ON THE SELECTION OF OPTIMAL ESTIMATES OF RANDOM VARIABILITY OF GEOLOGICAL FEATURES

L. M. PETROVSKII

(Presented by the scientific seminar of the departments of mineralogy, petrography, deposits of useful minerals, and geology and exploration of deposits)

The shortcomings associated with the use of the standard deviation $\sigma$ and the coefficient of variation $v$ in the case of an undifferentiated characterization of geological variability are well known (1, 2, 3, 6, 7).

P. L. Kallistov [1] was the first to investigate the possibility of separately characterizing the random and coordinated components of the overall variability of content. For a quantitative estimate of the random variability of a feature he recommended using the value of the root-mean-square deviation $S_m$ of the feature values from the corresponding smoothed values obtained by the “moving statistical window” method. It was noted, however, that the value of the degree of random variability fluctuates noticeably depending on the conditions (the initial size of the “window” and the number of smoothing operations) of smoothing.

L. I. Chetverikov [6] proposed, for a quantitative estimate of regular variability, indicators important in practical terms, based on specific interval first and second differences of smoothed content values. Of particular importance is the classification of variability of the content of a useful mineral that he proposed, which makes it possible to take into account random variability differing in degree against the background (in the obligatory presence) of regular variability differing in complexity or “persistence.” For a quantitative estimate of random variability, in the presence of regular changes, he recommends, relying on the conclusions of B. S. Yastremskii [7], using the value of the root-mean-square deviation $H_{ck}$ from the first successive differences of the feature values.

The investigation showed that the value of $S_m$ in the general case depends on the adopted method of smoothing the feature values. The principal methods for determining the moving average, or smoothing methods, should be regarded as the so-called methods of successive doubling [7] and successive tripling or quintupling [1]. In this case the second, fourth, and other even moving averaged estimates of the feature by the method of successive doubling, and the moving averages of the first, second, etc. smoothing operations of feature values by the method of successive tripling, for example, will respectively be based equally on 3, 5, etc. terms of the original series in the “window” (with different specific influence of the individual terms in it). On this basis, as is known, determination of the smoothing-stage order $m$ common to all methods in the formula for calculating the value of $S_m$ is carried out according to the principle of successive doubling.

The character of smoothing by different methods proves to be different.

(Table 1). The calculated results of example A characterize sampling data known from the literature for a gold-quartz vein \[1, 2\]*. The regular change in gold content observed here, according to L. I. Chetverikov, is very inconsistent, or is characterized by a strongly sinuous line; the indices of random variability are \(\sigma = 24.9\) and \(H_{sk} = 26.8\). The results of example B characterize data from operational sampling of a gold-quartz vein at one of the deposits of Eastern Siberia (Fig. 1). The regular change observed here is characterized accordingly as inconsistent (by a sinuous line);

Table 1

Calculation of the standard-deviation value of individual gold contents
from their smoothed values

the indices of random variability are \(\sigma = 33.6\) and \(H_{sk} = 28.1\). From an analysis of the comparative data in Table 1, one may arrive at the following conclusions:

1. Regardless of the character of the regular component of the variability of content, the smallest values of \(S_m\) (the greatest approximation of the moving-average values of content to the individual terms of the series (Fig. 1)) at all smoothing stages of the same name occur with the method of successive doubling. The corresponding largest values of \(S_m\) (the least approximation of the moving averages (Fig. 1)) occur with the method of successive quintupling, and the mean values of \(S_m\) occur with the method of successive tripling.

2. As the number of smoothing operations increases, for all methods there is a regular overall approximation of the value \(S_m\) to the values \(\sigma\) and \(H_{sk}\), for

* Here the initial data used were taken from the graph and rounded to integer values.

Fig. 1. Construction of regression curves (example B). 1 — curve constructed from the initial analysis data (the numbers are the initial gold contents); 2 — regression curve obtained in one smoothing pass by the method of successive doubling; 3 — the same by the method of successive tripling; 4 — the same by the method of successive quintupling.

with the exception of individual random deviations from the general increase in the value \(S_m\) (toward decrease or increase, and more often for higher-order smoothing stages), caused by the influence of the corresponding extreme content values in the initial series (stages Nos. 10, 16, 18, 24 in example A and 6, 8, 22, 24 in example B). At the same time, in the second smoothing pass by the methods of successive quintupling and tripling, as a rule, a decrease is observed in the corresponding value \(S_m\) in comparison with the value \(S_m\) of the first smoothing pass.

The conclusions obtained make it possible to develop practical recommendations for determining the rational method and number of smoothing passes for the values of a feature. Of the smoothing methods considered, smoothing in the first three to five passes by the methods of successive tripling and doubling should be recognized as practically rational. The method of successive quintupling, its second smoothing pass, may serve as a control: the values \(S_{2-10}\), determined at smoothing stages Nos. 2–10 by the “doubling” or “tripling” methods, should in magnitude be less than the value \(S_8\) (all the more so than \(S_4\)) by the method of successive quintupling. As the optimal value \(S_m\) by the method of successive tripling, the value following the smallest value \(S_4\) may serve; as a rule, this value will be greater than all its preceding values and will approach the like-named value \(S_6\) by the “doubling” method. In example A, such an optimal value is \(S_6 = 22.5\) (Table 1). In example B, the value \(S_6 = 25.1\) cannot be taken as such, since the latter considerably exceeds not only its preceding values \(S_{2-4} = 21.9\), but also the “control” value \(S_8 = 23.8\); nor can the value \(S_8 = 21.2\), which is less than the value preceding it \(S_2 = 21.9\), be taken as optimal. Obviously, in this case the value \(S_{10} = 22.3\), which is greater than its preceding-

quantities; the largest of them approaches in magnitude the corresponding \(S_m\) by the “doubling” method (\(S_{10}=20.7\)), but is smaller than the “control” \(S_8=23.8\) (Table 1).

