A far-field ground motion prediction model for interface earthquakes at the hill zone of Mexico City

Abstract 

This study presents an updated attenuation model to predict the peak ground acceleration (PGA), peak ground velocity (PGV), 5% damped pseudo-spectral acceleration (SA), and the average spectral acceleration (AvgSA) at the hill zone of Mexico City for interface earthquakes. The strong-motion dataset comprises 33 earthquakes recorded at CU station, covering a moment magnitude (M_w) range from 6.0 to 8.1 and a source-to-site distance (R_rup) range from 240 to 490 km. Given the small number of available observations, a Bayesian regression scheme is used to obtain the coefficients of the ground-motion prediction model (GMPM). In addition, the epistemic uncertainty in the estimation of the regression coefficients is evaluated, showing its impact on the framework of a probabilistic seismic hazard analysis (PSHA). The results are compared with models previously developed for the CU station, discussing the differences observed between the median predictions and their standard deviations. Likewise, seismic hazard curves are computed and compared with empirical curves obtained by counting the number of times per year that a given value of ground-motion intensity is exceeded. The results show that the dispersion of the GMPM proposed is lower than the previous models for PGA and SA, which means better predictability and more reliable estimates of the seismic hazard at the site.

1 Introduction 

Mexico City, located more than 280 km inland from Mexico’s Pacific Coast, has been affected by several interface earthquakes throughout its history, which have caused a considerable number of casualties and severe damage to the civil infrastructure. Some examples of these major events are (1) the 1957 San Marcos earthquake (M 7.5) that damaged more than 1000 buildings and claimed the life of 39 people (Orozco and Reinoso 2007); (2) the 1979 Petatlán earthquake (M_w 7.4) that damaged more than 50 structures and caused the collapse of three buildings (Alonso et al. 1979); and (3) the 1985 Michoacán earthquake (M_w 8.1) that produced the collapse of 256 buildings and more than 10,000 deaths (Astiz et al. 1987), being the latter the most destructive earthquake in Mexico City’s history whose economic losses amounted to US$ 4 billion (at 1985 value) according to Mexican Insurance Association. The damage pattern observed in these interface events was attributed to the well-known local site effects in the Mexico City basin (MCB) (Meli and Miranda 1985). The ground motion amplification and duration at MCB are mainly attributed to the elastic behavior of the clayed soil deposits (Diaz-Rodriguez 1989; Díaz-Rodríguez 1992; Romo and Ovando-Shelley 1996; Taboada-Urtuzuástegui et al. 1999; Díaz-Rodríguez and Santamarina 2001) whose parameters, shear modulus and damping ratio, do not show a significant reduction at high values of shear strain. This condition amplifies seismic waves in these deposits (Mayoral et al. 2008, 2016, 2019).

In the early 1960s, the Mexico City Accelerometric Network (MCAN) began operating to study the dynamic amplification of the seismic waves in the lakebed deposits. Nowadays, the MCAN has over 149 strong-motion stations distributed throughout the city at three different geotechnical zones: zone I (hill), formed by volcanic tuffs and lava flows; zone II (transition), composed of 20 m sand and silt layers interbedded with lacustrine clay layers; and zone III (lakebed), consisting of 30–80-m clay deposit highly compressible interbedded with sand layers (Mayoral et al. 2019). The station Ciudad Universitaria (herein, CU), located within the hill zone of the National Autonomous University of Mexico (UNAM, by its Spanish acronym), is one of the most important and oldest stations of the MCAN. It has been in continuous operation since 1962 and has become the reference station to study the site effect at the lakebed zone (Ordaz et al. 1988; Singh et al. 1988a; Lermo and Chávez-García 1993; Reinoso and Ordaz 1999; Montalvo-Arrieta et al. 2002). Furthermore, its information has helped develop and calibrate the early earthquake damage assessment system available for the management of seismic risk in the city (Iglesias et al. 2007; Ordaz et al. 2017).

Since the 1980s, several studies have been conducted to predict the ground motion at the hill zone of Mexico City from interface earthquakes recorded at the station CU. For instance, Singh et al. (1987) used a set of 16 earthquakes (5.6 ≤ M_s ≤ 8.1 and 282 ≤ R_rup ≤ 466) recorded at this station to derive a ground-motion prediction model (GMPM) in terms of the peak ground acceleration (PGA) and velocity (PGV). Based on the same strong-motion dataset, Castro et al. (1988) developed a further GMPM to estimate the Fourier amplitude spectra (FAS) of the ground acceleration for frequencies between 0.4 and 5.0 Hz. This last model was improved by Ordaz et al. (1994) using more records and applying a Bayesian regression approach instead of the multiple linear regression (MLR) method used in the previous works. Years later, Reyes (1999) employed 17 earthquakes (6.1 ≤ M_w ≤ 8.1 and 280 ≤ R_rup ≤ 466) to derive a new GMPM for PGA and the 5% damped pseudo-spectral acceleration (SA) for 60 structural periods between 0.1 and 6.0 s applying the Bayesian regression approach proposed by Ordaz et al. (1994). This latest is updated by Jaimes et al. (2006) adding to the dataset 5 events that occurred between 1996 and 2004 (6.0 ≤ M_w ≤ 7.5 and 301 ≤ R_rup ≤ 526) and deriving the GMPM in terms of the PGA and SA for 30 structural periods between 0.2 and 6.0 s. These latest two GMPMs have been used over time to perform site-specific probabilistic seismic hazard analysis (PSHA).

In particular, the GMPMs proposed by Reyes (1999) and Jaimes et al. (2006) (herein referred as to R99 and J06, respectively) have been used in the current Mexico City Standard for Seismic Design (NTC-DS 2017) to define the design spectrum at the hill zone based on the uniform hazard spectrum (UHS) obtained at CU for 250-year return period. This UHS has been used as a reference to estimate seismic hazard and design spectra at other city sites through the spectral amplification functions pre-computed between soft sites and the reference station CU (Ordaz 2017). Despite the good performance of R99 and J06, in terms of their predictive capability of ground motion for Mexican interface earthquakes, it is essential to update the database to consider the moderate earthquakes (6.0 < M_w < 7.4) that have occurred during 2004–2021. Furthermore, it is necessary to evaluate other intensity measures (IMs) that better represent the ground motion’s severity and be a good predictor of the structural response.

