Comparing subduction ground-motion models to observations for Cascadia

Abstract

We evaluate Cascadia subduction ground-motion models (GMMs), considered for the 2023 US National Seismic Hazard Model (NSHM) update, by comparing observations to model predictions. The observations comprise regional recordings from intraslab earthquakes, including contributions from 2021 and 2022 events in southern Cascadia and global records from interface earthquakes. Since the 2018 NSHM update, new GMMs for Cascadia have been published by the Next Generation Attenuation (NGA)-Subduction Project that require independent evaluation. In the regional intraslab comparisons, we highlight a characteristic frequency dependence for Cascadia data, with short periods having lower ground motions and longer periods being comparable to other subduction zones. We evaluate differences in northern and southern Cascadia and find that the NGA-Subduction GMMs developed using southern Cascadia data perform better in this region than the model that did not consider these data. We compare ground-motion variability in Cascadia with the NGA-Subduction model predictions and find differences at short periods (T = 0.1 s) due to the use of global versus regional data in the development of these models. Moreover, the within-event component of aleatory variability from the GMMs overpredicts the standard deviation of Cascadia recordings at very short periods (T < 0.05 s). Using global interface earthquakes as a proxy to evaluate the Cascadia GMMs, we find long-period overprediction from a simulation-based GMM and some of the empirical GMMs. When comparing recent observations, we find a similar misfit to GMMs and the 2010 and 2022 Ferndale earthquakes. Finally, we observe different basin amplification factors arising in different subsets of the data, which indicate that differences in basin factors between empirical GMMs could arise from the data selection choices by the developers. As part of evaluating the regional basin terms, we apply basin amplification factors from the magnitude 9 Cascadia earthquake simulations to the empirical GMMs for interface earthquakes. The comparisons presented in this study indicate that the NGA-Subduction GMMs for Cascadia perform well relative to observations and older subduction GMMs.

Introduction

Each update of the US National Seismic Hazard Model (NSHM) requires evaluating and selecting appropriate ground-motion models (GMMs) that are used to estimate the shaking for each earthquake in a hazard calculation (Petersen et al., 2020). Since the 2018 update of the NSHM, new subduction GMMs from the Next Generation Attenuation (NGA)-Subduction Project (Bozorgnia et al., 2022) have been published for Cascadia. Implementation of these new subduction GMMs in the 2023 NSHM is discussed by Rezaeian et al. (2024). However, regional Cascadia observations are limited to a small number of recordings from a few moderate intraslab earthquakes (Mazzoni, 2021). In addition, the developers of the NGA-Subduction GMMs used different selections of recordings and methodologies in constructing their models.

Seismic hazard is high in Cascadia due to subduction and crustal earthquake sources (Frankel et al., 2015). A major contributor to this seismic hazard is the potential for a megathrust earthquake, such as the one that shook the region in 1700 (Satake, 2003). However, no recordings from such an event are available for Cascadia. In addition, fractures in the northeastern subducting Juan de Fuca plate are a source of deep intraslab subduction events, such as the 2001 ( M  = 6.8) Nisqually and the 1999 ( M  = 5.75) Satsop earthquakes (Frankel et al., 2009). Also, the southern end of the Cascadia subduction zone (CSZ), near the Mendocino Triple Junction, includes the subducting Gorda plate that is a source of shallow intraslab earthquakes, such as the 2010 Ferndale ( M  = 6.2), 2021 Petrolia ( M  = 6.2), and 2022 Ferndale ( M  = 6.4) earthquakes (Guy et al., 2015).

Another important factor in evaluating GMMs for application in the CSZ relates to their basin terms. The basin response to incoming seismic waves can amplify the ground motion for sites on deep basins due to multiple factors, including surface-wave conversions from S waves at the basin edge and focusing of the S waves from changes in basin geometry (Frankel et al., 2009). In northern Cascadia, the Puget Lowland region is a large forearc basin with multiple sub-basins, including the deep Seattle basin that is known to generate strong amplification at longer periods (Rekoske et al., 2022; Rezaeian et al., 2024). The basin terms in the NGA-Subduction GMMs are parameterized in a similar way, but the developers used different selections of recordings to develop their basin coefficients that result in varying levels of basin amplification. Previous studies that simulated interface earthquakes in Cascadia hypothesized that the shallower incidence angle of interface events may generate stronger basin amplifications (Wirth et al., 2019). This hypothesis is consistent with the high amplification factors in the Seattle basin from the M9 Cascadia earthquake simulations (Frankel et al., 2018b; Wirth et al., 2018b). The empirical basin terms in the NGA-Subduction GMMs were not adjusted to the higher basin amplification factors (BAFs) computed from the M9 simulations. The NGA-Subduction modelers utilized the M9 simulation results in their development process, but only to make comparisons to the model’s empirical basin terms (Abrahamson and Gulerce, 2022; Kuehn et al., 2023; Parker et al., 2022). The 2023 NSHM has incorporated BAFs based on the M9 simulations for interface earthquakes recorded in the deep Seattle basin (Rezaeian et al., 2024). We compare the M9 BAFs for consistency with the observed data and with the parameterizations of the NGA-Subduction GMMs.

The aleatory variability describes the unmodeled uncertainty in the prediction of the median ground motions and has a direct impact on the probabilistic ground motions in a probabilistic seismic hazard analysis. Larger values of aleatory variability increase ground motions at the longer return periods (Bommer and Abrahamson, 2006). Previous studies have looked at ways to use regional data to reduce the standard deviation for site-specific studies. Two of the NGA-Subduction GMMs used global recordings to develop their aleatory model. The global recordings are predominantly from earthquakes in Japan, which may have different variability in site effects compared with Cascadia recordings (Atkinson, 2003). However, the Abrahamson and Gulerce GMM (AG20) divided their data set into similar subduction groups based on their spectral shapes in a way that predicts smaller variability for Cascadia relative to Japan (Abrahamson and Gulerce, 2022). It is not known if the variability of ground motions from future Cascadia events will better reflect the global observations or the small number of regional recordings.

In this study, we evaluate data fits to GMMs for Cascadia subduction earthquakes. The data are divided into NGA-Subduction recordings that include both intraslab regional and global interface recordings, and recent data that are limited to the 2021 Petrolia and 2022 Ferndale earthquakes in southern Cascadia. We compare the average differences between the GMM predictions and observations and examine the standard deviations of the Cascadia GMMs. Finally, we compute the linear BAFs for the Seattle basin and integrate the “factor-of-two” adjustment that comes from the M9 Cascadia earthquake simulations into the Casadia GMM basin terms (Frankel et al., 2018b). The 2023 NSHM update incorporates this adjustment for interface earthquakes in the deep part of the Seattle basin (Moschetti et al., 2024; Rezaeian et al., 2024).

Cascadia subduction GMMs

The selection of GMMs for the CSZ is an important step in computing the seismic hazard for both site-specific studies and at regional scales. Subduction GMMs have historically been developed from global recordings that did not account for regional differences in ground motions, whereas the NGA-Subduction GMMs include region-specific terms for the source, path, and site terms. Cascadia has very few intraslab records and no interface records of significance ( M  > 5), so the applicability of global GMMs is difficult to assess. In contrast, it is possible to derive region-specific GMMs for data-rich areas, such as Japan (Si et al., 2022). Different strategies for developing GMMs have been applied to deal with the lack of data in Cascadia. For example, the Abrahamson et al. (2016) GMM was developed from Japanese data and other global events, but modified for application to Cascadia. For intraslab events, these adjustments were based on the limited empirical data. There have also been multiple efforts to use simulations to model the Cascadia ground motions (Frankel et al., 2018b; Wirth et al., 2018b) and develop Cascadia-specific GMMs (Atkinson and Macias, 2009).

