# A Site Amplification Model for Crustal Earthquakes

^{*}

## Abstract

**:**

_{rock}) which is also modified with between-event residual. Application of PSA

_{rock}simplifies the usage of the site model by diminishing the need of using the period-dependent correlation coefficients in hazard studies. The soil stiffness is expressed by a Gompertz sigmoid function which restricts the nonlinear effects at both of the very soft soil sites and very stiff soil sites. In order to surpass the effect of low magnitude and long-distant recordings on soil nonlinearity, the nonlinear site coefficients are constrained by using a limited dataset. The coefficients of linear site scaling and deep soil effect are obtained with the full database. The period average of site-variability is found to be 0.43. The sigma decreases with decreasing the soil stiffness or increasing input rock motion. After employing residual analysis, the region-dependent correction coefficients for linear site scaling are also obtained.

## 1. Introduction

_{S}

_{30}(time-based average of shear wave velocity of top 30 m soil media) after Boore et al. [8]. The site amplification is assumed to decrease linearly with increasing natural logarithm of V

_{S}

_{30}[6,7,8,9,10,11,12,13]. Some researchers uses a period-independent fixed reference V

_{S}

_{30}for linear scaling (e.g., 760 m/s in [12]) or period-dependent reference V

_{S}

_{30}(e.g., [13]). Using either period-independent or dependent values does not affect the slope of linear site scaling [14]. Due to the low number of recordings at high V

_{S}

_{30}sites, a constant amplification portion at the high V

_{S}

_{30}values is preferred. Some model developers use period-dependent limiting V

_{S}

_{30}. At short periods, it reaches 1500 m/s and at longer periods, this value decreases to 400 m/s [15,16,17]. The period independent constraining value can be used as well (e.g., V

_{S}

_{30}= 1000 m/s in [7] or 1130 m/s in [18]). The second behavior in site amplification is soil nonlinearity, which is again modeled using V

_{S}

_{30}, which represents soil stiffness and input rock motion [6,7,9,10,11,12,13,15,16,17,18,19,20,21]. The level of soil nonlinearity decreases as soil stiffness increases or input rock motion decreases. The final behavior is the deep soil effect which is modeled with the depth to rock parameter (e.g., Z1, depth to VS profile reaches 1 km/s [21]).

_{0}(e.g., [22,23]). Although the results might be changed for global data, for European data, Sandıkkaya and Akkar [24] show that the use of the V

_{S}

_{30}-f

_{0}pair in site scaling leads a slight decrease the within-event sigma when it is compared with the V

_{S}

_{30}-Z1 pair. As well, none of the reference databases (see next section) provides f

_{0}. Consequently, within the context of this study we cannot use f

_{0}as a site parameter.

_{S}

_{30}. Application of the second functional form seems more practical, especially in computing the variability of site amplification in terms of either V

_{S}

_{30}or input rock motion or both.

_{rock}and V

_{S}

_{30}. The paper continues with a comparison of the proposed model with some of the site models in the literature. Finally, we employ residual analysis to investigate the possible regional effects in the linear site response term.

## 2. Ground-Motion Database

_{S}

_{30}and accelerograms recorded within 300 km (Joyner-Boore distance, R

_{JB}is used). The lower moment magnitude (M

_{w}) limit of normal, reverse, and strike-slip earthquakes was 3.5. Only shallow crustal earthquakes with a maximum depth of 35 km are used. Finally, we used only events and stations with at least two recordings (only exceptions are made to M

_{w}+ 6.75 events recorded at soft sites with V

_{S}

_{30}< 360 m/s, due to not losing any possible nonlinear site effects). Generally, in single-station sigma studies (e.g., [31]), at least 10 recordings per station was used. We cannot follow such criterion because if it were applied, most of the stations would be removed, as well as we would lose some records that might have possible nonlinear site effects.

_{w}-R

_{JB}distribution of the database is shown in Figure 1a for larger seismic regions used in this study. The database is dominated by low-magnitude and long-distant records. The distribution becomes sparse as the magnitude increases and distance decreases. The effect of the low number of such recordings on nonlinear site effects is discussed in the following section. Figure 1b shows the depth distribution of normal, reverse, and strike-slip earthquakes. Almost all types of earthquakes were well distributed for depths less than 15 km. This database also overcomes the drawbacks of NGA-West2 and RESORCE databases, which have a sufficiently low number of normal and reverse earthquakes, respectively. Figure 1c shows the distribution of PSA at T = 0.01 s versus V

_{S}

_{30}of the database. The data was well sampled for stiff sites. The distribution loosened both in rock and very soft sites.

## 3. Ground Motion Prediction Equation for Rock Motion

_{S}

_{30}= 760 m/s was generated to compute site amplification values. The functional form used to compute the median natural logarithm of pseudo-spectral acceleration at 5% damping ratio, ln(PSA

_{rock}) was composed of event scaling (magnitude scaling and style-of-faulting, SoF terms), distance scaling (geometric and anelastic attenuation terms), and site scaling (linear site response term) (Equations (1)–(7)). The regression coefficients and between-event residuals (η

_{i}), between-site residuals (δ

_{j}) and within-event residuals (ε

_{ij}) were computed with the random-effects algorithm proposed by Bates et al. [34]. These residuals were assumed to have normal distributions with standard deviations of σ

_{e}, σ

_{s}and σ

_{w}with the total variability of σ

_{t}[35].

