The evaluation of a probabilistic seismic hazard analysis for a particular site requires identification of all possible earthquake sources capable of producing a significant seismic event. For this purpose, Zúñiga et al. (2017) proposed a seismic regionalization for Mexico, which is used in the present study. Figure 2a shows the shallow-depth seismic zones where interplate earthquakes occur due to the subduction of the Rivera and Cocos plates (SUB1–SUB4). Figure 2b illustrates the intermediate-depth seismic zones. This region corresponds to intraslab events that take place inside the subducted Rivera and Cocos plates below south-central Mexico (IN1 to IN3). Additionally, Fig. 2c displays the seismic zones for characteristic seismic events (C1 to C14) proposed by Ordaz and Reyes (1999). Seismic zones in Fig. 2c are also included in the present study to compute PSHA.

Seismic sources are capable of producing different earthquake sizes. Therefore, it is crucial to define the probability distribution of the earthquake magnitudes and corresponding rates of occurrence for each source. In this regard, the distribution of earthquake sizes is commonly described by the bounded Gutenberg–Richter recurrence law (Eq. 1). 1λm=νexp⁡-βMw-Mmin-exp⁡-βMmax-Mmin1-exp⁡-βMmax-Mmin, where λm is the mean annual rate of exceedance for earthquakes between a minimum magnitude Mmin and a maximum magnitude Mmax, and ν=exp⁡(α-βMmin) is the mean annual number of earthquakes of magnitude Mw≥Mmin, where α=2.303p and β=2.303q. The values of p and q are indicated in Fig. 2a and b, according to Zúñiga et al. (2017).

For the seismic sources related to characteristic earthquakes (Fig. 2c), the bounded Gutenberg–Richter recurrence law does not accurately describe the magnitude exceedance rates. Accordingly, for Mw>7, we employ a Gaussian probability distribution function (pdf) of magnitudes to account for the characteristic events in the Mexican subduction zones (see Eq. 2) (Ordaz and Reyes, 1999). 2λm=ν71-ΦMw-EMwσMw, where ν7 is the mean annual number of earthquakes of magnitude Mw>7; EMw and σMw are the mean and standard deviation of the magnitude, respectively, and Φ (.) is the normal distribution function. The corresponding parameters to evaluate the distribution are shown in Fig. 2c.

The present study assumes Mmin=4.5 and Mmax=6.9 for the interplate shallow-depth seismic zones SUB1–SUB3 (see Fig. 2a). In contrast, Mmin=4.5 and Mmax=7.2, 7.8 and 7.9 are assumed for IN1–IN3, respectively (intermediate-depth seismic zones, Fig. 2b). Finally, Mmin=7.0 and Mmax=8.1 are assumed for the 14 earthquake sources shown in Fig. 2c.

Once the earthquake magnitude distribution is established, the pdf of distances from the earthquake location to the site of interest must be characterized. A uniform pdf is generally assigned to any point in the seismic zone (McGuire, 1995; Kramer, 1996). Since the area sources, where earthquakes can occur, are well-delimited (Fig. 2a–c), it is straightforward to determine the source-to-distance distribution.

Attenuation relationships are fundamental for PSHA. They are commonly developed to predict the peak ground acceleration, PGA, or the spectral acceleration, Sa(T1). Unfortunately, attenuation models have not yet been devised to provide INp as a function of the vibration period (as is done with existing GMPEs); however, with GMPEs for Sa(T1) currently available, it is possible to perform PSHA using INp. Here we employ the GMPEs proposed by Reyes et al. (2002) and Jaimes et al. (2015) for interplate and intraslab events, respectively. They were developed using accelerometric data recorded at Ciudad Universitaria station (CU), which is located at the hill zone (firm ground) of Mexico City, basically conformed by a surface layer of lava flows and volcanic tuffs with a shear wave velocity in the upper 30 m of 750 m s-1 (Ordaz and Singh, 1992; Singh et al., 2018).

The final product of a PSHA can be expressed in different forms. Seismic hazard curves are used frequently to represent the seismic hazard. They indicate the annual rate of exceeding a variety of intensity levels of a ground motion parameter at a site of interest. The procedure to compute a ground motion hazard curve is based on the total probability theorem (Baker, 2013; Cornell, 1968; Esteva, 1968; McGuire, 1995; Kramer, 1996).

