What is ‘Extension Variance’?
Geostatistics provides the practitioner with tools, via the variogram, that enable the calculation of estimation variances.
Suppose that we want to assess the average value of the grade of a given domainV
using data from a smaller supportv .For example the domainV might be a block, and the supportvmight be a central core sample (see figure 8.1).
Figure 8.1 volumes (‘supports’) discussed in the text.
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8
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We wish to estimate the grade ofZ ( V ) but we only have data forZ ( v ). It seems natural to estimateZ ( V ) using the core sample gradeZ ( v ). This is, of course the polygonal estimation of the block by a central sample. We know that the grade of the block will not always be equal to the grade of a central sample, so when estimating the block grade in this manner this estimation, we will make an error. What is this error?
First, under the assumption of the intrinsic hypothesis,Z ( v ) is anunbiased estimator ofZ ( V ), i.e.
E Z v ( )
−
Z V ( )=
0In general, the error isnot zero. However, the average ofall over- and underestimations should be zero. Therefore, it is more interesting to characterise the error made when estimatingZ ( V ) byZ ( v ) using the variance, i.e.
E Z v ( )
−
Z V ( )=
Var− =
Z v ( ) Z V ( ) e( , )v V2 σ 2
where σ e v V
2
( , ) is theextension variance. This is the variance of the error that we incur in ‘extending’ to the domainV the grade measured on domainv . In other words, theextension varianceσ e 2 of the value of a sample taken in a block of ore is
the average squared error (or ‘error variance’) incurred in assuming that this sample value is thetrue value of the block. In our example, the extension variance is thus
the average squared error made in assuming that the value of the sample extends over the entire block 25.
Extension Variance and Estimation Variance Extension Variance and Estimation Variance
Conceptually,σ e2 v V ( , ) is simply the variance of the estimation ofZ ( V ) byZ ( v ).
Often, the two terms ‘estimation variance ’ and ‘extension variance’ are treated as synonymous. The same notation ( σ e
2
) is also used, and conceptually the two are equivalent. However, the term ‘estimation variance’ is generally used with a broader meaning: estimation variance is thesum of all error variances associated with estimating the average grade of a given volume (block, bench, orebody). In geostatistics, the term estimation variance is thus used for more general situations, where several samples are combined to estimate a given area or volume.
The Formula for Extension Variance The Formula for Extension Variance
The theoretical value of the extension variance can be obtained by the formula:
σ e v V γ v V γ V V γ v v
2
2
( , )
=
( , )−
( , )−
( , )25 This isexactly the error made when using a traditional polygonal estimator for the panel.
Extension Extension variance variance
The value of a sample taken in a block of ore is the average squared error (or ‘error variance’) incurred in assuming that this sample value is the true value of the
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where:
vis the small support (lets say, a sample or point) centrally located inV .
V is the larger support (block).
σ e v V
2( , )
is the extension variance just introduced.
γ ( , )v v is the mean value of the variogram between two points sweeping independently within the volumev. This is the dispersion variance of points within a support of sizev. This is easily obtained by the auxiliary function F( v ) introduced in the previous chapter.
γ ( , )V V is the mean value of the variogram between two points sweeping independently within the volumeV. This is the dispersion variance of points within a support of sizeV. This is easily obtained by the auxiliary function F( V ) introduced in the previous chapter.
γ ( , )v V is the mean value of the variogram between two points, one sweeping independently withinv and the other sweeping independently withinV. This is the dispersion variance of supportv within a support of
sizeV. This value can also be obtained via an auxiliary function. These concepts are illustrated in figure 8.2.
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Formula (i) applies for any shapes forv andV;in particularv need not be included inV . The factors influencing the variance of extension are:
(i) The regularity of the variable through the model ofγ
( )
h(ii) The geometry ofV throughγ ( , )V V
(iii) The geometry ofv throughγ ( , )v v
(iv) The location ofv with respect toV throughγ ( , )v V
Factors Affecting the Extension Variance Factors Affecting the Extension Variance
If we consider formula (i) again:
σ e v V γ v V γ V V γ v v
2
2
( , )
=
( , )−
( , )−
( , )we can re-write it, by rearranging into two terms, as:
{
(, ){
(}
, )}
( , ) ( , ) ) , ( 2 vV vV V V vV vv e γ γ γ γ σ=
−
+
−
This makes it clear that the extension variance σ e v V
2
( , ) decreases as:
The samplingv of the domain to be estimatedV becomes larger. We
can see this by considering the extreme case wherev=V. In such a case,γ ( , )V V is identical toγ ( , )v v . In this case the two terms is (ii) above are equal andσ e v V
2
( , ) is zero.
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With regard to the second factor, ahighly discontinuous variable would have a variogram approaching that of a pure nugget effect. In this case, a sample within
the block is not representative of the block gradeat all andno local estimation is possible. By estimating any block, we make a maximum average error in this case. This is an interesting result, because one sometimes hears someone exclaim ‘we use
a polygonal estimator26 for grade control blocks because the variogram is poorly
defined (i.e. pure nugget)’. In fact, this is the most disastrous circumstance in which to use this type of estimator!
