## 2 Derivation of Amplification Factors (_A_Fs)
### Site Model, Wave Propagation Solution, and Transfer Function T(f)  
In this study, the soil is idealized by one horizontally layered soil deposit resting on a bedrock substratum (see Figure 2).  
The layer is fully defined by its shear modulus, \(G\), or shear wave velocity, \(V\); thickness, h; damping ratio \(\zeta\); and mass density \(\rho\). The underlying half-space has shear wave velocity \(V_{bedrock}\). The vertical z-axis is oriented downwards. The response of the soil column to harmonic, vertically incident plane shear waves is governed by differential Equation (2) [68]:  
\[G_{i}\frac{\partial^{2}u_{i}}{\partial z_{i}^{2}}+\eta_{i}\ \frac{\partial^{3}u_{i}}{ \partial z^{2}{}_{i}\partial t}=\rho_{i}\frac{\partial^{2}u_{i}}{\partial t ^{2}} \tag{2}\]  
where \(u_{i}\) is the horizontal displacement in the \(i\)th layer (in this study, \(i=1\) for the monolayer soil deposit).  
with:  
\[\eta_{i}=2G_{i}\zeta_{i}/\omega \tag{3}\]  
where \(\omega\) is the angular frequency of the exciting harmonic, and \(\zeta_{i}\) is the damping ratio.  
The general solution of governing differential Equation (2) is a summation of up-going and down-going plane waves with unknown amplitudes, \(A_{i}\) and \(B_{i}\), for each layer, given in Equation (4).  
\[u_{i}(z_{i},f)=A_{i}e^{(i2\pi ft)}e^{(ik_{i}^{*}z_{i})}+B_{i}e^{(i2\pi ft)}e^{(- ik_{i}^{*}z_{i})} \tag{4}\]  
The transfer function is a complex function defined as the ratio of the layer surface amplitude to the layer bottom amplitude, and the procedure to drive such a function can  
Figure 1: Normalized (to \(PGA\) = 1 g) acceleration spectra of signals used.  
Figure 2: A schematic representation of the 1D site response.  
be found in the classical literature [68; 70; 71]. For a single layer of soil, it can be written, using Euler's Law, as follows:  
\[T(f)=\frac{1}{\cos K_{s}^{*}h+i\alpha_{z}^{*}\sin K_{s}^{*}h} \tag{5}\]  
where \(K_{s}^{*}\) is a complex wave number, defined as  
\[K_{s}^{*}=\frac{2\pi f}{V_{1}(1+i\zeta_{1})} \tag{6}\]  
and \(\alpha_{z}^{*}\) is the complex impedance ratio, defined as  
\[\alpha_{s}^{*}=\frac{\rho_{1}\ V_{1}(1+i\zeta_{1})}{\rho_{ bedrock}\ V_{ bedrock}(1+i \zeta_{bedrock})} \tag{7}\]  
where \(\zeta_{bedrock}\) is the damping ratio in the bedrock, which generally has a negligible effect on the results, and is taken as null in this study.  
For a particular soil profile, \(AF(T)\) is computed once the transfer function \(T(f)\) is known using the equivalent linear model procedure extensively described in the geotechnical earthquake engineering literature [35; 67; 68].  
As mentioned earlier, we used the simple shear modulus degradation model proposed by [PERSON] et al. (1988) [58] for clay and the model proposed by [PERSON] and [PERSON] (1970) [52] for sand (G/Gmax is a function of the shear strain). For damping variation with shear strain, the curves of [PERSON] (1990) are used for both clay and sand [59]. These selected models are illustrated in Figure 3, and their key parameter values are presented in Supplementary File, section (c).  
Based on these models, the equivalent shear modulus and damping ratio are obtained by an iterative process to ensure that the values used are consistent with the strain level obtained according to the procedure described above. At the interface of adjacent layers, the stress and displacement continuity equations are solved, and the relationships between these amplitudes for two adjacent layers are established. Afterward, the waves are propagated from the bottom (unit up-going amplitude) to the top layer using the free surface condition (shear stress = 0). The wave amplitudes and the transfer function can be derived with respect to the motion at the outcropping bedrock.  
Figure 3: Degradation curves for clay and sand. (**a**) Shear modules. (**b**) Damping ratios. Refs. [52; 58; 59].