## 2 Models Taking into Account the Directional Nature of Vibration Attenuation
This model will be referred to as the elliptical model in the remainder of the paper. \[log\hat{a}_{i}^{max}=\alpha_{1}logE_{i}+\alpha_{2}logR_{i}^{*}+\alpha_{3}R_{i}^ {*}+\alpha_{0}+\xi_{i} \tag{2}\]  
where:  
\[R_{i}^{*}=\sqrt{l^{2}+m^{2}};\]  
\[l=p(\left(x_{w}-x_{st}\right)cosq+\left(y_{w}-y_{st}\right)sinq);\]  
\[m=\left(x_{st}-x_{w}\right)sinq+\left(y_{w}-y_{st}\right)cosq;\]  
\[p,q-\text{parameters}.\]  
The model parameters were estimated [30] with the following constraints (3):  
\[ \begin{array}{c}-100\leq a_{0}\leq 100\\ a_{1}>0\\ -100\leq a_{2}\leq 0\\ -10\leq a_{3}\leq 0\\ 0\leq p\leq 100\\ 0\leq q\leq 2\pi\end{array}  \tag{3}\]  
The assumption of an elliptical shape for the isolines determines their symmetrical course with respect to each axis of the ellipse. In order to depart from this condition, the authors proposed a model that takes into account the anisotropy of vibration attenuation without making any assumptions about the shape of the isolines of vibration intensity parameters. Assume that we have a large number of registrations at a given seismic station. The focal point of the analysis is the point with coordinates (0,0), which is also the location of the station. In the first step, angle \(\alpha\), hereinafter referred to as the angle of penetration, was determined, which determines the dilation of the slice \(\Omega_{i}\), containing the epicenters of the registered tremor, from which the vibration attenuation relation will be determined. This slice is marked with the letter \(\Omega_{0}\) in Figure 1. The model parameters determined from these registrations represent the intensity of vibration attenuation for the direction \(\gamma_{i}=0\). In the subsequent steps, slice \(\Omega\) is rotated by \(1^{\circ}\) counterclockwise, thus determining a new set of tremors. The model parameters determined from these represent the vibration attenuation for the direction \(\gamma_{i}=1\). These steps are repeated until a full rotation around the seismic station is performed, that is, until the model parameters representing the vibration attenuation intensity for directions \(\gamma_{0}\), \(\gamma_{1}\),..., \(\gamma_{359}\) are determined. Figure 1 presents the situation in which a penetration angle of \(90^{\circ}\) is assumed. The shaded slice \(\Omega_{0}\) covers the epicenters of the tremors, representing the set of input data for which the ground vibration acceleration attenuation will be determined. This relation will represent the vibration attenuation for the direction \(\gamma=0^{\circ}\). To determine the vibration attenuation relation for direction \(\gamma=150^{\circ}\), for example, taking into account the penetration angle assumed earlier, it is necessary to take into account all the events covered by slice \(\Omega_{150}\). After performing a full rotation around the seismic station, the final result is a \(360^{\circ}\) ground vibration acceleration attenuation relation. This model is described by the relation shown in Equation (4) and will be referred to as the rotational model in the rest of the article. \[log\hat{a}_{ij}^{max}=\alpha_{1j}logE_{ij}+\alpha_{2j}logR_{ij}+\alpha_{3j}R_{ ij}+\alpha_{0j}+\xi_{ij} \tag{4}\]  
where \(i\) is the index of observations in the subsample; \(j\) is the index of the subsample. Figure 1: Graphical representation of the selection of shocks for determining the directional ground damping relation. The direction to the tremor \(\alpha\) is calculated from Equation (5):  
\[ \begin{array}{c}\varphi=arctg\Big{|}\frac{y_{w}-y_{st}}{x_{w}-x_{st}}\Big{|}\; for\;x_{w}\
eq x_{st}\\ \alpha=0\;for\;x_{w}\geq x_{st}\\ \alpha=270\;for\;x_{w}<x_{st}\\ \alpha=\varphi\;for\;y_{w}-y_{st}>0\;and\;x_{w}-x_{st}>0\\ \alpha=180-\varphi\;for\;y_{w}-y_{st}>0\;and\;x_{w}-x_{st}<0\\ \alpha=180+\varphi\;for\;y_{w}-y_{st}<0\;and\;x_{w}-x_{st}>0\\ \alpha=360-\varphi\;for\;y_{w}-y_{st}<0\;and\;x_{w}-x_{st}>0\\ \end{array}  \tag{5}\]  
The question of selecting an appropriate penetration angle still requires comment.