The theoretically significant limiting number of smoothing operations can be determined from the conditions of simple inequalities:

\[ (n-m)-2 \geq m+1 \text{ or } n-m \geq m+3 \tag{1} \]

for the methods of successive doubling and tripling, and

\[ (n-m)-4 \geq m+1 \text{ or } n-m \geq m+5 \tag{2} \]

for the method of successive quintupling. Thus, in the case of example A (\(n=28\)), this number of smoothing operations is determined respectively by the order \(m=12\) and \(m=10\). From the conditions of these inequalities one can also determine the minimum necessary number of initial measurements of content \(n\) in a separate series (under conditions of an equivalent character of variability of the feature and density of observations at each of the objects), which for \(m=6\) will be respectively \(n=15\) and \(n=17\).

For primary and placer gold deposits, for example, such a minimum necessary number of complete intersections of ore bodies in individual underground workings and borehole lines may be comparatively easily attainable already at the stage of preliminary exploration of medium and large objects. For most nonferrous-metal deposits and for the simplest rare-metal deposits, such a minimum necessary number of complete intersections of ore bodies in individual exploration lines can be obtained only at the detailed or operational exploration stage. Thus, only under the condition of a sufficient number of complete intersections of ore bodies along the main directions of their exploration is it possible and expedient to refine the quantitative estimate of the random variability of a feature on the basis of the indicator \(S_m\). In cases that do not satisfy this condition, a more or less accurate quantitative estimate of the random geological variability of a feature can be made either on the basis of the indicator \(H_{ck}\), or on the basis of the standard deviation \(\sigma\).

Comparison of the various quantitative estimates obtained for the random variability of gold content for the examples given above yields the following results (Table 2). The most correct estimate of the degree of random variability of content is given by the values \(S_m\): at practically

Table 2

Comparative quantitative characteristic of the variability of gold content

the same level of random variability, the overall variability of gold content in example A proves to be more complex, taking into account the more complex character of the manifestation here of coordinated variability. The opposite picture is observed from the data of the indicators \(\sigma\) and \(H_{ck}\)*: the overall variability of content that they essentially represent proves, in example B, contrary to the obvious evidence, to be more complex than in

* The same picture is also observed for the values \(H_{2ck}\), calculated from second successive differences \(\bigl(H_{2ck}(A)=26.8;\ H_{2ck}(B)=27.1\bigr)\).

example A. In this case, the greatest distortion in the estimate of the random variability of the feature may be given by both $\sigma$ and $H_{ck}$.

In cases where the coordinated variability has the character (according to L. I. Chetverikov) of a horizontal (first type) or strongly sinuous (fourth type) line, as in example A, the indicators $H_{ck}$ and $\sigma$ characterize the combined random and coordinated variability. In this case, as is known, the specific magnitudes of the estimates under consideration are determined by the ratio between the degree of change in the general mean between two successive observations of the feature and the measure of its fluctuation $\sigma$ (5). A practically small change in the general mean between two successive observations of the feature in comparison with the standard of fluctuation $\sigma$ will occur if the absolute value of the mean successive difference $\overline{d_i}=(x_{i+1}-x_i)$ for the given population of the feature is less than (or equal to) the corresponding standard $\sigma$. For this kind of sample population of the feature, the value $H_{ck}$ will be less than the value $\sigma$; precisely for this kind of sample population of the feature it is possible to apply (as proposed by L. I. Chetverikov (6)) the criterion “$r$” of the sum of squares of successive differences for judging the presence or absence of regular changes. In cases where the absolute value of the mean successive difference $\overline{d_i}$ for the given sample population is greater than the corresponding $\sigma$, the value $H_{ck}$ will be greater than $\sigma$. Just such a case occurs in example A ($d_i=30$). Since the value of “$t$” in these cases will be greater than unity, it is obvious that the criterion “$r$,” with its significantly small values, cannot be used to judge the presence or absence of regular changes in the feature. Thus, the presence of a coordinated variability of content in example A of the fourth type, more complex in character, will be denied on the basis of the criterion “$r$.”

In the case where, in the character of the general variability of the content, there is a noticeable tendency toward an increase or decrease of its values in the form of an inclined (second type) or sinuous (third type) line of coordinated changes, the indicator $H_{ck}$, unlike $\sigma$, gives an estimate of random variability closer to the true one (here, as a rule, $\overline{d_i}\leq\sigma$). In this case the indicator $H_{ck}$ may serve as an “express” approximate estimate of the true random variability of the feature. It is precisely for this kind of population of features, from examples of social and production activity, that B. S. Yastremskii [7] recommends using the indicator $H_{ck}$ as the optimal estimate of random variability; precisely for such populations it is also possible to apply the criterion “$r$” to judge the presence or absence of coordinated changes in the feature.

REFERENCES

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

2. V. M. Kreiter. Prospecting and exploration of mineral deposits, part 2. Gosgeoltekhizdat, 1961.

3. V. I. Kuzmin. Geometrization and calculation of reserves of deposits of solid minerals. Nedra Publishing House, 1967.

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

5. A. Hald. Mathematical Statistics with Technical Applications. IL, 1956.

6. L. I. Chetverikov. Quantitative assessment of the variability of the content of a useful mineral observed from sampling data. Izv. VUZ “Geol. i razv.”, No. 3, 1968.

7. B. S. Yastremskii. Some questions of mathematical statistics. Gosstatizdat, 1961.