Conditional site amplification factor of the NGA-East models

Natural-log mean and standard deviation of the site amplification factor $F$ that converts a spectral ordinate at the reference condition Vref into the ordinate at a site of time-averaged velocity VS30, given the peak ground acceleration at the reference condition and the oscillator period. Model "ST20" is the final NGA-East ergodic model (linear part of Stewart et al. 2020, nonlinear part of Hashash et al. 2020, coefficients of the USGS nshmp-lib table carried in SiteAmp2020); model "ST17" is the 2017 generation (PEER reports 2017/04 and 2017/05). The pair defines the lognormal $\ln F \sim N(\mu, \sigma)$ conditional on PGA, period, VS30 and Vref; it is the factor, not an amplified spectrum (see evaluateSiteSa(), integrateSiteMixture(), integrateSiteSpectrum() and simulateSiteSpectrum(), which integrate it over the rock demand).

Usage

evaluateSiteFactor(PGA, Tn, VS30, Vref, model)

Arguments

PGA: Peak ground accelerations in g at the reference condition, finite and non-negative.
Tn: One oscillator period in seconds, finite and non-negative; 0 is PGA.
VS30: One target time-averaged shear-wave velocity in m/s.
Vref: One reference velocity in m/s, 760 or 3000.
model: "ST20" or "ST17".

Value

A data.table with muLnF and sigmaLnF (natural-log mean and standard deviation of $F$, one row per PGA) and ID (the model), unrounded.

Mathematics

Model "ST20". The coefficients $c, V_1, V_2, V_f, V_c, f_3, f_4, f_5, \sigma_{vc}, \sigma_l, \sigma_u, \sigma_c, f_{760i}, f_{760g}, f_{760is}, f_{760gs}$ of the period are read from SiteAmp2020: the PGA row for $T_n \le 0.01$ s, the tabulated row at a tabulated period, and otherwise the linear interpolation in $\ln T_n$ between the bracketing rows; a period above 10 s is clamped to 10 s with a warning. With $w(v) = \min(\max(0.667 \ln(v/400)/\ln 1.5 + 0.1, 0.1), 0.767)$ and $F_{760}(v) = w f_{760i} + (1 - w) f_{760g}$, the linear term is $f_{lin}(v) = f_v(v) + F_{760}(v)$, where $f_v(v)$ is $c \ln(V_1/760)$ for $v \le V_1$, $c \ln(v/760)$ up to $V_2$, $c \ln(V_2/760)$ up to 2000 m/s, $c \ln(V_2/760) - (c \ln(V_2/760) + F_{760}(v)) \ln(v/2000)/\ln 1.5$ below 3000 m/s and $-F_{760}(v)$ from 3000 m/s (Stewart et al. 2020, Eqs. 3 and 5-7). The nonlinear slope is $f_2(v) = f_4 [e^{f_5(\min(v, V_{NL}) - 360)} - e^{f_5(V_{NL} - 360)}]$ for $v < V_c$ and 0 otherwise, with $V_{NL} = 760$ m/s for $T_n < 0.4$ s and 3000 m/s above (Hashash et al. 2020, Eq. 3), and it multiplies $F_{Rk} = \ln((PGA_{3000} + f_3)/f_3)$, where $PGA_{3000} = PGA/2.275$ when Vref is 760 m/s (PEER 2017/05, Eq. 2.4) and $PGA$ when it is 3000 m/s. The linear dispersion is $\sigma_l - 2 D x + D x^2$ with $D = \sigma_l - \sigma_{vc}$ and $x = (v - 200)/(V_f - 200)$ below $V_f$, $\sigma_{vc}$ up to $V_2$, $\sigma_{vc} + (\sigma_u - \sigma_{vc})((v - V_2)/(2000 - V_2))^2$ below 2000 m/s, $\sigma_u (1 - \ln(v/2000)/\ln 1.5)$ up to 3000 m/s and 0 above (Stewart et al. 2020, Eq. 4); the nonlinear dispersion is $\sigma_{f2} F_{Rk}$ with $\sigma_{f2} = \sigma_c$ below 300 m/s, $\sigma_c (1 - \ln(v/300)/\ln(1000/300))$ below 1000 m/s and 0 above. For Vref = 760 m/s $$\mu = f_{lin}(V_{S30}) - f_{lin}(760) + (f_2(V_{S30}) - f_2(760)) F_{Rk}, \quad \sigma = \sqrt{\sigma_{lin}^2(V_{S30}) + (\sigma_{f2}(V_{S30}) F_{Rk})^2},$$ the reference side adding no dispersion; for Vref = 3000 m/s $\mu = f_{lin}(V_{S30}) + f_2(V_{S30}) F_{Rk}$ and the weighted $F_{760}$ dispersion $w f_{760is} + (1 - w) f_{760gs}$ enters the root sum of squares.

Model "ST17". The coefficients are interpolated linearly in $T_n$ from the tables of PEER 2017/04 (linear part, $F_{760}$) and 2017/05 (nonlinear part, Table 2.1). The linear mean is $c \ln(V_1/760)$, $c \ln(V_{S30}/760)$ or $c \ln(V_2/760) + (c/2) \ln(V_{S30}/V_2)$ below $V_1$, up to $V_2$ and above; the nonlinear mean is $f_2 \ln(PGA/b + 1)$ for $V_{S30} < V_c$ with $f_2 = f_4 [e^{f_5(\min(V_{S30}, 3000) - 360)} - e^{f_5(3000 - 360)}]$ and $b = 2.275 f_3$ (Vref 760) or $f_3$ (Vref 3000), and $F_{760}(T_n)$ is added for Vref = 3000. The dispersion combines the linear term (quadratic below $V_f$ between 200 m/s and $V_f$, $\sigma_{vc}$ up to $V_2$, quadratic towards 2000 m/s above), the tabulated $F_{760}$ dispersion for Vref = 3000 and the nonlinear term $\sigma_{c} \ln(PGA/b + 1)$ tapered from 300 to 1000 m/s.

When VS30 equals Vref the factor is the deterministic unit factor, $\mu = 0$ and $\sigma = 0$. Every result is a conditional evaluation at the supplied PGA; the dispersion combines the epistemic uncertainties of the published median model, not the uncertainty of the rock demand.

References

Stewart, J.P. et al. (2020). Ergodic site amplification model for central and eastern North America. Earthquake Spectra 36(1). Hashash, Y.M.A. et al. (2020). Nonlinear site amplification model for ergodic seismic hazard analysis in central and eastern North America. Earthquake Spectra 36(1). Stewart, J.P. et al. (2017). PEER Report 2017/04. Hashash, Y.M.A. et al. (2017). PEER Report 2017/05.

Examples

evaluateSiteFactor(PGA = c(0.05, 0.2, 0.5), Tn = 0.2, VS30 = 300, Vref = 760,
                   model = "ST20")
#>         muLnF  sigmaLnF     ID
#>         <num>     <num> <char>
#> 1: 0.37682363 0.2525448   ST20
#> 2: 0.24385574 0.2595137   ST20
#> 3: 0.06985233 0.2788971   ST20
evaluateSiteFactor(PGA = 0.1, Tn = 1.0, VS30 = 760, Vref = 3000, model = "ST20")
#>        muLnF  sigmaLnF     ID
#>        <num>     <num> <char>
#> 1: 0.1530131 0.2524343   ST20