Empirical models to estimate horizontal Fourier acceleration spectra from earthquakes in Northern Gulf of California, Mexico
Modelos empíricos para estimar los espectros de aceleración de Fourier horizontal a partir de terremotos en el norte del Golfo de California, México
José Sequeira-Arguedas1
, Raúl R. Castro1
1 Departamento de Sismología, División Ciencias de la Tierra, Centro de Investigación y de Educación Superior de Ensenada, Baja California, México (CICESE). Carretera Ensenada-Tijuana 3918, Zona Playitas, 22860, Ensenada, Baja California, México.
* Corresponding author:
(R. R. Castro) This email address is being protected from spambots. You need JavaScript enabled to view it.
How to cite this article:
Sequeira-Arguedas J., & Castro, R. R. (2026). Empirical models to estimate horizontal Fourier acceleration spectra from earthquakes in Northern Gulf of California, Mexico: Boletín de la Sociedad Geológica Mexicana, 78(2), A051125. https://doi.org/10.18268/ BSGM2026v78n2A051125
Manuscript received: June 26, 2025. Corrected manuscript received: October 15, 2025. Manuscript accepted: October 27, 2025.
ABSTRACT
We propose two empirical predictive equations to estimate acceleration spectra at rock sites using epicentral distance and moment magnitude as input parameters, in the frequency range of 0.32 to 25.12 Hz. We considered 42 earthquakes located in the northern region (29.5°N to 31.7°N) of the Gulf of California (GoC) that occurred in the period 2015-2022 with magnitudes between 4.0 to 5.7 Mw. We compiled 180 acceleration records from eight stations of the Broadband Seismological Network of the Gulf of California and the Northwestern Mexico Seismological Network, operated by CICESE. We calculated the horizontal acceleration spectra selecting time windows with variable length according to epicentral distance. We consider two models, one linear and another quadratic for the magnitude, obtaining the regression coefficients through ordinary least squares. The linear model was found to be more appropriate for the region, as the quadratic model yielded physically unlikely results at low frequencies. Comparative analysis with ground motion models from other tectonic regions highlighted the relevance of our regional approach. Additionally, a scenario for a Mw 5.3 event was simulated to evaluate the performance of the linear model in predicting spectral amplitudes in rock. Both models are considered a mean regional attenuation value and give reliable amplitudes at low frequencies (f ≤ 10 Hz); therefore, their applicability across epicentral distances must be carefully used on prior knowledge of the site amplification function. It was also evident the fitting difference of Fourier acceleration spectra between the northern (above 29.5° of latitude) and central GoC (Canal de Ballenas region), possibly attributed to azimuthal coverage, geological heterogeneities, rock quality and different states of stress.
Keywords: empirical spectral models, Gulf of California, horizontal Fourier acceleration spectrum, data inversion.
RESUMEN
Proponemos dos ecuaciones predictivas para estimar el espectro de aceleración en roca usando la distancia epicentral y la magnitud momento como parámetros de entrada dentro del rango frecuencial de 0.32 a 25.12 Hz. Consideramos 42 sismos localizados en la región norte del Golfo de California (GdC) ocurridos entre 2015 y 2022, con magnitudes entre 4.0 y 5.7 Mw. Para ello se compilaron 180 registros de aceleración para ocho estaciones de la Red de Banda Ancha del Golfo de California y de la Red del Noroeste de México, operadas por el CICESE. Calculamos el espectro de aceleración horizontal en ventanas de tiempo dependientes de la distancia epicentral. Dos modelos se consideraron, uno lineal y otro cuadrático para la magnitud, obteniendo coeficientes de regresión a partir de inversión por mínimos cuadrados. El modelo lineal es considerado como el más apropiado para la región, debido a la dificultad de ajuste del modelo cuadrático, sobre todo a bajas frecuencias. Un análisis comparativo con otros modelos para otras regiones tectónicas muestra la relevancia de nuestro enfoque. Adicionalmente, se simuló un escenario para un sismo de 5.3 M para evaluar el comportamiento del modelo lineal prediciendo la amplitud espectral en roca. Ambos modelos consideran una atenuación promedio para la región y brindan amplitudes congruentes a frecuencias menores que 10 Hz; sin embargo, su aplicabilidad es restringida al conocimiento de la función específica del sitio y en un rango específico de distancia epicentral. También se encontraron diferencias de ajuste del espectro de Fourier de aceleración horizontal entre la sección norte (al norte de 29.5° de latitud) y central del GdC (región Canal de Ballenas), posiblemente atribuido a la cobertura azimutal, heterogeneidades geológicas, calidad de la roca y distribución de los esfuerzos tectónicos en la región.
