Derivation of Geostatistical Formulas for Determining the Total Error Associated with Random and Systematic Changes in Exploration Parameters
Abstract Generated abstract
The paper develops geostatistical expressions for estimating variability and aggregate error in mean exploration parameters of mineral deposits under linear interpolation. It argues that limiting first differences better represent observed variability in unimodal and multimodal frequency distributions, with separate treatments for symmetric, left-asymmetric, and truncated right-asymmetric samples. From these variability indicators, the author derives formulas for specific total error in one-dimensional profile interpolation and two-dimensional interpolation within reserve-calculation contours. The proposed formulas are presented as a supplement to existing methods that separately account for random and systematic components, expanding assessment of exploration reliability.
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Derivation of Geostatistical Formulas for Determining the Total Error Associated with Random and Systematic Changes in Exploration Parameters
L. M. Petrovsky
(Presented by the scientific seminar of the Department of Geology and Exploration of Mineral Deposits)
The observed total variability according to the curve of a unimodal (elementary) distribution function of successive measurements of a parameter in all cases of linear interpolation (at any stage of exploration) can be assessed most fully on the basis of its limiting first difference [1]. Its relative magnitude, representing the dissymmetry \(D=(y_{\max}-\overline{y}):\overline{y}\) of the mean value of the parameter \(\overline{y}\) relative to its maximum value \(y_{\max}\), rather rigorously reflects the character of variability according to curves of functions with symmetrical, right- and left-asymmetrical differential-frequency distributions of measurements (Fig. 1, a). At the same time, under conditions of a limited exploration sample, as a rule, the following frequency curves of the distribution of measurements of a characteristic are typical (Fig. 1, b, c). From Fig. 1, c it is evident that the limiting absolute first difference for truncated right-asymmetrical curves must be determined not as the difference between the maximum and mean values, but between the mean and minimum values \(y_{\min}\) of the parameter; the relative limiting first difference of the parameter in this case will be \((\overline{y}-y_{\min}):\overline{y}\). Doubling both parts of the latter expression and replacing its left part \((2\overline{y}-2y_{\min})\) by the practically more convenient and equivalent, for right-asymmetrical curves, form \((y_{\max}-y_{\min})\), we obtain the expression \((y_{\max}-y_{\min}):2\overline{y}\) for the relative limiting difference of the parameter in the case of a right-asymmetrical differential-frequency curve.
Thus, one may propose [3] the following geostatistical formulas for the indicators \(I\) and \(I_F\) of the observed variability of a parameter, respectively for unimodal and multimodal curves of functions with left and right asymmetry:
\[ I=(y_{\max}-\overline{y})\,\frac{100}{\overline{y}}, \tag{1} \]
\[ I=(y_{\max}-y_{\min})\,\frac{100}{2\overline{y}}; \tag{2} \]
\[ I_F=\frac{\displaystyle\sum_{i=1}^{F}\left(y'_{\max}-\overline{y}'\right)}{\overline{y}}\,100, \tag{1'} \]
\[ I_F=\frac{\displaystyle\sum_{i=1}^{F}\left(y'_{\max}-y'_{\min}\right)}{2\overline{y}}\,100, \tag{2'} \]
where \(y_{\min}, y_{\max}\) are the specific values of the parameter from the total set of measurements along an individual profile or from all its measurements within the boundaries of each conditionally homogeneous exploration block; \(\overline{y}'_{\min}, \overline{y}^{\,i}, y'_{\max}\) —
minimum, mean, and maximum values of the attribute for each \(i\)-th of the \(F\) recorded initial elementary functions in the overall curve of the multimaximum function along the given direction of anisotropy of variability. It is obvious that formulas (1), (2) are a special case of the general formulas \((1^1)\), \((2^1)\). The use of formulas \((1^1)\), \((2^1)\) from data of point exploration intersections is expedient only at the stages of detailed and operational exploration, and from data of tracing mine-exploration workings—at any stage of exploration.
Fig. 1. Schematic curves of the differential-frequency distribution of measurements of an attribute with the corresponding values of their dissymmetries \(D\) according to data from: a general population and large-volume exploration samples (a); limited exploration samples in the case of symmetric or “truncated” weakly asymmetric (b) and asymmetric (c) distributions
On the basis of the general geostatistical formulas \((1^1)\), \((2^1)\) for estimating the observed variability of parameters, it is not difficult to substantiate geostatistical formulas for determining the specific aggregate error of their means under conditions of one- and two-dimensional rectilinear interpolation. Thus, with the number \(n\) and \(n^1\) of attribute measurements respectively in profiles oriented along the strike (length) and dip (width) of a deposit, the aggregate error \(P^1\) in calculating the mean parameter for an individual profile (oriented, for example, along the dip) will be determined by one of the following expressions:
\[ P^1=\frac{\sum_{i=1}^{F}\left(y'_{\max}-\bar{y}'\right)} {\bar{y}\left(n^1-1\right)}\,100, \tag{3} \]
\[ P^1=\frac{\sum_{i=1}^{F}\left(y'_{\max}-y'_{\min}\right)} {2\bar{y}\left(n^1-1\right)}\,100, \tag{4} \]
and for a series of profiles in the given direction the aggregate error \(P_n'\) may be determined as
\[ P_n'= \frac{\sum_n\left[\sum_{i=1}^{F}\left(y'_{\max}-\bar{y}'\right)n'\right]} {\bar{y}\left[\sum_n\left(n^1-1\right)\right]}\,100, \tag{3^1} \]
\[ P_n'= \frac{\sum_n\left[\sum_{i=1}^{F}\left(y'_{\max}-y'_{\min}\right)n'\right]} {2\bar{y}\left[\sum_n\left(n^1-1\right)\right]}\,100. \tag{4^1} \]
In the special case, the expression in square brackets in the denominator of formulas \((3^1)\), \((4^1)\) is transformed into the form \([n(n^1-1)]\).
Under conditions of two-dimensional interpolation, the aggregate error \(P_{n,n'}\) in calculating the mean parameter within the reserve-calculation contour is determined by one of the following formulas:
\[ P_{n.n'}= \frac{ \sum_n\left[\sum_{i=1}^{F}\left(y'_{\max}-\bar{y}'\,n'\right)\right] + \sum_{n'}\left[\sum_{i=1}^{F}\left(y'_{\max}-\bar{y}'\right)n\right] }{ \bar{y}\left[\sum_n\left(n^1-1\right)+\sum_n\left(n-1\right)\right] }\,100; \tag{5} \]
\[ P_{n,n'}= \frac{ \sum_n\left[\sum_{i=1}^{F}\left(y'_{\max}-y'_{\min}\right)n'\right] + \sum_{n'}\left[\sum_{i=1}^{F}\left(y'_{\max}-y'_{\min}\right)n\right] }{ 2\bar{y}\left[\sum_n\left(n^1-1\right)+\sum_{n'}\left(n-1\right)\right] }\,100. \tag{6} \]
The geostatistical formulas presented for determining the total error in calculating average accounting parameters, together with the known methods \([2, 4]\) for separately accounting for errors due to the random and systematic components of variability, substantially supplement and broaden the possibilities for a complete assessment of the degree of exploration of mineral deposits.
LITERATURE
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