- Ground-motion model is (for median):
_{1}= -1.5065, c_{1a}= 0.1650, c_{1b}= -0.2550, c_{1c}= -0.1650, c_{1d}= 0.2550, c_{2}= 1.06, c_{3}= 1.9636, c_{4}= -2.1, c_{4a}= -0.5, c_{5}= 6.4551, c_{6}= 0.4908, c_{7}= 0.0352, c_{7b}= 0.0462, c_{8}= 0.0000, c_{8a}= 0.2695, c_{8b}= 0.4833, c_{9}= 0.9228, c_{9a}= 0.1202, c_{9b}= 6.8607, c_{11}= 0, c_{11b}= -0.4536, c_{RB}= 50, c_{n}= 16.0875, c_{M}= 4.9993, c_{HM}= 3.0956, c_{γ1}= -0.007146, c_{γ2}= -0.006758, c_{γ3}= 4.2542, ϕ_{1}= -0.5210, ϕ_{2}= -0.1417, ϕ_{3}= -0.007010, ϕ_{4}= 0.102151, ϕ_{5}= 0.0000, ϕ_{6}= 300, γ_{Jp-It}= 1.5817 (use for Japan and Italy), γ_{Wn}= 0.7594 (use for 2008 Wenchuan earthquake), ϕ_{1Jp}= -0.6846 (use for Japan), ϕ_{5Jp}= 0.4590 (use for Japan) and ϕ_{6Jp}= 800 (use for Japan). - Ground-motion model is (for aleatory variability):
_{1}= 0.4000, τ_{2}= 0.2600, σ_{1}= 0.4912, σ_{2}= 0.3762, σ_{3}= 0.8000 and σ_{2Jp}= 0.4528 (for Japan). - Use V
_{s,30}(F_{inferred}=1 for inferred values and F_{measured}= 1 for measured values) and depth to 1km∕s shear-wave velocity horizon (Z_{1.0}) to characterise sites. State model applicable for 180 ≤ V_{s,30}≤ 1500m∕s. Estimate Z_{1.0}for those sites lacking measured value by empirical relations linking Z_{1.0}and V_{s,30}. - Use 3 mechanisms:
- Normal
- Rake angle -120 ≤ λ ≤-60
^{∘}. 8 M_{w}< 5.9 Californian events and 3 M_{w}≥ 6 Italian events. F_{NM}= 1. - Reverse
- Rake angle 30 ≤ λ ≤ 150
^{∘}. F_{RV }= 1. - Strike-slip
- Other rake angles. F
_{NM}= F_{RV }= 0.

Use two locations w.r.t. vertical projection of the top of rupture:

- Hanging wall
- R
_{x}≥ 0km. F_{HW }= 1. - Foot wall
- R
_{x}< 0km. F_{HW }= 0.

