Data

From the SSN database we selected 86 earthquakes (of various focal mechanism) located between 90.5° W and 96.5° W and between 13° and 17° N, and which occurred between 1995 and 2017. From this database, we present a wide range of magnitudes (5.0 ≤ Mw ≤ 8.2), distances (52 ≤ R ≤ 618 km; hipocentral distances for Mw<=7.0 and rupture distances for Mw> 7.0) and depths (10 ≤ H ≤ 243 km) as shown in Figure 1 and Table 1. We obtained 261 three-components accelerograms for those events recorded at 9 stations located in Chiapas, Oaxaca, Tabasco and Veracruz. The respective stations are OXJM, SCRU, NILT, MIHL, PIJI, TGBT, VHSA, SCCB and TAJN and belong to the Seismic Network of Institute of Engineering-UNAM (Pérez-Yáñez et al., 2010). Date, depth, moment magnitude (Mw) and distance for each recording are indicated in Table 1. The farthest away stations (VHSA and MIHL) have less recordings, in contrast with those sited in the central part of the study area (PIJI, NILT, SCCB, OXJM, TAJN, TGBT and SCRU) where the Chiapas State capital city (Tuxtla Gutierrez) and the hidroelectric dams are located.

The spatial distribution of the 9 accelerometric stations (red squares), and of the 86 epicenters (white and black circles) is shown in Figure 1. Also indicated are the locations of the Chiapas (AVC) and Guatemala (AVCA) volcanic arcs. The Chiapas Massif is depicted in pink. Figure 1 also shows the Polochic-Motagua (continuous red line), Tonalá and Los Tuxtlas (discontinuous red line) fault systems. The epicenter of the Tehuantepec, September 7, 2017 (Mw8.2) earthquake is indicated with a green star. Green small dots represent aftershocks with magnitudes lower than 5. A comparison of the area covered by the aftershocks of the Tehuantepec earthquake with the localized seismicity of the last 17 years indicates that the Tehuantepec aftershocks cover that portion of the Tehuantepec Gulf that had been inactive.

In the right panel of Figure 1. The 261 hypocenters analyzed in this study were projected to the A-A´ cross-section (right panel of Figure 1), whose location is indicated with a white line. A dipping angle of the slab of about 45 degrees, as well as a plate thinning at 80 km depth can be observed.

Stars indicate the epicenters of the three earthquakes with Mw > 7.0 (see Table 1). For these events, the minimum distance to the rupture was considered. For the October 21, 1995 earthquake (Mw 7.2), according to the rupture model proposed by ^{Rebollar et al. (1999)}, the rupture depth (htop) lies at 80 km. For the September 8, 2017 event (Mw 8.2), we used the rupture model obtained by ^{Ye et al. (2017)}, which has a htop = 30 km. Finally, for the November 7, 2012 (Mw 7.3) event, for which there is no rupture model, we assumed a rupture model with a htop = 10 km, with its closest edge point (northwestern edge of the fault plane) at latitude 14º N and longitude 92º W.

Figure 2 shows the distribution of magnitudes (5.0 ≤ Mw ≤ 8.2) versus distance (52 ≤ R ≤ 618 km) of the analyzed records. A concentration of events with magnitudes in the range from 5.0 to 5.5 for distances between 52 and 300 km is observed.

Figure 2 Moment magnitude (Mw) distribution against distance of the analyzed seismic events. 

Data Processing and Analysis

For each of the three components of the 261 recordings associated to the 86 selected seismic events, both shear waves and surface waves (also known as coda) were selected. Data processing included homogenization of the signal sampling of all extracted signals. Subsequently, for each selected record, the Fourier amplitude spectrum (FAS) was computed and the spectral ratio of horizontal components with respect to the vertical one obtained. After that, the quadratic means were obtained from both ratios. These are the directional earthquake horizontal to vertical spectral ratios (EHVSR), which were computed for a frequency band between 0.1 and 10 Hz. In Figure 3 we show the 261 ratios (thin of colors continuous lines) distributed in the 9 stations. Averages are depicted as continuous red lines (dashed red lines: average ± one standard deviation). We assume that this average of directional EHVSR´s is an estimate of the spectral amplification, a kind of empirical transfer function (ETF) of average horizontal components with respect to the vertical component. The source effect is approximately removed. In a recent paper, ^{Kawase et al. (2018)} suggested to consider the amplification due to vertical motion to avoid over-reduction of FAS. However, this requires recordings both in soil and rock sites.

