closely spaced drillholes. This makes uncertainty lower in such areas compared to sparsely sampled areas. However, erroneous sampling can cause uncertainty even at sparsely accurately sampled locations (Wilde, 2010; Rossi & Deutsch, 2014). The influence of each sample at a particular location is weighted relative to other samples in estimating values at unsampled locations (Caers, 2011). The spacing of drillholes beyond the variogram range displays high uncertainty compared to drillholes within the variogram range (Wilde, 2010). The possibility of drillholes intersecting mineral veins depends on their dips or inclinations, spacing between drillholes and strike(s) of the mineral deposit. Therefore, uncertainties due to drillhole positioning include, among others, bias sampling in mineral-rich areas, probability of intersecting minerals, inclination and spacing between drillholes (Dominy & Johansen, 2004). Sometimes, drillholes are drilled perpendicular to mineral outcrops to increase the possibility of intersecting minerals and reduce uncertainty. These uncertainties can be assessed using different methods. For example, Rossi & Deutsch (2014) used the distance function to assess uncertainty due to drillhole distances and vein intersection. This uncertainty was expressed in terms of bandwidth range. Other methods include SGS via generation of realizations in different spacings and the application of probability distribution (Table 2).

Uncertainty at Unsampled Locations

Mineral resource estimation methods use variables in sampled areas to estimate variables in unsampled places. The initial conditions that should be specified to estimate resources include block size, amount of data to estimate a block and search ellipsoid radii (Nowak & Leuangthong, 2017). Block dimension definition must adhere to geological characteristics, mining method and equipment anticipated and the quest to estimate accurately using available data. Block size is influenced by distance between drillholes, mine planning needs and equipment used (Abzalov, 2016; Maleki et al., 2022). Compared to small blocks, large blocks are easy to estimate and their estimated grades are close to true grades (Rossi & Deutsch, 2014). Uncertainties that impact block size definition comes from domain boundaries, search neighborhood and conditional biases (Nowak & Leuangthong, 2017; Talesh Hosseini et al., 2023) and, hence, affect precision of estimates in unsampled areas.

Collected data from sampled locations are interpolated/extrapolated or simulated to estimate variables in unsampled areas (Rossi & Deutsch, 2014; Haldar, 2018; Afeni et al., 2021; Dumakor-Dupey & Arya, 2021; Bergström, 2022). In addition to uncertainty due to block size definition, uncertainty at unsampled areas that impact mineral resource estimation originate from the geology of a deposit, continuous variables, interrelation between different continuous variables, interrelation between continuous and categorical variables and applied mathematical models (Bárdossy & Fodor, 2001; He et al., 2011; Soares et al., 2017; Wang et al., 2017; Wellmann & Caumon, 2018; Wang et al., 2023). The uncertainty types or sources at unsampled locations are discussed in the following subsections.

Uncertainty Associated with the Geology of a Mineral Deposit

A mineral resource model requires a geological model that considers rock types and other deposit characteristics (Rossi & Deutsch, 2014). The geology of a deposit forms the basis for accurate geological and mineral resource model. The geology of a deposit of interest is examined using geophysical, geochemical and remote sensing data (He et al., 2011). These data are used to create a geological model using approaches that interpret geological variables on plan maps and cross sections before extending them to generate 3D volumes. This model represents the spatial layout of rock types and lithology of similar characteristics and it is considered a crucial component in mineral resource estimation (Fouedjio et al., 2021; Abildin et al., 2022). A geological model supports other subsequent mining processes such as mine design, planning, mine development, production scheduling and investment decisions (Emery, 2006; Li et al., 2008). However, geological models can be affected by sparse data, subsurface geology and knowledge of the person dealing with the deposit (Bárdossy & Fodor, 2001; Emery, 2006; Stoch et al., 2022). In addition, the insufficient resolution of geological continuity and assumptions may contribute to uncertainty in resource estimates (Dominy et al., 2002). Also, many deposits are made of complex geology with inherent uncertainty, which may lead to overestimation or underestimation of mineral resources (Abuntori et al., 2021; Soltani-Mohammadi et al., 2022; Zaki et al., 2022). Most mineral deposits are divided into geological domains to simplify study and estimation.

