## 1 Introduction
### Implementation of Non-Ergodic GMMs into PSHA  
The simplest non-ergodic model is the partially non-ergodic GMM that only includes the systematic effects of the site-specific site effects. The site term applies to all earthquakes, which makes the implementation into PSHA relatively simple: the PSHA software needs to include the median value of the site term, a logic tree for the epistemic uncertainty in the site term, and the aleatory variability for single-station sigma [22]. Because the site term applies to all scenarios, the calculation time for a partially non-ergodic PSHA for the site term only is similar to the calculation time for an ergodic GMM. Due to the ease of application, PSHA using the single-station-sigma approach has become common practice for critical infrastructure (e.g., [23; 24; 25]). In addition to critical infrastructure, the single-station-sigma approach can be used for national maps of ground motion. For example, ref. [26] applied a partially non-ergodic GMM to develop seismic hazard estimates for Europe and the Middle East. Including non-ergodic path effects in addition to the non-ergodic site effects can significantly change the hazard estimate. As an example, we use the T = 3 s pseudo-spectral acceleration (PSA) values from the CyberShake 3-D simulations as a representative dataset with complex wave propagation effects due to the 3-D velocity structure. The simulated data were used to develop both an ergodic GMM and a non-ergodic GMM. The hazard was computed for two sites in the Los Angeles region using the ergodic GMM and the non-ergodic GMM as shown in Figure 1.  
For the ergodic GMM, the hazard for site 1 is slightly larger than the hazard for site 2. For a 2500-year return period (annual hazard = \(4\times 10^{-4}/\mathrm{yr}\)), the PSA (T = 3) value is 0.27 g for site 1 and 0.22 g for site 2. Using the non-ergodic GMM, there is a large change in the hazard with a large decrease for site 1, and a small increase for site 2: the 2500-year return period PSA value for site 1 is reduced from 0.27 g to 0.10 g, whereas the PSA value is increased from 0.23 g to 0.27 g for site 2. The CyberShake simulations include the rates of the event, which allows the hazard to be computed directly from the simulation dataset, which we treat as the correct hazard. The simulation-based heard curves are shown by the black curves in Figure 1. The non-ergodic hazard curves are much closer to the simulation-based hazard curves than the ergodic hazard curves. This example shows that using non-ergodic GMMs can lead to large changes in the estimated hazard and significant improvements in the accuracy of the hazard. Non-ergodic GMMs can also incorporate ground-motion data from region-specific 3-D simulations, allowing better integration of seismic hazard modeling with seismology research. The implementation of non-ergodic GMM into PSHA software is ongoing. Currently, there is no publicly available software for PSHA that implements GMMs with non-ergodic effects. The example shown in Figure 1 is for fault sources only using SPHA software that is still in the evaluation stage. The calculation time for PSHA using GMMs with non-ergodic path effects for faults is longer than for ergodic GMMs but is still practical to run on desktop computers. In contrast, the calculation time for PSHA with non-ergodic path effects for areal sources can be over 100 times longer than for ergodic GMMs due to the large number of subsources in areal source zones. The long calculation times are a key limitation to the adoption of non-ergodic GMMs into PSHA practice. The epistemic uncertainties in the inputs to PSHA are typically modeled by logic trees [27]. The size and complexity of logic trees used in PSHA have been growing as more advanced models for the source and ground motion are included in the PSHA, increasing the calculation times. Using non-ergodic GMMs leads to a large number of branches on the ground-motion logic tree required to capture the spatially correlated epistemic uncertainty in the non-ergodic path terms.