Empirical ground-motion prediction equations [GMPEs] such as the Next Generation Attenuation-West2 [NGA-West2] GMPEs are limited in the number of recordings on hard-rock stations used to develop the models. Therefore, the site response scaling in the GMPEs cannot be reliably extrapolated to hard-rock conditions. The state of practice for the development of hard-rock adjustment factors involves the use of analytical methods that typically assign small values to the high-frequency small strain damping [ 0] for hard-rock sites resulting in large scaling factors at short periods. Alternatively, the hard-rock scaling factors developed in Ktenidou and Abrahamson (2016) [KA16] based on empirical ground- motion data are used. These empirical factors, developed for a broad rock site category, show that the average hard-rock scaling factors observed in ground- motion data are small in amplitude. To address the shortcomings in the current state of practice, we present a methodology to develop hard-rock linear site adjustment factors to adjust the NGA- West2 GMPEs from VS30of 760 m/sec to target hard-rock site conditions with VS30 ranging from 1000 to 2200 m/sec. These factors are analytically derived using the IRVT approach of Al Atik et al. (2014) but with inputs constrained using the empirical KA16 factors and normalized to the scaling of the NGA-West2 GMPEs for VS30 of 1000 m/sec. The proposed factors merge the results of the NGA-West2 site response scaling for VS301000 m/sec with the KA16 hard-rock category factors to produce a site factor model that is a continuous function of VS30. The epistemic uncertainty of these factors is evaluated.4 Keywords Site response, site adjustment, site ampliﬁcation, 0, hard rock, GMPE consistent, NGA-West25 1Linda Alatik Consulting, San Francisco, CA 2Consultant, Oakland, CA 3University of California, Berkeley, CA 4Paciﬁc Gas & Electric, San Francisco, CA Corresponding author: Linda Al Atik, San Francisco, CA Email: linda.alatik@gmail.com Prepared using sagej.cls [Version: 2017/01/17 v1.20] Al Atik et al. 3 Introduction 6 Modern ground-motion prediction equations [GMPEs] such as the Next Generation 7 Attenuation-West2 [NGA-West2] GMPEs characterize site response as a continuous 8 function of the time-averaged shear-wave velocity over the top 30 m of the site proﬁle 9 [VS30]. Other parameters, such as the depth to a shear-wave velocity of 1.0 or 2.5 km/s 10 horizon [Z1:0andZ2:5], are used to characterize the long-period site ampliﬁcation due 11 to basin effects. Nonlinear site response is modeled in the GMPEs as a function of 12 VS30and the median spectral acceleration or peak ground acceleration on rock. The 13 histogram of the number of recording stations in the different VS30bins used in the 14 development of the Abrahamson et al. (2014) GMPE [ASK14] is shown in Figure 15 1. This ﬁgure shows that the number of recording stations with VS30>1000 m/sec 16 is limited. The VS30dependence of the site factors is modeled in the GMPEs as a 17 linear function of ln(VS30). With the sampling of VS30in the NGA-West2 dataset, 18 the coefﬁcient for ln(VS30)is constrained by empirical ground-motion data for VS3019 values between 200 and 800 m/s. For hard-rock sites with VS30>1000 m/sec, the site 20 factor is based on an extrapolation of this slope to high VS30values with little empirical 21 constraints. To reﬂect the limited hard-rock data, some GMPEs limit the reduction of 22 the site factor at high VS30values (e.g. 1500 m/s). 23 Empirical hard-rock adjustment factors were developed in Ktenidou and 24 Abrahamson (2016) [KA16] to adjust GMPEs from VS30of 760 m/sec to average 25 hard-rock conditions based on the Next Generation Attenuation-East Project [NGA- 26 East] and the BCHydro British Columbia ground-motion datasets. These factors were 27 developed using the average of total ground-motion residuals with VS30greater than 28 1000 m/sec relative to median predictions from an interim NGA-East GMPE and the 29 Chiou and Youngs (2014) [CY14] GMPE with VS30of 760 m/sec for the NGA-East 30 and the BCHydro datasets, respectively. These scaling factors account for differences in 31 theVSproﬁles and the high-frequency small-strain damping [ 0] between the reference 32 site condition in the GMPEs with VS30of 760 m/sec and an average hard-rock site 33 in the NGA-East and the BCHydro datasets. Figure 2 presents the two hard-rock 34 scaling models proposed by KA16: model 2 is based on the residual analysis of the 35 BCHydro dataset and model 1 is based on an average of the scaling obtained for 36 the combined BCHydro and the NGA-East datasets. KA16 indicate that the average 37 VS30for the BCHydro and the NGA-East data used to develop the models is 2380 and 38 1975 m/sec, respectively. The KA16 models shown in Figure 2 were developed as an 39 interim measure to provide an alternative to the typically large scale factors computed 40 Prepared using sagej.cls 4 Journal Title XX(X) using analytical methods for hard-rock sites. These models show that observed ground- 41 motion data on hard-rock sites, on average, do not show the large scale factors at short 42 periods typically found in analytical studies that assign small 0values, on the order 43 of 0.006 sec, for hard-rock sites. 