Grade and Tonnage Uncertainty Analysis of an African Copper Deposit Using Multiple-Point Geostatistics and Sequential Gaussian Simulation

Spatial uncertainty analysis is a complex and difficult task for orebody estimation in the mining industry. Conventional models (kriging and its variants) with variogram-based statistics fail to capture the spatial complexity of an orebody. Due to this, the grade and tonnage are incorrectly estimated resulting in inaccurate mine plans, which lead to costly financial decision. Multiple-point geostatistical simulation model can overcome the limitations of the conventional two-point spatial models. In this study, a multiple-point geostatistical method, namely SNESIM, was applied to generate multiple equiprobable orebody models for a copper deposit in Africa, and it helped to analyze the uncertainty of ore tonnage of the deposit. The grade uncertainty was evaluated by sequential Gaussian simulation within each equiprobable orebody models. The results were validated by reproducing the marginal distribution and two- and three-point statistics. The results show that deviations of volume of the simulated orebody models vary from ? 3 to 5% compared to the training image. The grade simulation results demonstrated that the average grades from the different simulation are varied from 3.77 to 4.92% and average grade 4.33%. The results also show that the volume and grade uncertainty model overestimates the orebody volume as compared to the conventional orebody. This study demonstrates that incorporating grade and volume uncertainty leads to significant changes in resource estimates.     

Geological Modeling of Gold Deposit Based on Grade Domaining Using Plurigaussian Simulation Technique

Mineral resource evaluation requires defining grade domains of an ore deposit. Common practice in mineral resource estimation consists of partitioning the ore body into several grade domains before the geostatistical modeling and estimation at unsampled locations. Many ore deposits are made up of different mineralogical ensembles such as oxide and sulfide zone: being able to model the spatial layout of the different grades is vital to good mine planning and management. This study addresses the application of the plurigaussian simulation to Sivas (Turkey) gold deposits for constructing grade domain models that reproduce the contacts between different grade domains in accordance with geologist’s interpretation. The method is based on the relationship between indicator variables from grade distributions on the Gaussian random functions chosen to represent them. Geological knowledge is incorporated into the model by the definition of the indicator variables, their truncation strategy, and the grade domain proportions. The advantages of the plurigaussian simulation are exhibited through the case study. The results indicated that the processes are seen to respect reproducing complex geometrical grades of an ore deposit by means of simulating several grade domains with different spatial structure and taking into account their global proportions. The proposed proportion model proves as simple to use in resource estimation, to account for spatial variations of the grade characteristics and their distribution across the studied area, and for the uncertainty in the grade domain proportions. The simulated models can also be incorporated into mine planning and scheduling.     

Additional Samples: Where They Should Be Located

Information for mine planning requires to be close spaced, if compared to the grid used for exploration and resource assessment. The additional samples collected during quasimining usually are located in the same pattern of the original diamond drillholes net but closer spaced. This procedure is not the best in mathematical sense for selecting a location. The impact of an additional information to reduce the uncertainty about the parameter been modeled is not the same everywhere within the deposit. Some locations are more sensitive in reducing the local and global uncertainty than others. This study introduces a methodology to select additional sample locations based on stochastic simulation. The procedure takes into account data variability and their spatial location. Multiple equally probable models representing a geological attribute are generated via geostatistical simulation. These models share basically the same histogram and the same variogram obtained from the original data set. At each block belonging to the model a value is obtained from the n simulations and their combination allows one to access local variability. Variability is measured using an uncertainty index proposed. This index was used to map zones of high variability. A value extracted from a given simulation is added to the original data set from a zone identified as erratic in the previous maps. The process of adding samples and simulation is repeated and the benefit of the additional sample is evaluated. The benefit in terms of uncertainty reduction is measure locally and globally. The procedure showed to be robust and theoretically sound, mapping zones where the additional information is most beneficial. A case study in a coal mine using coal seam thickness illustrates the method.     

Selection of Optimal Thresholds for Estimation and Simulation Based on Indicator Values of Highly Skewed Distributions of Ore Data

