Introduction

The Valley of Mexico is in the Trans-Mexican Volcanic Belt (TMVB, Fig. 1a), a volcanic arc overlying other Cretaceous and Cenozoic volcanic providences (Ferrari et al. 2012). Specifically, Mexico City is in the Mexico Basin (Arce et al. 2019); together with its surroundings, it lies inside the Valley of Mexico. This complex geological framework is responsible for the seismic wave behavior in the Valley of Mexico and is considered a paradigmatic example worldwide. Distinctively, the \({M}_{\text{w}}\) 8.1 earthquake on 19 September 1985 marked a scientific watershed in understanding the structure and site effect contributions to the anomalous behavior. Multiple mechanisms contribute to the complex wavefield patterns observed in the recordings when an earthquake occurs.

Fig. 1
figure 1

a Geological and tectonic setting for the study zone. The red rectangle in the reference World map delimits our study zone. The brown line in the Mexico reference map is the delimitation of the Trans-Mexican Volcanic Belt (TMVB), according to Ferrari et al. (2012). The red lines represent the Cocos and Rivera plates depth under the North American plate (Ferrari et al. 2012). The blue numbers are the age of the plates in Ma, and the green ones are their convergence rates in cm/year. The triangles are the stations used in this study: the purple ones were used to retrieve receiver functions (RF) and perform their joint inversion with dispersion curves (Aguilar-Velázquez et al., 2023). In contrast, the mint ones were used to retrieve empirical Green functions (EGF) from seismic noise. The orange diamonds correspond with mountain ranges. The blue rectangles are the states’ names: H is for Hidalgo, T, for Tlaxcala, P, for Puebla, and CDMX, for Mexico City; they are separated by the political boundaries presented as solid black lines. b Temporary distribution of the stations used to compute the 433 receiver functions considered for the joint inversion with the dispersion curves. c Azimuthal distribution of the receiver functions per quadrant: 1 is for the Atlantic Ocean (northeast), 2 for South America, 3 for the Fiji region (southwest), and 4 for the Kamchatka and Alaska region (northwest). d Seismic noise curves. Solid gray lines are the New Low Noise Model (NLHM) and New High Noise Model (NHNM) curves (Peterson 1993). The frequency ranges in color shades denote the noise sources described in Sect. “Empirical Green functions retrieving” (the green region is for cultural noise, while the blue and yellow regions are for microtremors). Finally, the frequency range for body waves is denoted by a solid blue line, surface waves by solid and dashed red lines, and earthquake hazard studies by a solid purple line (Ammon et al. 2020)

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For example, the first documented phenomena were the amplification and the long duration of strong motions. Singh et al. (1988) reported amplification by a factor from 8 to 50 in seismic records inside the Mexico City basin compared to a station located in the hill zone. Ordaz and Singh (1992) quantified the regional amplification, considering earthquakes occurred at similar distances from Mexico City along the trench, leaving no doubts about the essential role of the crustal structure in the seismic wave behavior. Additionally, Cárdenas-Soto and Chávez-García (2003) showed an azimuthal dependence in the amplification and duration of ground motion in Central Mexico. This implies that different seismic wave paths can result in contrasting behaviors due to diffracted pulses in the crust. They observed significant variability in the group velocity dispersion curves for periods shorter than 5 s. Furthermore, they concluded that for periods longer than 5 s, the crustal structure plays a crucial role by acting as an “efficient waveguide.” Years later, Chávez-García and Quintanar (2010) observed lateral variations in group velocity within the TMVB, with velocities ranging from 2.0 to 2.7 km/s for a 5 s period. This confirms previous observations at 5 s (0.2 Hz) and suggests differences in dispersion curves and, consequently, in the structure. However, Cruz-Atienza et al. (2016) showed that not only do the regional effects contribute to the long duration of the strong motions in Mexico City, but the basin also increases them in the range from 0.3 to 0.5 Hz since the energy propagates long distance in the deep structure.

On the other hand, Pita Sllim (2019) showed that receiver functions were sensitive to complex patterns observed in the crustal structure beneath Mexico City, suggesting that it was more complex than previous velocity models proposed by Cruz-Atienza et al. (2010) and Espíndola et al. (2017), who also retrieved them from receiver functions. Therefore, Aguilar-Velázquez et al. (2023) obtained a high-resolution S-wave velocity model from joint inversion of receiver functions and dispersion curves named Velocity Model from Receiver Functions and Dispersion Curves (VMRFDC), with VM also referring to Valley of Mexico. The authors showed a remarkable correlation between the VMRFDC velocity anomalies and geological and geophysical anomalies reported in the literature. However, a numerical validation of this model is necessary. Consequently, we complemented the VMRFDC velocity model with information also retrieved from joint inversion of receiver functions and dispersion curves at ten additional stations outside of Mexico City (Fig. 1a, b, c), obtaining the VMRFDC v2.0 model. To validate this model, we compare empirical Green functions (EGF) computed from seismic noise with synthetic Green functions (SGF) from numerical simulations using this model.

