## 4 Results and Discussion  
A classical model of the attenuation relation of ground vibration acceleration was determined for the recorded vibration cases. The structural parameters of the model, estimated using the least squares method, are shown in Equation (6). \[\mathrm{log}\hat{\mathrm{a}}_{\mathrm{i}}^{\mathrm{max}}=0.5004\mathrm{log} \mathrm{E}_{\mathrm{i}}-0.2764\mathrm{log}\mathrm{R}_{\mathrm{i}}-0.00019 \mathrm{R}_{\mathrm{i}}-3.233+\xi_{\mathrm{i}} \tag{6}\]  
The \(p\)-values for all estimated model parameters were not greater than 0.001, which means that they were statistically significant. The standard error of the residual component was 0.2283. The standard error of the residuals was calculated as the square root of the sum of the squared residuals divided by the sample size reduced by number of estimated parameters. The properties of the residuals of the model were examined in terms of normality and homoscedasticity using appropriate statistical tests. For normality, the Jarque-Bera test was used. The \(p\)-value for the JB statistic is lower than 0.001. However, when referring the central limit theorem, it is sufficient that the residual distribution is asymptotically normal, considering the very large sample size. Residual homoscedasticity was tested using the Breusch-Pagan statistic. The \(p\)-value equalled 0.21, which means that there is no evidence to reject the hypothesis of homoscedastic residuals. Based on the same data set, the parameters of the elliptical model were determined. The results are shown in Table 3. The values of the parameters \(p=1.24\) and \(q=1.21\) mean that the root of the quotient of the length of the major axis to the length of the minor axis of the ellipse is 1.24, while the angle of rotation of the ellipse around the beginning of the coordinate system is \(69^{\circ}\) (the direction of the minor axis of the ellipse). The standard error of the residual component for the elliptical model is 0.2267. In the next part of the analysis, the parameters of the vibration attenuation relation for the rotational model were estimated. A penetration angle of \(180^{\circ}\) was assumed, which resulted in the size of the least numerous rotational subsample being 540 tremors. In addition, this angle made it possible to obtain parameter estimators that were significant at a level equal to 0.01 and consistent with their physical interpretation. Also noteworthy is the fact that only at a penetration angle of \(100^{\circ}\) was the abundance of the least numerous \begin{table}
\begin{tabular}{c c c} \hline
**logE** & **n—Abundance** & **f—Percentage** \\ \hline
[1.99; 2.62] & 29 & 0.72 \\ \hline
[2.62; 3.25] & 87 & 2.16 \\ \hline
[3.25; 3.87] & 225 & 5.58 \\ \hline
[3.87; 4.49] & 514 & 12.75 \\ \hline
[4.49; 5.12] & 1271 & 31.52 \\ \hline
[5.12; 5.74] & 988 & 24.50 \\ \hline
[5.74; 6.36] & 548 & 13.59 \\ \hline
[6.36; 6.98] & 295 & 7.32 \\ \hline
[6.98; 7.61] & 59 & 1.46 \\ \hline
[7.61; 8.23] & 11 & 0.27 \\ \hline
[8.23; 8.85] & 4 & 0.10 \\ \hline
[8.85; 9.48] & 1 & 0.02 \\ \hline \end{tabular}
\end{table} 
Table 2: Interval series for the logarithms of the energy of the analyzed tremors. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Parameter** & **a0** & **a1** & **a2** & **a3** & \(p\) & \(q\) \\ \hline Parameter value & \(-3.2170\) & \(0.5019\) & \(-0.2767\) & \(-0.00018\) & \(1.24382\) & \(1.2112\) \\ \hline \end{tabular}
\end{table} 
Table 3: Parameter estimation results for the elliptical model. subsample 40 observations. Figure 3 shows the dependence of the minimum subsample size on the adopted penetration angle. In the figure, the red line indicates a value of 40 units in the subsample. Figure 4 plots the values of the estimators of the models' parameters for each \(\gamma_{\rm i}\) direction. The values of the estimator associated with the free expression range from \(-3.975\) to \(-2.964\).