This study presents an updated GMPM for the hill zone in Mexico City, based on a dataset from interface earthquakes recorded at station CU with a moment magnitude range (M_w) from 6.0 to 8.1 and a source-to-site distance range (R_rup) from 240 to 490 km. Four IMs are used to characterize the ground motion: peak ground acceleration (PGA), peak ground velocity (PGV), 5% damped pseudo-spectral acceleration (SA), and the average spectral acceleration (AvgSA). The results are compared with previously derived GMPMs for SA, discussing the differences observed between the predicted median response spectra and their residual distribution. Finally, the GMPM derived is used to compute seismic hazard curves and compare the results with empirical curves obtained directly from counting the times a given IM value has been exceeded per year. The seismic hazard curve directly estimated from GMPM-AvgSA is compared with the exceedance rate of AvgSA values derived indirectly from existing GMPM-SAs (R99 and J06) using a consistent SA inter-period correlation model proposed by Rodríguez-Castellanos et al. (2021).

2 Strong-motion dataset

The strong-motion dataset used in this study corresponds to the Mexican interface earthquakes recorded at the station CU, located within the hill zone of Mexico City. The dataset includes 33 record pairs (two horizontal components) from 33 reverse-faulting events with M_w > 6, which occurred between 1965 and 2021 along Mexico’s Pacific Coast. The small events (M_w < 6) were excluded from the dataset because their ground-motion intensity levels (e.g., PGA < 1 cm/s²) are not representative of most engineering interests. Table 1 summarizes the seismic source parameters (i.e., location, magnitude, and focal mechanism) of the events analyzed. These parameters were taken from the recently published earthquake catalog by Sawires et al. (2019), which includes an extensive and careful review of all Poissonian independent earthquakes that occurred in Mexico during a period ranging from 1787 to 2018. The source parameters of the events that occurred after 2018 were obtained from the earthquake reports published by the Servicio Sismológico Nacional (SSN (Mexican National Seismological Service)) through its website (see SSN (2022)). Figure 1 shows the epicenters and focal mechanism of the dataset used, covering a magnitude range from 6.0 to 8.1 and a source-to-site distance range from 240 to 490 km.

Table 1 Reverse-faulting interface events with M_w > 6 recorded at CU station

Fig. 1

Location of the epicenters and focal mechanisms of earthquakes recorded at station CU (filled triangle) in Mexico City. The dashed line represents the boundary of the Trans-Mexican volcanic belt (TMBV). The epicentral data and source parameters are listed in Table 1

2.1 Local site conditions

According to the geotechnical zoning of the NTCDS (2017), the hill zone is formed by rocks and hard soils where sand deposits may be interbedded with volcanic flows. Based on this definition, it could be assumed that the local site conditions of CU correspond to firm soil. However, several studies carried out at this site (Singh et al. 1988b, 1995; Ordaz and Singh 1992; Montalvo-Arrieta et al. 2002, 2003) have shown the existence of a large ground motion amplification in a range of frequencies between 0.2 and 10 Hz. Montalvo-Arrieta et al. (2003) report that the spectral ratios of stations located in the southwestern hill zone (including CU) exhibit relative amplifications up to 4 times larger than stations located in the northern for frequencies between 0.2 and 3 Hz. Ordaz and Singh 1992 suggest that this amplification can be related to the shallow (< 1 km) sediments that lie below the volcanic rocks that cover the hill zone of the Valley of Mexico. As regards, several studies have been conducted near station CU to determine the shear wave velocity profile (see Singh et al. 1995; Kagawa 1996; Flores Estrella and Aguirre González 2003) through geophysical prospecting techniques, whose results indicate that the site can be classified as a soft rock with an average shear wave velocity in the upper 30 m (V_S 30) of 450 to 760 m/s.

2.2 Source-to-site distance

The source-to-site distance selected to develop the GMPM was the closest to the rupture plane (R_rup). This distance was computed following the procedure presented by Chiou and Youngs (2008), simulating the rupture dimensions for each event through the scaling relationship derived by Ramírez-Gaytán et al. (2014) for Mexican subduction earthquakes. For eight of the most significant events (M_w > 7) in the dataset, the distance was inferred directly from the finite-fault geometry obtained by kinematic inversion, whose information was taken from the SRCMOD database compiled by Mai and Thingbaijam (2014).

Figure 2 shows the magnitude-distance (left panel) and magnitude-focal depth (right panel) distributions for the compiled dataset (hereafter, CU21). As can be seen, CU21 covers a magnitude range from 6.0 to 8.1 and a source-to-site distance range from 240 to 490 km. Something to highlight in this figure is the low sample size of the dataset for magnitudes and distances beyond 7.5 and 350 km, respectively. This lack of information can be compensated through prior theoretical knowledge about certain seismological parameters related to the scaling of the ground motion intensity with the magnitude and distance, whose prior information can be formally integrated into the regression process using a Bayesian approach, obtaining reasonable results as it is shown by Ordaz et al. 1994; Jaimes et al. 2006; Wang and Takada 2009. Notably, the last-mentioned work indicates that using prior information in a Bayesian approach is particularly effective when the dataset is small.