Using data from 25 broadband stations distributed across the Valley of Mexico (Fig. 1a), we estimated EGF from the seismic noise time series cross-correlations of the three seismometer components (HHZ, HHN, and HHE), i.e., we retrieved the diagonal components of the Green’s tensor for each pair of stations. On the other hand, the VMRFDC v2.0 was adaptatively meshed, and we performed numerical simulations of seismic wave propagation using the impulse response as a source, allowing us to retrieve SGF. Having the noise EGF and the SGF, this article compares them. Our results suggest similarities between both EGFs, explaining the previously reported lateral variations at 0.2 Hz and establishing the VMRFDC v2.0 as a well-validated model suitable for multiple applications. Additionally, we observed prominent phase changes between horizontal EGF and SGF at 0.2 Hz; these differences were quantified in this article for the first time and contribute to the knowledge of seismic wave behavior at this frequency.

Empirical Green functions retrieving

A Green function (GF) is defined as the wavefield response retrieved in one station due to a single force applied to the location of another station. At the beginning of the XXI century, many studies showed that the cross-correlation between seismic noise recorded in two stations may be an empirical way to retrieve de GF (i.e., EGF); this is true by assuming that the noise is a diffuse wavefield (Weaver and Lobkis 2001a, 2001b, 2004; Snieder 2004; Shapiro and Campillo 2004).

An important assumption when we work with seismic noise time series is that they are stationary, meaning that their mean and variance do not change with time (i.e., the statistical characteristics are not time-dependent; McNamara and Buland 2004). McNamara and Buland (2004) also described the main contributors to the seismic noise signals. The first contributor is the “cultural noise” due to anthropogenic activity, which can be found in the frequency range of 1–10 Hz (green region in Fig. 1d). The second contributor is geology, water, and wind. The third is the “microseisms,” defined by two peaks in the seismic noise curves: the first one, from 0.0625 to 0.1 Hz (blue region in Fig. 1d), known as the single-frequency peak, and the second one, from 0.125 to 0.25 Hz (yellow region in Fig. 1d), known as the double-frequency peak. The fourth contributor is the “instrumental noise,” which occurs due to the electromechanical operation of the instruments. The last one is the “earthquakes”; teleseisms may produce surface waves with dominant frequency values lower than 0.1 Hz, while local and small events may produce waves with dominant frequencies greater than 1 Hz. Figure 1d summarizes the frequency range of these noise bands, the body and surface waves, and the earthquake hazard frequency range (Ammon et al. 2020).

According to Stehly et al. (2006), if the seismic noise sources are homogeneously distributed in the space, the EGF will be symmetric, meaning their causal and the anticausal parts have identical waveforms. However, according to these authors, if this requirement is not fulfilled, the EGF can still be recovered, but it will not be symmetric. This last case is essential for the Valley of Mexico. We used two stations from the SSN network for the seismic noise cross-correlations: PZIG, located at Ciudad Universitaria, Mexico City, and YAIG, located at Yautepec, Morelos. Montoya Quintanar (2018) evaluated the temporary variations of the seismic noise in the stations of the Servicio Sismológico Nacional (SSN, National Seismological Service of Mexico) network. For PZIG, he observed that the seismic noise curves exceed the New High Noise Model (NHNM; Peterson 1993; McNamara and Boaz, 2005) in the frequency range from 1.0 to 5.0 Hz; the author attributes this behavior to the anthropogenic activity since the station is in the main campus of the Universidad Nacional Autónoma de México (UNAM, National Autonomous University of Mexico). However, in the case of YAIG, the seismic noise curve is within the limits established by the New Low Noise Model (NLHM) and the NMHM (Montoya Quintanar 2018). Specifically, for the 0.2 Hz (an important frequency analyzed in this article), the average seismic noise spectral amplitude for the YAIG station was approximately − 130 dB, while for the PZIG station was − 125 dB. This analysis suggests that stations inside Mexico City might be noisier than those outside due to anthropogenic activity. Another evidence that supports this hypothesis can be found in Saldaña-Tejada et al. (2022). They presented a relationship between the county population and the median noise level and concluded that the bigger the population, the higher the median noise level. Consequently, symmetric EGF is expected not to be recovered since the seismic noise sources are mainly concentrated within Mexico City with respect to the stations outside.

We used data from 23 broadband stations of the Red Sísmica del Valle de México (RSVM; Quintanar et al. 2018) and two from the Instituto de Geofísica network (MX; SSN 2023); see Fig. 1a. We used seismic noise data for nine months, from January 1 to September 6, 2017. First, we identified the days with complete records (i.e., we removed the gaps). The next step was to homogenize the data in SAC format (per day and then per station). Then, we removed the mean, the trend, and the instrument response from the records. Finally, we decimated the signals from 100 to 5 Hz, thus the Nyquist frequency of the seismic noise records was now 2.5 Hz. This procedure was done for each station’s three components (HHZ, HHN, and HHE). Then, the cross-correlations were performed using the Python notebook called “Ambient Seismic Noise Analysis,” from the project “seismo-live” (Krischer et al. 2018). This code follows the processing workflow established by Bensen et al. (2007); we summarize the processes in the following sequence:

  1. 1.

    Seek for standard days per pair of stations to correlate per day.

  2. 2.

    Band-pass filtering from 0.01 to 2.00 Hz.

  3. 3.

    Time domain normalization using the one-bit algorithm (i.e., positive values of the signal were modified to 1.0, while the negative ones were changed to − 1.0).

  4. 4.

    Spectral whitening in the 0.01–2.00 Hz frequency range (i.e., the same used for the band-pass filter).

  5. 5.

    Cross-correlations computation using 200-s size windows.

  6. 6.

    Stacking using the mean.