This study strives to outline a geostatistics model for estimation and simulation of the Qolqoleh gold ore deposit located in Saqqez, NW of Iran. Considering that this gold deposit contains high-grade values, accurate evaluation of such values is of high importance, and therefore different methods based on indicator values, such as full indicator kriging (FIK) and sequential indicator simulation (SIS), have been employed to improve the accuracy of estimation and simulation of high-grade values. FIK and SIS cover the full range of grades based on several thresholds on the indicator data. The cumulative distribution function (CDF) is typically used for selection of threshold values. Given the highly skewed distribution of gold grade and its intense fluctuations, the number of thresholds is increased using CDF, which in turn results in a whole lot of calculations. To reduce the volume of calculations, the number–size (N–S) fractal model has been used to select thresholds. From such a model, all optimal thresholds are chosen with respect to geology and the unnecessary thresholds are excluded from selection. Thus, a study of the selection of optimal thresholds for estimation and simulation of a gold ore resource by means of FIK and SIS, respectively, based on thresholds selected using the N–S fractal model is presented. Finally, it is proved that results of these geostatistical methods based on thresholds selection from the N–S model appear to be better-positioned to explain ore grade variability compared to thresholds selected from the CDF and threshold selection from the N–S model is more effective for reducing the volume of required calculations.     

河流水质风险评价的灰色-随机风险率方法

提出了量化影响河流水质的随机不确定性与灰色不确定性的水质超标灰色－随机风险率概念，建立了水质超标灰色－随机风险率评价模型。在水质单项参数评价模型中，将河流污染物浓度变量的分布处理成灰色概率分布，将污染物浓度超过水质类别标准值的风险率处理成灰色概率，即水质超标灰色－随机风险率。在水质综合评价模型中，河流水环境系统被考虑为担任某一使用可能的可靠性系统，而任意一种水质参数超标意味着河流水体使用功能不能得到应有的保证，也即表明水体综合评价超标，最后借鉴系统可靠性分析的理论和方法计算水质综合超标率。该方法应用于黄河花园口断面重金属污染风险评价。     

A context-sensitive correlated random walk: a new simulation model for movement

Computational Movement Analysis focuses on the characterization of the trajectory of individuals across space and time. Various analytic techniques, including but not limited to random walks, Brownian motion models, and step selection functions have been used for modeling movement. These fall under the rubric of signal models which are divided into deterministic and stochastic models. The difficulty of applying these models to the movement of dynamic objects (e.g. animals, humans, vehicles) is that the spatiotemporal signal produced by their trajectories a complex composite that is influenced by the Geography through which they move (i.e. the network or the physiography of the terrain), their behavioral state (i.e. hungry, going to work, shopping, tourism, etc.), and their interactions with other individuals. This signal reflects multiple scales of behavior from the local choices to the global objectives that drive movement. In this research, we propose a stochastic simulation model that incorporates contextual factors (i.e. environmental conditions) that affect local choices along its movement trajectory. We show how actual global positioning systems observations can be used to parameterize movement and validate movement models and argue that incorporating context is essential in modeling movement.     

Geostatistical Algorithm Selection for Mineral Resources Assessment and its Impact on Open-pit Production Planning Considering Metal Grade Boundary Effect

Mine planning is influenced by many sources of uncertainty. Significant sources of geological uncertainty in mine planning include uncertainty in layout of geological domains and uncertainty in metal grades. These two sources of uncertainty cannot be modeled separately because the distribution of the grade is controlled usually by geological domains. Two approaches exist for combining these two sources of uncertainty: the joint simulation approach and the cascade approach. In this paper, these two approaches were compared using a real case study. To this end, uncertainty in iron grade (quantitative variable) and ore zones (qualitative variable) was modeled using both approaches. There were some considerable differences in the results obtained by each approach, which confirm the importance of choosing the most appropriate approach with consideration of the dominate features of a deposit.

     

Managing Technological and Financial Uncertainty: A Decision Science Approach for Strategic Drilling Decisions

This paper presents a framework to improve the quality of investment decisions in petroleum. The model presented enables the decision-maker to explicitly consider two major objectives when evaluating new petroleum opportunities—financial and technological gain. We utilize MultiAttribute Utility Theory (MAUT) to consider simultaneously the technological challenges of petroleum exploration into the capital budgeting process of an exploration and production firm. The MAUT methodology presented in this work demonstrates that in some mature areas the advantages to exploration are restricted further only to financial gain, based upon the present economic potential of the basin. On the other hand, other seemingly less attractive areas, such as deep horizons in deep-water basins, may represent attractive targets for new exploration as a result of the interaction of financial gain and technological advancement. This advantage reflects the technological gain as a key factor for future operations for oil discoveries in areas with big geological potential. The model presented in this work enables the decision-maker to consider explicitly the risk and rewards associated with both financial and technological payoffs, the decision-maker's tolerance for those types of risks, and the relative importance of each of those objectives in the context of ongoing petroleum exploration decisions.     