Fig. 2

Magnitude-distance and magnitude-depth distributions of the dataset used in the regression analysis

3 Ground-motion intensity measures

The intensity measures (IMs) commonly used to derive a GMPM are peak ground acceleration (PGA), peak ground velocity (PGV), and spectral acceleration (SA). However, several studies (Giovenale et al. 2004; Luco and Cornell 2007; Eads et al. 2015; Kohrangi et al. 2016a, b) have demonstrated that these IMs used in performance-based assessments can be inefficient and insufficient to predict the structural demand, especially for tall or long-period buildings whose higher modes contribute significantly to the structural response. This condition has led to the development of new IMs better correlated with the structural response of nonlinear systems, such as the average spectral acceleration (AvgSA). Particularly, Eads et al. (2015) define AvgSA as the geometric mean of SA ordinates in the range of 0.2T₁–3T₁, where T₁ is the fundamental period of vibration of the structure. In Eads et al. (2015), the authors used almost 700 buildings of different heights (4 to 16 stories) and structural systems (moment-resist frame and shear wall structures) to evaluate the efficiency and sufficiency of AvgSA as IM to predict the collapse intensities. Their results suggest that AvgSA is a better option than SA since it reduces the variability of the predicted structural demand and the IM dependency with the ground-motion characteristics, such as magnitude (M), source-to-site distance (R), and fault mechanism.

On the other hand, Kohrangi et al. (2017) mention that selecting the upper period limit (T*) to compute AvgSA is not a trivial matter since it depends on the objective of the analysis. In that study, the authors recommend a value of T* equal to 1.5 T₁ or 2 T₁ to predict the peak inter-story drift ratio (IDR) based on the results obtained in (Kohrangi et al. 2016a, b), in which the nonlinear response of three reinforced concrete buildings (3, 5, and 8 stories) subjected to both horizontal components of the ground motion are evaluated. Its results show that using T* = 1.5T₁ gets the lowest dispersion in response estimation of the IDR, which engineering demand parameter (EDP) correlates well with the collapse state. Likewise, they show the efficiency and sufficiency of predicting the IDR using AvgSA [0.2T₁–1.5T₁].

Based on the above, it was decided to include the AvgSA as an IM to derive the GMPM. The intention is not to replace SA as IM to define design loads in the standard for seismic design but rather to have an IM better correlated with the nonlinear structural behavior that allows for obtaining more accurate seismic risk estimations. Therefore, it was decided to keep the typical IMs used in previous works to derive the GMPM such as PGV, PGA, and SA, which metrics have been used with different objectives over time (e.g., seismic risk assessment, calibration of the early warning system, obtaining the design spectrum, among others).

In this study, the AvgSA is estimated for 40 periods, T₁, equally spaced between 0.1 s and 4 s, considering a period range of 0.1T₁–1.5T₁ and a uniform spacing (Δ) of 0.1 in the averaging process through the Eq. (1). This period range corresponds to the one proposed in the NTCDS (2017) to evaluate the nonlinear response of tall buildings in Mexico City. Therefore, the GMPM-AvgSA derived here could be used to obtain the conditional mean spectrum (CMS) and select acceleration time histories that can be used in dynamic analysis.

$$\mathrm{AvgSA}\left({T}_{1}\right)={\left[\prod_{i=1}^{N}\mathrm{SA}\left({C}_{i}{T}_{1}\right)\right]}^{{~}^{1}\!\left/ \!{~}_{N}\right.}$$

(1)

where SA stands the 5% damped spectral acceleration value, N is the number of periods used to compute AvgSA, in this case, N = 15; C_i is the ith coefficient of the period range used in the averaging process, ${C}_{i}=i\Delta$ and T₁ is the fundamental period of vibration of the structure.

Likewise, three IMs were used to characterize the ground motion: PGV, PGA, and SA. For these IMs, the quadratic mean (QM) of the maximum response of both horizontal components was computed to derive the GMPM, that is, SA_QM = [(SA_N² + SA_E²) / 2]^0.5. The QM was selected to be consistent with the definition adopted in other GMPMs developed for Mexico City (e.g., Reyes 1999; Jaimes et al. 2015) and used in the PSHA. For SA, the 5% damped response spectra at 102 structural periods between 0.05 s and 6 s were computed. The processing of the acceleration time histories used in the analysis is described as follows.

4 Data processing and analysis

4.1 Data processing

The records were processed applying a baseline correction and high-pass filter using two values of corner frequency (f_hp) depending on the type and resolution of the recording instrument: (1) f_hp = 0.30 for the events recorded at analog and digital accelerographs with low resolution (< 16 bit) and flat response in the range of 0.1 to 35 Hz and (2) f_hp = 0.05 for the events recorded at modern digital accelerographs. The minimum usable oscillator frequency (f_o) to compute SA was determined for each record as 1.25 times the corner frequency of the high-pass filter (Abrahamson and Silva 1997; Ancheta et al. 2014; Bindi et al. 2014; Kohrangi et al. 2018). This condition implies that for AvgSA the lowest usable f_o on the data selection is constrained by the upper limit of the period range 0.1T₁–1.5T₁ selected, which corresponds to 1.875 f_hp. Thus, the number of record pairs in the final dataset varies as a function of the oscillator period, ${\text{T}}$ (where T = 1/f_o), including a maximum of 33 strong-motion records to estimate SA and AvgSA for a T ≤ 2.7 s and T ≤ 1.8 s, respectively, and 24 to evaluate the response for both IMs at longer periods.

The PGA value was read directly from each record processed, and the 5% damped SA for 102 structural periods between 0.05 s and 6 s was computed. The PGV values were obtained by integrating the acceleration-time histories after applying the baseline correction and high-pass filter. For these IMs, the quadratic mean of the maximum response of both horizontal components was computed to derive the GMPM. Finally, the AvgSA values were estimated for 40 periods, T₁, equally spaced between 0.1 s and 4 s using Eq. (1).

