ON V. V. BOGATSKY’S MATHEMATICAL METHOD FOR DETERMINING THE DENSITY OF AN EXPLORATION NETWORK

L. M. PETROVSKY

(Presented by the scientific seminar of the departments of mineralogy, petrography and geology and exploration of mineral deposits)

One of the mathematical methods currently considered progressive for determining the density of an exploration network in the exploration of various mineral deposits is V. V. Bogatsky’s statistical-analytical method [1], which proceeds from the probabilistic calculation, from a sample exploration series, of the limiting error of analogy (interpolation). The latter, differing for the cases of the simplest (unimodal continuous function) and maximum (piecewise-continuous function) geological variability, is determined respectively [1] by formulas (1) and (2):

\[ \Delta_{\mathrm{pr}}^{\min} = \pm \frac{4(P_{\max}-P_{\min})}{\bar{P}(N-1)^2}, \tag{1} \]

\[ \Delta_{\mathrm{pr}} = \pm \frac{2(P_{\max}-\bar{P})}{\bar{P}(N-1)}, \tag{2} \]

where

\[ \Delta_{\mathrm{pr}}^{\min} \]

is the limiting error of analogy of continuous unimodal functions (the minimum value of the limiting error);

\[ \Delta_{\mathrm{pr}} \]

is the limiting or maximum possible error;

\[ P_{\max} \text{ and } P_{\min} \]

are the maximum and minimum values of the variable quantity;

\[ \bar{P} \]

is the mean value of the variable quantity, calculated on the basis of observations;

\[ N \]

is the total number of uniformly distributed observations.

With the same number of observations, \(\Delta_{\mathrm{pr}}\) must always be greater than \(\Delta_{\mathrm{pr}}^{\min}\), or, from another point of view, with the same error of analogy the required number of observations in an area with maximum geological variability must be greater than the required number of observations in an area with minimum geological variability.

To check the actual relationship between the values of \(\Delta_{\mathrm{pr}}^{\min}\) and \(\Delta_{\mathrm{pr}}\), we carried out calculations of them for the parameters of coal seams of the Baidaevsky and Leninsky districts of Kuzbass and of the Chernigov deposit (according to the data of V. V. Bogatsky [1]). Study of the analogy errors calculated by the formulas given showed that when

\[ \frac{P_{\max}}{\bar{P}} > 2\Delta_{\mathrm{pr}}, \]

act—

respectively, always greater than $\Delta_{\mathrm{pr}}^{\min}$ for any number of observations $N$. If

\[ \frac{P_{\max}}{\overline{P}}<2, \]

then $\Delta_{\mathrm{pr}}$ remains greater than $\Delta_{\mathrm{pr}}^{\min}$ only for a sufficiently large value of $N$ (more than $4^*—16$); at a small value of $N$, often sufficient for studying the attribute in areas with its minimum variability, $\Delta_{\mathrm{pr}}$ proves to be less than $\Delta_{\mathrm{pr}}^{\min}$. The excess of $\Delta_{\mathrm{pr}}^{\min}$ over $\Delta_{\mathrm{pr}}$ becomes the greater, the closer the ratio

\[ \frac{P_{\max}}{\overline{P}} \]

is to unity. Obviously, in this case the value of $N$, calculated for one and the same specified analogy error by formula (1) for cases with minimum variability of the attribute, will be greater than the value of $N$ determined by formula (2) for cases with maximum variability of the attribute. Let us specify the limits of existence of the inconsistency characterized above.

Expressing the analogy errors by formulas (1), (2) and carrying out reductions in both parts of the inequality, we obtain

\[ (N-1)(P_{\max}-\overline{P})>2(P_{\max}-P_{\min}) \quad \text{or} \]

\[ (N-1)(P_{\max}-\overline{P})-2(P_{\max}-P_{\min})>0. \]

