Systematic ic VVariogram ariogram InterpretationInterpretation
Some geostatistical textbooks may leave the reader with the impression that fitting variograms is a fairly easy task. In many cases, especially when dealing with
precious metals data, this is far from true. Having a systematic approach is important when first setting out to perform variographic analysis. We'll tackle seriously troublesome variography later, but for the moment we consider here the usual procedure for modelling variograms.
In fact, fitting variogram models is as much a ‘craft’ as a science, in the sense that it isn't an activity that can be completely reduced to a formula. Repeated experience fitting variograms increases the practitioners’ skill. However, we present some guidelines intended to help you with variogram modelling.
Obviously, not all the following comments will apply to every experimental variogram you'll encounter. However, it's especially useful when first dealing with
structural analysis to have a few ‘pointers’.
Ten Key Steps When Looking at a Variogram Ten Key Steps When Looking at a Variogram
We consider key points to look for when examining experimental variography. Always remember that the experimental variogram is an estimate of the
‘underlying’ variogram. As such some irregularity is generally expected, due to statistical fluctuation.
Black boxes Black boxes
Software can't be expected to mimic skilled variogram modelling. An intelligent operator with site-specific geological knowledge is required Skill & Skill & Experience Experience
Fitting variogram models is as much a ‘craft’ as a science, in the sense that it isn't an activity that can be completely reduced to a formula
CH 6 –
CH 6 – VARIOGRAPHYVARIOGRAPHY
Quantitative Group Short Course Manual
1. The number of pairs for each lag in the experimental variogram. 1. The number of pairs for each lag in the experimental variogram.
The number of pairs contributing to the first lag (the first point of the experimental variogram) can often be quite low. This will depend upon the exact sampling
pattern and search used. Consequently, the first point may not be very representative. Some software annotates the experimental variogram with the number of pairs used. So long as a listing is available, we can check to see that points at short lags are reliable.
Although rules are difficult to apply strictly, fewer than 30 pairs is likely to be unreliable in any mining situation. Note that the number of pairs should be considered in proportion to the size of the data set, for an experimental variogram with 2,000 pairs at lag No. 2, lag 1 might be considered dubious with 100 pairs. See
figure 6.11a.
Similarly, at distal lags, the number of pairs decreases. It's easy to see why: with increasing distance there comes a point where only samples at the very edges of our area can be used. It can be shown theoretically that the variogram can become dangerously unreliable at lags beyond 1/3 of the maximum sample separation. See Figure 6.11b.
2. Smoothness of the experimental variogram 2. Smoothness of the experimental variogram
The smoothness of the experimental variogram in figure 6.11a and 6.11b can be contrasted to the more erratic behaviour in figures 6.11c and 6.11d. Many factors can contribute to erratic variograms, and we'll discuss these in detail, in the section on troublesome variograms below.
However, two types of erratic variography can be distinguished. In figure 6.11c, the experimental variogram is saw-toothed in a regular up-and-down fashion. This may indicate poor selection of lags or possible inclusion/exclusion of a very high value. In any case, thestructure is still visible. If we exclude other sources of irregularity, this variogram might be modelled in an ‘averaged’ way, as shown in the figure. Some of the techniques discussed below in the section on troublesome variograms might result in a ‘cleaner’ variogram.
On the other hand, figure 6.11d shows a noisy variogram with no obvious structuring, nor is there clearly evident the kind of ‘saw-toothing’ behaviour seen in figure 6.11c. In this case we have to resort to some kind of robust variography (relative variograms) or transformation (logs, etc., see further, below).
CH 6 –
CH 6 – VARIOGRAPHYVARIOGRAPHY
Quantitative Group Short Course Manual
CH 6 –
CH 6 – VARIOGRAPHYVARIOGRAPHY
Quantitative Group Short Course Manual
3. Shape near the srcin 3. Shape near the srcin
It's critical to assess the shape near the srcin correctly. As we've already said, the first points are sometimes suspicious or unrepresentative. In mining applications, especially for grade variables, the shape at the srcin is nearly alwayslinear. This is one reason that the spherical model is so popular.