44 The KA16 hard-rock adjustment factors suffer from shortcomings stemming from 45 the poor characterization of site conditions at the recordings stations in the NGA- 46 East and BCHydro datasets and the low sample rates at some seismic stations that 47 limit the useable frequency band. Most of the hard-rock stations in the NGA-East 48 dataset do not have measured VS30values, and the NGA-East ground motions suffer 49 from frequency-bandwidth limitations affecting data quality at the low and the high 50 frequencies. Similarly, all rock stations in the BCHydro dataset are classiﬁed based on 51 surface geology (Ktenidou and Abrahamson 2016). Errors in the VS30estimates for the 52 hard-rock sites could lead to misclassiﬁcation of the sites and affect the resulting scale 53 factors and average VS30values. Another limitation of the KA16 factors is that they 54 are not a continuous function of VS30and instead apply to a broad hard-rock site class 55 withVS30>1000 m/sec. This is a result of the limited recordings on hard-rock sites in 56 the empirical datasets. 57 The current state of practice for the development of site adjustment factors for hard- 58 rock site conditions involves three different approaches: (1) use of VS-0correction 59 factors developed using analytical methods with assigned target 0values andVS60 proﬁles; (2) use of empirical hard-rock factors such as the KA16 factors which are 61 developed for a broad category of hard-rock sites; and (3) use of the VS30scaling in 62 the NGA-West2 GMPEs extrapolated to hard-rock VS30despite the limited empirical 63 data on hard-rock site conditions available to constrain the V30scaling and the lack 64 of explicit0scaling in the GMPEs. To address the shortcomings in the current state 65 of practice, we present a methodology to develop hard-rock site factors to adjust the 66 NGA-West2 GMPEs from VS30of 760 m/sec to target hard-rock site conditions with 67 VS30ranging from 1000 to 2200 m/sec. These factors are analytically derived using 68 the IRVT approach of Al Atik et al. (2014) but constrained using the empirical KA16 69 factors and also normalized to the NGA-West2 site factors for VS30of 1000 m/sec. 70 These empirical constraints allow for the calibration of the hard-rock properties in the 71 analytical study so that they are consistent with the observed ground-motion scaling for 72 these sites. The proposed factors merge the results of the NGA-West2 VS30scaling for 73 VS301000 m/sec with the KA16 hard-rock category factors to produce a model for 74 the site factors that is a continuous function of VS30and consistent with the empirical 75 hard-rock factors. 76 Prepared using sagej.cls Al Atik et al. 5 Our approach starts with an evaluation of the average hard-rock site conditions 77 representative of the KA16 scaling factors, by inverting for the average VS30,VS78 proﬁle, and 0implicit in the KA16 models. Next, the target site conditions ( VS(z) 79 proﬁle and0) are deﬁned for a range of hard-rock site conditions with VS30of 1000 80 to 2200 m/sec based on a literature review of hard-rock VSproﬁles and 0estimates 81 in Western North America [WNA]. Target site parameters are adjusted using empirical 82 constraints, and hard-rock site adjustment factors are derived and presented for a suite 83 of thirteen target VS30values between 1000 and 2200 m/sec. 84 The site factors presented in this paper extend the NGA-West2 linear site response 85 scaling to hard-rock site conditions with target hard-rock sites deﬁned based on WNA 86 average rock site properties. Therefore, our derived hard-rock site adjustment factors 87 are applicable for target sites in WNA. We use the KA16 models derived from empirical 88 NGA-East and BCHydro data as constraints because of the scarcity of empirical 89 ground-motion data on hard rock sites in WNA. Moreover, KA16 showed that average 90 hard-rock scaling outside of WNA is not drastically different than what would be 91 expected for WNA sites. We note that the hard-rock factors presented in this paper are 92 intended for use at sites with limited site characterization, such as sites with measured 93 or estimated VS30values, but without measured VS(z)proﬁles. For hard-rock sites 94 with measured VS(z)proﬁles, site-speciﬁc hard-rock adjustment factors should be 95 computed based on the site proﬁle with its appropriate epistemic uncertainty. 96 Vs Proﬁle and 0Inversion for the KA16 Models 97 The use of the KA16 models to constrain the analytical hard-rock site adjustment 98 factors requires an evaluation of the implied average site conditions representative of 99 these scaling factors. Because a large number of the stations used in the KA16 analysis 100 did not have measured VS30and0values, we use the 0andVSproﬁle inversion 101 methodology of Al Atik and Abrahamson (2021) to invert for representative VS30, 102 VS(z)proﬁle, and 0for the average site conditions implied by the KA16 hard-rock 103 scaling factors. 104 The inversion is performed using the CY14 GMPE because KA16 model 2 is based 105 on the residual analysis of the BCHydro dataset with respect to CY14 and because 106 the spectral shape of the CY14 GMPE generally falls in the center of the range of 107 spectral shapes from the NGA-West2 GMPEs. The ﬁrst step involves converting the 108 KA16 hard-rock scaling factors from pseudo-spectral acceleration [PSA] domain to 109 Fourier amplitude spectra [FAS] domain. As such, CY14 median response spectra for 110 strike-slip scenarios with magnitude 5, 6, and 7, distance of 5, 10, and 20 km and 111 Prepared using sagej.cls 6 Journal Title XX(X) VS30of 760 m/sec are computed. These scenarios are selected to capture the short- 112 distance0scaling from a range of hazard-signiﬁcant magnitudes. The CY14 median 113 response spectra are corrected to hard-rock conditions by multiplying them with the 114 KA16 model 1 and model 2 factors. Next, the IRVT approach of Al Atik et al. (2014) 115 is used to convert the GMPE’s response spectra for VS30= 760 m/sec and the spectra 116 corrected to hard-rock conditions into corresponding FAS. Duration estimates for the 117 different scenarios are calculated using estimates of source and path durations with 118 generic Western US [WUS] parameters based on Campbell (2003). The peak factor of 119 Vanmarke (1975) is used in the IRVT method. The FAS for the scenarios with VS30= 120 760 m/sec and those corrected to hard-rock conditions are presented in Figure 3. These 121 IRVT-based FAS show a change in their spectral shape at frequency of about 50 Hz. 122 This is likely due to saturation effects in the IRVT process discussed in Al Atik et al. 123 (2014). For the hard-rock FAS, the sharp change observed around 50 Hz is caused by 124 the sharp changes in the KA16 factors in the same frequency range, particularly for 125 KA16 model 2. 