Theoretical relationship between the horizontal-to-vertical response and Fourier spectral ratios of ground motions

Similar to the horizontal-to-vertical Fourier spectral ratio (HVF) of ground motions, the horizontal-to-vertical response spectral ratio (HVR) is a valuable tool for evaluating site effects. Although these two spectral ratios often exhibit similar behaviours, each possesses its own set of properties, prompting increased attention to their relationship. Previously, the relationship between HVF and HVR has been statistically investigated to explore which is more reasonable for predominant period estimation. However, the theoretical link between them remains unexplored. To clarify their theoretical relationship, in this study, an expression relating HVR to HVF based on random vibration theory was derived. The accuracy of the derived expression was confirmed through a comparison with the results obtained via direct numerical integration using real seismic records. Subsequently, based on the derived expression, the theoretical relationship between HVF and HVR was systematically explored. HVR was found to be the result of smoothing the square of the HVF, and the spectral window for this smoothing operation was determined using the Fourier amplitude spectrum of the vertical ground motion and the oscillator transfer function.