Crustal velocity model

We used the VMRFDC model introduced by Aguilar-Velázquez et al. (2023). Figure 6 by Aguilar-Velázquez et al. (2023) shows the regions where S-wave velocity is estimated. Additionally, stations outside that domain were not included in the first version of the model. Consequently, we added the information on ACIG, AMVM, ATVM, INVM, MAVM, PPIG, TOVM, VTVM, YAIG, and ZUVM stations (Fig. 1a) to create an updated version of the VMRFDC model (VMRFDC v2.0). We retrieved 433 velocity records of 258 teleseismic events, with M ≥ 6.0 between January 2010 and January 2023, recorded in the ten stations mentioned above; Fig. 1b, c present the temporary and azimuthal distribution of the receiver functions. We performed the joint inversion scheme used by Aguilar-Velázquez et al. (2023), which we briefly summarized as follows:

  • 1. Identify the P-wave arrival manually; if it was unclear, TauP (Crotwell et al. 1999) was used to identify it, but it was manually adjusted.

  • 2. Cut windows from 5 s before the P-wave arrival time and 20 s after. Then, the mean and trend were removed, and a cosine taper of 5% was applied.

  • 3. Rotate the records from the vertical-northeast (ZNE) system to the vertical-radial-transverse (ZRT) one.

  • 4. Retrieve radial receiver functions using the iterative deconvolution algorithm by Ligorría and Ammon (1999). The spectral content of the retrieved receiver functions is controlled by a Gaussian filter defined as:

    $$G(\omega)=\exp\left(\frac{-{\omega }^{2}}{4{a}^{2}}\right),$$
    (1)

    where \(\omega\) is the angular frequency and \(a\) is the Gaussian width. As Aguilar-Velázquez et al. (2023), we retrieved three different radial receiver functions per teleseismic-event record using three values for \(a\): 2.5 (1.2 Hz), 5.0 (2.4 Hz), and 10.0 (4.8 Hz); the values in parentheses are the maximum frequency reached per Gaussian width. Finally, we kept only the receiver functions with reproducibility larger than 70%.

  • 5. Cluster of the events, per station, by ray parameter and back Azimuth to ensure that the receiver functions in a cluster map the same crustal structure.

  • 6. Collect the dispersion curves for each station in the Valley of Mexico from the work by Spica et al. (2016); they were retrieved using noise and coda cross-correlations in Mexico and southern USA. We used their dispersion curves retrieved for the Valley of Mexico.

  • 7. Jointly invert the receiver functions and the dispersion curves per station, using the algorithm by Julià et al. (2000, 2003). The starting model was the same as in Table 1 by Aguilar-Velázquez et al. (2023). The weight value for the joint inversion was 0.3, meaning that the receiver functions have more influence than dispersion curves.

  • 8. Keep only the models generated when the semblance (\(S\)) between the data (\(D\)) and inverted (\(I\)) receiver functions, defined as:

    $$S=\left(0.5-\frac{\text{corr}\left(D,I\right)}{\text{auto}\left(D\right)+\text{auto}\left(I\right)}\right)\times 100\text{\%},$$
    (2)

    was lower than 20%. We also established a criterion for dispersion curves: a mean absolute error lower than 10%. Finally, we only considered models with a maximum S-wave velocity value of 5.5 km/s, eliminating unstable solutions.

  • 9. Project the velocity models to the ray path using the pierce points calculated each 2 km with the retrieved velocity models, obtaining the S-wave velocity distribution per depth.

Aguilar-Velázquez et al. (2023) performed a geostatistical analysis after the joint inversion to estimate the S-wave velocity using kriging. However, this was impossible for the above stations due to the great distances between them. Instead, we performed a classical interpolation using MATLAB’s “natural neighbor interpolation” algorithm. Once the interpolation was done, a smoothing algorithm was applied to mitigate the effect of non-realistic velocity contrasts. Figure 2 shows the VMRFDC v2.0 S-wave model retrieved from this work, extending it outside Mexico City. P-wave velocity (\(\alpha\)) was retrieved using the \({V}_{P}/{V}_{S}\) ratio equal to \(\sqrt{3}\), i.e., a Poisson solid. Then, the density (\(\rho\)) is calculated using the Berteussen (1997) relationship:

Fig. 2
figure 2

Updated version of the VMRFDC velocity model (VMRFDC v2.0). This model includes information from stations outside Mexico City. The numbers in purple in the upper left corner indicate the depth of the slice

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$$\rho =0.32\cdot \alpha +0.77,$$
(3)

except in the first layer (the TMVB), where it was fixed to 2.2 g/cm3 (Shapiro et al. 2001).

Numerical validation

When a velocity model is retrieved using, for example, full-waveform inversion, it is assumed that the model will reproduce the seismograms. However, as discussed in the section above, the velocity model was retrieved from a joint inversion of receiver functions and dispersion curves; consequently, as discussed, it was necessary to validate it using seismograms and not only geological and geophysical information.