126 For each earthquake scenario, the ratio of FAS for the hard-rock site condition 127 relative to FAS for VS30= 760 m/sec is computed. Figure 4 presents these ratios for 128 each of the nine scenarios considered for model 1 and model 2 along with the average of 129 the ratios over the nine scenarios. These ratios approximate the FAS linear site factors 130 for hard rock relative to the reference site condition with VS30of 760 m/sec. These 131 relative site factors represent the differences in the VSproﬁle and0scaling between 132 the average hard-rock site implied by the KA16 models and the average site condition 133 for CY14 for VS30of 760 m/sec. The FAS ratios are stable over all nine scenarios for 134 frequencies up to about 20-30 Hz as shown in Figure 4. For frequencies greater than 135 20 Hz, the FAS ratios start diverging due to potential saturation effects in the IRVT- 136 derived FAS discussed in Al Atik et al. (2014). The average relative site factors are 137 smoothed as shown in Figure 5. We note that these average relative FAS site factors are 138 considered reliable for frequencies between 0.6 than 30 Hz. The upper limit is imposed 139 to avoid potential saturation effects in the IRVT-based FAS and the lower limit is based 140 on the KA16 factors being constrained by data for frequencies up to 0.6 Hz. 141 The next step involves converting the hard-rock site factors relative to 760 m/sec 142 to total site factors (relative to the VSand density at the source depth). As such, we 143 use the CY14-compatible VSproﬁle and0of Al Atik and Abrahamson (2021) as 144 representative of the reference site condition with VS30of 760 m/sec. The reference 145 VSproﬁle and corresponding quarter-wavelength [QWL] linear site ampliﬁcation for 146 CY14 forVS30of 760 m/sec are shown in Figure 10. The QWL site ampliﬁcation is 147 computed according to Boore (2003) with a zero angle of incidence and source VSand 148 Prepared using sagej.cls Al Atik et al. 7 density set at 3.5 km/sec and 2.75 g/cm3, respectively. The 0for CY14 for VS30of 760 149 m/sec is 0.039 sec. The total FAS site factors for the average hard-rock site condition 150 representative of the KA16 models 1 and 2 are obtained by multiplying the relative 151 hard-rock site factors with the site factors of the CY14 reference site condition and are 152 shown in Figure 6. 153 Inversion of KA16 Model 1 154 The total linear site factors represent the combined effects of the linear site 155 ampliﬁcation of the VSproﬁle and the attenuation due to damping, parameterized by 156 0. To reduce the trade-off between the VSproﬁle and0at high frequencies, we 157 assume that the depth dependence of the VSproﬁle follows a power law (e.g., azb). 158 With this assumption, we have use an analytical solution for the combined effects of 159 the site ampliﬁcation of the VSproﬁle in the top 30 m and the 0attenuation given the 160 VS30value. The methodology is described in Al Atik and Abrahamson (2021). 161 Using the total linear site factors for KA16 model 1 shown in Figure 6, the inversion 162 is performed to estimate the average 0andVSproﬁle representative of the average 163 hard-rock site condition in the model. A zero angle of incidence and a source VS164 and density of 3.5 km/sec and 2.75 g/cm3, respectively, are used in the inversion. 165 The density- VSrelationship used in Al Atik and Abrahamson (2021) is used in this 166 inversion. Because VS30is unknown for the KA16 models, the inversion is performed 167 to estimateVS30as well as for different assumed VS30values. Using the frequency 168 range of 10 to 20 Hz (10 Hz roughly corresponds to the frequency associated with 169 QWL ampliﬁcation for the top 30 m of the proﬁle and 20 Hz was chosen to avoid the 170 unreliable higher frequencies in the IRVT-based FAS), 0,VS30and theVSproﬁle 171 in the top 30 m are estimated analytically by ﬁtting the site response function in 172 the 10-20 Hz frequency range assuming that the top 30 m of the VSproﬁle follows 173 a power law function. The estimated 0andVS30are 0.032 sec and 1300 m/sec, 174 respectively. Figure 7(a) shows the high-frequency ﬁt compared to the total site factors 175 for frequencies >10 Hz. For frequencies <10 Hz, the ﬁt uses the initial site factors as 176 shown in the pink curve. 177 The site factors modiﬁed for frequencies greater than 10 Hz to follow the high- 178 frequency ﬁt (pink curve in Figure 7(a)) are divided by the 0operator to obtain the 179 linear site ampliﬁcation function due only to the VSproﬁle which is subsequently 180 smoothed as shown in Figure 7(b). The inverse QWL approach outlined in Al Atik 181 and Abrahamson (2021) is then applied to invert for the VSproﬁle working from high 182 to low frequencies of the site ampliﬁcation and solving for the shallow to deep layers 183 Prepared using sagej.cls 8 Journal Title XX(X) of the proﬁle. The inverted VSproﬁle, which is subsequently smoothed, is shown by 184 the pink curve in Figure 7(c). Because linear site ampliﬁcations are considered reliable 185 for frequencies >0.6 Hz, theVSproﬁle could only be inverted to a depth of 1.06 km. 186 A comparison of the initial relative site factors of KA16 model 1 to those obtained 187 using the