Effects of frequency on SdS reflection coefficients

S^dS reflection coefficients for a two-layered medium are fully described by Eq. (¹) and are not frequency dependent. However, Δβ and Δρ profiles from thermodynamic modeling usually show both abrupt and gradual changes, leading to the frequency-dependent nature of S^dS reflection coefficients. Due to nonlinear effects and/or phase interference, it is difficult to calculate the effect on reflection coefficients.

We use synthetic waveform modeling to derive (empirically) an equivalent depth interval, over which the total changes in density and V_S can predict the observed amplitudes of SS precursors via Eq. (¹). Supplementary Fig. ⁹ shows the procedure of deriving an equivalent depth interval using two different models: (a) a harzburgite model along a 1400 °C adiabat with gradual changes in density and velocity near 410 and 660 and (b) the PREM reference model. We first measure amplitude ratios of S^dS/SS on synthetic waveforms and then correct them for geometrical spreading and intrinsic attenuation. For the gradual 1400 °C harzburgite model, the observed S⁴¹⁰S/SS and S⁶⁶⁰S/SS amplitude ratios are equivalent to those predicted by total changes of density and V_S over a depth interval of ~10 km and ~25 km, respectively. In contrast, for the PREM model, the observed S⁴¹⁰S/SS and S⁶⁶⁰S/SS are well predicted by the first order discontinuities at 400- and 670-km depth. We apply the above procedure to all thermodynamic models. The results are shown in Fig. 6b, d.