ASSESSMENT OF THE DEGREE OF EXPLORATION OF A DEPOSIT BY THE METHOD OF SECOND DIFFERENCES WITH A VARIABLE SAMPLING INTERVAL

A. I. VOLKOV

(Presented by the scientific seminar of the Department of Mine Surveying and Geodesy)

The method of second differences, proposed by Prof. D. A. Kazakovskii [1] for assessing the degree of exploration of a deposit and determining the error of analogy, assumes that observations of changes in one or another element of a deposit (thickness, grade, etc.) are carried out on an exploration grid.

The exploration grid may be square, rectangular, rhombic, etc. A necessary condition for every exploration grid is the equality of its corresponding sides.

The method of second differences may be applied to determine the degree of exploration of the deposit element of interest to us not only within the limits of some area, but also along some specified direction, for example, along the direction of an individual working, an exploration line, etc. The principal condition in this case is likewise the observance of equality of the distances between sampling points.

However, it must be borne in mind that, when carrying out exploration work, strict observance of equality of the distances between exploration workings is difficult. Deviations from the designed interval between exploration workings may be caused by various reasons, for example: the presence, in the area being explored, of artificial structures, ravines, bodies of water, curvature of boreholes. Bearing in mind only the curvature of boreholes, one may already assert that the distance between exploration workings (sampling points) never remains constant, and this nonconstancy may reach considerable magnitudes.

Naturally, the question arises whether, in this case, the method of second differences can be applied for revealing the nature of the variability of a deposit and assessing the degree of its exploration. We think that this question should be answered in the affirmative.

We shall demonstrate the validity of this opinion first of all with respect to the possibility of revealing the nature of the regularity of the deposit element under study. To reveal the nature of the regularity means to determine whether the change in the deposit element of interest to us proceeds according to the law of a straight line or the law of a curve. As is known, this problem is solved

is very simply solved graphically. Analytically it can easily be solved by the method of reduced differences.

Reduction of differences to a single interval may be carried out by the formula

\[ \Delta_i'=\frac{\Delta_i}{L_i}\cdot L_0, \tag{1} \]

where \(\Delta_i'\) is the first difference reduced to an interval equal to \(L_0\);
\(\Delta_i\) is the first difference corresponding to the interval \(L_i\).

If the reduced differences calculated by formula (1) are equal to one another, this will indicate the rectilinear character of the variation of the element of the deposit under study; otherwise the regularity will be curvilinear. In solving the problem posed, \(L_0\) may be chosen arbitrarily.

The example given in Table 1 illustrates the application of the method of reduced differences for revealing the character of the regularity of variation of the element of the deposit under study.

Table 1

The fact that the reduced differences in this example proved to be equal to one another, and that the second differences calculated from the reduced differences turned into zeros, indicates the rectilinear character of the change in thickness in the case under consideration.

When deciding on the possibility of applying the method of reduced differences to the assessment of the degree of exploration of a deposit with a variable sampling interval, it should be borne in mind that reducing the first differences to a single sampling interval is carried out by the method of rectilinear interpolation, which is equivalent to admitting the existence of a rectilinear regularity between each pair of sampling points. Such an assumption is unavoidable, since the actual character of the variation of the property under study between each pair of sampling points always remains unknown to us. The reduced differences, therefore, are quantities that are to a considerable extent theoretical, making it possible to pass from an exploration curve in a given direction, with a nonuniform arrangement of sampling points, to a reduced curve with a uniform arrangement of the latter.

Bearing in mind that the reduced curve preserves between corresponding points the rectilinear regularity of the explored curve, we may speak of the similarity of these curves (see Fig. 1) and, on this basis, in resolving the question of the degree of exploration of the deposit

the explored curve can, with a certain degree of reliability, be replaced by the reduced curve. The distortion from such a replacement in the assessment of the degree of exploration of a deposit will be the smaller, the smaller the fluctuations of the sampling interval are and the more regular the change in the property under study is.

When calculating reduced differences by formula (1), the question inevitably arises as to what should be taken as \(L_0\). As \(L_0\) one should always take the sampling interval that is dominant in the given case, or the mean value of the interval for the explored area.

Fig. 1.

When choosing \(L_0\), it must be remembered that this quantity should characterize the density of sampling of the explored area; that an artificial decrease in this quantity, just like an artificial increase of it, will lead to distortion of the index of the degree of study of the deposit. This becomes clear if one takes into account that a decrease in \(L_0\) is associated with a decrease in the sum of second differences, while an increase in \(L_0\) always leads to an increase in this sum. The sum of second differences, as is known, is inversely related to the coefficient of study of the deposit \(R\) and directly related to the error of analogy \(M\).

If the first differences calculated from the direct data are actual differences, then the reduced differences are only probable values corresponding to the interval \(L_0\).

