## 2 Materials and Methods
### Assumptions for the Analytical Assessment of Tremor-Prone Formations Activation Possibility
The Coulomb-Mohr failure criterion recalled in the analysis relies on the effort factor (\(\Omega\)), expressed in terms of the function of elastic strain energy components in the rock strata (\(A_{f}\), \(A_{v}\), \(A_{c}\)) and is defined as [44; 45; 46]  
\[\Omega=\frac{3E_{s}}{(1+\
u_{s})R_{c}R_{r}}A_{f}+sign\big{(}\sigma_{x}+\sigma_{ y}+\sigma_{z}\big{)}\frac{R_{c}-R_{r}}{R_{c}R_{r}}\sqrt{\frac{6E_{s}}{(1-2\
u_{s} )}A_{v}} \tag{1}\]  
\[A_{f}=\frac{1+\
u_{s}}{6E_{s}}\Big{[}\big{(}\sigma_{x}-\sigma_{y}\big{)}^{2}+ \big{(}\sigma_{y}-\sigma_{z}\big{)}^{2}+\big{(}\sigma_{z}-\sigma_{x}\big{)}^{2 }+6\Big{(}\tau_{xy}^{2}+\tau_{zx}^{2}+\tau_{yz}^{2}\Big{)}\Big{]} \tag{2}\]  
\[A_{v}=\frac{1-2\
u_{s}}{6E_{s}}\big{(}\sigma_{x}+\sigma_{y}+\sigma_{z}\big{)}^ {2} \tag{3}\]  
\[A_{c}=\frac{1}{2E_{s}}\Big{[}\sigma_{x}^{2}+\sigma_{y}^{2}+\sigma_{z}^{2}-2\
u _{s}\big{(}\sigma_{x}\sigma_{y}+\sigma_{y}\sigma_{z}+\sigma_{z}\sigma_{x}\big{)} +2(1+\
u_{s})\Big{(}\tau_{xy}^{2}+\tau_{zx}^{2}+\tau_{yz}^{2}\Big{)}\Big{]} \tag{4}\]  
where  
\(A_{f}\), \(A_{v}\), \(A_{c}\)--elastic strain energy density (shear \(A_{f}\), volumetric \(A_{v}\) and total strain energy density, respectively) in the secondary state of stress, J/m\({}^{3}\),  
\(\sigma_{i}\), \(\tau_{ij}\)--secondary stress tensor components (\(i,j=x,y,z\)), Pa,  
\(R_{c}\), \(R_{r}\)--instantaneous compressive and tensile strength, Pa,  
\(E_{s}\), \(\
u_{s}\)--Young modulus and Poisson ratio of the medium, Pa, --. Assuming the ultimate effort to be equal to all physical and structural processes within the material during the induced deformation (under the generalised loads), the formula can be recalled that defines the quoted effort factor in terms of possible values obtained in assessments. As the densities of elastic strain energy components are positively defined (alongside geomechanical parameters), two cases can be distinguished. When \(-1<\Omega<1\), then the actual effort in the analysed section of the rock strata is less than the critical value and the structural failure will not occur. When \(\Omega\geq 1\) or \(\Omega\leq-1\), then effort in the considered strata exceeds the critical value, the failure process (fracturing) will be initiated and a tremor is likely to occur. To analyse the stress-strain characteristics of burst-prone strata in the context of seismic events registered in Year 3 and Year 6, a numerical procedure was applied, based on the following assumptions:  
* It was assumed that fracturing (failure) took place on the horizon level of the contact zone between the dolomite and anhydrite layers (Figure 1); this level appears to be most likely in terms of the pre-supposed error in locating the vertical coordinate of registered high-energy events;
* We were cognisant of the state and progress of mining operations at the time when the two events were registered (December in Year 3--Figure 2 and in December Year 6--Figure 3), and of impacts produced by advancing panelling works, liquidation of working zones and the presence of the existing gobs and development openings in neighbouring plots, as well as the applied methods of roof control (deflection, backfilling);
* The presence of faults with relatively low downthrow (in relation to the height of the face entry drivage) and their impacts on the state of stress within the burst-prone layer were neglected; hence, the lower value of the derived safety factor in regions in the vicinity of faults, characterised by the presence of additional stress concentration zones and areas of elastic strain energy concentration,
* The effects of potential changes in the structure of burst-prone formations (and consequent changes in their geomechanical properties) due to previous mining-induced seismicity were neglected; a quantitative analysis of such changes is in fact not feasible.