## 2 Materials and Methods
### Site Effect
#### 2.3.2 Empirical HVSR  
The site effect coefficients and the empirical HVSR analysis, using single station measurements, have so far only been studied using translational motion recordings but not for rotation. Therefore, we adapted the HVSR method to rotation in this study, considering the opposite direction of motions. The rotational motion is described as the curl of three components: the torsion (around the vertical axis) and two components of rocking (around the horizontal axis). The planar orientation of the rotational components allowed us to estimate the rotational HVSR as the torsion-to-rocking spectral ratio (TRSR) technique to find the local site effects for rotational data [21; 23; 24]. The H/V ratio curves were calculated using the J-SESAME (Site Effects Assessment Using Ambient Excitations) software (version 1.08) [36]. In our case, mining tremors were the source of data. Each recorded tremor was cataloged as a translational (acceleration) and rotational (velocity) waveform, and the values of peak ground acceleration (PGA) in \(\left[\frac{\text{m}}{\text{s}^{2}}\right]\) and peak rotational velocity (PRV) in \(\left[\frac{\text{rad}}{\text{s}}\right]\) were also found. The ambient noise was removed. The H/V ratios were calculated according to the SESAME criteria, which allowed us to select a reliable amplification peak. The main criteria correspond to the number of time windows and their duration, which are selected according to the expected values of the resonant frequency. The length of the windows is inversely proportional to the minimum expected resonance frequency value. Consequently, the longer time windows should be used at the sites where low resonance frequency is expected. The number of time windows corresponds with the minimum requirement to ensure statistical stability, and according to the SESAME criteria, the Gaussian statistic is achieved for the 200 significant cycle numbers at least. The number of significant cycles is determined by multiplying the window length, number of windows, and resonance frequency. Moreover, these criteria are called criteria I and II of reliability. The following two criteria, criteria III and IV of reliability, define the frequency ranges of resonant frequency with a measurement error. First, the waveforms were windowed. In our case, the windows overlap due to the short duration of each event (a few seconds). Considering that peak values of the signals were noticed at the S-wave phase, which is an effect of the double-couple source mechanism of the analyzed events, the HVSR estimations procedure was carried out only for the S-wave content. Next, the cut signals were transformed using the Fast Fourier Transform (FFT) and smoothed using the Konno-Ohmachi logarithmic window function [44]. In the end, the estimated HVSRs for each event were averaged [36; 38]. The time series of the windows allowed us to estimate signal curves after S-wave generation. The microtremor signals were a vestigial element whose potential impact on the H/V spectrum was insignificant. The following equation gives the averaged H/V estimations of the \(i\)-th event registered by translational signals:  
\[log_{10}HVSR_{AVi}(f)=0.5\big{[}log_{10}(S_{Txi}(f))+log_{10}\big{(}S_{Tyi}(f) \big{)}\big{]}-log_{10}(S_{Tzi}(f)) \tag{1}\]  
where \(S\) denotes the spectrum result obtained from horizontal \(T_{x}\) and \(T_{y}\) and vertical \(T_{z}\) components of the translational motion. Estimation of the H/V for the horizontal component of the translational motion can be described by the following formula:  
\[log_{10}HVSR_{ki}(f)=[log_{10}(S_{Tki}(f))]-log_{10}(S_{Tzi}(f)) \tag{2}\]  
where \(k\) determines the analyzed components of the recordings. The H/V of the rotations is estimated under the assumption that the torsion tilts are in a parallel plane to the horizontal recordings of the translations, while the rocking tilts are in a parallel plane to the vertical recordings of the translations [11; 21; 23; 24] (Figure 3b). In consequence, rotational H/V is the opposite of translational H/V.