Palabras clave: modelos espectrales empíricos, Golfo de California, espectro de Fourier de aceleración horizontal, inversión de datos.
1. Introduction
Empirical ground-motion predictive equations (GMPE) allow us to estimate the spectral acceleration in the frequency domain and constitute a worthful tool for seismic hazard and civil engineering. These equations have been successfully used for evaluating separately the seismic source, the attenuation due to propagation of body-wave path and the site effect in the surface (Iwata and Irikura, 1988). The GMPE estimate representative parameters of ground motion such as peak ground acceleration (PGA), spectral absolute acceleration (SA), absolute pseudo acceleration (PSA) or relative pseudo velocity (PSV) with engineering purposes (Schmidt-Díaz and Esquivel-Salas, 2022).
In the same way, frequency content evaluation of earthquakes is fundamental in seismic engineering, therefore the ground response estimation and plausible related effects are critical. For this, in every area is required an adequate predictive equation (Villalobos-Escobar and Castro, 2013) which represents the body-wave propagation medium, with previous knowledge of the seismogenic sources and the local subsurface role in seismic amplification, due to free-surface and mechanical behavior of materials.
According to Meenakshi et al. (2023), the availability of GMPE in terms of the Fourier amplitude acceleration spectra (FAAS) is somewhat lower compared to those models that consider PSA or PGA. Bayless and Abrahamson (2019) highlight that the usage of GMPE in terms of the FAAS allows for ease in the calibration of stochastic simulations of earthquakes and linear response for a wide range of frequencies.
Given this backdrop, it is possible to determine the empirical seismic response in terms of the Fourier spectra considering acceleration data from stations of the Broadband Seismological Network of the Gulf of California (RESBAN) and the Northwestern México Seismological Network (RESNOM), both operated by Ensenada Center for Scientific Research and Higher Education (CICESE), Baja California, Mexico.
Therefore, we propose a simple two-step model for estimating the horizontal FAAS based on the Joyner and Boore (1981) method, which uses as independent variables the moment magnitude (Mw) and epicentral distance to seismic source (repi). The entry data for the model corresponds to northern Gulf of California (GoC) earthquakes that occurred between 2015 and 2022, with a magnitude range of 4.0 to 5.7 Mw.
One relevant contribution of this study is the empirical evaluation of hypothetical characteristic earthquakes linked to the seismic potential of the region even in non-instrumented sites and also, it is the first one of its kind for the northern region of the GoC (NRGoC). At this region, there is no other frequency-based GMPE model, which makes this study a valuable extension of seismic hazard analysis beyond peak ground motion parameters, especially if the input parameters are commonly available for most seismic catalogs (i.e. distance and magnitude). This GMPE constitutes a practical framework that potentially could be applied to other tectonic regions.
1.1 TECTONIC SETTING
The Gulf of California comprises a transform zone between Pacific and North America plates (Figure 1), which includes a series of normal faults systems, en-échelon strike-slip faults and a series of active and inactive basins (Castro et al., 2021) capable of generating considerable big earthquakes and its related effects. Given this, future earthquakes related to the regional seismic potential could generate critical ground motion and compromise civil structures at main localities in the NRGoC, causing potential damage to human and economic activity. This study will focus on seismicity greater than 4.0 Mw magnitude located in the NRGoC region with a singular cortical thickness and tectonic heterogeneities (Castro et al., 2017a). The NRGoC is characterized by N to N 30°E strike oblique faults (Figure 1) with dips from 60° to 80° (Persaud et al., 2003). The Wagner, Consag, Delfín and Tiburón basins (Fuentes-Bustillos, 2020; Martín-Barajas et al., 2013) are the most prominent tectonic features in the region, where seismicity concentrates.
Beyond the northern Wagner basin, the seismicity centroids align with fault zones and depths from 15 to 22 km, with other minor pull-apart basins filled with recent Quaternary sediments (Castro et al., 2021). The Consag basin has predominantly normal fault styling, where plate motion generates a wide deformation zone, with the seismogenic zone varying from 3 to 22 km depth (Castro et al., 2017a; Cruz-Hernández et al., 2023). The Delfin basin developsdiffuseanddisperseextensionalzoneswith high segmentation in its faults (Castro et al., 2021; Goff et al., 1987). This pull-apart basins develop from extensional deformation accommodation between the Cerro Prieto zone (NW from the studied area, in the continental zone) and the interaction with the Canal de Ballenas and Agua Blanca strike-slip faults, respectively, located to the south and southwest of the studied area (Castro et al., 2021).
