The effects of various extraction and blending processes can then"be tested by simulating them on these numerical models. A mineralized phenomenon can be characterized by the spatial distribution of a certain number of measurable quantities called "regionalized variables". Examples are the distribution of grades in the three-dimensional space, the distribution of vertical thicknesses of a sedimentary bed in the horizontal space, and the distribution of market price of a metal in time.

Geostatistical theory is based on the observation that the variabilities of all regionalized variables have a particular structure.

The variability of gold grades in a placer deposit will differ from that of gold grades in a massive deposit. The direct study of the mathematical function z x is excluded because its spatial variability is usually extremely erratic, with all kinds of discontinuities and anisotropies. Both the random and structured aspects of the regionalized variable are expressed by the probabilistic language of "random functions".

The next section will deal with the operational aspect of this formalization by describing how the spatial variability of the regionalized variable z x can be characterized. The variog ram. To overcome this problem, the intrinsic hypothesis is introduced.

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This hypothesis is that the variograrn function 2y x, h depends only on the separation vector h modulus and direction and not on the location x. The intrinsic hypothesis is not as strong as the hypothesis of stationarity of the random function 2 x itself. The variogram as a tool for structural analysis "Structural analysis" is the name given to the procedure of characterizing the structures of the spatial distribution of the variables considered e.

It is the first and indispensable step of any geostatistical study. The variagram model acts as a quantified summary of all the available structural information, which is then channelled into the various procedures of resource and reserve evaluation. Thus y h can be regarded as injecting extensive geological experience into the sequence of studies involved in a mining project. In the proving of any mining project, it is imperative to have good communication between all the departments involved geology, mining techniques and economy, processing.

It is a waste, for instance, to have a geologist carry out an excellent study of geological structures, only to be followed by a project engineer evaluating resources by such arbitrary procedures as polygons of influence or inverse-squared distance weighting. Such evaluations are independent of the deposit and its geology, and make no distinction, for example, between a gold placer deposit and a porphyrycopper deposit.

The variogram model 2y h can help in this essential communication between the geologist and the project engineer. Continuity In the definition of the variograrn 2y h , h represents a vector of modulus Ih I and direction a. Consider a particular direction a.

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This is simply an expression of the fact that, on average, the difference between two grades taken at two different points increases as the distance Ih I between them increases. The manner in which this variogram increases for small values of Ih I characterizes the degree of spatial continuity of the variable studied. Figure The four corresponding semi-variograms have markedly different shapes, from which a geostatistician would be able to deduce the degree of continuity of the four piezometer profiles.

Beyond this distance a, the mear.

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The range a gives a precise meaning to the intuitive concept of the zone of influence of a sample z x. However, there is no reason for the range to be the same in all directions a of the space. In Fig. For a given distance Ihl, the horizontal variograrn presents a weaker variability than the vertical variograrn: this expresses the horizontal sedimentary character of the phenomenon considered. The estimation variance The variogram defined in 1. This estimation variance is expressed as 1.

The mean grades Z; and Zv can be defined on any supports, e. The general formula 1. This is embodied in the term y V, v. This is embodied in the term y V, V. Thus, the theoretical formula 1. Confidence interval.

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The estimation variance itself is not enough to establish a confidence interval for the proposed estimate, the distribution of the errors must also. Figure II. The arithmetic mean of the errors is O. The experimental variance of the errors is equal to 0. Change of support volume. The elementary variogram 2y h , estimated from experimental data, is defined on the support of these data, e.

By means of formula 1. This formalism for the change of support has great practical applications. Since future mining will be carried out on the basis of blocks and not drill cores, it is of great benefit to be able to evaluate the spatial variability of the mean grades of these blocks by means of the variogram 2Yv h defined on the block support. The dispersion variance.

There are two dispersion phenomena well known to the mining engineer. The first is that the dispersion around their mean value of a set of data collected within a domain V increases with the dimension of V. This is a logical consequence of the existence of spatial correlations: the smaller V, the closer the data and, thus, the closer their values. The second is that the dispersion within a fixed domain V decreases as the support v on which.

These two phenomena are expressed in the geostatistical concept of dispersion variance. Let V be a domain consisting of N units with the same support v. If the N grades of these units are known, their variance can be calculated. This formula can be used, for example, to calculate the dispersion variance of the mean grades of production units when the size of the units v varies or when the interval of time considered varies i.

The concepts of variogram and various variances, which so far have been defined for the regionalization of one variable, can be generalized to spatial coregionalizations of several variables. In a lead-zinc deposit, for example, the regionalizations of the grades in lead, ZI x , and in zinc Zz x , are characterized by their respective variograms 2YI h and 2Y2 h. However, these two variabilities are not independent of each other and we can define a cross-variograrn for lead and zinc:.

This coregionalization matrix can then be used to estimate the unknown lead grade Zj x from neighbouring lead and zinc data. Sometimes, two different types of data relating to the same metal can be used in block estimation.

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One set may be the precise data taken from core samples, while the other may be imprecise data taken from cuttings and blast-holes. A study of coregionalization of these two types of data will determine the proper weights to be assigned to them during evaluation. The regionalized variable z x , which is the subject of the geostatistical structural analysis, is interpreted as one possible realization of the random function Z x. It is possible to generate other realizations zs x of this random function Z x by the process of simulation.

This simulated realization zs x , will differ from point to point from the real realization z x , but, statistically, will show the same structure of variability same histograms and variograms ; moreover, at the experimental data locations Xi, the simulated values will be equal to the true values, i. The simulated and actual realization will, thus, have the same concentrations of rich and poor values at the same locations.

The two realizations can be viewed as two possible variants of the same genetic phenomenon characterized by the random function Z x. The simulated realization, however, has the advantage of being known at all points and not only at the data points Xi. Various methods of extraction, stoping, blending, stockpiling, etc.

Using feedback, the effect of these simulated processes on recovery, fluctuations of ore characteristics, and on the running of the mine and. The techniques for the simulation of a regionalization z x can be extended to simulations of coregionalizations, i. This was a study of the regionalization of water heights measured in rain gauges after rain showers, but it could equally well apply to the regionalization of the vertical thickness of a sedimentary bed.

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