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:
|
γ*(100)= |
[ (40-42)² + |
(42-40)² + |
(40-39)² + |
(39-37)² |
|
|
+ (37-36)² + |
(43-42)² + |
(42-39)² + |
(39-39)² |
|
|
+ (39-41)² + |
(41-40) ² + |
(40-38)² + |
(37-37)² |
|
|
+ (37-37)² + |
(37-35) ² + |
(35-38)² + |
(38-37)² |
|
|
+ (37-37)² + |
(37-33) ² + |
(33-34)² |
(35-38)² |
|
|
+ (35-37)² + |
(37-36) ² + |
(36-36)² + |
(36-35)² |
|
|
+ (36-35)² + |
(35-36) ² + |
(36-35)² + |
(35-34)² |
|
|
+ (34-33)² + |
(33-32)² + |
(32-29)² + |
(29-28)² |
|
|
+ (38-37)² + |
(37-35)² + |
(29-30)² |
|
|
|
+ (30-32)² ] |
¸ (2 ´ 36) |
|
|
|
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 |
|
|
+ (37-36)² + |
(36-35) ² + |
(36-36)² + |
(35-35)² |
|
|
+ (36-34)² + |
(35-33) ² + |
(34-32)² + |
(33-29)² |
|
|
+ (32-28)² + |
(38-35)² + |
(35-30)² + |
(30-29)² |
|
|
+ (29-32)² ] |
¸ (2 ´ 33) |
|
|
|
γ*(200)= |
3.30(% )² |
|
|
|
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.
|
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
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. A horizontal line has been drawn onto
the graph at this level.
A more
difficult parameter to ‘eyeball’ is the range of influence a. It can be shown that if a spherical model is to be
used -- as seems to be indicated by the flat nature of the sill -- then a line
drawn through the first few points of the experimental semi-variogram will
intersect the sill at a distance equal to two-thirds of a. Doing this on Fig 2.9 produces a value of 33m for
the intersection, giving a range of influence of approximately 50m.
Fig 2.9. First
estimation of model and parameters for the silver semi-variogram.
Indications
are that we need a spherical model with a range of influence of 50m and a sill
of 11(m%)². Since there is no objective
(statistical) way of deciding whether a model fits an experimental
semi-variogram, the only simple method is to draw the model curve onto the same
graph as the experimental one. The equation for this model is:
This curve has been drawn onto the same graph as the
experimental points, and the result is shown in Fig 2.10. The numerical values
for various points on the model curve are given in Table 2.6.
|
Fig 2.10. Fitted
spherical model to silver semi-variogram. |
Table 2.6. Spherical semi-variogram model for silver values up to h
= 75m
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This seems to give a fairly good fit. It is difficult to see how it
might be improved. Sometimes the two parameters require adjustment before an
adequate fit is found. Note that the model has only been fitted for distances
up to 75m. Beyond this the trend must be taken into account. In this case we
were very lucky, in that the trend does not ‘interfere’ until after the range
of influence is passed. This is not always so, and the closer the parabolic
behaviour is to the origin the more heed must be paid to the trend.
It might be argued that a more suitable model for this semi-variogram would be
the exponential model.
For interest, let us take the sill again at 11(m%)². For an exponential model the straight line through
the origin intersects the sill at a distance equal to the range of influence. That is,
if we try an exponential model the range will be 33m.
Figure
2.11 shows the model, alongside the data points.
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Fig 2.11. Exponential model with
same parameters as fitted spherical (for silver semi-variogram). |
Table 2.7
Attempts to fit exponential models to silver semi-variogram
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The slope
at the origin is correct but the rest of the curve is far too low. We can
increase the sill to bring up the values, but we also need to increase the
value of the range of influence, so that the behaviour near the origin is still
correct. Table 2.7 shows the ‘model’ values given by various sets of parameters
-- sill and range of influence.
Round figures have been used for
simplicity, but the ‘best’ exponential fit seems to be the last one, with a=50m and C=16(m%)². Figure 2.12 compares the fit of this
curve with the previous spherical model, and with the experimental
semi-variogram. I prefer the spherical model because it seems to fit the data
between 15 and 40m better than the exponential. Only a minority of the observed
points fall below the exponential curve. A shortening of the range of influence
to compensate for this results in a marked change in slope at the beginning of
the curve.
Fig 2.12. Comparison
of final models -- exponential and spherical -- for silver semi-variogram.