A key question in Cascadia relates to the smaller short-period ground motions observed in past intraslab earthquakes relative to other subduction zones and whether this should be expected for future events. All of the developers recognized the potential to constrain site and basin effects from Cascadia observations, but took different approaches in constraining the path and source effects. Some of the NGA-Subduction modelers pooled the data from different subduction regions, whereas others used global observations to constrain the source terms of the GMMs. We distinguish between “unadjusted GMMs,” which directly fit the Cascadia observations and predict smaller short-period ground motions, to “adjusted GMMs,” which adjust the short-period ground-motion level of the predictions by modifying the source terms. The size of the adjustment can be derived from the global average, a subset of specific events, or restricting the adjustment to the average from subduction zones with similar spectral shapes.

The NGA-Subduction GMMs for Cascadia intraslab earthquakes comprise the unadjusted GMM—Abrahamson and Gulerce (2022, hereafter AG20 Unadjusted) and three adjusted NGA-Subduction GMMs—Abrahamson and Gulerce (2022, hereafter AG20), Kuehn et al. (2023, hereafter KBCG20), and Parker et al. (2022, hereafter, PSBAH20) (Table 1). AG20 Unadjusted is based on the smaller short-period observed ground motions in Cascadia, whereas the adjusted AG20 GMM adjusts the ground motion to match the subduction data from other regions with similar spectral shapes (Central America, New Zealand, and Taiwan). The AG20 model developers do not recommend their unadjusted model for seismic hazard application in Cascadia (Abrahamson and Gulerce, 2022). In contrast, KBCG20 modified the intraslab ground motion to fit the data for the two largest events (2001 Nisqually and 2010 Ferndale), but did not incorporate the smaller magnitude Cascadia earthquakes that would have resulted in a smaller adjustment (Kuehn et al., 2023). PSBAH20 adjusted the intraslab constant to increase the smaller short-period ground motions so that the global level is matched at $\mathbf{M}>7.2$ for a rupture distance of 75 km (Parker et al., 2022).

Table 1.

List of GMMs used in this study with their region of applicability, magnitude range, distance range, site term, basin term, and number of events and records used in development. The events are limited to intraslab for Cascadia, but combine interface and intraslab for the global models.

GMM	Region	Magnitude	Distance (km)	Site	Basin	Events; records
AG20 CAS	Cascadia	5.0–8.5 (intra a )	$35<{R_{\mathrm{rup}}}^{\mathrm{b}}<800$	$V_{S30}$	$Z_{2.5}$	6; 177
AG20 Unadjusted	Cascadia	5.0–9.5 (inter c )				6; 177
AG20 Global	Global		$35<R_{\mathrm{rup}}<500$			202; 8764
KBCG20 CAS	Cascadia	5.0–8.5 (intra)	$10<R_{\mathrm{rup}}<1000$	$V_{S30}$	$Z_{2.5}$	12; 604
KBCG20 Global	Global	5.0–9.0 (inter)	$10<R_{\mathrm{rup}}<1000$			238; 16,035
PSBAH20 CAS	Cascadia	4.5–8.5 (intra)	$35<R_{\mathrm{rup}}<1000$	$V_{S30}$	$Z_{2.5}$	12; 432
PSBAH20 Global	Global	4.5–9.5 (inter)	$35<R_{\mathrm{rup}}<1000$			122; 6374
Zhao06	Japan	5.0–8.3 (intra)	$10<R_{\mathrm{rup}}<300$	Categorical	No	30; 3233
Zhao16	Japan	5.0–8.3 (intra)	$10<R_{\mathrm{rup}}<300$	Categorical	No	76; 6482
AM09	Cascadia	7.5–9.0 (inter)	$35<R_{\mathrm{rup}}<800$	No	No	Simulation

a

Intraslab earthquake; ^brupture distance (km); and ^cinterface earthquake.

In the case of interface earthquakes, no Cascadia observations are available for $\mathbf{M}>5$ . AG20 published both unadjusted and adjusted versions of their Cascadia interface models that are partly based on the regional intraslab records and predict smaller Cascadia interface ground motions relative to other subduction zones (Abrahamson and Gulerce, 2022). It is not known if the physical cause for the lower intraslab ground motions observed in Cascadia, such as low stress drop, is applicable to Cascadia interface earthquakes. KBCG20 adjusted their Cascadia interface source term so that it is correlated with the intraslab source term, resulting in a slightly negative adjustment relative to their global interface model (Kuehn et al., 2023). PSBAH20 adjusted the source term such that they match the global ground motions at large magnitudes, but based on the corner magnitude of the Cascadia magnitude-scaling term (Parker et al., 2022).

Table 1 provides the recommended magnitudes, distances, and the predictive parameters for shallow site response ( $V_{S30}$ , defined as the time-averaged shear-wave velocity to a depth of 30 m) and deep basin response ( $Z_{2.5}$ , defined as the depth to the shear-wave horizon where $V_S=2.5$ km/s) for each GMM considered in this study. The Cascadia-specific coefficients for the NGA-Subduction GMMs are a regionalized constant term, magnitude breakpoint, anelastic attenuation, and $V_{S30}$ and $Z_{2.5}$ scaling. The last column provides the number of earthquakes and recordings that were used to develop the regional coefficients for each GMM. Note that AG20 used a smaller data set that excludes the shallower intraslab earthquakes in southern Cascadia. KBCG20 and PSBAH20 incorporated selected recordings from southern Cascadia as additional data to constrain their models.

In addition to the NGA-Subduction GMMs, we evaluate two older subduction GMMs due to their use in the NSHM (Petersen et al., 2020). The Zhao (2006, hereafter Zhao06) GMM is based primarily on Japanese recordings, and this results in larger short-period predictions relative to the regional Cascadia observations. The Atkinson and Macias (2009, hereafter AM09) is a simulation-based GMM that was developed for interface earthquakes with M  = 7.5–9.0 in Cascadia. The rupture model is based on the M  = 8.1 Tokachi-Oki Japanese earthquake, and adjustments were made for the source, attenuation, and site parameters of the Cascadia region (Atkinson and Macias, 2009). Finally, the updated Zhao et al. (2016, hereafter Zhao16) GMM is included in the Cascadia intraslab comparisons, despite its exclusion from the 2023 NSHM update, to highlight differences between Zhao06 and Zhao16.

Both Zhao06 and Zhao16 use a binned $V_{S30}$ model that result in constant site amplification at $V_{S30}$ values corresponding to different site classes. The NSHM AM09 GMM was implemented with the Boore and Atkinson (2008) site-amplification model to allow scaling of the GMM with $V_{S30}$ . The NSHM basin amplification models for Zhao06 and AM09 use the Campbell and Bozorgnia (2014, hereafter CB14) basin-depth scaling to incorporate basin effects (Powers et al., 2021). This basin amplification model is developed from crustal earthquakes that are not located in the sedimentary basins of Cascadia. There are potential differences in amplification between crustal basin models and the empirical subduction basin models, but this study uses the NSHM-implemented CB14 basin model for AM09 and Zhao06 to be consistent with the NSHM application. Both Zhao06 and Zhao16 are not defined outside of $0.05\le T\le 5$  s. The 2023 NSHM implementation of Zhao06 extrapolated the model predictions to capture the full 22 spectral periods (Powers et al., 2021), but only the published spectral periods for Zhao06 are used in this study. Zhao16 uses the path length through the volcanic arc as an input parameter that is developed for Japan, but is not modeled in a way that is appropriate for the CSZ. We assume this parameter to be 0 for the comparisons.