_{N}and F

_{R}are dummies to represent the style-of-faulting effects of normal and reverse events with respect to strike-slip events, respectively. The Joyner-Boore distance metric, R

_{JB}was used in the regressions to surpass the hanging-wall effect as in Boore and Atkinson [9] predictive model. Both anelastic and geometric attenuation terms were included in distance scaling. The geometric attenuation term is magnitude dependent. The fictitious depth term, R

_{0}, was taken constant (10 km) in the regressions. The linear site response term with V

_{S}

_{30}was employed and the site amplification for high V

_{S}

_{30}values (1000 m/s) was constrained.

_{JB}< 20 km) the median estimations tended to be underestimated, at this stage of the study we did not apply any near-field correction. The between-site residuals were not shown in these plots, because they were very similar to the distribution observed in Figure 9 (please see the section entitled “Regional Effects”).

## 4. Site Amplification Model

_{w}> 4.5) recorded within 80 km. At the second stage (the full database is used), the nonlinear site coefficients were constrained from the first stage analysis; then the linear and deep soil coefficients were computed. A similar approach was also used in SS14.

_{1}, γ

_{2}and γ

_{3}. The soil nonlinearity was computed with the multiplication of nonlinear site coefficient, γ

_{1}with the input rock motion term and soil stiffness term. In order to represent the seismic demand on rock, SS14 preferred peak ground acceleration at the rock site (PGA

_{rock}) as an input parameter. They also fixed the γ

_{2}coefficient to 0.1 g, to fulfill a smooth transition in input rock-motion levels. The soil stiffness was expressed by an exponential function in terms of V

_{S}

_{30}. This term is decreasing with increasing V

_{S}

_{30}and for rock sites (V

_{S}

_{30}> 760 m/s) it becomes zero. The γ

_{3}coefficient was adapted from CY14 site model.

_{rock}at 0.01 s (by assuming PGA

_{rock}is equal to PSA

_{rock}at 0.01 s) to Alternative II which considers PSA

_{rock}at 0.01 s and between-event residuals. The linear site coefficients with two alternatives were comparable (Figure 3a). The SS14 model had lower coefficients than both alternatives resulting in higher amplification. Alternative I yielded similar nonlinear coefficients with the SS14 model at the short-period range (0.1–0.5 s) where strong nonlinearity is observed (Figure 3b). However, the Alternative II that considers between-event residuals, imposed lesser soil nonlinearity in this interval period. The results from the Alternative II were more reliable. When employing non-reference site amplification method, the median estimate for rock motion from a GMPE could be overestimated or underestimated. Thus, this bias should be removed [19]. Since the influence of soil nonlinearity diminishes at longer periods, all coefficients were similar.

_{rock}at 0.01 s and PSA

_{rock}(between-event residuals are taken into account) in the regressions, and negligible differences were observed in median site amplification estimates. Besides, the choice did not affect the site variability. This was parallel to findings of Sandıkkaya et al. [7] and Kamai et al. [13]. We preferred to continue with PSA

_{rock}because it diminished the need to correlation coefficients between the period of interest and PSA

_{rock}. This simplifies applications in the hazard analysis.

_{S}

_{30}-dependence) of the nonlinear functional form in SS14 site model linearly decreased with increasing logarithm of V

_{S}

_{30}for soft sites having V

_{S}

_{30}< ~350 m/s where nonlinearity was more pronounceable (Figure 3c). Instead of this formulation, we preferred to use a Gompertz sigmoid function. This function scales the soil nonlinearity linearly within a range of 200–400 m/s. This function is also capable of capping the rate of increase below 200 m/s. This enabled us to remove unwanted bias in very soft sites where the number of stations (or recordings) is limited.

_{S}

_{30}= 760 m/s and the site amplification was constrained for rock sites having V

_{S}

_{30}> 1000 m/s. The Z1 scaling of the proposed model was different from SS14 and CY14 site models. The deep soil effect was not considered in SS14 for periods shorter than 0.65 s. This period was 0.25 s in CY14 site model. In both models, the difference between measured and estimated Z1 values was used. However, Rodriguez-Marek et al. [30] study, which uses Z0.8 (depth at which shear-wave velocity attains 800 m/s), gives coefficients for shorter periods. The deep soil effect in the proposed model expressed in terms of the natural logarithm of the Z1 in meters. Since the number of stations with measured Z1 values is quite limited in the database, for the stations with unknown Z1 values, we employed V

_{S}

_{30}− Z1 relations given by CY14. Using estimated Z1 values in the regression did not cause any increase in between-site sigma. The period-independent b

_{4}and b

_{5}values were tuned before regression analysis to diminish the soil stiffness effect in high V

_{S}

_{30}values and constrain the soil nonlinearity at very soft sites.

