Multivariate Geostatistical Simulation Research Articles

Multivariate Geostatistical Simulation and Deep Q-Learning to Optimize Mining Decisions

Multivariate Geostatistical Simulation and Deep Q-Learning to Optimize Mining Decisions

Application of risk assessment to improve sustainability in bauxite mining

Uncertainty and risk management are essential for decision-making in the sustainable extraction of mineral resources. Uncertainty in mine planning typically starts with the geostatistical simulation of the variables that influence the decision process, which produces a series of equally probable models. Then a transfer function is applied to each model. The result is a distribution of the response function, which calculates the probability (or risk) that a given area achieves the product quality standards. High-risk operations often fail to achieve the ore quality standards and decrease the mineral resources value. In this context, failure in managing risk may collapse mining operations, leading to unrepairable economic, environmental, and social damages. However, risk assessment is a challenge in the case of multivariate deposits such as bauxite deposits. In these deposits, the response function depends on multiple variables simultaneously. The variables that characterize bauxite deposits also exhibit complex relationships and sum and fraction constraints. The geostatistical simulations should reproduce these peculiarities. This work proposes a workflow to measure risk and uncertainty in complex multivariate deposits aiming at sustainable gains for the mining operations. The workflow starts with a multivariate geostatistical simulation using Projection Pursuit Multivariate Transform. Then these simulated models are combined with the initially planned dig lines and mining scheduling. The results informed the dig lines and periods with the highest risks and uncertainty for all variables, allowing the decision team to take measures focusing on sustainable goals. The methodology is illustrated in a case study of a significant bauxite deposit located in northern Brazil.

Source apportionment of potentially toxic elements in soils of the Yellow River Delta Nature Reserve, China: The application of three receptor models and geostatistical independent simulation

The Yellow River Delta (YRD) wetland, the most important estuary wetland in eastern China, has an important ecosystem service function. Rapid and intensive development has inevitably led to the accumulation of potentially toxic elements (PTEs) in soils. Therefore, identifying quantitative sources and spatial distributions of PTEs is essential for soil environmental protection in the YRD. A total of 240 topsoil samples (0–20 cm) were collected in the Yellow River Delta Nature Reserve (YRDNR) and analyzed the PTE contents. To avoid the biases of the single receptor model, positive matrix factorization, factor analysis with nonnegative constraints, and maximum likelihood principal component analysis-multivariate curve resolution-alternating least squares were used for source apportionment of soil PTEs. To promote the efficiency of multivariate geostatistical simulation, a minimum/maximum autocorrelation factor-sequential Gaussian simulation was built to map the spatial patterns of PTEs. Three factors were derived by the three receptor models, and their contributions to the source explanation were similar. As, Cr, Cu, Mn, Ni, and Zn originated from natural sources, with contributions of 85.6%–96.4 %. A total of 61.5 % of Hg was associated with atmospheric deposition of coal combustion and wastewater from upstream. Agricultural activities and oil exploitation contributed 33.5 % and 15.9 % of the Cd and Pb concentrations. Spatial distributions of soil PTEs were controlled by sedimentary grain size. A total of 47.2 % of the total study area was identified as hazardous area for Cd, 10.3 % for As, and 5.4 % for Hg. This work is expected to provide references for soil pollution assessment and management of YRDNR.

Application of multivariate geostatistical simulation and fractal analysis for detection of rare-earth element geochemical anomalies in the Esfordi phosphate mine, Central Iran

The present study was conducted to determine different geochemical anomalies of rare-earth elements (REEs) using a combined approach of stepwise factor analysis (SFA), sequential Gaussian simulation (SGS), and concentration–area (C-A) fractal modeling based on surface lithogeochemical samples obtained from the Esfordi phosphate mine (Central Iran). The Esfordi mine is one of the important mines in the Bafq metallogenic zone due to average and maximum grades of 0.5 and 1.7%, respectively for REEs. With SFA operating in two steps, the REEs were placed in the first factor of the second stage (F1-2). Then, SGS and C-A fractal modeling were performed on F1-2 factor scores for classification of anomalies. A log-ratio matrix was used to evaluate the correlation of these results with anomalous lithogeochemical samples, as well as to determine the relationship of anomalies with rock types and mineralized units and, finally, to validate the results of the SGS–fractal modeling. The results confirmed an appropriate correlation between F1-2 anomalies and high concentration in further rock samples. The main anomalies were found to have good correlation with an apatite–iron unit and in general with other apatite-bearing units based on overall accuracy values. The apatite-bearing units with high values of REEs were located in the northern and central parts of the mine. The results of the SFA, SGS and C-A fractal modeling show that this hybrid approach can be useful in determining anomalies with high accuracy.

