On the Use of Mode Frequency in the Comparative Assessment of the Magnetism of Rocks

G. K. Avtenyev

(Presented by the scientific seminar of the Department of Geophysical Methods of Prospecting)

In processing the results of measuring the magnetic properties of rocks and ores in production organizations, the construction of probability-density graphs—variation curves or histograms—is most often resorted to [2, 4, 6].

However, when the distribution patterns of the values of the magnetic properties of individual petromagnetic varieties under study are sufficiently similar, the variation curves are indistinguishable in appearance. In practice such examples occur quite often (Fig. 1). In this case the modal values and the corresponding mean values, as well as the dispersions, are very close. It is not possible to separate the rocks from one another into distinct petromagnetic varieties using only the known simplest calculation techniques. Under these conditions, in order to obtain stable quantitative quantities that objectively characterize the degree of magnetism of rocks, more complex statistical calculations are necessary [1]. In most cases such calculations are laborious.

In the present article we propose a fairly simple and fairly reliable method, tested in practice, for separating rocks according to the degree of magnetism, using only the frequency of the modal value, determined from variation curves and histograms.

The essence of the proposed method is as follows. When the distribution patterns of random variables of individual or related processes are sufficiently similar, and the grouping intervals are identical, the modal values of these variables are grouped fairly close to one another. Thus, for weakly magnetic rocks, under a “semi-normal” distribution law, the modal values, as a rule, are grouped in the region of values close to zero; and for a lognormal distribution law, the modal values are close to the geometric mean value and, for the overwhelming majority of petromagnetic varieties, are close to the geometric mean value and for the overwhelming majority of petromagnetic varieties.

Fig. 1. Variation curves of the magnetic properties of rocks of the Murok–Suraz anomalous zone (northern part of the Kuznetsk Alatau). 1 — hornfels (467); 2 — spilites (300); 3 — porphyrites (468); 4 — diorites, granites, microgranites (136); albitophyres (105). The number of samples analyzed is given in parentheses.

rocks are localized in one and the same initial interval (Fig. 1). In this case, in terms of the absolute value of the mode they are not separated, but they are clearly differentiated by the magnitude of the frequency, or frequency of the modal value (Table 1).

The possibility of obtaining a sample of random variables with a specified probability density is well known [5]. Here it is only necessary to note that, in the general case, the frequency of the mode for one and the same homogeneous set of random values, depending on the sizes of the partial samples, will be a variable quantity tending to a certain constant limit when the sample becomes representative.

This gives grounds for using the quantity reciprocal to the frequency of the modal value (in a representative sample of random values of the measured parameters of magnetic properties) as a stable statistical parameter that is directly related to the degree of magnetism of rocks. Here, in general terms, the larger this parameter, the higher the degree of magnetism of the rocks under study.

The common magnitude of the grouping interval when constructing variation curves for several rock varieties can be determined from the sample with the smallest range of the measured magnetic quantities, using the graph of E. M. Kzyatkovsky [2], I. P. Sharapov [6], or calculated directly by Sturges’ formula, which relates the grouping interval to the range and size of the sample.

Using, as an example, the data of measurements of the magnetic properties of rocks from several areas of the northern part of the Kuznetsk Alatau, it was shown that the results obtained by the proposed method agree well with the results of other statistical methods and with geological data (Fig. 2; Tables 1 and 2). Thus, according to the reciprocal frequency of the modal value—

Table 1

Here \(M_o\) is the modal value. The grouping interval \(\Delta \chi\) for magnetic susceptibility was taken equal to \(500 \cdot 10^{-6}\) CGS, and for residual magnetization, \(250 \cdot 10^{-6}\) CGS.

Table 2

Physical properties of rocks of the Murok–Surazov zone

Note: The median in the lognormal distribution law with probability 99% is taken as the geometric mean value of the magnetic parameters: \((M_o \pm 3s)\).

Fig. 2. Cumulative-frequency plots for the magnetic properties of rocks of the Muryuk-Surazov anomalous zone. 1—albitophyres, 2—porphyrites, 3—hornfels

...tion close to zero, the magnetism of the rocks of the Muryuk-Surazov zone decreases from hornfels (total reciprocal frequency 6.22) to albitophyres (the total reciprocal frequency is 3.08). In the same sequence there decrease the arithmetic mean values of magnetic susceptibility and residual magnetization, and the mean values of the vertical component of the magnetic-field intensity observed over the rocks under study. The only exception is the spilites, which have a distribution of magnetic susceptibility different from that of the other rocks (Fig. 1).

Knowledge of the relative degree of magnetism of each petromagnetic variety of rocks separately is of great importance for the practice of interpreting magnetic-prospecting data. As applied to ore deposits, especially in calculating reserves of magnetic ores, knowledge of the relative degree of magnetism of rocks makes it possible, to some extent, to estimate the share contributed by each petromagnetic variety to the general magnetic background of the enclosing rocks.

In conclusion it should be noted that, in the mass processing and generalization of stock material on measurements of the physical properties of rocks—genetically or spatially related to one another—of individual geological formations, the proposed method for assessing the relative degree of magnetism of individual petromagnetic rock varieties may be regarded as a rapid method.

LITERATURE

1. V. N. Lugovenko, N. N. Lugovenko. Determination of the direction of the magnetization vector of magnetic rocks by calculating certain statistical parameters of the anomalous magnetic field. Geomagnetism and Aeronomy, vol. IV, No. 3, 1967.

2. “Methodological Guide to the Determination of the Physical Properties of Rocks and Useful Minerals.” Gosgeoltekhizdat, Moscow, 1962.

3. N. K. Razumovskii. Logarithmically Normal Law of the Distribution of a Substance and Its Properties. Proceedings of the Leningrad Mining Institute, vol. 20, 1948.

4. G. G. Tarkhov. On the statistical processing of the results of mass determinations of the physical properties of rock samples. Applied Geophysics, issue 20. Gostoptekhizdat, 1958.

5. D. Hudson. Statistics for Physicists. Mir Publishers, Moscow, 1967.

6. I. P. Sharapov. Application of Mathematical Statistics in Geology. Nedra Publishers, 1965.