We adopt a novel processing strategy for ambient noise correlations proposed by . It is based on the premise that variable incident ambient noise conditions result in different correlation waveforms. By grouping single-window autocorrelations into clusters and stacking them selectively, we aim to increase their temporal coherence and to reduce the effect of temporally varying noise sources. Clustering is performed by applying Gaussian mixture models after reducing the data dimensionality through principal component analysis (PCA), and the Bayesian information criterion (BIC) is used to determine the optimal number of clusters, balancing misfit and model complexity .

We modify the original approach in several ways in order to adapt it to long-term urban noise. Details and rationales are provided in the Supplement. The modifications can be summarized as follows: (i) we apply clustering per octave frequency band to account for narrow-band cultural sources; (ii) we pre-determine the principal component axes on a subset of the data due to the large data volume; (iii) we normalize by dividing each waveform by its maximum absolute value; (iv) we fix the number of clusters a priori at an optimum determined in a preliminary clustering run, here four clusters; and (v) we rely on cultural patterns with respect to local time to label the clusters as day, night, noise and other. Here, noise refers to the smallest cluster of sporadically appearing, transient disturbances. The other cluster is required to fulfill the optimum number of four clusters. It mostly coincides with the transition between day- and nighttime. The result for the N–N component of the seismometer at G.UNM is shown in Fig. . The urban day–night rhythm emerges for all stations at frequencies of 0.5 Hz and above, as well as for several stations inside the basin for frequencies between 0.25 and 0.5 Hz. We tested the effect of using the amplitude-unbiased phase cross-correlation on clustering and found that, although the clustering results change, the day–night rhythm is still clearly visible (Fig. S1 in the Supplement). This suggests that processing schemes aimed at equalizing noise amplitudes fail to suppress the effect of temporally varying noise sources in an urban environment. We therefore use the clustering approach for the subsequent analysis. We stack the short-term correlations for the daytime clusters, which are the largest in number and the most consistent with time, over a duration of 10 d. We handle the short-term correlation data, filtering, clustering and stacking with the Python module ruido (see the code and data availability section).

We use the stretching method to measure relative changes in velocities e.g.,. A preliminary comparison to the moving window cross-spectral method e.g., showed good overall agreement. We use a multiple-reference approach due to the lack of long-term waveform coherence in our observations; details are provided in the Supplement. Multiple-reference approaches have been previously used (in somewhat different forms), e.g., by and . The stretching is performed in four frequency bands (0.5–1, 1–2, 2–4 and 4–8 Hz) and in two windows on each stack, one of which extends from 4 to 10 times the longest period of interest (defined by the frequency band) and the other from 8 to 20 times the longest period of interest (e.g., for a 0.5–1 Hz observation, we measure velocity changes in the 8–16 and 16–40 s windows). Figure  shows results at station G.UNM (0.5–1 Hz) using the coda window at 16–40 s. We note that the information contained in the single-station cross-correlations (mixed components) is visually consistent with the pure autocorrelations (Fig. ). We thus proceed with the pure autocorrelations, mostly because their interpretation is conceptually simpler; this also reduces computational requirements for the following inversion. We also note that data quality as measured by the correlation coefficient between reference trace and current trace after stretching (CC_best; shown by color hue) increases markedly after the sensor at G.UNM was set to high-gain mode in 2008. For further analysis, we discard data points with CC_best <0.6.

