Modal Analysis of Rocks and Ores in Thin Sections

. The article is devoted to the history and justi ﬁ cation of the modal analysis of rocks and ores with a microscope. It is shown that the Delesse-Rosiwal-Glagolev ratios do not follow from the Cavalieri principle. They do not allow one to ﬁ nd the exact volume of the minerals in rocks or ores, but give only their average estimates. It is also shown that the volume fractions of convex mineral grains in rocks and ores, taken equal to the fractions of their ﬂ at sections, are always underestimated if compared with the matrix. Due to the wide variety and complexity of forms of mineral grains, the methods of stereological reconstruction lead to integral equations with a dif ﬁ cult to de ﬁ ne form factor. Most likely, tomography methods should come to replace the modal analysis of rocks and ores in thin sections.


Introduction
Modal (quantitative mineralogical) analysis of rocks and ores in thin sections is one of the first fundamental quantitative methods of mineralogy (including technological mineralogy), petrography (in classifications of rocks and ores, and petrological reconstructions), and lithology (i.e., petrography of sedimentary rocks). That is why its rigorous justification is of fundamental importance owing to the Delesse-Rosiwal-Glagolev ratios, as well as to stereological reconstruction. It makes sense to summarize the history of these methods in Russia and abroad, and to formulate the conclusion about their prospects.

The Cavalieri Principle and the Delesse-Rosiwal-Glagolev Ratios
First of all, we point out that the relations suggested by Delesse (1848) dV i = dS i , Rosiwal (1898) dS i = dL i , and Glagolev (1932) dL i = dN i , decreasing the dimension of space (namely, equating the volume fractions of minerals to areal, areal to linear, linear to point-like), have no relation to the Cavalieri principle: S 1i = S 2i ! V 1 = V 2 (if the areas of all arbitrarily close parallel sections of two bodies are pairwise equal, then their volumes are also equal, Fig. 1).
In recent notation, this principle, historically preceding integral calculus, has a clear meaning: is a continuous function of the areal fraction of a certain mineral along the z axis being normal to the sections. But, the modal analysis of rocks and ores in thin sections, which accumulates the statistics of areal, linear or point fractions of minerals from section to section, has nothing to do with the integration procedure. It only leads to an assessment of their average values. In this case, it can be argued that the volume of any mineral in a rock or ore is within the certain interval: where Dz is the thickness of rock sample under study. Despite of this contradiction, which was also considered in the works (Krumbein 1935;Chayes 1956), the method became firmly established in practice because of its apparent simplicity and was step by step automated (Shand 1916;Wentworth 1923;Hunt 1924;Dollar 1937;Hurlbut 1939) up to the use of modern computers for image analysis of thin sections. The list of parameters characterizing the cross-section of minerals, and the speed of processing have grown many times. But, in terms of the reconstruction of the true metric characteristics of mineral grains from those of their flat or even linear sections, the ideology remains the same.
Companies that produce computer structure analyzers offer software packages without discussing the fundamental problems. The analysis of 2D images does not use the available chapters of mathematics. For example, the distances between the mineral grains in thin section are replaced by the Euclidean distances between the points taken within the grains, whereas there is a more complicated, but easily programmable Hausdorff's metric which allows one to do this procedure correctly. In turn, it gives us the possibility of calculating space covariograms between the mineral grains of different species and their various clusters in the rock.

Stereological Reconstruction
A new line of research, i.e. stereological reconstruction, arose from the obvious observation that an arbitrary cross section of a spherical shape is always less than its characteristic cross section (Fig. 2, above). And it follows from this that the volume fraction of the convex mineral phase in the rock and ore, equal to the fraction of its flat sections, is always underestimated if compared with the host matrix. The corresponding general problem -finding the distribution of true particle sizes from the size distribution of their random sectionsbelongs to the inverse problems typical for geophysics. It is analytically solved only for spherical and ellipsoidal particles due to relatively simple description of these forms (Wicksel 1925(Wicksel , 1926. But, even in this case, the practical use of the theory requires the selection of the best solution and an estimate of the errors (Fig. 2, below). For more complex forms of mineral grains this can't be done without mathematical modeling on powerful computers. Ivanov O.P., Ermakov S.F., Kuznetsova V.N. Improving the accuracy of determining the weight particle size distribution of ore minerals from measurements in thin Fig. 2. Left: the size of the cross section of a convex grain is always less than the characteristic one, in the modal analysis its volume fraction is underestimated. Right: to recognize the true size of spherical particles by the size of circular sections; R = 100particles with a radius of 100 arbitrary units; R = 100, r = 50two sets of particles of the same type, indistinguishable in sections (for example, two generations of one mineral); horizontal scalesection radii from 0 to 100 bits per 10 classes; vertical scalefrequencies by classes (lower curves) and accumulated frequencies (upper curves).

Conclusions
Thus, due to the extraordinary diversity and complexity of the forms of mineral grains in rocks and ores, the methods of stereological reconstruction lead to integral equations with an analytically difficult-to-define form factor. The practical application of the theory is drowning in the selection of the best solution to the inverse problem and complex estimates of measurement errors. It seems that the modal analysis of rocks and ores in thin sections should be replaced by tomography methods. Standardizing modal analysis of rocks and ores in thin sections by creating their artificial counterparts with previously known volume fractions of mineral grains and a wide range of petrographic structures can serve as an inter-laboratory comparison of the accuracy of the method. But it does not solve the problems in essence.
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.
The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.