Recently, Cruz-Hernández et al. (2023) using focal mechanisms, determined seismogenic zones deeper than 22 km near the Pacific-North America interface. They also note normal-type focal mechanisms concentration for the Delfin basin (Figure 1) and oblique-type for its northern area (i.e., Wagner basin). In addition, these authors indicate that the seismicity concentrates between the Consag and Delfín basins, usually between 3 to 22 km depth, and establish clear differences between the northern region of the GoC with the central one, near Canal de Ballenas, where the seismicity diminishes in amount but not in potential, because it was this central region, where the 6.9 Mw event from August 3rd, 2009. This region also hosted other historical greater earthquakes (> 6.0 magnitude, see Figure 1), in contrast with the NRGoC where no greater earthquake is known (Castro et al., 2021).
2. Data
We used eight stations, from the RESBAN and RESNOM networks, located in the northern region of the GoC (Table 1 and Figure 1). These stations have wide-range triaxial sensors, time-integrated Global Positioning System (GPS) devices and 24-bit digitizer, and they are installed in concrete shelters that isolate each sensor from environmental-related noise (Castro et al., 2018).
The data were base line corrected and filtered with a band-pass 4th order Butterworth filter between 0.20 Hz and 30.00 Hz.
We selected the available earthquakes from the RESNOM (CICESE Data Centre, 2025) and RESBAN (Castro et al., 2018) catalogs that occurred between June 2015 and December 2022 with latitude greater than 29.5° N and greater than 4.0 Mw magnitude, considering only recordings with epicentral distance below 350 km. We selected 42 events (Table 2) with a good quality signal-to-noise ratio (S/N) for the stations selected, setting a cut-limit value of 2.0 for this ratio. The distribution of the events selected is shown in Figure 2.
The magnitude and epicentral distances distribution for the empirical prediction models are shown in Figure 3, where most event magnitudes are concentrated between 4.0 Mw to 4.8 Mw, and only seven earthquakes have magnitudes equal or greater than 5.0 Mw. For the epicentral distance, most of the data concentrate between 95 km to 240 km (62%), while 21% of the data correspond to distances less than 95 km and only 17% of the data correspond to distances greater than 240 km. Considering the number of recorded observations per frequency, Figure 3 shows a maximum at 2.00 Hz and a decreasing trend for frequencies greater than 5.01 Hz, reaching a minimum of 44 records at 25.12 Hz. For the spectral amplitudes, only those with a S/N ratio greater than 2.0 were considered.
3. Methods
We considered a simple point geometric spreading seismic source with a constant Q-value to scale acceleration spectra, modifying the two-step inversion technique proposed by Joyner and Boore (1981) focused on estimated peak ground motion parameters, such as PGA and PGV. This functional form is linearized and based on distance and moment magnitude, assuming that spectral amplitude for body waves can be approximated as the square of the energy radiated from a point source, considering the anelastic attenuation, as the following equations show, where i corresponds to the event recorded at station j (e.g., Castro, 1998):
Where:
f: frequency (Hz) a1 , a2 and a3 : coefficients of the magnitude
Mi : moment magnitude for event i, after Hanks and Kanamori (1979)
dij: distance in km
h: pseudo-depth parameter in km
rij: pondered distance in km
b: apparent anelastic attenuation coefficient
c: bimodal term (0 or 1) for station site, 0 for rock and 1 for sediment
Sj : site amplification term
Uij: observed horizontal spectral amplitude (the vector sum of components E-W and N-S)
Given that Sj = 0 when the site effect is removed (rock response), the equation for the model is:
Also, it is relevant to note that regressions were carried out using hypocentral distances and considering several values for the h parameter from 0.1 km to 31.0 km (maximum depth of events), and neither of those considerations significantly diminish the global error of the models, so we considered the epicentral distance (repi) for the models. An additional consideration for selecting epicentral distance is the network array and the distribution of the seismogenic zones in the studied area and the fact that for distant stations, epicentral distance equals Joyner and Boore distance ( Joyner and Boore, 1981) if a point source is considered.
