# Toward Unraveling Sub-Seismic Frequency Signals Using Stacked Global Broadband Seismological Data

Piero Poli, Josipa Majstorović, T. Dylan Mikesell, Martin Lott

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Piero Poli, Josipa Majstorović, T. Dylan Mikesell, Martin Lott. Toward Unraveling Sub-Seismic Frequency Signals Using Stacked Global Broadband Seismological Data. Geophysical Research Letters, 2025, 52, pp.2024GL114211. 10.1029/2024GL114211. insu-05080924

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# Geophysical Research Letters®

## RESEARCH LETTER

10.1029/2024GL114211

**Key Points:**

* A novel stacking approach at sub-seismic frequencies is presented

* Seismic stacking reveals sub-seismic signals at nanogal resolution, enabling insights into gravity perturbations

* Seismic data capture potential atmospheric tidal effects, offering a new path to infer the coupling with the solid Earth

**Supporting Information:**

Supporting Information may be found in the online version of this article.

**Correspondence to:**

P. Poli,
piero.poli@unipd.it

**Citation:**

Poli, P., Majstorović, J., Mikesell, T. D., & Lott, M. (2025). Toward unraveling sub-seismic frequency signals using stacked global broadband seismological data. *Geophysical Research Letters*, *52*, e2024GL114211. https://doi.org/10.1029/2024GL114211

Received 17 DEC 2024

Accepted 14 FEB 2025

© 2025. The Author(s).
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

<figure><img src="image_2.png" /></figure>

## Toward Unraveling Sub-Seismic Frequency Signals Using Stacked Global Broadband Seismological Data

Piero Poli¹ ID, Josipa Majstorović² ID, T. Dylan Mikesell³ ID, and Martin Lott⁴

¹Department of Geoscience, University of Padova, Padua, Italy, ²Université Paris Cité, Institut de Physique du Globe de Paris, CNRS, Paris, France, ³Norwegian Geotechnical Institute, Oslo, Norway, ⁴Université Grenoble Alpes, CEA, Leti, Grenoble, France

**Abstract** Seismological data from broadband seismometers have been used for decades to study various signals in the seismologically interesting frequency band between 0.0003 and 30.0 Hz. However, broadband seismometers are seldom used to explore gravity variations in sub-seismic periods larger than 1 hr because seismometers are exceeded in performance by superconducting gravimeters when considering a single sensor. In this study, we demonstrate how to leverage the existing network of global broadband seismometers to resolve tidal harmonics as small as 1 nanogal, equivalent to modern superconducting gravimeters. The presented seismic array analysis, based on the stacking of coherent signals in the frequency domain, is robust and can be used to track the spatial and temporal evolution of solid earth and oceanic tides. Our findings indicate that high-quality long-term seismological data provide a new and complementary data set for tracking gravitational changes over time, offering novel insights into geodynamics and atmospheric processes.

**Plain Language Summary** High-quality broadband seismometers have traditionally been used to study seismic waves in the frequency band 0.0003–30.0 Hz. This frequency band contains signals generated from tectonic phenomena such as earthquakes, as well as anthropogenic phenomena like traffic-related vibrations. Few studies have used broadband seismometers to study lower frequency signals generated by non-tectonic processes, such as variations in gravity due to the earth-moon-sun system. It is more common to use superconducting gravimeters to study these non-tectonic processes because they provide higher-quality data compared to a single seismic sensor. However, in this study, we demonstrate that by averaging over a global array of seismic sensors, we can achieve as accurate or better signals compared to superconducting gravimeters. The benefit here is that often seismic sensors have a more continuous-time recording, thus providing a new data set with which to study these non-tectonic processes and complement existing gravimeter data sets.

## 1. Introduction

Superconducting gravimeters are widely used to study sub-seismic (which we define as the range from 1 hr to 7 days) gravity (acceleration) perturbations at the surface of the Earth (Crossley et al., 2013). These instruments measure gravity changes on the order of nanogals, primarily induced by solid Earth and oceanic tides (Agnew, 2010; Crossley et al., 2013), but are also affected by Newtonian attraction from changes in the atmospheric mass and free air effect (Boy et al., 2002; Warburton & Goodkind, 1977; Zürn & Wielandt, 2007; Zürn et al., 2007). The atmospheric influence represents the largest source of *noise*, limiting the detection of faint signals of geodynamic origin (Crossley et al., 1992; Rosat et al., 2003).

