1 Hill zone amplitude Fourier spectra

Site response using spectral ratios for only one hill zone site should be avoided (^{E. Field, et al., 1992}) because minor differences among hill zone sites could yield significant lakebed amplifications challenging to explain in terms of physical reasons. As ^{Reinoso and Ordaz (1999)} suggested, an average Fourier spectrum for hill-zone stations must be used. Likewise, in the same article there is a discussion of the smoothing of the Fourier spectra and recommendations for their use for spectral quotient purposes. Figure 3 shows, with grey lines, the smoothed amplitude Fourier spectra of hill zone accelerometric stations for different earthquakes and, with a dark line, the mean spectra of the EW component. As shown in Figure 3, amplitude differences are relatively small, but there are peak values, due to very local site conditions, that can affect the spectral ratios, so using the average spectra softens the hill zone response and, therefore, the spectral ratio. The stations chosen to compute the hill zone mean spectra (^{Reinoso and Ordaz 1999}) are those which are located in the South-West zone of Mexico City: 78, 74, 50, 34, 07, 13, TACY and CU (Figure 1)

Figure 3 Smoothed amplitude Fourier spectra of hill zone accelerometric stations for different earthquakes and mean Fourier amplitude spectra at hill zone 

Figure 4 shows the average Fourier spectra for hill zone stations and all thirteen earthquakes presented in Table 1. Two groups of seismic events that follow similar behavior are noticed: interslab and intraslab earthquakes, shown in Figure 4a and Figure 4b, respectively. Figure 4c shows a comparison of the two most important and intense earthquakes of each kind; both occurred in September 19, the inslab one in 1985, and the intraslab one in 2017. Both earthquakes show different amplitudes, the 19/09/2017 earthquake presents peak values for frequencies larger than 1.0 Hz, while the 09/19/1985 one shows larger amplitudes for frequencies smaller than 1.0 Hz.

Figure 4 Average Fourier spectra for hill zone accelerometric stations and for the thirteen events reported in Table 1 (EW direction): a) Interslab earthquakes, b) intraslab earthquakes and c) comparison between average Fourier spectra of both events occurred in September 19 (1985, inslab, and 2017, intraslab). 

2 Lakebed zone spectral ratios

Mexico City has some of the most pronounced site effects known. For some frequencies, the amplification can be up 500 times to epicentral sites, and up to 100 times the observed at hill zone. As stated by ^{Ordaz et al. 1988}, ^{Singh et al. 1988} and ^{Reinoso and Ordaz 1999}, the lakebed zone soil responds approximately as the one-dimensional theory predicts, however, there are some sites where evidence of 2D or 3D behavior, or at least a non 1D one has been captured by spectral ratios (^{Reinoso and Ordaz 1999}). Spectral ratios have been computed since the 1985 earthquake and have shown to be relatively constant for different sources, epicentral distances, and magnitudes. The spectral ratios theoretically remove most of the effects from the source and path, and only leaves the site effect. However, there is evidence that in recent years, the soft-soil response in some regions of the lakebed zone has changed, mostly due to anthropogenic conditions, in particular, ground subsidence produced by groundwater withdrawal (^{Avilés and Pérez-Rocha 2010}, ^{Arroyo et al. 2013}).

In order to present the main changes and observations, we have selected five accelerometric stations that are considered as representative of the site amplification: SCT, 37, 12, 20, and 84. The Fourier spectral ratios were computed for each one of these stations, considering the mean spectrum for the hill zone, as explained above.

3 Soil dominant vibration period behavior

Estimating the soil vibration period (Tg) with the smallest uncertainty is important since it is directly correlated to the seismic response of structures. The Mexico City Building Code takes this into account as the design spectra are based on Tg. The code contains a very detailed map of Tg, built a few years ago with information from spectral ratios, geotechnical studies, and weak motions. However, some evidence proves that this Tg has changed over time (^{Avilés and Pérez-Rocha 2010}, ^{Arroyo et al. 2013}). The variation of Tg was examined for all earthquakes of Table 1 and Figure 2, and all accelerometric stations of the lake-bed zone (Figure 1). Figure 13 shows for the most representative twenty-five accelerometric stations the variation of the dominant soil period as a function of time, with an open circle, and the corresponding trend with dotted-line. It can be seen that there are sites with a clear trend of shortening the soil period over time. The rates of period-change vary from site to site being almost null for some stations, to up to 20% for others.

