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Taking into account axial symmetry in the covariance function of a Gaussian random field is essential when the purpose is modelling data defined over a large portion of the sphere representing our planet. Axially symmetric covariance functions admit a convoluted spectral representation that makes modelling and inference difficult. This motivates the interest in devising alternative strategies to attain axial symmetry, an appealing option being longitudinal integration of isotropic random fields on the sphere. This paper provides a comprehensive theoretical framework to model longitudinal integration on spheres through a nonparametric Bayesian approach. Longitudinally integrated covariances are treated as random objects, where the randomness is implied by the randomised spectrum associated with the covariance function. After investigating the topological support induced by our construction, we give the posterior distribution a thorough inspection. A Bayesian nonparametric model for the analysis of data defined on the sphere is described and implemented, its performance investigated by means of the analysis of both simulated and real data sets.

The paper provides a way to model axially symmetric random fields defined over the two-dimensional unit sphere embedded in the three-dimensional Euclidean space. Specifically, our strategy is to integrate an isotropic random field on the sphere over longitudinal arcs with a given central angle. The resulting random field is shown to be axially symmetric and to have the arc central angle as a tuning parameter that allows for isotropy as well as for longitudinal independence as limit cases. We then consider multivariate longitudinally integrated random fields, having the same properties of axial symmetry and a tuning parameter (arc central angle) proper to each random field component. This construction allows for a unified framework for vector-valued random fields that can be geodesically isotropic, axially symmetric, or longitudinally independent. Additionally, all the components of the vector random field are allowed to be cross-correlated. We finally show how to simulate the proposed axially symmetric scalar and vector random fields through a computationally efficient algorithm that exactly reproduces the desired covariance structure and provides approximately Gaussian finite-dimensional distributions.

A model has been produced for block cave mining production scheduling for short-term mine planning under a range of uncertainties. The model considered tonnage and grade as daily production targets and operational constraints including drawpoints and ore pass design, draw rate, mine production, and transportation capacities in different operational levels. The model can handle decision variables for both production and transportation levels that have non-linear relationships. A real-case copper block cave mining dataset was used to check the performance of the model under the uncertainty of demand and operational failures for three different scenarios: the base case, target grade, and drawpoint failure. The dataset consisted of two sectors, one with 250 drawpoints and 81 ore passes and the other with 234 drawpoints and 43 orepasses with the limited available tonnage for a period of 17 days. The results showed that the proposed model could minimise deviations from operational targets, allowing targets to be achieved., A model has been produced for block cave mining production scheduling for short-term mine planning under a range of uncertainties. The model considered tonnage and grade as daily production targets and operational constraints including drawpoints and ore pass design, draw rate, mine production, and transportation capacities in different operational levels. The model can handle decision variables for both production and transportation levels that have non-linear relationships. A real-case copper block cave mining dataset was used to check the performance of the model under the uncertainty of demand and operational failures for three different scenarios: the base case, target grade, and drawpoint failure. The dataset consisted of two sectors, one with 250 drawpoints and 81 ore passes and the other with 234 drawpoints and 43 orepasses with the limited available tonnage for a period of 17 days. The results showed that the proposed model could minimise deviations from operational targets, allowing targets to be achieved.

A virtual geometallurgical laboratory has been developed to support the computational requirements linking data acquisition with data-driven business decisions. A particular focus has been the development of user-guided workflow ensembles to enable comparison between plausible geometallurgical hypotheses. A case study was carried out using data from the Rocklea Dome iron deposit in Western Australia. A hypothetical mining schedule was produced from geochemical and hyperspectral drill core/chips data using: data preprocessing, exploratory data analysis, geochemical interpolation, block modelling, ultimate open pit design, and scheduling. The methodology showed a considerable impact on both shape and value of the final pit when computing it over different conditional simulations and with the introduction of reliability measures. Lower economic value was obtained with higher reliability constraints with the difference in terms of value being the cost of reducing risk between final pits., A virtual geometallurgical laboratory has been developed to support the computational requirements linking data acquisition with data-driven business decisions. A particular focus has been the development of user-guided workflow ensembles to enable comparison between plausible geometallurgical hypotheses. A case study was carried out using data from the Rocklea Dome iron deposit in Western Australia. A hypothetical mining schedule was produced from geochemical and hyperspectral drill core/chips data using: data preprocessing, exploratory data analysis, geochemical interpolation, block modelling, ultimate open pit design, and scheduling. The methodology showed a considerable impact on both shape and value of the final pit when computing it over different conditional simulations and with the introduction of reliability measures. Lower economic value was obtained with higher reliability constraints with the difference in terms of value being the cost of reducing risk between final pits.