Figure 3 Earthquake horizontal to vertical spectral ratios EHVSR at the nine studied stations. In thin of colours continuous lines, the quadratic mean spectral ratios (continuous red lines) of each earthquake are plotted in log-log scale (dashed red lines: average ± one standard deviation). The trends are clear and the averages at each station, depicted in red, are assumed to represent the site effect. The value of two suggested by SESAME (2000), continuous line, and the theoretical H/V in a Poissonian half-space are given as reference (discontinuous line). 

It has been proposed that the site effect is significant for ETF larger than two (SESAME, 2015) as it is the case of scalar amplification in a half-space. However, under the assumption of a diffuse field, ^{Sánchez-Sesma et al. (2011a)} found theoretically that the MHVSR at the surface of a half-space is approximately given by

where v=Poisson ratio. If v =0.25 then the H/V is about 1.332.

Thus, according to criteria from SESAME (continuous line in Figure 3), and to that of ^{Sánchez-Sesma et al. (2011}) (discontinuous line in Figure 3), the 9 accelerometric stations present site effects in the analyzed frequency band. Stations NILT and PIJI reach amplifications of more than ten times at 6 and 4 Hz, respectively. Stations OXJM, SCRU and TGBT present lower amplifications.

Finally, to suppress site effects from the 261 records, with the ETF computed previously (red continuous line, Figure 3) and the estimated FAS for each record, we compute the deconvolution between the FAS for the nine sites and ETF in the frequency domain for this site, subtracting their respective site effect. The corresponding intensities (spectral acceleration and peak values) are estimated using random vibration theory (RVT) (i.e., ^{Arciniega-Ceballos 1990}; ^{Reinoso and Ordaz 1999}), which comprises the following steps: (1) de-amplifying the FAS by dividing them by the EHVSR obtaining a proxy of ETF; (2) multiplication of the spectrum by the oscillator transfer function or by one if PGA is seek; (3) calculation of the moments m ₀ and m ₂ of the power density which are given by mk= ∫0∞ωkA(ω).2 dω; (4) estimate the root mean square in terms of m ₀ and the strong motion duration, D, for each event; (5) computation of the peak factor F _p , according to RVT, in terms of the number of extrema N occurring during the duration D and from the moments by means of N=(D/π)(m ₂ /m ₀ )^0.5 , and get the expected peak by multiplying the FAS by F _p . The peak factor is asymptotically given by F _p = (2ln N) ^0.5 +0.577(2ln N)^-0.5. Regarding D, we use the expression developed by Herrmann (1985), D=fc-1+0.05R where f _c = corner frequency (in Hz) and the R=distance (in km). As for response spectral ordinates, both the oscillator transfer function and the additional duration have to be accounted for (see e.g. ^{Boore, 1983}).

In this way, the estimated ground motion intensities (e.g. acceleration response spectra) for each event and site will be essentially free of site effect. In Figure 4 this reduction is illustrated for the July 6th, 2007 (M_w=6.2) event. The complex Fourier spectra were deconvolved by the corresponding average EHVSR (ETF) and then transformed back to time domain. The N-S accelerograms of the nine stations are displayed in the left panel. The corresponding distance and PGA are indicated. In the right panel the recordings with suppressed site effect are displayed (as if the corresponding stations were located in hard rock). This suppression gives significantly lower PGA values (for TAJN a factor of 7.5 was obtained). We claim that the use of average EHVSR may be adequate to correct GMPE for site effects, leading to a more realistic attenuation model.