Meshing the velocity model

We mesh the velocity model using TetGen (Si 2015). This software allows the generation of tetrahedral meshes for any 3D domain using Delaunay algorithms. We decided to use an adaptative mesh to optimize the computing time of the simulations. To do this, we defined the maximum frequency of the simulation as 0.5 Hz; this value justification can be found in the following sections. Then, the minimum wavelength, \({\lambda }_{\text{min}}\), is calculated as:

$${\lambda }_{\text{min}}=\frac{\beta }{0.5},$$
(4)

where \(\beta\) is the S-wave velocity at each node. We employ an iterative procedure to ensure the accurate propagation of the minimum wavelength, \({\lambda }_{\text{min}}\), in the model with the fewest elements. This process starts with an initial coarse mesh and progressively refines it iteratively. In each iteration, we identified the nodes with an adequate size since, to simulate the wavefield accurately, DGCrack requires that the elements’ size warranties the presence of at least three elements per minimum wavelength (Etienne et al. 2010). The elements that do not fulfill this size condition are updated in the next iteration until all the elements are sized correctly. Figure 3 shows a vertical profile of the resulting mesh for 0.5 Hz using the updated VMRFDC model presented in Fig. 2.

Fig. 3
figure 3

Vertical cross section of the mesh generated from the VMRFDC v2.0 model. If the S-wave velocity decreases, the element size becomes smaller

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As it was explained by Aguilar-Velázquez et al. (2023), the receiver functions and dispersion curves were not sensitive to the sedimentary basin; this was due to the frequency range reached by their receiver functions (up to 4.8 Hz); they can only be sensible to layer up to 1 km, and the deepest point estimated for the sedimentary basin was around 0.5 km. Therefore, solving its structure with the joint inversion was impossible, meaning this model could only be validated for low frequencies in this first approach. Consequently, the basin was not included in the numerical simulations since we want to prove that the pulse duration could be explained only by the effect of the complex crustal structure. Also, not introducing the basin saves computational time, crucial for these simulations.

Simulations with DGCrack

DGCrack (Etienne et al. 2010; Tago et al. 2012) is a software that solves the velocity-stress weak formulation of the elastodynamic equations in the 3D domain using a hp-adaptative discontinuous Galerkin method. The code provides the possibility to model the wavefield produced by single forces acting in the coordinate system directions (i.e., in the vertical, north, or east directions), allowing us to recover the Green tensor’s diagonal. The p-adaptative, interpolation order per element, was unnecessary in solving our problem since the mesh does not change the element sizes drastically. Instead, all the elements used were second order, i.e., quadratic interpolation functions.

At the surface, the free boundary conditions were applied. However, for the rest of the boundaries, we decided to use convolutional perfectly matched layers (CPML) to satisfy the absorbing boundary conditions since we have a finite medium. Following Collino and Tsogka (2001) and Etienne et al. (2010), we constructed CPML of three times the maximum value of all the minimum wavelengths calculated according to the largest frequency in our simulation (0.5 Hz). To ensure that the waves travel at least three minimum wavelengths in all depths, we choose the maximum S-wave velocity (5.1 km/s) to estimate the thickness of CPML as:

$$L_{{{\text{CPML}}}} = \frac{{5.1\left[ {\frac{{{\text{km}}}}{{\text{s}}}} \right] \times 3}}{{0.5\left[ {{\text{Hz}}} \right]}} = 30.6\,{\text{km}}.$$
(5)

However, we decided to round this value to 32 km to ensure we fulfilled the criteria. The 32 km layer was added at the boundaries by extending the S-wave velocities in the borders of our model; the CPML elements also had an adaptative size, depending on such velocities, to reduce the computation time. According to Collino and Tsogka (2001) and Etienne et al. (2010), the damping profile (\({d}_{\theta }\)) varies from 0 at the beginning of the CPML, up to \({d}_{{\theta }_{\text{max}}}\) at the end as:

$${d}_{\theta }={d}_{{\theta }_{\text{max}}}{\left(\frac{{\delta }_{\theta }}{{L}_{\text{CPML}}}\right)}^{2},$$
(6)

where:

$${d}_{{\theta }_{\text{max}}}=-3\alpha \frac{\text{log}\left({R}_{\text{coeff}}\right)}{2{L}_{\text{CPML}}},$$
(7)

\({\delta }_{\theta }\) is the perpendicular distance from the model region to the CPML in which the element barycenter is located, \({R}_{\text{coeff}}\) is the theoretical reflection coefficient, and \(\alpha\) is the maximum P-wave velocity in the model. For our numerical simulations, \({R}_{\text{coeff}}=0.01\) and \(\alpha =\left(\sqrt{3}\right)\left(5.1 \frac{km}{s}\right)\approx 8.8 \frac{km}{s}.\) Finally, the CPML elements also used a second-order interpolation.

Additionally, we added spatial support to the source. This strategy distributes the source over a 1.5 km ratio (i.e., almost half of the minimum wavelength for the TMVB) and helps to avoid numerical noise.

DGCrack works in parallel and calculates the time step for the numerical simulation, ensuring the Courant’s condition is fulfilled, meaning the time step (\(\Delta t\)) for the numerical simulation must satisfy:

$$\Delta t\le \frac{{e}_{s}}{\alpha },$$
(8)

where \({e}_{s}\) and \(\alpha\) are the element size and its P-wave velocity, respectively. Since both values are taken from the same element, the one with the lowest ratio is taken for the \(\Delta t\) calculation. The time step is penalized for our case due to the S-wave velocity contrast between the TMVB and the upper crust (from ~ 1.65 to ~ 2.85 km/s) since the refined mesh produces smaller elements than required at the discontinuity.