A methodology is described for calculating the footprint and economic envelope of an underground mine under geological uncertainty. The economic level where the undercut level should be placed is calculated in a similar way to the footprint finder methodology, based on the profitability of the blocks discounted by when they will be extracted given the position of the block in the block column. The results are then used to calculate the economic envelope, which represents an approximation to the mining reserves in the orebody. The method is based on the ultimate pit algorithm and is applied with some modifications in order to resemble the caving geometry. The geological uncertainty is introduced using conditional simulations to generate different resource models. The simulations are constructed with the TBSIM program. For each simulation or block model scenario the optimum footprint and economic outline of the mine can be computed. Quantification of the uncertainty is subsequently carried out by applying the value at risk evaluation. The footprint tool was validated against commonly used PCBC software, resulting in differences of about 10% near the maximum economic level, which is a good approximation. The results of 100 simulations of an actual orebody showed that there is a probability of about 36% of finding the economic footprint in the deepest elevation and only 14% probability of finding it in level 36, where a kriging model indicated that it should be., A methodology is described for calculating the footprint and economic envelope of an underground mine under geological uncertainty. The economic level where the undercut level should be placed is calculated in a similar way to the footprint finder methodology, based on the profitability of the blocks discounted by when they will be extracted given the position of the block in the block column. The results are then used to calculate the economic envelope, which represents an approximation to the mining reserves in the orebody. The method is based on the ultimate pit algorithm and is applied with some modifications in order to resemble the caving geometry. The geological uncertainty is introduced using conditional simulations to generate different resource models. The simulations are constructed with the TBSIM program. For each simulation or block model scenario the optimum footprint and economic outline of the mine can be computed. Quantification of the uncertainty is subsequently carried out by applying the value at risk evaluation. The footprint tool was validated against commonly used PCBC software, resulting in differences of about 10% near the maximum economic level, which is a good approximation. The results of 100 simulations of an actual orebody showed that there is a probability of about 36% of finding the economic footprint in the deepest elevation and only 14% probability of finding it in level 36, where a kriging model indicated that it should be.

A method is described for the identification of different levels of seismic hazard using the agglomerative hierarchical clustering technique. The use of cluster analysis for pre-processing data allows simplification of large databases and reduces calculation times for further analysis. The approach was applied to the Esmeralda Sur section of the El Teniente Cu-Mo mine in Chile where block and panel caving are employed. Filters were used to reduce the number of seismic events in the exploratory database to 5 654 and relocation of the position of the events was carried out using the statistical collapse algorithm, which employs uncertainty in the estimation of the hypocentre as a neighbourhood search for global optimisation of the position. The events were then grouped using a hierarchical agglomerative technique called clustering by complete linkage. Clustering of the seismic events was then performed in seven groups to obtain the distribution of the number of events per group. The results were compared using two concepts, goodness of fit of the linear portion of the cumulative frequency distribution and association with mining in relation to the temporal trend. A seismic hazard array was constructed, and two zones with different hazards were identified in the production and undercut levels., A method is described for the identification of different levels of seismic hazard using the agglomerative hierarchical clustering technique. The use of cluster analysis for pre-processing data allows simplification of large databases and reduces calculation times for further analysis. The approach was applied to the Esmeralda Sur section of the El Teniente Cu-Mo mine in Chile where block and panel caving are employed. Filters were used to reduce the number of seismic events in the exploratory database to 5 654 and relocation of the position of the events was carried out using the statistical collapse algorithm, which employs uncertainty in the estimation of the hypocentre as a neighbourhood search for global optimisation of the position. The events were then grouped using a hierarchical agglomerative technique called clustering by complete linkage. Clustering of the seismic events was then performed in seven groups to obtain the distribution of the number of events per group. The results were compared using two concepts, goodness of fit of the linear portion of the cumulative frequency distribution and association with mining in relation to the temporal trend. A seismic hazard array was constructed, and two zones with different hazards were identified in the production and undercut levels.