Cascadia basin terms

Ground-motion amplification by sedimentary basins is modeled through both the shallow site predictor (i.e., $V_{S30}$ scaling) and the deeper basin predictor (i.e., $Z_x$ scaling, where x is the depth to the shear-wave horizon where $V_S=x$  km/s). The Cascadia basin terms are parameterized using $Z_{2.5}$ . The reason $Z_{2.5}$ is the predictor in this region as opposed to $Z_{1.0}$ is because the Puget region is covered with glacial sediments that result in unusually shallow $Z_{1.0}$ values (Wirth et al., 2018b).

The global GMMs do not include a basin amplification term. The Cascadia GMMs use $Z_{2.5}$ (input in m for PSBAH20 and in km for AG20 and KBCG20) as the parameter to capture the deep basin response. The resonance effects from basin amplification were primarily developed using data from well-defined basins, such as the Seattle and Tacoma basins in northern Cascadia. The $Z_{2.5}$ parameter is normalized to avoid double counting the amplification from the shallow site predictor $V_{S30}$ and the amplification from the deeper basin predictor $Z_{2.5}$ . This normalization uses the GMM-dependent reference basin depth for an average $V_{S30}$ for the region. The reference basin depth for Cascadia for each of the NGA-Subduction GMMs is defined in the equations below:

$$\ln ({\mu }_{Z_{2.5}}^{\mathrm{AG}20})=\{\begin{array}{cc}8.52\hfill & V_{S30}\le 200\mathrm{m}/\mathrm{s}\hfill \\ \multicolumn{1}{c}{8.52-0.88*\ln (V_{S30}/200)}& 200\le V_{S30}\le 570\mathrm{m}/\mathrm{s}\hfill \\ \multicolumn{1}{c}{7.6}& V_{S30}\ge 570\mathrm{m}/\mathrm{s}\hfill \end{array}$$

(1)

$$\ln ({\mu }_{Z_{2.5}}^{\mathrm{KBCG}20})=8.294+(2.302-8.294)\frac{\exp \left(\frac{\ln (V_{S30})-6.396)}{0.271}\right)}{1+\exp \left(\frac{\ln (V_{S30})-6.396}{0.271}\right)}$$

(2)

$$\ln ({\mu }_{Z_{2.5}}^{\mathrm{PSBAH}20})=\ln (10)+-0.42(1+\mathrm{erf}(\frac{\underset{10}{\log }(V_{S30})-\underset{10}{\log }(200)}{0.2\sqrt{2}})+\ln (10)*3.05$$

(3)

Figure 1 plots the above functional forms as well as the $Z_{2.5}$ and $V_{S30}$ estimates for Puget Lowland basin sites in gray and the Seattle basin sites in orange. In considering the relationship between $Z_{2.5}$ and $V_{S30}$ for sites only in the Seattle basin, the Pearson correlation coefficient is $-0.39$ with a p value of $0.000027$ . This indicates the parameters are weakly negatively correlated, but the majority of sites in the Seattle basin have $Z_{2.5}>5$ km and this subset of sites has a Pearson correlation coefficient of $0.05$ with a p value of $0.6$ , which indicates little to no correlation between the parameters. KBCG20 assumed no correlation between $V_{S30}$ and $Z_{2.5}$ and developed a period-dependent basin model for the Seattle basin that is independent of $V_{S30}$ (Kuehn et al., 2023). The reference basin depth for a given $V_{S30}$ (blue line in Figure 1) is distinct for KBCG20 partly because they excluded Seattle basin sites in defining their function.

Figure 1.

$Z_{2.5}$ estimates against $V_{S30}$ estimates from the NGA-Subduction Cascadia data. The colored lines represent the functional forms used for normalizing the $Z_{2.5}$ parameter for each of the NGA-Subduction GMMs.

Data used in comparisons

We compare GMMs for application to the CSZ using three data sources: Cascadia intraslab recordings from the NGA-Subduction database, global interface recordings from the NGA-Subduction database, and independently processed records from earthquakes near the Mendocino Triple Junction that occurred after the end of the NGA-Subduction project. We do not compare the GMMs using the log-likelihood score, stochastic area metric, or similar criteria, because the independent observations are insufficient to provide robust estimates of these metrics. Moreover, ranking GMMs for regions with sparse data may result in unwarranted confidence when selecting GMMs for hazard applications.

NGA-Subduction database

The NGA-Subduction Project developed a database of earthquake observations for subduction regions (Mazzoni, 2021). The database includes a total of 214,020 individual records from 1880 subduction events. For the CSZ, there are 35 regional earthquakes with 2543 regional recordings. No significant subduction earthquakes in northern Cascadia have been recorded since the M  = 4.7 Vancouver Island Earthquake on December 30, 2015, was added to the database. Therefore, this study is largely dependent on the records provided in the NGA-Subduction database.

Cascadia intraslab recordings

Cascadia intraslab recordings from the NGA-Subduction database are used for the regional GMM comparisons. When selecting Cascadia recordings from the database, we gave consideration to the recommended magnitude and distance range as defined in Table 1. Records were removed if they had problematic flags (bad quality, late P-trigger, multiple events, and non free-field stations). The sampling bias was accounted for by only selecting records with rupture distances less than $R_{\max }$ , defined as the minimum distance to non-triggered recordings for each event (Kuehn et al., 2023). This resulted in a total of 322 recordings from 7 earthquakes with a minimum magnitude of 5.0 and a maximum rupture distance of 792 km. The earthquake and station locations are shown in Figure 2a. The data in magnitude–depth space highlight the difference in depth between intraslab events in northern and southern Cascadia (Figure 2b).

Figure 2.

Intraslab earthquakes, sites, and recordings from Cascadia used in this study: (a) earthquakes shown using their focal mechanism and seismic stations shown as inverted triangles and (b) data in magnitude–distance space and magnitude–depth space.

Global interface recordings

The lack of recordings from interface earthquakes in Cascadia resulted in selecting global interface recordings as a proxy for the ground motion from a future Cascadia interface earthquake. An important question for the interface GMMs is whether they are consistent with global observations of subduction interface events; certainly, the NGA-Subduction GMMs are developed from these records, but other GMMs were developed from alternative data sets, and this comparison provides an evaluation of these GMMs. The seismic hazard in Cascadia from interface events in the NSHM is based on large-magnitude ( $\mathbf{M}>7.5$ ) sources (Rezaeian et al., 2024). In addition, the inclusion of AM09 in the comparisons restricts the selection of interface earthquake recordings to $\mathbf{M}>7.5$ . The two largest events in the database are the M  = 9.0 Tohoku and the M  = 8.8 Maule earthquakes, with the majority of recordings (980) coming from the Tohoku earthquake (Mazzoni, 2021). The global data set consists of 1938 recordings from 14 interface earthquakes (Figure 3a). Figure 3b shows the records in magnitude–distance space and distinguishes the three subduction zones featured in the $\mathbf{M}>7.5$ global interface recordings.

Figure 3.

Global interface earthquakes and recordings used in this study: (a) global earthquakes with M  > 7.5 and (b) data in magnitude–distance space.

