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 |
|
|
|
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
|
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.
|
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.