Figure 13 Variation of the soil dominant period over time for 25 selected accelerometric stations in Mexico City 

A map with the dominant soil period decreases is shown in Figure 14. This figure shows the decrease (in percentage) of the dominant soil periods between those computed during the April 25, 1989, and September 19, 2017 earthquakes, using the spectral ratios shown in this article. The results shown in Figure 14 do not correspond to any approximate model and are based on real data and earthquakes recorded by the accelerometric network of Mexico City. However, it should be considered that results could have minor changes if a special interpolation were used, similar to those proposed by ^{Perez-Rocha (1998)} and ^{Worden et al. (2018)}.

Figure 14 Decrease in percent of soil dominant periods in the lake area of Mexico City 

4 Fourier spectral amplifications

A similar analysis was performed for the maximum spectral amplification of each station at the lakebed zone. That is, if, according to the previous section, the dominant period at a given station changed, maximum amplification is the observed for this changed period. Figure 15 shows the time-variations maximum spectral amplitude for 25 accelerometric stations. This amplification is the maximum computed in the spectral ratio, no matter at which period, for each station and earthquake, and both orthogonal components (EW amplification, open circles, NS component, filled triangles); trends are plotted with dotted lines. The observed variations as a function of time are different for both orthogonal components, and no overall trend is observed.

Figure 15 Fourier spectral maximum amplitude for several accelerometric stations and for both orthogonal directions 

In most cases, the amplitude has decreased over time, however stations 37 (NS), 84 (EW), 15 (EW) and 12 (NS) show an increase in their amplitude. Similar to what was observed above about station 84, stations 37 and 15 are located very close to it (Figure 1), so similar conclusions of 3D effects could explain this amplification. However, station 54, also within the area, does not exhibit any amplification over time. Station 12 has been one of the most affected by the settlement of the city, and it is also located very close to a hill zone so that similar 3D effects could be present there.

Figure 16 shows the maximum amplification maps for the EW component of April 25, 1989 (top row), and the September 19, 2017 (bottom row) earthquakes. The amplification shown is for vibration periods of 1.5, 2.0, and 4.0s. Figures 16a and 16b show the peak amplifications maps for a period of 1.5s. When comparing these figures, similar amplifications (less than 10) are observed for the zone with soil periods between 0.5 and 1.8s, except for station 84 that shows the largest amplification for both earthquakes, with values up to 20 times for the 2017 earthquake. On the other hand, a decrease is observed close to "Basilica de Guadalupe" (north of the city), as well as in the deepest clay deposits (stations 20, 31, and 35). The amplification for periods of 2.0s (Figure 16c and Figure 16d) are larger for the 1989 earthquake, especially in sites located close to the Condesa and Roma colonies, where severe damage was observed during both September 19 earthquakes. Finally, for sites in zones with the deepest soil deposits of lakebed with periods around 4.0s (Figure 16e and Figure 16f), there is a decrease in the amplification, as can be seen in the "Xochimilco" area and around the Mexico City airport (31 and 35 stations). For other periods not shown in Figure 16, the trend indicates that there is a decrease in the peak amplification, except for station 84 for the EW component, where there is an increase close to 50% for the 2017 earthquake. As mentioned previously, this station has a particular behavior due to possible basin-edge effects that increase its response in only one direction.

Figure 16 Maximum spectral amplification maps for the EW direction. a) 25/04/1989 and Te=1.5s, b) 19/09/2017 and Te=1.5s, c) 25/04/1989 and Te=2.0s, d) 19/09/2017 and Te=2.0s, e) 25/04/1989 and Te=4.0s, f) 19/09/2017 and Te=4.0s. 

Figure 17 shows the amplification decrease, in percentage, between the 1989 and 2017 earthquakes for both orthogonal components and the amplitude corresponding to the dominant soil period of each accelerometric station. As can be seen in Figure 17, there is a considerable decrease in amplification in many stations, especially in the area with soil periods between 1 and 3s, coinciding, for most sites, with the soil dominant period changes shown in Figure 14.

Figure 17 Map of amplification differences between the 1989 and 2017 earthquakes (in percentage). 