_{s}was found between 0.34 and 0.51. It was noted that this sigma value was obtained with the assumption of a homoscedastic model. This assumption was removed and how the variability of site amplification changes with V

_{S}

_{30}and PSA

_{rock}bins was investigated for T = 0.2 s and 1 s (Figure 4). The sites were grouped as very soft sites (V

_{S}

_{30}≤ 200 m/s), soft sites (200 < V

_{S}

_{30}≤ 300 m/s), stiff sites (300 < V

_{S}

_{30}≤ 400 m/s), moderate stiff sites (400 < V

_{S}

_{30}≤ 550 m/s), very stiff sites (550 < V

_{S}

_{30}≤ 800 m/s), and rock sites (V

_{S}

_{30}> 800 m/s). We used PSA

_{rock}bins to classify very weak motion, weak motion, moderate motion, strong motion, and very strong motion (please read the caption of Figure 4 for PSA

_{rock}bins).

_{rock}increases, the variability decreased, which is common in practice. Similar capping can also be made for lower and upper bounds of PSA

_{rock}values. The V

_{S}

_{30}and PSA

_{rock}dependent site variability is given in (Equations (10)–(12)).

_{0}is the site variability constant and c

_{1}and c

_{2}represent the slope of PSA

_{rock}and V

_{S}

_{30}terms, respectively. The coefficients for the site model and standard deviation model are given in Table 2.

_{S}

_{30}= 180 m/s (very soft soil), V

_{S}

_{30}= 360 m/s (stiff soil) and V

_{S}

_{30}= 550 m/s (very stiff soil) for weak and strong input rock motion levels. At weak motion levels, our estimations were lower than estimates of the SS14 and CY14 site models, on the other hand, as input rock motion level increased due to the lower nonlinearity imposed in our model, our estimations became higher than SS14 and CY14 site models at short periods. Generally, the proposed model produced lower amplifications at long periods. The amplification estimations were very close to the SS14 and CY14 site models at very stiff sites. As sites get softer, the difference in amplification becomes more visible in this period interval. The observed differences are due to modeling approaches and database features (especially including more data from Japan) in the regressions.

_{rock}and V

_{S}

_{30}dependence of site amplification at 0.2 s (right column) and 1.0 s (left column), respectively. In Figure 6, three site conditions (V

_{S}

_{30}= 150 m/s, 255 m/s and 450 m/s) were considered. We selected weak, moderate and strong PSA

_{rock}for the comparisons in Figure 7 (read the figure caption for the PSA

_{rock}values). At T = 0.2 s, the site amplification estimations of the proposed model matched with the SS14 and CY14 site models at low seismic demands. As PSA

_{rock}increases, our model tended to estimate lower amplification demonstrating less soil nonlinear behavior. At V

_{S}

_{30}= 450 m/s, although CY14 and SS14 models had some nonlinearity, the proposed model was not very sensitive to PSA

_{rock}. At T = 1 s, for stiff sites, amplifications were generally lower than the CY14 site model. Due to the lower nonlinear site behavior of our model, as PSA

_{rock}increased, the amplification became higher when compared to the SS14 site model. For the V

_{S}

_{30}-scaling of the proposed model, SS14 and CY14 site models were similar at stiff site condition. However, as V

_{S}

_{30}decreased the differences became more prominent. The 1 s amplification was matched with the SS14 site model, with both models producing lower amplification when compared to the CY14 site model.

## 5. Regional Effects

_{S}

_{30}scaling. To do so, we added a region-dependent correction factor, c

_{k}, to the linear site term (Equation (13)).

_{k}values to determine whether the regional site amplification scaling was similar to the global site amplification trend or not. That is, the hypothesis c

_{k}was equal to zero is tested for each region. Figure 8a shows the test results. The site amplification trend for CH (the acronyms for each region is given in Table 1) was found to be statistically insignificant whereas the hypothesis is rejected for WA in all period range. At only one or two periods, the slopes were found to be statistically significant for GRTR, WMT and NWE regions. These results for these regions were thus not given in Figure 8a. For periods lower than 0.1 s, the test results for region-dependent V

_{S}

_{30}-scaling of TW sites indicated that different slopes to the global model should be used. At periods of 3 s to 4 s, both TW and JP had significant different slopes. At short periods (0.2–0.8 s), TW, JP, and USNZ slopes were found to be statistically significant. We give all c

_{k}values in Table 3 but the statistically significant ones are highlighted with bold font. The period variation of the correction factor is shown in Figure 8b.

_{rock}and between-event residuals) for each region is plotted for V

_{S}

_{30}(left panel) and PSA

_{rock}(right panel) bins for T = 0.2 s (top row) and T = 1.0 s (bottom row) in Figure 10. The intervals of V

_{S}

_{30}and PSA

_{rock}are given in the figure caption. The regional variability also had a wide range for both V

_{S}

_{30}and PSA

_{rock}bins. For example, soft-soil site standard deviation was 0.2 and 0.55 for the TW region and JP region, respectively. Rock-site standard deviation was generally lower than very stiff site sigma. There is a decreasing trend from very stiff sites to very soft sites. However, for some regions this observation did not hold, with larger variability being computed at very soft sites. The similar observation for PSA

_{rock}could also be made. The fact that the site bins did not have uniform station distribution, and also an unbalanced number of recordings (besides not uniform magnitude distance scatters) for each region, were the main reasons for violating the general observation.

