## 2 Models Taking into Account the Directional Nature of Vibration Attenuation  
For a complete description of the ground vibration distribution of an area affected by induced tremors, it is necessary to have a relation called the vibration attenuation relation. This relation is a function that relates a selected parameter, characterizing the intensity of ground vibrations (e.g., PGA, PGV) to the energy (magnitude) of the tremor and its distance (epicentral, hypocentral) from the source. As a rule, this relation is determined by regression analysis. This is due to the significant complexity of the analytical description of the phenomena occurring between the source of the vibrations (tremor hypocenter) and the surface area. The intensity of ground vibrations caused by mining tremors depends on diverse factors: the seismic energy of a tremor; the depth of the focus and the epicentral distance of an observation point; the source mechanism; the centre's geological structure, especially local ground conditions at the point of observation--the type and thickness of loose overburden; its watering; as well as the area topography. Assuming a known tremor energy, the vibration attenuation relations make it possible to predict the intensity of surface vibrations at different distances from the source. Additional parameters that are taken into account in studies of ground motion prediction equations are: amplification of ground vibrations, directivity of vibration propagation and dominant frequencies of ground vibrations. A wide overview of the applied models can be found in the literature [4]. Models of vibration attenuation relations estimated from data obtained from measuring stations located on the surface are used to predict vibration parameters averaged for the analyzed area. In this paper, the model proposed by [PERSON] [5] was first used to describe the relationship between PGA, tremor energy and distance from the source to the seismometer station (Equation (1)):  
\[log\hat{a}_{i}^{max}=\alpha_{1}logE_{i}+\alpha_{2}logR_{i}+\alpha_{3}R_{i}+\alpha _{0}+\xi_{i} \tag{1}\]  
where: \(a_{i}^{max}\) is the maximum value of ground vibration acceleration, [m/s\({}^{2}\)]; \(E_{i}\) is the energy of a tremor, [J]; \(R_{i}\) is the epicenter distance, [m]; \(\alpha_{0},\alpha_{1}\alpha_{2}\alpha_{3}\) are the estimated parameters of the model; \(\xi_{i}\) is the random component;  
\[R_{i}=\sqrt{\left(x_{w}-x_{st}\right)^{2}+\left(y_{w}-y_{st}\right)^{2}};\]  
\(x_{w},\ y_{w}\) are the coordinates of the tremor; and \(x_{st},\ y_{st}\) are the coordinates of the measuring station. The parameters of the above model are estimated using the least squares method. The form of attenuation relation model (1) is most often used for estimating local attenuation relations in Polish mines. The model described by Equation (1) will be referred to as the classical model in the remainder of this article. This model assumes the isotropic nature of vibration attenuation, which results in the isolines of PGA and PGV taking the shape of circles. This is a rather broad simplification of the problem, if only due to the fact that there is a phenomenon of vibration amplification, which, as a rule, causes a differentiation of the recorded parameters of vibration intensity at similar distances from the source of the tremor. Unfortunately, this variation in observation results cannot always be explained by the phenomenon of vibration amplification. Numerous studies on the phenomenon of ground vibration attenuation indicate that the directional nature of attenuation can have a significant effect on the shape of isolines of PGA and PGV [9; 12; 13]. In the case of the second model considered in the paper, in order to take into account the anisotropy of vibration attenuation, the authors assumed that the isolines take the shape of ellipses [5; 9; 10]. The directionality of vibration damping was introduced into the model by transforming the variable associated with the epicenter distance, as shown in Equation (2).