## 2 Materials and Methods
### Methods
#### 2.3.2 Spectral Methods  
The Horizontal-to-Vertical Spectral Ratio (HVSR) method is a site response analysis technique used to estimate the fundamental resonance frequency of soil or site conditions, utilizing ambient vibrations (microtremors) or earthquake data [29]. It is beneficial foridentifying site amplification effects without needing a borehole or detailed geotechnical data. For a given frequency \(f\), the HVSR is defined as  
\[HVSR(f)=\sqrt{\frac{S_{H}(f)}{S_{V}(f)}} \tag{4}\]  
where \(S_{H}(f)=\frac{1}{2}{\left[\left|F_{\mathrm{z}}(f)\right|^{2}+\left|F_{\mathrm{ y}}(f)\right|^{2}\right. }\) is the average horizontal spectral amplitude (from two orthogonal components \(x\) and _y_), \(S_{H}(f)=\left|F_{\mathrm{z}}(f)\right|^{2}\) is the vertical spectral amplitude, and \(F_{\mathrm{x}}(f)\), \(F_{\mathrm{y}}(f)\), and \(F_{\mathrm{z}}(f)\) are Fourier amplitude spectra of ground motion in the x, y, and z directions. A clear peak in the HVSR curve corresponds to the site's fundamental frequency, as the amplitude of the peak indicates relative amplification. The HVSR approach is useful for seismic microzonation and site classification (e.g., identifying soft vs. hard soil layers). As a non-invasive geotechnical method, it is applied for site resonance frequency identification, urban seismic hazard mapping, and input for 1D site response modeling. Data processing to obtain the HVSR on the free field according to Equation (4) was performed as follows: the recorded time series were visually inspected to identify possible erroneous measurements and stronger transient noise. Each record was then split into 20-30 s long windows tapered with a 5% cosine function. Fast Fourier Transform (FFT) was calculated for each window in each seismometer component. The Fourier spectra were smoothed using the Konno and Ohmachi function [49] with 40 smoothing constants. HVSR was computed as the geometric average of both horizontal component spectra divided by the vertical spectrum for each window. The floor spectra ratio (FSR) method is a technique for determining the natural and resonant frequencies of buildings, which describes the characteristics of buildings concerning earthquakes [50; 51]. This ratio helps identify resonance frequencies, mode shapes, and amplification effects within the structure. For a given frequency \(f\), the FSR is defined as  
\[HFSR(f)=\frac{S_{\mathrm{T}}(f)}{S_{B}(f)}, \tag{5}\]  
where \(S_{\mathrm{T}}(f)\) is the spectral amplitude (e.g., Fourier or response spectrum) at the top floor, \(S_{B}(f)\) is the spectral amplitude at the base (typically the ground or first floor), and \(f\) is the frequency [Hz]. The Fourier amplitude spectrum is computed for each floor using the acceleration time series  
\[HFSR(f)=\frac{\left|F[a_{\mathrm{T}}(t)\right|}{\left|F[a_{\mathrm{B}}(t) \right|}, \tag{6}\]  
where \(\left|F[a_{\mathrm{T}}(t)\right|\) is the Fourier transform of acceleration \(a(t)\) at time \(t\), \(a_{\mathrm{T}}(t)\), and \(a_{\mathrm{B}}(t)\) are acceleration signals at the top and base floors. Peaks in FSR indicate natural frequencies or mode shapes, as a high FSR value implies strong amplification at that frequency. Any changes in FSR over time can indicate structural damage or degradation of the stiffness. The RDM is a signal processing technique used to extract the damped natural frequency, damping ratio, and mode shape of a structure under random excitation, such as ambient vibrations or seismic events [39; 52]. This method is advantageous in structural health monitoring and damage detection. It is applied to non-periodic (random) excitations, allowing for the analysis of structures without requiring knowledge of the exact forcing function.