## 6. Conclusions

_{S}

_{30}and PSA

_{rock}. As sites get softer or PSA

_{rock}increases, the variability of the site amplification decreases. This study also focused on the regional differences in site effects. Regional differences are found to be statistically insignificant for CH, GRTR, WMT, and NWE. However, the site amplification observed in WA, USNZ, JP, and TW are different from the global V

_{S}

_{30}-scaling.

_{S}

_{30}< 1200 m/s. The proposed site model can be served as a site-scaling term in the future ground motion prediction equations. We also emphasize that when it is the case, only nonlinear site coefficients should be used and the coefficients for linear site response and deep soil effects should be computed in the regression steps. The site model can also be considered as a candidate site model in the computation of the regional site factors for seismic design codes (e.g., [37]).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Borcherdt, R.D. Effects of Local Geology on Ground Motion near San Francisco Bay. Bull. Seismol. Soc. Am.
**1970**, 60, 29–61. [Google Scholar] - Stewart, J.P.; Liu, A.H.; Choi, Y. Amplification Factors for Spectral Acceleration in Tectonically Active Regions. Bull. Seismol. Soc. Am.
**2003**, 93, 332–352. [Google Scholar] [CrossRef] - Field, E.H. A modified ground-motion attenuation relationship for southern California that accounts for detailed site classification and a basin-depth effect. Bull. Seismol. Soc. Am.
**2000**, 90, S209–S221. [Google Scholar] [CrossRef] - Lee, Y.; Anderson, J.G. Potential for improving ground-motion relations in southern California by incorporating various site parameters. Bull. Seismol. Soc. Am.
**2000**, 90, S170–S186. [Google Scholar] [CrossRef] - Steidl, J.H. Site response in southern California for probabilistic seismic hazard analysis. Bull. Seismol. Soc. Am.
**2000**, 90, S149–S169. [Google Scholar] [CrossRef] - Choi, Y.; Stewart, J.P. Nonlinear Site Amplification as Function of 30 m Shear Wave Velocity. Earthq. Spectra
**2005**, 21, 1–30. [Google Scholar] [CrossRef][Green Version] - Sandıkkaya, M.A.; Akkar, S.; Bard, P.Y. A Nonlinear Site-Amplification Model for the Next Pan-European Ground-Motion Prediction Equations. Bull. Seismol. Soc. Am.
**2013**, 103, 19–32. [Google Scholar] [CrossRef] - Boore, D.M.; Joyner, W.B. Site Amplifications for Generic Rock Sites. Bull. Seismol. Soc. Am.
**1997**, 87, 327–341. [Google Scholar] - Boore, D.M.; Atkinson, G.M. Ground-Motion Prediction Equations for the Average Horizontal Component of PGA, PGV, and 5%-Damped PSA at Spectral Periods between 0.01 s and 10.0 s. Earthq. Spectra
**2008**, 24, 99–138. [Google Scholar] [CrossRef] - Chiou, B.S.J.; Youngs, R.R. An NGA Model for the Average Horizontal Component of Peak Ground Motion and Response Spectra. Earthq. Spectra
**2008**, 24, 173–215. [Google Scholar] [CrossRef] - Walling, M.; Silva, W.; Abrahamson, N. Nonlinear Site Amplification Factors for Constraining the NGA Models. Earthq. Spectra
**2008**, 24, 243–255. [Google Scholar] [CrossRef] - Seyhan, E.; Stewart, J.P. Semi-Empirical Nonlinear Site Amplification from NGA-West2 Data and Simulations. Earthq. Spectra
**2014**, 30, 1241–1256. [Google Scholar] [CrossRef] - Kamai, R.; Abrahamson, N.A.; Silva, W.J. Nonlinear Horizontal Site Amplification for Constraining the NGA-West2 GMPEs. Earthq. Spectra
**2014**, 30, 1223–1240. [Google Scholar] [CrossRef] - Salic, R.; Sandıkkaya, M.A.; Milutinovic, Z.; Gulerce, Z.; Duni, L.; Kovacevic, V.; Markusic, S.; Mihaljevic, J.; Kuka, N.; Kaludjerovic, N.; et al. Reply to “Comment to BSHAP project strong ground motion database and selection of suitable ground motion models for the Western Balkan Region” by Carlo Cauzzi and Ezio Faccioli. Bull. Earthq. Eng.
**2017**, 15, 1349–1353. [Google Scholar] [CrossRef] - Abrahamson, N.A.; Silva, W.J.; Kamai, R. Summary of the ASK14 ground motion relation for active crustal regions. Earthq. Spectra
**2014**, 30, 1025–1055. [Google Scholar] [CrossRef] - Boore, D.M.; Stewart, J.P.; Seyhan, E.; Atkinson, G.M. NGA-West2 Equations for Predicting PGA, PGV, and 5% Damped PSA for Shallow Crustal Earthquakes. Earthq. Spectra
**2014**, 30, 1057–1085. [Google Scholar] [CrossRef] - Campbell, K.W.; Bozorgnia, Y. NGA-West2 ground motion model for the average horizontal components of PGA, PGV, and 5%-damped linear acceleration response spectra. Earthq. Spectra
**2014**, 30, 1087–1115. [Google Scholar] [CrossRef] - Chiou, B.S.J.; Youngs, R.R. Update of the Chiou and Youngs NGA Model for the Average Horizontal Component of Peak Ground Motion and Response Spectra. Earthq. Spectra
**2014**, 30, 1117–1153. [Google Scholar] [CrossRef] - Abrahamson, N.A.; Silva, W.J. Empirical Response Spectral Attenuation Relations for Shallow Crustal Earthquakes. Seismol. Res. Lett.
