The horizontal-to-vertical spectral ratio (HVSR)

The HVSR, or commonly referred to as the Nakamura method (^{Nakamura, 1989}), is used to evaluate the site effect, i.e., the relative amplification of the subsoil due to the presence of a soft soil layer and the frequency at which this amplification occurs. The method makes use of ambient seismic noise by assuming the wave field is predominantly formed by surface waves (^{Bonnefoy et al., 2006}). If the layer is present, the spectral ratio of the average Fourier amplitude spectra of horizontal components, among the amplitude spectrum of vertical component, provides the seismic response of the soft layer. The frequency where the maximum response appears is called the fundamental frequency of vibration of the soil layer (f₀). The Nakamura method represents a suitable approach to amplification due to S-wave resonance (^{Lermo & Chávez, 1994b}). Therefore, it is possible to approximate the thickness of the soft soil layer (if it exists) by the formulation f₀=β/4H.

Where:

f₀ = fundamental frequency

β = shear-wave velocity and

H = layer thickness

In the assessment of the HVSR, we used 30 min of ambient noise recording to obtain the average spectral ratio result of windows of 40s duration with an overlap of 50 %. In this estimate, the spectra were smoothed by a Konno-Ohmachi window (^{Konno & Ohmachi, 1998}) with a smoothening fraction of 20 %.

Figure 2 shows the average HVSR curves of the 16 measurement points shown in Figure 1. Except for points P and L, the results show well-defined spectral ratios with an average relative amplification between 2 and 3. These ratios convincingly indicate the distribution of unconsolidated materials overlaying a more competent layer (interpretation criterion according to the SESAME guidelines; ^{Bard and SESAME-Team, 2004}). The range of vibrating frequencies that define these ratios is 4.5 to 7.0 Hz, indicating the variability of the thickness of the unconsolidated materials or the irregularity of the sublayer. If we consider that the average fundamental frequency of the 16 measuring points is 5.4 Hz and an average Vs velocity of 350 m/s (which we will demonstrate later), then the substratum depth is about 16 m. Some of the stations (F, G, H, J, and P) show a second peak near the frequency of 20 Hz with amplitude almost similar to the first peak. We can attribute that this second peak is related to an impedance contrast caused by a surface layer, which can behave independently of the entire soil column (^{Guéguen et al., 1998}; ^{Macau et al., 2015}). The presence of unconsolidated materials in the study area seems extraordinary because we have described an area of basaltic materials. However, site recognition shows volumes of highly fractured volcanic material with an irregular distribution of filling areas, as is showed in the image of Figure 1.

Figure 2: Mean HVSR curves of each of the recording points within the study area (Figure 1). The number within each graph indicates the maximum frequency of the HVSR, which can be interpreted as the proper frequency of vibration of the soil layer. In the upper right corner of each graph is indicated the letter denoting each measurement point 

Ambient noise tomography

Noise records are usually preprocessed to retain the stationary part of the background noise (^{Bensen et al., 2007}). In this process, an equalization of the Fourier spectrum (spectral whitening) of the noise recorded is performed so that there are contributions of the energy associated with the distinct phases. From the normalized records, we carry out cross-correlations between all pairs of receivers. In this work, we have chosen to cross-correlate windows of 60 s of overlapping duration 50 % over the 60 min of time record. In Figure 3a, we show the virtual source section 01, the result of the cross-correlation of the noise record of the receiver 01 with the remaining records of the 48 receivers. In this figure, it is observed that wave trains emerge whose amplitude is remarkable in time delays less than 1 s. The pulses have a high signal to noise ratio and emerge consistently either in the anti-causal part (times less than zero) or causal (times greater than zero). The correlation pulses between the receivers 01 and 10 show a sudden change in velocity from receiver 06. Between receivers, 10 and 19 phases and amplitude are lost in some receiver. In Figure 3b, we show the virtual source section of the receiver 25. Variations in the correlation pulse and the presence of secondary wave trains are observed in receivers 01 to 08 and 41 to 48. Moreover, at short distances (receivers 30 to 33), the correlation pulse loses its shape. Results similar to those shown in Figure 3 can be observed for the other 48 virtual source sections.

Figure 3: Virtual source sections result from the correlation of the record in receiver: a) 01 and b) 25, concerning the remaining 48. Traces are filtered between 5 and 15 Hz 

The symmetry of the pulses in the cross-correlation function is related to the equipartition condition, which must be completely satisfied to obtain the EGF. This condition is fundamental in IS under the view that the wavefield in a closed system should consist of arbitrary contributions of all possible modes of propagation, with equal weight on average (Weaver & Lobkis, 1982; ^{Schuster, 2014}). If the medium contains a sufficient distribution of scatters, then it is possible to satisfy the condition of equipartition with an unique source because each scatter acts as a secondary source. In open systems, the most appropriate definition of equipartition states that the flow of energy must be equal in all directions (e.g., ^{Snieder et al., 2007}). For practical purposes, the emergence of EGF either in the causal or anti-causal part can be used to characterize the subsoil, as it is enough to add the anti-causal part to the causal part to increase the coherence of the pulse. The extraction of the correlation pulse or EGF) is because the primary source of seismic noise is due to vehicular traffic (^{Halliday et al., 2011}) that occurs in the road circuit near the study area.