Exploring Spatially Non-stationary Relationships in the Determinants of Mineralization in 3D Geological Space

Exploring the spatial relationships between various geological features and mineralization is not only conducive to understanding the genesis of ore deposits but can also help to guide mineral exploration by providing predictive mineral maps. However, most current methods assume spatially constant determinants of mineralization and therefore have limited applicability to detecting possible spatially non-stationary relationships between the geological features and the mineralization. In this paper, the spatial variation between the distribution of mineralization and its determining factors is described for a case study in the Dingjiashan Pb–Zn deposit, China. A local regression modeling technique, geological weighted regression (GWR), was leveraged to study the spatial non-stationarity in the 3D geological space. First, ordinary least-squares (OLS) regression was applied, the redundancy and significance of the controlling factors were tested, and the spatial dependency in Zn and Pb ore grade measurements was confirmed. Second, GWR models with different kernel functions in 3D space were applied, and their results were compared to the OLS model. The results show a superior performance of GWR compared with OLS and a significant spatial non-stationarity in the determinants of ore grade. Third, a non-stationarity test was performed. The stationarity index and the Monte Carlo stationarity test demonstrate the non-stationarity of all the variables throughout the area. Finally, the influences of the degree of non-stationary of all controlling factors on mineralization are discussed. The existence of significant non-stationarity of mineral ore determinants in 3D space opens up an exciting avenue for research into the prediction of underground ore bodies.

     

Spatial Pair-Copula Modeling of Grade in Ore Bodies: A Case Study

A real-world mining application of pair-copulas is presented to model the spatial distribution of metal grade in an ore body. Inaccurate estimation of metal grade in an ore reserve can lead to failure of a mining project. Conventional kriged models are the most commonly used models for estimating grade and other spatial variables. However, kriged models use the variogram or covariance function, which produces a single average value to represent the spatial dependence for a given distance. Kriged models also assume linear spatial dependence. In the application, spatial pair-copulas are used to appropriately model the non-linear spatial dependence present in the data. The spatial pair-copula model is adopted over other copula-based spatial models since it is better able to capture complex spatial dependence structures. The performance of the pair-copula model is shown to be favorable compared to a conventional lognormal kriged model.     

Uncertainty Estimate in Resources Assessment: A Geostatistical Contribution

For many decades the mining industry regarded resources/reserves estimation and classification as a mere calculation requiring basic mathematical and geological knowledge. Most methods were based on geometrical procedures and spatial data distribution. Therefore, uncertainty associated with tonnages and grades either were ignored or mishandled, although various mining codes require a measure of confidence in the values reported. Traditional methods fail in reporting the level of confidence in the quantities and grades. Conversely, kriging is known to provide the best estimate and its associated variance. Among kriging methods, Ordinary Kriging (OK) probably is the most widely used one for mineral resource/reserve estimation, mainly because of its robustness and its facility in uncertainty assessment by using the kriging variance. It also is known that OK variance is unable to recognize local data variability, an important issue when heterogeneous mineral deposits with higher and poorer grade zones are being evaluated. Altenatively, stochastic simulation are used to build local or global uncertainty about a geological attribute respecting its statistical moments. This study investigates methods capable of incorporating uncertainty to the estimates of resources and reserves via OK and sequential gaussian and sequential indicator simulation The results showed that for the type of mineralization studied all methods classified the tonnages similarly. The methods are illustrated using an exploration drill hole data sets from a large Brazilian coal deposit.     

Radial Basis Function Network for Ore Grade Estimation

This paper highlights the performance of a radial basis function (RBF) network for ore grade estimation in an offshore placer gold deposit. Several pertinent issues including RBF model construction, data division for model training, calibration and validation, and efficacy of the RBF network over the kriging and the multilayer perceptron models have been addressed in this study. For the construction of the RBF model, an orthogonal least-square algorithm (OLS) was used. The efficacy of this algorithm was testified against the random selection algorithm. It was found that OLS algorithm performed substantially better than the random selection algorithm. The model was trained using training data set, calibrated using calibration data set, and finally validated on the validation data set. However, for accurate performance measurement of the model, these three data sets should have similar statistical properties. To achieve the statistical similarity properties, an approach utilizing data segmentation and genetic algorithm was applied. A comparative evaluation of the RBF model against the kriging and the multilayer perceptron was then performed. It was seen that the RBF model produced estimates with the R ² (coefficient of determination) value of 0.39 as against of 0.19 for the kriging and of 0.18 for the multilayer perceptron.     

