CORRESPONDENCE OF THE DENSITY OF EXPLORATION POINTS TO THE NATURE OF REGULAR VARIATIONS IN A PARAMETER¹ AS THE MAIN FACTOR IN THE NUMERICAL ASSESSMENT OF THE ACCURACY OF MINERAL DEPOSIT EXPLORATION

L. M. PETROVSKII

(Presented by the scientific seminar of the Department of Geology and Exploration of Mineral Deposits)

The diversity and complexity of the variability of parameters of mineral bodies, as is known, are determined by: 1) the nature and intensity of the manifestation of regular (coordinated) variations, and 2) the degree of manifestation of random variations.

A quantitative assessment of the variability of a parameter may be based on both regular [14] and random [4, 5, 8, 13, 16] variations; however, in any case, the principal and indispensable condition for assessing variability must be a sufficiently complete determination of the nature of the regular variations of the parameter. To calculate numerical indicators of random or regular variations of a parameter, it is also necessary to separate the random variations sufficiently completely from the regular ones.

Despite the exceptionally great applied significance of questions connected with the quantitative assessment of parameter variability, their study still remains insufficient. Thus, although there are several special investigations [2, 4, 7, 8, 11, 14, 16] concerning the numerical assessment of random variations of a parameter and their separation from regular ones, the main question—the criteria for the completeness of determining, in the course of exploration, the nature of regular variations of a parameter—has practically not been studied. It is therefore no accident that very often, in the course of exploration, the question remains unclear: at what density (beginning with what minimum density) of exploration points can one obtain a sufficiently objective assessment of the degree of random variations of a parameter from the magnitude, for example, of the standard deviation \(S\) of smoothed values of the parameter [4], or of the nature of its regular variations on the basis of constructed regression curves. In this connection, below an attempt is made to solve the problem of establishing, in the course of exploration, a sufficient correspondence between the density of exploration points and the nature of the deposit being explored.

A numerical assessment of the error in calculating the mean value of a parameter within the contour of a given category of reserves is based on the primary role of the number of observations made [2, 4, 5, 6, 7, 13, 14]. In this case, the probabilistic-statistical error in calculating the mean is determined by the number only of more or less uniformly distributed individual exploration points, forming a quite definite number

of elementary interpolation elements and elementary nodal exploration blocks [2, 7, 9]. Depending on the specific number \(N\) of such elementary exploration blocks and on the magnitude of the specified maximum permissible error \(\Delta_{\mathrm{pr}}\) in calculating the general mean or the total reserve of the given category, the maximum permissible error in calculating the mean or determining the reserve of the same category in an elementary block, \(\Delta\), is estimated from the expression known from probability theory and mathematical statistics [2]:

\[ \Delta_{\mathrm{pr}}^{\mathrm{bl}}=\frac{\Delta_{\mathrm{pr}}}{3}\sqrt{N}. \tag{1} \]

It is obvious that, as the number of elementary exploration blocks \(N\) increases or decreases, the magnitude of the permissible error in calculating the mean in an elementary block \(\Delta_{\mathrm{pr}}^{\mathrm{bl}}\), for a specified total permissible error \(\Delta_{\mathrm{pr}}\), may accordingly be increased or decreased (the required accuracy of determining the reserve in a block will accordingly decrease or increase). A consequence of this, as is known, is the need to regulate the optimally sufficient actual accuracy of determining mean values of the attribute in an elementary exploration block by changing the distance between workings forming the elementary block in the dip and strike directions for mineral bodies of different sizes: the necessary reduction of the distance in exploration of small bodies and the possible increase of it in exploration of large bodies. In doing so, it must be borne in mind that the increase in the detail of study from the beginning of exploration to its completion occurs not only in proportion to the necessary increase in the accuracy of the estimate of reserves as a whole, but also in connection with the transition to a smaller study area, owing to the so-called regression of reserve categories [1].

