“Complexity–entropy” diagrams and their application to the study of coal tectonic disturbance

Annotation. It is investigated the possibility of using so–called "complexity–entropy" diagrams for the quantitative description of the degree of coal disturbance using coal images obtained by means of scanning electron microscope (SEM). These diagrams plot structural complexity measure (vertical axis) versus entropy measure (horizontal axis) for distribution of probability given in some way on the image. In this paper, the values of both measures were calculated on the basis of the shearlet transform, and the Jensen divergence was used as the basic divergence for calculating the complexity measure. All calculations were performed for more than 140 images of coal specimens with various degrees of disturbance, obtained from the quiet zone of the seam and the outburst zone. As a result of research, it was found that two-dimensional distributions for measures of complexity and entropy in most cases are informative data sets for differentiating coals by degree of complexity. Moreover, such characteristics of these distributions as mathematical expectation and, to a less degree, mode can be used as simple quantitative descriptors of coals with various degrees of disturbance. These characteristics can be used to show the closeness of the spatial structure for the analyzed coal specimens to strictly periodic or absolutely chaotic ones. On the basis of the obtained results, conclusions about the possibility to separate coals according to the degree of their outburst hazard were done.


Introduction
Natural coal is a product of a complex chain of transformations of heteromolecular substances of plant remains under the influence of biological, chemical and tectonic factors at high temperatures and pressures. Coal seams may contain a large amount of gas, mainly methane, most of which is in the absorbed state in the coal substance [1]. The complex irregular structure of the pore space of fossil coals is largely determined by the distribution of voids (pores, microcracks) filled with gas by their number, size and directions. The sizes * Corresponing author: olga_malinnikova@mail.ru of these voids vary in rather wide limits: from 0.3 nm to 10 cm, and the directions depend on the dip and strike of rock layers and movement of material particles in coal seams [2]. It is known that the outburst-hazardous coal seams generally have a more disturbed structure, which is often a superposition of several systems of exogenous and endogenous fracture, and have increased microporosity. Thus, the study and quantitative assessment of natural and man-made disturbance of coal can contribute to the understanding of the mechanism of rapid release of sorbed methane, the participation of methane in the destruction of coal and improve the methods of preliminary prediction of dangerous gas-dynamic phenomena in the development of coal seams [3].
In general, the processes of multiple destruction of fossil coal (accumulation of destructions in coals) are the processes of spatial self-organization of their structure. The formation of the main macrofracture in the process of destruction of coal seams is preceded by the process of development (origin, movement, growth and aggregation) of microdefects (pores, microfractures, dislocations, etc.), which is stochastic in nature [4]. Not so long ago, the stochastic behavior of the system could be explained only by the influence of random forces. However, in recent years, the chaotic behavior of nonlinear deterministic systems has become widely known. Chaos is a rather unusual form of behavior of a deterministic system in a steady state. Although the evolution of chaotic system is uniquely determined by dynamic laws and no random forces acting on her, the dynamics of the system in some area of the phase space is stochastic [5]. Chaos easily occurs in many natural and living systems where nonlinearity exists. In this regard, the question arises: does the image of the surface structure of the coal show the result of a chaotic or random process? One of the methods for solving this problem is the use of "complexity-entropy" diagrams, proposed in [6], which are planes of values of a measure of complexity (vertical axis) depending on the corresponding values of the entropy of the probability distribution (horizontal axis). When constructing the "complexity-entropy" diagrams, the key issue is the choice of the method of forming the measure. In the case of studying the disturbance of coals by their SEMimages, methods of forming a measure that take into account the high degree of anisotropy of the surface of the coal samples are interesting.
Over the past twenty years, various methods have been proposed to recognize anisotropic objects, among them: directional wavelets, complex wavelets, contourlets, curvlets, etc. In turn, Donoho D., Labate D. and Kutinek G. [7][8][9][10][11] suggested a slightly different approach to the analysis of anisotropic components based on shearlettransformation. Unlike wavelets or curvlets, the system of shearlets is built in the class of affine systems and has the ability to recognize the directionality due to the additional shift parameter. Shearlets have characteristics that favorably distinguish them from a number of similar functions: a finite number of generating functions; optimal representation of the anisotropic characteristics of the analyzed data; fast algorithmic implementation; unified approach to the decomposition of continuous and discrete data [12]. For the first time, the use of a shearlet transform to create "complexity-entropy" diagrams was proposed by Brazhe A. in [13]. In this paper, we studied the possibility of using "complexity-entropy" diagrams based on shearlet transform to quantitatively describe the degree of disturbance of coals by their SEM-images. Thus, given scale a , shear s , and translation t , it is possible define the "mother" shearlet function