Intraslab recordings independent of NGA-Subduction database

Two notable earthquakes were recorded in southern Cascadia after the final version of the NGA-Subduction database was published—the 2021 Petrolia earthquake and the 2022 Ferndale earthquake. Both events occurred inside the Gorda plate and feature shallower depths of 27.0 km and 17.9 km, respectively, compared with typical intraslab earthquake depths of 40–50 km. These events are important because they are independent of the NGA-Subduction database that was used by the model developers and are appropriate recordings for comparing GMMs as described by Mak et al. (2017). We compile and process records from these two earthquakes that are independent of the NGA-Subduction database and also the 2010 Ferndale earthquake that is included in the NGA-Subduction database. We used the US Geological Survey Python-based package gmprocess to automate the downloading and processing of recordings from the these two earthquakes in a uniform manner (Hearne et al., 2019). Recordings are restricted to latitudes greater than 38.5° to exclude recordings from northern California that were within the 500 km search radius, but recorded on California sites that had $V_{S30}$ scaling or were located in basins that are not part of the CSZ. The resulting data set comprises 152 recordings from the 2010 Ferndale earthquake, 170 recordings from 2021 Petrolia earthquake, and 189 recordings from the 2022 Ferndale earthquake. Figure 4a shows the earthquakes and corresponding station locations, and Figure 4b displays the processed records from the three earthquakes in magnitude–distance space. The details of the processing and data selection criteria steps used by gmprocess are described in the Appendix and can also be found by Rekoske et al. (2020).

Figure 4.

Earthquakes and recordings from the independent data set: (a) independent data relative to Ferndale 2010 and (b) independent data in magnitude–distance space.

Residual analysis

When evaluating the difference between the GMM predictions and the earthquake observations, a mixed-effects approach is used so that repeatable effects from an earthquake or site location can be handled in the analysis and for its ability to account for irregular sampling in sparse data sets (Abrahamson and Youngs, 1992). The prediction for each GMM for each record in the data set is made using the Python OpenQuake package (Pagani et al., 2014). The required input parameters for the NGA-Subduction GMMs are as follows: M, $Z_{\mathrm{tor}}$ (depth to top of rupture, km), $R_{\mathrm{rup}}$ (closest distance to the rupture plane, km), $V_{S30}$ (m/s), and $Z_{2.5}$ (km). The total residuals are calculated using the (natural log) difference between the observations and the predictions:

$$R_{\mathrm{tot},\mathrm{es}}=\ln (S{A}_{\mathrm{obs},\mathrm{es}})-\ln (S{A}_{\mathrm{pred},\mathrm{es}})$$

(4)

where $R_{\mathrm{tot},\mathrm{es}}$ is the total residual for event $e$ and station $s$ ; $S{A}_{\mathrm{obs},\mathrm{es}}$ is the observation for a given spectral acceleration; and $S{A}_{\mathrm{pred},\mathrm{es}}$ is the predicted median ground motion for a given spectral acceleration. The mixed-effects regression was performed using the Python statsmodel package (Seabold and Perktold, 2010) to break down the total residuals for each GMM into the bias (average difference), event terms, and within-event terms as defined below:

$$R_{\mathrm{tot},\mathrm{es}}=c+\delta B_e+\delta W_{\mathrm{es}}$$

(5)

where $c$ is the bias; $\delta B_e$ are the event terms; and $\delta W_{\mathrm{es}}$ are the within-event terms.

The bias $\left(c\right)$ is used to assess the average difference of the GMM predictions to regional recordings across different periods. The event terms $\delta B_e$ are zero-mean random variables with standard deviation $\tau$ that account for how much each event shifts from the observed ground motion compared with the average predicted ground motion across all earthquakes in the data set. The within-event terms $\delta W_{\mathrm{es}}$ are zero-mean, normally distributed random variables with standard deviation $\varphi$ that represent the misfit between observation at a specific site from an earthquake median prediction after accounting for the event term for that earthquake.

The within-event terms $\delta W_{\mathrm{es}}$ can further be divided into the between-site terms and within-site terms:

$$\delta W_{\mathrm{es}}=\delta S2S_s+\delta W{S}_{\mathrm{es}}$$

(6)

where $\delta S2S_s$ are the between-site terms and $\delta W{S}_{\mathrm{es}}$ are the within-site terms. The between-site terms represent the systematic deviation of the observations at a given site from the GMM predictions and will be used when computing the linear BAFs.

This study includes comparisons of the standard deviations of the within-event terms from regional records. The modeling groups made different choices with respect to whether to use regional data (or analog regional) data or global data for developing the aleatory variability models. To account for the $V_{S30}$ and distance dependence used in the GMM’s within-event variability models $\left(\varphi \right)$ , we standardize the within-event residuals by the GMM-predicted ${\varphi }_{\mathrm{es}}$ values following Worden et al. (2018). Analysis of the variability of $\delta W_{\mathrm{es}}$ without consideration for the modeled trends with $V_{S30}$ and distance would not be consistent with the GMM predictions. The standardized within-event terms are computed using the following equation:

$$Z_{\delta W_{\mathrm{es}}}=\frac{\delta W_{\mathrm{es}}}{{\varphi }_{\mathrm{es}}}$$

(7)

where $\delta W_{\mathrm{es}}$ is the within-event residual and ${\varphi }_{\mathrm{es}}$ is the predicted GMM $\varphi$ for a particular event and site. Then, $Z_{{\varphi }_{\mathrm{es}}}$ , the normalized within-event standard deviation, is computed as follows:

$$Z_{{\varphi }_{\mathrm{es}}}=\frac{1}{n-1}\sum _{i=1}^n(Z_{\delta W_{\mathrm{es}}}-\overline{Z_{\delta W_{\mathrm{es}}}}{)}^2$$

(8)

We emphasize that the reason to compare $Z_{{\varphi }_{\mathrm{es}}}$ instead of ${\varphi }_{\mathrm{es}}$ is so that we account for the effect of $V_{S30}$ and $R_{\mathrm{rup}}$ on ground-motion variability because they are not the same for all records in the data set.

GMM performance against observations

GMMs against Cascadia intraslab observations

2001 Nisqually earthquake

We compare the GMMs to data from a relatively well-recorded earthquake. For this purpose, we use the 2001 ( M  = 6.8) Nisqually earthquake that originated in the upper part of the Juan de Fuca plate at a depth of approximately 50 km and had a maximum peak ground acceleration (PGA) of 0.3 g recorded at Seward Park in Seattle (Frankel et al., 2009). A comparison between the Cascadia GMMs and observations from the Nisqually earthquake at two spectral periods (0.2 and 2.0 s) can be seen in Figure 5. The GMM predictions are for $V_{S30}=600m/s$ , whereas the bias (Figure 6) takes into account the specific $V_{S30}$ value at each station. The Seattle basin sites, depicted by diamond symbols in Figure 5, highlight the basin amplification at a period of 2 s.

Figure 5.

Nisqually data against GMM predictions for two spectral periods. The prediction is an average site condition of $V_{S30}=600m/s$ with no basin term.

Figure 6.

Nisqually bias computed using site-specific $V_{S30}$ and $Z_{2.5}$ .

The Nisqually bias, shown in Figure 6, incorporates the basin terms of the NGA-Subduction GMMs by including the site-specific $Z_{2.5}$ parameter. This means that sites located in the Seattle (and other) basins incorporate the estimated basin response from each GMM. The range of GMM predictions at 0.2 s shows greater variability compared with 2 s, which can clearly be seen when examining the bias (Figure 6). PSBAH20 features the largest predictions at short periods (0.01–0.5 s) because their adjustment was based on the global average ground motion (Parker et al., 2022). Zhao06 features an increasing bias in the period band of 0.2–2 s, which could be a result of the difference in the site response between Japan and Cascadia (Atkinson and Macias, 2009). AG20 Unadjusted is close to matching the observations at periods <2 s, whereas GMMs show a slight overprediction and a reduced range of variability at the longest periods (T < 2 s).

GMMs against NGA-Subduction intraslab observations including Nisqually

We next evaluate the mixed effects using the data set of selected intraslab recordings, including the Nisqually observations previously discussed, for Cascadia (Figure 2a). Figure 7 displays the bias for nine GMMs. At short periods $(T<1s)$ , the adjusted Cascadia GMMs (top row of Figure 7) have a bias in the −1.0 to −0.5 range, indicating that these GMMs overpredict the regional observations. At longer periods, the GMMs are closer to matching the observations. AG20 Unadjusted displays a flat bias that is close to zero across the period band.