If the experimental variogram suggests a parabolic shape near the srcin (like the Gaussian model introduced above) be very cautious. This will nearly always be a statistical feature when dealing with grades. Resist the temptation to fit Gaussian models! The consequences for kriging are quite profound: the Gaussian model represents extraordinarily continuous, smooth short-scale behaviour of a type not seen for mineral grades. In the case of topographic variables (depth to a geological surface, the water table, vein width, etc.) caution is still advisable.
If you are convinced that a topographic variableis Gaussian, then always fit the model with a nugget effect (in order to avoid instability in subsequent kriging). In many cases, a cubic model is preferred, but this model is not generally available in mining software.
Theslope of the variogram near the srcin is a critical factor in subsequent kriging. Greater weight is generally given to the experimental points closest to the srcin when assessing this slope (given that these points have a reasonable number of
pairs contributing). Note that the slope is relative to the ratio of the range to the proportion of nugget effect.
4. Discontinuity at the srcin—nugget effect 4. Discontinuity at the srcin—nugget effect
Along with the shape and slope at the srcin, the proportion of nugget effect is a critical factor in modelling the variogram. Most grade variables have some nugget effect. The proportion of nugget effect relative to the sill is often called therelative nugget effectε ,andis measured as a ratio to the sill:
ε
=
+
Co Co C
The relative nugget effect is often expressed as a percentage.
The nugget effect is the same in any direction (being defined at very small distances relative to the sample spacing). Because of this, in mining we will generally use the down hole direction to set the nugget effect and then use this value for each of the other directions.
Note also that the relative nugget effect is dependent upon compositing length (i.e., for a given spatial grade distribution, as we use longer composites,ε is reduced). We discuss this phenomena (related to ‘support effect’) in more detail in the next
chapter.
5. Is there a sill?—transitional phenomena 5. Is there a sill?—transitional phenomena Important Note:
Important Note:
If the experimental variogram suggests a
parabolic shape near the srcin (like the Gaussian model introduced above) be very cautious
Co is Isotropic Co is Isotropic
Because of this, in mining we will generally use the
down hole direction to set the nugget effect and then use this value for each of the other directions.
CH 6 –
CH 6 – VARIOGRAPHYVARIOGRAPHY
Quantitative Group Short Course Manual
Answering this question is sometimes not as easy as you might expect. For example, take figure 6.11e. Here we have an example of an experimental variogram (I) that clearly has a sill. However, variogram II seems to continue to rise. We may have a linear (or unbounded) variogram, but equally, we may not have yet reached the range of a transitional model. For example, we have been restricting the zone upon which we calculate the variogram too severely. Then again, it is possible we have a drift (see further below).
Note that, so long as the shape of the function we choose fits the experimental data well, especially near the srcin, the difference between choosing linear or spherical (with a very long range) is negligible.
If the sill level is not clearly defined (for example figure 6.11c) then we often use the ‘average’ level of fluctuation. If this corresponds to the variance of the data (as it should in the stationary case), our confidence is increased.
Although the sill should coincide with the overall variance (in conditions of stationarity), this is not always the case, for example because of the presence of long-range trends in the data. Note that the level of the sill for the longest structures in a nested model has little impact upon kriging weights, so in most cases fixing it with great precision is not necessary.
6. Assess the range 6. Assess the range
If wedo have a transitional model, then we need to assess the range. In general, the range is assessed visually, as the distance at which the experimental variogram stabilises at a sill.
In many cases, the range is fairly clear, especially for experimental variograms that closely approximate a spherical scheme. In other cases, it may not be so easy. Firstly, bear in mind that there are some mechanisms for specifying range inherent in the functional forms of the model we choose. In particular, the linear extrapolation of the slope at the srcin should cross the sill at 2/3 of the range for a spherical variogram. If the sill is clear, this rule of thumb can be quite helpful. In this case, the first few reliable points of the experimental variogram, not those close to the range, should control the slope.
Careful definition of the ranges of shorter structures (when more than one regionalised structure is evident) is very important.
