CHAPTER 2: The Semi-Variogram
We have seen in Chapter 1 how the definition of a semi-variogram arises out of
the notions of ‘continuity’ and ‘relationship due to position within the
deposit’. The semi-variogram, g, is a graph (and/or formula) describing the expected
difference in value between pairs of samples with a given relative orientation.
We also discussed the ideal forms which semi-variograms might take. We are now
going to discuss calculated or ‘experimental’ semi-variograms.
Consider the data shown in Fig 2.1.
Fig 2.1. Example of data on a grid for the calculation of an experimental semi-variogram -- iron ore.
We have here a stratiform iron orebody,
through which a set of drill-holes have been bored, perpendicular to the dip of
the ore. The value given at each location is the average value of Fe (% by
weight) over the intersection of the borehole with the ore (see Fig 2.2).
Essentially this is a two-dimensional problem, so that the h in our definition of the semi-variogram depends on
the distance between the pair of samples, and their relative orientation in a
two-dimensional plane.
Fig 2.2. Cross-section through the iron ore deposit.
Let us consider the east-west direction,
and try to construct an experimental semi-variogram for this relative
orientation. The grid on which the holes have been so conveniently placed is
100ft by 100ft, so that we can only calculate values of the experimental
semi-variogram, g*, for distances which are multiples
of this. At zero we know that g*(0) is equal to zero. At 100ft we need
to find all pairs of samples at a separation of 100ft in the east-west
direction. These are shown in Fig 2.3.
Fig 2.3.
Identifying all the pairs at 100ft apart in the east-west direction.
The calculation as defined says: take each
pair; measure the difference in value between the two samples; square it; add
up all the squares; divide this sum by twice the number of pairs. In our
example:
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γ*(100)= |
[ (40-42)² + |
(42-40)² + |
(40-39)² + |
(39-37)² |
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|
+ (37-36)² + |
(43-42)² + |
(42-39)² + |
(39-39)² |
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+ (39-41)² + |
(41-40) ² + |
(40-38)² + |
(37-37)² |
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|
+ (37-37)² + |
(37-35) ² + |
(35-38)² + |
(38-37)² |
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+ (37-37)² + |
(37-33) ² + |
(33-34)² |
(35-38)² |
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|
+ (35-37)² + |
(37-36) ² + |
(36-36)² + |
(36-35)² |
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|
+ (36-35)² + |
(35-36) ² + |
(36-35)² + |
(35-34)² |
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+ (34-33)² + |
(33-32)² + |
(32-29)² + |
(29-28)² |
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+ (38-37)² + |
(37-35)² + |
(29-30)² |
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+ (30-32)² ] |
¸ (2 ´ 36) |
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γ*(100)= |
1.46(% )² |
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This gives us one point which we can plot
on a graph of the experimental semi-variogram g* versus the distance between the samples (h), that is [100ft,1.46(%)²]. Now let us consider a distance between
samples of 200ft.
Fig 2.4. Identifying all the pairs 200ft apart in the east-west direction.
Figure 2.4 shows the pairs which lie at this distance in
the east-west direction, and the calculation becomes:
|
γ*(200)= |
[ (44-40)² + |
(40-40)² + |
(42-39)² + |
(40-37)² |
|
|
+ (39-36)² + |
(42-43)² + |
(43-39)² + |
(42-39)² |
|
|
+ (39-41)² + |
(39-40) ² + |
(41-38)² + |
(37-37)² |
|
|
+ (37-35)² + |
(37-38) ² + |
(35-37)² + |
(38-37)² |
|
|
+ (37-33)² + |
(37-34) ² + |
(38-35)² + |
(35-36)² |
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|
+ (37-36)² + |
(36-35) ² + |
(36-36)² + |
(35-35)² |
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|
+ (36-34)² + |
(35-33) ² + |
(34-32)² + |
(33-29)² |
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|
+ (32-28)² + |
(38-35)² + |
(35-30)² + |
(30-29)² |
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|
+ (29-32)² ] |
¸ (2 ´ 33) |
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γ*(200)= |
3.30(% )² |
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which we
can plot on the graph versus 200ft.
The
question now arises of where to stop. We could obviously continue up to
distances of 800ft, for which we would have 7 pairs. In practice, we rarely go
past about half the total sampled extent -- in this case, say, 400ft. Table 2.1
shows the calculated points for the experimental semi-variograms in the
east-west and in the north-south direction, and Fig 2.5 shows a plot of the two
g*s.
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Fig 2.5.
Experimental semi-variograms in the two major directions for the iron ore
example. |
Table 2.1.
Calculation of experimental semi-variogram values in two major directions for
iron ore example on square grid |
There
seems to be a distinct difference in the structure in the two directions. The
north-south semi-variogram rises much more sharply than the east-west,
suggesting a greater continuity in the east-west direction. To verify this, we
should then calculate the semi-variogram in at least one ‘diagonal’ direction,
e.g. northwest-southeast. These figures are shown in Table 2.2, and Fig 2.6
shows the three experimental semi-variograms plotted on the same graph.
|
Fig
2.6. Experimental semi-variograms including a diagonal for the iron ore
example. |
Table
2.2. Calculation of semi-variogram in diagonal direction for iron ore |
Of
course, the intervals at which the diagonal semi-variogram values are
calculated are now multiples of 100√2= 141 ft. The new g* seems to verify the difference between
the other two, since it lies between them -- although it seems to be closer to
the north-south than to the east-west. The conclusion which must be drawn is
that more information is needed to determine the ‘true’ axis of the anisotropy.
It would be rather optimistic to suppose that our drill grid was laid down in
the exactly correct direction for the different structures. Secondly, we must
decide whether, say, the last point on the diagonal semi-variogram is reliable.
