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A mineral resource is an accumulation
of naturally occurring materials in or on the earth's crust. Accurately determining the boundaries of this resource requires
investigating the geology via mapping, geophysics and conducting geochemical or intensive geophysical testing of the surface
and subsurface. In most cases drilling may be performed directly as a mechanism for surveying content composition, including
calculation of recoverable amount of mineral at a given grade and/or quality, and determining the worth of the mineral resource.
In engineering, when a number of data points can
be obtained by sampling and experimentation, it is possible to construct
a function that closely fits those data points. Fortunately, numerical techniques exist that can be applied to the estimation
of a function over the range covered by a set of points (as in core drill samples) at which the function’s values are known.
Interpolation is the process of finding unknown values where the simplest method requires knowledge of two point’s constant
rate of change. For instance, take any function y=f(x) where the process of estimating any value of y, for any intermediate
value of x, is called interpolation.
One method of estimating missing values is
by using the Lagrange interpolation polynomial. In its simplest form the degree
of the polynomial is equal to the number of supplied points minus 1. Basically, there are three numerical algorithms widely
used to compute Lagrange interpolation: Newton’s algorithm, Nevilles’s algorithm and a direct Lagrange formula. The algorithm
of choice varies based on efficiency characteristics such as number of sample points, complexity and degree of estimation of
numerical errors.
Another often used method of interpolation is
the Bulirsch-Stoer interpolation. This approach uses a rational function,
that is, a quotient of two polynomials, like R(x) = P(x) / Q(x). The extrapolation in numerical integration is superior to
using polynomial functions because rational functions are able to approximate functions with sample points rather well
(compared to polynomial functions), given that there are enough higher-power terms in the denominator to account for nearby
sample points. This type of function can have remarkable accuracy.
The Cubic Spline interpolation is also heavily
used in mining reserve estimation. In numerical analysis, the spline
interpolation is a form of interpolation using a special type of piecewise polynomial called a spline. This method
provides a great deal of smoothness for interpolations with significantly varying data. As a matter of fact, in the old
days people drew smooth curves by sticking nails at the location of computed points and placing flat bands of metal
between the nails. The bands were then used as rulers to draw the desired curve. These bands of metal were called splines,
which is where the name of this interpolation algorithm comes from.
With distinct types of interpolation
techniques available, which method to choose? there is often difficulty in choosing
among these algorithms and there are indeed many ways to skin a cat . One often accepted selection criteria is based on
the number of sample points where the Cubic Spline algorithm would be preferable when not enough sampling points are
available. If a function is hard to reproduce then the Bulirsch-Stoer interpolation may be appropriate. Lagrange
interpolation is useful when medium to large number of sample points are available.
The above represents a first step in mining
reserve estimation. Several other tasks - minimizing estimation errors,
calculating optimal sampling distances, block grade estimates, contour mapping, estimation of the size of the recovery
area are also part of the process of reserve estimation. Each task has a numerical solution and algorithms are available
to compute results.
Mineral prospecting in general is the
process undertaken in the attempt of finding commercially viable concentrations of
ore to mine at a profit. In this process reserve estimation is a much more intensive, organized and efficient form of
mineral prospecting. The use of applied mathematics interpolation algorithms in mining reserve estimation provides the
industry with computer competence, reduced time-consuming tasks to manageable units, and solutions which otherwise would
be very difficult to accomplish.
Practical samples and further interesting
discussions on mineral engineering can be found at http://www.keystoneminingpost.com
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