RESERVE ESTIMATION
9.4 GEOSTATISTICAL APPLICATIONS
These problems of grade/tonnage mismatch and wider grade variances of estimated blocks have been resolved by developing regionalized stationarity and variability of metal distribution within the deposit (Matheron, 1971, Hans, 2013).
9.4.1 Block Variance
Let us assume that we have a large number of evenly spaced sample points distributed within the orebody. The body is divided into a number of blocks of equal size. An average grade can be computed for each block by taking the arithmetic or weighted average of all the sample points falling in that block. The simplest case would be an equal number of blocks with one sample in each block. The other extreme case would be only one block representing the whole deposit that contains all sample points. If the vari- ances of the block grades are plotted against size of the block, the block variance relationship is obtained (Fig. 9.17).
Obviously, the variance of block grades decreases with increase in block size. As each block value is an average of all sample points in the block and the mean of the ore deposit is the mean of all block values, the mean of the deposit remains constant. The variance of the errors be- tween the true grade of a block and that estimated from a center sample point generally decreases with increase in block size.
The estimation of block size will depend on mine planning and it is necessary to select an estimation tech- nique that will take into account all surrounding samples to reduce the variance of the estimation error. An orebody model has to be developed based on geological and
statistical information. Some geological concepts that need to be considered are:
1. The better the orebody continuity, the smaller the error associated with extrapolating a given grade over a larger area.
2. Influence of a sample over an area may be related to a geological zone of influence that may vary in different directions (anisotropy).
3. Some ore deposits, such as gold, silver, and platinum, may exhibit a nugget effect (large variation in grade over a small distance) and therefore larger samples may be required since a sample is a volume (support) and not a point.
4. Demarcate geologically distinct areas within a deposit so that different estimation procedures may be required (nonstationarity).
9.4.2 Semivariogram
The most natural way to compare two values is to consider their differences. Let us assume that two samples Z(x) and Z(xþh) are located at two points, x and xþh, with two different % metal grades. The second sample is h meters away from thefirst. This value expressing the dissimilarity of grade between two particular points is insignificant. The average difference for all possible pairs of samples at h meters apart throughout the deposit will be geologically significant. The difference between any pair of samples will be either positive or negative, and the sum of all pairs will be misleading. It is therefore logical to square the differ- ences, sum them, and divide by the number of pairs to understand the average variance of paired samples at certain distances apart. This dissimilarity function is expressed by the model equation:
2g (h)¼AverageS{Z(x)Z(xþh)}2
FIGURE 9.17 Concept of mutual relationship between sample volume size and variances.
where 2g (h) is thesemivariogramor simplyvariogram function. It is the function of a vector; in other words, a distance and the orientation of that distance. It articulates how the grades differ on average according to the distance in that direction (Fig. 9.18).
The semivariogram is thus stated as:
gðhÞ ¼ 1=2NXN
i1
fZðxÞeZðxþhÞg20
where g(h)¼semivariogram function value, N¼number of sample pairs, Z(x)¼value at point (x), may be grade, width, accumulation, etc., and Z (xþh)¼value at point (xþh), i.e., h distance away from point (x).
The simplest semivariogram computation can be by sliding mineralization in one direction along a borehole (Fig. 9.18). Let there be a set offive samples of 1 m length each along a borehole as 2%, 6%, 3%, 9%, and 12% Zn.
The sample string slides by one step (1 lag), i.e., at 1, 2, 3 m lag, to compute the average variance and continue.
This can be represented in a semivariogram plot of lag againstg(h) function along the x- and y-axes, respectively (Fig. 9.19). The freehand or fitted curve is extended downward to intersect the variance axis. If it touches above the origin, this part is known as thenugget effect (C0). The curve then rises up to the maximum variance (population variance d2) at a particular lag equivalent distance and levels out orflattens. This distance is known asrange (a).
Each sample value has an influence up to about 2/3 of the range. The extension variance C1is the difference between population variance and nugget effect.
In the case of 2D grid sample data the variogram is computed in different directions. Sample points on either side of the variogram line, within acceptable limits, are projected on it. The variance between points is computed and grouped under a similar lag range. This process is repeated on all samples and arranged in serial order as per lag. The number of participating pairs contributing to the
FIGURE 9.18 Concept of semivariogram by sliding sample data string at certain space intervals.
Sample Lag Difference Difference2
2 1 m
6 2 4 16 Ʃdifference ¼ 702
3 6 3 9 No. of pairs ¼ 4
9 3 6 36 Average variance ¼ 70/(24)
12 9 3 9 Semivariance ¼ 8.75
2 2 m
6
3 2 1 1 Ʃdifference2 ¼ 91
9 6 3 9 No. of pairs ¼ 3
12 3 9 81 Average variance ¼ 15.16
2 3 m
6 Ʃdifference2 ¼ 85
3 No. of pairs ¼ 2
9 2 7 49 Average variance ¼ 21.25
12 6 6 36
FIGURE 9.19 Drawing of standard semivariogram along the drill hole samples.
sum of differences square is known, and a variogram is computed. Similarly, variograms in other directions within the study area are computed and compared.
In practice, a theoretical semivariogram is never real- ized. The gamma function g(h) is estimated from limited points and is called the experimental variogram (Fig. 9.20).
