Revisiting the determination of the deposit exploration index and the analogy error using the second-difference method
Abstract Generated abstract
The paper critiques a proposed formula for determining the absolute index of variability of a deposit property and the analogy error using second differences on a square exploration grid. It argues that reducing second differences from diagonal intervals to side intervals assumes an unjustified linear relation and artificially increases the apparent accuracy of exploration, since side and diagonal differences represent variability in different directions and do not have the fixed geometric relationship of square sides and diagonals. The author compares alternative formulations, discusses weighting by sampling interval and by reliability of direct versus reduced differences, and concludes that the contested formula should be excluded from practice, while a simple average of directly calculated second differences is often sufficient for practical calculations.
Full Text
REVISITING THE DETERMINATION OF THE DEPOSIT EXPLORATION INDEX AND THE ANALOGY ERROR BY THE METHOD OF SECOND DIFFERENCES
A. I. Volkov
(Presented by the scientific seminar of the departments of mine surveying and geodesy)
In article [1] I showed the erroneousness of the formula of Prof. D. A. Kazakovsky, proposed by him for determining the absolute index of variability of the studied property of a deposit with allowance for the second differences computed along the diagonals of the squares of the exploration grid,
\[ I_a=\frac{\sum q_i \Delta''}{\kappa}, \tag{1} \]
where \(q_i\) are the weights of the second differences, inversely proportional to the sampling intervals (as Prof. D. A. Kazakovsky called these quantities in a number of his works, including work [2]);
\(\Delta''\) are the second differences computed along the sides and diagonals;
\(\kappa\) is the total number of second differences along the sides and diagonals;
\(I_a\) is the absolute index of variability of the studied property of the deposit.
In work [3] Prof. D. A. Kazakovsky no longer calls \(q\) weights, in their usual sense, but coefficients that make it possible to pass from second differences corresponding to one sampling interval (the diagonal of the exploration grid) to second differences corresponding to another sampling interval (the side of a square of the exploration grid).
In this work no fundamental explanation is given as to why it is necessary to pass specifically from second differences corresponding to the larger sampling interval to second differences corresponding to the smaller sampling interval, and not the reverse, although this is by no means immaterial: by passing, by a desk method, to a smaller sampling interval, we artificially increase the accuracy of exploration. As proof of the validity of applying formula (1), Prof. D. A. Kazakovsky considers the fact that this formula is suitable for computing the mean value of the side of a square when all its sides and diagonals have been measured with the same relative error. Indeed, in this particular case formula (1) is suitable for computing the mean value of the side of a square with allowance for the diagonals, but we see absolutely no analogy between this problem and the determination of the absolute index of variability of the deposit property that interests us.
Our objections to the use of formula (1) for comput-
...of \(I_a\) and the arguments in its defense that are presented in [3] are as follows.
The absolute index of variability of a given property of a deposit and the analogy error are determined sufficiently reliably only when the number of samples is relatively large. With a large number of samples, as follows from formulas (2), (3), and (4) and Table 1, the number of second differences determined along the sides and along the diagonals of the exploration grid is almost identical (the exploration grid is square in form).
\[ \kappa_1 = 2n(n - 2), \tag{2} \]
\[ \kappa_2 = 2(n - 2)^2, \tag{3} \]
\[ d = \frac{\kappa_1}{\kappa_2} = \frac{n}{n - 2}, \tag{4} \]
where
\(\kappa_1\) is the number of second differences calculated along the sides of the exploration grid;
\(\kappa_2\) is the number of second differences calculated along the diagonals of the exploration grid;
\(d\) is the ratio of the number of second differences calculated along the sides of the exploration grid to the number of second differences calculated along the diagonals;
\(n\) is the number of sampling points in one row of the exploration grid.
Table 1
| Number of sampling points \(n \times n = n^2\) | Number of second differences along sides \((\kappa_1)\) | Number of second differences along diagonals \((\kappa_2)\) | Difference \(\kappa_1 - \kappa_2\) | Ratio \(d = \frac{\kappa_1}{\kappa_2}\) |
|---|---|---|---|---|
| \(3 \times 3 = 9\) | 6 | 2 | 4 | 3 |
| \(4 \times 4 = 16\) | 16 | 8 | 8 | 2 |
| \(5 \times 5 = 25\) | 30 | 18 | 12 | 1,67 |
| \(6 \times 6 = 36\) | 48 | 32 | 16 | 1,50 |
| \(7 \times 7 = 49\) | 70 | 50 | 20 | 1,40 |
| \(8 \times 8 = 64\) | 96 | 72 | 24 | 1,33 |
| \(9 \times 9 = 81\) | 126 | 98 | 28 | 1,29 |
| \(10 \times 10 = 100\) | 160 | 128 | 32 | 1,25 |
| \(15 \times 15 = 225\) | 390 | 338 | 52 | 1,15 |
| \(20 \times 20 = 400\) | 720 | 648 | 72 | 1,11 |
The fact that, as the number of sampling points increases, the number of second differences along the diagonals of the exploration grid is close to the number of second differences along the sides of this grid indicates their equal importance in solving the problem posed. This proposition is further strengthened by the circumstance that the second differences along the sides and diagonals express the variability of the studied property of the deposit in different directions.
