CFSG Module A: Linear Geostatistics for Local Resource Estimation

Content

This is the first module of the CFSG series. Module A is the program’s core module, covering the fundamental principles and workflows of mining geostatistics for mineral resource estimation.

WEEK 1 – STATISTICS FOR MINERAL RESOURCES

Theory

The different types of quantities
– Quantitative variables, such as grade, density, or metal quantity.
– Categorical variables, such as geological facies and rock types.
– Missing information, Limit of Detection (LOD), and Limit of Quantification (LOQ).
– Spatially defined variables, including drillholes, maps, and block models.
– Additive variables.
– Support of information, including its size and shape, and the volume of selection, such as the Selective Mining Unit.

Sampling for spatialized variables
– Clustered and preferential sampling.
– Sampling geometries, including scattered data, seismic lines, drillholes, and regular grids.
– Declustering and weighted statistics.

Univariate statistics
– Histograms.
– Measures of central tendency: mean, median, and mode.
– Summary statistics: mean, median, mode to capture the centrality; variance, interquartile range, coefficient of variation to capture the dispersion; minimum, maximum, quantiles, box plots to capture the extremes.
– Base maps and swath plots.
– Transformation of the variable: logarithm, log, indicator, capping, ranking, proportional effect
– Continuous and discrete distributions: Gaussian, lognormal, uniform, triangular, exponential, gamma, Bernoulli, Binomial, Poisson.

Selectivity curves
– ules for selection: cutoff and support (sample vs. Selective Mining Unit)
– Tonnage, Average Grade, Metal, and Conventional benefit
– Support effect
– Information effect.

Practice

Introduction to Isatis.neo Mining Edition. Learn how to create and manage a project, work with different types of datasets, and apply the statistical concepts introduced during the theoretical sessions.

WEEK 2 – MODELING SPATIAL CONTINUITY

Theory

Exploratory Data Analysis
– Stationarity analysis using swath plots.

Measuring spatial continuity
– Spatial covariance and variograms.
– Variogram clouds and variogram maps.
– Calculations in one-, two-, and three-dimensional spaces.
– Other empirical structural tools: robust variogram, madogram, rodogram.

Variogram modeling
– The basic models: Nugget Effect, Exponential, Spherical, Gaussian, Cubic, and Linear.
– Model parameters and properties.
–  The nested model and its multi-scale interpretation.
–  Anisotropies: geometric, zonal, separable.
–  Variogram fitting strategy.

Practice

Several exercises to learn to import data, perform exploratory data analysis, compute experimental variograms, and adjust variogram models with Isatis.neo.

WEEK 3 – KRIGING FOR LOCAL RESOURCE ESTIMATION

Theory

Estimator
– Examples: Moving Mean, Nearest Neighbor, and Inverse Distance.
– Understanding precision versus accuracy.
– Dichotomy between (deterministic) Drift and (stochastic) Residuals: (strictly) stationary, intrinsic, or non-stationary.
– Linear, unbiased, and optimal estimation.
– Estimation quality and estimation error.

Kriging – Best Linear Unbiased Estimation
– Simple Kriging with a known mean.
– Ordinary Kriging for intrinsic cases.
– Block Kriging and change of support.
– Extensions such as Kriging with measurement error variance and filtering.

Neighborhood parameters
– The Neighborhood: Moving vs. Global.
– Kriging Neighborhood Analysis (KNA).

Validating resource models
– Cross-validation: leave-one-out and K-fold.
– Validation of kriging estimates and block models.

Practice

Complete several exercises using Isatis.neo. Learn how to build block models using ordinary and simple kriging and validate estimation results through cross-validation.

WEEK 4 – MULTIVARIATE GEOSTATISTICS & INTRODUCTION TO SIMULATIONS

Theory

Multivariate statistics
– Experimental statistics: scatter plots, correlation table, regressions (linear and non-linear).
– Marginal and conditional distributions.
– Linear and empirical regression.
– Transforms: Principal Component Analysis (PCA), Minimum/Maximum Autocorrelation Factors (MAF), indicator residuals.

Multivariate modeling
– Direct and cross-variograms.
– Linear Model of Coregionalization.

Multivariate estimation
– Simple and Ordinary Cokriging.
– Collocated Cokriging.
– Rescaled Cokriging.
– Factorial Kriging Analysis.

Non-stationary modeling
– Dichotomy between (deterministic) Drift and (stochastic) Residuals: (strictly) stationary, intrinsic, or non-stationary.
– Exploratory analysis: swath plots, cross plots, experimental variograms (quadratic behavior, or more).
– Non-stationary models: drift and stationary residuals.
– Extension to complex drifts: Intrinsic Random Function of order k (Generalized covariances).

Estimation
– Kriging with local anisotropies.
– Universal Kriging.
– Kriging with External Drift.
– Factorial Kriging Analysis.

Introduction to geostatistical simulations
– Understanding the smoothing effect of kriging.
– Introduction to spatial uncertainty assessment.
Participants interested in more detailed simulations can proceed to Module C: Simulation of Continuous Variables for Uncertainty and Risk Analysis.

Practice

Complete several exercises using Isatis.neo. Learn how to analyze contacts between domains, calculate and fit multivariate variograms, apply different cokriging methods, and work with non-stationary modeling techniques.