Expressing the total number of elementary nodal exploration blocks \(N\), formed by boreholes (shafts), as the product of the mean number \(n\) of interpolation elements in the dip direction and the mean number \(m\) of interpolation elements in the strike direction of the mineral body, we present formula (1) in the form

\[ \Delta_{\mathrm{pr}}^{\mathrm{bl}}=\frac{\Delta_{\mathrm{pr}}}{3}\sqrt{n\cdot m}. \tag{2} \]

Then the permissible error in determining the mean value of the attribute in an exploration profile in the dip direction \(\Delta_{\mathrm{pr}}^{n}\) and in the strike direction \(\Delta_{\mathrm{pr}}^{m}\), in the case of exploration by boreholes, will be determined as

\[ \Delta_{\mathrm{pr}}^{n}=\frac{\Delta_{\mathrm{pr}}^{\mathrm{bl}}}{\sqrt{m}} =\frac{\Delta_{\mathrm{pr}}}{3}\sqrt{n},\qquad \Delta_{\mathrm{pr}}^{m}=\frac{\Delta_{\mathrm{pr}}^{\mathrm{bl}}}{\sqrt{n}} =\frac{\Delta_{\mathrm{pr}}}{3}\sqrt{m}. \tag{3} \]

In the case of exploration of a deposit by underground mine workings, i.e., when there are mining-exploration calculation blocks, calculation of the permissible error in determining the mean value of the attribute in a mine working becomes more complicated. The point is that, in exploration by mining systems, the mean value of the attribute within the contour of one or another reserve category is a mean of the third and higher order. Denoting by \(\Delta_{\mathrm{pr}}\) the magnitude of the specified maximum permissible error in calculating the general mean of the third order, in accordance with formula (2) one can determine the permissible error of the mean of the second order, i.e., the permissible error in determining the mean value of the attribute in an individual mining-exploration block \(\Delta_{\mathrm{pr}}^{\mathrm{bl1}}\), for the number of such blocks \(N\), as

\[ \Delta_{\mathrm{pr}}^{\mathrm{bl1}}=\frac{\Delta_{\mathrm{pr}}}{3}\sqrt{N}. \]

In turn, the permissible error in determining the value of the attribute in a conditional unit sampling block \(\Delta_{\mathrm{pr}}^{\mathrm{bl}}\) may be determined as

\[ \Delta_{\mathrm{pr}}^{\mathrm{bl}_0}=\frac{\Delta_{\mathrm{pr}}^{\mathrm{bl}_1}}{3}\sqrt{n\cdot m}, \]

where \(m\) is the number of unit interpolation elements on the section of the mine working that bounds the mining-and-exploration block along the strike of the mineral body;

\(n\) is the same along the dip of the mineral body.

Then the permissible error in determining the mean value of the attribute on the section of the mine working that bounds the mining-and-exploration block in the dip direction, \(\Delta_{\mathrm{pr}}^{n}\), and in the strike direction, \(\Delta_{\mathrm{pr}}^{m}\), in the case of exploration by systems of mine workings will be determined as

\[ \Delta_{\mathrm{pr}}^{n}=\frac{\Delta_{\mathrm{pr}}\sqrt{N}}{9}\sqrt{n}, \qquad \Delta_{\mathrm{pr}}^{m}=\frac{\Delta_{\mathrm{pr}}\sqrt{N}}{9}\sqrt{m}. \tag{4} \]

The necessity of a differentiated assessment of the accuracy of exploration of mineral deposits has been emphasized repeatedly. The most successful attempt in this direction, in our opinion, was made by A. I. Ossetsky [7] with respect to bedded deposits. However, A. I. Ossetsky’s method has not become widely used, primarily because it did not substantiate the choice of the permissible magnitude of the error in calculating the mean in a unit exploration block (the permissible magnitude of the “index of correspondence of the achieved density of exploration points to the character of the deposit being explored”). In addition, the value of the “index of correspondence of the achieved density of exploration points to the character of the deposit being explored” (the value of the “red number”) as determined by A. I. Ossetsky gives practically no idea of the degree of such correspondence from the standpoint of regular variations in the attribute. Finally, it is impossible to assess the degree of correspondence of the achieved density of exploration points to the character of the deposit being explored by means of any single general “correspondence index,” just as this cannot be done with respect to assessing the variability of an attribute: repeated attempts by researchers in this direction, as is known [8], have not been successful.