From the analyses presented above, it can be noticed, since 1989, a change in the behavior of the local soil effects in large part of the lakebed of Mexico City. As other authors have shown (^{Arroyo et al. 2013}, ^{Avilés and Pérez-Rocha 2010}), this is associated with the groundwater pumping that has contributed to the consolidation of the aquitards underlying the Valley of Mexico causing exceptional ground subsidence levels.

1 Pseudoacceleration demands

Figure 18 shows the pseudoacceleration demands in terms of response spectra (Sa) for selected accelerometric stations, each one plotted with the official design spectra of the Mexico City code (NTCD-2004). The design spectra are different for most stations since these spectra are computed from the dominant period at each site. Also shown are the buildings that collapsed during the earthquake (Figure 18, red dots). As can be seen in the spectra, the highest demands were computed where the dominant soil period is between 1.0s and 1.8s, coinciding roughly with the severely damaged area. Note that station 84 exhibits very large Sa demand, exceeding for one horizontal component the design spectrum by approximately 30%, while the SCT spectra remain below 50% of its corresponding design spectrum. Likewise, the spectra in other sites of the hill zone (30, 47, 74) also exceeded the design values. Station 35 is over the deepest soil deposits of lakebed zone, and the demands for the second and third vibration modes are large, reaching intensities that exceeded the design spectrum in periods lower than 1.0s. As has been stated, some of the spectra exceeded the design one but only by one horizontal component of motion. If the average horizontal spectra are computed, the resulting spectra never exceeded, in any station, the design one.

Figure 18 Pseudoacceleration spectra for the September 19, 2017 earthquake for both orthogonal directions and compared with the design spectra included in the seismic regulations in 2017 (NTCD-2004). 

The maximum computed demands and reported damage during the September 19, 2017 earthquake, occurred in areas considerably different from those traditionally observed with inslab earthquakes, such as that of September 19, 1985. This because the intraslab earthquakes at R<250 km from Mexico City have shown to have more energy for shorter frequencies than those of the inslab events. In particular, the 1985 earthquake had peaks around 2s while de 2017 ones had them in 1-1.5s.

Figure 19 shows Sa maps for the September 19, 2017 earthquake for SDOF systems with periods of 1.0, 1.5, 2.0, and 3.0s, and for the geometric mean spectra of both orthogonal components, together with the accelerometric stations, damage buildings and some reference sites. There are intensity differences in areas with largest demands as well as in the spatial distribution of the damage. All the maps where computed using a bilinear interpolation without considering additional soil characteristics to interpolate the intensities.

Figure 19 Pseudoacceleration maps for the September 19, 2017 earthquake for different periods. a) 1.0s, b) 1.5s, c) 2.0s, and d) 3.0s 

From Figure 19, it can be observed that the lakebed zones with periods between 1 and 1.5 had the most significant intensities, coinciding roughly with the zones where damage to buildings was present (Figure 19a and Figure 19b). The map shown in Figure 19c, which corresponds to the maximum intensity for 2s, shows a better correlation to damage in the downtown area. Figure 19d exhibit that, despite large soil dominant periods in Mexico City, the intensity for periods larger than 2.5 s is not large, and slight damage is expected in those areas; however, second and third vibration modes, with periods around 1s, may present relatively high intensities. What these maps show is that there is a strong correlation between the dominant period, soil amplification and damage, but is not strong enough to explain all damage patterns and, also, no damage patterns. This apparent complexity arises from many aspects such as 1) most of the damaged structures were built before 1985. Therefore, the seismic regulations at that time did not represent the real seismic hazard of the area, 2) although there is a considerable density of accelerometric stations in that area, the soft-soil behavior in Mexico City varies significantly within dozens of meters due to the clay depth, so important intensity information was simply not captured by the array. A special interpolation scheme (^{Perez-Rocha 1998}, ^{Worden et al. 2018}) should be used to obtain a reliable and definite intensity map.

Figure 20 illustrates the variation of the pseudoacceleration spectra for this earthquake and the design spectra for the lakebed clay depth of the ancient "Texcoco" lake for the cross-section A-A shown in Figure 18. For most sites in the valley, the soil vibration period is directly correlated to the depth of the clay strata shown in Figure 20 and, consequently, to the intensities recorded at ground level.