**1997**, 68, 94–127. [Google Scholar] [CrossRef] - Zhao, J.X.; Hu, J.S.; Jiang, F.; Zhou, J.; Rhoades, D.A. Nonlinear site models derived from 1-D analyses for ground-motion prediction equations using site class as the site parameter. Bull. Seismol. Soc. Am.
**2015**, 105, 2010–2022. [Google Scholar] [CrossRef] - Abrahamson, N.A.; Silva, W. Summary of the Abrahamson and Silva NGA ground motion relations. Earthq. Spectra
**2008**, 24, 67–98. [Google Scholar] [CrossRef] - Cadet, H.; Bard, P.-Y.; Rodriguez-Marek, A. Defining a standard rock site: Propositions based on the KiK-net database. Bull. Seismol. Soc. Am.
**2010**, 100, 172–195. [Google Scholar] [CrossRef] - Hassani, B.; Atkinson, G.M. Applicability of the site fundamental frequency as a VS 30 proxy for central and eastern North America. Bull. Seismol. Soc. Am.
**2016**, 106, 653–664. [Google Scholar] [CrossRef] - Sandıkkaya, M.A.; Akkar, S. A Detailed Investigation on Akkar et al. (2013) pan-European Ground-Motion Prediction Equations and Proposals for Future Versions. In Proceedings of the 2nd European Conference on Earthquake Engineering and Seismology, Istanbul, Turkey, 24–29 August 2014. [Google Scholar]
- Ancheta, T.D.; Darragh, R.B.; Stewart, J.P.; Seyhan, E.; Silva, W.J.; Chiou, B.S.J.; Wooddell, K.E.; Graves, R.W.; Kottke, A.R.; Boore, D.M.; et al. NGA-West2 Database. Earthq. Spectra
**2014**, 30, 989–1005. [Google Scholar] [CrossRef] - Akkar, S.; Sandıkkaya, M.A.; Şenyurt, M.; Sisi, A.A.; Ay, B.Ö.; Traversa, P.; Douglas, J.; Cotton, F.; Luzi, L.; Hernandez, B.; et al. Reference Database for Seismic Ground-Motion in Europe (RESORCE). Bull. Earthq. Eng.
**2014**, 12, 311–339. [Google Scholar] [CrossRef][Green Version] - Luzi, L.; Puglia, R.; Russo, E.; Orfeus, W.G. Engineering Strong Motion Database, version 1.0.; Istituto Nazionale di Geofisica e Vulcanologia, Observatories & Research Facilities for European Seismology: Milano, Italy, 2016. [Google Scholar]
- Kale, Ö.; Akkar, S.; Ansari, A.; Hamzehloo, H. A ground-motion predictive model for Iran and Turkey for horizontal PGA, PGV and 5%-damped response spectrum: Investigation of possible regional effects. Bull. Seismol. Soc. Am.
**2015**, 105, 963–980. [Google Scholar] [CrossRef] - Dawood, H.M.; Rodriguez-Marek, A.; Bayless, J.; Goulet, C.; Thompson, E. A Flatfile for the KiK-net Database Processed Using an Automated Protocol. Earthq. Spectra
**2016**, 32, 1281–1302. [Google Scholar] [CrossRef] - Sandıkkaya, M.A.; Aghaalipour, N.; Gülerce, Z. Türkiye kuvvetli yer hareketi veri tabaninin genişletilmesi: Bir ön çalişma. In Proceedings of the 4th International Conference on Earthquake Engineering and Seismology, Eskişehir, Turkiye, 11–13 October 2017. (In Turkish). [Google Scholar]
- Rodriguez-Marek, A.; Montalva, G.A.; Cotton, F.; Bonilla, F. Analysis of single-station standard deviation using the KiK-net data. Bull. Seismol. Soc. Am.
**2011**, 101, 1242–1258. [Google Scholar] [CrossRef] - Flinn, E.A.; Engdahl, E.R.; Hill, A.R. Seismic and Geographical Regionalization. Bull. Seismol. Soc. Am.
**1974**, 64, 771–992. [Google Scholar] - Boore, D.M. Orientation-independent, non geometric-mean measures of seismic intensity from two horizontal components of motion. Bull. Seismol. Soc. Am.
**2010**, 100, 1830–1835. [Google Scholar] [CrossRef] - Bates, D.M.; Maechler, M.; Bolker, B. Lme4: Linear Mixed-Effects Models Using S4 Classes, R Manual, 2013. Available online: http://CRAN.R-project.org/package=lme4 (accessed on 8 December 2016).
- Al Atik, L.; Abrahamson, N.; Bommer, J.J; Scherbaum, F.; Cotton, F.; Kuehn, N. The variability of ground motion prediction models and its components. Seismol. Res. Lett.
**2010**, 81, 794–801. [Google Scholar] [CrossRef] - Akkar, S.; Sandıkkaya, M.A.; Bommer, J.J. Empirical ground-motion models for point- and extended-source crustal earthquake scenarios in Europe and the Middle East. Bull. Earthq. Eng.
**2014**, 12, 359–387. [Google Scholar] [CrossRef] - Sandıkkaya, M.A.; Akkar, S.; Bard, P.Y. A probabilistic procedure to describe site amplification factors for seismic design codes. Soil Dyn. Earthq. Eng.
**2018**. [Google Scholar] [CrossRef]

