# Site-Specific Spectra for the City of Mexicali, Mexico, Obtained from April 2010 Earthquake Records
## 2 Determination of the Site-Specific Spectra
### Transfer Function  
For modeling purposes, the seismic record observed at the surface can be viewed mathematically as the result of the convolution of the function describing the seismic energy propagating through the earth's interior and the function describing the response of the soil structure. One of the most widely used methodologies to obtain it is through the propagation matrix or the so-called Thomson-Haskell method (proposed by [PERSON] [31] and corrected by [PERSON] [32]). The method describes the transformation \(H\) that an input signal \(e(t)\) undergoes as it travels through a medium and generates an output signal \(s(t)\), which is the movement of the ground at the surface. The medium is modeled by a set of homogeneous layers superimposed on each other (vertically heterogeneous, isotropic, and elastic medium), and the properties that define the medium do not vary continuously but discretely. The method works by assuming a linear behavior of the soil. Under these conditions, the equations of motion and the constitutive relations can be combined so that only the first-order derivatives of the stress and displacement concerning the depth ordinate (\(z\)) are needed to describe the motion according to Equation (5). \[\frac{d}{dz}f(z)=A(z)f(z) \tag{5}\]  
Figure 6: Spectral ratios from seismic noise measurements at three points at the same site where the geotechnical survey was conducted. Three measurements (red, blue, and black lines) were taken at each point. The arrow indicates the position of the dominant period. where \(f(z)\) is a vector containing the relation of stresses and strains to depth, and \(A(z)\) is a matrix \(n\times n\) (in the case of SH waves \(n=2\)), which depends on the elastic parameters of the medium according to:  
\[A(Z)= \begin{pmatrix}0&\mu^{-1}\\ \mu\omega p^{2}-\rho\omega^{2}&0\end{pmatrix}  \tag{6}\]  
Since the method works with homogeneous layers, the elastic parameters do not vary within a layer, so it does not depend on the \(z\)-depth within each layer. Once the matrix incorporates the dynamic properties from the base of the deposit to the point where the elastic response of the deposit is to be known, a propagation method can be applied [33], described by (7) and (8). \[\mathrm{f}(Z)=(Z,Z_{0})\mathrm{f}(Z_{0}) \tag{7}\]  
\[P(Z,Z_{0})=e^{(Z,Z_{0})A} \tag{8}\]  
We used the Degtra software 9.3 [34] to obtain the transfer function (Figure 7) of the stratigraphic profile described in the previous section. Density and damping were chosen according to the material found in the geotechnical study (Figure 5). Comparing this function with the spectral quotients estimated from seismic noise, it can be seen that the results are similar up to 5 Hz. We consider that spectral ratios can be considered as empirical transfer functions up to this frequency. Once the transfer function is obtained, a synthetic seismic record can be generated using any input signal (Green's function) and a numerical simulation of propagation through the medium (convolution). This record will represent the movement at the surface due to the site effect. One goal of calculating transfer functions is to use them to estimate the response of a site due to the occurrence of an earthquake where seismic records are not available. As Green functions, we chose the horizontal acceleration records of the nearer stations to the April 2010 event. Since these stations are on sediments and therefore are affected by the site, we first had to remove this effect. As we do not have a stratigraphic profile for these sites, we used empirical transfer functions obtained by spectral quotient [29], as we did for the site in Mexicali, and applied a deconvolution process with the corresponding record. The stations considered for this analysis were CHI, TAM, SAL, GEO, MDO, and RII, the closer stations to the epicenter.