Figure 20 Behavior of seismic demands in relation with clay depth of lake-bed zone 

2 Hysteretic energy demands

The damage suffered by the structures over the soft-soil in Mexico City depends not only on the acceleration demands and the frequency content but also on the very long duration of the strong ground motion. In the case of lakebed zone of Mexico City, where the structures are subjected to long-duration earthquakes, the disadvantage of using spectral acceleration as structural design parameter is that the potential damage due to cumulative damage effects is not considered (^{Fajfar, 1992}; (^{Kunnath and Chai, 2004}) ^{Quinde et al. 2019}). The explicit consideration of cumulative damage (low-cycle fatigue) during seismic design is usually based on energy concepts (^{Housner, 1956}; ^{Kuwamura and Akiyama, 1994}).

Structural damage is associated with the plastic behavior of the system, which could be studied from the hysteretic energy dissipated by a structure during an earthquake (E _Hμ ). This energy corresponds to the total area of hysteretic cycles that a structure develops during an intense earthquake, and for its calculation, the entire duration of the seismic record is considered. Figure 21 shows the hysteretic energy (E _Hμ ) demands from the September 19, 2017 earthquake, for SDOF systems with elasto-perfectly-plastic behavior. The maps represent the E _Hμ for periods of 1.5s as well as those associated with the peak energy of each site no matter at which period (maximum envelope demands).

Figure 21 Hysteretic energy maps for the September 19, 2017 earthquake: E _Hμ for a SDOF of 1.50s, left, and E _Hμ for the maximum demands envelope, right.  

As shown in Figure 21, the E _Hμ demands are large for site with soil period close to 1.5s, with station 84 (Tg = 1.4s) being the one with the largest demands. In the map of the envelope of maximum demands, it is shown that the E _Hμ is reasonably associated with damage area.

From the hysteretic energy, the accumulated plastic demands could be known. However, this energy does not contain enough information to associate it with structural damage, since the total dissipated energy could be similar for two different structural responses. Due to this, it is convenient to use the normalize hysteretic energy (^{Darwin and Nmai 1986}); (^{Krawinkler, H. and Nassar 1992}):

where F _y and x _y are the yielding strength and displacement, respectively.

This normalized hysteretic energy, NE _Hμ is related to the number of times the accumulated elastic energy must be dissipated and, therefore, can be associated with structural damage. As shown in Figure 22, the largest NE _Hμ (maximum) match with part of the area with reported damage. These NE _Hμ demands were computed for an elasto-perfectly-plastic model using Mexican regulations to calculate the single-degree-of-freedom (SDOF) lateral strength. The maps correspond to a value of relatively large ductility of the structure (Q = 3.0 from the NTCD-2004). The damage reported in areas near downtown is associated with the structural performance of the buildings rather than the intensity of the seismic demands.

Figure 22 Normalized hysteretic energy maps for the September 19, 2017 earthquake, for the maximum demands envelope 

The largest NE _Hμ intensities are concentrated to the south of the lakebed area (Xochimilco, Tlahuac) and near the international airport. However, the downtown area, where severe damage was observed, shows relatively low demands (less than 10). This is because the maps have been computed with data from stations, but the interpolation did not accounted for soil characteristics at sites without recorded data, where larger amplifications (not recorded) could have been present.

Changes in the dominant soil period may have implications for the structural performance of buildings since the initial design conditions would be different by changing the soil-foundation behavior. This change of period has been considered in the update of the construction standards of Mexico City, with predictive models (^{Arroyo et al. 2013}). However, continuous monitoring of this period change should be taken, since it is no related to a natural phenomenon.

On the other hand, in soft soil sites similar to those in Mexico City, where every 10 to 20 years are subjected to intense long-duration and narrow-banded earthquakes, it is not enough to analyze the intensities associated with peak values, such as pseudoacceleration and spectral displacement. In such sites, it is necessary to analyze the seismic demands that consider the accumulated damage and dissipated energy. As can be seen in Figures 21 and 22, damage to sites with soil periods similar to Tg=1.5s may be explained from the high plastic demands, although in areas near downtown and Roma-Condesa neighborhoods, the damage is related to structural characteristics, such as the year of construction, structural pathologies or even accumulated damage over time.