A geological domain refers to an area or volume in which the characteristics of the mineralization are more similar than outside it (Glacken & Snowden, 2001; Sterk et al., 2019). The decision on where to establish domain boundaries is based on lithology, mineralized zones, alteration, structural boundaries or other unique properties (Harrison et al., 2009; Rossi & Deutsch, 2014). A geological domain helps to determine areas for further exploration, plan how and which material to extract and time of extraction, and set the needs for other mining operations (Leuangthong & Srivastava, 2012; McManus et al., 2020, 2021b; Fouedjio et al., 2021). The source of uncertainty in a spatial domain model is the geological setting interpretation from data, expert interpretations, their boundaries and shortage of knowledge in areas with little or no data (Karpekov, 2016; Fouedjio et al., 2021; McManus et al., 2021a, 2021b).

The uncertainties encountered in data, the geology of a deposit and domaining introduce uncertainty to resource estimates, which may cause financial loss or even premature mine closure (Li et al., 2008). Therefore, assessing the uncertainty of deposit geology and created domains is important to estimate accurate mineral resources. Geodomain uncertainty is assessed mostly using different modeling approaches such as SIS, MPS, truncated Gaussian simulation, plurigaussian simulation, extreme gradient boosting (XGBoost) and Bayesian approach (Chiles & Delfiner, 2012; Maleki et al., 2016; McManus et al., 2020; Fouedjio et al., 2021). The latter approach uses probability to show uncertainty about unknowns being assessed and is particularly useful in the early stages because of shortage of data for geostatistical simulation (McManus et al., 2020, 2021a). The main approaches used to assess uncertainty (i.e., first type of uncertainty) due to deposit geology and domain are listed in Table 3, which shows numerous uncertainty types, causes of uncertainty and methods used for assessment.

Table 3 Types of uncertainty associated with mineral resource estimation at unsampled locations
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Uncertainty Associated with Continuous Variables

Continuous variables used to estimate mineral resources include grade, density, coordinates and elevation. Estimated grade and tonnage are the most important continuous variables for deciding the viability of mining projects. This is because they determine the profit that can be obtained (Emery & Silva, 2009). The accuracy of grade and tonnage can be affected by, among others, nugget effect, continuity, the mineral element being primary (element of interest) or secondary, accuracy of grade testing equipment and amount of data in some highly/less sampled parts of the deposit (Hosseini et al., 2017; Talebi et al., 2019). These factors cause uncertainty to models developed from continuous variables such as mineral resource (Emery et al., 2006). For example, overestimated ore grade due to uncertainty can lead to overinvestment of a project and have high expectations and vice versa for underestimation (Li et al., 2008). Therefore, these uncertainties are assessed to reduce their negative impact on resource models.

Many kriging methods can be used to quantify the grade and tonnage of a mineral deposit and associated uncertainty. For example, the spatial distribution of continuous variables across regions of interest and its uncertainty can be assessed using KED and quantile regression forest (QRF) given auxiliary information in sampled areas (Fouedjio & Klump, 2019). However, they are incapable of capturing the spatial complexity of a deposit. One of the approaches that can be used to overcome this weakness is MPS, which can generate multiple equiprobable orebody models and assess associated tonnage uncertainty, while SGS can be used to assess grade uncertainty (Paithankar & Chatterjee, 2018). SGS can be used to model continuous variables and assess uncertainty within a model in any scale through generation of multiple solutions across a study area (Harrison et al., 2009). Other methods for assessing mineral resource uncertainty resulting from continuous variables include MCS (Manchuk et al., 2015), CS (Emery et al., 2006), mining selectivity and information effect (Chiquini & Deutsch, 2020) (Table 3, types of uncertainty in continuous variables).