COMPLEX MODELS
Now let us try some real semi-variograms, rather than
these hand-picked simple ones. Figure 2.13 shows the experimental
semi-variograms for three metals in another complex base-metal sulphide. The
metal of economic importance is the copper, but the other two metals are of
sufficient value to warrant investigation. The semi-variograms are
‘down-the-hole’ in direction, and contain information from about 50 boreholes
perpendicular to the plane of the ore-body.
Fig 2.13. Experimental
semi-variograms for a complex base-metal sulphide deposit.
My interpretation of the
lead and zinc semi-variograms is pure nugget effect. That is, the ‘model’ is a
horizontal line at a value equal to the sample variance. There appears to be
very little relationship even between neighbouring cores! On the other hand,
the copper semi-variogram appears to be a combination of a nugget effect
(constant) and a parabola. As in the previous example, the parabola implies a
polynomial trend, in this case acting on pairs of samples even at 1m spacing.
The nugget effect implies completely random behaviour. So we have a trend with
random variation; an ideal case for Trend Surface Analysis.
The next example concerns a nickel ore-body disseminated in peridotite,
which has been ‘proved’ by means of about 45 vertical boreholes. The average
spacing between the boreholes was about 60m and they were not regularly spaced,
so that only the ‘down-the-hole’ experimental semi-variograms were calculated.
Altogether
approximately 4000m of core was recovered and assayed in 2m core sections. In
this case the logarithm of the grades was used, rather than the grades; the
reason for this has no relevance here. The experimental semi-variogram is shown
in Fig 2.14, and the numerical values are given in Table 2.8.
Fig 2.14. Experimental
semi-variogram for a nickel deposit -- logarithms of grade values.
Table 2.8 Experimental semi-variogram from a disseminated nickel deposit
(logarithm of grade)
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There
appears to be a definite flat sill at about 2.55(log %)². However, drawing a
straight line through the first two points, as we did in the silver
semi-variogram produces two odd results. First, the line intersects the
semi-variogram axis at 0.40(log%)² not at zero. This suggests that there
is a component of each value which is ‘random’ or unpredictable. Samples very
close together still have a reasonably large difference in value. Remembering
that the sill (if it exists) is equal to the sample variance, we can see that
0.40/2.55=0.156
suggests that about 16% of the variation in the sample values is
random and unpredictable. Thus, no matter how closely we sample, this
unpredictability will still exist. The semi-variogram model will need to be of
the form:
where g’(h) is the usual sort of model (e.g.
linear). In effect, the nugget effect is a simple constant raising the whole
theoretical semi-variogram 0.4 units. Thus we now seek a model with a sill of
2.15(log %)². Now, we
saw in the silver example that extending the initial straight line slope up to
the sill gave a value of two-thirds of the range of influence, when using a
spherical model. In this case the intersection produces a value of 13m implying
a range of influence of about 20m. On the other hand, the curve does not even
approach the sill until some distance past 45m. Clearly neither of our ideal
models will cope with this sort of situation. Let us look again at the
experimental curve. There seems to be an ‘intermediate’ sill, reached at about
14m and a value on the g-axis of 1.95-0.40=1.55 (to allow for nugget effect).
We seem to have a mixture of two spherical type models, one with a shortish range and one with a range of about 50m. Let us
try out this tentative model and see how it fits the experimental
semi-variogram. We have a fairly complex model:
Putting these values
into the proposed model produces the following:
For distances (h) between 14 and 50m, the model is
given by:
and when the distance between the two samples is greater than 50m,
the model semi-variogram takes the form:
In order
to compare the theoretical model with the experimental points we must evaluate
the model at various distances, and draw the resulting curve onto the same
graph. For example, for distance h equal to 2m:
and for a distance h equal to 40m:
A set of values was selected for h and the theoretical curve constructed. The values are
shown in Table 2.9, and the resulting model has been plotted in Fig 2.15. The
experimental points are also shown for comparison.
|
Fig 2.15. First attempt
to fit a mixture of spherical models to the nickel semi-variogram. |
Table 2.9 First
attempt to fit a mixture of Spherical models to the experimental nickel
semi-variogram (parameters in text)
|
The ‘model’ curve fits fairly well to the beginning and end
of the experimental semi-variogram, but does not seem too good in the middle.
The kink in the curve is at far too high a level -- it needs to occur at g=1.95.
