Basic Linear Geostatistics

Frontmatter

1. Introduction

Summary

After outlining the types of problems in mining that geostatistics can be used to solve, an introductory exercise is presented. It illustrates the need for good estimators, particularly when selective mining is being used. The economic impact of the support and information effects on reserve calculations is stressed. Lastly some case studies comparing geostatistics with other estimation methods are reviewed.

Margaret Armstrong

2. Regionalized Variables

Summary

In this chapter the basic definitions in geostatistics including the concepts of random function and regionalized variable are presented. The underlying hypotheses (second order stationary and the weaker intrinsic hypothesis) are introduced. The variogram and the spatial covariance are defined. The problem of how to decide whether to treat a variable as stationary, intrinsic or nonstationary is discussed. Some of the basic properties of the spatial covariance are introduced in this chapter as they are helpful in deciding on the degree of stationarity. The relationship between the variogram and the spatial covariance is derived but the rest of the variogram properties are left to the next chapter.

Margaret Armstrong

3. The Variogram

Summary

This chapter and the following one are devoted to the variogram. In this one, after defining the variogram, its theoretical properties are discussed (e.g. zone of influence, behaviour near the origin, anisotropies, presence of a drift, etc.). The common variogram models are presented. Images of variables having some of these variograms have been simulated to highlight the differences between the models. The formula for calculating the variance of a linear combination of regionalized variables in terms of the variogram is proved. The reason why only positive definite functions can be used as models for the variogam is stressed.

Margaret Armstrong

4. Experimental Variograms

Summary

Like the preceding chapter, this one is on the variogram. The reader is shown how to calculate experimental variograms in 1D, 2D and 3D, and how to fit models to them. Several exercises are provided. The practical problems encountered with troublesome experimental variograms are discussed. These include outliers, almost regularly spaced data, and so on.

Margaret Armstrong

5. Structural Analysis

Summary

Several case studies showing how to carry out a structural analysis are presented in this chapter. Firstly, the principal decisions that have to be made by the geostatistician are reviewed. Are the data stationary? Are they isotropic? Should we work with the variables themselves or their accumulations? Should the study be carried out in 2D or 3D?

The first case study is a relatively simple 3D study of an iron ore deposit. As the horizontal and vertical variograms are well structured, it provides a clear illustration of a straightforward variographic analysis. The second study concerns an Archaean gold deposit that is being mined by opencut methods. So the grades of closely spaced blastholes are available in addition to the more widely spaced exploration drillholes. The third deposit presented is a sedimentary gold deposit with a periodic variogram in one direction because the gold was deposited by stream action. In contrast to the other two deposits, this one is only about im thick and so the study was carried out in 2D rather than 3D.

Margaret Armstrong

6. Dispersion as a Function of Block Size

Summary

This chapter deals with the effect of the support of a regionalized variable (i.e. its physical shape and volume) on its histogram and its variogram. In an introductory exercise, the histograms and the basic statistics are calculated for two support sizes: 1m x 1m blocks and 2m x 2m blocks. Although the means of the two distributions are identical, the variance of the larger support is much smaller and its histogram isalmost bell shaped whereas the other one is skewed.

The formulas for the variance of a point within a block and of one support v inside another V are then given. Krige’s additivity relation is proved. Then we see how the regularized variogram is related to the point support variogram. An exercise illustrates the effect that regularizing has on the variogram.

Margaret Armstrong

7. The Theory of Kriging

Summary

This chapter presents the theory of kriging. Kriging is an estimation method that gives the best unbiased linear estimates of point values or of block averages. Here “best” means minimum variance. Three types of kriging estimators are discussed: ordinary kriging (OK) used when the mean is unknown, kriging the unknown mean value and simple kriging (SK) used when the mean is known.

The equations for these three estimators are derived for the stationary case, and are extended to the case of intrinsic variables for ordinary kriging. The additivity theorem which gives the links between the OK and SK estimators is proved. For ordinary kriging, the formula for the slope of the linear regression of the true grade on its estimate is given, and its importance in relation to conditional unbiasedness is discussed. Lastly, kriging is shown to be an exact interpolator.

Margaret Armstrong

8. Practical Aspects of Kriging

Summary

This chapter is designed to give an overview of the practical aspects of kriging: negative kriging weights, the impact of the choice of the variogram model on the kriging weights, crossvalidation, the screen effect and last but not least, some criteria for testing the quality of a kriging.

Margaret Armstrong

9. Case Study using Kriging

Summary

The case study in Chapter 5 presented the structural analysis for an iron ore deposit. We now show how to use the fitted 3D variogram model to krige point values then block grades. As the model has a high nugget effect, a large kriging neighbourhood is required. The fourth section shows what happens when smaller neighbourhoods are used. The last section illustrates why it is not advisable to krige small blocks from sparse samples, in order to calculate the recoverable reserves.

Margaret Armstrong

10. Estimating the Total Reserves

Summary

Once a suitable prospect has been found, an exploratory drilling campaign is carried out to establish the limits of the mineralization (if they are not already known), and to determine the total ore tonnage and the average grade. As well as knowing the total reserves, it is very important to know how accurate the estimates are. Provided there are not too many samples, kriging can be used to estimate the reserves and the kriging variance will give a measure of its accuracy. However when there are too many points to invert the kriging system a different approach is required.

This chapter presents several approximations for estimating the variance associated with the total reserves when kriging cannot be used. The variance depends on whether the limits of the orebody are known a priori or not. After presenting the concept of extension variance, the first part of the chapter treats different approximations for evaluating the global estimation variance while the second half considers the question of “optimal” drilling grids.

Margaret Armstrong

Backmatter