At present, practically the only² method of control, in the course of exploration, of the correspondence of the density of exploration points to the character of the deposit being explored is the establishment, for each category of reserves, of an analytical value for the actual randomness of the error \(\Delta(\Delta^{\mathrm{bl}}, \Delta^{n}, \Delta^{m})\), i.e., the fulfillment of the well-known requirement

\[ \Delta \leq \Delta_{\mathrm{pr}}, \qquad \Delta^{n} \leq \Delta_{\mathrm{pr}}^{n}, \qquad \Delta^{\mathrm{bl}} \leq \Delta_{\mathrm{pr}}^{\mathrm{bl}}, \qquad \Delta \leq \Delta_{\mathrm{pr}}^{m} \tag{5} \]

Accordingly, it is assumed that the assessment of the variability of an attribute can likewise be carried out on the basis of taking into account only random variations [4, 5]. However, the single control requirement (5), for revealing in the course of exploration the character of variability of an attribute, usually proves sufficient only in the exploration of the simplest objects, in which the character of regular variations, according to [14], will be determined by horizontal or inclined lines. In most other cases, including under conditions of obtaining category \(A\) reserves \((\Delta_{\mathrm{pr}}=10\text{—}15\%)\), fulfillment of this requirement alone does not guarantee the corresponding revelation of the character of regular variations and, consequently, does not give a correct idea of the variability of the attribute as a whole.

² Unless one counts D. A. Zenkov’s method for assessing the accuracy of geological sections [3], which does not provide for a numerical expression of the assessment.

The calculation of the absolute value of the actual random error \(\Delta^{n,m}\) of the sample mean, as is known, is carried out by the formula

\[ \Delta^n=\frac{s_n}{\sqrt{n}}, \qquad \Delta^m=\frac{s_m}{\sqrt{m}} . \tag{6} \]

The shortcomings associated with the use of the standard deviation \(\sigma\) (the coefficient of variation \(V\)) to characterize the random component of the variability of an attribute are well known [4, 5, 6, 8, 14]. In view of these shortcomings, the most rigorous numerical assessment of random changes in an attribute is possible with the aid of the root-mean-square deviation \(S\) of smoothed attribute values from the initial ones [4, 5, 14]. The study showed that the possibility and advisability of refining the numerical assessment of random changes in an attribute on the basis of the index \(S\) is conditioned by the presence of a sufficient number of complete sections of the mineral body in the main directions of its exploration (not fewer than 15 sections when no more than three smoothing operations are performed on the initial attribute values). In cases not satisfying this condition, a more or less accurate quantitative assessment of random changes in the attribute may be obtained either on the basis of the standard deviation \(\sigma\), or on the basis of the root-mean-square deviation \(H\) of the first successive differences of the attribute values.

L. I. Chetverikov [14] recommends using, as a refined (in relation to the standard \(\sigma\)) and also the simplest and most convenient in practice quantitative assessment of random changes in the attributes of mineral bodies, the value of the root-mean-square deviation \(H\) of the first successive differences. Indeed, the use of the index \(H\), which is more expedient than \(\sigma\), may occur under conditions in which a tendency toward increase (decrease) in the attribute values is manifested in the initial sample series [16], or where there is little change, on the average, between two successive observations of the attribute in it, in the absence of such a tendency [15]. From the standpoint of both of these conditions, preferential use of the index \(H\) for assessing random changes in an attribute is possible only for deposits of groups I and II [6].

It is important to emphasize, however, that the very concept “variability of an attribute of a deposit,” as was shown by E. I. Popov [11], is relative. “Its numerical expression is not constant; it is determined by the complexity of the attribute being studied and by the density of exploration points, i.e., it reflects the degree of our knowledge of the object under study.”