Now, we compare the EGF recovered from seismic noise with the synthetic seismograms. Let the EGF retrieved using seismic noise (\({\text{EGF}}_{k}\)) defined as:

$${\text{EGF}}_{k} \left( t \right) = W_{{N_{k} }}^{{S_{c} }} \left( t \right) \otimes W_{{N_{k} }}^{R} \left( t \right),$$
(9)

where both, \({W}_{{N}_{k}}^{{S}_{c}}\left(t\right)\) and \({W}_{{N}_{k}}^{R}\left(t\right)\) are seismic noise wavefields recorded in the \(k\) component; \({S}_{c}\) indicates the station assumed as the virtual source, and \(R\) is the receiver and \(\otimes\) means cross-correlation. As discussed in Sect. “Empirical Green functions retrieving,” \({W}_{{N}_{k}}^{{S}_{c}}\left(t\right)\) and \({W}_{{N}_{k}}^{R}\left(t\right)\) were preprocessed using the Bensen et al. (2007) methodology.

We performed numerical simulations to compute synthetic seismograms using impulsive point forces, which can be interpreted as synthetic Green functions (\({\text{SGF}}_{k}\)). As Chen et al. (2014), we assume that:

$${\text{EGF}}_{k}\left(t\right)\approx {\text{SGF}}_{k}\left(t\right);$$
(10)

the \({\text{SGF}}_{k}\) were retrieved using synthetic seismograms of the impulse response wavefield recorded in the \(k\) component if a point source is also applied in the \(k\) direction. These \({\text{SGF}}_{k}\) are valid up to the maximum frequency reached in the study, \({f}_{\text{max}}\) (0.5 Hz for our case) since our source time function has a planar spectrum up to \({f}_{\text{max}}\). Finally, since the impulsive source is applied in the same direction as the records, we only analyze the diagonal of the empirical and synthetic Green tensors.

Results

Figure 4 shows the source time function and its amplitude spectrum; note that the spectrum is planar up to ~ 0.50 Hz. Additionally, Fig. 4 shows the wavefield snapshots, considering that the source is located at the YAIG station. Further, Fig. 5 shows an example of the cross-correlations retrieved from seismic noise and the numerical simulations in the Valley of Mexico if the source were at the YAIG station and using the HHZ component; seismograms are band-pass filtered from 0.10 to 0.22 Hz. This figure illustrates that the SGF surface wave pulses arrive at the expected time compared to the EGF ones. Additionally, there is no observable azimuthal dependence of the travel times between the EGF and the SGF, indicating that the VMRFDC v2.0 represents a reliable 3D velocity structure. Upon examining the seismograms in Fig. 5, it becomes evident that the EGF lacks high-frequency content compared to the SGF. This discrepancy is acceptable since this article focuses solely on comparing large wavelengths associated with the crust.

Fig. 4
figure 4

Examples of a numerical simulation of a vertical source applied at the YAIG station location. The source time function and its amplitude spectrum can be found on top; the solid green line in the spectrum indicates the 0.5 Hz frequency. The wavefield in the snapshots is amplified by a factor of 10 to see all the changes in the propagation

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Fig. 5
figure 5

Left: EGF of the HHZ component (\({\text{EGF}}_{Z}\left(t\right)\)); they were retrieved using seismic noise and supposing the virtual source is at YAIG. Solid gray lines are the EGF recovered from the cross-correlation, whereas solid blue lines are those EGF filtered in the frequency range 0.10–0.22 Hz. Center: SGF of the HHZ component (\({\text{SGF}}_{Z}\left(t\right)\)+\({\text{SGF}}_{Z}\left(-t\right)\)) retrieved from the numerical simulations. Right: azimuth regarding YAIG location; 0° is for North, whereas 90° is for East. The y-axis in this figure is not to scale; it only represents the distance

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On the other hand, an interesting observation can be found at the Green tensor diagonal. Figure 6 shows some examples. The tensor’s diagonal for the AOVM-PZIG pair of stations reveals that the pulses in the horizontal components (HHN and HHE) are similar. However, if we analyze pairs separated for a greater distance (as the YAIG-TXVM, the VTVM-PZIG, or the ZUVM-PZIG pair), we observe that the pulse in the HHN arrives before the one in the HHE. A similar effect can be observed in the TOVM-TXVM pair, but the HHE pulse arrives first this time. This observation suggests the presence of a time shift. Additionally, the pulse duration is also different in the components. In this work, these effects may be explained by the complexity of the crustal structure observed in the VMRFDC v2.0 model.

Fig. 6
figure 6

Green tensor diagonals for different source (S) and receiver (R) combinations; they were retrieved using seismic noise; thus, we show the \({\text{EGF}}_{{k}}\left(t\right)\). Dotted lines correspond with the trajectories and follow the nomenclature color of their respective tensor. The regions delimited by the solid lines correspond with mountain ranges: brown for Sierra de las Cruces, green for Sierra Chichinautzin, and pink for Sierra Nevada

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Knowing that there is a time shift presumably produced by the crustal structure complexity, and the next step is quantifying it, we performed the continuous wavelet transform (CWT) analysis introduced by Kristeková et al. (2006), a misfit criterion for comparing seismograms. We aimed to contrast the HHN and HHE Green pulses in a frequency-time domain. This comparison is crucial to also understanding the frequency content differences. The steps to perform this analysis are described as follows:

  1. 1.

    Transform the HHN and HHE using the CWT kernel, resulting in two time–frequency matrices (one for each component).

  2. 2.

    Calculate the amplitude and phase of the CWT matrices since the CWT produces complex numbers like the Fourier transform.

  3. 3.

    Subtract the amplitude and phase matrices of HHE from HHN, which will produce two new matrices: the differences in amplitude and phase.