4.2 Regression analysis

The regression of the dataset was performed using the Bayesian approach proposed by Ordaz et al. (1994), which was considered adequate to obtain the GMPM given the small number of available observations. Besides, it is not possible to apply a two-step or random-effect approach because the observation comes from a single station. The Bayesian approach makes it possible to incorporate theoretical knowledge about the phenomenon being studied to constrain the values of some seismological parameters that cannot be directly derived from the dataset. This prior information, combined with the empirical data through the Bayes theorem, provides a rational and stable numerical solution, as shown by Ordaz et al. (1994), Jaimes et al. (2006), and Wang and Takada (2009). The functional form adopted in the regression analysis to develop the GMPM is shown in Eq. (2).

$$\mathit{ln}Y={c}_{0}+{c}_{1}\left({M}_{W}-6\right)+{c}_{2}{\left({M}_{W}-6\right)}^{2}+{c}_{3}\mathit{ln}R+{c}_{4}R+\varepsilon$$

(2)

where Y is the predicted median of a ground-motion parameter: PGA, PGV, SA, or AvgSA. The predicted values for acceleration and velocity are in cm/s² and cm/s, respectively. M_w is the moment magnitude; R_rup is the closest distance to the rupture plane in kilometers; c_i are the coefficients to be determined by the regression analysis, and $\varepsilon$ is the random error normally distributed with zero mean and variance σ_e². In this case, being a single-station analysis, σ_e² represents the variance of the interevent residuals (Atkinson 2006). Note that SA and AvgSA coefficients depend on the fundamental vibration period of the structure, T; however, for simplicity, this term has been removed from Eq. (2).

The prior information is an intrinsic part of a Bayesian approach and reflects our state of knowledge about the phenomenon analyzed prior to the observation and analysis of data. In this study, the prior probability distributions of the coefficients, c_i, are obtained from the ground-motion intensity (GMI) values estimated theoretically at CU without using the strong-motion recordings. For that purpose, a far-field point source model proposed by Singh et al. (1989) and the random vibration theory (RVT) was used to compute the Fourier amplitude spectrum (FAS) and predict the GMI in terms of PGA, PGV, and SA. The AvgSAs were obtained from the SAs using Eq. (1). The frequency-dependent attenuation parameters required to compute the FAS were taken from Ordaz and Singh (1992). Likewise, the strong-motion duration parameter used in the RVT method (Liu and Pezeshk 1999) was computed through the relationship proposed by López-Castañeda and Reinoso (2021). The expected values, E′ [c_i], and standard deviations, σ′ [c_i], of prior coefficients, were established following the procedure suggested by Reyes (1999), whose steps are briefly described below.

• 1. The prior expected values of c₀, c₁, and c₂ were estimated by applying a least squared fit to GMIs predicted for a specific source-to-site distance (R_rup = 290 km) and a magnitude (M_w) range between 6.0 and 8.2. This procedure is considered adequate since, as shown in Eq. (2), these coefficients are independent of the distance; c₀ represents the site-effects amplifications, while c₁ and c₂ control the magnitude scaling. The standard deviation of c₁ and c₂ was assigned according by Ordaz et al. (1994), adopting σ′ [c_i] = E′ [c_i] / 1.7.

• 2. The prior expected value of c₃ was fixed at -0.5 for all periods. This value is consistent with the geometrical attenuation of surface waves at source-to-site distances greater than 100 km (Ordaz et al. 1992). A low standard deviation was assigned (σ′ [c₃] ∝ 0) to ensure that the value remained constant after incorporating the observations.

• 3. The prior expected value of c₄, related to the quality factor Q (anelastic attenuation), was obtained as follows: (1) a magnitude value was set, M_w = 7.0; (2) for this value, the GMI was computed for two different source-to-site distances, R₁ = 260 km and R₂ = 480 km; at last (3) the ratio between GMI_S estimated, which allows combining the effect of the geometrical spreading and the anelastic attenuation (Castro et al. 1990), was determined by the following equation. A sufficiently large variance was adopted for this coefficient to obtain the best value from the regression, σ′ [c₄] = E′ [c₄] / 1.7.

  $${E}^{^{\prime}}\left[{c}_{4}\right]=\frac{\mathrm{ln}\left({Y}_{1}/{Y}_{2}\right)-{E}^{^{\prime}}\left[{c}_{3}\right]\mathrm{ln}\left({R}_{1}/{R}_{2}\right)}{{R}_{1}-{R}_{2}}$$

  (3)

• 4. Finally, the prior expected value of the σ_e model was assigned based on the results reported in several studies (e.g., Joyner and Boore 1988; Ordaz et al. 1994; Jaimes et al. 2006), adopting E′ [σ_e] = 0.7 and σ′ [σ_e] = E′ [σ_e] / 1.7.

The posterior expected values of E″ [c_i] and E″ [σ] were obtained through a Bayesian linear regression using the optimization algorithm provided by Salvatier et al. (2016).

4.3 Uncertainty in the regression coefficients

As mentioned earlier, the dataset used in the regression analysis presents magnitude-distance regions poorly sampled, particularly for M_w > 7.5 and R_rup > 350 km (see Fig. 2). This lack of data raises concerns regarding the reliability of GMPM when used to predict the GMI value for events whose parameters magnitude and distance are beyond the well-sampled region of the dataset. Therefore, in this study, it was decided to use the method proposed by Arroyo and Ordaz (2011) to quantify how the uncertainty associated with GMPM is affected due to the quality of the dataset. This method evaluated the epistemic uncertainty using the predictive variance, σ_p², which depends on the sample, on the regression technique, and on the adopted functional form. Equation (4) shows that the σ_p² of a forecasted value depends on two factors: (1) the earthquake-to-earthquake variability, σ_e², contained in the database and (2) the uncertainty in the regression coefficients given by $\text{Z COV}\left(\widehat{\text{c}}\right) \, {\text{Z}}^{\text{T}}$.

$${{\upsigma }_{p}}^{2}={{\upsigma }_{e}}^{2}+Z COV\left(\widehat{c}\right) {Z}^{T}$$

(4)

where σ_p² is the total forecasted variance considering the extra uncertainty due to the quality of the dataset, σ_e² the variance of the interevent residuals, Z is a n_p row vector with the GMPM parameters (see Eq. 2) for which a specific value, Y, is being forecasted and $COV\left(\widehat{c}\right)$ is the covariance matrix of the coefficients given by,

$$\mathrm{COV}\left(\widehat c\right)=\frac1{n_o-n_p-2}\left[\left(X^T\Phi^{-1}X\right)^{-1}\left(\left(Y-X\widehat c\right)^T\Phi^{-1}\left(Y-X\;\widehat c\right)\right)\right]$$

(5)

where X is a ${n}_{o}\times {n}_{p}$ matrix with ${n}_{o}$ observations and ${n}_{p}$ parameters considered in the model, Y is a ${n}_{o}\times 1$ matrix with the predicted values for the parameter given by X, $\widehat{c}$ is a ${n}_{p}\times 1$ matrix with the coefficients estimated from the regression analysis, and $\Phi$, in this case, is an identity matrix of size ${\text{n}}_{e}$ due to only the interevent variability being analyzed (see Arroyo and Ordaz 2011). Finally, ${\text{n}}_{e}$ is the number of earthquakes in the dataset.