inversion results is shown in Figure 7(d). This plot shows that the inverted VS188 proﬁle and0representative of the hard-rock condition for KA16 model 1 used along 189 with the reference VSproﬁle and0for CY14 at VS30of 760 m/sec can approximate 190 reasonably well the initial relative site factors of KA16 model 1 for frequencies up to 191 30 Hz. 192 Next, the inversion of KA16 model 1 described in this section is repeated using 193 different assumed VS30values instead of inverting for VS30as shown above. This 194 sensitivity analysis allows for a more robust estimation of 0from the high-frequency 195 site factors as well as an evaluation of the range of VS30and0values that can ﬁt 196 the hard-rock site factors of KA16 model 1 relative to the reference site condition 197 withVS30of 760 m/sec. Assumed VS30values of 1500, 1700, and 1975 m/sec are 198 used in this sensitivity analysis. The value of 1975 m/sec is used because it represents 199 the average VS30of the NGA-East hard-rock data used in KA16. Figure 8 shows a 200 comparison of the initial relative site factors of KA16 model 1 to those obtained using 201 the inversion for the derived and assumed VS30values. The inversion results for the 202 different assumed VS30values indicate that, as the assumed VS30increases, the derived 203 0value decreases and the slope of the inverted VSproﬁle in the top 30 m becomes less 204 steep approaching a single constant layer. The sum-of-squared errors (SSE) between 205 the inversion-based relative site factors and the initial site factors in the frequency range 206 of 0.6 to 30 Hz are calculated and listed in the plots of Figure 8. An evaluation of the 207 SSE values for the different inversion cases as well as the corresponding shapes of the 208 invertedVSproﬁles indicates that the assumed VS30of 1975 m/sec does not represent 209 the average hard-rock site conditions of KA16 model 1. The average VS30of 1975 210 m/sec obtained using the NGA-East hard-rock data in KA16 is likely biased high due 211 to the large number of stations with estimated or assigned VS30values. As a result, we 212 conclude that, within the context of the QWL approach used in these inversions and the 213 related assumptions made, a VS30of 1300 m/sec (with a range of 1300 to 1500 m/sec) 214 and0of 0.032 sec (with a range of 0.03 to 0.032 sec) are representative of the average 215 site conditions of KA16 hard-rock model 1. 216 Prepared using sagej.cls Al Atik et al. 9 Inversion of KA16 Model 2 217 An inversion approach similar to that described in the previous section is applied to 218 estimate the average hard-rock site characteristics representative of KA16 model 2. The 219 ﬁrst inversion case is performed to estimate VS30along with0for the total site factors 220 of KA16 model 2 for the high-frequency range of 12 to 25 Hz shown in Figure 6. The 221 frequency range of 12 to 25 Hz is chosen to capture the smaller- 0scaling expected for 222 this model while staying below the high-frequency limit of 30 Hz. The inversion for 223 KA16 model 2 results in an average VS30estimate of 1600 m/sec and 0of 0.025 sec. 224 We note that, for KA16 model 2, the inverted VS30value is sensitive to the frequency 225 range used to ﬁt the site factors with the analytical function that assumes that the top 226 30 m of the VSproﬁle can be approximated with a power-law function. Moreover, the 227 inversion of KA16 model 2 generally required more smoothing than that of model 1 228 due to the shape of the KA16 model 2 hard-rock factors with bigger jumps in the site 229 factors in the high-frequency range and less smooth transitions. 230 Next, KA16 model 2 is inverted using different assumed VS30values of 1500, 1700, 231 1850, 2000, and 2380 m/sec. The VS30of 2380 m/sec is reported in KA16 as the 232 averageVS30of the BCHydro data used to derive model 2 scaling factors. Inverted 0233 values and calculated SSE values for the different inversion cases are listed in Table 1. 234 Similar to the trends observed for model 1, the inversion results for KA16 model 2 235 indicate that the inverted 0value decreases with increasing VS30and that KA16 model 236 2 cannot be well represented with hard-rock conditions with large average VS30values, 237 particularly greater than 2000 m/sec. Based on a qualitative evaluation of the inversion 238 results as well as the SSE values for the different cases, we conclude that the inversion 239 results forVS30of 1700 m/sec (range of 1600 to 1850 m/sec) and 0of 0.024 sec 240 (range of 0.022 to 0.025 sec) best represent the average hard-rock site conditions of 241 KA16 model 2. The best-case inversion results for KA16 model 2 in terms of 0ﬁt, 242 site ampliﬁcation, and inverted VSproﬁle are shown in Figure 9. 243 Discussion of Vs Proﬁle and 0Inversions of KA16 Models 244 The inverted VSproﬁles and 0values presented in this section are representative of 245 the average hard-rock site conditions of KA16 models 1 and 2 within the context of 246 the QWL method used in the inversion and the assumptions employed to solve for 247 the multiple unknowns in this process. These assumptions are related to the assigned 248 half-spaceVSand density values, density- VSrelationship, vertical angle of incidence, 249 smoothVSproﬁles, and the representation of the top 30 m of the VSproﬁle with a 250 power-law function. While these assumptions are reasonable, they do introduce a level 251 Prepared using sagej.cls 10 Journal Title XX(X) of uncertainty in the resulting inverted VSproﬁles and 0values. Moreover, due the 252 frequency limitations of the KA16 hard-rock factors and their jagged appearance, the 253 invertedVSproﬁles are limited in their depth range. 