Multivariate geostatistical simulation with PPMT: an application for uncertainty measurement

ABSTRACT Grade models built by traditional estimation or simulation methods often fail to reproduce the complex relationships between the variables. This work investigates the use of the multivariate transformation called Projection Pursuit Multivariate Transform (PPMT), which fully decorrelates the multiple variables of interest, allowing the independent conditional simulation of each variable in the transformed space. Finally, the simulated variables are back-transformed, reproducing the initial correlations of the data. The PPMT workflow was applied to a nickel laterite deposit considering five variables: nickel, iron, silica, magnesium, and calcium grades. Conditional simulations of each variable were run and validated. The back-transformed realisations reproduced the multivariate relationships of the data. To calculate the uncertainties, mining panels equivalent to two and four weeks of production were generated using the k-means clustering technique. Uncertainties were summarised by the coefficient of variation (CV) and the results were used to define classes of mineral resources.

Combining finite mixture distribution, receptor model, and geostatistical simulation to evaluate heavy metals pollution in soils: Source and spatial pattern

AbstractSoil heavy metals (HMs) pollution has caused significant land degradation by affecting soil properties and functions. Clarifying sources and spatial distributions of soil HMs is necessary for the investigation of land degradation, but lacks accurate and efficient methods. This study proposes a combined method for improving source apportionment and spatial prediction of soil HMs. Finite mixture distribution modelling (FMDM) was used to explore the backgrounds and contamination thresholds, and can verify the source contribution calculated by positive matrix factorization (PMF). To improve the efficiency of traditional multivariate geostatistical simulation, independent component analysis and sequential Gaussian simulation were integrated into geostatistical independent simulation for revealing the spatial patterns and the polluted areas. The combined method was applied to an HMs dataset of Longkou City, eastern China as a case‐study. Both PMF and FMDM produced three sources of HMs. Cr and Ni were dominated by natural sources. Agricultural inputs mainly affected Cd, Cu, and Zn concentrations. Hg, 66.8%, and Pb, 8.1%, originated from industrial, mining, and traffic emissions. Hazardous areas exceeding moderate pollution cutoffs comprised 203.88 km2, accounting for 23.2% of the study area. Less than 0.1% of the areas for all HMs were identified as hazardous zones exceeding the high pollution thresholds. The reasonable management of industrial and agricultural activities, traffic emissions, and mining could have an important influence on the protection of land resources. Combining FMDM, PMF, and geostatistical independent simulation proved valid for investigating the land degradation due to HMs pollution.

Integrated receptor models and multivariate geostatistical simulation for source apportionment of potentially toxic elements in soils

Adequately understanding the source characteristic of potentially toxic elements (PTEs) in soils plays an important role in soil environmental protection and regulation. To improve the accuracy of quantifying source contributions, US-EPA positive matrix factorization (EPAPMF), weighted alternating least squares positive matrix factorization (WALSPMF), absolute principal component analysis/multiple linear regression (APCA/MLR), and UNMIX were applied and compared. Spatial distribution maps were delineated by using sequential Gaussian cosimulation (SGCS) with a linear model of coregionalization (LMC) fitting, and can be used to demarcate natural or polluted sources through superposing with auxiliary environmental data. Polluted area of PTEs exceeding a given threshold was determined using uncertainty analysis. The combination of receptor models and SGCS was used to a dataset consisting of As, Cd, Cr, Hg, Ni, Pb, and Zn as a case study. The four receptor models yielded three factors with comparable contributions to seven PTEs, but APCA/MLR produced some negative contributions. The average contributions were calculated based on EPAPMF, WALSPMF, and UNMIX. 85.3% of As, 81.3% of Cr, and 86.7% of Ni originated from natural source, and these three PTEs indicated consistent distributions with parent material map. Cd, Pb, and Zn were contributed by both industrial-traffic emissions and parent materials, with hotspots related to pollution sources and lacustrine deposits. Hg exhibited large-scale high value area around industrial sites, and exclusively derived from atmospheric deposition. A critical probability of 0.95 was adopted to determine polluted areas with PTE content exceeding 1.5 times its background. More than 40.0% of the total area was classified as contaminated for Hg, followed by Cd (3.1%) and Pb (2.9%). The combination of receptor models and SGCS proved to be an effective integrated approach for source apportionment.