**Figure 1.**Seismological features of the database (

**a**) R

_{JB}-M

_{w}scatters, (

**b**) Depth vs. style-of-faulting, and (

**c**) V

_{S}

_{30}vs. PSA at T = 0.01 s. Note that the larger seismic regions are shown with the same color-code in each plot.

**Figure 2.**Between-event and within-event residual distributions of the reference rock motion predictive equations at T = 0.2 s (

**left column**) and T = 1.0 s (

**right column**). The same color-coding with Figure 1 is applied.

**Figure 3.**Comparison of (

**a**) linear and (

**b**) nonlinear site model coefficients computed with alternative functional forms. The coefficients given in BSSA14 are also included in these plots; (

**c**) the soil stiffness term in the BSSA14 functional form and the proposed Gompertz sigmoid function are compared.

**Figure 4.**Variability of site amplification for T = 0.2 s and 1 s (

**a**) for site bins (

**b**) PSA

_{rock}bins. The input rock motion bins for T = 0.2 s are: very weak motion (PSA

_{rock}< 0.05 g), weak motion (0.05 ≤ PSA

_{rock}< 0.15 g), moderate motion (0.15 ≤ PSA

_{rock}< 0.45 g), strong motion (0.45 ≤ PSA

_{rock}< 0.75 g), and very strong motion (PSA

_{rock}≥ 0.75 g). For T = 1.0 s they are: very weak motion (PSA

_{rock}< 0.01 g), weak motion (0.01 < PSA

_{rock}< 0.05 g), moderate motion (0.05 < PSA

_{rock}< 0.10 g), strong motion (0.10 < PSA

_{rock}< 0.20 g), and very strong motion (PSA

_{rock}> 0.20 g).

**Figure 5.**Period-dependent comparison of the site models for weak and strong motions in very soft soil (V

_{S}

_{30}= 180 m/s), soft soil (V

_{S}

_{30}= 360 m/s), and stiff soil (V

_{S}

_{30}= 550 m/s). Deep soil effect is not included in the plots.

**Figure 6.**The comparison of the site models for very soft soil (V

_{S}

_{30}= 150 m/s), soft soil (V

_{S}

_{30}= 255 m/s), and stiff soil (V

_{S}

_{30}= 450 m/s) at 0.2 s and 1.0 s. Deep soil effect is not included in the plots.

**Figure 7.**The comparison of the site models for weak, moderate, and strong input rock levels at 0.2 s and 1.0 s. Weak motion (PSA

_{rock}

_{@T0.2s}= 0.1 g, PSA

_{rock}

_{@T1.0s}= 0.05 g), moderate motion (PSA

_{rock}

_{@T0.2s}= 0.4 g, PSA

_{rock}

_{@T1.0s}= 0.15 g), and strong motion (PSA

_{rock}

_{@T0.2s}= 0.8 g, PSA

_{rock}

_{@T1.0s}= 0.30 g). Deep soil effect is not included in the plots.

**Figure 10.**Variability of site amplification for V

_{S}

_{30}and PSA

_{rock}at T = 0.2 s and 1.0 s. The site bins are: very soft sites (V

_{S}

_{30}≤ 200 m/s), soft sites (200 < V

_{S}

_{30}≤ 300 m/s), stiff sites (300 < V

_{S}

_{30}≤ 400 m/s), moderate stiff sites (400 < V

_{S}

_{30}≤ 550 m/s), very stiff sites (550 < V

_{S}

_{30}≤ 800 m/s), and rock sites (V

_{S}

_{30}> 800 m/s). The input rock motion bins for T = 0.2 s are: very weak motion (PSA

_{rock}< 0.05 g), weak motion (0.05 ≤ PSA

_{rock}< 0.15 g), moderate motion (0.15 ≤ PSA

_{roc}

_{k}< 0.45 g), strong motion (0.45 ≤ PSA

_{rock}< 0.75 g), and very strong motion (PSA

_{rock}≥ 0.75 g). For T = 1.0 s they are: very weak motion (PSA

_{rock}< 0.01 g), weak motion (0.01 < PSA

_{rock}< 0.05 g), moderate motion (0.05 < PSA

_{rock}< 0.10 g), strong motion (0.10 < PSA

_{rock}< 0.20 g), and very strong motion (PSA

_{rock}> 0.20 g).