The measure of random changes in an attribute, characterized by the indices \(H\), \(S\), or \(\sigma\), determines, in accordance with formula (5), the so-called principal random component of the total random component of the error of analogy. Let us call this principal random component of the total random component of the error of analogy, known in the literature as the random error \(\Delta\) of the sample mean, the random error of the first kind \(\Delta_1\left(\Delta_1^{\mathrm{bl}}, \Delta_1^n\right);\ \Delta^{n,m}=\Delta^{n,m}\cdot\Delta_1\).

The second random component of the total random component of the error of analogy, in our opinion, should be connected with the random determination of the mean due to the random disposition of the system of exploration sections relative to the mineral body [12], i.e., it should be determined by the random character of the sample of attribute measurements [11]. This component of the total random component of the error of analogy, due to the random variant of the disposition of the system of exploration sections, we shall call the random error of the second kind \(\Delta_2\left(\Delta_2^{\mathrm{bl}}, \Delta_2^n, \Delta_2^m\right)\). An approximate estimate of the magnitude of the actual random error of the second kind may be car-

known from comparison of the variant quantities \(H\) or \(S\), calculated on a grid twice as sparse. In practice, it is always more convenient to determine the random error \(\Delta_2\) on the basis of a comparison of variant values of the index \(H\). For this purpose it is necessary to carry out a two-variant comparative assessment of the quantities \(H^{n/2}\) or \(H^{m/2}\), obtained from the data of a twice reduced number of measurements of the attribute in the original sample series. The root-mean-square intervariant increment of the first successive differences of the attribute values \(H_\Delta^{n/2}\), for example, reduced by \(\sqrt{n}\) times, will be the sought absolute value of the actual random error \(\Delta_2^n\)

\[ \Delta_2^n=\frac{H_\Delta^{n/2}}{\sqrt{n}} = \sqrt{ \frac{ \sum_{\max}(X_{i+1}-X_i)^2-\sum_{\min}(X_{i+1}-X_i)^2 }{n} }, \tag{7} \]

where \(\sum_{\max}(x_{i+1}-x_i)^2\) and \(\sum_{\min}(x_{i+1}-x_i)^2\) are the larger and the smaller sums of squares of the first successive differences of the attribute values for the twice thinned series.

The total actual random component of the error of analogy \(\Delta^{n,m}\) in determining the mean value of the attribute in the principal directions of exploration, apparently, may be determined by the sum of the corresponding random errors of the first and second kinds:

\[ \Delta^n=\Delta_1^n+\Delta_2^n,\quad \Delta^m=\Delta_1^m+\Delta_2^m . \tag{8} \]

In this case, those quantities \(\Delta_1^n\) and \(\Delta_2^n\), or, respectively, \(H^n\) and \(H_\Delta^{n/2}\), for example, must be taken into account which satisfy the conditions of sufficient correspondence between the attained density of intersection points in the exploration line and the character of the regular variation of the attribute in the given direction. The necessity of such a requirement in the case of an analytical determination of the error of analogy has been pointed out in a number of recent works [10].

Proceeding from the concept of the total actual random component of the error of analogy, condition (5), which restricts the actual random errors relative to the maximum permissible ones, should be rewritten in the following form:

\[ \Delta<\Delta_{\mathrm{pr}},\quad \Delta^n<\Delta_{\mathrm{pr}}^n,\quad \Delta^{\mathrm{bl}}<\Delta_{\mathrm{pr}}^{\mathrm{bl}},\quad \Delta^m<\Delta_{\mathrm{pr}}^m . \tag{9} \]

It is obvious that, for the same values of the maximum permissible errors of analogy, requirement (9), in contrast to requirement (5), makes it possible to control more fully, in the course of exploration, the identification of the character of the variability of the attribute; however, it too is insufficient for an unambiguous establishment of the actual correspondence between the attained density of points and the character of the regular changes of the attribute in exploration profiles.

