David F Machuca Mory - Academia.edu (original) (raw)

Papers by David F Machuca Mory

Research paper thumbnail of Ore Grade Prediction Using a Genetic Algorithm and Clustering Based Ensemble Neural Network Model

Mathematical Geosciences, 2010

Abstract Accurate prediction of ore grade is essential for many basic mine operations, including ... more Abstract Accurate prediction of ore grade is essential for many basic mine operations, including mine planning and design, pit optimization, and ore grade control. Preference is given to the neural network over other interpolation techniques for ore grade estimation ...

Research paper thumbnail of Locally Stationary MultiGaussian Kriging with Local Change of Support Model

The incorporation of location-dependent distributions and statistics in multiGaussian kriging is ... more The incorporation of location-dependent distributions and statistics in multiGaussian kriging is proposed to overcome the limitations imposed by the strict stationarity assumption. These are obtained using distance weighting functions and are defined specifically for every location to be estimated. Local distributions require local Gaussian transformations. These are modeled using Hermite polynomial series, which are efficient and allow implementing the discrete Gaussian change of support model with locally varying parameters. Location dependent variogram models complement the definition of the location-dependent distributions and its statistics offering the capability of adjusting to local changes of the spatial continuity. The Locally Stationary multiGaussian Kriging algorithm uses the local Hermitian Gaussian transformation and local change of support model to produce point and block support estimates.

Research paper thumbnail of Location Dependent Variograms

Research paper thumbnail of On the challenge of using sequential indicator simulation for the estimation of recoverable reserves

International Journal of Mining, Reclamation and Environment, 2008

Despite its flexibility when compared with parametric methods, there are significant problems wit... more Despite its flexibility when compared with parametric methods, there are significant problems with sequential indicator simulation (SIS) for continuous variables. These come from the successive construction of the local conditional distributions by indicator kriging and the use of Monte–Carlo simulation for sampling the simulated values from such distributions. Some problems of SIS have been addressed satisfactorily; order relation deviations and the extrapolation of the distribution tails are manageable. The most important outstanding challenges are: (1) there is no resolution within a grade range, and (2) there is no correlation between different grade ranges. These problems lead to an over-smoothing of grades and a reduced, but misleading, uncertainty when point scale simulations are average up at the selective mining unit size. Attempting to account for the interclass correlation by cokriging fails because there is no adequate model for fitting the complete matrix of indicator variograms. Also, attempting to improve the within class resolution by using the class edge distance function does not provide a complete solution. In regard of these unresolved challenges, the practitioner is cautioned about the use of SIS of continuous variables for Resource Evaluation. Instead, a combination of categorical variables simulation methods (including SIS) for modeling the mineralisation domains, and Gaussian simulation of grades within these domains is recommended.

Research paper thumbnail of Grade Modelling with Local Anisotropy Angles: A Practical Point of View

Research paper thumbnail of Simulation of a Structurally-Controlled Gold Deposit using High-Order Statistics

The algorithm for conditional simulation based on spatial high-order statistics is applied to a d... more The algorithm for conditional simulation based on spatial high-order statistics is applied to a drilling dataset obtained from a structurally complex gold deposit, the Apensu deposit in Ghana. Spatial high-order statistics allow capturing nonlinear spatial features of the gold mineralization that variograms and covariances cannot. Since robust spatial high-order statistics cannot be inferred only from scattered samples, they are borrowed from a training image. In this case, sequential Gaussian simulation with local variograms within domain boundaries is used to build a training image. At different locations HOSIM uses the spatial high-order statistics to approximate non-Gaussian distributions of possible values conditioned by neighboring data. The effect of sampling clustering in the probability distribution and its statistics is taken into account by incorporating declustering weights in the inference of low and high-order statistics required by high-order simulation. The resulting...