254 Boore (2013) compared site ampliﬁcations calculated using the QWL method to 255 those obtained from theoretical simulations of wave propagation in layered media 256 accounting for the constructive and destructive interference of all reverberations in the 257 layers (full resonant [FR] method). For velocity models made up of gradients, Boore 258 (2013) found that the QWL method systematically underestimates the theoretical FR 259 site ampliﬁcation over a wide frequency range. This underestimation can be on the 260 order of 20%. Based on that, the QWL-based inversion can potentially underestimate 261 the derivedVSproﬁles compared to those expected from the FR method for the same 262 site ampliﬁcation. The use of the QWL method in the inversion is, however, consistent 263 with the approach used to develop analytical site adjustment factors presented in the 264 next section. Therefore, we consider the inverted proﬁles and 0values presented in 265 this section as appropriate values for use with the QWL method to represent the average 266 hard-rock site conditions of the KA16 factors. We use these inverted proﬁles and 0267 values to constrain the inputs to the analytical calculations for the hard-rock factors. 268 Development of GMPE-Consistent Analytical Hard-Rock Site 269 Adjustment Factors 270 The inversion of the KA16 empirical hard-rock factors indicates that these factors can 271 be used to scale response spectra from a reference VS30of 760 m/sec to target VS30272 of about 1300 (model 1) or 1700 m/sec (model 2). To develop rock site adjustment 273 factors that are a continuous function of VS30between 1000 and 2200 m/sec, we use 274 the analytical IRVT method of Al Atik et al. (2014) with empirical constrains based on 275 the KA16 scaling factors for VS30of 1300 and 1700 m/sec and the NGA-West2 scaling 276 factors forVS30of 1000 m/sec. Because the spectral shape for CY14 generally lies 277 in the center of the range of spectral shapes of the NGA-West2 GMPEs, we develop 278 the rock scaling factors using the CY14 GMPE and assume the resulting factors are 279 applicable to the other NGA-West2 GMPEs. 280 The development of analytical site adjustment factors requires the deﬁnition of host 281 and target site conditions in terms of VSproﬁles and 0values. For the host site 282 condition, the VSproﬁle and0value of 0.039 sec inverted for CY14 for VS30of 283 760 m/sec in Al Atik and Abrahamson (2021) are used. Thirteen target site conditions 284 are deﬁned having VS30ranging between 1000 and 2200 m/sec. The VSproﬁles for 285 the target sites are obtained using Boore (2016) based on a VS30-based interpolation 286 Prepared using sagej.cls Al Atik et al. 11 between generic WUS and Eastern US proﬁles with VS30of 618 and 2780 m/sec, 287 respectively. Figure 10 presents host CY14 VSproﬁle along with the target VSproﬁles 288 and their corresponding QWL site ampliﬁcations. Figure 10 shows that there is a 289 signiﬁcant difference between the host VSproﬁle for CY14 for VS30of 760 m/sec 290 and the target proﬁle for VS30of 1000 m/sec. This difference is due to CY14 having a 291 relatively high VSscaling from 1000 to 760 m/sec resulting in higher site ampliﬁcation 292 and softerVSproﬁle forVS30of 760 m/sec compared to the target proﬁle VS30of 1000 293 m/sec. These effects are discussed in Al Atik and Abrahamson (2021). 294 Target 0295 The estimation of site-speciﬁc 0is a complex process that often involves a large 296 degree of uncertainty and trade-offs. The origins of 0and the relationship between the 297 observed high-frequency attenuation in FAS ( 0scaling) and the low-strain damping at 298 a site are subject of ongoing debate . The current paradigm assumes that 0, estimated 299 with the source, path, and site effects removed, is due only to damping at the site 300 (EPRI). As a result, a low 0implies low damping that must lead to an increase in 301 the high-frequency ground motion. Hard-rock to soft-rock site factors of 2-3 at the 302 frequency range of 20-40 Hz are common (Biro and Renault 2012). When the current 303 paradigm was established in the 1990s, there were only a four hard-rock recordings 304 with low0values and they were consistent with the large factors of 2-3 ampliﬁcation 305 for hard-rock sites relative to soft-rock sites. The current data sets for hard-rock sites 306 are much larger with over 100 recordings, and they do not show the large site factors at 307 high frequencies that are predicted for hard-rock sites with 0in the 0.006-sec range. 308 This indicates that estimated 0values are not just the result of damping; they also 309 reﬂect the errors in the assumed source, path, and site effects on the slope of the FAS 310 used to estimate 0. The negative values of 0estimated for some sites also indicate 311 that there is more than just damping controlling the 0values (Ktenidou et al. 2021). 312 To avoid the common tendencies for underestimating 0, We use target 0values 313 that are consistent with the observed ground-motion scaling at high-frequencies for 314 rock site conditions. By using the amplitude of the ground motion and not just the 315 high-frequency slope of the FAS, the 0values can be interpreted as effects of damping 316 and used in the traditional 0scaling methodology. We note that our resulting target 0317 values are not site-speciﬁc; they are average values that can be expected for hard-rock 318 sites with different VS30values. We also account for the uncertainty in the average 0319 value for a rock site condition as described below. 