Scale dependence in habitat selection: the case of the endangered brown bear (Ursus arctos) in the Cantabrian Range (NW Spain)

Animals select habitat resources at multiple spatial scales. Thus, explicit attention to scale dependency in species–habitat relationships is critical to understand the habitat suitability patterns as perceived by organisms in complex landscapes. Identification of the scales at which particular environmental variables influence habitat selection may be as important as the selection of variables themselves. In this study, we combined bivariate scaling and Maximum entropy (Maxent) modeling to investigate multiscale habitat selection of endangered brown bear (Ursus arctos) populations in northwest Spain. Bivariate scaling showed that the strength of apparent habitat relationships was highly sensitive to the scale at which predictor variables are evaluated. Maxent models on the optimal scale for each variable suggested that landscape composition together with human disturbances was dominant drivers of bear habitat selection, while habitat configuration and edge effects were substantially less influential. We found that explicitly optimizing the scale of habitat suitability models considerably improved single-scale modeling in terms of model performance and spatial prediction. We found that patterns of brown bear habitat suitability represent the cumulative influence of habitat selection across a broad range of scales, from local resources within habitat patches to the landscape composition at broader spatial scales.     

A systematic sensitivity analysis of constrained cellular automata model for urban growth simulation based on different transition rules

Cellular automata (CA) have emerged as a primary tool for urban growth modeling due to its simplicity, transparency, and ease of implementation. Sensitivity analysis is an important component in CA modeling for a better understanding of errors or uncertainties and their propagation. Most studies on sensitivity analyses in urban CA modeling focus on specific component such as neighborhood configuration or stochastic perturbation. However, sensitivity analysis of transition rules, which is one of the core components in CA models, has not been systematically done. This article proposes a systematic sensitivity analysis of major operational components in urban CA modeling using a stepwise comparison approach. After obtaining transition rules, three stages (i.e. static calibration of transition rules, dynamic evolution with varied time steps, and incorporation with stochastic perturbation) are designed to facilitate a comprehensive analysis. This scheme implemented with a case study in Guangzhou City (China) reveals that gaps in performance from static calibration with different transition rules can be reduced when dynamic evolution is considered. Moreover, the degree of stochastic perturbation is closely related to obtain urban morphology. However, a more realistic (i.e. fragmented) urban landscape is achieved at the cost of decreasing pixel-based accuracy in this study. Thus, a trade-off between pixel-based and pattern-based comparisons should be balanced in practical urban modeling. Finally, experimental results illustrate that models for transition rules extraction with good quality can do an assistance for urban modeling through reducing errors and uncertainty range. Additionally, ensemble methods can feasibly improve the performance of CA models when coupled with nonparametric models (i.e. classification and regression tree).     

Conditional simulation as a tool for measuring uncertainties in petroleum exploration

A fundamental task for petroleum exploration decision-making is to evaluate the uncertainty of well outcomes. The recent development of geostatistical simulation techniques provides an effective means to the generation of a full uncertainty model for any random variable. Sequential indicator simulation has been used as a tool to generate alternate, equal-probable stochastic models, from which various representations of uncertainties can be created. These results can be used as input for the quantification of various risks associated with a wildcat drilling program or the estimation of petroleum resources. A simple case study is given to demonstrate the use of sequential indicator simulation. The data involves a set of wildcat wells in a gas play. The multiple simulated stochastic models are then post-processed to characterize various uncertainties associated with drilling outcomes.     

Categorization of Mineral Resources Based on Different Geostatistical Simulation Algorithms: A Case Study from an Iron Ore Deposit

Mineral resource classification plays an important role in the downstream activities of a mining project. Spatial modeling of the grade variability in a deposit directly impacts the evaluation of recovery functions, such as the tonnage, metal quantity and mean grade above cutoffs. The use of geostatistical simulations for this purpose is becoming popular among practitioners because they produce statistical parameters of the sample dataset in cases of global distribution (e.g., histograms) and local distribution (e.g., variograms). Conditional simulations can also be assessed to quantify the uncertainty within the blocks. In this sense, mineral resource classification based on obtained realizations leads to the likely computation of reliable recovery functions, showing the worst and best scenarios. However, applying the proper geostatistical (co)-simulation algorithms is critical in the case of modeling variables with strong cross-correlation structures. In this context, enhanced approaches such as projection pursuit multivariate transforms (PPMTs) are highly desirable. In this paper, the mineral resources in an iron ore deposit are computed and categorized employing the PPMT method, and then, the outputs are compared with conventional (co)-simulation methods for the reproduction of statistical parameters and for the calculation of tonnage at different levels of cutoff grades. The results show that the PPMT outperforms conventional (co)-simulation approaches not only in terms of local and global cross-correlation reproductions between two underlying grades (Fe and Al₂O₃) in this iron deposit but also in terms of mineral resource categories according to the Joint Ore Reserves Committee standard.

