## 3 PSHA for Non-Ergodic GMMs
### Polynomial Chaos Expansion  
Following the approach of [31], we use polynomial chaos (PC) to propagate both the variability in the \(\Delta\mu\) within a distance and epistemic uncertainty of the \(\Delta\mu\) at each subsource location.  
The basic idea of the PC expansion is to replace the direct computation of the hazard for each branch of the logic tree with a set of polynomial basis functions that approximate the hazard at a given \(z\) value for a range of variable input parameters (here, \(\Delta\mu\)). The hazard curve can be approximated by its PC expansion given in Equation (11) [31].  
\[Haz(IM>z)\simeq\sum_{k=0}^{p}C_{k}(z)\Psi_{k}[\xi] \tag{11}\]  
in which \(C_{k}(z),k=0,...,P\) are the PC coefficients evaluated at ground-motion value z and the \(\Psi_{k}[\xi],k=0,...,P\) are the Hermite polynomials given in Table 2.  
If the variability in the \(\Delta\mu_{es}\) is assumed to be normally distributed and fully correlated for the different subsources, then the PC method gives an analytical approximation of the hazard that depends on a single normally distributed random variable, \(\xi\). The \(m\)th sample of the marginal hazard from a single scenario can be written as follows:  
\[\begin{split} Haz_{m}(IM>z,M_{l},R_{j})&=Rate(M_{ l},R_{j})\left[1-\Phi\left(\frac{\ln(z)-(\mu_{erg}(M_{l},R_{j},S)+\overline{ \Delta\mu}+\sigma_{\Delta\mu}\xi_{m})}{\sigma_{NE}(M_{l},R_{j})}\right) \right]\\ &\approx\sum_{k=0}^{p}C_{k}(z)\Psi_{k}[\{\xi_{m}\}]\end{split} \tag{12}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Degree** & **Hermite Polynomial** \\ \hline \(i=0\) & \(1\) \\ \hline \(i=1\) & \(\xi\) \\ \hline \(i=2\) & \(\xi^{2}-1\) \\ \hline \(i=3\) & \(\xi^{3}-3\xi\) \\ \hline \(i=4\) & \(\xi^{4}-6\xi^{2}+3\) \\ \hline \(i=5\) & \(\xi^{5}-10\xi^{3}+15\xi\) \\ \hline \(i=6\) & \(\xi^{6}-15\xi^{4}+45\xi^{2}-15\) \\ \hline \end{tabular}
\end{table}
Table 2: Hermite Polynomials up to Order 6.  
in which \(\overline{\Delta\mu}(M_{i},R_{j})\) is the mean of the \(\Delta\mu\) in the \(j\)th distance bin, \(\sigma_{\Delta\mu}\) is the epistemic uncertainty of the \(\Delta\mu\) in the \(j\)th distance bin, \(C_{i}\) are the PC coefficients, \(\mathbf{\Psi}_{i}[\{\xi\}]\) are the basis functions, and \(\xi_{m}\) is a standard normal random variable.  
Given the PC coefficients, the epistemic fractiles of the hazard can be computed in a post-processing step by sampling the \(\xi\) and computing the hazard for each sample. A large number of samples can be generated efficiently to estimate the epistemic fractiles. The equations for the polynomials, \(\Psi_{i}\), and the PC coefficients, \(C_{i}\), are given in the Supplementary Material.  
The PC method is a numerically efficient method; however, the PC approach assumes that the \(\Delta\mu\) are normally distributed. For the areal source, there are two parts of the standard deviation of the \(\Delta\mu\): the variability in the mean \(\Delta\mu\) within a distance bin and the epistemic uncertainty of the \(\Delta\mu\) at each subsource within the distance bin. The assumption of a normal distribution is appropriate for the epistemic uncertainty of the \(\Delta\mu\) for a single subsource, but it does not hold for the distribution of the mean \(\Delta\mu\) at different locations within a distance bin. Also, the \(\sigma_{\Delta\mu}\) depends on the number of available local data to constrain the non-ergodic term and is not constant for all the subsource locations within a distance bin.  
One approach to the non-normal distribution of the \(\Delta\mu\) is to apply the PC method to each subsource in a given distance bin. That is, the PC terms are computed for each subsource within a distance bin. In this approach, there are computational savings compared to the direct sampling of the logic tree branches of the \(\Delta\mu\) due to not having to directly sample a large number of branches of the \(\Delta\mu\), but it needs to be applied to each of the subsources in a distance bin. In the following example, we call this the PC method.