On the Issue of Assessing the Accuracy of Geological Exploration Results by Their Stages

L. M. Petrovsky

(Submitted by the Department of Geology and Exploration of Mineral Deposits)

Assessment of the accuracy (error) in determining mineral reserves or in calculating the mean values of estimated parameters, as is known, must be based on a classification of the errors that arise during exploration. The currently widespread division of errors [4, 5] into technical errors and errors of analogy (geological errors) does not meet the increased demands for assessing the accuracy of exploration results at its various stages.

Geological errors, which are the principal ones in their significance, depending on the degree of detail of exploration of a deposit, successively appear as errors of analogy and of interpolation (Table). The need for such a division of geological errors is explained not only by the difference in the initial conditions for estimating the mean values of estimated parameters, but also by the practically different methodological approach to estimating their magnitude.

The initial data for assessing the geological error in determining the mean values of a parameter at the prospecting-and-exploration stage and

Table

Classification of geological errors in determining mean values of estimated parameters

the preliminary exploration stage is usually served by the results of “point” measurements (as applied to the areal projection of bodies of useful minerals) of its values from boreholes and pits, arranged on a sparse grid and covering one or another part of the object being explored with unequal completeness. Under these conditions, as a rule, it proves impossible to identify the regular component of the observed variability of the parameters and, consequently, the assessment of the errors of analogy can be carried out only on the basis of errors of the random component of variability. For this same reason, as is known, limited tasks are set for the preliminary exploration stage: to reveal in general outline the geological structure of the deposit and to establish approximate mean values of the estimated parameters for the entire area (reserve estimation in category \(C_1\)). The error \(\Delta\) of the random component of the variability of a parameter for a sample with a certain number \(N\) of point measurements of its values, proceeding from the theory of random errors, can be calculated as the statistical error of the mean:

\[ \Delta=\frac{\sigma}{\sqrt{N}}, \tag{1} \]

where \(\sigma\) is the standard deviation of the means.

It follows from formula (1) that, with an increase in the number \(N\) of point measurements of a parameter, all other conditions being equal, the error \(\Delta\) of the random component of its variability decreases in magnitude. It is also known that a change in the share of participation of the random component of the observed variability of a parameter is accompanied by an ever greater disclosure and corresponding increase in the share of participation of its regular (coordinated) component. If, according to data from the detailed stage of deposit exploration, the share of participation of the regular component in the overall measure of the assessment of observed variability, as a rule, predominates over the share of participation of its random component, then the largest random changes in parameters observed from data of preliminary exploration and the stage of prospecting-and-exploration work usually veil their regular changes. Thus, the limiting actual error of analogy \(\Delta_{\text{an}}\) may be estimated at least as \(2\Delta\) (with probable allowance for expected regular changes), i.e.,

\[ \Delta_{\text{an}}=\frac{2\sigma}{\sqrt{N}}. \tag{2} \]

The actual limiting error of analogy calculated by formula (2) (when converted to a relative value) must not exceed the permissible relative limiting error in calculating the mean values of the estimated parameters according to data from the preliminary exploration stage (for category \(C_1\) reserves). Levels of permissible errors in reserve estimation (calculation of mean estimated parameters) have at various times been proposed by many researchers (V. V. Bogatskii, V. M. Kreiter, P. A. Shekhtman, and others), but there are still no generally approved requirements. As the average level of permissible relative errors of analogy (category \(C_1\)), it is proposed to adopt a value of \(\pm 50\%\) with a probability of reliable assessment of its value of 67% [1].

The geological error of interpolation must be determined within each of the conditionally homogeneous exploration blocks distinguished. In this connection, the concept of a conditionally homogeneous exploration block is given a broader meaning than in the definition of an “ele-

“elementary block,” according to L. I. Chetverikov [6], is that conditionally homogeneous part of an explored object (at the detailed or production stage) which is bounded by the contour of a reserves category of the same name for a definite natural type of mineral deposit and which combines calculation blocks explored by one type of (“point” or “tracing”) principal workings.

