Lecture Notes - Introduction to Geostatistics 2021 PDF

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INTRODUCTION TO GEOSTATISTICS

Anna KORRE Wenzhuo CAO Department of Earth Science and Engineering

[email protected] [email protected] office tel. 020 759 47372

Introduction to Geostatistics

3

TABLE OF CONTENTS Chapter 1 1.1 1.2 1.3

Chapter 2 2.1

2.2

4.3 4.4 4.5 4.6

7.4 7.5 7.6

7.7

Statistical Problems............................................................................................................... 79

THEORETICAL BASIS FOR THE COMPARISON OF SAMPLE SERIES WITH DIFFERENT SUPPORT ................. 79 GEOSTATISTICAL METHODS FOR DETERMINING THE DRILL SPACING .................................................... 83

Chapter 7 7.1 7.2 7.3

Kriging .................................................................................................................................... 54

THE THEORY OF KRIGING ........................................................................................................................ 54 VARIOGRAM MODEL PARAMETERS ......................................................................................................... 57 SEARCH NEIGHBOURHOODS ................................................................................................................... 62 CROSS-VALIDATION ................................................................................................................................ 63 RESERVE CLASSIFICATION ....................................................................................................................... 64 5.5.1 Size of the blocks and drill grid .................................................................................................... 64 5.5.2 Calculation of the geostatistical estimation variance ................................................................... 65

Chapter 6 6.1 6.2

Spatial Description ............................................................................................................... 26

DATA POSTING AND CONTOURING ........................................................................................................ 26 SPATIAL CONTINUITY ............................................................................................................................. 28 4.2.1 h-scatterplots ................................................................................................................................ 28 4.2.2 Correlation functions, covariance functions and variograms ...................................................... 30 4.2.3 Cross h-scatterplots ...................................................................................................................... 33 THE HYPOTHESIS OF STATIONARITY ....................................................................................................... 34 MODELS OF ANISOTROPY ........................................................................................................................ 38 RELATIVE VARIOGRAMS.......................................................................................................................... 41 MODELLING THE SAMPLE VARIOGRAM .................................................................................................. 44 4.6.1 Estimating the grades of individual blocks using the sample variogram ..................................... 49

Chapter 5 5.1 5.2 5.3 5.4 5.5

Introduction to ....................................................................................................................... 18

APPLICATIONS OF GEOSTATISTICS IN THE MINING INDUSTRY ............................................................... 18

Chapter 4 4.1 4.2

Descriptive Statistics ............................................................................................................ 11

UNIVARIATE DESCRIPTION ..................................................................................................................... 11 2.1.1 Frequency and cumulative frequency tables and histograms ....................................................... 11 2.1.2 Summary statistics ...................................................................................................................... 13 BIVARIATE DESCRIPTION......................................................................................................................... 16

Chapter 3 3.1

Introduction.............................................................................................................................. 5

ESTIMATION .............................................................................................................................................. 6 SCALE AND REPRESENTATION OF THE OREBODY ..................................................................................... 7 STATISTICAL REPRESENTATIONS ............................................................................................................... 9

Simulation .............................................................................................................................. 86

INTRODUCTORY EXAMPLE ...................................................................................................................... 86 SOME OBJECTIVES OF SIMULATION IN MINERALS APPLICATIONS .......................................................... 88 DEFINITION OF CONDITIONAL SIMULATIONS (AFTER CHILES AND DELFINER 2000).................... 89 7.3.1 Use of conditional simulations ..................................................................................................... 90 7.3.2 How many simulations should we generate?............................................................................... 92 7.3.3 Should we simulate the uncertainty on the parameters? ............................................................. 92 CLASSIFICATION OF THE METHODS ........................................................................................................ 92 DIRECT CONDITIONAL SIMULATION OF A CONTINUOUS VARIABLE ...................................................... 94 7.5.1 Sequential Simulation .................................................................................................................. 95 CONDITIONING BY KRIGING ................................................................................................................... 95 7.6.1 Conditioning on the data ............................................................................................................. 95 7.6.2 Matching the histogram............................................................................................................... 97 TURNING BANDS ..................................................................................................................................... 98 7.7.1 Presentation of the method in the plane ....................................................................................... 99

Introduction to Geostatistics 7.8

4

ESTIMATING THE GRADES OF INDIVIDUAL BLOCKS WITH KRIGING ....................................................... 73

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Introduction to Geostatistics