Figure 7.

Bias for Cascadia GMMs with 95% confidence intervals based on the regional Cascadia data set. The bias for the global versions of the GMMs is in the bottom right subplot.

The NGA-Subduction global GMMs are shown in the bottom right-hand plot in Figure 7. At short periods, the difference between the bias from the global GMMs and the adjusted regional models can be explained by the developers’ different approaches to the adjustment terms for Cascadia. At longer periods, all of the global GMMs underpredict observations due to differences between the global and Cascadia GMMs. In addition, some of the underprediction may be explainable because of recordings in deep basins, where the site response is not fully captured by the global GMMs.

The event terms against magnitude are shown for a representative short period, $T=0.2\mathrm{s}$ , and long period, $T=2\mathrm{s}$ , in Figure 8 for both Cascadia and global NGA-Subduction GMMs. Nisqually ( M  = 6.8) is the only earthquake that shows noticeable difference between the global and regional versions for KBCG20 for $T=2\mathrm{s}$ . There is an apparent slight trend of events terms with increasing magnitude; however, events are limited, and we are not confident that this trend will hold for future larger-magnitude $(\mathbf{M}>6)$ earthquakes.

Figure 8.

Cascadia versus Global NGA-Subduction event terms for Cascadia for regional earthquakes.

We also assess distance trends by evaluating the within-event terms as a function of rupture distance. The within-event terms for $T=0.2\mathrm{s}$ are depicted in Figure 9. We bin the within-event terms to investigate average trends in the data. The Cascadia versions of the NGA-Subduction GMMs show good agreement with within-event terms across the full distance range $(<500\mathrm{km})$ . However, the global versions of the NGA-Subduction GMMs reveal a slight negative trend beyond 100 km, indicating that the GMMs slightly overpredict distance attenuation at larger distances. This result is likely due to the difference in anelastic attenuation between Cascadia compared with what is captured for the global average. Table 2 provides the binned values for five additional periods; the mean residual values for the Cascadia models tend to be closer to 0 compared with the global models.

Figure 9.

Within-event terms against rupture distance for $T=0.2\mathrm{s}$ . Binned means are shown as squares with their corresponding error bars. The top row is the Cascadia versions of the NGA-Subduction GMMs, and the bottom row is the global versions.

Table 2.

Average binned within-event residuals for five periods comparing the Cascadia models with the global models for the regional intraslab data.

Mean within-event residuals for intraslab events
		Cascadia model			Global model
Period (s)	Bin range (km)	AG20	KBCG20	PSBAH20	AG20	KBCG20	PSBAH20
0.2	35–100	−0.083	0.067	−0.02	0.023	0.276	0.121
	400–800	−0.002	0.009	0.041	0.009	0.053	0.093
0.5	35–100	0.091	−0.116	−0.044	−0.059	−0.441	−0.294
	400–800	−0.006	0.165	0.071	0.196	0.432	0.246
1.0	35–100	−0.04	−0.062	−0.032	−0.017	0.003	0.04
	400–800	0.038	−0.096	−0.036	−0.167	−0.4	−0.265
2.0	35–100	0.05	0.196	0.117	0.245	0.427	0.281
	400–800	−0.087	−0.13	−0.101	−0.087	−0.084	−0.034
5.0	35–100	0.031	−0.044	−0.003	−0.112	−0.25	−0.178
	400–800	0.12	0.206	0.171	0.168	0.325	0.278

Assessment of regional and global aleatory models

We next compare the predictions of the GMM aleatory models to the variance in the regional data, using a standardized-residual approach (Equation 8). There are not sufficient earthquakes to constrain a regional between-event standard deviation term $\left(\tau \right)$ , so we limit the comparison with the standard deviation of the within-event variability $\left(Z_{\varphi }\right)$ . These values are plotted against 20 spectral periods for the NGA-Subduction Cascadia GMMs in Figure 10. AG20 Unadjusted uses the same aleatory model as the adjusted AG20.

Figure 10.

Z _ϕ for the three adjusted NGA-Subduction Cascadia GMMs. The unadjusted AG20 GMM uses the same aleatory model as the adjusted AG20 and so is not shown. The vertical lines represent the standard error. The symbols for the different GMMs are plotted with minor offsets in the period values for better visualization.

In interpreting $Z_{{\varphi }_{\mathrm{es}}}$ , if the value is below 1, then the model-predicted ${\varphi }_{\mathrm{es}}$ overestimates the mixed-effects estimate using the regional data. Likewise, values above 1 indicate the model-predicted ${\varphi }_{\mathrm{es}}$ underestimates the within-event variance relative to the regional data. KBCG20 and PSBAH20 predict higher ${\varphi }_{\mathrm{es}}$ values from periods of 0.05–0.2 s; KBCG20 underpredicts observed variability for 0.4–2.0 s and overpredicts the variability near 2–4 s. PSBAH20 matches the observed variability from 0.5–2 s and overpredicts variability near 3–4 s. AG20 overpredicts variability for $T<0.05\mathrm{s}$ , matches the variability from 0.05–2.0 s, and overpredicts variability for $T>2.0\mathrm{s}$ . KBCG20 and PSBAH20 incorporated Japanese recordings (60% of the data) in their aleatory models, unlike the AG20 GMM, which used “analogous” subduction zones. Site-to-site variability in Japan has been recognized to be higher than the global average, which may give rise to the short-period overprediction (Abrahamson and Gulerce, 2022). The cause of the under- and overpredictions for longer periods $(T>1s)$ is unclear.

Regional Cascadia influence on GMM comparisons

We examine geographical influences on the GMM predictions by dividing the recordings between northern and southern Cascadia, using a latitude of 45.5° as the dividing threshold (Figure 2a). Seismicity in the data set is clearly distinct, with deeper intraslab earthquakes in northern Cascadia beneath the Puget Lowlands and Olympic Peninsula and shallower intraslab events in southern Cascadia near the Mendocino Triple Junction. The northern Cascadia bias (Figure 11) shows a strong overprediction at short periods (0.01–1 s), whereas the overprediction is less (but more variable) for southern Cascadia. The unadjusted AG20 model notably underpredicts shaking for southern Cascadia across all 22 periods, whereas the adjusted AG20 performs well relative to the observations for $T<1\mathrm{s}$ , but underpredicts the ground motion for $T>1\mathrm{s}$ . Zhao06 shows the strongest overprediction at short periods for both northern and southern Cascadia data sets.

Figure 11.

Bias for northern Cascadia data compared with southern Cascadia data.

The event terms against hypocentral depth are shown for $T=0.2\mathrm{s}$ and $T=2.0\mathrm{s}$ in Figure 12. At $T=0.2\mathrm{s}$ , there is a positive trend in the between-event terms with hypocenter depth, except for the shallowest $(z<25\mathrm{km})$ earthquakes, which have positive event terms. The shallowest intraslab earthquakes in southern Cascadia have the largest positive event terms, meaning that the event-specific deviation in ground motion from these earthquakes is greater than the average earthquake predictions for all of the NGA-Subduction GMM. Note that the events in southern Cascadia generally show high positive event terms, and the event terms for northern Cascadia are negative, with the exception of Nisqually. The large negative event term for AG20 is from an earthquake with only two records in the database that is located in the far northwest of the subduction zone (Figure 2a) and illustrates the uncertainty in using limited recordings to constrain event terms. Note that data from this event were not used to develop AG20 because their quality control required a minimum of three recordings per earthquake, whereas the residual analysis performed in this study considered events with only two recordings.