  4. 4.

    Average the amplitudes of all the time samples per frequency and obtain the amplitude and phase differences as a function of the frequency.

  5. 5.

    Average the amplitudes of all the frequency samples per time and obtain the amplitude and phase differences as a function of the time.

  6. 6.

    Obtain the single value misfit for the amplitude and phase by calculating their integrals in frequency and time.

The central hypothesis when using this technique is that if the pulses at the HHN and HHE components are equal, all the differences should be zero. Therefore, when the pulses are different, we can characterize where the changes are in time and frequency. Figure 7 presents the comparison between the CWT analyses of the Green functions retrieved from both data (EGF) and numerical simulations (SGF) using the YAIG-TXVM station pair (separated ~ 65 km; the red path in Fig. 6). We observe a change in the phase around 25 s and 0.20 Hz in the EGF (region delimited by the black rectangle in Fig. 7k). This change is also observed in the SGF but at ~ 0.25 Hz (region delimited by the black rectangle in Fig. 7n). The differences in amplitude (Fig. 7c, f, g) may have the same shape. Both observations suggest that the VMRFDC v2.0 model may reproduce the EGF. We repeated this analysis using the TOVM-TXVM station pair (separated ~ 80 km; purple path in Fig. 6); please note that the analysis was done at TXVM for a fair comparison; quasi-perpendicular paths were selected (see Fig. 6). Figure 8k reveals that the phase shift in the EGF also occurs around 0.20 Hz, but now at ~ 30 s (region delimited by the black rectangle in Fig. 8k); it makes sense that the shift appears after the YAIG-TXVM pair since the distance is larger in this example. In this case, the SGF suggests that the phase shift is at ~ 0.23 Hz, which is again consistent with the EGF. Figure 8c, f, g suggests that the differences in relative amplitude are similar for the EGF and the SGF (see the regions delimited by the black rectangles). Figures in the Supplementary Information show examples of the other paths observed in Fig. 6 (see Figs. S1 to S3 in the Supplementary Information). Furthermore, Fig. 8h reveals that in the TOVM-TXVM path, the observed EGF obtained from the HHN component has a longer motion duration than in the HHE component; the numerical simulation also shows that behavior, meaning that the VMRFDC v2.0, can reproduce such a feature.

Fig. 7
figure 7

Example of a CWT analysis performed using YAIG and TXVM stations. a Amplitude spectrums of the EGFs retrieved from observed data. b Average amplitude misfits between horizontal EGFs retrieved from observed data, per frequency, calculated using the amplitude differences matrix in c (solid black line delimited region for a fair comparison). c Amplitude differences matrix of the horizontal EGFs retrieved from observed data. d Amplitude spectrums of the horizontal SGFs retrieved from numerical simulations. e Average amplitude misfits between the horizontal SGFs retrieved from numerical simulations, per frequency, calculated using the amplitude differences matrix in f (solid black line delimited region for a fair comparison). f Amplitude differences matrix of the horizontal EGFs retrieved from numerical simulations. g Average of the amplitude misfits per time, calculated using the amplitude differences matrices; the mint line is for observed data, and the brown line is for numerical simulations. h Normalized horizontal (HHN and HHE) Green pulses comparison in time. i Phase spectrums of the EGFs retrieved from observed data. j Average phase misfits between horizontal EGFs retrieved from observed data, per frequency, calculated using the phase differences matrix in k (solid black line delimited region for a fair comparison). k Phase differences matrix of the horizontal EGFs retrieved from observed data. l Phase spectrums of the horizontal SGFs retrieved from numerical simulations. m Average phase misfits between the horizontal SGFs retrieved from numerical simulations, per frequency, calculated using the amplitude differences matrix in n (solid black line delimited region for a fair comparison). n Phase differences matrix of the horizontal SGFs retrieved from numerical simulations. o Average of the phase misfits per time, calculated using the phase differences matrices; the mint line is for observed data, and the brown line is for numerical simulations. The figure also presents the single-valued envelope (amplitude) and phase misfits (EM and PM, respectively) calculated using the windows delimited by the solid black lines (i.e., using the Green pulse only for a fair comparison)

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Fig. 8
figure 8

Example of a CWT analysis performed using TOVM and TXVM stations. a to o have the same labels as Fig. 7

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Discussion

Before discussing our results, we consider it important to remember that a recent study by Hernández-Aguirre et al. (2023) supports the decision not to include the basin in the numerical simulations in this work. They also performed 3D numerical simulations, but they included the sedimentary basin and retrieved synthetic seismograms of the 17 July 2019 \({M}_{\text{w}}\) 3.2 earthquake instead. Their results suggest that the amplification occurs in the frequency interval from 0.4 to 0.7 Hz. Additionally, they establish that the dominant wavefields in the lake-sediments zone are the surface waves with predominant prograde particle motion; according to their results, the transition from retrograde to prograde particle motions occurs from 0.53 to 0.65 Hz (depending on the site). As reported before, our phase anomalies are detected around 0.2 Hz, meaning that the basin is meaningless to the seismogram, positioning the crustal structure as the primary contributor.

