On the Relationship between Fourier and Response Spectra: Implications for the Adjustment of Empirical Ground-Motion Prediction Equations (GMPEs)

The functional form of empirical response spectral ground-motion prediction equations (GMPEs) is often derived using concepts borrowed from Fourier spectral modeling of ground motion. As these GMPEs are subsequently calibrated with empirical observations, this may not appear to pose any major problems in the prediction of ground motion for a particular earthquake scenario. However, the assumption that Fourier spectral concepts persist for response spectra can lead to undesirable consequences when it comes to the adjustment of response spectral GMPEs to represent conditions not covered in the original empirical data set. In this context, a couple of important questions arise, for example, what are the distinctions and/or similarities between Fourier and response spectra of ground motions? And, if they are different, then what is the mechanism responsible for such differences and how do adjustments that are made to Fourier amplitude spectrum (FAS) manifest in response spectra? The present article explores the relationship between the Fourier and response spectrum of ground motion by using random vibration theory (RVT). With a simple Brune (1970, 1971) source model, RVT-generated acceleration spectra for a fixed magnitude and distance scenario are used. The RVT analyses reveal that the scaling of low oscillator-frequency response spectral ordinates can be treated as being equivalent to the scaling of the corresponding Fourier spectral ordinates. However, the high oscillator-frequency response spectral ordinates are controlled by a rather wide band of Fourier spectral ordinates. In fact, the peak ground acceleration, counter to the popular perception that it is a reflection of the high-frequency characteristics of ground motion, is controlled by the entire Fourier spectrum of ground motion. Additionally, this article demonstrates how an adjustment made to FAS is similar or different to the same adjustment made to response spectral ordinates. For this purpose, two cases: adjustments to the stress parameter (Delta sigma) (source term), and adjustments to the attributes reflecting site response (V-S - kappa(0)) are considered.