Uncertainty Associated with Interrelation Between Different Continuous Variables

Continuous variables such as grade, tonnage, density and coordinates are interrelated, and they influence the mineral resource estimate. Their interrelation comes from the presence of similar conditions during geological formation and deposition (Paravarzar et al., 2015; Hosseini et al., 2017) and dependency on each other for estimation purposes. The best example of interrelation among continuous variables can be demonstrated by a multi-element deposit. In this kind of deposit, few minerals are often the main target, and others are secondary minerals that might not be economical to mine. However, because they are interrelated, the estimation of one variable can be influenced by other variables, which makes their simultaneous consideration necessary. Failure to include multiple interrelated elements in the estimation process add uncertainty to a mineral resource. Another interrelation between continuous variables is vividly seen in a grade of an orebody, which determines mineable tonnage. Inaccurate grade often lowers or increases deposit tonnage and hence uncertain mineral resource. Therefore, the estimation of mineral resources should be performed in consideration of multiple minerals in a deposit, their interrelationships and other variables. Multivariate and joint simulation methods are preferred in deposits with more than one mineral of interest. For example, min/max autocorrelation factors (MAF) was applied to simulate Cu, Mo and Ag grades simultaneously and assess uncertainty via multiple realizations. Also, Cu grade–tonnage curve was used to assess uncertainty in using different cutoff grades (Tajvidi et al., 2015). Mery et al. (2017) used multi-Gaussian simulation to assess the uncertainty of multiple elements of interest in an Fe ore deposit, and Hosseini et al. (2017) used block-support sequential Gaussian co-simulation to assess the uncertainty associated with a deposit with Cu and Mo grades. Other methods used to assess uncertainty from continuous interrelated variables are shown in Table 3.

Uncertainty Associated with Interrelation Between Categorical and Continuous Variables

An ore deposit is modeled through categorical variables (e.g., rock types, lithology) and continuous variables (e.g., geochemical and mineral compositions). Both sets of variables are interdependent and crucial for estimating mineral resources and modeling their joint uncertainty (Harrison et al., 2009). For example, the creation of grade block models in the Kittilä gold deposit mine in Finland was affected by undulating structures present in the deposit (Bergström, 2022). Geostatistical simulation for categorical data can be used to create domains and assess uncertainty in the location of geological boundaries via the generation of multiple realizations. Then, continuous variables can be simulated independently in each created domain (Emery & Silva, 2009; Talebi et al., 2017). Domains simulated from continuous variables are defined as mineralized zones based mainly on grade variability (Sterk et al., 2019; Abildin et al., 2022). The boundaries connecting constructed domains can be categorized as soft or hard. A domain boundary is termed hard when the parameters on both sides of the boundary change suddenly. Due to the discontinuity of parameter variability in adjacent domains, the estimation or simulation process is straightforward for hard boundary domains. In contrast, when the parameter variability or grade across the boundary is gradual and makes a smooth graph, the boundary is termed soft. In this case, the estimation process is difficult because grade variability across boundaries must be taken into account. Therefore, determining domain boundaries and their uncertainty is important for accurate estimation (Kasmaee et al., 2019; Abildin et al., 2022).

Some studies assessed the uncertainty emanating from interrelation between continuous variables (multi-element deposit) and deposit geology. Mery et al. (2017) used an Fe deposit with six grades of elements of interest to assess geological uncertainty using plurigaussian and multi-Gaussian simulations. The approach was hierarchical because the rock types were simulated first before the simulation of continuous variables in each rock-type domain. Plurigaussian assessed lithology uncertainty via realizations and multi-Gaussian assessed uncertainty of continuous variables (Emery & Silva, 2009). Talebi et al. (2019) used MPS with a direct sampling algorithm to joint model continuous and categorical data. Also, Jones et al. (2013) used MPS in combination with SGS to assess combined geological and grade uncertainty in mineral resource estimation. More methods used to assess uncertainty emanating from interrelation between continuous and categorical variables are shown in Table 3.

Uncertainty Associated with Mathematical Models Applied in Estimation

Despite many methods and developed models being acceptable to quantify uncertainty in mineral resource estimation, they are still susceptible to ease/difficulty of applicability, assumptions made and weaknesses of each method. Model uncertainty entails the inadequacy of mathematical functions and associated assumptions used to generate mineral resource estimates (Talebi et al., 2022). The model quality strongly depends on the type of integrated geological data and its associated uncertainties (Schweizer et al., 2017). For instance, in the case of model parameters, the simulated variables/data depend on conditioning data that can be affected by scarcity and irregular sampling (Montoya et al., 2012). The final model is sometimes created from multiple sub-models of different variables in a deposit. Used sub-models can be associated with uncertainties due to uncertain experimental data and assumptions made, which propagate to output/final models (Abdelfatah et al., 2018).