Some studies have shown that surface waves retrieved from EGF can be used to characterize azimuthal velocity changes. Castellanos et al. (2018) obtained a high-resolution S-wave velocity and radial anisotropy model of the Eastern TMVB zone using ambient seismic noise correlations. According to the authors, the discrepancy between Rayleigh and Love waves allowed them to produce anisotropic images for central Mexico. On the other hand, Granados-Chavarría et al. (2022) obtained a high-resolution anisotropic S-wave velocity model for the geothermal system of the Los Humeros caldera in Mexico also using Rayleigh and Love wave dispersion curves retrieved from seismic noise EGF. Both examples showed that surface waves are sensitive to anisotropic effects, and their interpretations followed that line of research. However, as we said before, in this article, we propose that the seismic wave behavior change observed in the Valley of Mexico may not be due to an anisotropic crust. Instead, we postulate that these effects may be explained by the complexity of the crustal structure observed in the VMRFDC v2.0 model since these velocity anomalies might influence the surface wave behavior. We must bear in mind that even transverse isotropic layers or discontinuities between isotropic materials may produce time-shifted observations (Stein and Wyssession, 2003), meaning that anisotropy is not the only feature that can induce them. Understanding this concept is crucial for contextualizing our findings in relation to previous studies. We hypothesize that the observations by Cárdenas-Soto and Chávez-García (2003) and Chávez-García and Quintanar (2010) of a prominent change in the seismic wave behavior at 5 s period (i.e., 0.2 Hz) may be explained by the 3D structure variations mapped by VMRFDC v2.0. However, we also suggest that these changes could potentially lead to time and phase shifts in the horizontal components.

Following the discussion on time and phase shifts, Figs. 7 and 8 also show the single-valued envelope (amplitude) misfit (EM) and the single-valued phase misfit (PM) values (left bottom values; the mint ones for the observed and the brown ones for synthetic), calculated as the integrals of the amplitude and phase differences matrices in the domain delimited from 0.10 to 0.35 Hz and 10–60 s (delimited by the solid black lines). The next step was to verify if those values were sensitive to the time shifts. To do this, we performed a sensitivity test by shifting the wavelet presented on top of Fig. 9. The shifting was done in two ways: the simple one considered only an amplification factor for the shifted signal and the time shift. The complex shift is considered a more sophisticated and realistic amplification factor controlled not only by a factor since it is also multiplied by a negative-power exponential function. The envelope and phase diagrams obtained as a function of the amplitude variations (Amp. v. in Fig. 9) and the time shifts (\({\delta }_{\text{shift}}\) in Fig. 9) revealed that the single-valued misfits (SVM) for the envelope may produce non-unique solutions, inducing imprecise interpretations. Instead, the SVM for the phase may be the value to interpret since, independently of the amplification factor, the minimum of the misfits’ function is always at \({\delta }_{shift}=0\) s, i.e., when there is no time shift. However, the criteria cannot distinguish between positive and negative \({\delta }_{\text{shift}}\). Taking this into account, the maps in Fig. 9 show that if the source is at YAIG or TOVM, the SVM for the phase is more significant as long as the distance increases, meaning that the time shifts between the HHN and HHE components should be more evident because the influence of the complex structure is more considerable as well since when the impulse response is registered in greater distances, the influence the crustal velocity model is more “accumulated” (or contributes more to the record) than in those seismograms recorded at shorter distances.

Fig. 9
figure 9

Sensitivity analysis of the CWT technique. Simple shift means that the signal is only time-shifted and multiplied by a scalar. In contrast, the complex shift means it is not only shifted since its amplitude is also modified using an exponential function. The single value misfit (SVM) is presented for envelope and phase in each case as a function of the amplitude (Amp. v.) and the time shift (\({\delta }_{shift}\)). The maps show the SVM for the phase using the observed EGF. As discussed in Sect. “Results,” this value is the most sensitive misfit for time shifts. Left: source (black star) at YAIG station. Right: source at TOVM station

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As discussed, the EGF retrieved from seismic noise is dominated by surface wave pulses. Since the anomalous behavior occurs in the frequency interval from 0.15 to 0.30 Hz, the next step was to calculate the radial and transverse eigenfunctions, using the VMRFDC v2.0 model, to investigate which part of the crustal structure is responsible for the observations reported above. We used the Computer Programs in Seismology (Herrmann 2013) and 1D velocity models retrieved at TOVM, TXVM, and YAIG station locations; two points were also included to have a representative 1D velocity model for the midpoint between the station’s paths (M1 and M2 in Fig. 10). Figure 10 presents the S-wave velocity profiles between the station pairs explored in Figs. 7 and 8. Additionally, Fig. 10 shows the radial (UR) and transverse (UT) normalized displacement of surface waves as a function of the depth and frequency. It also reveals that surface waves are sensitive until 20 km depth for the frequency range reported above. Consequently, the shallow crust may be responsible for the time shifts and anomalous behavior reported before. The velocity profiles support this asseveration since we see prominent changes in the shallow-crustal structure between the two paths in that depth interval; the YAIG-TXVM path has faster velocities than the TOVM-TXVM one.

