## 2 Derivation of Amplification Factors (_A_Fs)
### Database
#### 2.4.2 Selection of Site Parameters Studied  
Each soil profile can be partially described by a few site parameters. In this paper, we investigate seven of them. Extensive studies of seismic site response have been performed in recent decades. In 1994, [PERSON] developed intensity-dependent, short- and long-period amplification factors based on the average shear wave velocity measured over the upper 30 m of a site [10]. [PERSON] et al. (1988) developed a geotechnical site classification system based on the shear wave velocity, the depth to bedrock, and general geotechnical descriptions of soil deposits [72]. [PERSON] et al. (1991) [73] then developed intensity-dependent site amplification factors that modify the baseline \"rock\" peak ground acceleration (_PGA_) to account for site effects. With such a site, _PGA_ value, and site-dependent normalized acceleration response spectra, [PERSON] and [PERSON] (1994) derived site-dependent design spectra (primarily based on the site classification system and amplification factors) [11], which were incorporated in the 1997 Uniform Building Code (UBC) [30]. The National Building Code of Canada (NBC) (2005, 2015a, b) and a later version (2020) adopt the same philosophy as the UBC, where the soil classification as well as site design spectra are based on the shear wave velocity in the upper 30 m supporting the foundations [32; 33; 34]. However, two important limitations are associated with such an approach.  
First, it requires a relatively extensive field investigation, and second, it overlooks the potential importance of other site parameters, such as the fundamental frequency or depth to bedrock, among others, which have been identified by many researchers as pertinent for site classification [26; 27; 28; 35; 74].  
To overcome the above limitation, our study considers the following parameters in addition to the \(V_{30}\): the depth of deposit to bedrock (depth); the average shear velocity (\(V_{sm}\)) over that depth; the velocity contrast, that is, the ratio between the shear wave velocity in the bedrock and at the surface (Cv); the second velocity contrast, that is, the ratio between the shear wave velocity in the bedrock and the average shear wave velocity over the upper 30 m (Cv2); the soil profile's fundamental frequency (\(f_{0}\)); and the _PGA_ in the bedrock. The inclusion of the _PGA_ as a parameter of the study is justified by the well-established fact that site amplification for nonlinear soil is fundamentally dependent on the intensity of the input wave. The site proxies considered are calculated using Relations (8) to (13) as follows:  
\[\text{Depth}=\text{h}_{1} \tag{8}\]  
which is the thickness of the first layer (deposit layer), h\({}_{1}\), with the bedrock being considered as a half-infinite space (see Figure 2).  
\[V_{sm}=V_{1} \tag{9}\]where \(V_{1}\) is the shear wave velocity in the first (deposit) layer:  
\[V_{s30}=\sum_{i=1}^{l_{30}}\frac{h_{i}}{30} \tag{10}\]  
where \(l_{30}\) is the number of distinct layers found in the top 30 m:  
\[\text{Cv}=\frac{V_{Bedrock}}{V_{1}} \tag{11}\]  
\[\text{Cv2}=\frac{V_{Bedrock}}{V_{s30}} \tag{12}\]  
The fundamental soil frequency \(f_{0}\) is determined by using the simplified Rayleigh procedure, described by [PERSON] et al. (1976) [75], based on Equations (13) and (14):  
\[f_{0}=\frac{\sqrt{\left(4\left(\sum_{i=1}^{n}\frac{\left(z_{i}+z_{i+1}\right) ^{2}}{V_{i}^{2}}\;h_{i}\right)/\left(\sum_{i=1}^{n}\left(X_{i}+X_{i+1}\right)^ {2}h_{i}\right)\right)}}{2\pi} \tag{13}\]  
where \(\frac{z_{i+1}+z_{i}}{2}\) is the depth of the midpoint of layer (\(i\)), and the \(X_{i}\) values correspond to the estimated deformations due to the fundamental mode shape at the top of each layer (\(i\)), derived according to [PERSON] et al. (1976) as follows [75]:  
\[X_{n}=0;\;X_{i-1}=\;X_{i}+\frac{z_{i}+z_{i-1}}{V_{i}^{2}}\;h_{i} \tag{14}\]  
To better illustrate the domain of validity of this study, the covered site parameters domain is summarized in Table 1 through 10%, 50%, and 90% fractiles for all site parameters.