Conditional spectral acceleration under site amplification

The site-response model transforms spectral acceleration at a reference condition into spectral acceleration at a target site condition. At oscillator period $T_n$, $S_a^o(T_n)$ denotes the reference spectral acceleration and $S_a^s(T_n,V_{S30})$ the site spectral acceleration. The target condition is described by $V_{S30}$, the travel-time-averaged shear-wave velocity in the uppermost $30\,\mathrm{m}$. The random amplification factor $F$ relates the accelerations through

$$S_a^s(T_n,V_{S30})=F(T_n,V_{S30},\mathrm{PGA}^o)S_a^o(T_n).$$

Here, $\mathrm{PGA}^o$ is the peak ground acceleration at the reference condition; it conditions the nonlinear part of the factor. The superscript $o$ identifies that reference condition. Spectral acceleration and PGA are expressed in $\mathrm{g}$, period in $\mathrm{s}$, and velocity in $\mathrm{m/s}$. At fixed period, velocities and input PGA, the factor is lognormal, with natural-log mean $\mu_{\ln F}$ and natural-log standard deviation $\sigma_{\ln F}$. This lognormal distribution is the probabilistic representation that the library applies to the factor. Its natural-log mean combines the adjustment between the $760$ and $3000\,\mathrm{m/s}$ references, a linear term that depends on $V_{S30}$ and a nonlinear term that depends on the input PGA. The library assigns its natural-log variance as a sum of contributions from the linear term, the change of reference and the nonlinear term, without covariances between them; the published model characterizes the uncertainties of its median components as epistemic (Stewart et al. 2020, 1, 6–7, 9). The standard deviation $\sigma_{\ln F}$ is therefore the dispersion assigned by this formulation and does not by itself represent record-to-record variability.

Fixed reference acceleration and PGA

For fixed values $S_a^o=s>0$ and $\mathrm{PGA}^o=pga^*$, multiplication by the factor gives

$$\ln S_a^s\mid s,pga^*\sim\mathcal{N}\!\left(\ln s+\mu_{\ln F},\sigma_{\ln F}^{2}\right).$$

The notation $\mathcal{N}(\mu,\sigma^2)$ specifies a normal distribution by its mean and variance (Kroese et al. 2011, sec. 4.2.11, p. 122). The factor parameters are evaluated at the same period, target velocity and conditioning PGA. By the expectation of the lognormal distribution (Kroese et al. 2011, Table 4.18, p. 121), the mean acceleration under this conditional distribution is

$$E[S_a^s\mid s,pga^*]=s\exp\!\left(\mu_{\ln F}+\tfrac12\sigma_{\ln F}^{2}\right).$$

Thus the returned mean of the logarithm and the mean acceleration are different quantities. The reference acceleration and PGA in this case are conditioning values; their uncertainty is absent from the conditional distribution.

evaluateSiteFactor() returns $\mu_{\ln F}$ and $\sigma_{\ln F}$ as muLnF and sigmaLnF. evaluateSiteSa() receives aligned Sa and PGA values and returns muLnSaF and sigmaLnSaF, the mean and standard deviation of the logarithm of site acceleration. Both functions receive the period Tn, target velocity VS30, reference velocity Vref, and factor identifier model. The supported identifiers are ST17 and ST20, with reference velocities of $760$ or $3000\,\mathrm{m/s}$. These identifiers designate formulations based on the 2017 and 2020 editions of the site amplification model, respectively (Stewart et al. 2017, 13; 2020, 1, 6). When target and reference velocities coincide, the factor is deterministic and equal to one, so this conditional acceleration equals $s$.

Uncertain reference demand within a mixture component

A reference-demand component $k$ instead specifies random spectral acceleration and PGA. Their logarithms have means $\mu_k$ and $\mu_{P,k}$, standard deviations $\sigma_k$ and $\sigma_{P,k}$, and correlation $\rho$. For a jointly normal logarithmic pair with positive standard deviations and $|\rho|<1$, the logarithm of reference spectral acceleration conditional on $q=\ln pga^*$ is normal. The model adopts independence between the amplification residual and the reference-acceleration residual once $k$ and $q$ are fixed. The conditional site acceleration then satisfies

$$\ln S_a^s\mid(k,q)\sim\mathcal{N}\!\left(\mu_k(q),\varsigma_k^2(q)\right).$$

Its conditional logarithmic mean is

$$\mu_k(q)=\mu_k+\rho\frac{\sigma_k}{\sigma_{P,k}}(q-\mu_{P,k})+\mu_{\ln F}.$$

Its conditional logarithmic variance is

$$\varsigma_k^2(q)=(1-\rho^2)\sigma_k^2+\sigma_{\ln F}^2.$$

The factor parameters in these expressions are evaluated at PGA $e^q$. The variance retains the part of reference-demand variability left after conditioning and adds the factor variance. If $\sigma_{P,k}=0$, the PGA is constant: the logarithmic mean becomes $\mu_k+\mu_{\ln F}$ and the variance becomes $\sigma_k^2+\sigma_{\ln F}^2$, without a division by $\sigma_{P,k}$. At $T_n=0$, the reference ordinate is the PGA itself; for positive PGA dispersion, its correlation with itself is one and the conditional reference variance is zero.

integrateSiteMixture() approximates the integration of these conditional distributions over each component’s PGA distribution by Gauss-Hermite quadrature when target and reference velocities differ. Its input contains component labels k, period Tn, logarithmic parameters muLn and sigmaLn, and weights w summing to one at each period. The quadrature requires a matching PGA component at Tn = 0 for each spectral component. At node $m$ of component $k$, with PGA logarithm $q_{km}$, the site mixture receives a lognormal component when $\varsigma_k(q_{km})>0$ and a point mass at $\exp(\mu_k(q_{km}))$ when $\varsigma_k(q_{km})=0$; its weight equals the input component weight times the normalized node weight. When VS30 equals Vref, the function returns the validated input mixture without quadrature or component expansion. summarizeMixture() obtains the mean and requested quantiles from the returned mixture. These are marginal statistics at each period; they do not specify dependence between periods or a single realized spectrum. The quantile at probability $p$ is the generalized inverse of the mixture distribution function, the infimum of the accelerations at which that function reaches $p$(Kroese et al. 2011, eq. 3.1, p. 45). When $p$ lies within the jump of that function at a point mass, the quantile is the location of the point mass. The search for the quantile is numerical and uses a fixed interval of the logarithmic acceleration; that interval is not guaranteed to bracket the quantile for every probability in $(0,1)$.

References

Kroese, Dirk P., Thomas Taimre, and Zdravko I. Botev. 2011. Handbook of Monte Carlo Methods. Wiley Series in Probability and Statistics. John Wiley & Sons. https://doi.org/10.1002/9781118014967.

Stewart, Jonathan P., Grace A. Parker, Gail M. Atkinson, David M. Boore, Youssef M. A. Hashash, and Walter J. Silva. 2020. “Ergodic Site Amplification Model for Central and Eastern North America.” Earthquake Spectra, ahead of print. https://doi.org/10.1177/8755293019878185.

Stewart, Jonathan P., Grace A. Parker, Joseph A. Harmon, et al. 2017. Expert Panel Recommendations for Ergodic Site Amplification in Central and Eastern North America. 2017/04. Pacific Earthquake Engineering Research Center.