Assessment of an ore body internal dilution based on multivariate geostatistical simulation using exploratory drill hole data

Dilution can best be defined as the proportion of waste tonnage to the total weight of ore and waste in each block. Predicting the internal dilution based on geological boundaries of waste and ore in each block can help engineers to develop more reliable long-term planning designs in mining activities. This paper presents a method to calculate the geological internal dilution in each block and to correct the ultimate grade of each geological block according to the internal dilution values that have already been calculated for each one of them. In this regard, the input data is first indexed based on the lithological logs of drill holes. The occurrence probabilities of ore and waste in each block are calculated via 100 realizations using the sequential indicator simulation. Dilution is computed as the ratio of waste rock tonnage to the total tonnage of ore and waste. Furthermore, joint simulation of the continuous variables is performed for each mining block using the minimum/maximum auto-correlation factors. In the next step, for each block, the final grade variables including iron and iron oxide are computed by considering the calculated internal dilution. These analyses are applied to the Gohar Zamin iron ore deposit, and the actual internal dilution calculated based on the lithological logs of blast holes is compared with the same values obtained based on the proposed method in each block. The results obtained were found to be satisfactory.

Multivariate geostatistical simulation with sum and fraction constraints

ABSTRACTGeostatistical modelling of grades in mineral deposits often requires the simulation of multiple related variables that have sum and fraction constraints. Sum constraints occur when the sum of some variables may not exceed or must equal a given constant. Fraction constraints occur when a variable may not exceed another variable. In this case, the geostatistical simulations should reproduce the histograms, variograms, multivariate relationships and honour the constraints. We present a methodology for the geostatistical simulation of multiple related variables that considers sum and fraction constraints. The methodology is illustrated in a case study. The data were obtained from a bauxite deposit. First, the original variables were re-expressed as ratios. Second, the variables were transformed using the Projection Pursuit Multivariate Transform (PPMT). Then the PPMT transformed variables were simulated independently using sequential Gaussian simulation and back-transformed. The simulations reproduced the histograms, variograms and bivariate relationships and honoured the sum and fraction constraints.

Multivariate Geostatistical Simulation on Block-Support in the Presence of Complex Multivariate Relationships: Iron Ore Deposit Case Study

A reliable and realistic subsurface resource spatial modeling is a common and critical task for geosciences projects (mining, petroleum and environmental) because the models are then integrated with downstream processes to evaluate process performance. Coregionalized variables considered in resource modeling are often related through compositional constraints and complex dependence relationships. Satisfying these constraints and relationships in spatial modeling is a practical requirement to obtain accurate predictions of mineral resources in the subsurface. This paper presents a multistage method that addresses these issues and allows the multivariate simulation of cross-correlated variables with complex features directly on block-support, conditionally to the information of drill hole data at a quasi-point support. At a first stage, a chained transformation is used for removal multivariate complexities and decorrelation of the variables. Then, this chained transformation is adapted within the direct block-support sequential Gaussian simulation algorithm. The back-transformation of the chained transformation reintroduces complex features and correlations. A proof of the concept using a data set from the Gole-Gohar iron ore deposit in Iran demonstrates the performance of the proposed approach, the results of which are then compared against a common modeling approach.

Multivariate geostatistical simulation by minimising spatial cross-correlation

Joint simulation of attributes in multivariate geostatistics can be achieved by transforming spatially correlated variables into independent factors. In this study, a new approach for this transformation, Minimum Spatial Cross-correlation (MSC) method, is suggested. The method is based on minimising the sum of squares of cross-variograms at different distances. In the approach, the problem in higher space (N×N) is reduced to N×N−1/2 problems in the two-dimensional space and the reduced problem is solved iteratively using Gradient Descent Algorithm. The method is applied to the joint simulation of a set of multivariate data in a marble quarry and the results are compared with Minimum/Maximum Autocorrelation Factors (MAF) method.