Figure 4 Example of site effects correction for a Mw=6.2 event of July 6th , 2007 for the studied stations. Left panel depicts the N-S accelerations indicating the epicentral distance and the PGA in gals (cm/s2). The right panel shows the time series with the site effects removed with the procedure described herein. 

Peak ground accelerations (PGA) are obtained from these corrected records. Also, response spectra with 5 % critical damping are obtained for 24 structural periods ranging between 0.3 and 40 Hz. Finally, peak ground velocities are obtained by integrating these recordings after correcting them for base line (^{Boore, 2005}) and band-pass filtering between 0.3 and 40 Hz.

Each parameter (i.e., PGA, PGV, or the spectral ordinates) is separately calculated for both horizontal components, then the quadratic mean is obtained from both ortogonal components (^{Boore, 2005}). Other alternatives would include the geometric mean, or other no geometrical means (^{Boore, 2010}). We used the quadratic vector mean as it is a common practice in the development of attenuation models, since from the physics point of view it is more rational than other means. All recordings were processed in the same form.

Regression analysis

To estimate the spectral accelerations with a damping of 5 %, as well as the PGA and PGV, the regression analysis of the data set was made using the maximal verisimilitude method of one stage proposed by Boore (1993), which constitutes the most direct form to predict the response spectra of observed data. We use a simpler functional form proposed by ^{Ordaz et al. (1989)}, and ^{García-Soto and Jaimes (2017)} to estimate the spectral ordinates, PGA, and PGV for seismic events from southeastern Mexico.

where Y(T) represents the horizontal spectral ordinate based on a quadratic mean of the horizontal components, T in seconds is the period of the single degree of freedom system, Mw is the moment magnitude, R is the closest distance from site to fault surface for larger events (Mw > 6.5) or the hypocentral distance for the rest, both in km,  _i are the coefficients estimated by the regression analysis, and ε₁ is the error estimation by assuming a normal distribution.

Noteworthy is that in previous studies (i.e., ^{Arroyo, 2010}; ^{García-Soto and Jaimes, 2017}) no important dependence with the focal depth was found, consequently it was not considered in excluded from this study. Even more, the quadratic mean was used since the use of the geometric mean and was the development of the attenuation relationship is slightly less conservative than the use of the quadratic mean to evaluate seismic risk (i.e., ^{Hong and Goda, 2007}). The horizontal geometric dispersion can be obtained as G(R)=R^α ₃ ^(T), where α₃(T) is the geometric attenuation coefficient, which controls the amplitude decay with the distance, R. By applying the natural logarithm to both sides of the last expression, it can be linearized as ln(G(R))=α₃(T)∙lnR, which corresponds to the third term of equation (2). In this study, it was considered that α₃(T) is very well constrained by seismic observations in intraplate events (i.e., ^{Ordaz et al. 1994}; ^{Reyes, 1999}; ^{Jaimes et al. 2006}; ^{Garcia-Soto and Jaimes, 2017}). Further, by fixing the geometric dispersion coefficient at -0.5 for all the ordinates in both components (i.e., ^{Ordaz et al., 1994}; ^{Reyes, 1999}; ^{Jaimes et al., 2006}), unrealistic values are avoided (i.e., non-negative values of α₃(T),) that physically have no sense (^{Ordaz et al. 1994}).

Regression Coefficients and Residuals

Regression coefficients, α_i(T), and the standard deviation σ_i(T), were estimated for periods T between 0.1 and 10 s, for recordings of 4 groups: Group 1, considers all recordings without site effects; Group 2, includes the recordings with site effects; Group 3, comprises recordings of earthquakes with depths less than 80 km but without site effects; while Group 4, is constituted by recordings of earthquakes with depths less than 250 km and no corrected for site effects. Figure 5 shows, in a logarithmic scale, the values of the regression coefficients α₁(T), α₂(T) and α₄(T) for period values between 0.1 and 10 s. No differences significant are found between coefficients α₂ and α₄ among the 4 groups. However, coefficient α₁ of group 2 (which includes site effects) shows an evident divergence, from the others groups. Confirmation of the validity of the attenuation model is indicated by the standard deviation σ_i(T) obtained for the cases with and without site effects. Table 2 summarizes the regression coefficients α_i(T) and the standard deviation obtained from the analyzed recordings by considering the quadratic mean of the horizontal components.

