Site spectral acceleration given the rock ordinate and PGA
evaluateSiteSa.RdNatural-log mean and standard deviation of the spectral acceleration at a
site of velocity VS30, conditional on the rock ordinate Sa and on the
rock PGA of the same realization at the reference condition, under a site
factor of evaluateSiteFactor(). Applied to the mean ordinates of a
spectrum with the mean PGA, it is the site spectrum of a deterministic
rock spectrum.
Arguments
- Sa
Rock spectral accelerations in g at period
Tn, finite and positive.- PGA
Rock peak ground accelerations in g, aligned with
Sa.- Tn
One oscillator period in seconds.
- VS30
One target velocity in m/s.
- Vref
One reference velocity in m/s, 760 or 3000.
- model
Site factor model of
evaluateSiteFactor(),"ST20"or"ST17".
Value
A data.table with muLnSaF and sigmaLnSaF (natural-log mean and
standard deviation of the site ordinate, one row per Sa) and ID
(the model), unrounded.
Mathematics
With \(\ln F \mid PGA \sim N(\mu_F, \sigma_F)\) from
evaluateSiteFactor(),
$$\ln S_{aF} \mid S_a, PGA \sim N(\ln S_a + \mu_F, \sigma_F),$$
row by row; its quantile at probability \(p\) is
\(\exp(\ln S_a + \mu_F + \sigma_F \Phi^{-1}(p))\) and its mean
\(\exp(\ln S_a + \mu_F + \sigma_F^2/2)\). The dispersion is that of the
site factor alone: Sa and PGA are conditioning values, not
distributions, so the result does not carry the uncertainty of the rock
demand. When VS30 equals Vref the result is \(\ln S_a\) with zero
dispersion.
Examples
evaluateSiteSa(Sa = VancouverSpectra[Tn == 0.5][order(TR), Sa], Tn = 0.5, VS30 = 300,
PGA = VancouverSpectra[Tn == 0][order(TR), Sa], Vref = 760, model = "ST17")
#> muLnSaF sigmaLnSaF ID
#> <num> <num> <char>
#> 1: -1.9928918 0.2289710 ST17
#> 2: -1.7557468 0.2309299 ST17
#> 3: -1.4685714 0.2343195 ST17
#> 4: -1.2545218 0.2379519 ST17
#> 5: -1.0778820 0.2420694 ST17
#> 6: -0.9074465 0.2468905 ST17
#> 7: -0.7613664 0.2518012 ST17
#> 8: -0.6177185 0.2577204 ST17
#> 9: -0.4919898 0.2635989 ST17
#> 10: -0.4103219 0.2677460 ST17