The 2017 Mw 7.1 Puebla earthquake caused strong ground motion in Mexico City, reaching peak ground accelerations over 15 % g and a sharp acceleration of subsidence in various locations . We observe velocity drops coinciding with this earthquake, followed by approximately logarithmic recovery at most stations. More subtly, a similar signal is observed at several stations following the 2020 Mw 7.4 Oaxaca (Sta. María Xadani) earthquake – see Fig. . Physical processes causing the co-seismic velocity drop may be caused by plastic deformation (fractures opening due to shaking-induced dynamic stresses that exceed the geologic material yield strength; ) or by a non-linear mesoscopic elasticity that describes the loss and reestablishment of chemical bonds or capillary bridges that change frictional contacts . proposed the following model for the long-term post-seismic recovery described as the superposition of exponential relaxation mechanisms with different relaxation timescales τ, ranging from τmin to τmax, after the time of the earthquake tquake: 1c(t)-c0c0=sc0ln⁡τmaxτmin-1∫τminτmax1τexp⁡(-t/τ)dτfor t≥tquake. Here, s is a negative value indicating the drop of seismic velocity from its previous value c0. For intermediate timescales, (τmin≪t≪τmax), this model captures the approximately logarithmic time dependence, while it tends towards c0 for large times and is finite for t=0 where a purely logarithmic recovery is not defined. Hypothesizing that the minimum timescale τmin is below the temporal resolution of our measurements, we set τmin=0.1 s and fit both τmax and the co-seismic step drop s through the inversion described below. Velocity changes due to non-linear elasticity are depth dependent e.g.,. For the earthquake-induced non-linearity, we account for depth dependence only insofar as we invert data from different frequency bands separately .

The following sections describe how we model the effects of surface temperature (Sect. ) and precipitation (Sect. ) on Rayleigh wave phase velocity. In both cases, we use analytical solutions to diffusion-like problems for a homogeneous half-space to model their effect on vs from the surface to 2 km depth in terms of (a) thermoelastic stress and (b) pore pressure. In particular, we use (a) the solution of as formulated by and (b) the solution of as formulated by . In the thermoelastic model, annual and sub-annual periodic surface temperature changes diffuse through the shallow subsurface. In the pore pressure model, rain leads to a sudden increase in pore pressure near the surface, which then diffuses towards depth. We compute surface wave sensitivity kernels with surf, a Python package based on the solutions to the surface wave eigenproblem in layered, anisotropic elastic media . This is done using station-specific 1-D velocity profiles and assuming an isotropic medium (see Sect. ). Finally, we integrate the depth-dependent vs change to obtain the predicted surface wave phase velocity (c) changes.

The solutions at depth in the homogeneous half-space depend on (a) thermal diffusivity and (b) hydraulic diffusivity of the sediments, which are not well known and in the case of (b) can vary by several orders of magnitude . Inverting for these parameters probabilistically would be challenging because it requires re-evaluating the diffusion terms at all depths for each iteration. Instead, we run the inversion repeatedly for six homogeneous half-spaces characterized by (a) three and (b) two trial diffusivities in the ranges of (a) 0.15 to 2 mm2s-1 and (b) 0.0001 to 4 m2s-1 . We retain the diffusivity for each site that produces the best fit across all components and frequency bands.

Considering the superposition of the temperature, precipitation, earthquake and linear trend, we aim to model the phase velocity change dcc as 5dcc=ftemp(κt,st)+frain(κh,p0)+fseismic(Δc,τmax)+flin(a,b), where κt,κh are the thermal and hydraulic conductivity, respectively; st is the sensitivity of the shear wave velocity to thermoelastic stress; p0 is the minimal overburden pressure; Δc is the co-seismic drop in observed velocity (which we assume is surface wave phase velocity); τmax is the maximum timescale of exponential recovery in the slow-dynamics model of ; and a and b are the slope and offset of the linear trend. Values for the remaining variables are taken from the literature. In particular, we use fluid volume fractions of 0.6 and 0.2 for lacustrine sediments and all other materials, respectively, from as an approximation for porosity. Furthermore, we use approximate bulk moduli of 2.5 and 35 GPa for the pore water and rock matrix, respectively. We use these values to estimate Skempton's B following , and we adopt their estimate of the undrained Poisson ratio. We conduct an MCMC inversion to determine the parameters st, p0, Δc, τmax, a and b using the Euclidean distance between the observed and modeled dcc with the emcee Python package . Details of the initialization and convergence of the emcee runs are given in Sect. S4 in the Supplement. Median models are shown in Fig.  for the lag window of 4–8 s of the 2–4 Hz frequency band; results in terms of median models and model ranges for all frequency bands and lag windows are shown in Figs. S5 and S6 in the Supplement.