Fig. 10
figure 10

S-wave velocity profiles between YAIG-TXVM (red path in Fig. 6) and TOVM-TXVM (purple path in Fig. 6) stations. Additionally, the radial (UR) and the transverse (UT) normalized displacements in depth are presented as a function of the frequency; they were retrieved using the Computer Programs in Seismology (Herrmann 2013), considering 1D velocity models beneath each station. Distance equal to 80 km is a common point for both profiles (TXVM station). M1 and M2 are the midpoints between each station pair

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The comparisons presented in this paper were made using normalized amplitudes since our goal was to show only the crustal structure contribution to the seismograms. However, this has an essential implication in our work: we can only interpret spectral changes between normalized signals, but not amplification factors; once we normalized the seismic noise records using the one-bit method, we lose the actual amplitudes of the EGF, meaning that no absolute amplitude changes can be interpreted. Instead, relative amplitudes can be interpreted. For example, Fig. 7h shows that for the YAIG-TXVM path, the HHN component oscillates before the HHE starts doing it. Figure 8h reveals that for the TOVM-TXVM path, the HHN wavefield vibrates when the HHE ones are attenuated. Both examples reveal that the wavefield behavior can be comparable using only the relative amplitudes. Fig. S4, in the Supplementary Information, presents that these differences can also be observed in the radial-transverse system for these analyzed paths.

However, the limitation described before does not mean we do not recognize the influence of the crustal structure in the amplification. As is well-known, the surface waves generated by Mexican subduction earthquakes’ wavefield and the impedance contrast between the crust and the TMVB and between the TMVB and the sedimentary basin are responsible for the significant amplification in the Valley of Mexico. Let us provide a simple analysis of body waves to prove this. We approximated the velocity of the layer just beneath the TMVB, from VMRFDC v2.0, as \({\alpha }_{\text{crust}}=\) 4.94 km/s, \({\beta }_{\text{crust}}=\) 2.85 km/s, and \({\rho }_{\text{crust}}=\) 2.35 g/cm3. Then, the TMVB properties were approximated as \({\alpha }_{\text{TMVB}}=\) 2.86 km/s, \({\beta }_{\text{TMVB}}=\) 1.65 km/s, and \({\rho }_{\text{TMVB}}=\) 2.20 g/cm3. Finally, an average velocity structure for the basin was retrieved from the Shapiro et al. (2001) velocity model, approximating the basin parameters as \({\alpha }_{\text{Basin}}=\) 2.00 km/s, \({\beta }_{\text{Basin}}=\) 0.40 km/s, and \({\rho }_{\text{Basin}}=\) 2.05 g/cm3. Assuming plane layers, which is realistic for the crust-TMVB discontinuity but may not be for the TMVB-basin one, we calculated the P-SV and SH reflection and transmission coefficients using Table 13.1 by Ammon et al. (2020) equations and the Snell’s law. Figure S5, in the Supplementary Information, presents the reflection and transmission coefficients for both seismic discontinuities as a function of the incidence angle.

We see that if, for example, a P-wave is incident at an angle of 38.0° (a value obtained averaging incidence angles of some Mexican subduction earthquakes) from the crust to the TMVB, it may produce a P-wave amplified ~ 20%, if this transmitted wave is incident at the basin, it may produce an extra amplification of ~ 20%, meaning that if the P-wave amplitude was unitary at the beginning, the amplitude at the surface should be, at least, 1.44. On the other hand, if an SV-wave is incident at an angle of 39.0° (a value also obtained by us averaging incidence angles of some Mexican subduction earthquakes) from the crust to the TMVB, it may produce an SV-wave with no amplification; however, if this wave is incident at the basin, it may produce an amplification of ~ 50%, meaning that if the SV-wave amplitude was unitary at the beginning, the amplitude at the surface should be, at least, 1.50. Finally, the SH case reveals an interesting result since if an SH-wave is incident at an angle of 39.0°, the amplification due to the TMVB is ~ 25%. Then, the basin gives an extra amplification of ~ 62.5%, meaning that if the SH-wave amplitude was unitary at the beginning, the amplitude at the surface should be at least 2.03. This simple example may help in understanding the influence of the TMVB in the amplification and in having a simplified idea of the basin’s effect. The average amplification factors due to the TMVB should give a precise idea of the shallow crust influence in the amplification. However, the actual amplification factors reported in the literature for stations in the sedimentary basin may differ from those reported in this paper since they also include the attenuation, the site effects, and the basin’s 3D model.

Conclusions

We introduced the VMRFDC v2.0 model, an upgraded version of the VMRFDC model obtained by Aguilar-Velázquez et al. (2023) using stations outside Mexico City that extends from Toluca, Estado de México to Sierra Nevada in the east–west direction, and from Yautepec, Morelos to Zumpango, Estado de México in the north–south direction (see Figs. 1a, 2, 6). The authors previously compared the VMRFDC model with geological and geophysical features reported in the literature; however, in this work, we present the validation of the entire model using numerical simulations of the wave propagation employing a discontinuous Galerkin program, DGCrack. An anomalous of the seismic wave behavior at 5 s period was previously reported by Cárdenas-Soto and Chávez-García (2003) and Chávez-García and Quintanar (2010). Our synthetic EGF obtained from numerical simulations and our velocity model proposed here showed that the first 20 km of the crustal structure is responsible for the time shifts and the phase and amplitude changes observed in the EGF at 0.2 Hz. Our numerical simulations are enough to prove the influence of the crustal structure in the frequency range analyzed in this work because the crustal structure is positioned as the primary wavefield contributor, validating our results. Consequently, the VMRFDC v2.0 was thoroughly tested, and the results suggest that it is a good velocity model to be used by the seismological community; this is the main contribution of this work.

However, we recognize a limitation of the results presented in this article. Since we compare large wavelengths (or low frequencies), the use of VMRFDC v2.0 for higher-frequency studies may still be tested; for example: 1) the numerical simulations of small earthquakes that occurred inside the Valley of Mexico, or 2) its use for earthquake location and moment tensor retrieving.