     

A framework for uncertainty assessment in simulation models

In this article, we introduce a conceptual framework for systematic identification and assessment of sources of uncertainty in simulation models. This concept builds on a novel typology of uncertainty in model validation and extends the GIScience research focus on uncertainty in spatial data to uncertainty in simulation modelling. Such a concept helps a modeller to interpret and handle uncertainty in order to efficiently optimise a model and better understand simulation results.

To illustrate our approach, we apply the proposed framework for uncertainty assessment to the TREE LIne Model (TREELIM), an individual-based model that simulates forest succession at the alpine tree line. Using this example, uncertainty is identified in the modelling workflow during conceptualisation, formalisation, parameterisation, analysis and validation. With help of a set of indicators we quantify the emerging uncertainties and assess the overall model uncertainty as a function of all occurring sources of uncertainty.

An understanding of the sources of uncertainty in an ecological model proves beneficial for: (1) developing a structurally valid model in a systematic way; (2) deciding if further refinement of the conceptual model is beneficial for the modelling purpose; and (3) interpreting the overall model uncertainty by understanding its sources. Our approach results in a guideline for assessing uncertainty in the validation of simulation models in a feasible and defensible way, and thus functions as a toolbox for modellers. We consider this work as a contribution towards a general concept of uncertainty in spatially explicit simulation models.     

Development Optimization and Uncertainty Analysis Methods for Oil and Gas Reservoirs

Uncertainty complicates the development optimization of oil and gas exploration and production projects, but methods have been devised to analyze uncertainty and its impact on optimal decision-making. This paper compares two methods for development optimization and uncertainty analysis: Monte Carlo (MC) simulation and stochastic programming. Two example problems for a gas field development and an oilfield development are solved and discussed to elaborate the advantages and disadvantages of each method. Development optimization involves decisions regarding the configuration of initial capital investment and subsequent operational decisions. Uncertainty analysis involves the quantification of the impact of uncertain parameters on the optimum design concept. The gas field development problem is designed to highlight the differences in the implementation of the two methods and to show that both methods yield the exact same optimum design. The results show that both MC optimization and stochastic programming provide unique benefits, and that the choice of method depends on the goal of the analysis. While the MC method generates more useful information, along with the optimum design configuration, the stochastic programming method is more computationally efficient in determining the optimal solution. Reservoirs comprise multiple compartments and layers with multiphase flow of oil, water, and gas. We present a workflow for development optimization under uncertainty for these reservoirs, and solve an example on the design optimization of a multicompartment, multilayer oilfield development.     

Responding to the consequences of uncertainty in geographical data

The presence of uncertainty in geographical data has the potential to expose users to undesirable consequences in their decision making. Accordingly, our efforts to understand uncertainty seek to (a) avoid the use of data that are not suitable for their intended purpose (that is, data whose consequences are unacceptable), (b) to reduce any undesirable consequences to an acceptable level, and (c) to devise ways of living with undesirable data when the adverse consequences caused by poor data do not alter our ultimate decision choice. To assist this task, we propose an approach where the adverse consequences of uncertainty caused by the use of unsuitable geographical data are expressed in terms of risk. In this paper we first show that risk management offers geographical data users a range of options for responding to the adverse consequences of data uncertainty, and secondly we present and discuss the various risk response options.     

Multi-scale Quantitative Risk Analysis of Seabed Minerals: Principles and Application to Seafloor Massive Sulfide Prospects

The potential for mining hydrothermal mineral deposits on the seafloor, such as seafloor massive sulfides, has become technically possible, and some companies (currently not many) are considering their exploration and development. Yet, no present methodology has been designed to quantify the ore potential and assess the risks relative to prospectivity at prospect and regional scales. Multi-scale exploration techniques, similar to those of the play analysis that are used in the oil and gas industry, can help to fulfill this task by identifying the characteristics of geologic environments indicative of ore-forming processes. Such characteristics can represent a combination of, e.g., heat source, pathway, trap and reservoir that all dictate how and where ore components are mobilized from source to deposition. In this study, the understanding of these key elements is developed as a mineral system, which serves as a guide for mapping the risk of the presence or absence of ore-forming processes within the region of interest (the permissive tract). The risk analysis is carried out using geoscience data, and it is paired with quantitative resource estimation analysis to estimate the in-place mineral potential. Resource estimates are simulated stochastically with the help of available data (bathymetric features in this study), conventional grade–tonnage models and Monte Carlo simulation techniques. In this paper, the workflow for a multi-scale quantitative risk analysis, from the definition to the evaluation of a permissive tract and related prospect(s), is described with the help of multi-beam data of a known hydrothermal vent site.