## 3 Seismic Response Spectra in Mexico
### Deterministic Spectrum According to MDOC Sismo CFE 2015  
A deterministic spectrum implies a procedure to obtain the spectral accelerations as a function of a specific earthquake event, particularly of high magnitude with a shorter distance between the seismic source and the specific site. Therefore, we need the seismic records data near the site of interest to identify the appropriate seismic events and perform the analysis.  
The initial inputs required to perform a deterministic spectrum are the magnitude of the earthquake, \(M\), the focal length, \(R\), the fault type related to the seismic event, and the distance from the hypocenters to the surface. The general analysis process is explained in MDOC Sismo CFE 2015 in its comments section [11].  
The deterministic method suggests the application of the local seismicity with an appropriate attenuation of the seismic waves, by using the ground motion prediction equations (GMPE), to obtain the spectral acceleration, \(S_{a}\), at the site of interest.  
Local seismicity shows the recurrence of earthquakes according to the magnitudes registered at a particular site. The recurrence is measured with the exceedance rate \(\lambda(M)\) and can be obtained by applying the modified Gutenberg-Richter relationship [12] for most seismic sources in Mexico and the characteristic seismic model [13], which works adequately for the subduction fault zone located between the Cocos and North American plates in southern Mexico. The parameters for both models are mapped for different seismic-generator zones across the country and can be found in the MDOC Sismo CFE 2015 in its comments section [11].  
The ground motion prediction equations (GMPE) are models developed for seismic zones with a particular geological fault type, considering a decrease in the amplitudes of the seismic waves by getting away from the epicenter. For Mexico, three types of models are used, depending on the fault type and the hypocenter's depth. The model of [PERSON] and [PERSON] [14] is applied to shallow earthquakes, while the model of [PERSON] et al. [15] is applied to earthquakes of intermediate depth, and the model of [PERSON] et al. [16] is related to interplate earthquakes according to the subduction fault. These models can be modified according to data on recent earthquakes.  
Once the local seismicity has been assessed for an earthquake of magnitude, \(M\), an appropriate GMPE model, and a focal length, \(R\), the maximum spectral acceleration, \(A_{m}\), can be obtained deterministically. However, there are uncertainties in the magnitude, \(M\), and the focal length, \(R\), that affect the spectral acceleration, \(S_{a}\). Therefore, a log normal distribution must be applied due to the nature of earthquakes, according to the following equation:  
\[S_{a}=A_{m}(M,R)e^{\alpha\tau_{InA}} \tag{21}\]  
where \(A_{m}(M,R)\) is the mean spectral acceleration for an earthquake of magnitude, M, and focal length, R, with an appropriate GMPE model, \(\sigma_{InA}\) is the standard deviation, and \(\varepsilon\) is the percentile used, with a value of \(\varepsilon=1\) for an 84% percentile according to the log-normal distribution.