Figure 5 - Regression coefficients and natural logarithmic standard deviation logarithmic for horizontal components in the sites of Chiapas State, Mexico. a) Group 1: all records without site effects. B) Group 2 all records presenting site effects, c) Group 3 including records with depths less than 80 km, and d) Group 4 with records with depths less than 250 km and no site effect. 

For the purpose of this analysis, the residual is defined as:

where ln(Y_i) is the natural logarithm of i-eth observed value Y _i and lnY- is the corresponding predicted value. The attenuation model in order to non-biased estimations, the residual must have a zero mean, and do not present any correlation with the regression model parameters, i.e., the magnitude (Mw) and distance R. Figure 6 shows the residuals δ_i obtained from the regression of the horizontal components as functions of magnitude (upper panel), and of the distance (lower panel) for PGA and spectral ordinates for T= 0.06, 0.5, and 1 s. These figures consider a) all recordings without site effects, b) all recordings with site effects, c) recordings from earthquakes with depths less than 80 km and without site effect, and d) recordings from earthquakes with depths less than 250 km and without site effects. These figures shows that the regression model is not biased neither towards magnitude nor distance. The tendency lines are shown with a thick line.

Figure 6 Residual values obtained from the regression of the horizontal components according to magnitude (upper part of panel), and with respect to distance (lower part of each panel) for peak ground acceleration (PGA) and spectral pseudoaceleration, Sa, at T values of 0.06, 0.5, and 1 s. Panel a comprises all records without site effect, panel b includes all records with site effect, panel c considers records with depths less than 80 km and without site effect. Finally, panel d corresponds to records with depth less than 250 without site effect. 

Comparison with Other Studies

To the author's knowledge, there are no GMPEs in the region under study, which makes the outcome more necessary to compute the seismic hazard in the region. For that reason, the Figure 7 compares the attenuation model obtained in this study (for magnitudes Mw of 5.5, 6.5, and 7.5) with those of ^{García et al., (2005)}, ^{Arroyo et al., (2010)} and ^{García-Soto and Jaimes (2017)}, based on earthquakes located at the Pacific Ocean coasts (discontinuous lines). Attenuation models obtained in this study present a slower decay than previous ones, and models that include site effects (yellow lines) present larger amplitudes than those without site effects (black lines). Previous attenuation models are clearly different to the attenuation models here presented for southeastern Mexico. These differences could be due to the fact that in southeastern Mexico earthquakes attain depths up to about 243 km, while in the Guerrero coast (southern Mexico) depths are less than 80 km. In general, models by ^{Arroyo et al. (2010)} and ^{Garcia-Soto and Jaimes (2017)} follow a similar pattern in all cases, and for Sa, for T = 0.5 s and 1 s, they decay almost in the same way as the model by ^{Garcia (2005)} for Mw 5.5 and 6.5. For PGA, the model by ^{Garcia (2005)} present larger amplitudes (for the three magnitudes) than those by the models by ^{Arroyo et al. (2010)} and ^{Garcia-Soto and Jaimes (2017)}. In particular, the PGA models corrected for site effects (obtained in this study) have smaller amplitudes than the models by ^{Arroyo et al. (2010)} and ^{Garcia-Soto and Jaimes (2017)} only for distances smaller than 130, 90 and 50 km for Mw 5.5, 6.5 and 7.5, respectively. On the contrary, Sa models corrected for site effects present lower amplitudes than those of the previous models. However, and as it was expected, the corresponding models that include site effects (yellow lines) present the largest amplitudes at all distances. In other words, a model comprising site effects presents overestimated values.

Figure 7 Regression curves for the horizontal component for PGA and spectral pseudo-acceleration, Sa, for periods of 0.5 and 1 s for earthquakes with magnitudes Mw of 5.5, 6.5, and 7.5. Group 1 comprises regression of all records without site effect, while group 2 includes all records with site effect.