Research paper thumbnail of High-Order Spacial Direct and Cross-Statistics for Categorical Attributes

The characterization of the spatial continuity of categorical variables, such as geological units... more The characterization of the spatial continuity of categorical variables, such as geological units, is a longstanding subject in geostatistics. Indicator covariances and variograms are used to measure spatial relationships of categorical data between pairs of points. Alternatively, transition probabilities, or transiograms, have been proposed to measure the probability of transition from one category to another as a function of distance. Recently, high-order moments and cumulants built from them have been proposed as measures of complex non-linear spatial relationships for arrangements of multiple points in 3D space. This paper extends the spatial high order statistics, originally conceived for continuous data, to the analysis of categorical spatial datasets. In addition the concept of two-point conditional transition probabilities is expanded to multiple point conditioned probabilities. The algorithm for high-order statistics, HOSC, has been updated to allow for the proposed high-or...

Research paper thumbnail of Short Note on Cokriging in Sequential Indicator Simulation : The Adjacent cut-off Alternative

A drawback of Sequential Indicator Simulation is the uncontrolled transitions between classes, wh... more A drawback of Sequential Indicator Simulation is the uncontrolled transitions between classes, which translates in the patchiness of high and low values areas in the resulting realizations. The full cokriging approach has been proposed to solve this disadvantage; all direct and cross indicator variograms would be used. This approach should introduce some order in the interclass transitions by including the interclass cross correlation information; however, the Linear Model of Corregionalization does not provide a satisfactory fitting for indicator cross variograms of extremely separated thresholds. The alternative proposed in this paper is to use only the corregionalization information of the two closest thresholds to the one that is been used for the conditional CDF estimation. This alternative has been implemented in the SISIM_adj program. The implementation details, the results using synthetic and real data and the performance comparison of this alternative with the direct indica...

Research paper thumbnail of Tonnage Uncertainty Assessment of Vein-Type Deposits Using Distance Functions and Location-Dependent Correlograms

Modelling the geometry of a vein is a crucial step in resources estimation. The resulting models ... more Modelling the geometry of a vein is a crucial step in resources estimation. The resulting models are used as mineralization domain boundaries and have a direct impact on the tonnage of estimated resources. Deterministic models are often built using time consuming wireframing techniques usually based on hand interpretation of the drillhole intercepts. Another approach consists in coding the drillhole samples by a function of their distance to the veins contacts. The coding is subsequently used for modelling the vein contacts away from drillholes. This is a more efficient approach and is able to provide a measure of tonnage uncertainty. The use of location-dependent variograms improves the modelling by incorporating local changes in the anisotropy of the vein structure. This combined approach results in more realistic vein models, particularly when the geometry of the vein has been altered by folding, shearing and other structural processes. This numerical approach is illustrated on a...

Research paper thumbnail of Locally Stationary MultiGaussian Kriging with Local Change of Support Model

The incorporation of location-dependent distributions and statistics in multiGaussian kriging is ... more The incorporation of location-dependent distributions and statistics in multiGaussian kriging is proposed to overcome the limitations imposed by the strict stationarity assumption. These are obtained using distance weighting functions and are defined specifically for every location to be estimated. Local distributions require local Gaussian transformations. These are modeled using Hermite polynomial series, which are efficient and allow implementing the discrete Gaussian change of support model with locally varying parameters. Location dependent variogram models complement the definition of the location-dependent distributions and its statistics offering the capability of adjusting to local changes of the spatial continuity. The Locally Stationary multiGaussian Kriging algorithm uses the local Hermitian Gaussian transformation and local change of support model to produce point and block support estimates.

Research paper thumbnail of Sequential Gaussian and Indicator Simulation with Location- Dependent Distributions and Statistics

The use of location-dependent distributions and statistics is proposed for geostatistical simulat... more The use of location-dependent distributions and statistics is proposed for geostatistical simulation under the assumption of local stationarity. The local distributions and statistics are obtained using distance weighting functions. For Sequential Gaussian Simulation, the Gaussian transformation of each local distribution embeds the local changes in the local mean, variance and histogram shape. The same weights used for inferring the local distribution modify the local measures of spatial continuity, which adapt to local variations informed by data. The local Gaussian transformations are modelled by Hermite polynomial series and the resulting coefficients are stored. The local measures of correlation are fitted semiautomatically and, as for the Hermite coefficients, the resulting parameters are stored at the resolution of the simulation grid. The sequential simulation algorithms read these local parameters, update the local distribution, retransform the data, and recalculate the cov...