The first step considers distance fitting, and the problem can be formulated as a matrix being solved by least squares with QR decomposition, resulting in a standard deviation by frequency for this regression step (σ1). Once the distance coefficients for each frequency were estimated, the second stage of the inversion process was carried out, considering the solution from the first stage (Equation 3) and using linear or quadratic regression fits for the source term. Similarly to the first step, a standard deviation for the adjustment is obtained, denoted as σ2. This decoupled method provides a separation of the errors obtained in each stage, and the overall standard error of the model (σT), consists in the sum of the squares of the error obtained at each stage, according to Equation 4:
The estimation of coefficients was performed through a least-squares method using QR decomposition, which is very useful in over-determined inverse problems (m > n). The matrix factorization of observed data (A) by a Q-matrix (square unit matrix of orthogonal factors) and a R-matrix (superior triangular matrix) avoids the singularity in ordinary least squares inversion (Press et al., 1992). This matrix decomposition gives linear regression coefficients with values as small as they can be, in contrast to the ordinary least squares inversion (Press et al., 1992).
Since the model considered is based on the spectral amplitude for each frequency, it is necessary to detail the discretization and refinement process. Initially, S-wave time windows are selected according to the criteria of Torres-Sánchez (2022), who, when studying attenuation in the GoC, determined optimal windows dependent on the epicentral distance (repi) of each record (Table 3). Additionally, a variable-duration pre-event noise window was selected for each record, to later estimate the signal-to-noise ratios and site effects considering spectral ratios using the S-wave windows (e.g., Lermo and Chávez-García, 1993; Nakamura, 1989).
The time-window selected was cosine tapered with a 5% duration and Fast Fourier Transform (FFT) was calculated to obtain the amplitude spectra for the three components between the frequency range from 0.1 to 50 Hz. Since the records are sampled at 100 samples per second, 50 Hz is the maximum available value of Nyquist frequency. Thirty-one frequencies were selected for further analysis spaced every 0.1 Hz in the logarithmic space.
The spectral amplitudes were smoothed using a moving average of 1/3 of an octave centered on the previously selected frequency value, aiming to preserve energy and reduce spectral variance, as stipulated by Parseval’s theorem (Hsu, 1989). Later, the smoothed amplitude spectra were discretized in 30 frequency bands using the Slog2 routine (Press et al., 1992), which divides the spectra and centers every band around the pre-stablished frequencies. The continuous amplitude spectra values are weighted point by point, resulting in amplitudes that are then averaged to obtain the representative value of the band, which preserves the energy of the original continuous spectrum of the selected time window.
From the resulting discretized spectra, we selected a frequency range from 0.32 Hz to 25.12 Hz. Since the noise windows also underwent the previous processing, S/N ratios were calculated, and only amplitudes greater than or equal to 2.0 were included in the database. Because both horizontal ground-motion components are considered, the vector sum of the E-W and N-S of the discretized components was calculated to obtain the horizontal resulting amplitude, as shown in Equation 5.
Finally, the database for each of the 20 frequencies was conformed, containing for each earthquake i, the epicentral distances (rij), the Mw magnitude, the discretized horizontal amplitude (Uij) and the station code.
The site effects were evaluated for each of the 20 frequencies following the spectral method proposed by Lermo and Chávez-García (1993), using the same events that are indicated in Table 2 and the same S-phase window spectra for the database elaboration process. These site effects values are specific for each of the eight stations, and they were used for correcting the discretized spectral amplitudes to obtain the equivalent rock response, using the resulting average horizontal to vertical (H/V) spectra as a transfer function.
We calibrated the model with two approaches. Firstly, we generated a synthetic database using the coefficients reported by Castro (1998) for 1.00 Hz to 19.95 Hz frequency range obtained for the northern Baja California, Mexico. With these coefficients and 15-random selected earthquakes, a synthetic amplitude spectrum was computed for each of the eight stations (120 observations in total), even if the synthetic value was not included in the observed records. The inversion results were verified by obtaining the exact coefficients proposed by Castro (1998) in the data inversion. The second calibration approach of the two-step model was through the exclusion of one event from the database (e.g., Castro, 1998; McGuire, 1978), where the resulting empirical model must provide a congruent adjustment to the observed spectral amplitudes at each of the eight stations selected. In this case, we exclude event #28 (see Table 2), being the only earthquake greater than 4.5 Mw magnitude registered by all the stations.