Through Eq. (5) was possible to quantify the uncertainty in the regression coefficients for the GMPM derived. As will be seen later, the total forecasted variability obtained was larger for the cases where the dataset is poorly sampled and smaller enough for the opposite cases.

5 Results and discussion

5.1 Regression coefficients

Table 2 lists the attenuation coefficients and the standard deviation values obtained from the regression for PGA, PGV, SA, and AvgSA. For simplicity, Table 2 shows the coefficients for 41 and 30 structural periods for SA and AvgSA, respectively. The complete table of the regression coefficients is provided as an electronic supplement. Figures 3a and b compare the prior and posterior period-dependent coefficients estimated for SA and AvgSA, respectively. In these figures, it is noted that the prior and posterior coefficients follow a similar shape with significant differences in their amplitude, especially in the coefficients c₀, c₂, and c₄, where the updated values are so far from those computed theoretically, which means that our prior knowledge of these coefficients is vague and wrong. Regarding the total standard deviation, it can be noted that the dispersion (σ) calculated for AvgSA is lower than that from SA at the period range between 0.1 s and 4.0 s. Therefore, AvgSA has a better predictive capacity than SA, which has an important impact on PSHA as it contributes to a more accurate estimation of the hazard curves at small rates of exceedance.

Table 2 Coefficients obtained (see Eq. 2)

Fig. 3

Regression coefficients of the GMPM for a SA and b AvgSA. The solid and dashed lines represent the posterior and prior coefficients, respectively, used in this study

5.2 Covariance matrixes and comparison of variances

The covariance matrixes needed to compute the forecasted variability, see Eq. (5), are shown in Table 3 for some IMs such as PGA, PGV, SA (0.2, 0.5, 1.0, 1.5, 2.0), and AvgSA (0.1, 0.2, 0.5, 1.0, 1.5, 2.0). As stated before, with these matrixes and the parameters to be forecasted, the epistemic uncertainty due to the sampling of the dataset could be considered in the estimation of the hazard. The supplementary material shows an extended version of Table 3 for all IMs reported in Table 2.

Table 3 Covariance matrixes obtained (see Eq. 5)

Moreover, to see the impact of adding the epistemic uncertainty due to the dataset sampling, a ratio between the standard deviation from the total forecasted and the one coming from the regression analysis, $s={\sigma }_{p}/{\sigma }_{e}$, is computed for some intensity measures of SA and AvgSA, and it is shown in Fig. 4a, b, respectively. The dash lines correspond to the contour of s values, while the dots are the observations of the dataset (see Table 1). As it can be seen, s values close to 1 occur in the vicinity of the well-sampled region of the dataset, while higher s values (which means higher values of total forecasted variability) occur when there is a lack of data. As will be seen later, this uncertainty increase directly impacts the hazard assessment estimated for the site.

Fig. 4

Contours of s values, the ratio between the standard deviation of the total forecasted variability and the variability of the regression ($s={\upsigma }_{p}/{\sigma }_{e}$). a Intensity measures PGV, PGA, and SA (0.2, 0.5, 1.0, 1.5, 2.0). b Intensity measures AvgSA (0.1, 0.2, 0.5, 1.0, 1.5, 2.0). The dashed lines represent, s contours, and the dots are the observed values used in the regression analysis

5.3 Residual analysis

The residuals as a function of magnitude (M_w) and distance (R_rup) are shown in Fig. 5a, b, respectively, for PGA, PGV, SA, and AvgSA. For the last two IMs, the residuals are shown for four different periods of vibration that correspond to 0.5 s, 1 s, 1.5 s, and 2 s. The plotted residuals represent the difference between the natural logarithms of the observed data and the predicted SA (circles) and AvgSA (triangles) values. The solid and dotted lines correspond to the linear fit between the residuals computed and the M_w and R_rup values, respectively. As can be seen, the mean residuals obtained for M_w and R_rup are almost negligible for PGA, PGV, and SA, which means that the estimated coefficients of the GMPMs are adequate and provide a reasonable estimation of the GMI. Similar trends are obtained for the rest of the periods of vibration considered in this study for SA. For the case of AvgSA, the slope decreases slightly as M_w increases, especially for T > 1 s. This behavior may be related to the number of events used in the regression process to obtain the coefficients for long periods since, as mentioned earlier, the number of events in the dataset is reduced by approximately 23% to estimate AvgSA at T ≥ 1.8 s.

Fig. 5

Residuals in natural logarithm of PGV, PGA, SA (circles), and AvgSA (triangles) as a function of a M_w and b R_rup. The solid black line represents the linear fit for PGV, PGA, and SA residuals. Similarly, the dotted red line represents the linear fit for AvgSA

5.4 Median predictions

Figure 6 compares the observations against the predicted values of PGV, PGA, SA, and AvgSA for two different magnitudes (M_w) as a function of the source-to-site distance (R_rup). For SA and AvgSA, the comparison is carried out for four different periods of vibration that correspond to 0.5 s, 1 s, 1.5 s, and 2 s. In this figure, the black and red solid lines represent the median prediction (50th percentile) for a magnitude value (M_w) of 6.5 and 7.5, respectively, and the dashed lines correspond to the 16th and 84th percentiles. Likewise, the filled circles correspond to the 6.3 ≤ M_w < 6.7 observations and the filled triangles for 7.3 ≤ M_w < 7.7. Significant data variability for the evaluated magnitude ranges can be observed, especially for 6.3 ≤ M_w < 6.7, whose dispersion increases as the period increases. This dispersion is also observed in PGV, whose intensity measure is more sensitive to the low-frequency amplitudes of the ground motion. Likewise, it is noted that the dispersion between the observations and predicted values for AvgSA is lower than the one from SA, which leads a better predictive capacity of the GMPM-AvgSA. This dispersion increase over long periods is likely related to the characteristics of the seismic recording equipment operated at the station CU since most have been short-period accelerometers with low sensitivity at frequencies below 1 Hz. Despite this, it is observed that the empirical data is relatively well represented by the GMPM proposed.