In this section a methodology to perform PSHA using INp is proposed. First, INp is defined as follows (Bojórquez and Iervolino, 2011): 3INp=SaT1⋅NPα,4NP=SaavgT1…TNSaT1, where INp is the scalar intensity measure, α is a parameter that should be calibrated according to the structure and the earthquake demand parameter selected (in this study α=0.5 is adopted, as recommended in Bojórquez and Iervolino, 2011), and Saavg(T1 … TN) is the geometric mean of the spectral acceleration at N numbers of structural vibration periods considered. Saavg(T1 …TN) takes into account the vibration period lengthening due to structural damage and is expressed as 5SaavgT1…TN=∏i=1NSaTi1/N. Substituting Eqs. (4) and (5) into Eq. (3), applying the natural logarithm, results in 6ln⁡INp=(1-α)ln⁡SaT1+αN∑i=1Nln⁡SaTi. Then, the expected value and the variance of ln⁡(INp) can be expressed as in Eqs. (7) and (8), respectively. 7Eln⁡INp=(1-α)Eln⁡SaT1+αN∑i=1NEln⁡SaTi8Varln⁡INp=α2Varln⁡SaavgT1…TN+(1-α)2Varln⁡SaT1+2α(1-α)ρln⁡SaavgT1…TN,ln⁡SaT1σln⁡SaavgT1…TNσln⁡SaT1 The values of ln⁡[Sa(Ti)] are obtained from existing attenuation models (e.g., the GMPEs described in Sect. 3.4). On the other hand, ln⁡[Sa(Ti)] terms are commonly assumed to have a joint Gaussian pdf; consequently, the summation also has Gaussian distribution. Therefore, the variance Var{ln⁡[Saavg(T1…TN)]} and the correlation coefficient ρln⁡[Saavg(T1…TN)]ln⁡[Sa(Ti)] can be obtained by Eqs. (9) and (10), respectively: 9Varln⁡SaavgT1…TN=1N2∑i=1N∑j=1Nρln⁡SaTi,ln⁡SaTjσln⁡SaTiσln⁡SaTj,10ρln⁡SaavgT1…TN,ln⁡SaT1=∑i=1Nρln⁡SaTi,ln⁡SaT1σln⁡SaTi∑i=1N∑j=1Nρln⁡SaTi,ln⁡SaTjσln⁡SaTiσln⁡SaTj, where the term ρln⁡[Sa(Ti)], ln⁡[Sa(Tj)] represents the correlation between spectral acceleration values at periods Ti and Tj. The correlation coefficients have been obtained by the authors of the present study (Rodríguez-Castellanos et al., 2019, 2020).

Among the parameters that define the intensity measure INp, the geometric mean, Saavg(T1…TN), has a crucial role when computing the uniform hazard spectra (UHS). The TN value (Nth structural vibration period) takes into account the level of nonlinearity developed by the structure. Bojórquez et al. (2008) and Bojórquez and Iervolino (2011) recommend using TN=2.0T1. Nevertheless, we consider that there is no optimal period range for Saavg(T1…TN) that meets the entire range of structural vibration periods; therefore, here we propose that TN should depend on the structural vibration period, which is in agreement with Tsantaki et al. (2012, 2017).

It has been pointed out that the stiffer the structure, the larger the period lengthening. Accordingly, for structures with short vibration periods, we adopt TN=2.0T1, which agrees with recommendations made by Bianchini et al. (2009), Katsanos and Sextos (2015), and Tsantaki et al. (2017), for relatively stiff structures, and assuming a ductility demand between 2 and 3.

At short to moderate vibration periods, the structural period lengthening diminishes somewhat linearly until it reaches a semi-constant behavior (which is independent of the level of nonlinearity developed by the structure) (Katsanos and Sextos, 2015). In this regard, Di Sarno and Amiri (2019) quantified the fundamental period lengthening of structures by the ratio of response spectra corresponding to the lengthened and the elastic structural vibration period (Tin/Tel). They suggested dividing the response spectra into two main regions: the first associated with short to moderate period structures, whose period shift ratio Tin/Tel decreases with increasing the elastic period, and the second region related to long-period structures, where the ratio period Tin/Tel behaves practically constant. Consequently, there must be a certain bound where the period shift ratio switches to remain constant; therefore, we propose TN=Ts as that bound from which the lengthening of the structural vibration period remains almost constant. In this context, Miranda and Ruiz-Garcia (2002), Ruiz-Garcia and Miranda (2003), and independently Terán-Gilmore and Espinosa Johnson (2008), found that strength demands between degrading and non-degrading systems are similar when the structural period and dominant soil period are comparable, which means that the mean ratio value should be approximate to one when Tn≈Ts.

For vibration periods longer than the dominant soil period, it is assumed TN=1.25T1, which is, on average, the period shift ratio value for structures with a short to moderate nonlinearity level, that is, with a ductility ratio around 2 to 3 (Katsanos and Sextos, 2015; Di Sarno and Amiri, 2019).

Summarizing, we used TN=2.0T1 in this study for structural systems with a short fundamental period; TN=Ts for those with an intermediate period and TN=1.25T1 for systems with a long fundamental period. It is possible to get a better approximation of TN bounds, by means of a parametric study of the ratios of the equivalent period of SDOF degraded systems and that of the elastic systems (Tin/Tel), as a function of Tel, for a given ductility. Such a study can consider both ground motion characteristics and structural properties (such as degrading stiffness ratio, pinching factor, accumulated damage factor, etc.), as was done by Di Sarno and Amiri (2019). They proposed a mathematical expression for estimating the lengthening of the fundamental period as a function of the structural elastic period and the significant structural parameters, which is applicable to systems in site classes D and C according to ASCE 7–10 (2010), with shear wave velocities 182.88<Vs30<365.76 and 365.76<VS30<762 m s-1, respectively. However, the TN bounds used here lead to reasonable results, as is verified below.

