Computers & Geosciences, Vol. 3, pp. 341-346. Pergamon Press, 1977. Printed in
REGULARIZATION OF A
SEMIVARIOGRAM
ISOBEL
Department of Mineral Resources Engineering, Imperial College of Science
and Technology,
(Received 1 October 1976)
Abstract - The production of estimates using the technique of kriging,
and the evaluation of the accuracy of these estimates depends completely on the
production of a model for the semivariogram of the deposit. The process of choosing such a model can be
complicated in practice by the bulk and geometry of the samples taken. Some aspects of this problem are discussed
and some formulae and a FORTRAN IV subroutine are presented to ease the task of
modeling.
Key Words: FORTRAN, Kriging, Regionalized variables, Semivariogram.
INTRODUCTION
It has been said that the semivariogram is the basic tool of the Theory
of Regionalized Variables. The computation
of an experimental semivariogram may be tedious, but rarely difficult. It is in the interpretation and fitting of a
model to a practical semivariogram that the real difficulties arise. For instance, to carry out a kriging exercise
of any type, we require for the deposit a theoretical model of the
semivariogram. This model produces for
us the relationship between two points a certain distance apart. Note, however, that we said points. Seldom in practice can we measure a value,
say the grade of a mineral, at a point.
At the feasibility stage of mine development we are concerned mostly
with borehole cores; during development and production we may have many
different types of samples, but all of them have one thing in common. They have a certain bulk, a certain
geometrical shape and volume. In some
instances, they may be assayed using different techniques. All of these factors combine to form the
"support" of the samples. A
semivariogram which has been calculated on one type of support will not be the
same as that calculated on another. The
process, of averaging the grade over the bulk of a sample, will influence the
shape and behavior in such a manner that we must use an indirect method to
derive the theoretical model for the point semivariogram necessary to further
analysis and estimation.
REGULARIZATION
Suppose that we had a line of n point samples at regular intervals 
along a line (say a drive or a
raise). For example in Figure 1 points on
the experimental semivariogram may be calculated with ease:
These points may be plotted on a graph of g* vs h, and a model for the point semivariogram chosen and fitted directly
to the data points.
Now suppose that, instead of a line of point samples, we are concerned
with a borehole which has been assayed in sections of length 1. We no longer
know the assay value at some point x,
but rather the average value over a length (x-/2, x+/2). How will this affect the
semivariogram? Let us consider the problem
from an intuitive point of view.
Consider two segments, or "cores", of length a distance h apart as in Figure
2. Suppose that we knew the grade at every point along both of those
segments. Then we could calculate g*(h) for all the pairs of points, that distance h apart, by computing
the difference in grade for each pair, squaring the result, and averaging
through all the pairs. However, suppose
that we do not know the grade at every point, but only the average grade of
each segment. The semivariogram
calculated then must be denoted by
g
*(h), and this will be
computed by taking the difference in the average grades, squaring and averaging
those. However, to obtain the average
grade of the segment, all the variation of grades within the segment has been
eliminated. Thus, the average grade of
the segment will be a more regular variable than the average grade of
individual points. If the variable is
more regular, this implies that the differences between any two of the values
will be less. That is g*(h) will of necessity be
less in value than the corresponding g*(h),
whatever the form of g*(h). Now consider those models for g*(h) which possess a sill,
and therefore a range of influence. Two
points which are farther apart than the range of influence, a, may be said to
be independent. But, two segments whose
centers are a distance a apart will not be independent of one another, because
one-half of one segment will be within the range of influence of one-half of
the other segment. It is not until h = a + , that is the ends of the segments are a apart, that no point in one
segment influences the other. Thus, for
a point model g*(h) with a certain range of influence, it is manifest that g*(h) will have a range of influence units larger. This phenomenon of the change in character of
the semivariogram as the support of the sample changes is known as regularization, and g*(h) is termed the regularized semivariogram.
REGULARIZATION OF THE USUAL MODELS
The mathematics of the process of regularization are in Matheron (1971),
and need not be discussed here. However,
it may be of advantage to outline the effect of regularization on the more
usual models of the semivariogram. The
linear model of g*(h) is the most widely used of those without a sill, and is
characterized by:
For samples of length , we determine that:
That is, for distances greater than - which is all we have in practice - the
regularized semivariogram is a straight line with the same slope as the point
model, but with an intercept on the g
axis of - p/3 (see Fig. 3), if the semivariogram is not complicated by a nugget
effect. Thus, the model for the point
semivariogram may be obtained by estimating the slope for the regularized
semivariogram.