The discrete Gaussian model is used to estimate the local block grade distributions in a lateritic nickel deposit at Cerro Matoso in Colombia. The expected profit of each selective mining unit can be derived from this along with its best destination (processing plant or dump), depending on the probabilities that the true unknown grades exceed given cut-offs and on the ratio between silica and magnesium grades (which plays an important role in the metallurgical processing of Fe and Ni). A comparison with the grade estimates obtained by ordinary co-kriging shows differences in the destination of 24.6% of the selective mining units and a discrepancy of US, The discrete Gaussian model is used to estimate the local block grade distributions in a lateritic nickel deposit at Cerro Matoso in Colombia. The expected profit of each selective mining unit can be derived from this along with its best destination (processing plant or dump), depending on the probabilities that the true unknown grades exceed given cut-offs and on the ratio between silica and magnesium grades (which plays an important role in the metallurgical processing of Fe and Ni). A comparison with the grade estimates obtained by ordinary co-kriging shows differences in the destination of 24.6% of the selective mining units and a discrepancy of US

The current methods of estimating regional ore compositions cannot deal with zero data values and do not provide uncertainty measures other than estimation error variances. A new geostatistical model, an extension of the plurigaussian model, is being developed to represent a copper ore containing bornite, chalcopyrite and other sulphide minerals. The specific objectives are to design a geostatistical model for compositional data, to determine the model parameters that fit a given data set, and to predict the mineralogy feeding the plant for a given production plan. Once the composition is fitted to the model parameters, conditional simulation of the composition at any location without data can be performed. The model is applied to a porphyry copper deposit that has been recognised by a set of exploration drill holes and is used to simulate the percentages of bornite and chalcopyrite. These variables have a major influence on metallurgical recovery, and on the subsequent optimisation of an open pit mining sequence, considering 14 phases, 150 000 tpd of sulphide ore sent to flotation and 42 period horizons. The expected values of the sulphide mineral percentages can be calculated, and also the probability of successfully achieving the predicted mineralogy on the first periods of the production plan., The current methods of estimating regional ore compositions cannot deal with zero data values and do not provide uncertainty measures other than estimation error variances. A new geostatistical model, an extension of the plurigaussian model, is being developed to represent a copper ore containing bornite, chalcopyrite and other sulphide minerals. The specific objectives are to design a geostatistical model for compositional data, to determine the model parameters that fit a given data set, and to predict the mineralogy feeding the plant for a given production plan. Once the composition is fitted to the model parameters, conditional simulation of the composition at any location without data can be performed. The model is applied to a porphyry copper deposit that has been recognised by a set of exploration drill holes and is used to simulate the percentages of bornite and chalcopyrite. These variables have a major influence on metallurgical recovery, and on the subsequent optimisation of an open pit mining sequence, considering 14 phases, 150 000 tpd of sulphide ore sent to flotation and 42 period horizons. The expected values of the sulphide mineral percentages can be calculated, and also the probability of successfully achieving the predicted mineralogy on the first periods of the production plan.