The uniform hazard spectra are computed, first, for the CU site, which is in firm ground. Figure 3a shows the UHS if only interplate, or alternatively intraslab, earthquakes occur. It also displays when both types of events are considered simultaneously (Total). Figure 3b shows the total UHS of Sa(T1) and INp, both associated with a 250-year return period. It can be seen that the spectra are quite similar; practically, they reach the same acceleration levels, and slight differences occur at long periods.

Estimating the seismic hazard at firm ground allows us to proceed with a technique to assess the seismic hazard at soft-soil sites. In this regard, Esteva (1970) presented a formulation in which through a known hazard curve at a reference site it is feasible to estimate a hazard curve at a recipient site. In this study, we used CU station as the reference site because, since 1964, it has recorded all the significant ground motions that have struck Mexico City. In addition, different studies have taken CU as a reference site (Ordaz et al., 1988; Reinoso and Ordaz, 1999; Singh et al., 1988). Therefore, it is viable to perform a hazard analysis for CU station and then to compute the annual rate of exceedance at other sites located in soft or medium soils, as follows: 11νY(y)=∫0∞νXyτfτ(τ)dτ=Eτνxyz, where νY(y) is the mean annual rate of exceedance of a seismic IM, for the recipient site. νx(y/τ) is the mean annual rate of exceedance of a seismic IM for the reference site, divided by the variable τ. τ represents the response spectral ratios between the response spectra corresponding to the recipient site and the reference site (Y/X). fτ(τ) is the pdf of τ.

Therefore, to evaluate the previous function, firstly, the spectral ratios are estimated, and then they are coupled with the seismic hazard curves via Eq. (11). In this respect, Fig. 4a–f show the mean response of the spectral ratios for Sa(T1) (solid line) and INp (dashed line) for one representative station located in each of the zones listed in Table 1. In this sense, the spectral ratios roughly represent the spectral amplification of soft soil with respect to firm ground. It is observed how the peak values shift towards increasingly longer periods, which, approximately, match with the dominant soil period (see Table 1). For this analysis, more than 1100 ground motion records corresponding to the different recording stations were used. The stations are grouped depending on the dominant soil period where these are located, as follows: zone A: Ts<0.5 s; zone B: 0.5 s < Ts < 1.0 s; zone C: 1.0 s < Ts < 1.5 s; zone D: 1.5 s < Ts < 2.0 s; zone E: 2.0 s < Ts < 2.5 s; and zone F: 2.5 s < Ts < 3.0 s. Additionally, Fig. 5 shows the location of the recording stations in Mexico City, which are represented with circles of different colors associated with each of the proposed zones (see Table 1).

Next, in order to compute the mean annual rate of exceedance of Sa(T1) and INp, the seismic hazard curves corresponding to CU station are coupled with the response spectral ratios, using Eq. (11). Figure 6a to f show the hazard curves (λ) of Sa(T1) and INp, associated with different vibration periods, corresponding to CU and the same recording stations of Fig. 4a to f. First, as expected, the rates of exceedance for all the recording stations analyzed are higher than the corresponding ones of CU (up and down). Additionally, concerning the CU site, the hazard curves for both intensity measures INp and Sa(T1) are very similar, and differences are barely visible at long return periods. Now, for the rest of the recording stations, Fig. 4c and d show noticeable variations between exceedance rates of Sa(T1) and INp; nevertheless, Fig. 4e and f display almost no contrast between the rates of exceedance of the two intensity measures. The previous is relative, because to fully characterize the variations between exceedance rates of Sa(T1) and INp, a wide range of periods need to be covered; for this reason, we estimate the UHS in the following.

Then, having the mean rates of exceedance for each recording station site (see Table 1), the UHS are estimated for a given return interval. Figure 7a to f show the UHS of Sa(T1) and INp for the same stations of Fig. 6a to f, for a 250-year return period. It is observed that, at vibration periods shorter than the dominant soil period, the spectral ordinates corresponding to firm ground (zones A–C) are comparable for both IMs. However, at soft soil (zones D–F), the spectral ordinates of INp are notably higher than those of Sa(T1) (up to 30 %). In contrast, at vibration periods longer than Ts, they are smaller than those corresponding to Sa(T1) (5 % to 20 %, depending on the soil type). The same can be appreciated for different sites of the city in the maps shown in Fig. 8a to d, which corresponds to Sa(T1) (left side) and INp (right side), for T1=0.5 s (up side) and T1=1.0 s (down side), for a return interval of Tr=250 years.