## 3 PSHA for Non-Ergodic GMMs  
Consider an areal source over which we would like to perform a non-ergodic hazard calculation. To calculate the seismic hazard at a site, the scenarios are discretized into magnitude bins and source location grids. The hazard for each magnitude and location bin is calculated and summed to obtain the total hazard at the site. Typically, the earthquake magnitudes are evaluated at discrete values (e.g., \(M=5.0,\ 5.1,\...\)). For an ergodic GMM, the distances bin widths are often increased at larger distances to improve the calculation speed with negligible loss of accuracy. An example of variable distance bins used in PSHA with minimal loss of accuracy is given in Table 1. For large areal source zones, the number of source locations that fall into a distance bin can be large. For example, for the 95-99 km distance bin, the number of 1 km \(\times\) 1 km source grids that fall into the distance can be up to 600. Therefore, grouping the source locations into distance bins with a single median ground motion can greatly reduce the calculation time for ergodic GMMs. The ergodic hazard given in Equation (1) can be written in terms of the rate of occurrence of events within a given M and R bin:  
\[Haz(IM>z)=\sum_{l=1}^{N_{MBin}}\sum_{j=1}^{N_{MBin}}rate(M_{l},R_{j})\left[1- \Phi\left(\frac{\ln(z)-\mu_{erg}(M_{l},R_{j})}{\sigma_{erg}(M_{l},R_{j})}\right)\right] \tag{7}\]  
in which \(N_{MBin}\) and \(N_{MBin}\) are the numbers of magnitude bins and distances bins, respectively. The hazard can be written in terms of the sum of the marginal hazards from each magnitude and distance bin:  
\[Haz(IM>z)=\sum_{l=1}^{N_{MBin}}\sum_{j=1}^{N_{MBin}}Haz(IM>z,M_{l},R_{j}) \tag{8}\]  
in which the marginal hazard is given by the following:  
\[Haz(IM>z,M_{l},R_{j})=\sum_{i=1}^{NS_{j}}rate_{i}(M_{l})\left[1-\Phi\left( \frac{\ln(z)-\mu_{erg}(M_{l},R_{j})}{\sigma_{erg}(M_{l},R_{j})}\right)\right] \tag{9}\]  
and \(NS_{j}\) is the number of subsources in the \(j\)th distance bin. For an ergodic GMM, the median ground motion will be the same for all subsources within an M, R bin, so the \(P(IM>z|M,R)\) term can be removed from inside the summation and computed only once for each \(M_{l},R_{j}\) bin, which greatly speeds up the hazard calculation. \[Haz(IM>z|M_{l},R_{j})=\left(\sum_{i=1}^{NS_{j}}rate_{i}(M_{l},R_{j})\right) \times\left[1-\Phi\left(\frac{\ln(z)-\mu_{erg}(M_{l},R_{j})}{\sigma_{erg}(M_{l },R_{j})}\right)\right] \tag{10}\]  
For a non-ergodic GMM, \(\Delta\mu_{es}\) will vary for each subsource location within the distance bin, so the rate and ground motion are not separable. A direct computation of the hazard for each discrete subsource will significantly increase the hazard calculation time. The current approach to the long calculation times is to limit the epistemic uncertainty to a small number of cases (a small number of branches in the logic tree) and to use high-performance computing. As an alternative, we develop an analytical approximation to the hazard accounting for the distribution of \(\Delta\mu_{es}\) terms within a distance bin which greatly reduces the calculation times. An added advantage of using analytical approximations for commuting the non-ergodic hazard is that the full epistemic uncertainty can be included without affecting the calculation time compared to using a few discrete branches on the logic tree to limit the calculation time. For example, ref. [30] showed that using approximations to the hazard calculation based on polynomial chaos provided a more accurate approximation of the hazard fractiles compared to using a small number of branches for the ground-motion model logic tree combined with an exact hazard calculation for each branch. To do so, we first refer to the polynomial chaos (PC) method from [30] to efficiently propagate the epistemic uncertainty in the logic tree of the hazard calculation. For each