As well as the total copper grade, resource models for copper oxide deposits should consider the acid soluble Cu grade, which represents the fraction of total Cu grade recoverable by heap leaching. A methodology is presented which estimates the bivariate distribution of total and soluble Cu grades, based on explicit modelling of the soluble Cu grade distributions conditional to the total Cu grade and to the elevation. Once such a bivariate distribution is fitted, the total and soluble Cu grades can be converted into two unconstrained Gaussian variables via a stepwise conditional transformation and co-simulated by means of a Gaussian simulation algorithm. The method is applied to an oxide Cu deposit located in northern Chile. Drill hole samples have been assayed for the total Cu grade, whereas the soluble Cu grade has been measured for less than 30% of them. The realisations of total and soluble Cu grades are regularised onto a block support of 10×10×10 m3 and used to define the destination of each block (flotation plant, leaching plant, low-grade ore pile, or dump), depending on its expected total Cu grade and solubility ratio. The results showed considerable differences to those from co-kriging in the number of blocks sent to dump and to flotation plant, probably due to the smoothing effect of co-kriging. The proposed methodology would therefore be better for characterising the mineral type (oxide, mixed or waste) and for optimising the material sent to the metallurgical plant., As well as the total copper grade, resource models for copper oxide deposits should consider the acid soluble Cu grade, which represents the fraction of total Cu grade recoverable by heap leaching. A methodology is presented which estimates the bivariate distribution of total and soluble Cu grades, based on explicit modelling of the soluble Cu grade distributions conditional to the total Cu grade and to the elevation. Once such a bivariate distribution is fitted, the total and soluble Cu grades can be converted into two unconstrained Gaussian variables via a stepwise conditional transformation and co-simulated by means of a Gaussian simulation algorithm. The method is applied to an oxide Cu deposit located in northern Chile. Drill hole samples have been assayed for the total Cu grade, whereas the soluble Cu grade has been measured for less than 30% of them. The realisations of total and soluble Cu grades are regularised onto a block support of 10×10×10 m3 and used to define the destination of each block (flotation plant, leaching plant, low-grade ore pile, or dump), depending on its expected total Cu grade and solubility ratio. The results showed considerable differences to those from co-kriging in the number of blocks sent to dump and to flotation plant, probably due to the smoothing effect of co-kriging. The proposed methodology would therefore be better for characterising the mineral type (oxide, mixed or waste) and for optimising the material sent to the metallurgical plant.

Discussion is presented of the use of the Final Pit model for mine planning under conditions of geological uncertainty. Rather than running new models at each stage of mining, Final Pit uses algorithms which take account of the original plan and knowledge gained during mining. A floating cones algorithm is included in which a new cone is accepted in the model if it definitely adds value or if it does not destroy too much value. Use of Final Pit for a hypothetical copper mine showed that floating cones with simulated annealing could be more efficient for big block models or strong nugget effect situations. Its use is cheaper than other methods, making it possible to run models for several different simulations of block models., Discussion is presented of the use of the Final Pit model for mine planning under conditions of geological uncertainty. Rather than running new models at each stage of mining, Final Pit uses algorithms which take account of the original plan and knowledge gained during mining. A floating cones algorithm is included in which a new cone is accepted in the model if it definitely adds value or if it does not destroy too much value. Use of Final Pit for a hypothetical copper mine showed that floating cones with simulated annealing could be more efficient for big block models or strong nugget effect situations. Its use is cheaper than other methods, making it possible to run models for several different simulations of block models.

Drill-hole samples from the deposit contain information on the spatial coordinates, total iron grade, silica grade, Fe recovery ratio and rock density. There is an under-sampling of the Fe recovery ratio as this information is only available for 17 out of 70 drill holes. If the resources are estimated by co-kriging the total Fe grade, silica grade and Fe recovery ratio can be used to derive rock density and the total Fe, silica and recoverable Fe estimated on a block-by-block basis. This is likely to provide inaccurate estimates of the true contents, however, as the product of the grade, density and recovery ratio estimates may be biased. A second approach consists of directly co-kriging the contents of total Fe, silica and recoverable Fe, using as data the values of these contents calculated at the drill hole samples. Unlike the grades and recovery ratio, the contents are additive, which allows total resource calculations without bias. A comparison of the two approaches shows differences of about 55 000 000 tons in the estimated ore tonnage and of 4% in the estimated mean Fe grade. It is suggested that resource estimation should be carried out using metal contents rather than non-additive variables such as Fe grade, and that more attention should be given to the measurement of the rock density., Drill-hole samples from the deposit contain information on the spatial coordinates, total iron grade, silica grade, Fe recovery ratio and rock density. There is an under-sampling of the Fe recovery ratio as this information is only available for 17 out of 70 drill holes. If the resources are estimated by co-kriging the total Fe grade, silica grade and Fe recovery ratio can be used to derive rock density and the total Fe, silica and recoverable Fe estimated on a block-by-block basis. This is likely to provide inaccurate estimates of the true contents, however, as the product of the grade, density and recovery ratio estimates may be biased. A second approach consists of directly co-kriging the contents of total Fe, silica and recoverable Fe, using as data the values of these contents calculated at the drill hole samples. Unlike the grades and recovery ratio, the contents are additive, which allows total resource calculations without bias. A comparison of the two approaches shows differences of about 55 000 000 tons in the estimated ore tonnage and of 4% in the estimated mean Fe grade. It is suggested that resource estimation should be carried out using metal contents rather than non-additive variables such as Fe grade, and that more attention should be given to the measurement of the rock density.