Fig. 6

Regression data compared with attenuation curves computed for two different magnitudes: M_w = 6.5 (black lines) and M_w = 7.5 (red lines). Solid and dashed lines represent the median and the 16th–84th percentiles. The circles and triangles filled correspond to the data observed in the range of 6.3 ≤ M_w < 6.7 and 7.3 ≤ M_w < 7.7, respectively

Figure 7 compares the response spectra observed (solid black line) with the median estimated (dotted red line) at the station CU for SA and AvgSA from the GMPM derived. Only 12 of the most intense earthquakes in the dataset are compared. As expected, the suitability of the fit varies from event to event, showing a good agreement for most of them. In the case of the events: 5 (M_w 7.7), 12 (M_w 6.9), 15 (M_w 7.3), and 26 (M_w 7.25), there is a significant difference between the observed and predicted values for both SA and AvgSA; however, the fitting is reasonable given the uncertainty of the GMPM. On the other hand, it can be noted that the spectral shape of SA is strongly influenced by the site effects present at station CU, which yield a large spectral amplification at the period range from 1 to 3 s and, consequently, widen the spectral shape of AvgSA to long periods. This condition is particularly atypical at sites classified as firm soils, where it is expected that the peak amplitude has a place at short periods below 1 s. However, in the firm zone of Mexico City, this amplification is caused by the rather shallow (< 1 km) sediments that lie below the volcanic rocks that cover the hill zone of the Valley of Mexico (Ordaz et al. 1992).

Fig. 7

Comparison between observed and computed response spectra at the station CU for a SA and b AvgSA, using the GMPM derived. The solid black line represents the response spectrum observed, and the dotted red line the response spectrum estimated

5.5 Comparison of standard deviations

Figure 8a compares the total standard deviation of PGA and SA (σ_lnIM) obtained from the GMPM proposed with those obtained from two previously developed models for the station CU, corresponding to R99 and J06. This figure shows that the σ_lnSA computed in this study is less than estimated by R99 and J06 models at all periods of vibration. Note that the PGA value is associated with T = 0 s. On the other hand, Fig. 8b clearly shows the percentage reduction achieved for σ_lnIM regarding the values reported at the two GMPMs evaluated. This percentage varies from 5 to 60% for R99 and from 0.3 to 58% for J06. For both models, the lowest value occurs in the period range from 1 to 3 s. In general, a greater reduction of the dispersion of PGA and SA is obtained for the R99 model compared to that of J06. These differences are related to the number of earthquakes used in the regression approach since CU21 contains 18 more events than R99 and 13 more than J06. These results indicate a more significant predictive capacity of the proposed GMPM than previous models.

Fig. 8

Comparison between standard deviation values from R99, J06, and the GMPM proposed for PGA and SA

5.6 Comparison with previous models

To evaluate the performance of the proposed GMPM concerning previous models developed for the station CU, Fig. 9 presents the differences in natural logarithm between the predictions of GMPM-SA proposed and the R99-SA model. Similarly, Fig. 10 shows the differences regarding the J06-SA model. In both figures, the results are presented through surface plots that allow to quickly evaluate the behavior of the residuals obtained for a wide range of magnitude values (6.0 ≤ M_w ≤ 8.5) and periods of vibration (0 ≤ T ≤ 6), associated with a set of four different distances. These distances correspond to 250, 300, 350, and 450 km. Likewise, a color scheme ranging from blue to red facilitates interpreting the results; the blue tones indicate that the predicted values are lower regarding the evaluated model, and the red tones indicate the opposite. In both figures, it can be observed that for T < 3 s, the GMPM proposed predicts intensities below R99 and J06 for the magnitude range of 6 to 8 and distances less than 350 km. In this zone, the differences are less than 0.2 log units. For T > 3 s, the model predicts intensities above R99 and J06 for the entire range of magnitude and distance. The differences are greater than 1 log unit in this area, especially at T > 5 s. In general, a smaller difference is obtained regarding the J06 model than that of J06. These differences are mainly attributed to the dataset size since CU21 includes thirteen earthquakes not considered in J06 and eighteen more than R99, which provide essential information about the GMI for events with magnitudes (M_w) between 6.0 and 7.4, and distances (R_rup) between 260 and 400 km.

Fig. 9

Differences in terms of natural logarithm between predictions of GMPM-SA proposed and R99-SA model. The differences are presented for a set of four distances (R_rup) as a function of the period of vibration (T) and magnitude (M_w)

Fig. 10

Differences in terms of natural logarithm between predictions of GMPM-SA proposed and J06-SA model. The differences are presented for a set of four distances (R_rup) as a function of the period of vibration (T) and magnitude (M_w)

Another point that was evaluated of the GMPM was its performance in estimating the ground motion in other sites of the hill zone of Mexico City. For that purpose, a dataset of 24 events recorded at nine stations located southwest of Mexico City was used. These stations present a similar condition of the ground motion amplification that agrees with the reported by Montalvo-Arrieta et al. (2002, 2003). Table 4 lists the stations and earthquakes employed in the analysis. In total, 105 pairs of records (two horizontal components) were obtained from these stations.

Table 4 Stations and earthquakes used in the analysis

The maximum likelihood methods LH and LHH proposed by Scherbaum et al. (2004, 2009) were chosen for testing the GMPMs against the data observed. These methods allow quantifying in statistical terms the similarity between the observed and predicted ground-motion data and have been used in several studies to identify suitable models for specific seismic regions (Beauval et al. 2012; Delavaud et al. 2012). These methods were used to evaluate the GMPM performance for PGA and SA at nineteen periods between 0.1 and 6.0 s.