320 Prepared using sagej.cls 12 Journal Title XX(X) For this study, target 0values are estimated based on a review of Silva and Darragh 321 (1995) with additional empirical constrains. Silva and Darragh (1995) analyzed 49 rock 322 sites in WNA and 22 rock sites in Eastern North America [ENA]. Table 5-3 of Silva 323 and Darragh (1995) lists the median and range of 0values for average site conditions 324 in WNA and ENA. It indicates that average 0values for WNA rock site conditions 325 are not small and are larger than those for ENA. Silva and Darragh (1995) interpreted 326 the0to be the result of damping in the top 1-2 km below the site and proposed 327 two Q models ( Q= VS) with = 0.007 and 0.029 sec/m for soft-rock and hard- 328 rock sites, respectively. Their soft-rock and hard-rock sites are representative WNA 329 and ENA generic VSproﬁles, respectively, and are shown in Figure 10(a). 330 For each of the 13 target VSproﬁles in this study, we estimate 0by summing up 331 the damping in the proﬁle layers over the top 1 and 2 km of the proﬁle as shown in 332 Equations 1 and 2. Two proﬁle depths are used to capture the uncertainty in the total 333 depth of the proﬁle contributing to damping. Two alternative Q models are used: a 334 linear Q model with gamma = 0.007 sec/m representative of WNA soft-rock condition 335 and a bilinear Q model with gamma of 0.007 sec/m for the proﬁle layers with VS 336 2700 m/sec and 0.029 sec/m for larger VS. This results in a total of four 0estimates 337 for each target VSproﬁle. The alternative target 0estimates as a function of VS30338 are shown in Figure 11 (a) and are compared to empirical 0estimates inferred from 339 ground-motion data. Empirical 0estimates shown in Figure 11 are based on 0340 estimates for the 4 NGA-West2 GMPEs in Al Atik and Abrahamson (2021) for VS30341 of 760 and 1000 m/sec and on 0andVS30inverted for the KA16 models. The upper 342 estimates of target 0values for this study shown in Figure 11 are the result of using 343 = 0.007 sec/m and a proﬁle depth of 2 km contributing to damping while the lower 344 estimates are the result of the bilinear Q model with a proﬁle depth of 1 km contributing 345 to0. 346 0=X iHi VS;iQi(1) 347 Q= VS (2) Figure 11(a) indicates that the target 0values have a similar trend with VS30as the 348 empirical0estimates, but with the average target 0values falling below the average 349 empirical0estimates, indicating an underestimation of the target 0values compared 350 to the empirical data. Because this study uses CY14 to develop analytical hard-rock 351 site adjustment factors, we constrain the average target 0forVS30= 1000 m/sec to 352 match that of CY14 (0.0345 sec). As a result, the target 0values are scaled up by a 353 Prepared using sagej.cls Al Atik et al. 13 constant factor and the adjusted target 0values are shown in Figure 11(b). We note 354 that the trend of the empirical 0values as a function of VS30is still different from 355 that of the scaled target 0values for this study. Our ultimate goal is not to match the 356 exact empirical 0values but to have a good match between the analytical and the 357 empirical rock site adjustment factors. We aim to match the hard-rock scaling observed 358 in empirical data reﬂecting the combined effects of 0and andVSproﬁle scaling. We 359 also note that the upper estimates of the scaled target 0are within the range of 0360 values for WNA rock from Silva and Darragh (1995) and are considered reasonable. 361 Table 2 lists the four 0values for the different target VSproﬁles along with their 362 average and standard deviation. 363 Hard-Rock Site Adjustment Factors 364 For each of the target VS30values ranging from 1000 to 2200 m/sec, four sets of 365 adjustment factors are developed using the IRVT approach of Al Atik et al. (2014) 366 corresponding to the four target 0values listed in Table 2. Strike-slip earthquake 367 scenarios with magnitude 5, 6, and 7, distance of 5, 10, and 20 km, and VS30of 760 368 m/sec are used in the IRVT approach. CY14 median response spectra are computed for 369 the nine scenarios considered for the linear site response. These response spectra are 370 converted into compatible FAS using the IRVT approach as described in the previous 371 sections. Then, each FAS is scaled to adjust for the differences in the linear site 372 ampliﬁcation and 0scaling between the host and target VSproﬁles and 0values. 373 TheVS-scaled FAS are then converted into a VS-scaled response spectra using 374 random vibration theory. The VS-scaling factors are calculated as the ratio of the 375 scaled response spectra to the initial GMPE response spectra and averaged over the 376 nine scenarios considered. 377 For each target VSproﬁle, four sets of VS-scaling factors are computed 378 corresponding to the four target 0values. Average VS-scaling factors are derived 379 assuming equal weights for the four target 0values. Figure 12 shows the VS-scaling 380 factors for the individual target 0values as well as the average scaling factors for 381 VS30of 1700 m/sec compared to the empirical hard-rock factors of KA16. Figure 12 382 indicates a good agreement between the average analytical factors for VS30of 1700 383 m/sec and the KA16 model 2 factors which have a representative VS30of about 1700 384 m/sec. Figure 13 compares the set of average analytical hard-rock adjustment factors 385 for the range of VS30of 1000 to 2200 m/sec to the CY14 empirical site factors for VS30386 of 1000 m/sec and the KA16 hard-rock factors. While some mismatch can be observed 387 in Figure 13 between the analytical factors for VS30of 1300 m/sec and the KA16 388 Prepared using sagej.cls 14 Journal Title XX(X) model 1 factors, there is good agreement between the analytical hard-rock factors for 389 VS30of 1000 m/sec and the corresponding CY14 site factors for frequencies less than 390 20 Hz and between the analytical factors for VS30of 1700 m/sec and the KA16 model 2 391 factors for 15-30 Hz. We conclude that, on average, the analytical hard-rock factors are 392 reasonable based on their comparison with empirical scaling for rock site conditions 393 (CY14 forVS30= 1000 m/sec and KA16 factors). 