An extension is proposed to the approach of Ortiz and Deutsch (2004). The key aspect is to infer the conditional distribution and then to update it only at a few thresholds, preserving its shape as much as possible. The proposed method is implemented in a case study, where point- support grades in the lower two benches of a copper mine are simulated from exploration drill-hole data, using sequential Gaussian and sequential indicator simulation, and then updated using multiple-point statistics extracted from blast-hole data from the two mined-out upper benches., An extension is proposed to the approach of Ortiz and Deutsch (2004). The key aspect is to infer the conditional distribution and then to update it only at a few thresholds, preserving its shape as much as possible. The proposed method is implemented in a case study, where point- support grades in the lower two benches of a copper mine are simulated from exploration drill-hole data, using sequential Gaussian and sequential indicator simulation, and then updated using multiple-point statistics extracted from blast-hole data from the two mined-out upper benches.

Conditional bias in kriging resource estimates with too few samples may be introduced purposely to match variance to the true block dispersion variance. Its influence on ultimate optimum pit design is evaluated in relation to shape, size and net present value with reference to a carrot-shaped gold porphyry deposit in Greece and a manto-type exotic copper deposit in Chile. For each deposit, a real block model and four resource estimates were generated by conditional simulation and four different kriging plans applied to the drillhole databases. In the worst case, conditional bias overestimated the project's net present value by 32% for the Au and 5% for the Cu deposit. Overestimation of high grades was more relevant than that of low grades, resulting in tonnage over-extraction of 148% for the Au but only 1% for the Cu deposit. The smoothing effect of kriging, without or with very little conditional bias, produced pits very different from the optimum and underestimated NPV by 10% and 6% respectively. Other factors influencing design and controlling the effect of conditional bias were cut-off grade, orebody geometry, grade distribution and amount of overburden., Conditional bias in kriging resource estimates with too few samples may be introduced purposely to match variance to the true block dispersion variance. Its influence on ultimate optimum pit design is evaluated in relation to shape, size and net present value with reference to a carrot-shaped gold porphyry deposit in Greece and a manto-type exotic copper deposit in Chile. For each deposit, a real block model and four resource estimates were generated by conditional simulation and four different kriging plans applied to the drillhole databases. In the worst case, conditional bias overestimated the project's net present value by 32% for the Au and 5% for the Cu deposit. Overestimation of high grades was more relevant than that of low grades, resulting in tonnage over-extraction of 148% for the Au but only 1% for the Cu deposit. The smoothing effect of kriging, without or with very little conditional bias, produced pits very different from the optimum and underestimated NPV by 10% and 6% respectively. Other factors influencing design and controlling the effect of conditional bias were cut-off grade, orebody geometry, grade distribution and amount of overburden.

The effect was investigated of using imprecise measurements in resource/reserve evaluation with the aid of two case studies involving Au and porphyry Cu deposits. The results show that imprecise measurements should not be discarded since they help to improve the estimates, and that simple methods such as kriging are as efficient as more sophisticated ones. The diagnosis and modelling of data imprecision are discussed., The effect was investigated of using imprecise measurements in resource/reserve evaluation with the aid of two case studies involving Au and porphyry Cu deposits. The results show that imprecise measurements should not be discarded since they help to improve the estimates, and that simple methods such as kriging are as efficient as more sophisticated ones. The diagnosis and modelling of data imprecision are discussed.

