Orientation-Independent HVSR Calculation
The horizontal-to-vertical spectral ratio (HVSR) is computed for each horizontal component of seismic recordings at each station. This involves calculating spectral ratios, similar to signal-to-noise ratios, by selecting S-wave windows unaffected by tapering. Acceleration FAS (Fourier Amplitude Spectra) are computed and smoothed using a Konno and Ohmachi filter (b=40). The mean HVSR per site is determined as the logarithmic average of all events, considering that empirical spectral ratio ordinates are lognormally distributed. The mean is calculated at each frequency using recordings within their legitimate bandwidth, meaning the number of contributing earthquakes varies with frequency.
To achieve an orientation-independent estimate, the FAS of the two horizontal components are combined using the square root of the sum of squares (SRSS). For example, Figure 4 (top) displays rotation-invariant mean HVSRs, showing the number of usable recordings per frequency. Curves and their standard deviation are drawn only when at least 5 usable events contribute, ensuring robust estimation.
A reference site is expected to show a relatively flat HVSR close to unity. Significant departure from reference conditions is identified by various thresholds, such as HVSR > 2, > 2√2 (approximately 2.8), or > 3. HVSR is generally considered to underestimate the "true" site transfer function compared to the standard spectral ratio (SSR). The premise of HVSR is that the vertical component is expected to amplify frequencies higher than those of horizontal components, allowing identification of at least the first S-wave resonant peak, albeit with an underestimated amplitude. Given the SRSS computation for horizontal components, a threshold of HVSR > 2√2 is used to identify significant resonant peaks, and HVSR < 2 is used for selecting potential reference sites. For instance, station APE exhibits a clear resonance at 6 Hz with an amplitude > 2.8, while station IMMV shows a flat HVSR with an amplitude < 2, indicating a potential reference site.
Rotational Sensitivity of HVSR
Reference sites should ideally not exhibit strong directional dependence, meaning ground motions should not be sensitive to sensor orientation within the frequency range of interest. To assess the variability of site response to azimuth, the technique of rotating each time series by 10° increments (from 0–90°) is employed, recomputing FAS and HVSR each time. This directional dependence indicates a departure from 1D behavior, likely due to local geomorphological features.
Directionality is quantified by computing the mean HVSR per component as it is rotated, and then determining the standard deviation (SD) of HVSR values across all angles for each frequency. This SD value serves as an index of directional variability. Key parameters extracted include the resonant frequency (f0), corresponding amplitude (A0), and the same metrics for higher modes. The mean of the SD function is calculated across two frequency ranges: 0.3–30 Hz (SD_0.3-30) and 1–10 Hz (SD_1-10), along with the SD value around the resonant frequency (SD_fo). These three values are proposed as indicators of azimuthal stability.
For example, station APE, identified as poor due to strong peaks, also exhibits non-negligible directional variability (SD_fo of 1.20 at 6.1 Hz). In contrast, station IMMV appears to be a good reference site candidate, with no identifiable peak and low directional sensitivity (SD=1.07). SD values less than 1.06 are considered low, more than 1.15 high, and more than 1.20 very high. Despite some stations being geologically labeled as rock or used as seismological stations, observations show cases of low-frequency (IACM, NISR2) or high-frequency (VLS) amplifications up to 6–8. Additionally, an SD > 1.20 suggests a strong dependence of reference ground motion on sensor orientation, potentially by factors of 2 or 3 at certain frequencies.
Correction of HVSR for the Vertical Component
HVSR may underestimate the actual amplification level, partly due to potential amplification of the vertical component used as a reference. To investigate this, an additional calculation is performed to correct HVSR for implicit vertical component amplification. A Vertical Amplification Correction Function (VACF), developed for Japan by Ito et al. (2020), is applied as a first approximation, despite its origin. This VACF estimates horizontal S-wave amplification from HVSR, based on diffuse field theory and generalized spectral inversion techniques for over 1600 strong-motion sites in Japan. The VACF is proposed for the 0.12–15 Hz range, and corrections are constrained to this applicability range.
For station APE, the 6 Hz resonant peak becomes more prominent after VACF correction, with amplification doubling from 3 to 6. At station IMMV, where HVSR did not show significant peaks, a peak becomes visible at 1 Hz after VACF correction. These results provide an idea of possible amplification levels up to 15 Hz, but are noted as approximations with inherent shortcomings, such as the Japan-derived VACF and the possibility of unexpected vertical component amplification mapping onto the horizontal. Maximum amplitudes after VACF correction (A0_corr) and newly identified resonant frequencies (f0_corr) are provided as rough indications of possible absolute amplification levels. Nonlinearity in recordings is another potential source of uncertainty, though less probable for rock or stiff conditions.