Methodology of research
and on its basis a continuous shearlet-transformation of a digital image, as a convolution of the image with scaled, sheared and translated copies of the "mother" shearlet function.
The discrete system of shearlets associated with shearlet  , is the set of functions , In general, the scale parameter is selected from the set   This leads to the following shearlet transform [13]: Thus, a discrete shearlet transform is defined as a mapping from the original image to a set of shearlet coefficients: , It was shown in [13] that because the translation grid (dimensionality m ) is scale independent and redundant, the shearlet coefficients can also be represented as i S x y are images, obtained by convolution of the function ( , ) f x y with directional filters of different spatial scales of the same size as the function ( , ) f x y .

Entropy and complexity measures based on shearlet transform
Both entropy and complexity measures are defined as functionals of some probability distribution

"Complexity-entropy" diagrams
In this paper, the above-mentioned interrelated measures of entropy and complexity of digital images were used to construct of "complexity-entropy" diagrams. Thus  To implement an algorithm for constructing the above diagrams, a computer program in Python was developed using the free program shearlexity [17], standard Python libraries scipy and matplotlib for scientific calculations and visualization, as well as fast discrete transform library PyShearlets [18].
The interpretation of these diagrams is based on the following intuitive considerations: zero entropy and complexity correspond to a completely regular structure, and high entropy and zero complexity correspond to a completely random spatially independent noise [19,20].

Source data
In this study, imaging of coal specimen surfaces was performed using scanning electron microscopes JEOL JSM 5910-LV and Jeol-6610-LV. The spatial resolution of the microscopes is more than 10 and 100 nm for secondary and reflected electrons, respectively. Low-energy secondary electrons are used in imaging surface topography. Natural-shape coal specimens were placed in a work camera via a gate. In the mode of registration of secondary electrons, the work camera was vacuumized (with >10 -6 mm hg vacuum). Secondary electrons were recorded by a standard detector, which a type of a sweeping-field photomultiplier tube connected to scintillator.
As the source data for this study was IPKON collection of coal specimens from the Zapolyarnaya mine (Vorkuta) and the Kirov mine (Leninsk-Kuznetsk), obtained from outburst-nonhazardous zones and outburst zones. For our research, we analyzed microstructure of coal surface in the images magnified 1000 times which showed coal grains with a characteristic size from 0.5 to a few microns. Methane can desorb from such grains, diffuse and flow in fractures as free gas [21].
Processed regions of these images were same as in [22], which further allowed to compare obtained results with the results from [22].

Research results
In Figs. 1, d and 2, d, examples of typical "complexity-entropy" diagrams constructed for selected regions of test coal surface images (Figs. 1, a and 2, a) are shown. These diagrams are based on values for measures of local entropy H and complexity C , which are demonstrated in Figs. 1, b, c and 2  At the same time, the images of coals from the outburst zone have in most cases a higher mean local complexity and a lower mean local entropy compared to the images of coals from the quiet zone of the seam (Fig. 3, a). Modal values, as seen in Fig. 3, b, practically do not find this separability. Comparing the obtained results with the results of multifractal analysis [22,23] it can be noted that both approaches indicate a more complex surface structural organization for coals from the outburst zone compared with coals from the quite zone of the seam. This is manifested by the mean values of measures of local statistical complexity and in the spectra of fractal dimensions. The discovered regularities require further study and confirmation on a larger series of test samples to reveal the potential, which, as our studies have shown, is inherent in the described methods of SEM-image processing.

Conclusions
The following conclusions can be drawn from the present study:  In most cases, the two-dimensional distributions for measures of complexity and entropy are informative data sets for differentiating coals by degree of complexity. Moreover, such characteristics of these distributions as mathematical expectation and, to a less degree, mode can be used as simple quantitative descriptors of coals with various degrees of disturbance. These characteristics can be used to show the closeness of the spatial structure for the analyzed coal specimens to strictly periodic or absolutely chaotic ones.  By analyzing more than 140 test images of coal specimens from the quite zone of the seam and the outburst zone, it was found that most of the studied specimens exhibit a random (not chaotic) nature of the distribution of anisotropic properties.  In most cases, images of coals from the outburst zone have a higher mean local complexity and a lower mean local entropy compared to images of coals from the quite zone of the seam.
This work was supported by the Russian Foundation for Basic Research, project no. 19-05-00824.