There are two usual models for semivariograms with a sill. The exponential model:
When regularized by , this becomes:
With a somewhat more complex form for the parabolic part with h<. The sill of this regularized
semivariogram obviously will be lower than the sill of the corresponding point
model. This can be shown to be:
We have experimentally values of the regularized semivariogram, that is g*(h) for various values of h<. We need to be able to estimate the parameters of the point
semivariogram, a and C, from g* in some manner. Having done
this, we may substitute these values into the equation (1) to give
'theoretical' points on g, for comparison with g*.
The first step is to plot a graph of g* against h (see Fig. 4). From this we can estimate C immediately as the asymptote of the curve. In practice, of course, g* will be more irregular. A first
approximation for a, now may be determined.
Draw a straight line through the first few points on g* and extend it until it meets the sill level C. The value of h at which the two meet should be
approximately a. We have seen that a = a + , so an estimate for a follows immediately.
Having estimated C
and a (if only approximately) equation (2) may be used to determine
an approximation for C. These values, a and C, then may be used
in equation (1) to produce values of g(h). Plotting these points on the
same graph as g*(h) will give a visual
comparison between the model and the data.
The parameters can be adjusted if necessary to improve the fit, and the
process repeated. For example, if all
the values of g were slightly higher
than the corresponding g*, either the estimate of the range of influence is too small or the
estimate of the sill is too high, or both.
Visual inspection should reveal which.
The process is repeated until we produce a g(h) which seems "close enough" to g*(h) for our
satisfaction. The final values of a and C then can be used to characterize the point semivariogram for the
deposit.
The third model - and perhaps the one in widest use in mineral valuation
- is the spherical or Matheron model.
This semivariogram, expressed as:
is the popular one which exhibits linear behavior near the origin,
gradually curves off, until at h = a
it reaches its sill and stays there permanently. Many mineral deposits seem to conform to this
sort of behavior, but the discontinuity in the formula renders the mathematical
handling more complex than the other models.
This discontinuity also divides the resulting formula for g(h) into a series of alternatives given various boundary conditions on h
and a. These formulae will not be given in the text, but are presented in
Appendix I in the form of a FORTRAN IV subroutine which will be described more
fully in a later section. However, we
can say that the relationship between the sill of g(h) and that of g(h) is given by:
If the linear behavior for the first few points of g(h) is extended to meet the sill, the distance at which this occurs is 2a/3 (see Fig. 5). So, we may approximate a and C as follows:
·
guess C, the sill of the regularized semivariogram;
·
produce the line through the
first few points up to C, to give h = 2a/3, and hence a;
·
a = a - as before;
·
calculate C from the appropriate part of equation
(3);
·
use attached subroutine to
produce values of g(h) using a and C;
·
plot or graph; compare with g*(h);
·
adjust a and C if necessary;
repeat, etc.
When a satisfactory fit has been obtained, we have the point spherical
model for the deposit and may proceed with estimation and kriging.
SUBROUTINE REGSPH
The subroutine included in Appendix I is a FORTRAN IV routine to
evaluate points of a regularized spherical semivariogram given the following
information;
NH - number of points to be evaluated.
A - range of influence of the point semivariogram.
C - sill of the point semivariogram
EL - length of samples producing regularization.
H - array containing values of h for
which g(h) is to be calculated
The subroutine is evoked by:
CALL REGSPH (GL, H, NH, A, C, EL)
and returns in array GL the values of the semivariogram g(h) for the corresponding values of h
in H. Note that the distances in H need not be in ascending order. To assist the user in implementing the
subroutine and checking for possible mispunches, a program segment and printout
are included in Appendix II.
Acknowledgments The subroutine REGSPH was developed on the CDC 6400 installation at
Imperial College, London and the program and subroutine implemented on the DEC
system-10 at Syracuse University, Syracuse, New York, while the author was
Visiting Associate Professor in the Department of Geology.
REFERENCE
Matheron, G., 1971, The
theory of regionalized variables and its applications: Cahiers du Centre de
Morphologie Mathematique de Fontainebleau, no. 5, 211 p.
APPENDIX I