Research paper thumbnail of Flexible change of support model suitable for a wide range of mineralization styles

Mining Engineering, 2008

A change of support model consists of a procedure to change a stationary histogram at a small dat... more A change of support model consists of a procedure to change a stationary histogram at a small data scale to represent a larger scale; typically a distribution of sample drillhole data is changed to represent a chosen selective mining unit (SMU) scale. Most grade variables average arithmetically. Thus, the mean stays the same for different scales and the variance changes according to well established theory using average variogram values. The longstanding challenge has been to predict how the shape of the histogram changes. The discrete Gaussian model is widely used because it appears reasonable and introduces few artifacts; there are no artificial minimum/maximum values and the target mean and variance are reproduced exactly. The resulting distribution shape, however, is strongly dependent on a multivariate Gaussian distribution. We generalize the approach by taking advantage of a property of the isofactorial model, which allows us to specify where the random function falls on the s...

Research paper thumbnail of High-order simulation at block support scale

The use of spatial high-order statistics has been previously proposed as an alternative to introd... more The use of spatial high-order statistics has been previously proposed as an alternative to introduce richer information about complex spatial patterns in the simulation of continuous attributes. These statistics are normally inferred from exhaustive quasi -support training images. Spatial high-order statistics values are combined within series of orthogonal polynomials to approximate local conditional distributions that can be used for the drawing of simulated point-support values. This paper extends this formalism to direct blocksupport simulation. This is achieved by inferring block-point high-order statistics from up-scaled training images and incorporating these statistics in the orthogonal polynomials approximation of the conditional distributions. This methodology is computationally expensive, so a reasonable option is to approximate all the required local conditional distributions only once. These can be subsequently sampled by different fields of correlated probabilities to ...

Research paper thumbnail of A Program for Robust Calculation of Drillhole Spacing in Three Dimensions

A robust algorithm and program to calculate the drillhole spacing in 3-D is developed. This progr... more A robust algorithm and program to calculate the drillhole spacing in 3-D is developed. This program overcomes issues such as artefacts in regular grids, irregularity and over-smoothing of the resultant maps when a very small or very large constant search radius is used. Grade continuity and anisotropy are taken in account. Program performance is tested with synthetic and real data sets. The relation between geometric and probabilistic criteria is shown and used to define the limits for the resources categories.

Research paper thumbnail of On the Challenge of Estimating Recoverable Reserves with Continuous Variable Sequential Indicator Simulation

Indicator Kriging or Multiple Indicator Kriging received significant attention as a non-linear ap... more Indicator Kriging or Multiple Indicator Kriging received significant attention as a non-linear approach to estimate recoverable reserves. The basic idea is to discretize the range of variability and directly predict the conditional distribution at unsampled locations. These point-scale distributions are sometimes corrected to account for a selective block size, which provides a direct estimate of recoverable reserves. A related idea is sequential simulation, which is a well-established paradigm of simulation. A multivariate distribution is sampled via sampling a succession of conditional distributions. Sequential Indicator Simulation (SIS) was proposed in the 1980s as a flexible simulation approach to categorical and continuous variables. The conditional distributions are built with indicator kriging. Simulation is done at a point scale: no volume variance change is applied.