Figure 12.

Cascadia event terms against hypocentral depth. Northern Cascadia events are plotted in blue, and southern Cascadia events are shown in red.

GMMs against global interface events

The regional data comparison was restricted to intraslab earthquakes because we lack interface recordings from Cascadia. As an alternative, a subset of NGA-Subduction ground-motion recordings of global interface records with $\mathbf{M}>7.5$ is used to evaluate GMMs that were developed prior to the compilation of this database and to ensure consistency between these older GMMs and observations. We acknowledge that the comparison between a Cascadia GMM and a global data set is not suitable for evaluating the application of the GMM to Cascadia. However, an important question for seismic hazards is how the Cascadia interface GMMs perform when compared with data from larger-magnitude earthquakes, for which we need to use global observations. Some differences between the global observations and regional Cascadia GMMs are expected by definition of developing a regionalized model. The comparisons relative to regional interface GMMs are not meant to provide insight into which regional GMM performs best for Cascadia, but to examine GMM performance relative to a small global data set of large-magnitude ground motions.

Figure 13 compares the Tohoku observations at two spectral periods with all GMMs—0.2 and 2.0 s. There is good agreement betweeen the NGA-Subduction GMMs and AM09 at shorter periods, but the GMMs exhibit significant (order of magnitude) differences in their predictions at longer periods.

Figure 13.

Tohoku observations for SA (0.2 s) and SA (2.0 s). GMM predictions are uniform $V_{S30}=760m/s$ .

We checked for distance trends for the Cascadia and global GMMs by plotting within-event terms as a function of distance (Figure 14 and Table 3). AG20 and PSBAH20 assume interface path attenuation is the same for Cascadia as it is for the global case. KBCG20 uses the intraslab-derived stronger path attenuation in Cascadia. The difference is not clearly visible in Figure 14, but there is a subtle upward trend in KBCG20 Cascadia residuals at the farthest distances compared with the global version of that model. At longer periods, the binned values in Table 3 show little and no trends with distance. Figure 15 uses the global data set to compare the bias for the NGA-Subduction GMMs, along with AM09 and Zhao06. The long-period behavior was examined for various subsets of the large global interface events (one example is removing all Tohoku records), and the results were consistent with Figure 15.

Figure 14.

Within-event terms against rupture distance for Cascadia (top row) and Global interface NGA-Subduction GMMs (bottom row). Binned means are shown as squares with their corresponding error bars.

Table 3.

Average binned within-event residuals for five periods comparing the Cascadia models with the global models for the regional interface data.

Mean within-event residuals for interface events
		Cascadia model			Global model
Period (s)	Bin range (km)	AG20	KBCG20	PSBAH20	AG20	KBCG20	PSBAH20
0.2	35–100	0.028	−0.021	0.034	0.052	0.020	0.031
	400–800	0.087	0.147	0.112	0.043	0.054	0.113
0.5	35–100	−0.008	−0.054	−0.007	0.020	−0.009	−0.008
	400–800	0.080	0.141	0.116	0.018	0.058	0.115
1.0	35–100	−0.006	−0.048	−0.024	0.012	−0.013	−0.024
	400–800	0.074	0.133	0.140	0.029	0.073	0.140
2.0	35–100	0.017	−0.022	−0.014	0.011	−0.005	−0.016
	400–800	0.061	0.113	0.145	0.073	0.095	0.149
5.0	35–100	0.028	−0.011	−0.004	−0.005	−0.014	−0.005
	400–800	0.010	0.045	0.057	0.070	0.066	0.058

Figure 15.

Bias for GMMs and global NGA-Subduction GMMs, as well as AM09 and Zhao06. The data used are interface records with M  > 7.5.

There is a clear period-dependent trend in the bias for the suite of GMMs, with all GMMs performing well at short periods $(T<1\mathrm{s})$ and a large dispersion in bias at longer periods (Figure 15). PSBAH20 slightly underpredicts long periods $(T>1\mathrm{s})$ , and AG20 and KBCG20 slightly overpredict long periods. AM09 exhibits a prominent negative gradient at long periods, which indicates that the GMM overpredicts the average ground-motion level of interface observations by factors of up to about three at long periods $(T>1\mathrm{s})$ . The comparison between global and Cascadia GMMs can be used to assess the choices made by the developers in their Cascadia interface GMMs.

GMMs against NGA-Subduction independent observations

We next use the observations from the 2010 ( M  = 6.5) Ferndale earthquake to the 2021 Petrolia ( M  = 6.2), and the 2022 ( M  = 6.4) Ferndale earthquakes, to compare the GMMs using data independent of the NGA-Subduction project. Figure 16 compares the observations to the predictions at SA (0.2 s) and SA (2.0 s) for all three earthquakes. Figure 17 shows the bias computed for the three events as a function of period. Bias calculations use the available site parameters in the NGA-Subduction database or the USGS global $V_{S30}$ database otherwise (Heath et al., 2020). At $T=0.2\mathrm{s}$ , the GMMs match or slightly overpredict the observed ground motions. At longer periods both Ferndale events are underpredicted. The overall trend in the bias for the 2010 and 2022 Ferndale events is highly similar, as are their distance trends. The 2022 Ferndale event includes more data (50 additional records). The 2021 Petrolia sequence featured two overlapping events 10 s apart in slightly different locations and this complex earthquake rupture may cause the distinctive bias plot, with minima near $T=1.0\mathrm{s}$ (Yeck et al., 2023). The complex rupture that resulted in the 2021 Petrolia sequence is different from the empirical data that was used in GMM development, and so differences in the misfit may be attributed to the observed waveforms rather than the GMM performance. The 2010 Ferndale earthquake shows a similar misfit to the 2022 Ferndale earthquake, which indicates the GMMs are predicting similar median ground motion for an event that was considered by the model developers and an event that is independent of the database used to develop Cascadia GMMs.

Figure 16.

2010 Ferndale, 2021 Petrolia earthquake, and 2022 Ferndale earthquake observations compared with GMM predictions against rupture distance for two spectral periods. The GMM predictions are for a uniform site condition $(V_{S30}=600\mathrm{m}/\mathrm{s})$ with no basin term.

Figure 17.

Bias for the three earthquakes used in the independent data evaluation.

Average Seattle BAFs

We compute linear BAFs for the Seattle basin with respect to a reference region defined outside of the Puget Lowlands basin and restricted to the Pacific Northwest observations. The Seattle basin is defined based on geological constraints and the estimated $Z_{2.5}$ depth, which reaches a maximum value of nearly 8 km in the deepest parts of the basin (Stephenson et al., 2017). The BAFs are computed for each period using the following equation:

$$\ln (\mathrm{BA}{F}_s)=〈\delta S2S_{s(\mathrm{basin})}〉-〈\delta S2S_{s(\mathrm{ref})}〉$$

(9)

where the 〈〉 brackets indicate to compute the mean of what is enclosed, $\delta S2S_{s(\mathrm{basin})}$ are the site terms for sites inside the Seattle basin, and $\delta S2S_{s(\mathrm{ref})}$ are site terms for stations located outside of the Puget Lowlands basin (Equation 6). The site-specific $V_{S30}$ is accounted for in the predictions so that the amplification can be related to the deep-basin effect and not the site effects modeled by $V_{S30}$ .

