## 4 Example Application  
To demonstrate the methodology, we show an example of a non-ergodic hazard calculation with an areal source from the South of France. The areal source is discretized into 1 km \(\times\) 1 km subsources and is represented in Figure 2. The discretization of the area results in 21,000 subsources. The subsources at distances of 0 to 200 km are grouped into 62 distance bins for the hazard calculation using the variable distance bins shown in Table 1. The logic tree includes 100 branches for the alternative maps of the \(\Delta\mu\). For this example, only a single magnitude (M = 6) is used with the rate of 0.0004 events/year based on the rate of events between M5.5 and M6.5 for this source zone. The contours of the mean \(\Delta\mu\) and \(\sigma_{\Delta\mu}\) values for subsources falling in the 71-75 km distance bin are shown in Figures 3 and 4. This distance bin was chosen to illustrate the proposed approach with a significant amount of seismic sources and with a broad range of non-ergodic median adjustment terms and epistemic uncertainty. The \(\Delta\mu\) values in this distance bin range from \(-\)0.15 to 0.1 LN units. The histogram of the non-ergodic median adjustment terms, \(\Delta\mu\), for this distance bin is shown in Figure 5. The shape of the distribution of the \(\Delta\mu\) terms is not close to a normal distribution, which confirms that the normal assumption is not a reasonable assumption for characterizing the distribution of the \(\Delta\mu\) for the hazard calculation. This prevents using the PC method to estimate the hazard due to the range of \(\Delta\mu\) within a distance bin with a single PC expansion for the distance bin. The histogram of the standard deviation of the epistemic uncertainty of the \(\delta\mu\) within the 71-75 km distance bin is shown in Figure 6. The distribution of the epistemic uncertainty for the sources in this distance bin is also not close to a normal distribution. Figure 4: Epistemic uncertainty in the non-ergodic median adjustment terms for the subset of source locations with rupture distances between 71 km and 75 km from the site (black triangle). The red line shows the boundary of the areal source zone. Figure 3: Non-ergodic median adjustment terms for the subset of source locations with rupture distances between 71 km and 75 km from the site (black triangle). The red line shows the boundary of the areal source zone. Figure 2: The areal source used for the example hazard calculation is shown by the blue shaded region. The zone is discretized into approximately 21,000 1 km \(\times\) 1 km grids. The site location is shown by the black triangle  
For this example, three methods are used to calculate the total hazard from the areal source under the non-ergodic assumption, including the epistemic uncertainty in the median adjustment terms: the logic tree method, the PC method, and the Taylor expansion method. For the logic tree method, 100 samples of the spatially correlated \(\Delta\mu\) are generated. The hazard is computed for each of the 21,000 subsources for each of the 100 samples of the \(\Delta\mu\). For the PC method, five PC terms (PC order 4) are calculated for each of the 210,000 seismic sources, which results in the calculation of about 100,000 PC terms to obtain the PC expansion of the total hazard. The 100 samples of the Hermite polynomials are then generated for all the subsources. The samples are then multiplied with the PC terms of each seismic source and summed up to obtain 100 samples of the total hazard. For the Taylor Expansion method, five PC terms are developed for the reference hazard curves, and the Taylor Expansion coefficients of order 2 are calculated for each of the 62 distance bins, resulting in the calculation of \(5\times 6\times 62=1860\) PC/TE terms. The PC terms of the seismic sources inside each distance bin are then approximated using the Taylor Expansion of the reference PC terms.