Drawbacks and limitations are considered of indicator kriging applied to continuous variables in the scope of the lognormal random function model. Particular attention is given to precision problems, inconsistencies in the results when one sample is added or removed and biases generated by the post-processing steps (tail extrapolation and change of support) and by the use of a non-bias condition in the kriging system. The selectivity of the local distributions is systematically overestimated when performing a change of support based on the global variance reduction factor, and underestimated when using ordinary kriging instead of simple kriging. This situation may lead to strong biases in the evaluation of the resources and reserves in ore deposits. To solve the change-of-support problem, a local variance reduction factor is presented for the lognormal case. The proposed approach may be generalised to other models to improve the estimates., Drawbacks and limitations are considered of indicator kriging applied to continuous variables in the scope of the lognormal random function model. Particular attention is given to precision problems, inconsistencies in the results when one sample is added or removed and biases generated by the post-processing steps (tail extrapolation and change of support) and by the use of a non-bias condition in the kriging system. The selectivity of the local distributions is systematically overestimated when performing a change of support based on the global variance reduction factor, and underestimated when using ordinary kriging instead of simple kriging. This situation may lead to strong biases in the evaluation of the resources and reserves in ore deposits. To solve the change-of-support problem, a local variance reduction factor is presented for the lognormal case. The proposed approach may be generalised to other models to improve the estimates.

Geostatistical techniques such as kriging are applied to forecasting the annual copper price from one to five years ahead, in place of the conventional econometric or statistical modelling methods. These techniques allow the incorporation of monthly and multivariate information in the analysis and provide more accurate results than traditional approaches, especially as far as the one-year forecast is concerned. A second advantage is the ability of geostatistics to assess the uncertainty associated with the unknown future prices by means of estimation variances or confidence intervals, depending on the technique used. Such an assessment has a crucial importance in decision-making, from both macro- and microeconomic points of view. The method was tested by ex ante forecasts, using as the "current" date each successive year from 1975 to 1999., Geostatistical techniques such as kriging are applied to forecasting the annual copper price from one to five years ahead, in place of the conventional econometric or statistical modelling methods. These techniques allow the incorporation of monthly and multivariate information in the analysis and provide more accurate results than traditional approaches, especially as far as the one-year forecast is concerned. A second advantage is the ability of geostatistics to assess the uncertainty associated with the unknown future prices by means of estimation variances or confidence intervals, depending on the technique used. Such an assessment has a crucial importance in decision-making, from both macro- and microeconomic points of view. The method was tested by ex ante forecasts, using as the "current" date each successive year from 1975 to 1999.

The evaluation of process performance within mining operations at Mehdiabad complex deposit, Central Iran, requires modelling the spatial variability of six cross-correlated grade variables: zinc (Zn) and lead (Pb) as the main products, silver (Ag) and copper (Cu) as byproducts, and iron (Fe) and manganese (Mn) as contaminants. To this end, the variables are transformed first into spatially uncorrelated factors, using the minimum/maximum autocorrelation factors approach, then the factors are simulated directly at the target block support. The result is a set of realisations (equiprobable scenarios) of the grade distribution within the deposit, which reproduce the joint dependence relationships and natural variability at all spatial scales. This model can decrease the level of uncertainty in Mehdiabad project development and consequently in production. (Authors.), The evaluation of process performance within mining operations at Mehdiabad complex deposit, Central Iran, requires modelling the spatial variability of six cross-correlated grade variables: zinc (Zn) and lead (Pb) as the main products, silver (Ag) and copper (Cu) as byproducts, and iron (Fe) and manganese (Mn) as contaminants. To this end, the variables are transformed first into spatially uncorrelated factors, using the minimum/maximum autocorrelation factors approach, then the factors are simulated directly at the target block support. The result is a set of realisations (equiprobable scenarios) of the grade distribution within the deposit, which reproduce the joint dependence relationships and natural variability at all spatial scales. This model can decrease the level of uncertainty in Mehdiabad project development and consequently in production. (Authors.)