Larger Flinn-Engdahl Seismic Regions | Acronym |
---|---|

Oregon, California and Nevada * | USNZ |

New Zealand Region | USNZ |

Guam to Japan | JP |

Japan-Kuril Islands-Kamchatka Peninsula | JP |

Southwestern Japan and Ryukyu Islands | JP |

Eastern Asia | JP |

Taiwan | TW |

Indıa-Xizang-Sichuan-Yunnan | CH |

Western Asia ** | WA |

Middle East-Crimea-Eastern Balkans | TRGR |

Western Mediterranean area *** | WMT |

Northwestern Europe | NWE |

**Table 2.**Site model coefficients. The period independent b

_{4}and b

_{5}are 2 and 11, respectively.

T (s) | b_{1} | b_{2} | b_{3} | σ_{s} | c_{0} | c_{1} | c_{2} |
---|---|---|---|---|---|---|---|

0.01 | −0.53307 | −0.46412 | 0.02105 | 0.47096 | 1.24013 | 0.09542 | −0.05865 |

0.025 | −0.50842 | −0.3904 | 0.02023 | 0.47508 | 1.24682 | 0.09906 | −0.05951 |

0.04 | −0.45025 | −0.31255 | 0.01858 | 0.48906 | 1.33552 | 0.12324 | −0.06481 |

0.05 | −0.38023 | −0.23187 | 0.02029 | 0.50412 | 1.6779 | 0.18762 | −0.08741 |

0.07 | −0.3505 | −0.18413 | 0.02376 | 0.50892 | 1.57403 | 0.12994 | −0.0791 |

0.1 | −0.42752 | −0.37652 | 0.03221 | 0.49777 | 1.52282 | 0.12604 | −0.07408 |

0.15 | −0.55919 | −0.53679 | 0.03248 | 0.47977 | 1.31863 | 0.11085 | −0.05612 |

0.2 | −0.6673 | −0.6571 | 0.02956 | 0.46896 | 1.21025 | 0.10065 | −0.04777 |

0.25 | −0.73135 | −0.69189 | 0.02516 | 0.45698 | 1.13978 | 0.07837 | −0.03958 |

0.3 | −0.7884 | −0.68208 | 0.03152 | 0.45065 | 1.05645 | 0.04621 | −0.03245 |

0.35 | −0.8332 | −0.69252 | 0.03233 | 0.44141 | 1.01481 | 0.05533 | −0.02765 |

0.4 | −0.8681 | −0.74537 | 0.03521 | 0.43589 | 1.00182 | 0.05914 | −0.02363 |

0.45 | −0.88575 | −0.73547 | 0.03923 | 0.42954 | 0.94803 | 0.06557 | −0.0179 |

0.5 | −0.89944 | −0.69269 | 0.04159 | 0.42699 | 0.94724 | 0.06067 | −0.0171 |

0.6 | −0.91493 | −0.6348 | 0.0458 | 0.41593 | 0.95504 | 0.07576 | −0.01606 |

0.7 | −0.93236 | −0.63204 | 0.04993 | 0.40303 | 1.01362 | 0.08323 | −0.01527 |

0.75 | −0.93217 | −0.6378 | 0.04989 | 0.40219 | 1.03634 | 0.08203 | −0.01622 |

0.8 | −0.92975 | −0.65092 | 0.05114 | 0.39766 | 1.05807 | 0.08385 | −0.01434 |

0.9 | −0.92777 | −0.57775 | 0.05266 | 0.38861 | 1.11036 | 0.09388 | −0.01658 |

1 | −0.93815 | −0.60041 | 0.05421 | 0.3815 | 1.16634 | 0.09095 | −0.01502 |

1.2 | −0.93377 | −0.56801 | 0.05576 | 0.36982 | 1.29484 | 0.08078 | −0.01434 |

1.4 | −0.93847 | −0.48684 | 0.05782 | 0.35868 | 1.32222 | 0.08353 | −0.00681 |

1.6 | −0.92242 | −0.40484 | 0.05645 | 0.35713 | 1.30431 | 0.07158 | −0.00268 |

1.8 | −0.91608 | −0.29053 | 0.05615 | 0.34643 | 1.35426 | 0.07341 | 0 |

2 | −0.90369 | −0.18149 | 0.05307 | 0.34133 | 1.38763 | 0.0679 | 0 |

2.5 | −0.89442 | −0.04175 | 0.05954 | 0.3396 | 1.41986 | 0.08582 | 0 |

3 | −0.87386 | 0 | 0.05596 | 0.35349 | 1.37795 | 0.10208 | 0 |

3.5 | −0.8551 | 0 | 0.05469 | 0.35286 | 1.34678 | 0.07501 | 0 |

4 | −0.8468 | 0 | 0.05469 | 0.36845 | 1.2583 | 0.05876 | 0 |

T (s) | c_{k,USNZ} | c_{k,JP} | c_{k,TW} | c_{k,CH} | c_{k,WA} | c_{k,GRTR} | c_{k,WMT} | c_{k,NWE} |
---|---|---|---|---|---|---|---|---|