The two-step empirical models, in both the linear and quadratic cases, were compared with the models proposed by Bayless and Abrahamson (2019); Castro et al. (1988); McGuire (1978) applicable to other regions or tectonic regimes across the world. Table 4 summarizes the applicability for each model. Finally, the best model was evaluated by computing the empirical horizontal amplitude spectra for events from the database at instrumented and non-instrumented sites, considering the eight stations.
Among the study limitations, it is noted that the functional form of the GMPE and its coefficients are specific to the NRGoC and are valid for epicentral distances less than 305 km, as the model may overestimate or underestimate the spectral response at greater distances. Smaller events may not generate sufficient energy to be classified as strong motions or may not even be detected in some stations, for instance, Castro et al. (2021) estimated that the minimum magnitude of completeness (Mc) is equal to 3.6 in the GoC. Larger events may generate extensive rupture areas that could be considered non-point sources.
4. Results and discussion
The resulting coefficients from the inversion process for both cases of the two-step model are shown in Figures 4a and 4b, respectively. Values are shown in Table 5 with their respective estimated errors.
The seismic site response of the stations used is shown in Figure 5, including one standard deviation and the number of S-windows used to compute the H/V ratio, according to Lermo and Chávez-García (1993) method. This specific function allows us to correct the spectral amplitudes from the database frequency by frequency, as measured at the surface to rock-response, assuming that the vertical component is not amplified, and the surface behaves linearly for the far-field. Considerable variance was noted for the DOCTX station, possibly related to azimuthal coverage, radiation pattern of events, number of registers and site type (the only one located at soft materials).
From the magnitude and distance scaling (Figures 6 and 7), we also evaluated the performance of the linear and quadratic two-step models. Firstly, there is a notorious anomalous magnitude scaling for the quadratic two-step model in the frequency range from 0.40 to 1.26 Hz and a fair performance for the linear case for the whole magnitude range in the database. Secondly, a congruent spectral decay for rock response in both models (Figure 7) was evident for a distance range between 1 to 350 km, with inconsistencies for 15.85, 19.95 and 25.12 Hz, because for these frequencies at distances greater than 107.0 km their spectral amplitudes increase due to the influence of the noise. This inconsistency at high frequencies in the distance scaling influences the magnitude scaling as well (Figure 6), in which we observed an amplitude peak for hypothetical spectra for both linear and quadratic cases of the two-step model, if the distance is set to be greater than 107.0 km. This behavior in both models could be explained as high-frequency noise predominance (i.e., needing to increase the S/N to 3.0, for example) and the number of observations considered in the regression process, that clearly decreases from 10.00 Hz to less than one hundred (see Figure 3).
Comparing the magnitude and distance scaling between the empirical fitting of both models (linear and quadratic) to observed data, the difference between the northern (DOCTX, PLIB, PPXB, SFX and SLGB) and the southern stations (BAHB, BKIRB and SFQB) around the 29.5° parallel is significant. For southern station the models perform poorly, especially for frequencies beyond 10.00 Hz, where the linear and quadratic models tend to overestimate the amplitude spectra for distances less than 100.0 km and underestimate the amplitude for distances beyond 200.0 km. Taking this in consideration, the spectra from events calculated beyond 107.0 km had to be restricted from 0.32 Hz to 10.00 Hz. Likewise, for northern stations there was a clear improvement in the data fit that could be related to more observations in this region in comparison to southern stations, the azimuth from the energy arriving and path effects such as geological heterogeneity, crustal thickness, rock quality or faulting density. This difference was suggested by several authors (e.g., Torres-Sánchez and Castro, 2023) and is also evident for estimating seismic potential, because no greater than 6.0 Mw magnitude earthquake has occurred in the NRGoC between 1901 to present (Castro et al., 2021), in contrast with the central region of the GoC (Canal de Ballenas region), where great earthquakes are more common.
The data fit is presented for a sample of stations and various distance range in Figure 8, considering other GMPE focused on amplitude acceleration spectra as a dependent variable, such as in Bayless and Abrahamson (2019) and McGuire (1978) applicable for cortical regions, and Castro et al. (1988) one-step model for subduction zone earthquakes. The main purpose of the comparison is to evaluate the pertinency of the models proposed in this research derived from Joyner and Boore (1981) GMPE in contrast to data mismatch for global models of other regions, in some cases predicting an horizontal amplitude spectra equal or greater than one magnitude order. MG-78 model was proposed for a different crustal region and Castro et al. (1988) model was proposed for estimating FAAS from subduction zone earthquakes at distances greater than 280 km obtained using a single station in rock and has an implicit assumption for a simple path effect and crust thickness variation.