The emphasis of this study is to characterize the lateral variations of subsoil velocity. For this, we exploited the position of the delay times of the envelope maximum of the correlograms filtered in different frequencies (^{Cárdenas et al., 2016}). The identification of travel time (maximum correlation pulse) at different frequencies can then be used to produce a tomography image that provides the distribution of the velocity of the medium separating two receivers (^{Shapiro et al., 2005}; ^{Sabra et al., 2005}).

We used the filtered virtual source sections and extracted the maximum delay times from the envelope of the correlation pulse for 36 central frequencies generated from a mobile filter whose width is a function of the frequency for the range of 4 to 24 Hz. From those times, we built travel time tomographies to know the distribution of the group velocity. The study area was discretized by a mesh whose cells were 4 m per side, where the number of pairs of trajectories (rays) that covered the mesh, for 48 receivers (n=48), was n(n-1)/2=1128. The inversion processes were carried out by Simultaneous Interaction Reconstruction Tomographic (SIRT) method (^{Nolet & Wapenaar, 2009}) to derive phase velocity distributions. The inversion process was further improved by incorporating the maximum slope method (^{Arfken et al., 1999}), which minimizes the objective function. The objective function is defined as the difference between the observed data and the synthetic calculated data produced by the forward modeling, and sometimes with another term of regularization or damping (^{Uieda et al., 2013}).

Figure 4 shows as an example, the images of the results at the frequencies of 6, 8, and 12 Hz. The color scale is different for each image to highlight the contrasts between the resolved phase velocity values. This contrast results between the mean phase velocity of 250 m/s and 500 m/s. The existence of velocity values of 250 m/s at low frequencies, such as 6 Hz, indicates that in-depth little consolidated materials prevail in the East and West of the study site. At 6 Hz, velocity values higher than 650 m/s are observed concerning the results of the images at 8 and 12 Hz. High-velocity values at low frequencies indicate the presence of more compact material in depth. A high-speed anomaly highlights at the center of the array on all three frequencies. The distribution of high velocities to the south, center, and northwest part of the array is consistent with the distribution of basaltic materials observed at the site. In general, the resolved velocity at low frequencies is higher than at intermediate and high frequencies. This behavior is characteristic of the dispersion property of surface waves, the variation of velocity as a function of frequency.

Figure 4: Seismic noise tomography at frequencies of 6, 8, and 12 Hz, a), b) and c) respectively. The phase velocity distribution (m/s) in a UTM reference system is displayed. The black stars indicate the position of the geophones. The black dots represent the Centers of the cells where the tomography method is solved. The color scale indicates the phase velocity values, and it is different for each image 

The frequency-dependent velocity variation at each node of the tomography model is shown in Figure 5a. These values describe phase velocity dispersion curves of Rayleigh waves. The curves show that at frequencies close to 4 Hz (limit of the characteristic frequency of geophones), the phase velocities can be between 400 and 1400 m/s. Most curves describe velocity values between 400 and 600 m/s. for frequencies higher than 6 Hz, the decay of the curves shows values between 200 and 500 m/s. The curves shape in Figure 5a indicates they can reasonably be inverted to obtain a shear wave velocity profile. Figure 5b shows an example of the inversion of a dispersion curve (^{Herrmann, 2013}). The procedure consisted of proposing a model of 11 layers whose thickness increases with the depth at a constant velocity (in this case, 600 m/s). After several interactions, the inversion method converges to provide a Vs velocity profile. The example in Figure 5b shows excellent convergence towards the final model.

Figure 5: a) Dispersion curves obtained from travel time tomography, b) shear wave velocity profile resulting from the inversion of a dispersion curve. The discontinuous black line is the initial velocity model. The colored lines represent the different solved models until they converge to the final model (black line) 

We have constructed a volume of shear wave velocity as a function of depth (Voxler 4.0 Software) from the 1D profiles resulting from the inversion of each of the dispersion curves. Figure 6 shows the 3D model of this volume, where we have highlighted two isosurfaces that describe in a general way the subsoil structure in the study area. The first of them has a value of 700 m/s and indicates the limit of poorly consolidated materials distributed heterogeneously between the surface and approximately 16 m deep. As discussed in the section on HVSR spectral ratios (Figure 2), this depth is consistent with that suggested by the 1D model of short-wave propagation (Equation 1). The depth at which the substratum is located was also determined by ^{Sanchez (2019)} from join inversion results of the HVSR and dispersion curves obtained by SPAC (^{Aki, 1957}) and f-k (^{Capon, 1969}) methods.

Figure 6: Shear wave velocity model produced by the interpolation of 1D Vs profiles obtained from the tomographic results. Two isosurfaces are shown; at 700 m/s (green) and 300 m/s (blue lines) 

The second isosurface corresponds to the lowest velocity values (close to 300 m/s). The distribution of these values should be located close to the surface of the terrain. However, a low-velocity anomaly is observed between 10 and 20 m deep, which extends inward of the array to a distance of approximately 30 m. The anomaly coincides with an old crack located below the adjacent building. Most likely, that crack or lava tube is filled with tuffs and fractured materials. We observe that the low-velocity anomaly emerges due to inconsistent time delays of the correlation pulses. The irregularity of the correlation pulses described above suggests it is a cavity-like structure that probably causes resonance in the wave field extracted by SI (^{Catheline et al., 2008}; ^{Hillers et al., 2014}, ^{Argote et al., 2020}).