## 4 Modification of the Gaussian Model
\[u_{l}(t)=\sqrt{\zeta_{s,l}}\sum_{k=1}^{N_{\zeta}}\left[A_{lR}cos(f_{k}t)+B_{ lR}sin(f_{k}t)\right],l=1,\ldots,L \tag{23}\]  
where \(A_{lR}\) and \(B_{lR}\) are the Fourier coefficients generated by the conventional Gaussian-model-based conditional methods, \(\sqrt{\zeta_{s,l}}\) can be directly calculated by \(\sqrt{\zeta_{s,l}}=v_{\zeta,l}\left( \begin{smallmatrix}1\\ \mathbf{\xi}_{\zeta}^{+}\end{smallmatrix} \right)\). Figure 10 shows the realization of the normalized periodograms' variance and coefficient of variance when \(N_{\zeta}\) equal to 12. The subscript \(g\) and \(m\) represent the results realized by the conventional Gaussian model and the modified model, respectively. As shown in this figure, the SCVs are smaller than the variances of the periodograms, which is consistent with the conclusion derived in the previous section. The SCV realized by the Gaussian model is smaller than one and obviously smaller than the actual ground motion. The variance and coherence between \(\zeta_{s}\) at different locations cannot be solved directly. In this study, a genetic algorithm is used to search for the coherence of \(\zeta_{s}\), such that the simulated NPLVs comply with the actual ground motions in Table 3. Monte Carlo simulations were carried out to calculate the NPLVs with the modified Gaussian model. In the SMART 1 array, 37 stations possess 36 unique separation distances. The coherence value for these separations should be updated, which should be smaller than one and greater than zero. The derivative of a coherence model with respect to separation distance should be smaller than zero because the coherence is expected to decrease with the increase  
Figure 10: Numerical realization of the normalized periodograms’ variance and SCV: (**a**) variance of Gaussian model, (**b**) variance of modified Gaussian model, (**c**) SCV of Gaussian model, and (**d**) SCV of modified Gaussian model. in separation distance. Therefore, the coherence of \(\zeta\), denoted by \(\gamma_{\zeta}\), is assumed to possess a form as shown by Equation (24). \[\gamma_{\zeta}=a_{0}+a_{1}\exp\Bigl{(}-a_{3}d_{ij}\Bigr{)}+a_{2}\exp\Bigl{(}-a_{ 4}d_{ij}^{2}\Bigr{)}-a_{5}d_{ij} \tag{24}\]  
where \(d_{ij}\) is the distance between two locations, and \(a_{i}\) is the function's parameter to be fitted. Since \(\gamma_{\zeta}\in[0,\ 1]\), and \(\gamma_{\zeta}\) is expected to decrease with the increase in \(d_{ij}\), thus, the following equations should be established. \[ \begin{array}{l}a_{0},\ a_{1},\ a_{2},\ a_{3},\ a_{4},\ a_{5}>0\\ a_{0}+a_{1}+a_{2}<1\end{array}  \tag{25}\]  
In the Monte Carlo simulations, when \(\gamma_{\zeta}<0\), \(\gamma_{\zeta}\) was forced to be zero, and when \(\gamma_{\zeta}>1\), \(\gamma_{\zeta}\) was forced to be one. The NPLVs of the modified Gaussian model are estimated with ten thousand simulation results. The difference between the simulated NPLVs and the actual ground motions is defined as Equation (26). \[J=\|\left(\sigma_{R,I}^{2}\quad\sigma_{R,M}^{2}\quad\sigma_{R,O}^{2}\right)- \left(\sigma_{R,I}^{2}\quad\sigma_{R,M}^{2}\quad\sigma_{R,O}^{2}\right)\| \tag{26}\]  
where \(\|\Box\|\) represents the Euclidean norm of the inner variable \(\Box\). The conditions when \(N_{\zeta}=6\) and \(N_{\zeta}=12\) are studied to show the influence of \(N_{\zeta}\) on the coherence of \(\zeta\). Table 4 lists the fitted results of the parameters in Equation (24). The fitness results when \(N_{\zeta}=6\) are better than when \(N_{\zeta}=12\). Table 5 lists the results of the averaged NPLVs and their error ratios relative to the actual records for different \(N_{\zeta}\). The averaged NPLVs of actual earthquakes can be referred to in Table 3. As detailed in Table 5, nearly all the results from the model when \(N_{\zeta}=6\) are better than those when \(N_{\zeta}=12\). Figure 11 demonstrates the fitted results of coherence model for different \(N_{\zeta}\) and different spectral coherence models.