Note also that, if only one sample is available for a given block it seems logical to assume that a central location for this sample is optimal, in the sense of minimising
σ e
2
. This is borne out by geostatistical theory: the extension variance attached to assigning the grade of a sample at one corner to a block, for example, is higher than that resulting from using a central sample27.
Other Properties of Extension variance Other Properties of Extension variance
An obvious, but important property of extension variances is that they involve only the geometry of the samples/blocks and the variogram model:they do not involve the actual samples involved in a particular situation. This means that determining the appropriate sampling geometry by comparing estimation variances different situations requires only an acceptable variogram model.
On the other hand, it does not take into account local conditions, for example, if we wish to estimate our block by a central sample that happens to be the maximum
grade for the deposit, we expect a worse error variance than the ‘average’ error variance provided by the extension variance.
Extension Variance & Dispersion Variance
Extension Variance & Dispersion Variance
The extension variance σ e v V
2( , )
should not be confused with the dispersion variance D v V 2( | )
, although we use dispersion variances in determining the extension variance.
26 Or inverse distance squared … which is very ‘polygonal’ in nature (i.e. similar to nearest neighbour). 27 If we have a "pure nugget effect" model for the variogram, the two extension variances are equal: this
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In summary:
Table 6.1 Extension Vs Dispersion Variance Table 6.1 Extension Vs Dispersion Variance
Dispersion Variance D v V 2( | ) Extensi on Variance σ
e v V
2( , ) This is the variance of grades defined on one support
v within another supportV e.g. the variance of points within a block
or
the variance of cores within the deposit
Physical Significance Physical Significance
This is the variance of the error we make when assuming that the grade of one volumev is the true
grade of a larger volumeV e.g. if we estimate a 5m x 5m x 5m block by acentral, 5m high drill hole sample
or
we estimate the average thickness of a coal seam in a 100m x 100m area by a single measurement located
at one corner
Useful Concept Useful Concept
To make it quite clear: the dispersion variance D v V 2( | )
has a physical significance: it measures the variance of samples of sizev within the domainV. In contrast, the extension variance is a usefulconcept that allows us to characterise the error associated with estimating a volume by a sample of given support.
Again, in many cases, the volume of our samples is very small in comparison to the blocks we consider. Consequently, we can consider the sample support to be point support . Our formula is then simplified to:
σ e o V o V γ V V γ 2 2 ( , )
=
( , )−
( , ) Dispersion vs Dispersion vs Extension ExtensionDispersion variance deals with support, while
extension variance deals with estimation errors.
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Practicalities Practicalities
Auxiliary functions are available to calculate all the terms required to evaluate various extension variances. The F-Function and other auxiliary functions do not
have convenient analytical forms, so they have historically been obtained via charts or tabulations of the values (see Journel and Huijbregts, 1978), but are now obtained through computer programs. These programs can give the necessary values to allow calculation of extension variances for a variety of situations (1D,
2D, 3D, point within block, point at margin of block, etc.)
Note that the resultant extension variance
σ
e2 applies to a single blockV .Combination of Elementary Extension Variances Combination of Elementary Extension Variances
In order to calculate the combined estimation variance
σ
E2 when estimating a volume V composed of N elementary blocks, each of volumevand possessing associated elementary extension varianceσ
e2 a method known ascombination ofelementary extension variances (Journel and Huijbregts, 1978, p.413) can be employed. The formula used is simple:
σ E σ e
N
2
=
2The resultant estimation variance
σ
E2 is that associated with estimating the mean grade of a volume V from N samples, each centrally located in one of N unit blocksv with elementary extension varianceσ
e2. It does not account for any error associated with estimation of the geometry of the mineralisation.A N I M P O R T A N T A S S U M P T I O N
The method of ‘combination of elementary extension variances’ also assumes that the error made for any block isindependent of the errors made for other blocks ; this is generally a valid assumption if blocks are fairly large and there is only one sample per block.
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Geometry of Mineralisation Geometry of Mineralisation
When estimating a zone from a number of samples, the error associated with estimating the geometry of this mineralisation must also accounted for.
Given that there aren regularly spaced drill holes in ‘mineralisation’ (however this is defined), the following formula gives agood estimate of the relative variance of estimation for a surface (David, 1977):
σ s s n N N N N N 2 2 2 2 1 2 2 2 1
1
6
0 061
=
(
+
.
)
≤
Derivation of this formula (due to George Matheron) was by means of calculating estimation variances of an indicator variable where each drill hole has a value 0 for waste and 1 for ore. There aren ‘positive’, i.e. above cut off, regularly spaced drill
holes, each central to a block ( v ) of dimensions lx L; there are thus n blocks.
There are N 1 blocks in one direction and N 2 in the other
(
N 2≤
N 1)
.Having calculated an extension term
σ
E2 and a surface termσ
s2 a ‘total’ error termσ
T2 can be calculated28: σ T σ E σ S m m s 2 2 2 2 2 2=
+
The geometric error
σ
s2 in this case would be assessed using a two dimensional methodology. A three dimensional problem would therefore be reduced to two dimensions and the results would be indicative only.28 An example of additive error terms is given by David (1977, example 8.6.1, pp.221-225). Various
formulae and examples are also given by Journel and Huijbregts (1978, section V.C. pp.410-443). Note
Note
This approximation is two dimensional.
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