We compute BAFs using the two different magnitude thresholds selected by the NGA-Subduction modelers for Cascadia. AG20 used data from $\mathbf{M}>5$ earthquakes for developing their GMM, while KBCG20 and PSBAH20 were both developed using recordings for earthquakes $\mathbf{M}>4.5$ . The BAFs computed using the $\mathbf{M}>5$ data set and the $\mathbf{M}>4.5$ data set are depicted in Figure 18c and 18d, respectively. A difference in the basin models from KBCG20 and PSBAH20, compared with AG20, is the presence of a deamplification at shorter-period ground motions $(T<0.1\mathrm{s})$ in KBCG20 and PSBAH20 and the presence of smaller long-period $(T>1\mathrm{s})$ amplification compared with AG20. These different features emerge in the empirical BAFs for the regions depending on the data selection criteria for earthquake magnitude and indicate that differences in the basin-depth scaling models of these GMMs may result from unexplained features of the ground motions relating to station sampling or other effects. We note that the recordings for $\mathbf{M}>5$ in the Seattle basin are limited to only one well-recorded event (Nisqually) and the applicability to future events is unclear. Moreover, the lower amplification from using data selection of $\mathbf{M}>4.5$ may result from using seismic waves that enter the Seattle basin from the northwest. Earlier studies indicated strong dependence of source azimuth on the degree of amplification for the Seattle basin (Frankel et al., 2015; Thompson et al., 2020) that would be consistent with these observations because the recordings from $\mathbf{M}>5$ events are at an azimuth entering the Seattle basin that produces higher ground motions relative to events occurring northwest of the basin.

Figure 18.

Figures (a) and (c) plot the station-event pairs inside the Seattle basin and the reference stations used to compute the average BAF shown in Figures (b) and (d). The red lines in (a) and (c) illustrate the azimuths for the source-site pairs used for the respective data sets. The gray shaded area in (b) and (d) shows the confidence intervals: (a) reference sites $\mathbf{M}>5$ ; (b) reference sites $\mathbf{M}>5$ ; (c) reference sites $\mathbf{M}>4.5$ ; and (d) average BAFs $\mathbf{M}>4.5$

M9-based basin adjustment

The M9 project for Cascadia simulated the ground motions from a M9 megathrust event for the Cascadia Interface, and the results were discussed by Frankel et al. (2018a). The simulated ground motions indicate an average BAF of two for $T\ge 2\mathrm{s}$ (Marafi et al., 2019; Wirth et al., 2018b) when defined as the ratio of ground motions inside deep basins $(Z_{2.5}>6\mathrm{km})$ relative to ground motions at a reference depth of $Z_{2.5}=3\mathrm{km}$ (Figure 19). Note that the average amplification factor for the Seattle basin can be as high as five if a different basin definition $(Z_{2.5}>5\mathrm{km})$ and a reference site condition $(Z_{2.5}<1\mathrm{km})$ are used in defining the ratio (Marafi et al., 2019). We evaluate the sensitivity of the ground-motion amplifications to different assumptions about the reference (i.e., outside-of-basin) site conditions in the M9 simulations. The M9-based BAFs are used by the Seattle Department of Construction and Inspection (SDCI) in their building codes and this study provides details about how to implement the M9-based BAFs into empirical NGA-Subduction GMMs for interface events recorded in the deepest parts of the Seattle basin, but the assumption here is that the ground motions from the M9 simulations are reasonable and accurate (Wirth et al., 2018a). The decision to examine the SDCI factor-of-two basin factors is based on personal preference and does not have a strong technical justification for adoption in seismic hazard for Cascadia.

Figure 19.

Spectral ratios for the NGA-Subduction GMMs for a 6 km basin site relative to 3 km reference site.

If we compare the NGA-Subduction empirical BAFs to the factors from M9 where the BAFs are defined using the ratio of sites with $Z_{2.5}=6\mathrm{km}$ to $Z_{2.5}=3\mathrm{km}$ , we find that the NGA-Subduction basin terms underestimate the M9 simulation basin effects for $T\ge 3\mathrm{s}$ , which are a factor of 2. AG20 basin terms produce an amplification factor of about 1.7; PSBAH20 has an amplification factor of approximately 1.4; and KBCG20 Puget basin model has a factor of approximately 1.3, compared with the 1.6 factor for their Seattle model at $T=4\mathrm{s}$ . The BAFs from the crustal-based CB14 (used to adjust the older subduction GMMs in the 2014 NSHM) are about 1.5 at longer periods. The simulation-based results from Wirth et al. (2019) found shallower crustal earthquakes produced higher amplifications in the Seattle basin compared with deeper intraslab earthquakes. The difference in the empirical BAFs compared with the M9 simulation basin factors motivates an adjustment to the NGA-Subduction GMM basin terms.

Multiple approaches could be used to adjust the basin terms in NGA-Subduction GMMs to match the M9 basin amplifications. One approach is to apply a factor-of-two with respect to a defined reference condition as the basin term. The M9 simulations were run using a minimum uniform $V_{S30}=600\mathrm{m}/\mathrm{s}$ , and the factor-of-two BAFs were computed using a reference condition of $Z_{2.5}=3\mathrm{km}$ . Therefore, we consider two reference conditions based on the M9 simulations for computing the adjustment:

1. The reference conditions are specified by the mean $Z_{2.5}$ value for $V_{S30}=600\mathrm{m}/\mathrm{s}$ conditions: $Z_{2.5}={\mu }_{Z2.5}(V_{S30}=600)\mathrm{m}/\mathrm{s}$ .

2. The reference conditions are specified by the $Z_{2.5}=3\mathrm{km}$ conditions used in the factor-of-two adjustment computed from the simulations.

The ground motions for Seattle basin sites with $Z_{2.5}\ge 6\mathrm{km}$ using the factor-of-two adjustment are as follows:

$$\ln (Y_s)=\mu (M,R,V_{S30}=V_{S30}^{\mathrm{site}},Z_{2.5}=Z_{2.5}^{\mathrm{ref}})+\ln (2)$$

(10)

where $V_{S30}^{\mathrm{site}}$ corresponds to site-specific estimates of these parameters. The two approaches differ in the reference conditions that are specified through the $Z_{2.5}^{\mathrm{ref}}$ parameter. Note that these adjustments are $V_{S30}$ independent and do not allow for additional effects of $V_{S30}$ on the basin term. We subtract the contribution of the $V_{S30}$ -scaling model that includes the deeper basin effects using the $V_{S30}$ - $Z_{2.5}$ function developed for each GMM (Figure 1).

In approach 1, the factor-of-two adjustment is relative to the average $Z_{2.5}$ for a site condition of $V_{S30}=600\mathrm{m}/\mathrm{s}$ :

$$\ln (Y_s)=\mu (M,R,V_{S30}=V_{S30}^{\mathrm{site}},Z_{2.5}={\mu }_{Z_{2.5}}(V_{S30}=600))+\ln (2)$$

(11)

KBCG20 average $Z_{2.5}$ for a given $V_{S30}$ is determined from basin data in the Puget Lowlands outside of the Seattle basin. However, if the same $V_{S30}$ scaling is used inside and outside the Seattle basin, then the $V_{S30}$ scaling should incorporate some basin effects from the Puget Lowlands region that need to be subtracted from the total ground motion. In approach 2, the reference condition is specified by $Z_{2.5}^{\mathrm{ref}}=3\mathrm{km}$ :

$$\ln (Y_s)=\mu (M,R,V_{S30}=V_{S30}^{\mathrm{site}},Z_{2.5}=3)+\ln (2))$$

(12)

The original basin terms for the NGA-Subduction GMMs and the adjusted factor-of-two basin terms using the two different reference conditions are shown as heat maps in Figure 20. The adjustment terms are computed for the uniform $V_{S30}$ values applied in the NSHM maps: 150, 185, 260, 365, 530, 760, 1080, and 1500 m/s. Note that AG20 and PSBAH20 have a constant (minimum) depth for $V_{S30}\ge 600\mathrm{m}/\mathrm{s}$ (Figure 1) that gives similar M9 adjustment factors for sites with $V_{S30}\ge 600\mathrm{m}/\mathrm{s}$ . AG20 basin terms are close to a factor of two ( $\ln (2)=0.69$ ) for $T\ge 4\mathrm{s}$ .