- Model derived within NGA West 2 project, using the project database (Ancheta et al., 2014).
- Update model of Chiou and Youngs (2008) w.r.t. faulting mechanism, hanging-wall effects, scaling with
the depth to top of rupture (Z
_{TOR}), scaling with sediment thickness (Z_{1.0}), fault dip (δ) and rupture directivity. Also account for regional differences in distance attenuation and site effects. - Use observations and simulations (Donahue and Abrahamson, 2014) to develop model.
- Since database consists mainly of Californian data initially focus on developing moel for California using primarily Californian data. Then supplement these data with records from large earthquakes elsewhere to refine magnitude-scaling and to derive more robust σ for larger events. Examine regional differences.
- Use same selection criteria as Chiou and Youngs (2008) except for these changes. Include only free-field
data from 18 well-recorded M
_{w}≥ 6 earthquakes (2587 records) from outside California. Assess maximum usable distance (R_{max}) for each earthquake using truncated regression with truncation level equal to second lowest PGA for each earthquake. Set R_{max}equal to distance where truncation level equals -2.5 standard deviations below fitted median from a event-specific model. This allows final model to be derived using non-truncated regression. Older earthquakes with high truncation levels have R_{max}< 70km but R_{max}for recent events is relatively large. Exclude Class 2 earthquakes (including 1999 Duzce event) located within 20km of Class 1 earthquake. - Functional form based on stochastic simulations, seismological arguments (e.g. change from body-wave spreading to surface/Lg-wave spreading) and examination of data for various periods. Mainly unchanged from Chiou and Youngs (2008).
- Assess variation of γ (anelastic attenuation) with T for 3 magnitude intervals. For each T and interval
compute variance-weighted average of fitted values of γ for individual events. Find variation in γ with
T is magnitude dependent. Examine regional differences in γ for non-Californian earthquakes, including
aftershocks not selected for final model. Find γ for New Zealand, Taiwan and Turkey are similar to those
for California, whereas those for Italy and Japan (only use data in range 6 ≤ M
_{w}≤ 6.9) indicate more rapid far-source attenuation and data for Wenchuan slower attenuation. Include regional differences in γ in final model. - Exploratory analysis of data indicates mechanism effect weaker for M
_{w}< 5 than for M_{w}> 6. Find similar effects for Z_{TOR}. Hence include these effects in final model using term that prevents undue influence on large-magnitude scaling by small earthquakes whose estimates of M_{w}, Z_{TOR}and mechanism are more uncertain than those for larger events. - Develop M-Z
_{TOR}relation to centre the Z_{TOR}adjustment. - Preliminary analysis indicates dependence of event terms for M
_{w}< 5 increase with δ but that there is no effect for M_{w}> 6. - Note very few observations for region inside surface projection of rupture (r
_{jb}= 0). Hence use simulations of (Donahue and Abrahamson, 2014) to develop hanging-wall model here using R_{x}, the horizontal distance from top of rupture measured perpendicular to strike. Foot-wall data for each simulation fit using simple functional form. Compute residuals at r_{jb}= 0 and plot w.r.t R_{x}for specific dip angle. Derive model using R_{x}trend excluding data for M_{w}6, which showed different behaviour. Find model matches simulations and empirical data for r_{jb}> 0. - Include directivity effects using direct point parameter (DPP), centred on its mean, as variable. Use
narrow-band formulation of directivity effects, excluding linear-magnitude dependence which is unstable
w.r.t T and statistically insignificant for many T. Assume directivity for M
_{w}< 5.5 is negligible because of absence of finite-fault information for M_{w}< 5.7 but note that this assumption may not be true. - Use centred Z
_{1.0}to investigate de-amplification for shallow sediment sites. Find evidence for differences in ΔZ_{1.0}scaling between Japan and California. State model applicable for Z_{TOR}≤ 20km and do not recommend using large depth for M_{w}> 7 because of lack of data. - Find nonlinear V
_{s,30}component does not need updating w.r.t. Chiou and Youngs (2008) but linear scaling does. Find evidence for difference in linear V_{s,30}between Japan and California, which include in model. - Normal-faulting term not well constrained because of limited data hence do not update coefficients of Chiou and Youngs (2008).
- Model developed through iterative process of regression for all Ts with some parts of model fixed, smoothing
a few coefficients w.r.t. T, then repeating regression using smoothed coefficients. Correct for sample bias
at long-periods smooth c
_{1}by imposing smooth variation in the slope of c_{1}w.r.t. T. - 2 earthquakes (2000 Tottori (M
_{w}6.61) and 1999 Chi-Chi (M_{w}7.6)) have large absolute event terms. Analysis of event-term distribution using robust regression suggest Tottori may be a outlier so remove it when assessing τ. Do not remove Chi-Chi event term because may lead to underestimate of τ. - Bin τ and σ in 0.5-magnitude-unit bins. Find magnitude dependency for most T. Use trilinear form. Allow
for discontinuity in σ at M
_{w}5 but not for τ. Find inclusion of data from events with < 5 records inflates τ, at least for small events, and hence derive aleatory variability model using only events with ≥ 5 records. Between-event residuals suggest dependence on r_{rup}but this largely explained for small T by nonlinear site amplification and increased intra-event variability for Japanese data. Observed dependence for large T may be due to unmodelled basin effects because of lack of Z_{1.0}for areas outside California and Japan. - Note that useful to include κ in future models because of potential influence on aleatory variability model.
- Examine inter-event residuals w.r.t. M
_{w}and do not find significant trends. Some outliers (> 2τ) for large non-California earthquakes (1999 Chi-Chi, 2000 Tottori and 2008 Wenchuan). Add loess fits to plot and find 95% confidence limits emcompass zero hence outliers not significant. Also using only California earthquakes results in similar event terms. - Examine intra-event residuals w.r.t. M
_{w}, r_{rup}, V_{s,30}and ΔZ_{1.0}. Find no significant trends except at edges of data. Using loess fits conclude that trends are not significant. - Plot intra-event residuals without V
_{s,30}term grouped by y_{ref}w.r.t. V_{s,30}. Compare to predicted site amplification. Find good agreement. For V_{s,30}model overestimates amplification for Japanese data suggesting deviation from linear lnV_{s,30}scaling for stronger nonlinearity at Japanese sites. - Note that, because all M
_{w}< 6 earthquakes are from California, model may not be applicable for small events in other regions. - Note that for application to regions with different anelastic attenuation may adjust γ model using estimates of Q for regions derived using geometric spreading models consistent with model.
- Note that amplification for V
_{s,30}> 1130m∕s constrained to unity. Little data in database to examine amplification for higher V_{s,30}, where κ may decrease. - Recommend setting ΔZ
_{1.0}= 0 when Z_{1.0}unknown. - When Z
_{1.0}is much lower than E(Z_{1.0}) recommend checking predictions not lower than predictions for reference condition of V_{s,30}= 1130m∕s.