To apply the LH method is necessary to obtain the normalized residual for the set of observed and predicted values and compute the likelihood of an observation to be equal to or larger than a z value through the equations,

$$\mathrm{LH}=Erf\left(\frac{\left|z\right|}{\sqrt{2}},\infty \right)$$

(6)

$$z=\frac{x-\mu }{\sigma }$$

(7)

where z represents the normalized residual, x is the observed data, and μ and σ are the mean and the standard deviation of the GMPM, respectively. Erf(.) is the error function while integrating both tails of the standard normal distribution. LH can take values between 0 and 1. An LH value of 0.5 means that the observed and predicted data match their mean and standard deviation perfectly. Likewise, Scherbaum et al. (2004) provide a ranking criterion to quantify the overall model capability for predicting the observed data, assigning a category A, B, C, or D as a function of the LH obtained (refer to Scherbaum et al. (2004) for more details).

For its part, the LLH method is based on the log-likelihood approach to measuring the distance between two continuous probability density functions f and g, log-normally distributed, where f represents the distribution of the observed values and g the distribution of the GMPM evaluated. This approach computes the average log-likelihood of the considered GMPM using the observed dataset through the equation,

$$\mathrm{LLH}=-\frac{1}{N}\sum_{i=1}^{N}{\mathrm{log}}_{2}\left(g\left({x}_{i}\right)\right)$$

(8)

where N represents the total number of observations, x and g is the probability density function predicted by the model. A small LLH value indicates a good fit between the observed and predicted data.

Figure 11 shows the general performance of the GMPMs evaluated at the selected period range, and Table 5 lists the corresponding LH and LHH ranking indices. From this figure, the R99 GMPM reflects the best fit between the observed and predicted data, followed by J06 and the model proposed in this study. The largest differences can be observed at short periods, below 1 s, where the model derived has the lower values of LH and highest values of LLH, which means a lower capability to predict the ground motion intensity at the entire hill zone of Mexico City. Table 5 shows that the models proposed for the period range between 0.3 and 0.7 s have a low low predictive capacity. The difference between the GMPMs is less evident at long periods and shows a more stable ranking result.

Fig. 11

Performance of the GMPMs evaluated at the selected period range. a LH values vs period and b LLH values vs period

Table 5 Ranking of the GMPM evaluated

The low predictive capacity of the proposed model, obtained from the LH and LLH methods, is likely related to the sigma of the model since its value is lower than that reported by the R99 and J06 models (see Fig. 8). Kale and Akkar (2013) point out that two GMPMs with a similar median estimation evaluated with the LH and LLH methods would benefit the one with the highest sigma. However, in terms of a PSHA, the models with larger sigma lead to a large seismic hazard at long periods.

Goodness-fit measures: median likelihood value ($\widetilde{\text{LH}}$), and the median, mean and standard deviation of the normalized residual ($\widetilde{\text{NR}}$, $\stackrel{\mathrm{-}}{\text{NR}}$, and σ_NR). $\widetilde{{\text{LH}}{\text{H}}}$ represent the median log-likelihood value and RK the ranking assigned. The capital letters A, B, C and D stands the predictive capability of the GMPM: high, medium, low, and unacceptable in that order.

5.7 Comparison of hazard curves and uniform hazard spectra

Figure 12 compares the empirical and theoretical hazard curves computed from a probabilistic seismic hazard analysis (PSHA) for PGA and SA at five structural periods that correspond to 0.2 s, 0.5 s, 1 s, 1.5 s, and 2 s. The empirical curve is obtained by counting the number of times per year a given IM value has been exceeded, dividing by the observation period that, in this case, corresponds to 57 years for PGA, SA (T ≤ 2.7), and AvgSA (T ≤ 1.8), and 37 years for PGV, SA (T > 2.7) and AvgSA (T > 1.8). For these IMs, the observation period decreases because only 24 events recorded from 1985 to 2021 were used to derive the GMPM. The PSHA analysis is performed just for interface earthquakes along the Mexican Pacific Coast. The characteristics of the seismogenic zones (i.e., geometry and seismicity) are taken from the current version of Mexico’s Seismic Design Code of the Federal Electricity Commission (MDOC CFE 2015). The attenuation models used correspond to R99, J06, and the one proposed in this study. The computations were made using R-CRISIS (Ordaz et al. 2021), based on the classic Esteva–Cornell approach (McGuire 2008).

Fig. 12

Comparison between empirical and computed hazard curves for SA(T). The circles denote the empirical curve obtained from observed data. The continuous black line represents the hazard curve computed considering the uncertainty in the regression coefficients (HCU). T and the dotted black line correspond to the hazard curve computed, disregarding this uncertainty (HDU). Both curves were obtained with the GMPM proposed in this study. Likewise, the dotted red line and the continuous gray line represent the hazard curve obtained with the GMPM proposed by J06 and R99, respectively

The empirical curves were computed for 30 intensity values logarithmically separated by equal intervals between the limits 1 to 1000 cm/s² for the SA curve (Fig. 11) and 1 to 100 cm/s² for the AvgSA curve (Fig. 12). Therefore, their ordinates are the same in the six figures associated with a specific IM. In these figures, most of the empirical hazard curves, compared with the theoretical hazard curves computed from PSHA, tend to saturate at intensities below 5 cm/s², which may be related to the fact that earthquakes with low ground-motion intensities were dismissed from the dataset. The same behavior occurs at intensities above 30 cm/s² because there is only one event in the dataset that generates intensities above this threshold, so its exceedance rate corresponds to the maximum observation period, equivalent to v(a) = 0.0175 for PGA, SA (T ≤ 2.7) and AvgSA (T ≤ 1.8), and v(a) = 0.027 for PGV, SA (T > 2.7) and AvgSA (T > 1.8). For moderate intensities, the differences between both hazard curves are not so large and seem fit for the observed data. Similar observations were made by Ordaz and Reyes (1999).