Figure 20.

Heat maps of basin terms where the three framed columns correspond to the type of adjustment (no adjustment, approach 1, and approach 2) and the three framed rows correspond to the NGA-Subduction GMMs. Yellow corresponds to smaller basin terms, and red boxes correspond to larger basin terms.

The first reference condition results in slightly lower ground-motion levels for AG20. Because it is not defensible to lower the ground motions based on the factor-of-two, we propose a modification to the adjustment such that the maximum of the original basin term and this adjustment is used as described in the following equation:

$$\begin{array}{c}\ln (Y_s)=max[\mu (M,R,V_{S30}=V_{S30}^{\mathrm{site}},Z_{2.5}=Z_{2.5}^{\mathrm{site}}),\hfill \\ \multicolumn{1}{c}{\mu (M,R,V_{S30}=V_{S30}^{\mathrm{site}},Z_{2.5}={\mu }_{Z_{2.5}}(V_{S30}=600))+\ln (2)]}\end{array}$$

(13)

KBCG20 period-dependent Seattle basin model uses a reference basin depth that is unusually shallow (200 m) compared with the other two GMMs, and this can result in factors greater than two for high $V_{S30}$ values. This behavior indicates that the factor-of-two adjustment should only be applied for $V_{S30}<760\mathrm{m}/\mathrm{s}$ (and the original basin terms are used for other site conditions).

In approach 2, the reference condition $(Z_{2.5}^{\mathrm{ref}}=3\mathrm{km})$ increases the ground-motion level for the NGA-Subduction GMMs for the full $V_{S30}$ range. However, factors >2 result because $Z_{2.5}=3>Z_{{2.5}_{\mathrm{ref}}}^{V_{S30}}$ for some $V_{S30}$ . If the adjustments are only applied for $V_{S30}<760\mathrm{m}/\mathrm{s}$ and the maximum of the factor-of-two and original basin terms are used, then the lower $V_{S30}$ values would result in higher adjustment factors that significantly increase the ground motion. However, further study is warranted because some sites built on artificial fill in the Seattle basin have recorded very high amplification factors (Frankel et al., 2009). Approach 1 was ultimately selected for the 2023 NSHM update (Petersen et al., 2023).

Note that Powers et al. (2021) used the basin-depth scaling models from the active crustal GMM of CB14 to model basin effects for the Zhao06 and AM09 GMMs in the 2018 NSHM. The basin terms for the CB14 are not defined using differential depth but directly use $Z_{2.5}$ . This means that we do not remove the basin part of the $V_{S30}$ scaling from these models, and their basin terms for the factor-of-two adjustment are defined to be a constant $\ln (2)$ across the period band and $V_{S30}$ range.

Considerations for seismic hazard analysis in Cascadia

The data comparisons are useful in evaluating Cascadia GMMs based strictly on data-fit criteria. Regional intraslab comparisons reveal that AG20 Unadjusted performs well when compared to the Casadia data set. The three adjusted NGA-Subduction GMMs overpredict the short-period data by factors of 0.7 to 3. Uncertainty is high in the GMMs for this region due to the low number of recorded intraslab earthquakes with moderate to large magnitudes and the absence of recordings of large-magnitude interface events that are included in hazard models. Whether future intraslab earthquakes will exhibit the characteristic frequency content observed in past events (i.e., lower-than-average short-period ground motions and consistent to higher-than-average long-period motions) is unclear. The global versions of NGA-Subduction intraslab GMMs do not perform as well compared with the data and are not able to match the long-period observations, which could in part be due to missing some of the deep site response from stations recorded in the sedimentary basins of the Pacific Northwest. Finally, Zhao06 and Zhao16 show poor fits to the regional intraslab data; based on data-fit criteria alone, these GMMs are not recommended for Cascadia-specific hazard applications.

The development of aleatory variability models from regional (versus global) data is an important consideration for hazard model development. Too few events are available for regional data to provide guidance on the development of $\tau$ models. For the development of $\varphi$ models, there is limited representation from different events, but only 322 regional recordings were used to constrain the variability of within-event terms. Comparisons of the standard deviation show that a regional $\varphi$ model performs better across the short period ranges (0.05–0.2 s), perhaps due to the higher site-to-site variability encountered in the global database compared with the regional database.

Because no interface records are available for testing Cascadia interface GMMs, we use the global data set to evaluate NGA-Subduction and other GMMs. Comparisons of the GMMs developed prior to the NGA-Subduction Project—Zhao06 and AM09—are useful because the NGA-Subduction data set comprises the best compilation of subduction ground motions and enables evaluation of these GMMs with current observations. Short-period $(T<1\mathrm{s})$ predictions from the Zhao06 and AM09 GMMs are consistent with observations; however, the long periods show overprediction indicating that the average long-period predictive skill of the AM09 GMM is poor (based strictly on data fit).

Evaluation of regionalization effects in Cascadia ground motions shows that GMMs that included southern Cascadia data (KBCG20 and PSBAH20) show a smaller misfit when compared to a GMM based strictly on northern Cascadia (AG20). Between-event terms reveal an intriguing correlation with hypocentral depth, except for the shallowest events, which produce much higher than predicted ground motions. Bias from the Ferndale 2010 and 2022 events is highly similar, as are their distance attenuations, indicating repeatable ground-motion effects. The 2021 Petrolia event, however, has a distinctive bias pattern. This effect may result from the complex rupture of this event, with two distinct, spatially separated slip patches.

Calculation of basin amplifications reveals that gross features in the basin-depth scaling models of the NGA-Subduction GMMs can be reproduced using different magnitude subsets of the regional data. Using a higher magnitude threshold $\mathbf{M}>5$ results in no short-period deamplification and greater long-period amplification than result from a magnitude threshold of $\mathbf{M}>4.5$ . The basin terms indicate that AG20 provides higher basin amplification on average and indicate amplification factors similar to the M9 simulations. The other GMMs may underestimate basin effects from a M9 megathrust event if the shallower incidence angle of incoming seismic waves from interface events produces higher basin amplifications (Thompson et al., 2020). We recommend minor modifications to the factor-of-two to disallow deamplification from this model for some site conditions. We also recommend applying the factor-of-two adjustment presented as approach 1 in this study, to the original basin terms of KBCG20 and PSBAH20 for interface earthquakes recorded in the deep parts of the Seattle basin ( $Z_{2.5}>6\mathrm{km}$ in Cascadia.

Conclusion

This study compared Cascadia subduction GMMs against a common but limited set of regional recordings, in addition to global interface recordings, and two recent earthquakes from southern Cascadia. The comparison highlighted differences in the Cascadia GMMs based on the data and the influence of the adjustment terms in these GMMs. The unadjusted AG20 GMM is consistent with stress drops and attenuation that result from a young and warm Juan de Fuca slab (Frankel et al., 2015) and fits the data the best for the regional intraslab data set. The limited data indicate an adjustment to increase the short-period ground motions to be more in line with other subduction zones is reasonable. Moreover, Cascadia interface events pose a high hazard for this region, and the adjusted regional NGA-Subduction GMMs are useful for hazard studies in Cascadia. The factor-of-two adjustment was incorporated into the NGA-Subduction GMMs for interface sources based on the minimum $V_{S30}$ used in the M9 earthquake simulations. In future studies, alternative approaches to adjusting the empirical GMMs that account for the full M9 basin scaling with depth and adjust the source term to match the average M9 ground motion from the simulations can be incorporated into be GMMs for Cascadia.