394 Implementation 395 The hard-rock site adjustment factors derived in this study are used to extrapolate the 396 average NGA-West2 empirical site factors to hard-rock conditions in a relative sense 397 to ensure a smooth transition in the scaling factors to hard-rock sites. As such, the 398 ratios of hard-rock analytical factors relative to those for VS30of 1000 m/sec are used 399 to model the scaling of the hard-rock site factors. These ratios are then applied to the 400 empirical site factors for VS30of 1000 m/sec relative to reference VS30=760 m/sec. This 401 normalization of the analytical site factors allows the site factors from the analytical 402 modeling to be centered on the GMPEs which provides a smooth scaling from soft-rock 403 to hard-rock site conditions. The empirical linear site factors for VS30of 1000 m/sec 404 are obtained by averaging the ratio of median response spectra for VS30of 1000 m/sec 405 relative to 760 m/sec for 4 NGA-West2 GMPEs (Abrahamson et al. 2014; Boore et al. 406 2014; Campbell and Bozorgnia 2014; Chiou and Youngs 2014). The resulting rock-site 407 adjustment factors are shown in Figure 14 and included as an electronic appendix to this 408 paper. Figure 14 also shows the average empirical linear site factors of the NGA-West2 409 GMPEs forVS30of 680 to 1000 m/sec relative to the reference 760 m/sec. The GMPEs 410 nonlinear site response is not included in the calculation of the average empirical site 411 factors. Figure 14 indicates a smooth extrapolation of the empirical average GMPE site 412 factors to hard-rock conditions based on the analytical factors described in this paper. 413 Figure 15 shows the linear VSscaling of the NGA-West2 GMPEs relative to VS30414 of 760 m/sec and extrapolated to hard-rock conditions. Also plotted in Figure 15 are 415 the average of the scaling from the 4 NGA-West2 GMPEs and the hard-rock scaling 416 proposed in this study. Comparisons of the linear VSscaling are shown for frequencies 417 of 0.2, 1, 5, and 25 Hz. These comparisons indicate that, for VS30values>1000 418 m/sec, linear VSscaling varies among the NGA-West2 GMPEs reﬂecting the different 419 hard-rock extrapolation constraints imposed in the models. The extrapolated hard-rock 420 scaling in the NGA-West2 GMPEs is unconstrained with empirical data for hard-rock 421 conditions and is, therefore, unreliable for application to hard-rock sites. In contrast to 422 the hard-rock factors proposed in this study, the NGA-West2 scaling does not follow 423 Prepared using sagej.cls Al Atik et al. 15 expected trends with for hard-rock sites at the high frequency of 25 Hz as shown in 424 Figure 15 (a). Therefore, the NGA-West2 linear VSscaling should not be extrapolated 425 to hard-rock sites and the factors presented in this paper should be used instead. 426 The average hard-rock adjustment factors from this study, presented in Figure 14 and 427 included as an electronic appendix to this paper, can be applied to correct the average 428 median ground motion predicted by the NGA-West2 GMPEs with VS30of 760 m/sec 429 to a hard-rock site with VS30between 1000 and 2200 m/sec. Nonlinear site response 430 should be disabled when calculating the NGA-West2 ground-motion predictions for 431 VS30of 760 m/sec before applying the hard-rock adjustment factors. For target VS30432 values not explicitly listed in the electronic appendix, hard-rock factors can be obtained 433 using a log-log interpolation of the provided factors for the neighboring VS30values. 434 For hard-rock sites with qualitative assessment of site conditions, hard-rock adjustment 435 factors for a range of target VS30values can be enveloped to estimate the median hard- 436 rock adjustment factors. 437 Site-to-Site Uncertainty 438 The adjustment of median ground-motion predictions for hard-rock sites is presented 439 in this paper. To evaluate the uncertainty in the hard-rock adjustment factors, we 440 examine the site-to-site variability [ S2S] in the NGA-West2 GMPEs for soil versus 441 rock sites. Site terms are obtained using a mixed-effects regression on the within-event 442 residuals of the NGA-West2 GMPEs with the station term as the random effect and 443 using earthquakes with magnitude >=5 and stations with a minimum of 3 recordings 444 as described in Al Atik (2015). Ground-motion data with magnitude <3 are not 445 used in this analysis to reduce the dependence of linear site factors on earthquake 446 magnitude. This effect was examined in Stafford et al. (2017) and was found to be 447 most pronounced at short periods and for small magnitude scenarios. Soil sites in the 448 NGA-West2 database are classiﬁed with VS30<680 m/sec while rock sites have VS30449 680 m/sec. Site terms for each NGA-West2 GMPE are divided in these two site 450 categories and the resulting S2Sare computed. 