Research paper thumbnail of Simulation of a Structurally-Controlled Gold Deposit using High-Order Statistics

The algorithm for conditional simulation based on spatial high-order statistics is applied to a d... more The algorithm for conditional simulation based on spatial high-order statistics is applied to a drilling dataset obtaine d from a structurally complex gold deposit, the Apensu deposit in Ghana. Spatial high- order statistics allow capturing nonlinear spatial features of the gold mineralizati on that variograms and covariances cannot. Since robust spatial high-order statistics cannot be inferred only from scattered samples, they are borrowed from a training image. In this case, sequential Gaussian simulation with local var iograms within domain boundaries is used to build a training image. At di fferent locations HOSIM uses the spatial high-order statistics to approximate no n-Gaussian distributions of possible values conditioned by neighboring data. Th e effect of sampling clustering in the probability distribution and its statistics is taken into account by incorporating declustering weights in the inference of low and high-order statistics required by high-order simulation. The re...

Research paper thumbnail of Excess Variability in Realizations of Sequential Indicator Simulation of Continuous Variables

Sequential Indicator Simulation (SIS) realizations often exhibit high and unrealistic short scale... more Sequential Indicator Simulation (SIS) realizations often exhibit high and unrealistic short scale variability; this is due to the uncontrolled transitions between classes and the randomness inside each class introduced by the Monte-Carlo drawing within classes. Despite these problems, SIS has some useful properties that most of the other simulation techniques have not; this motivates further research to overcome the problems of SIS. As a first step towards the improvement of SIS, the impact of this unwarranted short scale variability in the block scale uncertainty is analyzed and compared to Sequential Gaussian Simulation results in a numerical example, obtaining a reduced block scale uncertainty for SIS results. A path for subsequent research work to improve the algorithm and its results is also delineated.

Research paper thumbnail of Optimal weights for Location Dependent Moments

One approach for the estimation of location dependent moments is to weight the available samples ... more One approach for the estimation of location dependent moments is to weight the available samples according a function of their isotropic or anisotropic distance to a given location. These weights are then incorporated in the calculation of 1-point and 2-point moments. Several desirable properties must be fulfilled by them, such as smoothness, unbiasedness, positivity, global consistency and independency of units. The weights can be calibrated with declustering weights. The estimation of 2-point moments, such as the variogram, can be achieved from the weights assigned to sample pairs and an appropriate mixture rule of the sample weights involved (arithmetic average or geometric average). Several weighting functions can be used including inverse distance, Gaussian kernel and global ordinary kriging. The choice of the weighting function is more important than the choice of the mixture rule. Gaussian Kernel weighting proves to be very useful for location dependent moments estimation and...

Research paper thumbnail of Validation and Confirmation of Non-Stationary Models with the Ventersdorp Reef Data

Validation criteria for non-stationary parameters and the models generated using them within a lo... more Validation criteria for non-stationary parameters and the models generated using them within a locallystationary framework are presented and discussed. A smooth and unbiased adaptation of the local parameters to the variations informed by data values is important. The criteria for simulated models are the reproduction of the input data and the input models of spatial continuity. These validity criteria for parameters and numerical models are illustrated with the help of a sparse data set that mimics samples taken from drillholes interception the Ventersdrop Contact Reef. A denser data set is used for confirming the initial local models of spatial continuity and the realizations generated using them. Local data scarcity and short scale variations of the spatial distribution appear to be the main difficulties in the confirmation of the initial local variogram models. Simulated models satisfy minimum validity criteria, such as the reproduction of the global histogram and the local spat...

Research paper thumbnail of Estimation with Non Stationary First and Second Moments

The calculation and modelling of local variograms was developed in the previous paper. These loca... more The calculation and modelling of local variograms was developed in the previous paper. These local variograms and correlograms, as well as local mean and variance values are built by weighting the sample pairs used in variogram calculation. They prove to better reflect the local spatial behaviour of the variable under study and the results are represented as 2-D or 3-D maps of the model parameters values; there is no longer a stationary set of variogram model parameters. Once the locally varying first and second moments are available at every location, the next step is to use them in estimation. A non stationary Kriging approach and program are developed below. The variogram or correlogram are assumed stationary for all locations within the search of a particular point or block.