The combined [5] accounting for the errors of the coordinated and random components of the observed variability of a parameter under conditions of one- or two-dimensional rectilinear interpolation of its measurements may be carried out, respectively, on the basis of determining the linear (by profile) or areal (by section) interpolation error. The reduced (taking into account the relative sizes of the objects) actual limiting value of the relative (percentage) areal interpolation error \(P_{\mathrm{int}}\) for initial curves of functions with right- and left-hand asymmetry of parameter values relative to the mean, respectively, may be determined by the following formulas:

\[ P_{\mathrm{int}}= \frac{ \sum_m \left[ \frac{ \sum_{i=1}^{M}(y'_{\max}-y'_{\min})_n }{ n(m-1)+m(n-1) } \right] }{\bar{y}} \left(1+\frac{S_{\mathrm{sec}}-S_{\mathrm{opt}}}{S_{\max}-S_{\mathrm{opt}}}\right)100, \tag{3,a} \]

\[ P_{\mathrm{int}}= \frac{ 2\sum_m \left[ \frac{ \sum_{i=1}^{M}(y'_{\max}-y^{-1})_n }{ n(m-1)+m(n-1) } \right] }{ \bar{y} } \left(1+\frac{S_{\mathrm{sec}}-S_{\mathrm{opt}}}{S_{\max}-S_{\mathrm{opt}}}\right)100, \tag{3,b} \]

where \(y^{1}_{\max}\), \(y^{1}_{\min}\), \(\bar{y}^{1}\) are the maximum, minimum, and mean for each \(1\)-th among the \(M\) contrast elementary (single-maximum) functions recorded along a given profile in the general curve of a multi-maximum function within the boundaries of a conditionally homogeneous exploration block;

\(\bar{y}\) is the general mean value of the parameter within the boundaries of the conditionally homogeneous exploration block;

\(n, m\) are the numbers of exploration intersections (“point” measurements of the parameter) in profiles oriented, respectively, in the direction of dip (width) and strike (length) of the mineral body;

\(S_{\mathrm{sec}}\) is the area of the explored section;

\(S_{\mathrm{opt}}, S_{\max}\) are the optimal and maximum areas of sections for the entire series of sections explored in a given district.

It follows from formulas (3,a) and (3,b) that, with an equal number of “point” measurements of the parameter on areas analogous in variability, the actual interpolation error on the area with the larger size will be greater than on the area with the smaller size; to achieve the same exploration accuracy on the larger area, a larger number of intersections of the mineral body is necessary than on the smaller area.

Under conditions of a differentiated assessment of the degree of exploration of the reserves of a section, one has to operate with areas within the contours of individual reserve categories, more precisely with the areas \((S_{\mathrm{bl}}, S^{\mathrm{bl}}_{\max}, S^{\mathrm{bl}}_{\mathrm{opt}})\) of individual conditionally homogeneous exploration blocks. In this case the optimal area \(S^{\mathrm{bl}}_{\mathrm{opt}}\) of such blocks for each given section, for deposits differing in complexity, may be determined on the basis of the recommended ratios of reserves (areas of their distribution) of different categories according to the current reserves classification and the accepted optimal area \(S_{\mathrm{opt}}\) of the section. Thus, under conditions of differentiated exploration of deposit objects, the magnitude of the actual interpolation error can also be determined by the following formulas:

\[ P_{\text{in}}= \frac{ \sum_m\left[\sum_{l=1}^{M}(y'_{\max}-y'_{\min})_n\right] }{ \bar y\,[\,n(m-1)+m(n-1)\,] } \left(1+\frac{S^{\text{bl}}-S^{\text{bl}}_{\text{op}}}{S^{\text{bl}}_{\max}-S^{\text{bl}}_{\text{op}}}\right)100, \tag{4,a} \]

\[ P_{\text{in}}= \frac{ 2\sum_m\left[\sum_{l=1}^{M}(\bar y'_{\max}-\bar y')_n\right] }{ \bar y\,[\,n(m-1)+m(n-1)\,] } \left(1+\frac{S^{\text{bl}}-S^{\text{bl}}_{\text{op}}}{S^{\text{bl}}_{\max}-S^{\text{bl}}_{\text{op}}}\right)100. \tag{4,b} \]

