## 4 Modification of the Gaussian Model  
In this section, a modification is made to the Gaussian model to realize the periodograms' amplitude variation features of the SMART 1 records shown in Figure 7. The statistical characteristics of the modified model and its parameters are investigated. The modified Gaussian model is acquired by multiplying Equation (14) by a local power variation coefficient \(\zeta_{s}\), as shown in Equation (21). The modified Gaussian model retains the main structure of the conventional Gaussian model. \(\zeta_{s}\) is a random variable with a mean value of one and a variance of \(\sigma_{\zeta_{s}}^{2}\). \[\begin{array}{ll}G_{l}(f_{k})&=\zeta_{s}S(f_{k})R_{l}(f_{k})\\ &=\frac{1}{2}\zeta_{s}[\big{(}\ A_{rk}\quad\sqrt{S}\mathbf{\xi}_{a}^{*}\ \ \big{)}\mathbf{W}_{l}\bigg{(} \begin{array}{c}A_{rk}\\ \sqrt{S}\mathbf{\xi}_{a}\end{array} \ \big{)}+\big{(} \begin{array}{c}B_{rk} \quad\sqrt{S}\mathbf{\xi}_{b}^{*}\ \ \big{)}\mathbf{W}_{l}\bigg{(}\begin{array}{c}B_{rk}\\ \sqrt{S}\mathbf{\xi}_{b}\end{array}\ \bigg{)}\end{array}  \tag{21}\]  
In Equation (21), \(\zeta_{s}\) is a coefficient that serves as an amplifier of the periodogram. Since the expected value of \(\zeta_{s}\) is one, it neither affects the target spectrum \(S_{l}\) nor the spectral coherence relationship between any two realized periodograms. Moreover, \(\zeta_{s}\) should be greater than zero. \(\zeta_{s}\) at different locations are correlated, making the properties of \(\zeta_{s}\) similar to the periodograms at a particular frequency. On account of the previous requirements, assume \(\zeta_{s}\) is exhibiting a noncentral Wishart distribution [40]. The simulated \(\zeta_{s}\) at location \(l\) can be generated by a Gaussian quadratic form, as shown by Equation (22). \[\zeta_{s,l}=\frac{1}{N_{\zeta}}\sum_{i=1}^{N_{\zeta}}\big{(}1\quad\mathbf{\xi}_{ \zeta,i}\big{)}v_{\zeta,l}^{*}v_{\zeta,l}\bigg{(} \begin{array}{c}1\\ \mathbf{\xi}_{\zeta,i}^{*}\end{array} \ \bigg{)} \tag{22}\]  
where \(v_{\zeta,l}\) is the \(l^{th}\) row of matrix \(\mathbf{\mathrm{V}}_{\zeta}\), \(\mathbf{\mathrm{V}}_{\zeta}\) can be solved by any conditional simulating methods, such as Cholesky decomposition. Defining \(\mathbf{\mathrm{C}}_{\zeta}\) as the cross-variance matrix of \(\zeta_{s}\), then \(\mathbf{\mathrm{V}}_{\zeta}\) satisfies \(\mathbf{\mathrm{C}}_{\zeta}=\mathbf{\mathrm{V}}_{\zeta}^{*}\mathbf{\mathrm{V}}_{\zeta}\). \(\mathbf{\mathrm{\underline{\epsilon}}}_{\zeta}\) is a vector of random variables with elements that are independent and exhibit the standard normal distribution. The local power coefficient of the seed ground motion is assumed to be 1. \(N_{\zeta}\) is the number of quadratic forms in the summation of Equation (22). The larger \(N_{\zeta}\) is, the smaller the kurtosis of \(\zeta_{s}\) is. When \(N_{\zeta}\) approaches infinity, \(\zeta_{s}\) approaches a normal distribution. When the spectral coherence among locations approaches zero, \(N_{\zeta}\zeta_{s}\) approaches a chi-squared distribution, i.e., \(\chi^{2}(N_{\zeta})\). The variance and kurtosis of \(\zeta_{s}\) are \(2/N_{\zeta}\) and \(12/N_{\zeta}\), respectively. Figure 9: Simulated normalized power loop standard deviation with Gaussian model. \(N_{\zeta}\) should be a proper value that is not too large in case the variance of \(\zeta_{s}\) is not large enough to capture the expected NPLV nor too small in case the fourth moment of the periodogram become too large to have a stable estimation of the second moment with a small sample size. Regarding artificial ground motion simulation, the Fourier coefficients simulated by the conventional Gaussian model are multiplied by the square root of \(\zeta_{s}\). Equation (23) shows the formula of ground motions simulated by the modified Gaussian model.