Research paper thumbnail of Ore Grade Prediction Using a Genetic Algorithm and Clustering Based Ensemble Neural Network Model

Mathematical Geosciences, 2010

Abstract Accurate prediction of ore grade is essential for many basic mine operations, including ... more Abstract Accurate prediction of ore grade is essential for many basic mine operations, including mine planning and design, pit optimization, and ore grade control. Preference is given to the neural network over other interpolation techniques for ore grade estimation ...

Research paper thumbnail of Locally Stationary MultiGaussian Kriging with Local Change of Support Model

The incorporation of location-dependent distributions and statistics in multiGaussian kriging is ... more The incorporation of location-dependent distributions and statistics in multiGaussian kriging is proposed to overcome the limitations imposed by the strict stationarity assumption. These are obtained using distance weighting functions and are defined specifically for every location to be estimated. Local distributions require local Gaussian transformations. These are modeled using Hermite polynomial series, which are efficient and allow implementing the discrete Gaussian change of support model with locally varying parameters. Location dependent variogram models complement the definition of the location-dependent distributions and its statistics offering the capability of adjusting to local changes of the spatial continuity. The Locally Stationary multiGaussian Kriging algorithm uses the local Hermitian Gaussian transformation and local change of support model to produce point and block support estimates.

Research paper thumbnail of Location Dependent Variograms

Research paper thumbnail of On the challenge of using sequential indicator simulation for the estimation of recoverable reserves

International Journal of Mining, Reclamation and Environment, 2008

Despite its flexibility when compared with parametric methods, there are significant problems wit... more Despite its flexibility when compared with parametric methods, there are significant problems with sequential indicator simulation (SIS) for continuous variables. These come from the successive construction of the local conditional distributions by indicator kriging and the use of Monte–Carlo simulation for sampling the simulated values from such distributions. Some problems of SIS have been addressed satisfactorily; order relation deviations and the extrapolation of the distribution tails are manageable. The most important outstanding challenges are: (1) there is no resolution within a grade range, and (2) there is no correlation between different grade ranges. These problems lead to an over-smoothing of grades and a reduced, but misleading, uncertainty when point scale simulations are average up at the selective mining unit size. Attempting to account for the interclass correlation by cokriging fails because there is no adequate model for fitting the complete matrix of indicator variograms. Also, attempting to improve the within class resolution by using the class edge distance function does not provide a complete solution. In regard of these unresolved challenges, the practitioner is cautioned about the use of SIS of continuous variables for Resource Evaluation. Instead, a combination of categorical variables simulation methods (including SIS) for modeling the mineralisation domains, and Gaussian simulation of grades within these domains is recommended.

Research paper thumbnail of Grade Modelling with Local Anisotropy Angles: A Practical Point of View

Research paper thumbnail of Simulation of a Structurally-Controlled Gold Deposit using High-Order Statistics

The algorithm for conditional simulation based on spatial high-order statistics is applied to a d... more The algorithm for conditional simulation based on spatial high-order statistics is applied to a drilling dataset obtained from a structurally complex gold deposit, the Apensu deposit in Ghana. Spatial high-order statistics allow capturing nonlinear spatial features of the gold mineralization that variograms and covariances cannot. Since robust spatial high-order statistics cannot be inferred only from scattered samples, they are borrowed from a training image. In this case, sequential Gaussian simulation with local variograms within domain boundaries is used to build a training image. At different locations HOSIM uses the spatial high-order statistics to approximate non-Gaussian distributions of possible values conditioned by neighboring data. The effect of sampling clustering in the probability distribution and its statistics is taken into account by incorporating declustering weights in the inference of low and high-order statistics required by high-order simulation. The resulting...

Research paper thumbnail of High-Order Spacial Direct and Cross-Statistics for Categorical Attributes

The characterization of the spatial continuity of categorical variables, such as geological units... more The characterization of the spatial continuity of categorical variables, such as geological units, is a longstanding subject in geostatistics. Indicator covariances and variograms are used to measure spatial relationships of categorical data between pairs of points. Alternatively, transition probabilities, or transiograms, have been proposed to measure the probability of transition from one category to another as a function of distance. Recently, high-order moments and cumulants built from them have been proposed as measures of complex non-linear spatial relationships for arrangements of multiple points in 3D space. This paper extends the spatial high order statistics, originally conceived for continuous data, to the analysis of categorical spatial datasets. In addition the concept of two-point conditional transition probabilities is expanded to multiple point conditioned probabilities. The algorithm for high-order statistics, HOSC, has been updated to allow for the proposed high-or...

