On the Analytical Determination of Actual and Expected Limiting Interpolation Errors Taking into Account the Anisotropy of Observed Variability in Ore Mineralization Parameters
Abstract Generated abstract
This paper proposes an analytical method for estimating actual and expected limiting interpolation errors in ore exploration, with attention to asymmetric distributions of mineralization parameters, block geometry, exploration area, and anisotropy of variability. Building on the use of maximum first differences, it derives formulas for relative profile-mean and areal-mean errors in the principal directions of dip and strike, separately for right- and left-asymmetric frequency distributions. The method is presented as a way to compare calculated errors with permissible reserve-category thresholds and to estimate expected errors from two perpendicular exploration profiles. It also supports selecting an exploration grid density and the ratio of intersections in dip and strike directions according to the observed anisotropy of the parameter.
Full Text
ON THE ANALYTICAL DETERMINATION OF ACTUAL AND EXPECTED MAXIMUM INTERPOLATION ERRORS TAKING INTO ACCOUNT THE ANISOTROPY OF OBSERVED VARIABILITY IN ORE MINERALIZATION PARAMETERS
L. M. PETROVSKII
(Presented by the seminar of the Department of Geology and Exploration of Mineral Deposits)
The proposed methodology for the analytical determination of interpolation error from the results of detailed and operational stages of exploration is based on:
1) maximally probable (for right- and left-asymmetric differential-frequency distributions of measurements of a characteristic) combined accounting for errors in determining mean values of the parameter associated with the regular and random components of its variability;
2) accounting for the size and geometry of the area of the block;
3) accounting for the anisotropy of the variability of the parameter.
Basing ourselves on the theoretical ideas of V. V. Bogatskii [1] on the universal significance of maximum first differences of measurements of a characteristic and on the author’s further studies [2], the reduced (with allowance for the relative dimensions of objects) actual maximum value of the relative mean-areal (by section) interpolation error \(P_{n,m}\), for cases with right- and left-asymmetric differential-frequency distributions of parameter values, respectively, may be determined by the following formulas:
\[ P_{n,m}^{\mathrm{right}}= \frac{ \Sigma_m\left[\sum_{i=1}^{M}\left(Y'_{\max}-Y'_{\min}\right)_n\right] + \Sigma_n\left[\sum_{i=1}^{M}\left(Y'_{\max}-Y'_{\min}\right)_m\right] }{ 2\bar{Y}\left[\Sigma_m(n_j-1)+\Sigma_n(m_j-1)\right] } 100 \left(1+\frac{S_{\mathrm{uch}}-S_{\mathrm{op}}}{S_{\max}-S_{\mathrm{op}}}\right), \tag{1, a} \]
\[ P_{n,m}^{\mathrm{left}}= \frac{ \Sigma_m\left[\sum_{i=1}^{M}\left(Y'_{\max}-\bar{Y}'\right)_n\right] + \Sigma_n\left[\sum_{i=1}^{M}\left(Y'_{\max}-\bar{Y}'\right)_m\right] }{ \bar{Y}\left[\Sigma_m(n_j-1)+\Sigma_n(m_j-1)\right] } 100 \left(1+\frac{S_{\mathrm{uch}}-S_{\mathrm{op}}}{S_{\max}-S_{\mathrm{op}}}\right), \tag{1, б} \]
where \(Y'_{\max}\), \(Y'_{\min}\), \(\bar{Y}'\) are the maximum, minimum, and mean values of the parameter for each \(i\)-th of the \(M\) recorded, along a given profile, contrast elementary single-maximum functions in the general curve of the multi-maximum function within the boundaries of a conditionally homogeneous exploration block;
\(\bar{Y}\) is the overall mean value of the parameter within the boundaries of a conditionally homogeneous exploration block;
\(n_j, m_j\) are the number of exploration intersections (“point” measurements of the pa—
parameter) in profiles oriented respectively in the direction of dip (width) and strike (length) of the body of the useful mineral;
\(S_{\text{uch}}\)—area of the explored sector;
\(S_{\text{op}}, S_{\max}\)—optimal and maximum areas of sectors for the entire series of sectors explored in the given district.