Many methods can be used to model uncertainty associated with mineral resources under different circumstances. Despite being very useful, most of these methods have some weaknesses/limitations that contribute to uncertain estimates. For instance, the Bayesian method cannot be used as a standalone method but in combination with other methods. This method relies heavily on expert opinion (Singer & Mosier, 1981; McManus et al., 2020, 2021a). Uniform conditioning (UC) estimates mineral resources using conditional data and is affected by an incorrect variogram model (Rossi & Deutsch, 2014; Chiquini & Deutsch, 2020; Mery & Marcotte, 2022a). CS can assess regionalized variables uncertainty by generating multiple realizations; however, it often depends on stationarity, Gaussian distribution assumptions and a defined variogram model. Also, the CS uncertainty model depends on the random function (RF) model used and its estimates are affected by smoothed values produced by kriging (Bastante et al., 2008; Mery & Marcotte, 2022a). SGS is based on a strong assumption of stationarity, assumes multi-Gaussian distribution and poorly reproduces the connectivity of low and high values (De Souza et al., 2004; Harrison et al., 2009; Sadeghi & Cohen, 2023). SIS can reproduce connectivity between high and low values but reproduces only images/structures with two-point statistics (De Souza et al., 2004; Bastante et al., 2008). TBS is a versatile and fast non-conditional simulation but post-processing for conditioning is difficult (Rossi & Deutsch, 2014; Abildin et al., 2022). MPS can be used to reproduce structures with higher-order statistics using training image (TI), capable of representing variability and characterizing the relationship among multiple points. However, the quality of the result depends on the chosen TI, which represents the studied phenomenon and often assumes stationarity (Bastante et al., 2008; Muhammad & Glass, 2011; van der Grijp et al., 2021).

The ML models applied can likewise be uncertain. For example, Abdar et al. (2021) and Gawlikowski et al. (2021) studied uncertainty associated with neural networks and deep learning methods. In these studies, uncertainties associated with deep learning model must be quantified along the process as follows (Gawlikowski et al., 2021): (i) selection and collection of data for training of a model; (ii) accuracy and comprehensiveness of data for training; (iii) an awareness of the deep learning model performance limits; and (iv) uncertainties associated with model performance based on operational data.

Few developed models can be useful to fulfill a designed objective. Hence, their performance must be tested to enable selection of useful ones (Caers, 2011). Model performance is influenced by the amount of available data. This scenario was studied by Schweizer et al. (2017), who assessed model performance using information entropy. This was conducted by adding more information continuously while observing changes in uncertainty and resulting models at discrete locations. The applied approach examined how a model performs with the use of available information and associated uncertainty and the improvement of model quality after the incorporation of new information (Schweizer et al., 2017). Huang et al. (2022) used backdrop convolutional neural network to assess uncertainty due to predictive model. This uncertainty was expressed in variance range and was due to the fact that the entire test data area was not covered by the training data; just a subset of known gold deposits and non-deposit sites were included. Therefore, conceptual and mathematical models developed to estimate mineral resources and quantify uncertainty are prone to errors as well. The uncertainties due to deficiency of developed model to assess uncertainty has been investigated using other methods (Table 3, uncertainty in the mathematical model part).

Uncertainties in Recoverable Mineral Resources

Estimating recoverable mineral resources aims to establish mineable orebodies in which their grade, tonnage and continuity become known at acceptable CIs. A mineable part of a mineral resource is identified after introducing a cutoff grade, and size and shape of a deposit are subject to current and expected economic conditions (Paithankar & Chatterjee, 2018). The estimation is carried out after competent personnel confirms that all other dependent factors such as sampling, geological interpretation and geological modeling are correct (Dunham & Vann, 2007; Revuelta, 2017). Estimated resource models consist of tonnage and grade above a cutoff grade and continuity. Grade–tonnage curves help to determine suitable cutoff grades (Tajvidi et al., 2015; Mery & Marcotte, 2022a). The over-/underestimation of grade and tonnage impacts mining plans, and hence, their uncertainty is assessed and incorporated into mining planning (Paithankar & Chatterjee, 2018).