Research paper thumbnail of Short Note on Cokriging in Sequential Indicator Simulation : The Adjacent cut-off Alternative

A drawback of Sequential Indicator Simulation is the uncontrolled transitions between classes, wh... more A drawback of Sequential Indicator Simulation is the uncontrolled transitions between classes, which translates in the patchiness of high and low values areas in the resulting realizations. The full cokriging approach has been proposed to solve this disadvantage; all direct and cross indicator variograms would be used. This approach should introduce some order in the interclass transitions by including the interclass cross correlation information; however, the Linear Model of Corregionalization does not provide a satisfactory fitting for indicator cross variograms of extremely separated thresholds. The alternative proposed in this paper is to use only the corregionalization information of the two closest thresholds to the one that is been used for the conditional CDF estimation. This alternative has been implemented in the SISIM_adj program. The implementation details, the results using synthetic and real data and the performance comparison of this alternative with the direct indica...

Research paper thumbnail of Tonnage Uncertainty Assessment of Vein-Type Deposits Using Distance Functions and Location-Dependent Correlograms

Modelling the geometry of a vein is a crucial step in resources estimation. The resulting models ... more Modelling the geometry of a vein is a crucial step in resources estimation. The resulting models are used as mineralization domain boundaries and have a direct impact on the tonnage of estimated resources. Deterministic models are often built using time consuming wireframing techniques usually based on hand interpretation of the drillhole intercepts. Another approach consists in coding the drillhole samples by a function of their distance to the veins contacts. The coding is subsequently used for modelling the vein contacts away from drillholes. This is a more efficient approach and is able to provide a measure of tonnage uncertainty. The use of location-dependent variograms improves the modelling by incorporating local changes in the anisotropy of the vein structure. This combined approach results in more realistic vein models, particularly when the geometry of the vein has been altered by folding, shearing and other structural processes. This numerical approach is illustrated on a...

Research paper thumbnail of Locally Stationary MultiGaussian Kriging with Local Change of Support Model

The incorporation of location-dependent distributions and statistics in multiGaussian kriging is ... more The incorporation of location-dependent distributions and statistics in multiGaussian kriging is proposed to overcome the limitations imposed by the strict stationarity assumption. These are obtained using distance weighting functions and are defined specifically for every location to be estimated. Local distributions require local Gaussian transformations. These are modeled using Hermite polynomial series, which are efficient and allow implementing the discrete Gaussian change of support model with locally varying parameters. Location dependent variogram models complement the definition of the location-dependent distributions and its statistics offering the capability of adjusting to local changes of the spatial continuity. The Locally Stationary multiGaussian Kriging algorithm uses the local Hermitian Gaussian transformation and local change of support model to produce point and block support estimates.

Research paper thumbnail of Sequential Gaussian and Indicator Simulation with Location- Dependent Distributions and Statistics

The use of location-dependent distributions and statistics is proposed for geostatistical simulat... more The use of location-dependent distributions and statistics is proposed for geostatistical simulation under the assumption of local stationarity. The local distributions and statistics are obtained using distance weighting functions. For Sequential Gaussian Simulation, the Gaussian transformation of each local distribution embeds the local changes in the local mean, variance and histogram shape. The same weights used for inferring the local distribution modify the local measures of spatial continuity, which adapt to local variations informed by data. The local Gaussian transformations are modelled by Hermite polynomial series and the resulting coefficients are stored. The local measures of correlation are fitted semiautomatically and, as for the Hermite coefficients, the resulting parameters are stored at the resolution of the simulation grid. The sequential simulation algorithms read these local parameters, update the local distribution, retransform the data, and recalculate the cov...

