Problems of Classical Geostatistics – Introduction
Gold grades estimation requires more than using solid geostatistical techniques. Mineral resources analysts have worked at several gold deposits and in all of them none of the mathematical methods was able to avoid manual interventions, such as cutting high grades for local estimation and using information beyond the data to infer the variograms.
The geostatistical theory has a solid and coherent mathematical body, developed by Matheron (1963) which works satisfactorily for many other minerals and geosciences’ variables without the need of intense data manipulation. The question is why a theory constituted on solid basis is not able to provide good local estimation for variables having a strong positive asymmetry.
Linear kriging methods, such as ordinary or simple kriging, normally lead to poor estimates in these situations (Costa, 2003). The practice of cutting or capping high values to limit their influence during grade estimation is very common in the mining industry. Cutting (capping) of grades is generally undesirable because its arbitrary nature can lead to large uncertainties in grade and tonnage estimates (Sinclair and Blackwell, 2004). Indicator kriging (Journel, 1982) was developed initially to avoid both overestimations by using fully the very high grades of a deposit and underestimation by cutting or abandoning these high grades. It allows attributing to blocks the probability to have grade values above a chosen cutoff.
Indicator transform is an effective way of limiting the effect of very high values (Glacken and Blackney, 1998). But MIK does not work appropriately both for the median variogram and the percentiles variograms. Applying the median IK is easy and comfortable, a “simple and fast procedure” (Deutsch and Journel, 1992), but assumes that the same variogram is valid for all percentiles, which is a simplification in most circumstances incorrect. Moreover, it is often hard to define the continuity of grades for indicators above the 90th percentile, due to few samples and their sparse spatial distribution. If instead of the median IK the true variograms for all percentiles are used, it will bring almost always order relation problems which corrections are clearly artificial. And it is very difficult to produce a variogram for grades above the 90th percentile. An overestimation is expected as at the highest percentile the grades remain very high and their values are used for the estimation. Overestimation is exactly what IK method intended to avoid.
It is unacceptable that there is no way to develop a mathematical theory or method to explain and justify the practice of miners of lowering high grades. The assumption that this is not feasible is an underestimation of the power of the mathematics.
The Three Problems of Classical Geostatistics
Classical geostatistics is not able to deal with highly skewed distribution due to three main reasons:
When the practitioner lowers a high grade, for instance a 100 g/t of gold sample, indeed he is assuming that this grade, or another high grade like 90 g/t or 80 g/t, cannot be extended to one tonne. It is a logical and correct procedure. He has enough information about the deposit to proceed in this way. Classical geostatistics do not use the information available at the grade distribution. The cutting grade procedure can be mathematically justified assuming that the mineral resource analyst is reducing the spatial influence instead of cutting the grade.
The expression 100 ppm contains an a priori wrong inference, which is if any samples replicate will be extracted from the same 1 t there is no guaranty that the 100 ppm or another very high grade will be produced. Indeed, if the sample weights 10 kg its real grade would be 1 g/(10 kg). This initial mistake will cause error propagation in ore reserve estimation if the mineral resource analyst will no cap arbitrarily the grade. The question is for which amount of space (volume) the ratio 100 g/t remains true.
An axiom is a premise assumed as true without any explanation. By observing the way estimations are performed, two axioms can be easily inferred:
- Representativeness: every sample represents a spatial portion larger than itself; this is the principle of representativeness;
- Continuity: in the classical methods the portion represented by a sample fulfils some volume around it. It is the premise of continuity. In Classical Geostatistics the continuity is given by the abstract random function and by the concrete existence of the variogram.
In the polygons method the representativeness is given by the area of influence. For the polyhedra there is a volume or a tonnage of influence, and the same for estimation using vertical sections and for declustered cells. They are all piecewise continuous.
FPG call any area or volume of influence by a generic name – the extension.
These mineral resource estimation methods work with two variables. This pair, i. e., grade and extension are herein called the Field of the sample, as for an electron in Physics.
Classical Geostatistics treats all samples as if they have the same influence. Hence it does not follow the representativeness axiom.
Geostatistics works as if the samples were points and their precision is absolute. However, the samples have a size, and they are not points, but measures, in the Schwartz’s Theory of Distributions sense (Schwartz, 1957). Hence, a variogram can never be written as an integral. Even if the deposit is completely drilled, the number of samples will remain finite and for each size of samples the variogram will be different. The point variogram was needed as an abstraction to build the theory and logically produce different sizes variograms through convolutions (David, 1977). But a deconvolution is impossible to perform; a variogram is always a discrete summation of terms. In the same way no equipment produces an infinitely precise measurement, the grade has the accuracy of the measuring equipment. If the grade ranges from 0.00 g/t to 100.00 g/t there are only 10 001 possible values. The usage of the discrete spectrum was essential to the mathematical formulation of the FPG theory.