Comparison between observed dcc at the co-located seismometer and accelerometer of the station G.UNM shows overall strong consistency (Fig. ). Correlation coefficients between observed time series range from 0.83 to 0.99, with the exception of the East component at 0.5 and 2.0 Hz, where a gap in highly coherent observations results in an offset of dcc between the seismometer and accelerometer after the Puebla earthquake. Consequently, we discourage the analysis of single components; a synoptic analysis or weighted average of components, as suggested by, e.g., by , is preferable. We conclude that strong-motion stations yield overall good results for dcc studies in urban settings with high-amplitude anthropogenic noise comparable to G.UNM.

Results at 2–4 Hz for the strong-motion stations of RSVM and the seismometer at G.UNM (Fig. ) illustrate the seasonal dcc variations and the drops and recoveries for the 2017 Puebla (19 September 2017) and 2020 Oaxaca (23 June 2020) earthquakes. Despite its simplicity, our model captures the behavior of the observed velocity changes reasonably well. For RSVM and G.UNM at 2–4 Hz, with a lag window of 4–10 s, the mean correlation coefficient (CC) between modeled and observed dcc is 0.77, and the median CC is 0.84. (0.5–1.0 Hz: 0.68 and 0.75; 1–2 Hz: 0.67 and 0.73; 4–8 Hz: 0.69 and 0.76). The inversion failed to converge in approximately 10 % of cases, which we include nevertheless (see Figs. S5 and S6 in the Supplement). We exclude results above a misfit threshold set at the bottom quartile from further analysis. These models are shown by dashed white lines in Fig. .

Our model generally fits better for stations inside the basin, including the transition zone and lake zone stations, than for those outside. This is reflected by a consistently higher CC between observed and modeled dcc for stations inside the basin and by a slightly lower average misfit.

We propose three reasons for the better performance of the model inside the basin. First, incident ambient noise may be more stable near the city center due to repetitive traffic patterns and a generally higher amplitude. A detailed analysis of noise source conditions is out of scope of this work but may benefit future urban monitoring studies. Second, the assumption that observations are dominated by scattered surface waves may be more appropriate for basin stations located on stratified sediment with strong impedance contrasts. Third, better a priori information is available about the basin subsurface structure compared to areas outside of the basin thanks to the detailed thickness information of the low-vs aquitard and to shallow profiles of vs from well logs . This results in what are likely to be more accurate station-specific surface wave sensitivity kernels inside the basin, which illustrates the value of a priori site-specific geologic and hydrologic information for quantitative shallow-structure monitoring with ambient noise.

Several previous studies have applied detailed physics-based models to dcc, usually focusing on a small number of selected stations or time series . Previous array-wide analyses have mostly focused on empirical transfer functions between hydrologic and meteorologic parameters and observed dcc e.g.,. Here, we take a partially physics-based approach to the scale of a sedimentary basin and metropolitan seismic array, using a comprehensive set of time series in terms of frequency (0.5–8 Hz in octaves) and spatial components (E, N, Z) and integrating multiple processes influencing dcc (poroelastic stress, thermal stress, non-linear elasticity with co-seismic drop and recovery). It is straightforward to extend our approach to other surface wave modes (Love waves, higher modes). Here, we chose to limit our analysis to fundamental-mode Rayleigh waves, since limited information on subsurface structure is available, and we have limited knowledge of the surface wave modal representation in scattered waves.

A current limitation of our model is that we assume a linear superposition of different processes affecting dcc seefor a discussion on this point. Nevertheless, it is the first step to a comprehensive model for dcc based on simplified physics at the basin scale.

