The paper presents the considerations devoted to homomorphism analysis of processes: displacement and medium (rock) vibration, generating physical risk for the area of topological transformations. The aim of the study was to associate interactions: mining induced vibrations and displacement and their temporal relations. The basis for the analysis were both: sets of observed displacements -measured with GPS technology and recordings of acceleration process by using 3D vibration sensors, as well as mathematical modelling of these processes. It has been indicated, that the carrier of a formal process will be the probability space -which implies the stochastic nature of the phenomenon. Measures of the destruction occurrence -a critical process -there are the probability measures. In the estimated moment of time t, the results of observations of both processes -which are sets of real numbers -indicate that objects that are at risk of event are subjected to danger or failure. Fundamental here are mappings, used for modelling of concurrent processes, those communicate with each other. Features of timing relationships are: the structure and dynamics of processes.The emergence of the conflict -the occurrence of a critical event -has been estimated on the basis of the lemma for reaching of the outer extremes, in terms of equivalence relations in a fixed set of circumstances. The structural similarity between the two processes, their predictability and determination of characteristic frequencies was performed with using the relevant statistics, including the so-called Hurst exponent. An important element of proceeding the threats is the relationship in the domain of extreme events as a decisive influence -the difference compared to traditional analysis. The application layer contains specific examples of estimating the risk of a critical event arising.

Sinkholes that are often formed over shallow mining workings constitute a significant threat to buildings, infrastructure and-most of all-to the residents. These deformations often occur after a long time following the completion of mining works. A considerable number of sinkholes are formed above shallow headings due to the loss of bearing capacity of old wooden supports. Because of the above, the problem of predicting the formation of sinkholes gains significance. This paper presents the assumptions of Strzałkowski's method of predicting discontinuous surface deformations (sinkholes). The deterministic model reflects the essence of the mechanism of destruction of the rock mass surrounding the void. The presented study case-another example of using the method for providing an ex post prognosis indicates its practical usability. The computer programme used in this paper is helpful to laborious calculations.

The aim of this work is the estimation of the risk of mining damage occurrence, based on uncertain information regarding the impact of the concurrent processes of deformation and vibration. This problem concerns the experimental and theoretical description of the so-called critical phenomena occurring during the reaction mining area ↔ building object. Post-mining deformations of the rock mass medium and paraseismic vibrations can appear at a considerable distance from the sub-area of the mining operation -hence, the determination of the measures of their impacts is usually somewhat subjective, while the estimation of the mining damage based on deterministic methods is often insufficient. It is difficult to show the correlation between the local maximum of the impact of the velocity vector amplitude and the damage to the building -especially if the measures of interaction are not additive. The parameters of these impacts, as registered by measurements, form finite sets with a highly random character. Formally, it is adequate to the mapping from the probability space to the power set. For the purposes of the present study, the Dempster -Shafer model was used, where space is characterised by subadditive and superadditive measures. Regarding the application layer, the conclusions from the expert evaluations are assumed to be the values of random variables. The model was defined, and the risk of damage occurrence was estimated.Keywords: mining area, risk of damage, stochastic process, uncertainty, model, expert inference, imprecise probability Celem pracy jest szacowanie ryzyka powstania szkody górniczej, poprzez niepewne informacje dotyczące oddziaływania współbieżnych procesów deformacyjnych i drgań. Problem dotyczy doświad-czalnego i teoretycznego opisu tak zwanych zjawisk krytycznych, zachodzących podczas reakcji teren górniczy ↔ obiekt budowlany. Pogórnicze deformacje ośrodka oraz drgania parasejsmiczne ujawniają się również w znacznej odległości od podobszaru eksploatacji -stąd też wyznaczenie miar tych oddziaływań z reguły jest nieco subiektywne a szacowanie szkody górniczej metodami deterministycznymi często jest niewystarczające. Trudno jest wykazać, że istnieje skorelowanie pomiędzy lokalnym maksimum oddziaływania amplitudy wektora prędkości a szkodą w obiekcie -zwłaszcza, jeśli miary oddziaływania

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The paper analyses the process of post-mining displacements generated by underground mining. Innovative mathematical structures for the modeling of hazard field emission were developed as strong solutions to partial differential equations in R3+1. Moreover, a stochastic equation in L2(Ω) (probabilistic space) was defined and applied as a model that takes into account the randomness of the process. Monitoring of a mining area based on solutions in the GNSS technology and classical geodesy supports the analysis of topological transformations of a given subspace. The data was archived and stored in digital form and then analyzed in many ways. The quality of the representation (measurements and modeling) was estimated with the use of incremental statistics. Thus, obtained distributions of density function are not ranked as normal distribution. The performed analyses make it possible to predict the optimal scenarios for post-mining environmental hazards.