Chapter 1

INTRODUCTION

Estimation may be defined as the procedure of predicting the values of a characteristic at unsampled locations from measurements made at a number of locations within the same region. In most cases, the estimated value is not equal to the true value. This discrepancy between the estimated and the real value is called an estimation error and its size depends on the variability of the values to be estimated and the estimation method used. There are many widely used methods of ore reserve estimation and general grade estimation. One of these is the cross-sectional method as illustrated in Figure 1. It is common practise in many mining operations to drill holes on vertical cross-sections perpendicular to the strike of the orebody (Figure 1a). Geologists interpret and interpolate the characteristics of the orebody on these two-dimensional planes (Figure 1b) and then combine the two dimensional interpretations to provide a three dimensional estimation of such characteristics (Figure 1c) such as the shape and boundaries of the orebody and the local grades. To achieve such an interpolation, a number of assumptions are required. In this example the assumptions made are: -

that the grade, or any other estimated characteristic, is constant over a given volume, which is largely defined by the drilling pattern the variability and continuity of the mineralisation are largely a function of the drill hole spacing

Whether or not this is explicitly recognised, these assumptions build the model upon which the estimations are based. It may well be that for a particular orebody this model very accurately describes the mineralisation. However, at least at initial stages, there will be little information to validate the assumptions and the model employed. Of course, if the method gives wrong results it will be modified or even abandoned in favour of another method, i.e. some form of verification of the method is ultimately employed.

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Introduction to Geostatistics

a)

b)

A3

A2

A1

c)

Figure 1.

1.1

Ore reserve estimation with the cross-sectional method.

Estimation

In any type of grade, or reserve estimation method there are six stages as illustrated in Figure 2. Stages 1 and 2 need no further discussion. Stage three is the critical step in all analyses from the stage of feasibility study to the final stope, block and mining design and forms the basis of the estimation model in stage 4. In stage 4, a geologist doing manual interpretation on a cross section uses a model. The model may be implicit, such as a three dimensional solid model, or, more often, implicit as a picture build in his mind from his own and from accumulated past experience from this and similar orebodies. Reserve estimation based on polygonal methods or reserve distance weighting methods use models. The model is essentially based on assumptions, often unverified, about the degree of continuity of mineralisation from one intersection to another. Stage 5, the model validation, is of equal importance to stages 3 and 4. Notwithstanding the fact that very often estimated grades are consistently higher than indicated by the final recovered metal, it is often the case in operating mines that little attempt is made to rectify and correct the model. Assumptions, models and estimates are approximations to unknown reality, and as such, they inevitably incur errors. In general the two most important criteria to assess assumptions, models and estimates are: -

on average, models, concepts, assumptions and estimates are correct and any local or individual deviations from reality are as small as possible.

The first criterion means, for example, that the manually interpreted orebody outline will sometimes be to one side of the actual orebody, sometimes to the other and sometimes may even coincide with it. The manually interpreted outline of the orebody is then said to be an unbiased estimate of the true orebody outline, since the calculation has not consistently predicted the location of the ore body to the east or to the west of the actual location. Similarly, when calculating ore grades, the estimates are unbiased if they are neither consistently overestimated nor consistently underestimated.

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Introduction to Geostatistics

2.

measure the characteristics of the samples

Figure 2.

Stages of the estimation procedure.

Given that the grades are sometimes overestimated and sometimes underestimated, one estimation method may give estimates that on average deviate by 10% from the true values. Another estimation method may give estimates that on average deviate by 5% from the true grades. Both methods may be unbiased (honour the first criterion), however, the second method is clearly preferable. The second criterion states exactly that, because by using the second estimation method the deviation from reality is smaller. Similarly, an orebody outline within ±0.5 metre of the actual orebody is preferable to an outline within ±1 metre of the real orebody boundaries, provided that both outlines are unbiased. As assumptions increase the risk of error they should be: -

kept as simple as possible, kept to a minimum, reasonable and logical from every point of view (geological, structural, mathematical), verifiable.

It is important to point out here that both stages 1 and 2 introduce explicit errors of collection, measurement and observation. The estimation model cannot rectify these errors, however, these can be kept to a minimum by following good sampling practice and the principles and guidelines laid down by Pierre Gy (1977). 1.2

Scale and representation of the orebody

The orebody may be viewed on any scale. A geologist interested in ore genesis may be interested in the orebody on a microscopic scale, a mine geologist prefers a larger scale of observable structures, while the mining engineer requires a model on a scale comparable to the size of mining stopes or blocks. At the planning stage the only information available is on a scale the size of the samples which might be called the scale of observation.