Even considering a robust global model such as Bayless and Abrahamson (2019), that considers parameters such as Vs30, style of faulting, and complex representation of the source and path, it is not guaranteed the data would fit the global model for the bandwidth of interest. At certain ranges of frequencies, the FAAS predicted by the BA-18 model tends to fit better, that is the case for stations BKIRB and SFQB for frequencies less than the corner frequency, but not for DOCTX station, where amplitude is underestimated (rock response) but it is also fair to point out, that H/V site function could strongly influence the predicted response, due to its wide variance. Parameters for the BA-18 model were set according to site geology and hypocentral localization.
A residual analysis was conducted to evaluate which of the two-step models gives the best performance (Figure 9). We selected the 1.00 Hz frequency residuals (Figure 9a) to show the data dispersion in respect to the distance and magnitude, noting that it is complex to evaluate which model behaves the best along the distance and magnitude range. Therefore, the mean and standard deviation of the absolute value of the residuals were computed to ease the best model selection (Figure 9b), highlighting that the significant difference noted at lower frequencies (from 0.32 to 2.00 Hz) and the similarities at higher frequencies stand for most of the twenty frequencies of interest. In general, the linear and quadratic models have resemblances in terms of the root mean square error (RMSE), being more evident the difference at lower frequencies (i.e. < 1.00 Hz), in which the quadratic two-step model gives minor values of RMSE (Figure 9c). For higher frequencies (i.e. > 1.00 Hz), the RMSE tends to increase as well for both models being almost the same, but the quadratic case gives the minor RMSE value. Error values from MG-78, CA-88 and BA-18 models were included in Figure 9c to contrast the proficiency in the data available for the NRGoC, regression techniques and model performance.
Since, it was evident the deficiency for the quadratic two-step model in scaling the magnitude (Figure 6), we determined that between 0.32 Hz to 1.26 Hz range, the quadratic two-step model does not make physical sense and the linear two-step model was selected as the optimum model for the region, giving that is a simpler model, has physical congruency in representing the source, and gives a congruent distance scaling, at least for frequencies less than 12.59 Hz (see Figure 7).
A seismic scenario was performed for a single event located at the same site as the 5.3 Mw March 28th, 2016, event (event #25 from Table 2) considering the linear two-step model to forecast its effects in the NRGoC, in terms of the horizontal acceleration spectra (Figure 10) and including observed data where it was available. The empirical model predicts horizontal acceleration spectra with a plateau shape, as expected, and works properly for the selected magnitude at least for 0.32 Hz to 10.00 Hz bandwidth. Near to the source (SLGB site for instance), the amplitude spectra has greater amplitude at the plateau bandwidth but faster decaying for frequencies greater than 5.01 Hz. We also noticed strong attenuation for distances greater than 50 km, because the noise tends to dominate the amplitudes, and influencing the regression results for higher frequencies (> 10.00 Hz) as mentioned above (refer to Figures 6 and 7). It is plausible that if we had considered an event of lesser magnitude (for example, 4.5 Mw or less), the amplitude predicted for distant stations (around 200 km or more) could be smaller than the station limit detection due to attenuation along the path and noise could have greater influence on the amplitudes obtained.
The difficulty in the data fit for the north and south stations in the region could be related to geological, path and source complexity, with a clear differentiation between the northern sector of the GoC with the central section (around the parallel 29.5° N), known as Canal de Ballenas region (Castro et al., 2011; González-Fernández et al., 2005; Torres-Sánchez and Castro, 2023). These differences result in regional attenuation and a type of seismicity that has an impact in seismic potential (Castro et al., 2021; Cruz-Hernández et al., 2023; Torres-Sánchez, 2022) and were not considered in the analyzed models. Thus, the two-step models in their linear or quadratic cases considered a mean regional value for the attenuation of each frequency analyzed, and as well, stations with more observations will weigh more in the estimation of regression coefficients, being path effect the factor of the seismic phenomena over which there is less control in empirical models, if compared to source and site effect.