The actual relative limiting error of interpolation, calculated by formulas (3, a, b) and (4, a, b), must not exceed the permissible relative limiting error in calculating the mean values of the reserve-estimation parameters for an individual exploration stage. As the upper level (category \(B\)) of permissible relative errors of interpolation of measurements of the attribute in the detailed and production exploration stages, it is proposed to take a value half as large as for errors of analogy in the preliminary exploration stage, i.e., \(\pm 25\%\); as the lower level (category \(A\)) of permissible relative interpolation errors in the detailed and production exploration stages, it is proposed to take \(\pm 15\%\). According to the recommendations of A. B. Kazhdan [1], the permissible probability of a reliable assessment of the error in calculating the mean values of reserve-estimation parameters from detailed exploration data (for categories \(B\) and \(A\)) should be taken as equal to 87% (probability coefficient \(t=1.5\)), and from production exploration data (categories \(B\) and \(A\)) as 95% (\(t=2\)).

Taking into account the predominant asymmetric differential-frequency distributions of exploration data, the approximate actual probability of a reliable assessment of exploration error should [2] be determined on the basis of calculating the arithmetic mean value of the probability coefficient \(t\) from all its variant values \(t_i\), which can be calculated by the formula:

\[ t_i=\frac{\bar y_i-\bar y}{\sigma_i} \tag{5} \]

where \(\bar y_i\) and \(\sigma_i\) are the mean value of the parameter and its standard deviation, calculated for the \(i\)-th variant of a conditional exploration sample from among the exploration data for each “sliding” pair of mutually perpendicular exploration profiles.

As the second criterion for a reliable assessment of the interpolation error in calculating the mean values of reserve-estimation parameters, it is proposed to adopt the establishment of correspondence between the attained density of exploration points along an exploration profile (in the detailed or production stage) and the character of regular (coordinated) changes in the attribute. An unambiguous establishment of the actual correspondence of the attained density of intersections to the character of coordinated changes in the attribute in an exploration profile can be carried out [3] by identifying the similarity (assessing the closeness and character of the conditional correlation relationship) between two conditionally parallel coordinated curves of successive measurements of the attribute, obtained as a result of a twofold thinning of the original sample of measurements along the profile. It is obvious that, when correspondence is achieved between the attained density of intersections and the character of coordinated changes in the attribute in the given exploration profile, each thinned series of measurements must show a regular change in the magnitude of the successive first differences of the attribute values and a regular alternation of their signs. In this case the conditional correlation-

…the relation, by its nature, must be strictly direct (positive) and linear.

References

1. A. B. Kazhdan. Quantitative assessment of the degree of exploration of reserves of solid minerals. Razvedka i okhrana nedr, 1971, No. 1.

2. V. I. Kuzmin, V. V. Novitskii. On the methodology for assessing the reliability of mean values of exploration results by V. V. Bogatskii. Geologiya i razvedka, 1969, No. 1.

3. L. M. Petrovskii. Correspondence between the density of exploration points and the character of regular changes in a feature as the principal factor in the numerical assessment of the accuracy of exploration of mineral deposits. Izv. TPI, vol. 252. TPU Publishing House, 1974.

4. P. A. Ryzhov, V. L. Gukov. Application of mathematical statistics in subsurface exploration. Moscow, Nedra, 1966.

5. N. N. Ushakov. Mining Geometry. Gosgeoltekhizdat, 1962.

6. L. I. Chetverikov. Theoretical Foundations for Modeling Bodies of Solid Minerals. Voronezh, Voronezh University Press, 1968.