## 2 Methodology for Assessing the Integral Index of Dynamic Stability of a Structural Tectonic Block  
Information about the methodology that was developed by the Geophysical Center RAS is detailed below. Its algorithms are based on the language of fuzzy sets and fuzzy logic [26; 30]. DMA algorithms have successfully proven themselves in solving a wide range of geological and geophysical problems in the field of Earth sciences (strong recognition of earthquake-prone areas, monitoring of volcanoes and geomagnetic activity, etc.) [26; 31; 32; 33; 34]. In [26], a finite system of functions of geological and geophysical parameters was studied on a two-dimensional grid. The challenge of calculating the anomaly measurements for a group of geological and geophysical features was formalized. The mathematical component of the sustainability assessment methodology for the structural-tectonic block is described in detail in [1; 26] and tested at the HLRW disposal site. Its main provisions are as follows. The area of interest \(\Pi=\left\{o\leq x_{1}\leq t;u\leq x_{2}\leq v\right\}\) exists on the coordinate plane \(\mathbb{R}^{2}(x_{1},x_{2})\). A set of geological and geophysical fields \(F\) from \(m\) datasets (in the form of digital maps of various parameters: geographical, geological, geophysical, geodynamic, economic, and others) is selected for this area for use in evaluating the stability of the region \(\Pi\). \[F=\left\{f_{1},\ldots,f_{m}\right\}, \tag{1}\] \[f_{i}:\ \Pi\rightarrow\ \mathbb{R};\ i=1,\ldots,m\]  
The goal is to divide the region \(\Pi\) into relatively unstable (conditionally dangerous) and stable (conditionally safe) elements. This ranking is also called geodynamic zoning [10]. In the region \(\Pi\) a regular grid \(W=W(h_{1},h_{2})\) with nodes \(w\) is given:  
\[W=\left\{w=\left(o+ih_{1};u+ih_{2}\right)\left| \begin{array}{l}i=0,\ldots,N; \ h_{1}=\frac{t-o}{N}\\ j=0,\ldots,M;\ h_{2}=\frac{v-u}{M}\end{array} \right.\right\} \tag{2}\]On the grid \(W\) it is necessary to analyze the spatial distribution of systems of functions \(F\) in the neighborhoods of the node \(w\). To do this, we defined a fuzzy measure of activity \(\mu(w)\) in the range from 0 to 1, according to the rules described below. Step 1. Calculation of the dynamic indicator. Each parameter \(f\) from the set \(F\) is a distribution function on the grid \(W\). For each parameter \(f\), it is possible to determine the dynamic indicator \(D_{f}\), which is a functional characteristic of the measurement of \(f\). The value of \(D_{f}(w)\) is interpreted as a quantitative assessment of the behavior of the function \(f\) in the node \(w\in W\), calculated according to the specified rules. In terms of data analysis, the dynamic indicator \(D_{f}(w)\) is a feature. Step 2. Calculation of measure of dynamic activity index (MDAI). For each dynamic indicator \(D_{f}\), a measure of activity (anomaly) \(\mu D_{f}\) is determined in the range from 0 to 1. It shows the degree of expression of the property \(f\) in the node \(w\), defined by the indicator \(D_{f}\). The measure of dynamic activity index \(\mu D_{f}\) is calculated from the dynamic indicator \(D_{f}\) in the methodology of discrete mathematical analysis. The transformation \(D_{f}\rightarrow\mu D_{f}\) translates the analysis of the measurement of \(f\) into the language of fuzzy logic: measures of dynamic activity index \(\mu D_{f}\) for different dynamic indicators of \(D_{f}\) are fuzzy structures on the grid \(W\), and they can be combined in any compositions and any quantities using fuzzy logic operations and averaging. Step 3. Calculation the integral measure of activity \(\mu_{F}\). As the last step of the algorithm, all the measures of dynamic activity index \(\mu D_{f}\) are combined into a single integral indicator of \(\mu_{F}\). The formula of the combination is the arithmetic mean of all measures of dynamic activity index \(\mu D_{f}\). Depending on the research task, weighting coefficients for activity measures or other compound formulas can be used [35].