Research paper thumbnail of Flexible change of support model suitable for a wide range of mineralization styles

Mining Engineering, 2008

A change of support model consists of a procedure to change a stationary histogram at a small dat... more A change of support model consists of a procedure to change a stationary histogram at a small data scale to represent a larger scale; typically a distribution of sample drillhole data is changed to represent a chosen selective mining unit (SMU) scale. Most grade variables average arithmetically. Thus, the mean stays the same for different scales and the variance changes according to well established theory using average variogram values. The longstanding challenge has been to predict how the shape of the histogram changes. The discrete Gaussian model is widely used because it appears reasonable and introduces few artifacts; there are no artificial minimum/maximum values and the target mean and variance are reproduced exactly. The resulting distribution shape, however, is strongly dependent on a multivariate Gaussian distribution. We generalize the approach by taking advantage of a property of the isofactorial model, which allows us to specify where the random function falls on the s...

Research paper thumbnail of High-order simulation at block support scale

The use of spatial high-order statistics has been previously proposed as an alternative to introd... more The use of spatial high-order statistics has been previously proposed as an alternative to introduce richer information about complex spatial patterns in the simulation of continuous attributes. These statistics are normally inferred from exhaustive quasi -support training images. Spatial high-order statistics values are combined within series of orthogonal polynomials to approximate local conditional distributions that can be used for the drawing of simulated point-support values. This paper extends this formalism to direct blocksupport simulation. This is achieved by inferring block-point high-order statistics from up-scaled training images and incorporating these statistics in the orthogonal polynomials approximation of the conditional distributions. This methodology is computationally expensive, so a reasonable option is to approximate all the required local conditional distributions only once. These can be subsequently sampled by different fields of correlated probabilities to ...

Research paper thumbnail of A Program for Robust Calculation of Drillhole Spacing in Three Dimensions

A robust algorithm and program to calculate the drillhole spacing in 3-D is developed. This progr... more A robust algorithm and program to calculate the drillhole spacing in 3-D is developed. This program overcomes issues such as artefacts in regular grids, irregularity and over-smoothing of the resultant maps when a very small or very large constant search radius is used. Grade continuity and anisotropy are taken in account. Program performance is tested with synthetic and real data sets. The relation between geometric and probabilistic criteria is shown and used to define the limits for the resources categories.

Research paper thumbnail of On the Challenge of Estimating Recoverable Reserves with Continuous Variable Sequential Indicator Simulation

Indicator Kriging or Multiple Indicator Kriging received significant attention as a non-linear ap... more Indicator Kriging or Multiple Indicator Kriging received significant attention as a non-linear approach to estimate recoverable reserves. The basic idea is to discretize the range of variability and directly predict the conditional distribution at unsampled locations. These point-scale distributions are sometimes corrected to account for a selective block size, which provides a direct estimate of recoverable reserves. A related idea is sequential simulation, which is a well-established paradigm of simulation. A multivariate distribution is sampled via sampling a succession of conditional distributions. Sequential Indicator Simulation (SIS) was proposed in the 1980s as a flexible simulation approach to categorical and continuous variables. The conditional distributions are built with indicator kriging. Simulation is done at a point scale: no volume variance change is applied.

Research paper thumbnail of Simulation of a Structurally-Controlled Gold Deposit using High-Order Statistics

The algorithm for conditional simulation based on spatial high-order statistics is applied to a d... more The algorithm for conditional simulation based on spatial high-order statistics is applied to a drilling dataset obtaine d from a structurally complex gold deposit, the Apensu deposit in Ghana. Spatial high- order statistics allow capturing nonlinear spatial features of the gold mineralizati on that variograms and covariances cannot. Since robust spatial high-order statistics cannot be inferred only from scattered samples, they are borrowed from a training image. In this case, sequential Gaussian simulation with local var iograms within domain boundaries is used to build a training image. At di fferent locations HOSIM uses the spatial high-order statistics to approximate no n-Gaussian distributions of possible values conditioned by neighboring data. Th e effect of sampling clustering in the probability distribution and its statistics is taken into account by incorporating declustering weights in the inference of low and high-order statistics required by high-order simulation. The re...