Multivariate geostatistical simulation of coal quality data by independent components

Quality of a lignite deposit can be characterized by many variables such as calorific value, ash content, and moisture content. These variables show complex spatial relationships with each other. Multivariate geostatistical simulation allows to reproducing such complex relations between the quality variables.In this paper we present independent component analysis and introduce this method as a factorization tool in multivariate geostatistical simulation based on factor approach. The method is based on deriving independent components of multivariate data and simulating each one independently. Independent component simulation technique is applied to geostatistical simulation of three quality variables for a part of the lignite seam subject to severe tectonic movement and regularly variable in quality. The lignite seam belongs to the Lower Coal succession deposited in the Soma coal field, Manisa, Turkey. Ash content, lower calorific value and moisture content are quality variables under consideration. For simulation purposes the independent factors are derived from a linear combination of these quality variables by using independent component analysis, the variograms for the factors are calculated and modeled. After ensuring that the factors are spatially orthogonal, they are independently simulated by direct sequential simulation and the simulated values are back-transformed into original space. The application shows that input statistics such as mean, histogram, variogram and correlation coefficient for the quality variables are reproduced well and independent component simulation can be used in simulation of multivariate data.

The U-WEDGE Transformation Method for Multivariate Geostatistical Simulation

To speed up multivariate geostatistical simulation it is common to transform the set of attributes into spatially uncorrelated factors that can be simulated independently. Spatial decorrelation methods are usually based on the diagonalisation of the variance/covariance and semivariogram matrices of the set of attributes for a chosen family of lag spacings. These matrices are symmetric and there are several efficient methods for the approximate joint diagonalisation of a family of symmetric matrices. One of these is the uniformly weighted exhaustive diagonalisation with Gauss iterations (U-WEDGE) method. In contrast to the method of minimum/maximum autocorrelation factors (MAF), where a two structure linear model of coregionalisation is assumed, U-WEDGE can be applied directly to the set of experimental semivariogram matrices without having to place restrictions on the number of structures in the linear model of coregionalisation, thus removing one of the restrictions placed on the subsequent modelling of the spatial structure of the factors. We use an iron-ore data set to illustrate the method and present a comparison between the simulated attributes obtained from U-WEDGE and MAF with the full co-simulation of the attributes.

An Application of Multivariate Simulation in the Cement Industry

The profitability of a cement plant depends largely on its capacity to produce homogeneous cement with chemical composition close to specified targets for the cement type produced. One crucial step is the mixing of limestone with other raw materials in proportions calculated to meet these targets. Major design and operation decisions depend on the efficiency of this homogenizing step. The adequate modeling of the mixing process requires simulation of representative cross-correlated time series of chemical compositions of the raw materials involved. The chemical composition signals are obtained by multivariate geostatistical simulation using an LU (Cholesky) decomposition of the covariance matrix. Modifications to the usual LU method are presented. First, the effect on the raw covariance matrix of the closure property of chemical analysis is imposed. Second, the problem of memory space limitations in the LU method is tackled by using overlapping sliding neighbourhoods. The simulation algorithm is applied to the Joppa cement plant owned by Lafarge North America. The simulated raw material input streams are fed into the quality mix control (QMC), a proprietary software that models and controls the mixing operation to produce an output stream with cement characteristics as close as possible to desired targets. Two signal series are studied, one autocorrelated with a moderate temporal range and one with no autocorrelation. The QMC produces C3S output signals having comparable short scale periodic variograms except that the variance of the uncorrelated signal is four times greater than those of the autocorrelated signal and the real Joppa data. The raw material feeder variograms have the same sill for both the white noise and the autocorrelated signals. However, the autocorrelated signal feeder variogram presents lower short term dispersion variance, a characteristic feature of Joppa operations. Our results show the importance of simulating the right temporal structure of the raw materials to realistically forecast the behavior of the output signals. We also discuss some practical implications of these findings for the design and operation of a cement plant.