The parameters relevant to seasonal variations in our model are (i) thermal diffusivity, (ii) hydraulic diffusivity, (iii) the sensitivity of vs to thermal stress st (which summarizes non-linear and linear elastic properties), and (iv) the minimum effective pressure p0. For practical reasons, only a few values were tested for (i) and (ii). Therefore, we only present the inference of st and p0 by the inversion. We observe variability between the Z, N and E components for both parameters, as well as between the frequency bands. Depending on the station and frequency band, either temperature or precipitation emerges as the dominant seasonal control from our inversion; this is apparent from results shown in the Supplement.

In previous studies, single-station correlations have been interpreted as multiply scattered body waves e.g.,, whereas we interpret them as surface waves. This is supported by elliptical particle motions (Fig. S7). The coda of cross-correlation is usually interpreted in terms of surface waves (Table ). The main difference between the auto- and cross-correlation coda is receiver separation, which may not suffice to change from observing predominantly scattered surface waves to predominantly scattered body waves. Moreover, found in numerical experiments that surface waves tend to dominate single-station correlation sensitivity in scattering media in the presence of depth-varying velocity structure, which is the case in sedimentary basins. For these reasons, we consider an interpretation in terms of surface waves to be sensible. Apart from the study by , the single-station correlation sensitivity to velocity changes remains scarcely researched in comparison to cross-correlation coda sensitivity. This topic merits further attention.

Assuming that surface waves dominate our observations, we deduce that the variation of parameter values with frequency range is an effect of surface wave dispersion. As a working hypothesis for the differences in terms of parameters between components, we assume that waves recorded on E, N and Z components are dominated by different scattered arrivals or different wave modes and therefore are sensitive to different parts of the surrounding medium.

We present the co-seismic drop and maximum relaxation timescale τmax of the 2017 Puebla earthquake with respect to peak ground acceleration (PGA) in Fig. . Markers show median models of velocity drop and τmax, and error bars show the range between the 16th and 84th percentiles. Not all strong-motion stations of the RSVM network recorded the Puebla earthquake; where PGA is not available from the RSVM stations, we use the PGA at the nearest available station (taken from horizontal ground motions), including the triggered strong-motion stations shown by black rhombs in (Fig. ). We add a dither (random shift) of up to 0.025 ms-2 to the PGA values in order to ensure that all markers are visible. Gray-shaded rectangles indicate values of τmax that lie beyond the duration of observation (approximately 2.5 years post-earthquake). Velocity drops strongly by up to ≈10 %, and recovers slowly at most sites, with several τmax on the order of a decade. The base 10 logarithms of both τmax and the velocity drop show significant positive correlation with the logarithm of the PGA (p<0.05), but the observed variances are not well explained (R2=0.28 for τmax and R2=0.12 for the velocity drop). We note that both τmax and, to a lesser extent, the velocity drop are not very well constrained (wide bars).

Due to limited data coverage of the 2020 Oaxaca earthquake, we will not interpret results of its co-seismic velocity drop and recovery apart from stating that it caused a sudden drop in phase velocity at several sites in Mexico City which was smaller than the drop during the 2017 Puebla earthquake.

A velocity decrease is observed across all the studied locations, including those on lacustrine clay, supporting the conclusions by and , who stated that the lacustrine sediment behaves non-linearly.

Slow dynamics, the approximately logarithmic recovery of material mechanical properties after sudden changes induced by transient strains, is a subject of active research e.g.,. However, it is difficult to test current hypotheses in the metropolitan and geologically complex area of Mexico City given our sparse seismic network. To the extent that interpretation is possible, our results suggest that larger perturbations (in terms of higher PGA) may lead to a longer recovery time τmax. This is consistent with other in situ observations, notably of , who observed a slower recovery of velocities in the Kanto basin after the 2011 Tohoku-Oki earthquake at sites that experienced a higher strain rate during the mainshock. applied the model to the 2015 Gorkha earthquake and aftershocks and found a τmax of 846 d for the main shock, which is much longer than for the aftershocks (155 d). These observations run counter to laboratory experiments of slow dynamics, which suggest that the recovery of a particular material is independent of the perturbation amplitude . The τmax on the order of 10 years inferred from our study are uncharacteristically long compared to other studies, which reported 100 d for the Kanto basin , 250 d in Nepal , and 3 years in Parkfield .