7. Can we see a drift? 7. Can we see a drift?
Drift is not so easy to detect in many mining situations. Firstly, at lags beyond about 1/3 of the maximum available sample separation, theory indicates that the variogram becomes increasingly unreliable. So a continuously rising experimental variogram, such as that shown in figure 6.11f may be quite misleading. Again, look
at the representivity of the pairs.
Assessing a drift should also be made in conjunction with examination of a posting of the data, or a contour map. Look for trends that might clearly be responsible.
Slope Slope
The first few reliable points of the experimental variogram, not those close
to the range, should control the slope.
CH 6 –
CH 6 – VARIOGRAPHYVARIOGRAPHY
Quantitative Group Short Course Manual
Some software will print out, for each lag, the number of pairs, the mean of the pairs and the ‘drift’, this being the mean value of the pairs for this lag. Of course, systematic increase in this statistic for distal lags is still only significant if we have sufficient pairs.
Even where a convincing case can be made for a drift for larger lags, this may have little impact on subsequent kriging. This is because, as we have repeated regularly so far, the shape of the variogram at shorter lags is the critical factor in the results of any subsequent kriging.
In most mining situations, modelling of drift isnot required. If it is required, there are techniques available, but these are beyond the scope of this course (see Journel and Huijbregts, 1978, p.313).
8. Hole effect 8. Hole effect
A hole effect appears as a bump on the variogram. As stated in the previous chapter, most apparent "hole effects" are, in fact, an artefact of the sampling used, lack of pairs etc. Although hole effect models exist, they are beyond the scope of this course and their use is not common.
9. Nested models 9. Nested models
Given an interpretable experimental variogram we will usually need to model more than one structure. In the simple case, we assess the nugget effect and then fit a single, say spherical, model for the structured component.
In many cases, mining data present more than one range. Clear inflections in the experimental data indicate the ranges of nested spherical models. We see several examples of this in the case study at the end of this chapter.
Generally, where several models are nested, fitting the model with the shortest range will prove critical (from the point of view of subsequent kriging).
10. Anisotropy 10. Anisotropy
It is essential that the experimental variogram be calculated in at least four directions in the plane, and at geologically sensible orientations in the third dimension, in order to detect anisotropies. The procedure for fitting an anisotropic model was discussed in the preceding part of this chapter.
In the absence of any detected anisotropy, an isotropic model can be fitted. Fitting of zonal anisotropy is beyond the scope of this course.
Impact of Drift Impact of Drift
Even where there is a drift, , the shape of the variogram at shorter lags is usually the critical factor in the results of any subsequent kriging.
CH 6 –
CH 6 – VARIOGRAPHYVARIOGRAPHY
Quantitative Group Short Course Manual
‘Uncooperative’ or ‘Troublesome’ Variograms
‘Uncooperative’ or ‘Troublesome’ Variograms
1818If the experimental variograms encountered in practical situations were as well behaved as those often given as textbook examples, this section would be unnecessary! In fact, the reader is unlikely to avoid ‘horror variograms’ like the one shown in figure 6.11d for very long (if they are working in a gold mine, sooner rather than later).
We will approach the subject of uncooperative or troublesome variograms via two different angles:
1. Can we improve variography by choosing different calculation parameters?
2. Can we deal with the problem by some approach more sophisticated
than calculation of the traditional grade variogram? Calculation of the Experimental Variogram Calculation of the Experimental Variogram
We examine here a number of factors we should check first when confronted by a dreadful-looking experimental variogram. The initial experimental variograms are often highly erratic and time, effort and thought is required to establish why this is so. The paper of Armstrong (1984) gives some excellent examples (some of which are discussed here).
Theoretical Reasons Theoretical Reasons
The experimental variogram is anestimate of the spatial structure. As such, it is often highly variable for large values ofh . Various geostatisticians have demonstrated that, when we have a very large number of closely spaced data (for example from blast hole drilling or a simulation), subsets of this data can have widely varying histograms and variography.