In determining the weights of the reduced differences, it is necessary to bear in mind that their degree of reliability will be the smaller, the greater the magnitude of the interpolation (extrapolation). On this basis, we consider that the weights of the reduced differences should be taken as quantities inverse to the absolute differences between the actual and the reduced sampling intervals.

With such a choice of weights, the weights of the reduced differences should be taken equal to unity in all cases where the indicated differences between the sampling intervals lie within the range from zero to unity.

Although this remark points to a certain imperfection in the method of determining the weights of reduced differences, it nevertheless leads only to strengthening the influence of those samples whose sampling interval is close to the reduced one, which is entirely logical; moreover, this remark is consistent with the accuracy with which the sampling interval is actually determined, especially in exploration by drilling.

In accordance with the value of the weights of the reduced differences adopted by us, the weights of the second differences are found from the formula

\[ P_i=\frac{A}{\varepsilon_i+\varepsilon_{i+1}}, \tag{2} \]

where \(A\) is an arbitrary constant quantity,

\(\varepsilon_i,\ \varepsilon_{i+1}\) are quantities determined from the expressions

\[ \begin{aligned} \varepsilon_i &= |L_0-L_i|,\\ \varepsilon_{i+1} &= |L_0-L_{i+1}|. \end{aligned} \tag{3} \]

where \(L_i\) and \(L_{i+1}\) are the actual values of adjacent sampling intervals.

When computing the weights, \(\varepsilon\) is taken equal to one in all cases when it lies within the range from 0 to 1.

In the case when the reduced sampling interval cannot be taken as common along and across the exploration grid, the absolute variability index \(I_a\) should be calculated as the weighted mean of these indices along and across the exploration grid according to the formula

\[ I_a=\frac{\sum P_1\Delta_1''+\sum P_2\Delta_2''}{\sum P_1+\sum P_2}, \tag{4} \]

where \(\sum P_1\) and \(\sum P_2\) are the sums of the weights of the second differences along and across the exploration grid;

\(\sum P_1\Delta_1''\) and \(\sum P_2\Delta_2''\) are the sums of the products of the weights by the second differences along and across the exploration grid.

Formula (4) does not take into account the magnitudes of the sampling intervals along and across the exploration grid. Taking this into account, the absolute variability index \(I_a\) should be calculated by the formula

\[ I_a=\frac{I_{a1}L_{02}+I_{a2}L_{01}}{L_{01}+L_{02}}, \tag{5} \]

where \(I_{a1}, I_{a2}\) are the absolute variability indices of the property under study along and across the exploration line;

\(L_{01}, L_{02}\) are the reduced sampling intervals along and across the exploration grid.

Formula (5) should evidently be recommended when there is a large difference between the sampling intervals along and across the exploration grid, but only in the case when the sampling intervals were selected without taking into account the nature of the variation of the deposit property under study in these directions.

The degree of exploration of the deposit \(R\) is determined from the known expression

\[ R=\frac{n}{1000\cdot I}, \tag{6} \]

where \(n\) is the number of sampling points,

\(I\) is the relative variability index of the element of the deposit under study, equal to

\[ I=\frac{I_a}{I_0}, \]

where \(I_0\) is the mean value of the deposit index under study.

In order to illustrate the determination of the degree-of-exploration index of a deposit and the error of analogy, using the reduced differences, we shall process the results of measurements of thickness along a specified direction (Fig. 2 and Table 2).

The value of the mean thickness \(m_0\) and the error of analogy \(M\), for several equal intervals \(L\) of thickness measurement in the given example, are equal to

\[ \begin{aligned} L&=5\ \text{m} & m_0&=5.54\ \text{m} & M&=2\%,\\ L&=10\ \text{m} & m_0&=5.53\ \text{m} & M&=10\%,\\ L&=15\ \text{m} & m_0&=5.64\ \text{m} & M&=13\%,\\ L&=20\ \text{m} & m_0&=5.40\ \text{m} & M&=13\%\ (R=0.012). \end{aligned} \]

Table 2

\[ L_n = 10\ \text{m} \]

\[ m_0 = 5.34\ \text{m} \]

\[ M = \pm 12\% \]

\[ L_n = 15\ \text{m} \]

\[ m_0 = 5.58\ \text{m} \]

\[ M = \pm 13\% \]

\[ L_n = 20\ \text{m} \]

\[ m_0 = 5.55\ \text{m} \]

\[ M = \pm 13\%\quad (R^2 = 0.015) \]

\(A\)—in calculating the weights was taken equal to 6.

It follows from Table 2 that the divided-differences method gives results close to those which we obtained with a constant sampling interval.

Fig. 2.

Thus, the proposed divided-differences method, based on rectilinear interpolation between sampling points, significantly expands the range of application of the second-differences method and makes it possible in most cases to determine the error of the analogue in two ways, i.e., with a check.

REFERENCES

1. Kazakovskii, D. A., Assessment of the accuracy of results in connection with the geometrization and calculation of reserves of deposits. Ugletekhizdat, 1948.