The problem of unambiguously establishing the actual correspondence of the attained density of points to the character of the regular changes of the attribute in an exploration profile can be solved on the basis of revealing similarity between two conditionally parallel series obtained as a result of a twofold successive thinning of the measurements of the attribute in the original homogeneous sample population. From this point of view, of practical interest is the assessment of the closeness of the conditional correlation relation between adjacent paired values of the attribute in the original sample series. Since any of the fixed measurements of the attribute in the sample series may equally be considered in a correlated successive pair either with the right-hand or with the left-hand adjacent measurements, in accordance with these two possible variants it is necessary to calculate [15] two conditional

correlation coefficient \(r\) and two conditional correlation ratios \(\eta\).

Obviously, when a sufficiently complete correspondence is achieved between the density of intersections and the character of the regular variations of the parameter in the original sample series, then in each thinned sample there should be observed a regular change in the magnitude of successive differences and a regular alternation of their signs. In this case the conditional correlation relationship, by its character, must be strictly direct (positive) and linear. The presence of such a relationship may be considered established if one of the two coefficients \(r\) calculated for each exploration profile proves to be \(>+0.5\) (the other coefficient \(r\) may be \(<0.0\)). Depending on the specific number of measurements of the parameter \(\frac{n+1}{2}\) in the twice-thinned series, the minimum significant values of \(|r|\) should be as follows (Table 1).

Table 1

It should be noted that, with a simple character of regular variations of the parameter (of the inclined-line type [14]), characteristic of deposits of Groups I and II [5], obtaining significant values of the coefficient \(r\) close to unity is easily attainable with a limited number of borehole intersections of the mineral body in an exploration profile, i.e., already from the data of the preliminary stage of drilling exploration. At objects with a complex character of regular variations of the parameter (of the winding and strongly winding line type [14]), characteristic of deposits of Groups III and IV [5], obtaining significant values of the conditional coefficient \(r\) requires a considerably larger number of measurements of the parameter. Usually such a large number of measurements of the parameter along several separate exploration profiles, within the contour of one reserve category, can be obtained only with detailed sampling of preparatory-development mine workings. In addition, because of the mechanism of successive thinning in series with winding and strongly winding lines of regular variations of the parameter, a negative correlation relationship may be observed in the bends of the line and, as a consequence, a curvilinear character of the conditional correlation dependence of the parameter values in the original sample series. Therefore, the assessment of sufficient correspondence between the sampling density achieved in the course of exploration and the revealed character of regular variations of the parameter at deposits of Groups III and IV should be made on the basis of calculating the conditional correlation ratio \(\eta\). The presence of a sufficiently close positive correlation relationship between homogeneous values of the parameter in the original sample series may be considered established if both \(\eta_{xy}\) and \(\eta_{yx}\), for one of the two calculated variants (according to variant \(\eta'\) or \(\eta''\)), prove to be \(>0.5\) (the corresponding \(r'\) or \(r''\) may be \(<0.0\)).

Under conditions in which the character of the manifestation of regular variations of the parameter has been revealed with sufficiently high probability (from the data of at least two or three exploration profiles of the same direction), the first objective numerical assessment of its random variations becomes possible. The indicator of the first objective numerical assessment of random variations of the parameter, first, may be used as the basis for calculating the analytical magnitude of the total actual random compos-

...the error of the analogy \(\Delta_{n,m}\) in the principal directions of exploration. After satisfying requirement (8) with respect to the precalculated values \(\Delta_{\mathrm{pr}}^{\,n,m}(C_1)\) or \(\Delta_{\mathrm{pr}}^{\,n,m}(B)\), the correspondence between the achieved density of exploration points and the identified character of the deposit being explored, as applied to the conditions for obtaining reserves of categories \(C_1\) or \(B\), may be considered established. Secondly, the indicator of the first objective numerical assessment of random changes in the attribute may be used as the basis for calculations in the analytical determination of the density of exploration points required for detailed drilling exploration of deposits of groups I and II. Moreover, under conditions of detailed exploration of these deposits, for carrying out subsequent stages of checking the correspondence of the achieved density of drill holes to the character of variability of the attribute, as a rule, one requirement (8) proves sufficient. On the basis of the experience of exploration work [5] and theoretical considerations [2], the mean values of the permissible error \(\Delta_{\mathrm{pr}}\) in determining the total reserve (the overall mean of the most variable reserve-estimation parameter) are set at the following levels: for category \(A\), 15%; for category \(B\), 25%; and for \(C_1\), 40–50%.

