## 2 Materials and Methods
### Calculating Horizontal to Vertical Spectral Ratios  
An HVSR is computed by taking the RotD50 response spectral ordinate and dividing it by the vertical ordinate at each corresponding frequency. Station representative HVSRsare taken to be logarithmically distributed at each frequency, thus the mean HVSR per frequency is given as  
\[\log HVSR_{i}=\frac{\sum_{j}^{n_{i}}\log\bigl{(}H_{i,j}/V_{i,j}\bigr{)}}{n_{i}} \tag{2}\]  
where \(H\) is the RotD50 response spectra of the horizontal components, \(V\) is the response spectra of the vertical component, \(i\) is the index number representing an observation station, and \(j\) is the index number representing an earthquake event. Note that in this study there is a station representative HVSR for acceleration and for velocity. For each station representative HVSR, a peak amplitude, \(A_{\mathrm{P}}\), is selected. This peak amplitude is the maximum amplitude within the frequency range considered. The frequency at which \(A_{\mathrm{P}}\) occurs is called the peak frequency,\(f_{\mathrm{P}}\). Fortunately, previous studies on estimating \(V_{\mathrm{S30}}\) from HVSR characteristics have offered guidelines on selecting appropriate peaks and frequencies. Some suggest a minimum \(f_{\mathrm{P}}\) of 1 [42; 45] or 2 Hz [43; 44]. Some of these studies also suggest how a significant peak is defined, typically an amplitude that is either greater than 2 or at least 1.5 to 2 times the mean HVSR, HVSRavg, or both [42; 43; 44; 45; 54]. A compilation of these guidelines is shown in Table 2. For this study, a station will have an \(f_{\mathrm{P}}\) if it is greater than 1 and corresponds to an \(A_{\mathrm{P}}\) that is greater than 2.0 as well as 1.5 times the mean HVSR at frequencies above 1.0. If more than one absolute maximum is found within the specified frequency range, then the lowest frequency is selected. Resultant \(f_{\mathrm{P}}\) are regressed against \(V_{\mathrm{S30}}\) to estimate a pattern of behavior. Station \(V_{\mathrm{S30}}\) is taken from Table 1; however, only those stations with surface recordings will be used for regression. Ordinary least squares regression will be applied to a power law equation similarly used by other studies [42; 43; 44; 45], but other forms will also be considered. Any discrepancies between acceleration and velocity HVSR regressions will be considered with respect to usage or application. Moreover, an inspection of several factors affecting the derivation of HVSR characteristics, such as arithmetic or geometric mean, minimum frequency, and conditional peaks, will also be conducted to ascertain their influence on regression, if any [45; 55]. Additionally, the resultant regression and data will be compared with other relevant models [42; 43; 44; 45]. The metric for comparison will be a standard deviation or root mean square error:  
\[\sigma_{log,i}=RMSE_{i}=\sqrt{\frac{\sum_{j}^{n_{i}}\left(\log V_{\mathrm{S30 },i}-\log\hat{V}_{\mathrm{S30},i}\right)^{2}}{n_{i}}} \tag{3}\]  
where \(V_{\mathrm{S30}}\) is site-measured, time-averaged shear wave velocity in the upper 30 m of subsurface material, \(\hat{V}_{\mathrm{S30}}\) is the estimated time-averaged shear wave velocity in the upper 30 m of subsurface material, \(n\) is the number of seismic observatory stations, and \(i\) is the index number representing an observation station. Equation (3) considers a residual of the logarithms, as the models considered in this study actually estimate the logarithm of \(V_{\mathrm{S30}}\). For this study, a lower RMSE would signify a better fit to the data.