The actual limiting values of the relative inter-profile interpolation error for the principal directions of anisotropy may be determined from the following formulas:
\[ P_{n}^{\text{right}} = \frac{ \Sigma_m \left[ \sum_{i=1}^{M} \left(Y_{\max}'-Y_{\min}'\right)_n \right]100 }{ 2\overline{Y}\left[\Sigma_m(n_j-1)\right] } \left( 1+ \frac{S_{\text{uch}}-S_{\text{op}}}{S_{\max}-S_{\text{op}}} \right), \tag{2, a} \]
\[ P_{n}^{\text{left}} = \frac{ \Sigma_m \left[ \sum_{i=1}^{M} \left(Y_{\max}'-\overline{Y}'\right)_n \right]100 }{ \overline{Y}\left[\Sigma_m(n_j-1)\right] } \left( 1+ \frac{S_{\text{uch}}-S_{\text{op}}}{S_{\max}-S_{\text{op}}} \right); \tag{2, b} \]
taking into account the observed variability of the attribute in the direction of strike (length) of the body
\[ P_{m}^{\text{right}} = \frac{ \Sigma_n \left[ \sum_{i=1}^{M} \left(Y_{\max}'-Y_{\min}'\right)_m \right]100 }{ 2\overline{Y}\left[\Sigma_n(m_j-1)\right] } \left( 1+ \frac{S_{\text{uch}}-S_{\text{op}}}{S_{\max}-S_{\text{op}}} \right), \tag{3, a} \]
\[ P_{m}^{\text{left}} = \frac{ \Sigma_n \left[ \sum_{i=1}^{M} \left(Y_{\max}'-\overline{Y}'\right)_m \right]100 }{ \overline{Y}\left[\Sigma_n(m_j-1)\right] } \left( 1+ \frac{S_{\text{uch}}-S_{\text{op}}}{S_{\max}-S_{\text{op}}} \right). \tag{3, b} \]
The calculated value of the actual areal-mean and profile-mean limiting relative interpolation errors must not exceed the permissible relative error for the corresponding reserve category and exploration stage [2]. It is evident that, in order to carry out a complete differentiated assessment of the accuracy of exploration of a sector, it is sufficient to compare with the permissible value of the relative error either each of the quantities \(P_n\) and \(P_m\), or \(P_{n,m}\) and one of the preceding ones, or only \(P_n\), if in the latter case it is known that the direction of dip (width) of the body corresponds to the maximum anisotropy of variability of the parameter.
On the basis of the proposed method it appears possible to determine the expected limiting values of the profile-mean and areal-mean relative interpolation errors for the entire sector from data of only two mutually perpendicular profiles of exploration workings. Starting from the actual limiting sums
\[ \tfrac{1}{2}\sum_{i=1}^{M}\left(Y_{\max}'-Y_{\min}'\right)_n \quad \text{and} \quad \tfrac{1}{2}\sum_{i=1}^{M}\left(Y_{\max}'-Y_{\min}'\right)_m \]
of elementary first differences of measurements of the parameter, calculated from the initial data in the direction of dip and strike of the useful-mineral body for the case of a right-asymmetric differential-frequency distribution, the expected profile-mean interpolation errors, respectively, may be determined from the formulas:
\[ P_{n}^{\text{right}} = \frac{ m\sum_{i=1}^{M}\left(Y_{\max}'-Y_{\min}'\right)_n }{ 2\overline{Y}m(n-1) } \left( 1+ \frac{S_{\text{uch}}-S_{\text{op}}}{S_{\max}-S_{\text{op}}} \right)100, \tag{4, a} \]