Research paper thumbnail of Excess Variability in Realizations of Sequential Indicator Simulation of Continuous Variables

Sequential Indicator Simulation (SIS) realizations often exhibit high and unrealistic short scale... more Sequential Indicator Simulation (SIS) realizations often exhibit high and unrealistic short scale variability; this is due to the uncontrolled transitions between classes and the randomness inside each class introduced by the Monte-Carlo drawing within classes. Despite these problems, SIS has some useful properties that most of the other simulation techniques have not; this motivates further research to overcome the problems of SIS. As a first step towards the improvement of SIS, the impact of this unwarranted short scale variability in the block scale uncertainty is analyzed and compared to Sequential Gaussian Simulation results in a numerical example, obtaining a reduced block scale uncertainty for SIS results. A path for subsequent research work to improve the algorithm and its results is also delineated.

Research paper thumbnail of Optimal weights for Location Dependent Moments

One approach for the estimation of location dependent moments is to weight the available samples ... more One approach for the estimation of location dependent moments is to weight the available samples according a function of their isotropic or anisotropic distance to a given location. These weights are then incorporated in the calculation of 1-point and 2-point moments. Several desirable properties must be fulfilled by them, such as smoothness, unbiasedness, positivity, global consistency and independency of units. The weights can be calibrated with declustering weights. The estimation of 2-point moments, such as the variogram, can be achieved from the weights assigned to sample pairs and an appropriate mixture rule of the sample weights involved (arithmetic average or geometric average). Several weighting functions can be used including inverse distance, Gaussian kernel and global ordinary kriging. The choice of the weighting function is more important than the choice of the mixture rule. Gaussian Kernel weighting proves to be very useful for location dependent moments estimation and...

Research paper thumbnail of Validation and Confirmation of Non-Stationary Models with the Ventersdorp Reef Data

Validation criteria for non-stationary parameters and the models generated using them within a lo... more Validation criteria for non-stationary parameters and the models generated using them within a locallystationary framework are presented and discussed. A smooth and unbiased adaptation of the local parameters to the variations informed by data values is important. The criteria for simulated models are the reproduction of the input data and the input models of spatial continuity. These validity criteria for parameters and numerical models are illustrated with the help of a sparse data set that mimics samples taken from drillholes interception the Ventersdrop Contact Reef. A denser data set is used for confirming the initial local models of spatial continuity and the realizations generated using them. Local data scarcity and short scale variations of the spatial distribution appear to be the main difficulties in the confirmation of the initial local variogram models. Simulated models satisfy minimum validity criteria, such as the reproduction of the global histogram and the local spat...

Research paper thumbnail of Estimation with Non Stationary First and Second Moments

The calculation and modelling of local variograms was developed in the previous paper. These loca... more The calculation and modelling of local variograms was developed in the previous paper. These local variograms and correlograms, as well as local mean and variance values are built by weighting the sample pairs used in variogram calculation. They prove to better reflect the local spatial behaviour of the variable under study and the results are represented as 2-D or 3-D maps of the model parameters values; there is no longer a stationary set of variogram model parameters. Once the locally varying first and second moments are available at every location, the next step is to use them in estimation. A non stationary Kriging approach and program are developed below. The variogram or correlogram are assumed stationary for all locations within the search of a particular point or block.

Research paper thumbnail of Simulación Geoestadística

Guía de la teoría dictada en el curso corto "Simulación Geoestadística", el cual fue parte de la... more Guía de la teoría dictada en el curso corto "Simulación Geoestadística", el cual fue parte de la Segunda Jornada Internacional de Probabilidad y Estadística. Escuela de Posgrado de la Pontificia Universidad Católica del Perú. Lima, Perú, Febrero de 2012.