As is known, for cases of the simplest geological variability of an attribute there is a comparatively insignificant difference between $P_{\max}$ and $\overline{P}$ with a relatively high (nonzero) stable minimum value $P_{\min}$. In the general case, under these conditions the fluctuation of $\overline{P}$ may be expressed by limits from $0.5\,P_{\max}$ to $0.9\,P_{\max}$, and the stable value $P_{\min}$ may be taken equal to $0.25\,P_{\max}$. Then the inequality, preliminarily transformed into the form $(N-1)(P_{\max}-\overline{P})-1.5\,P_{\max}>0$, may accordingly have an expression within the limits from

\[ (N-1)0.5P_{\max}-1.5P_{\max}>0 \quad \text{or} \quad P_{\max}(N-4)>0 \]

to

\[ (N-1)0.1P_{\max}-1.5P_{\max}>0 \quad \text{or} \quad P_{\max}(N-16)>0. \]

Consequently, the inequality condition for these expressions (with a positive value of their left-hand side) will be preserved only when the number of observations $N$ is more than $4—16$. With a smaller number of observations, the subtrahend becomes greater than the minuend ($N$), i.e., $\Delta_{\mathrm{pr}}^{\min}$ exceeds $\Delta_{\mathrm{pr}}$ in absolute value.

It is known that as the number of observations $N$ in areas increases, beginning with some sufficiently large totality of them, the limiting analogy error practically ceases to decrease, stabilizing at an infinitesimally small value (earlier for objects with the simplest geological variability). The difference between the relative values $\Delta_{\mathrm{pr}}$ and $\Delta_{\mathrm{pr}}^{\min}$, or the ratio $\Delta_{\mathrm{pr}}/\Delta_{\mathrm{pr}}^{\min}$, with a regular decrease in the values of each of them, should also decrease. The calculations of the values $\Delta_{\mathrm{pr}}$ and $\Delta_{\mathrm{pr}}^{\min}$ by formulas (1) and (2) give a different result: with a continuous increase in the number of observations $N$, the ratio $\Delta_{\mathrm{pr}}/\Delta_{\mathrm{pr}}^{\min}$ grows uncontrollably because of the different structure of the formulas.

Formula (1), according to V. V. Bogatsky’s expression, may be used for calculating errors associated with mine-survey measurements of the volumes of stripping operations, dumps (waste heaps), extracted spaces, and in a number of other cases. In practice, therefore, for esti—

* The guaranteed critical number of observations $N$ is determined [1] by the expression

\[ N=2\frac{P_{\max}}{\overline{P}}-1. \]

When

\[ \frac{P_{\max}}{\overline{P}}\leq 2, \]

a sample of 2–3 observations may already be guaranteed.

...of the admissible error of analogy in studying changes in the characteristic of all natural geological formations, a single formula (2) is recommended. We consider it possible to approach the question under consideration in a different way.

The difference in the behavior of piecewise-continuous (maximum variability) and unimodal (the simplest variability) functions should be determined both by the ratio of the values of neighboring or maximally distant measurements of the characteristic and, to a certain extent, by the number of observations in each sample. The variability of a characteristic along the curve of a unimodal continuous function, with the number of observations in a line equal to \(n\), may be characterized by the mean first difference \(\left(\dfrac{P_{\max}-P_{\min}}{n-1}\right)\) or, in the general case, by any partial first difference \((P_n-P_{n-1})\). Obviously, both the mean and any partial first differences on a segment of a unimodal function cannot be greater than the difference between the maximum and the mean values of the characteristic. Then the value of the expression \((P_{\max}-\overline{P})\) may be regarded as the limiting discrepancy between the calculated mean value of the characteristic \(\overline{P}\) and the true mean.

The limiting variability of a characteristic along the curve of a piecewise-continuous or multimodal function can be characterized only by the maximum first difference \((P_{\max}-P_{\min})\).

The range of possible absolute interpolation errors of the characteristic for unimodal and piecewise-continuous functions, respectively, should be equal to \(\pm 2(P_{\max}-\overline{P})\) and \(\pm 2(P_{\max}-P_{\min})\), and the range of possible relative errors should be

\[ \pm \frac{2(P_{\max}-\overline{P})}{\overline{P}} \quad \text{and} \quad \pm \frac{2(P_{\max}-P_{\min})}{\overline{P}} . \]

To estimate a concrete value of the limiting error of analogy, it is necessary to reduce the range of possible relative errors in proportion to the number of interpolation elements attained at each stage of investigation of the area. In the general case, the number of interpolation elements in areas with different variability of the characteristic may be determined by the expression

\[ (n-1)(m-1), \]

where

\(n\)—the number of observations (intersections) in the direction of the greatest variability of the characteristic (usually along the dip–rise line of ore bodies);

\(m\)—the number of observations (intersections) in the direction of the least variability of the characteristic (usually along the strike of ore bodies).