Let us consider the practical application of the proposed method using the example of a sample borrowed from [4, 5]^3. According to analyses of 28 samples taken in a mine working at intervals of 2 m, the following sequence of the distribution of gold contents was established (Table 2). The indicators of random changes in content, \(\sigma\), \(H\), calculated from the data of the original and thinned series of samples, are given in Table 3. According to the results of the third smoothing procedure applied to the original content values by the method of successive weighting [16], a strongly sinuous line of regular changes was established (according to [14]), with an estimate of random changes \(S=22.5\).

Table 2

Successive distributions of gold content in the original and twice-thinned sample series

To resolve the question of whether the achieved density of samples sufficiently corresponds to the character of the regular changes in gold content, we shall perform a correlation calculation to assess the closeness of the relationship between two conventionally parallel thinned series of samples (Table 3). As can be seen from Table 3, none of the calculated correlation coefficients \(r\) indicates the presence of the sought close positive relationship, as should be expected in the case of a strongly sinuous line of regular changes in content. The calculated values of the correlation ratio \(\eta_{yx}\) and \(\eta_{xy}\), on the contrary, confirm the presence of a sufficiently close sought positive correlation relationship. On this basis it may be concluded that the adopted density of samples in the exploration mine working corresponds to the identified character of the regular changes in content. As an objective measure of random changes in content here one may adopt the estimate \(S=22.5\).

^3 Here the original data were used, taken from the graph and rounded to integer values.

Table 3

Numerical estimates of random variations in gold content
and indicators of correlation calculation

LITERATURE

1. Bryukov V. I. Regression of reserve categories. Izv. vuzov.—Geologiya i razvedka, No. 4, 1965.

2. Bogatskii V. V. Mathematical Analysis of the Exploration Grid. Gosgeoltekhizdat, 1963.

3. Zenkov D. A. Methods for determining the density of an exploration grid. Sov. geol., Coll. 61, 1957.

4. Kallistov P. L. Variability of mineralization and density of observations in exploration and sampling. Sov. geol., Coll. 56, 1956.

5. Kreiter V. M. Prospecting and Exploration of Mineral Deposits. Part 2. Gosgeoltekhizdat, 1961.

6. Kuzmin V. I. Geometrization and Calculation of Reserves of Deposits of Solid Minerals. Nedra, Moscow, 1967.

7. Osetskii A. I. Indicator of the correspondence of the density of exploration points to the character of the explored deposit. Issled. po vopr. marksheid. dela. Izv. VNIMI, vol. 30, 1956.

8. Petrov A. A. Application of mathematical statistics for solving fundamental problems of deposit exploration. Sov. geol., No. 9, 1963.

9. Petrovskii L. M. On determining the optimal density of the exploration grid at reference deposits. Izv. vuzov.—Geologiya i razv., No. 10, 1965.

10. Pogrebitskii E. O. et al. Prospecting and Exploration of Mineral Deposits. Nedra, Moscow, 1968.

11. Popov E. I. On assessing the accuracy of representing a mineral deposit from exploration data. Zap. LGI, vol. 36, issue 2, 1959.

12. Sergeev O. P. Analysis of the exploration grid by the method of comparing variants. Razvedka i okhrana nedr, No. 11, 1960.

13. Smirnov V. I. Geological Foundations of Prospecting and Exploration of Ore Deposits. Moscow State University Press, 1957.

14. Chetverikov L. I. Quantitative assessment of the variability of mineral content observed from sampling data. Izv. vuzov.—Geol. i razv., No. 3, 1968.

15. Hald A. Mathematical Statistics with Technical Applications. IL, Moscow, 1956.

16. Yastremskii B. S. Some Questions of Mathematical Statistics. Gosstatizdat, 1961.

1. By a parameter of a mineral body, here and below, are understood the thickness, content, or volumetric weight of the mineral. ↩