We
assumed that this level was equal to C0+C1. What was forgotten is that, even
at short distances, the second spherical component still contributes some value to the model, so that the
value 1.95 should actually be equal to:
In other words, we need to lower the value of C1 and raise the value of C2, and then try the fit again. After a few
tries, I got the following model:
This model is shown in Fig 2.16 alongside the experimental
semi-variogram, and seems to be a relatively good fit. Perhaps the reader would
like to try to improve upon it? Table 2.10 gives the corresponding numerical
values for the model curve.
|
Fig 2.16. Final attempt to fit a mixture of spherical
models to the nickel semi-variogram. |
Table 2.10 Final attempt to fit a mixture of Spherical models to the
experimental Nickel semi-variogram (parameters in text)
|
Examples of semi-variogram models which are mixtures of
spherical components abound in the geostatistical literature, and seem to be
about the most common type encountered, especially in low concentration
minerals such as cassiterite, copper veins, uranium
and so on.
LOG-NORMALITY
I should like, now, to turn to another problem which is often discussed in the
literature when samples are expected to follow a log-normal distribution. Whilst
the construction of the experimental semi-variogram and the estimation
procedures produced by geostatistics do not depend on what distribution the
samples follow, there are one or two ‘side-effects’ which become apparent when
dealing with log-normal samples. As every schoolboy knows, the standard
deviation of a log-normal distribution is directly proportional to its mean.
Consequently the sample variance -- and hence the sill of the semi-variogram --
is proportional to the square of the mean of the samples. If experimental
semi-variograms are constructed on different sets of samples within a deposit,
this ‘proportional effect’ can have a radical effect on the individual
experimental semi-variograms. Examples in the literature usually concern cases
where, in order to construct experimental semi-variograms in different
directions, it has been necessary to use different sets of, e.g. borehole data
in each. As an example of ‘proportional effect’ consider the following
situation. In Cornish tin lodes the assay values are usually assumed to follow
a log-normal distribution. Such veins are developed by means of horizontal
drives approximately 100ft apart. These drives are sampled every 10ft by taking
chip samples from the roof. In the example under consideration, nine levels
have been developed, from 600ft below surface to 1400ft (6-14 levels).
Semi-variograms were calculated for each level separately. For simplicity, Fig
2.17 shows only three of these experimental semi-variograms, for levels 6, 10
and 12. The other six lie scattered between levels 6 and 12. Figure 2.18 shows
a graph of the average assay value along each drive versus the standard
deviation of the samples along that drive.
Fig 2.17. Example
of supposed zonal anisotropy -- cassiterite
vein.
Fig 2.18. Illustration
of the proportional effect -- cassiterite.
The
averages vary between 35lb/ton (pounds of SnO2 per ton of ore) and 80lb/ton, and the
standard deviations vary between 35 and 110lb/ton. The relationship is
virtually perfect between the two, with a calculated correlation coefficient of
over 0.85. Since the sill of the semi-variogram is roughly equal to the
calculated sample variance, it is easy to see that the experimental
semi-variogram for level 6 will be (and is) the lowest, with a sill of about
1200(lb/ton)²; level 10
will be in the middle with a sill of about 5000; and 12 will be the highest
with a sill of 12000(lb/ton)². The question is, can we
make an overall semi-variogram for this deposit when the individual
experimental semi-variograms vary by an order of magnitude from area to area.
The published authorities state that the valid way to
combine these semi-variograms is to ‘correct’ each one for the proportional
effect. That is, to divide the individual experimental semi-variograms by the
square of the average of the samples which went into its
calculation. This produces a ‘relative semi-variogram’ -- implying that
all values given by the semi-variogram are now ‘relative to the local mean’. Applying this method to the above example results in nine
experimental relative semi-variograms which vary in sill between about 1.00 and
1.80. Notice that these values now have no units. To be converted into
meaningful figures they must be multiplied by the square of the local mean. We
can now (supposedly) combine these semi-variograms into one for the deposit as
a whole, and fit a model to it. If we do so we must remember that in all our
estimation procedures etc. we have to ‘uncorrect’ the
values calculated from the semi-variogram -- estimation variances, standard
errors and so on.
This process of correcting experimental semi-variograms for
the proportional effect is widely advocated as the ‘right thing to do’. No one
seems to have bothered to test whether it actually works in practice. In the one case, that described above, where I have been able to investigate
in depth and compare what happens if you use the ‘relative’ semi-variogram,
I found that correction by the local mean gave completely erroneous results.