The practice of exploration of various mineral deposits shows that the ratio \(m:n\) for mutually perpendicular lines of equal extent may vary from \(1:1\) (rarely \(1.5:1\)) to \(1:3\) and more. Expressing, in each specific case, the value of \(m\) through \(n\), one may represent the number of interpolation elements in areas as \((n-1)^2\), \((n-1)\left(\dfrac{n-2}{2}\right)\), \((n-1)\left(\dfrac{n-3}{3}\right)\), etc., and the concrete values of the limiting interpolation error as

\[ \Delta_{\mathrm{pr}}=\pm \frac{2(P_{\max}-P_{\min})}{\overline{P}(n-1)^2}, \]

\[ \Delta_{\mathrm{pr}}=\pm \frac{4(P_{\max}-P_{\min})}{\overline{P}(n-1)(n-2)}, \qquad \Delta_{\mathrm{pr}}=\pm \frac{6(P_{\max}-P_{\min})}{\overline{P}(n-1)(n-3)} \quad \text{etc.;} \tag{3} \]

\[ \Delta_{\mathrm{pr}}^{\min}=\pm \frac{2(P_{\max}-\overline{P})}{\overline{P}(n-1)^2},\quad \Delta_{\mathrm{pr}}^{\min}=\pm \frac{4(P_{\max}-\overline{P})}{\overline{P}(n-1)(n-2)}, \]

\[ \Delta_{\mathrm{pr}}^{\min}=\pm \frac{6(P_{\max}-\overline{P})}{\overline{P}(n-1)(n-3)} \quad \text{and so on.} \tag{4} \]

From expressions (3) and (4) it is clear that the magnitude of the interpolation error depends not only on the absolute variability of the attribute (the ratio \(P_{\max}:\overline{P}\)), but also to a considerable extent on the degree of anisotropy of the exploration object: with increasing anisotropy (the ratio \(n:m\)), the magnitude of the interpolation error, all other conditions being equal and for a given \(n\), increases; or, in order to achieve the same interpolation error at an object with greater anisotropy, a larger number of intersections \(n\) is required.

Denoting any given number of interpolation elements (unit exploration blocks) in a sector by \(N\), we obtain general expressions for the limiting errors of the analogy for the cases of piecewise-discontinuous \((3^1)\) and single-maximum \((4^1)\) functions:

\[ \Delta_{\mathrm{pr}}=\pm \frac{2(P_{\max}-P_{\min})}{\overline{P}N} \quad \text{and} \tag{3^1} \]

\[ \Delta_{\mathrm{pr}}^{\min}=\pm \frac{2(P_{\max}-\overline{P})}{\overline{P}N}, \tag{4^1} \]

where \(N\) may be equal to \((n-1)^2\), \((n-1)\left(\frac{n-2}{2}\right)\), \((n-1)\left(\frac{n-3}{3}\right)\), etc.

Formulas \((3^1)\) and \((4^1)\) are simple, free of the shortcomings considered above, and, most importantly, are characterized by quite definite conditions for their application. For evaluating exploration results at deposits with simple variability of the attribute, which apparently can be characterized by values of the ratio \(P_{\max}:\overline{P}\) up to 4–6 (coal deposits, most sedimentary ore deposits, etc.), formula \((4^1)\) should be used. At deposits with strong variability of the attribute, which can be characterized by values of the ratio \(P_{\max}:\overline{P}\) of 6 and more (most vein, skarn, and pegmatite deposits), formula \((3^1)\) should be used.

The calculation of the necessary and sufficient final number of unit exploration blocks, as well as of the density of the exploration network itself, in sectors of different area must be carried out differentially with respect to the latter. One of the methods for such a calculation for coal deposits was recommended by us [2].

References

1. V. V. Bogatsky. Mathematical Analysis of an Exploration Network. Gosgeoltekhizdat, 1963.

2. L. M. Petrovsky. On determining the optimal density of an exploration network at coal deposits. Proceedings of Higher Educational Institutions, Geology and Exploration, No. 10, 1965.