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Introduction to Geostatistics

The scale is particularly important in estimation and planning because it affects the way which we observe a very important property of any measured characteristic, the variability. Variability exists at all scales and it is often markedly different at each. The example of Figure 3 illustrates this principle. The drill core in the figure is a 12m long core from an iron ore deposit cut into 1m segments and analysed for %Fe. The concentrations for the same core cut into 2m segments can be obtained by taking the average of two consecutive segments and for the 3m segments by averaging three consecutive segments. (Strictly speaking the density of each 1m segment should have been taken into account, however, this is a distraction that will not alter the conclusions drawn here. It is shown that the average %Fe calculated for the three different scales is the same, however the variability is quite different as shown by the minimum and maximum values in each case. Taking larger samples has reduced the variability. % Fe 0.3

1.9 1.1

1.7

1.3

1.6

1.5 1.3

2.8 2.3

Figure 3.

4.0

0.5

1.9 1.2

1.6

1.4

1.8

1.5 1.3

0.4 1.1

1.2

mean

min

max

1.5

0.3

4.0

1.5

1.1

2.8

1.5

1.2

2.3

Drill core from an iron ore deposit split into 1, 2 and 3 m segments.

In estimating the value of a characteristic such as grade, variability is the most important aspect of the model. If there is no variability there is no estimation error and the higher the variability the more difficult estimation is. As scale increases variability decreases. To understand the importance of this property, consider a gold deposit and a 1 metre long segment of a drill core with an Au content of 5.5g/t. A different way of expressing the Au concentration is to say that 0.00055 % of all grains of material contained in the cylindrical sample volume are Au, assuming that all grains are of equal size. If the whole sample size is reduced to that of a grain then only two values can occur for each grain, 0% gold and 100% gold. Such a sample size will yield values with the maximum possible variability. On the other hand, the one metre long core sample is highly unlikely to be pure gold and the range of gold values of such samples will be considerably less than that of grain size samples. In other words, the variability of the grades of larger samples is smaller than that of smaller samples. Usually, the length of core splits and to some extent the core diameter are selected in relation to geological structures and/or grades that are considered of importance either for geological hypotheses or for mining purposes. Therefore, the scale of observation depends on both the geological structures and the grades concerned. In general, the average split length is longer in large disseminated orebodies mined by large open stopes or open pits (e.g. porphyry copper deposits) than by narrow stratigraphic orebodies mined by cut-and-fill (e.g. massive sulphide deposits). In the stratigraphic orebody each sample is shown as extending across the full width of the orebody and oriented perpendicular to dip. In the disseminated orebody they are shown as equal size and vertical (Figure 4). In fact these are only two simplistic representations out of many equally valid representations.

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Introduction to Geostatistics

a) stratigraphic orebody

Figure 4.

1.3

b) disseminated orebody

Representation of orebodies as the sum of sample sized volumes.

Statistical representations

Statistical solutions are posed for problems in which there is no definite answer either because there is incomplete information and/or the information is at least partly reliable. In statistical terms a group of objects with specific characteristics is called the population. A number of objects, selected from the population according to a number of rules, constitute the sample. The characteristics of the sample are measured and studied. The assumptions, hypotheses, made about the characteristics of the sample constitute the model. The remainder of the population is then inferred to have the same characteristics as the sample. The characteristics of the population in most cases differ from the sample and thus the inference introduces an error. In mining geostatistics the population is the orebody subdivided into sample size volumes. A block whose grade is to be estimated may be viewed as consisting of sample size volumes as shown in Figure 5. oooooooooooooooo ooooooooooooo ooooooooooo oooooooo ooooo ooo oo o o ooo ooooooooo ooooooooooooo ooooooooooooooo Figure 5.

Representation of a block consisting of sample sized volumes.

In mineralisation the values of characteristics of samples taken close together are on average usually more similar. This relationship between the values of the characteristics of samples measured at different locations can be quantified by a numerical calculation. When there is no relationship between the characteristics of samples the values are said to be independent. The problem of estimation is now reduced to: -

collecting the samples, measure the sample characteristics (e.g. grade), from the sample characteristics infer the values of the same characteristics for the rest of the sample size volumes, and finally, predict the mean values of the characteristics of the sample size volumes which comprise the block.

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Introduction to Geostatistics

EXERCISES You should know how to expand single and double summations. That is n

 x = x + x + ...+ x i

i =1 n

m

i =1

j= 1

1

 x

ij

2

n

= x11 + x12 + ... + x1m + x21 + ... + x2 m + ... + xn1 + ... + xnm

1. Express the following sum in summation notation: x1 x2 x + +... + n 2 2 2

2. Expand the following sum...