## 1 Introduction
### Non-Ergodic GMMs  
Studies of ground motions from single earthquakes have shown strong effects on the ground motion due to the 3-D velocity structure. For example, in the 1994 Northridge earthquake, ref. [4] showed that large ground motions observed in Santa Monica could be explained by rapid changes in the 3-D velocity structure. Similarly, ref. [5] showed that large ground motions observed during the 1995 Kobe earthquake could be explained by basin edge effects. These examples show that the large observed ground motions have a physical relation to a specific ray path. They are not random behavior that can occur at any site from any earthquake. Similar behavior was seen in the 2019 Ridgecrest earthquake, in which there were small regions in Los Angeles that strongly amplified the long-period ground motions [6].  
Over the past 20 years, there has been an effort to develop non-ergodic GMMs that account for the systematic source, path, and site effects that are not random. Ref. [7] evaluated residuals of ground motions at sites in southern California that had recorded multiple earthquakes and found that the standard deviation of the residuals at a single site was 10-20% smaller than the standard deviation from the ergodic GMMs (global models). This reduced standard deviation is called single-station sigma. Multiple studies of datasets with multiple recordings at each station have found similar reductions. Ref. [8] evaluated the single-station sigma for multiple regions. Refs. [9; 10] used expanded datasets for Italy and Taiwan, respectively, to estimate the single-station sigma. All three studies showed similar reductions in the single-station standard deviation as compared to the ergodic standard deviation.  
Ref. [7] also evaluated the ground-motion residuals from a subset of events in a small region recorded at a single station. Ref. [11] conducted a similar evaluation using ground motion data from Japan. Ref. [12] estimated systemic path effects using the within-site residuals from closely spaced events in the Guerrero, Mexico region. Ref. [13] evaluated ground-motion data from southern California and computed the combined site and path terms for individual earthquake clusters. Non-ergodic path effects have been modeled by [14] using ground-motion data from Japan and by [15] using ground-motion data from Taiwan. These studies found that the single-path standard deviation is about 40-50% smaller than the standard deviation from the ergodic GMMs.  
In addition to empirical data, 3-D simulations have been used to estimate path effects on the long-period ground motion due to the 3-D velocity structure. The large set of 3-D simulations from the CyberShake project included multiple realizations of earthquakes at a single location [16]. The total (ergodic) standard deviation and the single-path standard deviation were estimated by [17]. The single-path standard deviation from the 3-D simulations showed a reduction of about 50% compared to the ergodic standard deviation. Ref. [18] estimated non-ergodic path effects using both empirical ground-motion data and CyberShake simulations for Southern California. Datasets of 3-D simulations developed for other regions (e.g., Japan [19]; Utah [20]; Seattle [21]) have been evaluatedfor their single-path standard deviations. These simulations show similar reductions in the single-path standard deviations compared to the ergodic standard deviation as found for the CyberShake simulations.  
The non-ergodic GMM includes three changes from the ergodic GMM: a site-source specific term for the source, path, and site term, the epistemic uncertainty in each term, and the reduced aleatory variability.