## 4 Example Application
For the spatially correlated case, we chose to retain 95% accuracy in the representation of the correlation structure, which led to  
Figure 5: Histogram of the non-ergodic median adjustment terms between 71 km and 75 km from the site. Figure 6: Histogram of the epistemic uncertainty of the non-ergodic median adjustment terms between 71 km and 75 km from the site. 28 eigenfunctions retained in Equation (22). The 100 samples of the Hermite polynomials are then generated for the locations of the subsources. These samples are then multiplied with the Taylor Expansion approximation of the PC terms for each distance bin and summed up to obtain 100 samples of the total hazard. We show the percentiles of the total hazard from the three methods (Monte-Carlo (MC), Polynomial Chaos (PC) and Taylor Expansion (TE)) in Figures 7 and 8 for the case of full and partial spatial correlation, respectively. The results from all three methods are very similar, indicating that the approximation used in the more efficient Taylor Expansion method has only a small reduction in accuracy. the same at each seismic source location in both cases; however, the spatial correlation does not allow samples of the median ground motion to have the same deviation from their mean at all seismic source locations, which reduces the number of extreme realizations of the total hazard, and therefore, reduces the tails of the hazard's distribution. The computational time for the example hazard calculation for the logic tree, PC, and TE methods are shown in Figures 10 and 11 as a function of the number of branches of the \(\Delta\mu\) logic tree for the fully and partially correlated cases, respectively. The hazard calculation is based on a single magnitude, single source model, and single reference ergodic GMM. For the fully correlated case, the same samples of the median ground motion are used for all the seismic sources. The sampling of the hazard curves for the PC and TE methods can be performed outside of the hazard calculation in a post-processing phase. Therefore, the computational times for the PC and TE methods do not depend on the number of samples generated. Figure 10: Computational Time of the logic tree, Polynomial Chaos and Taylor Expansion Approaches for the Fully Correlated Cases. The x-axis is the number of branches of the \(\Delta\mu\) on the logic tree. Table 1.0 Percentiles of the probability of exceedance for M = 6 with the logic tree approach (100 Branches) with full and partial spatial correlation. For the partially correlated case, the computational time depends on the number of samples generated for the three methods, and they all scale linearly with the number of median ground-motion samples; however, the PC method remains significantly faster by several orders of magnitude compared to the logic tree method, and the TE method performs one order of magnitude faster than the PC method for an equal number of samples generated. The accuracy of the mean hazard and hazard fractiles for the TE method for the example is shown in Table 1.1 At the higher ground-motion values, the hazard (y-axis value) for the TE method is about 2% larger than for the PC method. This is smaller than the difference in the hazard found using different hazard programs. In this example, the error is due to the TE approximation of the PC coefficient, which can be reduced by increasing the order of the TE expansion (a TE expansion of order 2 was chosen here). Table 1.2 Computational Time of the logic tree, Polynomial Chaos and Taylor Expansion Approaches for the Partially Correlated Cases. The x-axis is the number of branches of the \(\Delta\mu\) on the logic tree. Table 1.3 Ratio of Fractiles of the Annual Hazard of the PC Method by the Taylor Expansion Method.