5. Conclusions
A simple model to estimate FAAS for the NRGoC based on Joyner and Boore (1981) regression technique was proposed, considering S-wave windows and being the first study of this kind for this complex tectonic region. The importance of promoting a model based on the FAAS as a response parameter lies in the fact that this type of model is closer to the physics of strong motion and allows for better control over stochastic simulations (Meenakshi et al., 2023), in contrast to the majority of models that estimate peak parameters (PGA or PGV) or engineering parameters (e.g., PSA). Furthermore, Bayless and Abrahamson (2019) highlight a linear response of the medium for the entire range of frequencies.
When the site functions are known or can be estimated independently, it is possible to remove the site response simplifying the regression process by removing the bimodal c term in the original Joyner and Boore (1981) method. Having the specific site transfer function, the expressions proposed in this research can be applied and give a more realistic model response; in this case we used H/V ratios from phase S windows (i.e., Lermo and Chávez-García, 1993) as a proxy of the site effect. We propose the linear two-step model as optimal to estimate the mean horizontal Fourier acceleration spectra in the northern GoC, at least for the frequency range from 0.32 Hz to 10.00 Hz. If a spectral amplitude needs to be performed for higher frequencies (> 10.00 Hz) other approaches could be used, such as including kappa corrections as suggested by other studies in the region (e.g., Azua et al., 2024; Castro and Ávila-Barrientos, 2015). The quadratic two-step model tends to give lower values of error, but did not have physical sense for the magnitude scaling between 0.40 to 1.26 Hz. The two-step models consider a regional mean value for the attenuation, being better the fit to data for the northern stations (DOCTX, PLIB, PPXB, SFX and SLGB) respect to southern ones located near the Canal de Ballenas region (BAHB, BKIRB and SFQB). This behavior in the data could be explained by the distribution of records azimuthal range, geological heterogeneities, stress regimes, crustal, and faulting density differences marked around the 29.5° N parallel (Cruz-Hernández et al., 2023).
Other models such as Bayless and Abrahamson (2019), Castro et al. (1988) and McGuire (1978) do not improve the data fit significantly because these are models applicable to different regions (e.g., MG-78 and CA-88), even if considering a widely range of magnitudes or similar distance range, robust model structure and global data (case of BA-18 model). The two-step models improve the data fitting in the region for most distance and magnitude combinations, given the seismic potential features of NRGoC, where no greater than 6.0 magnitude earthquake has occurred, at least for the period from 1901 to the present day (Castro et al., 2021).
Some difficulties were noted in linear and quadratic models, especially at higher frequencies (> 10.00 Hz), where amplitude spectra had to be limited up to 10.00 Hz for sites located at distances greater than 107.0 km. For near sites (r ≤ 107.0 km) better fits were obtained if the station has considerable azimuthal range (PLIB, PPXB, SFX and SLGB), being southern stations (BAHB, BKIRB and SFQB), being the more difficult to adjust since both models tend to underestimate spectral amplitudes.
It is plausible to estimate the rock response for near and distant sites (> 250 km). Using the mean model response (i.e., 50th percentile) plus one standard deviation gives a reliable spectral response; therefore, we considered it as a conservative statistic for seismic scenarios, regardless of the magnitude considered.
Supplementary data
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
Contributions of authors
(1) Conceptualization: RRC, JSA; (2) Data analysis or acquisition: JSA; (3) Methodological or technical development: JSA, RRC; (4) Drafting of the original manuscript: JSA; (5) Drafting of the revised and edited manuscript: JSA, RRC; (6) Graphic design: JSA; (7) Interpretation: JSA, RRC; (8) Funding: RRC; (9) Other contributions: RRC (supervision).
Funding
This work was supported by the Mexican Secretary of Science, Humanities, Technology and Innovation (SECIHTI).
Acknowledgments
This research was founded by the Mexican National Council of Humanities, Science and Technology (CONAHCYT). Antonio Mendoza and Arturo Perez Vertti provided technical support along this study. We also thank Óscar Castro Artola, Luis Yegres and Alejandra Núñez for the provided data for DOCTX and SFX stations from the RESNOM network. Authors acknowledge the reviewers Dr. F. Ramón Zúñiga and Dr. Gina P. Villalobos Escobar for their valuable feedback, which helped enhance the manuscript.
Conflicts of interest
The authors have no relevant financial or non-financial interests to disclose. Authors declare no potential conflicts of interest.
Handling editor
Carlos Alberto Vargas Jiménez.
References
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