All this indicates the inadmissibility of applying formula (1) in the interpretation given to it in [3]. Taking into account what was said above about second differences and ignoring the complex character of the variability of the studied property between sampling points,
one could have proposed with greater logic the reduction of those and other differences to the mean sampling interval. In this case, in formula (1) the second differences computed along the sides of the exploration grid would have to be multiplied by \(q_1 = 1.20\), and the second differences computed along the diagonals by \(q_2 = 0.85\). Taking these remarks into account, formula (1) will have the form:
\[ I_a=\frac{q_1\sum \Delta_i''+q_2\sum \Delta_j''}{k}, \tag{5} \]
where \(\sum \Delta_i''\) is the sum of the second differences along the sides of the exploration grid;
\(\sum \Delta_j''\) is the sum of the second differences along the diagonals of the exploration grid.
The value of the absolute coefficient of variability, as shown in [1], can be obtained from the formula
\[ I_a=\frac{\sum P\Delta''}{\sum P}, \tag{6} \]
where \(P\) are the weights of the second differences, inversely proportional to the sampling interval. For the second differences computed along the sides of the exploration grid, the weight is equal to unity, and along the diagonals it is
\[ \frac{1}{\sqrt{2}}\simeq 0.7. \]
Formula (6), therefore, takes into account the generally accepted criterion for characterizing the accuracy of exploration—the sampling interval. This is undoubtedly a positive aspect of this formula.
If, however, one keeps in mind that in order to achieve a specified accuracy of exploration the sampling interval may vary within certain limits, then on this basis the weights of the second differences along the sides and along the diagonals may be considered identical, and then expression (6) will take the form
\[ I_a=\frac{\sum \Delta''}{k}. \tag{7} \]
Above we said that, in order to prove the validity of formula (1), work [3] referred to the fact that by this formula one can find the most probable value of the side of a square from measured values of the sides and diagonals with the same relative error (after replacing, of course, \(\Delta''\) in formula (1) by \(l\)). This is entirely correct, since between the side and the diagonal of a square there exists a strictly definite dependence—the side of a square is always \(\sqrt{2}\) times smaller than its diagonal. Such a dependence, however, does not exist (even approximately) between the second differences computed along the sides and along the diagonals of the exploration grid, and consequently there is no analogy, of which work [3] speaks, between these two problems.
Thus, the use of formula (1) cannot be justified by the geometrically known regularity between the linear elements of a regular quadrilateral. The use of this formula is in fact based on another, nonexistent regularity—on the assumption that the change in the second differences proceeds according to a linear law. This is readily revealed from expression (8), by means of which in formula (1) the second differences along the diagonals are reduced,
\[ \Delta_{\mathrm{pr}}''=\Delta''\frac{l_0}{l}=\frac{\Delta''}{\sqrt{2}}\simeq 0.7\Delta', \tag{8} \]
where \(\Delta_{\mathrm{pr}}\) is the reduced value of the second difference to an interval equal to the side of the square;
\(\Delta''\) is the calculated value of the second difference along the diagonal of the exploration grid;
\(l_0\) is the side of the square of the exploration grid;
\(l\) is the diagonal of the square of the exploration grid.
Since the change in any property for which the analogy error is determined, and consequently also the change in the second differences, proceeds along a complex curve, the reduced second differences have only a fictitious character, approaching reality when the regularity of change of the property being studied approaches a straight line and, conversely, departing from the truth when this regularity becomes more complicated. It follows from this that the reduced differences have lower reliability, i.e., lower weight, than those calculated from direct sampling data. Taking these remarks into account, formula [1] should be replaced by formula [9]
\[ I_a=\frac{P_1\Sigma\Delta''_1+P_2q\Sigma\Delta''_2}{\Sigma P}, \tag{9} \]
where \(P_1\) is the weight of the second differences calculated along the sides of the exploration grid;
\(P_2\) is the weight of the second differences calculated along the diagonals of the exploration grid;
\(q\) is the coefficient for reducing the second differences calculated along the diagonal of the exploration grid, equal to 0.7.
The question of determining the weights in this case was considered by us in work [4].
From all that we have said above concerning the determination of the absolute index of variability of one or another property of a deposit, the following conclusions may be drawn.
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At the basis of formula (1) lies a regularity that does not exist in nature. The use of this formula always leads to an artificial increase in the accuracy of exploration. It does not take into account the different reliability of the second differences calculated from direct observations and of the reduced second differences. This formula must be excluded from practice.
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Formula (9) is the best variant of formula (1), since it takes into account the weights of the reduced second differences and of those calculated from direct data, while nevertheless retaining its other shortcomings.
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Formula (5) retains the main shortcoming of formula (1)—interpolation; however, because here the second differences along the sides and diagonals are reduced to the mean sampling interval, it deserves some attention.
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Formula (6) is well substantiated; however, we believe that in practice all calculations can sufficiently be carried out by formula (7).
LITERATURE
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On the question of determining the index of exploration of a deposit and the analogy error by the method of second differences, Izvestiya vysshikh uchebnykh zavedenii (“Mining Journal”), No. 6, 1959.
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Estimation of the accuracy of results in connection with the geometrization and calculation of reserves of deposits. Ugletekhizdat, 1948.
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On the question of determining the mean value of second differences in estimating the relative variability of deposit characteristics. Izvestiya vysshikh uchebnykh zavedenii (“Mining Journal”), No. 4, 1960.
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Estimation of the degree of exploration of a deposit by the method of second differences when the sampling interval is variable. Izvestiya TPI, vol. 113, 1960.