Fig. 9.20displays a number of sample points at equal distances apart with their sample values, say thickness next to each point. We wish to compute g(1), i.e., the semi- variogram value of all samples points at 1 unit distance (lag) apart, irrespective of direction. There are 17 sample pairs at 1 unit apart:
gð1Þ ¼ 1=2Nð12Þ2þ ð22Þ2þ ð23Þ2þ ð21Þ2þ ð14Þ2þ ð43Þ2þ ð21Þ2þ ð11Þ2þ ð12Þ2þ ð12Þ2þ ð22Þ2þ ð21Þ2þ ð11Þ2þ ð24Þ2þ ð41Þ2þ
ð33Þ2þ ð32Þ2
¼ 31=34 ¼ 0:91
The next shortest sample distance is ffiffiffi p2
unit distance apart with total sample pairs of 12:
rg ffiffiffi p2
¼ 26=24 org ffiffiffiffiffiffiffi 1:4 p
¼ 1:08 The gamma functions for all the possible lags are computed and plotted asg(h)against lag. The semivario- gram is shown inFig. 9.21.
9.4.2.1 Properties of a Semivariogram 9.4.2.1.1 Continuity
Continuity is reflected by the growth rate of g(h) for a small value of h. The growth curve exhibits the region- alized element of samples. The steady and smooth in- crease is indicative of the high degree of continuity of mineralization until it plateaus off at some distance. This is known as the structured variance or explained variance (C or C1) read on a g(h) scale. A typical semivariogram of a coal deposit showing good continuity is given in Fig. 9.22.
9.4.2.1.2 Nugget Effect (C0)
The semivariogram value at zero separation distance (lag¼0) is theoretically 0. However, a semivariogram often exhibits a nugget effect (>0) at an infinitely small distance apart. The complex mineral deposits, like base and noble metals, may occur as nuggets, blobs, and are often concentrated as alternate veinlets resulting rapid changes over short distances. The gamma functions are extrapolated back to intersect the y-axis (Fig. 9.23). The nugget effect (C0) is the positive measure ofg(0)representing random or unexplained elements of samples. Noble and precious metals like gold, silver, and platinum-group elements often exhibit very high nugget effects even up to total unex- plained variance. This creates a lot of uncertainty in the continuity of mineralization and grade estimation leading to extensive sampling.
FIGURE 9.20 Computation of experimental semivariogram of grid sample data along various directions such as 0 and 90 degrees.
FIGURE 9.21 Typical semivariogram plot with curvefitting following the population variance after reaching the zone of influence.
There are three possible reasons for the nugget effect of various magnitudes:
1. Sampling and assaying errors.
2. Smaller microstructures or nested variograms at shorter distances where no borehole data exist.
3. Combination of both.
9.4.2.2 Semivariogram Model
The computed experimental points of a semivariogram can be tailored to a standard type tofit a model. There are many semivariogram models that fulfill certain mathematical constraints (Rendu, 1981; Clark, 1982). The models can be broadly divided into two groups: (1) those with a sill and (2) those without sill. A sill implies that, once a certain distance h is reached, the values of g(h) do not increase, although they may oscillate around the population variance.
Models without a sill have growing values of g(h) with increasing values of h.
9.4.2.2.1 Features of a Semivariogram The sillðC1þC0Þ.
The growing semivariogram curve normally reaches a plateau after a certain lag, and equals the population vari- ance. This is known as the sill. In practice the gamma function may fluctuate on the sill line because of metal zoning, layering, and the hole effect:
SillðCÞ ¼ C1þC0
9.4.2.2.2 The Zone of Influence (a)
The zone of influence is that neighborhood beyond which the influence of a sample disappears and samples become independent of each other. The zone of influence in a given direction is characterized by the distance at which the semivariogram eventually reaches a plateau and levels out.
The first few points of g(h) are joined by freehand or polynomial regression, and extended to the sill. The inter- section with the sill is w2/3 of the zone of influence or range (a).
9.4.2.2.3 The Isotropic Anisotropies
The isotropic anisotropies are defined by computing semi- variograms in different directions. The semivariogram that points in the eastewest orientation ofFig. 9.20will be:
gð1Þ ¼ 15=18 gð2Þ ¼ 12=12 gð3Þ ¼ 5=6
Semivariograms in the northesouth or any other di- rection can be computed and compared. The semivariogram is isotropic if the underlying structure exhibits the same features irrespective of the directions. The semivariograms isanisotropicif the structure displays different features in various directions. An average semivariogram can be considered in such a condition.
The types of semivariogram with and without sills are given inFigs. 9.24 and 9.25and as follows:
1. Spherical model gðhÞ ¼C0þC1
1:5h=ae0:5 h3
a3
; for h<a; and gðhÞ ¼C0þC1;for ha
where C0¼nugget value, C1¼explained variance, C0þC1 ¼ sill, h¼distance between points, and a¼zone of influence or range.
2. Random model
gðhÞ ¼ C0þC1; for all h
FIGURE 9.23 Typical semivariogram from massive base metal deposits with moderate unexplained variance (C0).
FIGURE 9.22 Typical spherical semivariogram from coal deposits without any nugget effect.