451 The S2Sfor soil and rock sites obtained using the residuals of ASK14, Boore et al. 452 (2014) [BSSA14] and CY14 for magnitude 5 were examined and the comparison 453 using CY14 residuals is shown in Figure 16. We note S2Sfor Campbell and Bozorgnia 454 (2014) [CB14] is not shown due to the limited CB14 dataset as a result of restricting 455 the residuals to magnitudes >5 and stations with a minimum of 3 recordings. This 456 impacted the stability of the S2Sestimates for CB14. The large error bars in Figure 16 457 reﬂect the smaller subset of stations with VS30680 m/sec compared to the number 458 Prepared using sagej.cls 16 Journal Title XX(X) of softer sites in the NGA-West2 dataset. For example, using the CY14 residuals, 459 the average VS30is about 390 m/sec for soil sites and 830 m/sec for rock sites. The 460 comparison of S2Sfor soil and rock sites indicates that the NGA-West2 S2Svalues 461 are generally comparable for the two site groups at high frequencies as shown in 462 Figure 16. At periods greater than 1 sec, S2Svalues for rock sites are lower than 463 those for soil sites. We note that the subsets of data for rock sites are very limited 464 in number of stations for periods >4 sec. We conclude that, for hazard signiﬁcant 465 scenarios with magnitudes 5, S2Sobtained from the NGA-West2 residuals for all 466 VS30can be used to estimate S2Sfor hard-rock sites with modiﬁcations associated 467 with the expected spectral shapes of site variability for hard-rock. 468 The average S2Sobtained using residuals of ASK14, BSSA14 and CY14 for 469 magnitude 5 and for all VS30values is shown in Figure 17 (a). We note that the 470 peak in S2Sat frequency 5-10 Hz is likely related to the variability of the resonance 471 frequency of shallow layers for soil and soft-rock sites. For hard-rock sites, this peak is 472 expected to be shifted to higher frequencies reﬂecting the variability in kappa scaling 473 for hard-rock conditions. We examine this effect using S2Sobtained from a residual 474 analysis of Japanese surface and borehole data. A discussion of the residual analysis 475 of the Japanese dataset is presented in Goulet et al. (2018). Figure 17 (b) presents a 476 comparison of S2Sfor the surface and borehole Japanese data with magnitude >=5. 477 Borehole S2Svalues obtained using stations with VS>=1000 m/sec are also shown. 478 Figure 17 (b) shows a shift in the peak of S2Sto higher frequencies for the rock 479 borehole data compared to the surface data. As a result, we correct the average S2S480 for NGA-West2 to follow the high-frequency scaling of the Japanese borehole S2S481 for frequencies greater than 2.5 Hz. For frequencies less than 2.5 Hz, the S2Sshape 482 is based on the NGA-West2 data. The resulting proposed S2Smodel for use for hard- 483 rock sites is shown in Figure 17 (a) and listed in Table 3. This proposed model can 484 be used to characterize the epistemic uncertainty of the average rock-site adjustment 485 factors presented in this study if additional site-speciﬁc information is not available to 486 constrain the epistemic uncertainty of the site factors. 487 For hard-rock adjustments of the NGA-West2 GMPEs using the ergodic aleatory 488 variability model, the standard deviation models in the NGA-West2 GMPEs, calculated 489 forVS30of 760 m/sec without including effects of nonlinear site response, could be 490 used for hard-rock sites. The use of the ergodic NGA-West2 sigma models is likely 491 conservative for some frequency ranges and might not capture the expected peaks in 492 the variability for hard-rock sites. Alternatively, ergodic sigma for hard-rock sites can 493 be constructed using S2Sproposed in this study along with NGA-West2 Tau models 494 and published single-station sigma models for WUS ((Al Atik 2015)). We note that the 495 Prepared using sagej.cls Al Atik et al. 17 S2Smodel proposed in this study for hard-rock site adjustment factors is a simpliﬁed 496 model based on adjusting the NGA-West2 S2S. A more detailed study of the ground- 497 motion variability and its components for hard-rock sites is warranted. 498 Conclusions and Discussion 499 Hard-rock adjustment factors are derived to adjust the NGA-West2 GMPEs from 500 their average host site conditions with VS30of 760 m/sec to target sites with VS30501 ranging from 1000 to 2200 m/sec. These analytical factors are obtained using the IRVT 502 approach Al Atik et al. (2014) and are consistent with empirical scaling observed in 503 ground-motion data. These factors can be applied to adjust median NGA-West2 ground 504 motions atVS30of 760 m/sec to hard-rock conditions and can be assumed to have the 505 same overall site-to-site uncertainty inherent in the NGA-West2 GMPEs. 506 The site adjustment factors developed in this study are computed using generic VS507 proﬁles and 0values that would be representative of average site response in WUS 508 for rock site conditions. The KA16 Scaling factors obtained using ENA and BCHydro 509 data are used as empirical constraints for this study because of the scarcity of empirical 510 data on hard-rock sites in WUS and because KA16 showed that average hard-rock 511 scaling in ENA is comparable to what would be expected for WUS sites. The proposed 512 hard-rock factors are intended for use at sites with measured or estimated VS30or 513 sites with qualitative assessment of site condition. For hard-rock sites with site-speciﬁc 514 measurements of VSproﬁles extending below the shallow 20 to 30 m of the proﬁle, 515 the hard-rock adjustment factors presented here are not recommended to be used. For 516 such sites, site-speciﬁc adjustments need to be developed following a characterization 517 of the target site-speciﬁc conditions in terms of best estimates and uncertainty of VS518 proﬁles and 0. Also, the use of S2Sfor site-speciﬁc adjustments is conservative and 519 can potentially be reduced based on the uncertainty in the site-speciﬁc characterization. 520 Data and Resources 521 Thepyrvt program used to perform the inverse random vibration theory (IRVT) and 522 random vibration theory (RVT) calculations (Kottke 2020). An Excel ﬁle containing the 523 hard rock adjustment factors for the NGA-West2 GMPEs is included as a supplemental 524 material. 525 Prepared using sagej.cls 18 Journal Title XX(X) Acknowledgments 526 This study was sponsored by Paciﬁc Gas & Electric (PG&E). The authors gratefully 527 acknowledge this funding. Any opinions, ﬁndings, and conclusions or recommendations 528 expressed in this material are those of the authors and do not necessarily reﬂect those of the 529 sponsoring company. 530