Therefore, I would not recommend
this procedure, but rather that you should combine the original experimental
semi-variograms and try to fit a model to the ‘uncorrected’ data. In the study
mentioned above this was found to give the correct values at all times.
OTHER VARIABLES
It has been said time and again that geostatistics -- Kriging and so on -- can
be applied equally well to other variables which are spatially or temporally
distributed. This book has been more or less devoted to mining applications,
because this is still the major field. However, many other variables can be
handled, and I should like to give one or two examples here. Even in mining
applications, grade or economic value of the mineral is not always the sole
variable of interest. In many deposits the ‘thickness’ of the deposit is as
important as the grade, and in many sedimentary deposits this factor is far
more important. In the Cornish tin example described above, the width of the
vein is fully as important a variable as the cassiterite
content. Both variables are required to assess the economic viability of the
lode or portions of it. Figure 2.19 shows the overall experimental
semi-variogram calculated for the nine levels, 6-14. To this semi-variogram I
fitted a model which consisted of: a small nugget effect, which was slightly
surprising; one spherical component with a range of influence of about 30ft and
another with a range of 150ft.
Fig 2.19. Experimental
semi-variogram constructed on the lode widths in the cassiterite
vein.
As an example of other types of spatially
distributed data which might be considered, Fig 2.20 shows an experimental
semi-variogram which was produced during a study of the rainfall
characteristics and runoff in a catchment area in the
Pennines in England.
Fig
2.20. Experimental semi-variogram constructed on the measured rainfall at
rain gauge sites.
The data observations are the recorded
monthly rainfall figures at rain gauges scattered over the catchment
area. The semi-variogram has been constructed without regard to the direction
of the
pair of samples. That is, the author has assumed that his variable shows the
same continuity down the long axis (direction of flow of the major river) as it
does across the valley. The erroneous nature of this assumption is immediately
apparent when given the information that the catchment
area measures about 30km across the valley (short axis). The marked
discontinuity in the experimental curve suggests that there is a definite
difference between the two major directions. Semi-variograms ought to be
constructed for at least two different directions to check for this. The second
conclusion which can be drawn from this experimental semi-variogram is that if
the same shape is shown by the new individual strong trend is in evidence which
must be taken into consideration. When considering the nature of rainfall it
does seem sensible to expect different amounts of rain to fall on the tops of
mountains than lower down in the valleys. This is a good example of when the
‘trend’ cannot be ignored in the geostatistical estimation procedure.
And now for a completely different type of
application we can take a time series example, rather than one which is
spatially distributed. A series of readings have been taken at the same site in
a large river, of various different variables of interest. This is a
‘one-dimensional’ situation, in which the dimension is time rather than space.
Instead of distance between samples, we now have time between samples, so that
the horizontal axis of the experimental semi-variogram will now read ‘time
between observations’. The estimation procedure will be used to predict values
of these variables forward into the future, or to fill in gaps in the records
caused by machine failure. Figure 2.21 shows two experimental semi-variograms
calculated in one case for the temperature of the water, and in the other for
the amount of suspended solids contained in the water. The latter looks like an
ideal case for a spherical type of model, with a suggestion of a ‘trend’ at the
weekly scale i.e., fairly homogeneous within any specified week, but varying in
level from week to week. The experimental semi-variogram for the temperature
shows a perfect daily cycle in temperature, with a little drift coming in after
3 or 4 days.
Fig 2.21. Experimental
semi-variograms calculated on water quality variables measured over time.
CONCLUSION
To summarise this chapter, we have seen how to
calculate an experimental semi-variogram in one and two dimensions, and how to
relate this ‘practical’ semi-variogram to the ‘ideal’ models which exist. We
have seen that, whilst some deposits may follow fairly simple behaviour, many
others require a fairly complex mixture of models to describe the experimental
semi-variogram. I have briefly pointed out some problem areas such as strong
trends, random phenomena and proportional effect, and tried to indicate how
these might be tackled. There are those in authority who say that the fitting
of a semi-variogram model is out-moded and
unnecessary. To counter this I should like to give an analogy with ordinary
statistics. If you take a limited number of samples from an exceedingly large
population and construct a histogram, are you prepared to assume that that sample histogram describes exactly the
behaviour of the whole population? The process of inference -- drawing
conclusions about the population from a few samples -- demands the construction
of some sort of model for the behaviour of the whole deposit.