# Sensitivity Analysis of Numerical Coherency Model for Rock Sites
## 2 Numerical Coherency Evaluation Methodology
### Estimation of Plane-Wave Coherency  
We estimated plane-wave coherency following the methodology of [PERSON] [1]. First, the time window for the estimated ground motion is defined. If the time history is too long, it may introduce background noise or affect the Arias intensity. Therefore, a time interval from \(-10\) s to \(+10\) s around the PGV was extracted. To prevent the time window from being too short, the final time window was adjusted by subtracting 0.5 s and adding 1 s to the times corresponding to normalized Arias intensities of 0.1 and 0.75, respectively. The normalized Arias intensity I(\(\tau\)) at time \(\tau\) is calculated as follows:  
\[\mathrm{I}(\tau)=\frac{\int_{\mathrm{T_{P}-10}}^{\tau}\left\{\mathrm{V_{1}^{2 }(t)+V_{2}^{2}(t)}\right\}\mathrm{dt}}{\int_{\mathrm{T_{P}-10}}^{\mathrm{T_{P} +10}}\left\{\mathrm{V_{1}^{2}(t)+V_{2}^{2}(t)}\right\}\mathrm{dt}} \tag{2}\]  
where \(\mathrm{T_{P}}\) is the time corresponding to PGV, and \(\mathrm{V_{1}}\) and \(\mathrm{V_{2}}\) are the ground velocities in two horizontal directions, respectively. The endpoints of the extracted time window contain discontinuities that are not zero. Therefore, to prevent abnormal results during the Fourier transform, a 5% cosine-bell taper function is applied to both ends. The taper function is defined as follows:  
\[\mathrm{v}(t)= \begin{cases}0.5\Big{[}\mathrm{cos}\Big{(}\frac{\mathrm{tr}}{ 0.05\mathrm{W_{L}}}+\pi\Big{)}+1\Big{]},&t<0.05\mathrm{W_{L}}\\ 1,&0.05\mathrm{W_{L}}\leq t\leq 0.95\mathrm{W_{L}}\\ 0.5\Big{[}\mathrm{cos}\Big{(}\frac{\pi(t-0.95\mathrm{W_{L}})}{0.05\mathrm{W_{L }}}\Big{)}+1\Big{]},&t>0.95\mathrm{W_{L}}\end{cases}  \tag{3}\]  
where \(\mathrm{W_{L}}\) is the length of the final time window, and \(\mathrm{t}\) is time. The tapered velocity time history was differentiated to obtain the acceleration time history. Since calculation of coherency is performed in the frequency domain, the discrete Fourier transform was performed using the following equation:  
\[\mathrm{u_{j}}(\omega)=\sum_{k=1}^{T}\mathrm{v}(t_{k})\mathrm{u_{j}}(t_{k}) \mathrm{e}^{-\mathrm{i}\omega t_{k}} \tag{4}\]  
where \(\omega\) is the angular frequency, \(\mathrm{u_{j}}(t_{k})\) is the acceleration time history at recorded point j, and \(\mathrm{v}(t_{k})\) corresponds to Equation (4). Seismic coherency analysis is a method for quantifying the similarity between two seismic records. The mathematical definition of seismic coherency involves the smoothed cross-spectrum and smoothed auto-power spectra, which are computed using the following equations:  
\[S_{ij}(\omega)=\sum_{m=-M}^{M}a_{m}u_{i}(\omega_{m})\overline{u_{j}}(\omega_{ m}) \tag{5}\]  
\[S_{ii}(\omega)=\sum_{m=-M}^{M}a_{m}\big{|}u_{i}(\omega_{m})\big{|}^{2} \tag{6}\]  
\[S_{jj}(\omega)=\sum_{m=-M}^{M}a_{m}\big{|}u_{j}(\omega_{m})\big{|}^{2} \tag{7}\]  
where the overbar indicates a complex conjugate, and \(a_{m}\) is the weights used in frequency smoothing, with 2M + 1 indicating the number of smoothed discrete frequencies. A Hamming window of M = 5 was employed, according to [PERSON] [1], and the form is given in Equation (8). \[\omega(M+1-m)=\omega(m+M+1)=0.54+0.46\mathrm{cos}\Big{(}\frac{\mathrm{tr}}{ M}\Big{)} \tag{8}\]Plane-wave coherency is estimated as the real part of the complex coherency considering slowness. The complex coherency is defined as the ratio of \(S_{ij}(\omega)\) to the geometric mean of \(S_{ii}(\omega)\):  
\[\gamma_{ij}(\xi_{ij},\omega)=\frac{S_{ij}(\omega)}{\sqrt{S_{ii}(\omega)S_{jj}( \omega)}} \tag{9}\]  
where \(\xi_{ij}\) is separation distance between points \(i\) and \(j\). The plane-wave coherency is calculated using the following Equation (10):  
\[\gamma^{PW}\big{(}\xi_{ij},\omega\big{)}=\text{Re}\Bigg{[}\frac{S_{ij}(\omega )}{\sqrt{S_{ii}(\omega)S_{jj}(\omega)}}e^{-[(d_{x})(s_{x})+(d_{y})(s_{y})]\omega i} \Bigg{]} \tag{10}\]  
where \(d_{x}\) and \(d_{y}\) are the separation distances in the x and y directions, respectively, and \(s_{x}\) and \(s_{y}\) are slowness vectors in the x and y directions.