Correlation of spectral residuals between two oscillator periods
correlateResiduals.RdEvaluates the Baker and Jayaram (2008) correlation between the logarithmic
residuals of spectral acceleration at two oscillator periods, the
coefficient that couples the ordinates of a correlated spectral sample
(sampleSpectra()). It depends on the pair of periods only, not on
magnitude or distance.
Mathematics
With \(T_{min} = \min(T_n, T_0)\) and \(T_{max} = \max(T_n, T_0)\),
equations 5 and 6 of Baker and Jayaram (2008) are
$$C_1 = 1 - \cos\left(\frac{\pi}{2} - 0.366 \ln \frac{T_{max}}{\max(T_{min}, 0.109)}\right)$$
$$C_2 = 1 - 0.105 \left(1 - \frac{1}{1 + e^{100 T_{max} - 5}}\right) \frac{T_{max} - T_{min}}{T_{max} - 0.0099}$$
when \(T_{max} < 0.2\) and \(C_2 = 0\) otherwise,
\(C_3 = C_2\) when \(T_{max} < 0.109\) and \(C_3 = C_1\) otherwise,
$$C_4 = C_1 + \frac{1}{2} \left(\sqrt{C_3} - C_3\right) \left(1 + \cos \frac{\pi T_{min}}{0.109}\right)$$
and the correlation is \(C_2\) when \(T_{max} < 0.109\), \(C_1\)
when \(T_{min} > 0.109\), \(\min(C_2, C_4)\) when
\(T_{max} < 0.2\) and \(C_4\) otherwise, tried in that order. The
coefficient is symmetric in the two periods. The fit is calibrated for
periods from 0.01 to 10 s; a pair outside that range is still evaluated
and a warning is signalled. As in newmark 1.7.1, a period of zero or less
stands for peak ground acceleration and is evaluated at 0.01 s, a
convention of the package and not of the paper.
References
Baker, J. W. and Jayaram, N. (2008). Correlation of spectral acceleration values from NGA ground motion models. Earthquake Spectra, 24(1), 299-317. doi:10.1193/1.2857544
Examples
correlateResiduals(Tn = c(0, 0.3, 1.5), T0 = 0.01)
#> [1] 1.0000000 0.7953519 0.4204404