## 5 Discussion  
A measure of the standard deviation of the residual component and [PERSON]'s correlation coefficient of theoretical and empirical values were used to compare the results obtained for the three models. The results are shown in Table 4.  
Figure 6: Vector field of the gradient function of the vibration attenuation relation (\(5\times 10^{6}\) J).  
The presented values prove that models taking into account the directional nature of vibration attenuation better describe the actual distribution of values of the vibration intensity parameter than the classical model. The slight increase in the fit to the empirical data of the rotational model compared to the classical model is due to the selection of a fairly wide penetration angle. The applied angle of penetration (180\({}^{\circ}\)) ensured that the assumptions of the least squares method were met. A better model fit to the empirical data could be achieved by selecting a smaller penetration angle, which in turn is associated with sample size (occurred tremors). In addition, a better fit of the model to the empirical data can be sought by using other estimating methods. These can be nonlinear least squares methods or non-classical methods. It is possible that geostatistical methods would lead to similar results.  
Noteworthy is the fact that both the elliptical and rotational models indicate a high convergence in the direction of anisotropy of vibration attenuation. For the elliptical model, the directions of the minor axis and major axis of the ellipse are 69\({}^{\circ}\) and 339\({}^{\circ}\), respectively. For the rotational model, the direction of the fastest increase in the value of the vibration acceleration parameter was 67\({}^{\circ}\), while the direction of the slowest increase in this parameter was estimated at 331\({}^{\circ}\).  
In addition, these directions are closely in line with the directions for which the extreme values of the parameter \(\alpha_{2j}\) (see Equation (4)) were calculated. This parameter is related to the geometric scattering of the wave front for the rotational model. The maximum value of this parameter was \(-\)0.048 for a direction equal to 334\({}^{\circ}\), while the minimum value, in turn, was estimated at \(-\)0.3736 for a direction equal to 71\({}^{\circ}\). This may mean that this parameter explains the direction of anisotropy to the greatest extent, especially since, in terms of the empirical material analyzed, it was characterized by the greatest heterogeneity when taking into account the direction of the epicenter of the tremor--the seismic station. In order to compare the anisotropy determined based on the elliptical and rotational model, a measure of the mean square deviation of the predictions of the anisotropic model and the isotropic model can be used for predetermined energy levels and distance from the source [31]. These relationships are presented in Equations (8) and (9).  
\[A_{eli}=\sqrt{\frac{\sum(\hat{a}_{eli}-\hat{a}_{klas})^{2}}{N}} \tag{8}\]  
\[A_{rot}=\sqrt{\frac{\sum(\hat{a}_{rot}-\hat{a}_{klas})^{2}}{N}} \tag{9}\]  
If the vibration attenuation phenomenon does not show anisotropy, then the values of A\({}_{\rm eli}\) and A\({}_{\rm rot}\) should be zero. For the estimated models, for an energy equal to 5 \(\times\) 10\({}^{6}\) J and an epicenter distance of 1000 m, they are 6.08 and 14.42, respectively. This means that for the elliptical model, the predicted values deviate on average from the predictions of the isotropic model plus or minus 0.0061 m/s\({}^{2}\), and 0.0144 m/s\({}^{2}\) for the rotational model.