Definition of Stationary Definition of Stationary
Sometimes the reason for poor variography is that we are calculating the variogram from two mixed populations that possess differing statistical characteristics. In an extreme case this will show up as a bimodal histogram, but this is certainly not always the case (Armstrong, 1984). Since the variogram assumes intrinsic stationarity, mixed populations can impact severely on the experimental variogram. Where possible, our variogram should therefore be calculated for a single statistical
population.
GEOGRAPHICALLY DISTINCT POPULATIONS
If the populations are geographically distinct, i.e. they can be outlined on maps of the deposit, then our problem is to define the boundaries of our zones. This is partly iterative, in that the variogram is one aspect of the evidence we will use to split or lump geological zones. Combining zones with quite different geostatistical
18 i.e. the usual variety for mining examples…
Stationarity Stationarity
The stationarity decision is probably the most important decision in a geostatistical study.
CH 6 –
CH 6 – VARIOGRAPHYVARIOGRAPHY
Quantitative Group Short Course Manual
characteristics can result in poorly defined and sometimes uninterpretable variography.
Alternatively, if we split the zones into too many categories, we may end up with too few data in each zone, thus statistical fluctuations may overwhelm the underlying spatial structure. Obviously, some experience and trial-and-error is involved.
INTERMIXED POPULATIONS
The problem may be more intractable. For example, if the two populations are related to intimately intermingled, but statistically contrasting lithologies, then the only means to resolve the problem may be more detailed sampling (i.e. closing up the drilling further).
Another example of intermixed populations might be mixing two or more different drilling campaigns. It is often the case that older campaigns have smaller diameter drilling, poorer sample preparation etc. The result of this is artificially higher variance for the older drilling campaign. This may be revealed by higher sills, shorter ranges and in extreme cases, apparent pure nugget behaviour of the variogram.
How to Determine Appropriate Variogram Calculation How to Determine Appropriate Variogram Calculation Parameters
Parameters
The parameters relating to search tolerances and lag selection are sometimes very sensitive.
Lag Selection. Lag Selection.
A strongly pronouncedfigure 6.11c, is a warning that we may have poorly specified the lag increment. Ifsaw-toothing of the experimental variogram, as shown in the data spacing is irregular, the basic lag interval to choose may not be immediately obvious, and we may get a situation where successive lags include larger, then smaller numbers of pairs in a cyclic fashion. The lags with fewer pairs will be less robust to extreme values, and tend to have—on average—higher values ofγ
( )
h .Tolerances Tolerances
In particular, selection of the lag and angular tolerances can sometimes have a drastic impact on the variogram. If the variogram looks bad, try larger or smaller tolerances. In doing so we are, in a sense, varying the smoothing the data in order to lessen the impact of including or excluding particular pairs in a given lag. The angular tolerance can be especially sensitive to this type of effect.
Missing Values Missing Values
Most programs allow specification of a minimum value to consider in calculation of the variogram. It is common to flag missing values with a negative number, say - 1. If we do not test for these values the impact can be quite drastic on the experimental variogram, since we are adding artificial (often randomly located) ‘noise’.
CH 6 –
CH 6 – VARIOGRAPHYVARIOGRAPHY
Quantitative Group Short Course Manual
Extreme Values Extreme Values
We will discuss some approaches to modelling variograms with extreme values below (e.g. log variograms, relative variograms). However, one particular case is that where there is a single, very large value in a data set mostly comprised of very small values. The variogram may be severely impacted by such a value, refer to a study by Rivoirard (1987a).
Note that, since the richest values often determine the economics of a deposit, cutting them (or removing them) should be a last resort if we are taking a scientific approach.
Other Approaches to Calculating Variograms Other Approaches to Calculating Variograms In addition to the traditional experimental variogram of grades:
( )
∑( ) ( )[ {
}]
=−
+
=
N i i i h Z x x Z N h 1 2 2 1 ˆα γThere are a number of other approaches to calculating experimental variograms. Such approaches fall into two broad categories:
1. ‘Robust’ estimators of the variogram (e.g. relative variograms).
2. Variography of transforms (e.g. logarithmic and Gaussian variography, indicators, etc.).