This was calculated on only 13 pairs, as opposed to the next lowest of 21. Does
this mean we should place only two-thirds as much confidence on it? Some
theoretical work on simple cases has been done at Fontainebleau, but in
practice the only rule is: the fewer pairs, the less reliable.
The
east-west semi-variogram seems to be reasonably consistent, and suggests a
straight line with slope 6.5(%)²/400ft=0.01625(% )²/ft. Thus for the east-west direction:
For the north-south direction, the following seems
reasonable:
That is,
in the east-west direction, we ‘expect’ a squared difference of 0.01625(%)² for each foot between the samples. Put
another way, a difference in grade of √0.01625=0.1275%Fe is expected for
two samples 1ft apart, with a relative orientation of east-west. In the
north-south direction the corresponding figure is 0.2236%Fe. For samples 100ft
apart, we would expect differences of 1.275%Fe (east-west) and 2.236%Fe
(north-south) and so on. Thus we have built up a picture of the grade
fluctuations within this section of the deposit, and have a fairly simple model
to describe the differences in grade.
Table 2.3. Hypothetical borehole log
from lead/zinc deposit --- Zinc values
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Now let us turn to another example.
Table 2.3 shows a borehole ‘log’ for one drill hole through a lead/zinc
mineralisation which is disseminated in limestone. The first 45.40m go through barren
rock, and the rest of the core has been divided into regular sections 1.52m
(5ft) long. At one point, the core has been lost -- perhaps due to a solution
cavity in the limestone. As is the case in most three-dimensional deposits,
there is very detailed information ‘down’ the borehole, but the boreholes are
widely scattered over the deposit. The usual practice is to make
‘down-the-hole’ semi-variograms, and then to look at the horizontal directions
as we did in the first example. So, for practice, let us calculate the
experimental semi-variogram down this one borehole. Effectively the problem is
simpler than the first one since we have one long line of regularly spaced
samples with a single gap of 6.08m. Table 2.4 shows the calculated g*, and Fig 2.7 the plot of this
experimental semi-variogram versus the distance between the pairs.
Fig 2.7. Experimental semi-variogram calculated on one ‘borehole’ through a hypothetical lead/zinc ore-body.
Table 2.4. Calculated experimental
semi-variogram from Lead/Zinc deposit
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In this case the number of pairs of
points decreases steadily as the distance increases, from 58 pairs at h=1.52m
to 28 pairs at h=48.64m. Thus the most ‘reliable’ points on the graph are those
for small distances, and the reliability drops off slowly and regularly. The
semi-variogram seems to follow approximately the ideal shape discussed in
Chapter 1. It rises from the origin, seems to more or less level off at about
15m, and continues with some variation around the value, say, of 10.5(%)². We could probably fit a spherical
model to this semi-variogram without further ado. However, let us look at the
supposed variation around the sill. There is a dip in the curve at 25m, and
another at about 35m. There is less difference
between samples 25m apart than there is at 15m. If we go back to the drill log
we find that the grade values seem to rise and fall quite regularly. There is a
‘rich’ patch centred at about 47m below collar, another at 81m and a possible
third at 106m, where the core has been lost. The distances between these rich
patches are 34m and 25m respectively. Thus the experimental semi-variogram is
drawing our attention to the presence of localised rich areas down the
borehole. The implications of this would need to be viewed in the light of other boreholes and/or information about the
deposit. If the same sort of pattern occurs on many of the other boreholes then
we would suspect some sort of lenticular (or stratified) structure. If the
other boreholes do not reflect this regular rise and fall, this is probably
just local fluctuation. This particular set of data was taken from a deposit
with a marked (geologically) lenticular structure which had already been mapped
on-site. This is one manifestation of what happens to the semi-variogram if
‘trends’ -- in this case periodic trends -- are present within the deposit and
are ignored. On the other hand, for small scale estimation, say up to 20m in
the vertical direction, a spherical model would be quite adequate.
Both of these illustrative examples have been
carried out on small sets of data, so that the reader can check his
understanding of the calculation by trying to reproduce the answers. The
interpretation of an experimental semi-variogram is another matter, and is
something that becomes easier with practice. I should like, therefore, to give
a few examples of semi-variograms from my own experience.
Table 2.5
shows an experimental semi-variogram which was calculated on silver values from
samples taken in a tabular, heavily-disseminated base-metal sulphide deposit.
An access adit has been driven into the deposit and a vertical channel sample
taken every metre along one wall of the tunnel.
Since the
width of the ore is variable, the accumulation (grade times width) was
calculated for each sample. 400m of the adit was sampled in this way, giving an
unbroken succession of values. The units of accumulation are
metres-per-cent(m%), so that the units of the experimental semi-variogram are
(m%)². Figure 2.8 shows the graph of
this semi-variogram versus distance. Near the origin, the points form an almost
straight line. This is a characteristic of most of the common semi-variogram
models.
Fig 2.8. Experimental semi-variogram
constructed on the silver values from a complex sulphide deposit.
The curve rises, flattens off at about 11(m%)², and then rises again more and more rapidly. In fact, after a distance of about 75m, the curve is virtually parabolic. This is an indication of the presence of a polynomial-type trend within the deposit. There appears to be a smoothly varying large scale trend in operation here. If we wished to consider points more than, say, 75m apart in any estimation procedure, then we should have to take account of that trend (see Chapter 6). However, if we restrict consideration to areas within the deposit of no more than 75m in radius, the problem may be safely ignored. Let us, then, look at the semi-variogram only up to distances of 75m (see Fig 2.9). A ‘sill’ appears to exist at C=11(m%)²