0.01 | −0.0302 | 0.0117 | −0.0233 | 0.0158 | 0.1001 | −0.0118 | 0.0172 | 0.0314 |

0.025 | −0.0303 | 0.0135 | −0.0272 | 0.015 | 0.1013 | −0.01 | 0.0174 | 0.0264 |

0.04 | −0.0336 | 0.0298 | −0.0394 | 0.0111 | 0.1059 | −0.0148 | 0.0101 | 0.0178 |

0.05 | −0.04 | 0.0575 | −0.0541 | 0.0099 | 0.1071 | −0.024 | −0.0093 | 0.0038 |

0.07 | −0.0346 | 0.0508 | −0.056 | −0.0012 | 0.1119 | −0.019 | −0.0114 | −0.0206 |

0.1 | −0.0287 | 0.0199 | −0.045 | 0.022 | 0.1251 | −0.0095 | 0.0084 | −0.0222 |

0.15 | −0.0187 | −0.0228 | −0.0114 | 0.0143 | 0.1105 | 0.0044 | 0.0258 | −0.0307 |

0.2 | −0.0196 | −0.0439 | 0.0089 | 0.0056 | 0.1134 | 0.0133 | 0.035 | −0.0254 |

0.25 | −0.0227 | −0.0543 | 0.0222 | 0.0059 | 0.1016 | 0.0162 | 0.048 | 0.0274 |

0.3 | −0.0216 | −0.0583 | 0.03 | −0.00003 | 0.086 | 0.0153 | 0.058 | 0.0407 |

0.35 | −0.0187 | −0.0583 | 0.0301 | 0.0025 | 0.089 | 0.0135 | 0.0534 | 0.065 |

0.4 | −0.0239 | −0.0544 | 0.0313 | 0.008 | 0.09462 | 0.007 | 0.05177 | 0.0728 |

0.45 | −0.0254 | −0.0502 | 0.0327 | 0.0142 | 0.0999 | 0.0041 | 0.0519 | 0.0798 |

0.5 | −0.0322 | −0.0461 | 0.036 | 0.0156 | 0.1073 | −0.0022 | 0.0553 | 0.0879 |

0.6 | −0.0388 | −0.0389 | 0.0356 | 0.0163 | 0.1209 | −0.0125 | 0.0565 | 0.0978 |

0.7 | −0.0411 | −0.0333 | 0.0336 | 0.022 | 0.1246 | −0.0197 | 0.0483 | 0.1104 |

0.75 | −0.0416 | −0.0305 | 0.0339 | 0.0252 | 0.1224 | −0.0269 | 0.0485 | 0.1166 |

0.8 | −0.0436 | −0.0289 | 0.0346 | 0.0297 | 0.1244 | −0.0321 | 0.0512 | 0.1193 |

0.9 | −0.0412 | −0.0262 | 0.0289 | 0.0325 | 0.1239 | −0.0408 | 0.0574 | 0.1303 |

1 | −0.0397 | −0.0195 | 0.0146 | 0.0375 | 0.1273 | −0.0434 | 0.0673 | 0.1369 |

1.2 | −0.0395 | −0.0071 | −0.0025 | 0.0463 | 0.1376 | −0.0467 | 0.0668 | 0.0914 |

1.4 | −0.0365 | −0.0036 | −0.0115 | 0.0574 | 0.1397 | −0.0446 | 0.064 | 0.0893 |

1.6 | −0.0361 | 0.0073 | −0.0188 | 0.062 | 0.1319 | −0.0473 | 0.06 | 0.0914 |

1.8 | −0.0307 | 0.0108 | −0.0252 | 0.0609 | 0.1332 | −0.0452 | 0.0523 | 0.1062 |

2 | −0.028 | 0.0129 | −0.0328 | 0.0591 | 0.1408 | −0.0445 | 0.041 | 0.1092 |

2.5 | −0.0336 | 0.0277 | −0.0413 | 0.0588 | 0.1471 | −0.0316 | 0.0197 | 0.0509 |

3 | −0.0325 | 0.0369 | −0.0579 | 0.0566 | 0.1679 | −0.0268 | 0.0138 | 0.105 |

3.5 | −0.0272 | 0.0461 | −0.063 | 0.0525 | 0.1422 | −0.0294 | 0.0216 | 0.156 |

4 | −0.0203 | 0.0503 | −0.0641 | 0.0572 | 0.1945 | −0.0242 | 0.0138 | 0.2198 |

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## Share and Cite

**MDPI and ACS Style**

Sandıkkaya, M.A.; Dinsever, L.D.
A Site Amplification Model for Crustal Earthquakes. *Geosciences* **2018**, *8*, 264.
https://doi.org/10.3390/geosciences8070264

**AMA Style**

Sandıkkaya MA, Dinsever LD.
A Site Amplification Model for Crustal Earthquakes. *Geosciences*. 2018; 8(7):264.
https://doi.org/10.3390/geosciences8070264

**Chicago/Turabian Style**

Sandıkkaya, M. Abdullah, and L. Doğan Dinsever.
2018. "A Site Amplification Model for Crustal Earthquakes" *Geosciences* 8, no. 7: 264.
https://doi.org/10.3390/geosciences8070264