A possible explanation for the inferred long τmax is that the relaxation functions used to model dcc do not account for permanent damage. Permanent changes in the interferometric waveforms may lead to poor estimates of the relaxation timescale. Permanent damage can be assessed using the decorrelation 6Dcorr=1-CCi in Fig. . Here, CCi is the correlation coefficient of the stretched traces with respect to the average trace for 2017. Stations that were not operational during the 2017 Puebla earthquake are omitted, and data with poor overall CCs are excluded (amounting to 40 % of data at 0.5 Hz, 12 % at 1 Hz, 3 % at 2 Hz and 6 % at 4 Hz). In the frequency bands 1–2, 2–4 and 4–8 Hz, decorrelation increases after the time window containing the earthquake (gray rectangle). This increase is more pronounced for stations not located on the clay aquitard, as defined in , although those stations experienced, on average, a lower PGA. At 0.5–1 Hz, decorrelation is less marked and is similar between stations on and off the aquitard. Using decorrelation as a proxy for permanent damage, this suggests that, compared to other sediments, the shallow clay of the lake zone would suffer less permanent damage.

Inferred seismic velocity drops in our model are high and weakly, but significantly, correlated in double-logarithmic space with PGA, as well as with the strain rate proxy PGA/vs10. We use the first 10 m instead of the first 30 m here, as this distinguishes more accurately between hard sites and intermediate and soft sites. A dependence of the magnitude of the velocity change on peak ground motion amplitudes is expected based on laboratory studies and is consistent with earlier field studies .

Following the visual appearance of the dcc time series, we introduced a linear term in the dcc model (see Sect. ). The inversion results in strong slopes for this term, with phase velocity increases of up to 0.75 % per year (see Fig. ). A linear term in a dvv model was previously used by , who similarly introduced it on a heuristic basis as the data appeared to require it. Slope values were comparable to those we observe (0.27 % per year) and were hypothesized to relate to recovery from an earthquake that had occurred 5 years prior to the start of their observations, with modified Mercalli intensity (MMI) IV to V at the investigated site. Prior to the 2017 Puebla earthquake, the most recent earthquake that inflicted severe damage on Mexico City was the 1985 Michoacán earthquake , which caused an MMI of IX in Mexico City . Despite its large intensity, we find it unlikely that the subsurface would still be recovering 30 years later (in a logarithmic regime, a recovery rate of 0.5 % per year would require an initial velocity drop of at least 50 %, which is unrealistic). However, we cannot entirely rule out the possibility that the subsurface is recovering from the cumulative effect of multiple events.

We propose an alternative hypothesis. Mexico City has been undergoing rapid subsidence since more than a century ago due to groundwater extraction and sediment compaction and references therein. It has been suggested that another consequence of the subsidence process is the reduction of the fundamental resonance periods of the sediment strata throughout the basin , which may be caused by an increase in seismic velocity, as well as the compaction and resulting reduction of sediment strata thickness. Figure  shows the correlation between the slope of the linear trend of our models and the vertical displacement from InSAR, which we interpret as the effects of ground subsidence. Although the datasets cover different time ranges, subsidence rates have been approximately constant for decades so that the rates can be directly compared. Both rates are moderately and significantly correlated (R2=0.29, p<0.0001), indicating that sediment compaction may indeed be causing a velocity increase in Mexico City's underlying stratigraphy. Similar results from at the Salton Sea (California) showed a steady velocity increase at a much smaller rate of less than 0.1 % yr-1, which they interpreted as an effect of poroelastic contraction. Because of the importance of resonance frequency changes as estimated by for Mexico City's seismic hazard, the ongoing rapid subsidence and its associated hazard , and the magnitude of the changes, this topic merits further and more detailed investigation.