## 3 Seismic Response Spectra in Mexico
### Site-Specific Spectrum According to MDOC Sismo CFE 2015
A first approximation to study the dynamic behavior of the surface soil and the frequencies is made by applying a 1D model and taking as inputs the rock layer at the base and the thickness of each surface layer where the seismic waves can travel. The formulation used for 1D models considers the harmonic motion, where their amplitude is measured on the free surface. Figure 9 shows a scheme of wave propagation in the soil layer with thickness H, applying the functions \(Ae^{i(\omega t+kz)}\) and \(Be^{i(\omega t-kz)}\), which are related to the wave movement in the \(z\)-axis direction. The functions shown in Figure 9 have different variables that define the equations of motion, where \(\omega\) is the circular frequency of the ground, \(k\) is the wave number expressed as \(k=\omega/v_{s}\), \(A\) and \(B\) are the wave amplitudes in each direction, \(z\) is the height measured from the free surface layer, and \(t\) is the time. Considering the thickness of the surface soil as an equivalent layer, with damping and elastic rock at the base, the 1D model is modified according to the variables shown in Figure 10, where \(H\) is the equivalent layer thickness, the subscript \(s\) refers to the equivalent soil layer, and the subscript \(r\) refers to the rock layer at the base. Thus, the transfer function \(F_{1}(\omega)\) is defined as follows [6]:  
\[F_{1}(\omega)=\frac{1}{cos\left(\frac{\omega H}{v_{ss}^{*}}\right)+i\alpha_{z} ^{*}sin\left(\frac{\omega H}{v_{ss}^{*}}\right)} \tag{17}\]  
where \(v_{ss}^{*}\) is the complex wave velocity in the soil layer, defined as \(v_{ss}^{*}=v_{ss}(1-i\xi)\); \(\alpha_{z}^{*}\) is the complex impedances ratio, defined as \(\alpha_{z}^{*}=\rho_{s}v_{ss}^{*}/\rho_{r}v_{sr}^{*}\); \(\rho_{s}\) is the volumetric weight of the soil layer; \(\rho_{r}\) is the volumetric weight of the rock layer; and \(v_{sr}^{*}\) is the complex wave velocity in the rock layer, defined as \(v_{sr}^{*}=v_{sr}(1-i\xi)\). Figure 8: Path of vibrational waves from the seismic source to the surface layers. Figure 9: 1D model for the dynamic analysis between the rock and surface layers. Now, instead of a single layer with equivalent thickness, it is convenient to consider multiple layers as damped soil over the elastic rock, modifying the 1D model according to Figure 11 to improve the results. Therefore, the transfer function, \(F_{2}(\omega)\), related to the motion, \(u(z,t)\), at layer \(m\), is expressed according to the following equations:  
\[u(z,t)=Ae^{i(\omega t+k^{*}z)}+Be^{i(\omega t-k^{*}z)} \tag{18}\]  
\[F_{2}(\omega)=\frac{\left|u_{i}\right|}{\left|u_{j}\right|} \tag{19}\]  
where \(k^{*}\) is the complex wave value, defined as \(k^{*}=k(1-i\xi)\) and the subscripts \(i\) and \(j\) are related to the layer number. The amplitude is obtained with the absolute value of the transfer function represented as \(\left|F(\omega)\right|\). The described equations are applicable for a single-degree-of-freedom (SDF) system to perform a linear analysis. Now, with an iterative process, the equations can be used to perform a nonlinear analysis. There are programs available that can numerically obtain the soil transfer functions \(F(\omega)\) using 1D models, including multiple layers methods with the possibility to perform a linear or non-linear analysis. Some examples are DeepSoil, developed by the University of Illinois in Urbana, IL, USA [7], or ProShake, developed by EduPro Civil Systems Inc. in Sammamish, WA, USA [8], where the interface allows us to assign the soil layer properties using the variables shown in Figure 11 and the program calculates the transfer function, \(F(\omega)\), displaying the results with a \(\left|F(\omega)\right|\) vs. frequency (Hz) graph.