Figure 12 shows that the shape of the hazard curves estimated with the proposed model disregarding the uncertainty in the regression coefficients (herein, HDU, dotted black line) follows a similar shape to the estimated with the R99 (continuous gray line) and J06 (dotted red line) models. The three GMPMs compute similar exceedance rates, v(a), for low intensities. However, their differences increase as the intensity increases. Compared to the proposed model, the R99 model estimates higher exceedance rates for PGA and SA for all periods evaluated, except for T = 2 s. On the other hand, the J06 model estimates lower exceedance rates for PGA and SA at T = 0.2 s and T = 0.5 s, the opposite case occurs for T = 1.0 s and T = 1.5 s. Like R99, the forecasts of J06 at T = 2 s are the same as those of the proposed model.

Likewise, in Fig. 12, the hazard curves are also compared with the proposed model, considering the uncertainty in the regression coefficients (herein, HCU, continuous black line). As expected, for small and large intensities, the hazard level of HDU is larger than HCU. This increment at small intensities, with v(a) ≥ 0.1, is produced by the additional epistemic uncertainty for small magnitude events (M_w < 6) at short distances (R_rup < 240 km) whose predictive standard deviation, σ_p, is between 20 and 100% more than the obtain directly from the Bayesian regression, σ_e (see Fig. 4). The same behavior occurs for large intensities, with v(a) ≤ 0.01, since the hazard level is controlled by large events with M_w > 7.5 and R_rup > 280 km, in which σ_p is between 10 and 80% more than σ_e. The differences become less evident at moderate intensities, with 0.1 ≥ v(a) ≥ 0.01, whose hazard level is controlled by earthquakes with magnitudes (M_w) between 6.0 and 7.5 and distances (R_rup) between 250 and 350 km. These parameters fall in the range where σ_e² and σ_p² are similar (see Fig. 4) and correspond to the well-sampled region of the dataset. The same effect is observed in the hazard curves computed for AvgSA (see Fig. 12), although less evident at low intensities than in SA curves.

Finally, Fig. 13 compares the empirical and theoretical hazard curve computed from PSHA for AvgSA at six periods corresponding to 0.1 s, 0.2 s, 0.5 s, 1 s, 1.5 s, and 2 s. In this case, the hazard curves are estimated directly from GMPM-AvgSA proposed, disregarding the uncertainty in the regression coefficient (HDU), and compared with those obtained indirectly using the existing GMPM-SAs (i.e., R99 and J06) and a SA inter-period correlation model proposed by Rodríguez-Castellanos et al. (2021). The stepped shape of the empirical curve is due to the lack of events in the database that generate large intensities. Therefore, the exceedance rates computed for large accelerations are less reliable for comparison purposes. Despite this, it is noted that the shapes of the calculated hazard curves match reasonably well with the observed data at intensities below 20 cm/s². The three GMPMs compute similar exceedance rates, v(a), for low intensities. However, as observed for SA, their differences increase as intensity increases. Additionally, the better predictive capacity of AvgSA with respect to SA has an important impact on PSHA since it contributes to a more precise estimation of the hazard curve at low rates of exceedances. The R99 model estimates higher exceedance rates for all periods than the proposed mode. Likewise, the J06 model also estimates higher exceedance rates for most periods, except for T = 0.1 s and T = 0.2 s. There are no significant differences in the seismic hazard estimations computed from the proposed model compared to previous models. However, according to the available data, the suggested GMPM estimates the seismic hazard more adequately than the other models.

Fig. 13

Comparison between empirical and computed hazard curves for AvgSA(T). The dotted denotes the empirical curve obtained from observed data. The continuous black line represents the hazard curve computed considering the uncertainty in the regression coefficients (HCU), and the dotted black line corresponds to the hazard curve computed disregarding this uncertainty (HDU). Both curves were obtained directly with the GMPM proposed in this study. Likewise, the dotted red line and the continuous gray line represent the hazard curve obtained with the GMPM proposed by J06 and R99, respectively

6 Conclusions

This article presents an updated GMPM to estimate the peak ground acceleration (PGA), peak ground velocity (PGV), 5% damped pseudo-spectral acceleration (SA), and the average spectral acceleration (AvgSA) at the hill zone of Mexico City for interface earthquakes that occur along the Pacific coast of Mexico. The model is built as a function of the magnitude (M_w) and the closest distance to the rupture plane (R_rup), using thirty-three reverse-faulting events recorded at the station CU from 1965 to 2021.

The new strong-motion dataset (CU21) includes thirteen more earthquakes than the one used in J06 and eighteen more than in R99, providing valuable information on the GMI for events with a magnitude range (M_w) between 6.0 and 7.4 and a distance (R_rup) between 260 and 400 km. Furthermore, the results showed that the dispersion of the GMPM proposed is lower than the previous models for PGA and SA, which means better predictability and more reliable estimates of the seismic hazard at the CU site. However, when quantifying its predictive capability to estimate the ground motion at the entire hill zone of Mexico City, the lowest ranking is obtained compared to the other tested models. It is thought that the low predictive capacity of the proposed model, obtained from the LH and LLH methods, is related to the standard deviation (σ_e) of the model since its value is lower than that reported by the R99 and J06 models. The standard deviation obtained for GMPM-AvgSA shows that the dispersion is lower than that from GMPM-SA at the period range between 0.1 s and 4.0 s, except in the range of 0.22–0.38 s where GMPM-SA presents a lower dispersion than the GMPM-AvgSA. Despite this, it is thought that the GMPM-AvgSA model has a better predictive capacity.

The analysis carried out of the uncertainties due to the sampling of the dataset showed that there is a variability that is not always taken into account in the GMPMs, and then an underestimation of the hazard levels is presented for lower and higher annual rates. This behavior explains that the source model used to estimate the hazard curves considers events below or above the magnitude range in the dataset used to obtain the regression coefficients. Therefore, it is important to highlight these uncertainties and include them in the seismic hazard assessment.