The uncertainties in the different steps of estimation propagate and accumulate in recoverable resources. The grade and tonnage uncertainty in a recoverable resource is assessed by generating various realizations using simulation approaches (Chiquini & Deutsch, 2020). The varying grade and tonnage depict uncertain recoverable resources and, hence, the need to consider this phenomenon in production planning. Chiquini & Deutsch (2020) included information effect and mining selectivity in the simulation of grades to honor possible more information obtained during mining and grade control. Tajvidi et al. (2015) assessed grade–tonnage curve uncertainty using CS generated realizations. Mery & Marcotte (2022a) used a combination of CS and MLR to assess tonnage resource uncertainty. Paithankar & Chatterjee (2018) used MPS to generate orebody model realizations to assess tonnage uncertainty and SGS to assess grade uncertainty. Mery & Marcotte (2022b) used MLR and a multilayer neural network to assess grade–tonnage curve uncertainty. Emery et al. (2006) used CS to assess mineral resources at a given cutoff grade, Hosseini et al. (2017) used direct block-support sequential co-simulation to assess the uncertainty of recoverable resources by generating numerous realizations of grades in a deposit. Harrison et al. (2009) used SGS to model grade uncertainty due to soft domain boundaries as depth increases during drilling. The uncertainty in the variation of block tonnage and grade due to conditional biases was assessed using SGS (Nowak & Leuangthong, 2017). The uncertainty of multi-element deposit in an in situ mineral resource was assessed through realization generated by MAF and direct block simulation (de Freitas Silva & Dimitrakopoulos, 2016). Other methods used to assess these uncertainties in recoverable resources are summarized in Table 4, which indicates cause of uncertainty, assessment method and references.

Table 4 Uncertainty assessment of recoverable mineral resources
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Summary and Conclusions

Uncertainty quantification is important for accurate mineral resource estimation. Uncertainty quantification methods are scattered in the literature, which makes their selection and application cumbersome. This study assembled a knowledge base of these methods and demonstrated an assessment of uncertainty along the estimation process. The knowledge base informs on expertise about existing and applied methods in different geological contexts at any step during estimation. The selection of uncertainty quantification method depends on available data, type of deposit and user experience. The developed knowledge base advises the selection and adoption of an estimation method with the capability of evaluating mineral resources and assessing uncertainty, and thus helps to deepen knowledge about uncertainty assessment in resource estimation and reduce the accuracy gap between estimated mineral resources and updated resources after beginning mining production.

Mineral resource imprecision has been classified broadly into uncertainties at sampled locations and unsampled locations. At sampled locations, uncertainty is due to sampling, handling and processing of data. At unsampled locations, uncertainty results from interpolation/extrapolation or simulation to formulate geological model and estimate mineral resources. The uncertainty assessment applies to different phases such as data sampling, geological modeling and application of mathematical methods. Geological uncertainty sources are the most dominant in every step of the estimation process, and it heavily influences the accuracy of estimated resources. Uncertainty associated with mineral resource estimation can be assessed using various techniques such as kriging, probabilistic, simulation and ML. Among them, geostatistical simulations are the most applied methods compared to other approaches in assessing both local and global uncertainties through the generation of numerous realizations. So far, no single uncertainty quantification method has been proved to be the most accurate and to work in every type of deposit. This is because every method is constrained by its limitations and assumptions, the geology of the deposit and the varying objectives of uncertainty quantification.

As the estimation progresses, so does uncertainty accumulation and propagation because estimation steps are interdependent. Therefore, this study discussed uncertainty causes, propagation and assessment in every step of mineral resource estimation. Knowledge of uncertainty propagation and stepwise assessment can help to increase awareness and avoid or establish control of the damaging impacts in each step during estimation to ensure accurate mineral resource estimates. Additionally, it would help researchers to determine and develop new ideas and procedures for uncertainty assessment that require further investigation.