\[ P_m^{\text{right}}= \frac{n\sum_{i=1}^{M}\left(Y'_{\max}-Y'_{\min}\right)_m} {2\overline{Y}\,n(m-1)} \left(1+\frac{S_{\text{uch}}-S_{\text{op}}}{S_{\max}-S_{\text{op}}}\right)100. \tag{4, б} \]
or, after performing the reduction,
\[ P_n^{\text{right}}= \frac{\sum_{i=1}^{M}\left(Y'_{\max}-Y'_{\min}\right)_n} {2\overline{Y}(n-1)} \left(1+\frac{S_{\text{uch}}-S_{\text{op}}}{S_{\max}-S_{\text{op}}}\right)100, \tag{5, а} \]
\[ P_m^{\text{right}}= \frac{\sum_{i=1}^{M}\left(Y'_{\max}-Y'_{\min}\right)_m} {2\overline{Y}(m-1)} \left(1+\frac{S_{\text{uch}}-S_{\text{op}}}{S_{\max}-S_{\text{op}}}\right)100; \tag{5, б} \]
the expected root-mean-square interpolation error may be determined from the formula
\[ P_{n,m}^{\text{right}}= \frac{ m\sum_{i=1}^{M}\left(Y'_{\max}-Y'_{\min}\right)_n + n\sum_{i=1}^{M}\left(Y'_{\max}-Y'_{\min}\right)_m } {2\overline{Y}\,[m(n-1)+n(m-1)]} \left(1+\frac{S_{\text{uch}}-S_{\text{op}}}{S_{\max}-S_{\text{op}}}\right)100, \tag{6} \]
In the case of a left-asymmetric differential-frequency distribution of measurements of the parameter, the expected values of the limiting interpolation errors, respectively, may be determined from the formulas:
\[ P_n^{\text{left}}= \frac{\sum_{i=1}^{M}\left(Y'_{\max}-\overline{Y}'\right)_n} {\overline{Y}(n-1)} \left(1+\frac{S_{\text{uch}}-S_{\text{op}}}{S_{\max}-S_{\text{op}}}\right)100. \tag{5^{1}, а} \]
\[ P_m^{\text{left}}= \frac{\sum_{i=1}^{M}\left(Y'_{\max}-\overline{Y}'\right)_m} {\overline{Y}(m-1)} \left(1+\frac{S_{\text{uch}}-S_{\text{op}}}{S_{\max}-S_{\text{op}}}\right)100, \tag{5^{1}, б} \]
\[ P_{n,m}^{\text{left}}= \frac{ m\sum_{i=1}^{M}\left(Y'_{\max}-\overline{Y}'\right)_n + n\sum_{i=1}^{M}\left(Y'_{\max}-\overline{Y}'\right)_m } {2\overline{Y}\,[m(n-1)+n(m-1)]} \left(1+\frac{S_{\text{uch}}-S_{\text{op}}}{S_{\max}-S_{\text{op}}}\right)100. \tag{6^{1}} \]
Formulas (5, а, б; 6) and (5¹, а, б; 6¹) make it possible to solve objectively the questions of the optimal choice of the density of exploration points already at the initial stage of exploration of a site. In this case, the optimal choice of the ratio between the numbers \(n\) and \(m\) of intersections of the body in the direction of its dip and strike must be made in accordance with the specific anisotropy of the observed variability of the parameter, characterized by the ratio of the expressions \(\sum_{i=1}^{M}(Y'_{\max}-Y'_{\min})_n\) and \(\sum_{i=1}^{M}(Y'_{\max}-Y'_{\min})_m\) in the case of a right-asymmetric differential-frequency distribution of the parameter values, or \(\sum_{i=1}^{M}(Y'_{\max}-\overline{Y}')_n\) and \(\sum_{i=1}^{M}(Y'_{\max}-\overline{Y}')_m\) in the case of a left-asymmetric distribution.
REFERENCES
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V. V. Bogatskii. Mathematical Analysis of the Exploration Grid. Gosgeoltekhizdat, 1963.
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L. M. Petrovskii. “Toward a substantiation of a rational method for determining the density of the exploration grid.” Geology, materials of the conference dedicated to the 75th anniversary of the institute. TSU Publishing House, 1973.