Similarly, broadband seismometers (Rademacher, 2023; Wielandt & Streckeisen, 1982) operate with an inertial mass to measure acceleration perturbations. These seismometers are primarily sensitive to perturbations at frequencies corresponding to seismic waves generated by earthquakes and other tectonic phenomena (i.e., from ~0.0003 to ~30 Hz). However, high-quality broadband instruments, which make up most of the global seismic networks (Leroy et al., 2024; Wilson et al., 2024), are sensitive to lower frequencies and can locally resolve the strongest tidal perturbations within the diurnal, semidiurnal, and terdiurnal frequency bands (Davis & Berger, 2007; Freybourger et al., 1997; Lambotte et al., 2006; Martynov et al., 2020). Additionally, several studies have suggested that seismometers are sensitive to small vertical acceleration caused by barometric variations associated with atmospheric dynamics (Forbriger et al., 2021; Nishida, 2013; Zürn & Widmer, 1995; Zürn & Wielandt, 2007).
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Results of global analysis of seismic data in sub-seismic frequency range. (a) Map of the broadband seismic stations used in this analysis. (b) Time-averaged coherence of vertical acceleration recorded at all stations from 2010 to 2024 (blue line). The inset shows a zoom from 3.5 to 14 cycles per day (cpd). Black vertical lines are at integer fractions of 24 hr, while black dots mark the frequencies of the known solid Earth tidal constituents (Hartmann & Wenzel, 1995). The red shaded area represent the standard deviation of the mean spectrum estimated from bootstrap analysis.</figcaption></figure>

Unfortunately, the self noise of broadband seismic instruments at sub-seismic periods is generally higher compared to other instruments (Rosat et al., 2015). This makes measurements of signals smaller than tens of nanogals, such as higher tidal harmonics (Freybourger et al., 1997), with a single seismometer a challenge (Lambotte et al., 2006; Rosat et al., 2015). A further complication is that the instrument response of a seismometer is less accurate at sub-seismic periods, with several studies reporting anomalous admittance to both pressure perturbations (Lambotte et al., 2006; Rosat et al., 2015) and tidal deformation (Davis & Berger, 2007). Regardless of the problems associated with these sensors, the spatial distribution, and in general the amount of the existing, maintained, and operating broadband seismometers (Figure 1a), is incomparable to any other ground-based sensor or sensor array. There are hundreds of times series available spanning decades, making seismological data a complementary data set to gravimeters to study gravity and other global phenomena.

In this study, we precisely analyze the global seismological data set (Leroy et al., 2024; Wilson et al., 2024), focusing on resolving small signals which seismometers can record in the poorly explored sub-seismic period band. To overcome issues related to the high noise in single-station analyses, and to limit problems associated with the variable response of seismometers (Davis & Berger, 2007), we perform an array analysis based on
stacking coherent signals in the frequency domain (Bendat & Piersol, 2011). While several studies have proposed stacking global geophysical data (e.g., Ding et al., 2021, and references therein), little attention has been paid to broadband seismic data. Due to the large number of stations and long time series available, in this study we demonstrate that by stacking data from broadband seismic networks we are able to increase the signal-to-noise ratio and obtain stable sub-seismic period signals. Additionally, this approach mitigates the effects of transient signals (e.g., spikes) and uncertainty in the instrument responses (Davis & Berger, 2007), allowing for the first time the seismological resolution of tidal harmonics as small as 1 nanogal. Moreover, we detect a series of spectral peaks, previously unobserved in seismological data, at harmonics of 24 hr, which are reminiscent of the Newtonian attraction (Warburton & Goodkind, 1977) effect induced by atmospheric tides (Hedlin et al., 2018; Lindzen & Chapman, 1969).

## 2. Data Preparation and Coherence Estimation

We analyze 14 years of data (2010–2024) collected from approximately 180 seismological stations (Figure 1) associated with the Geoscope and Global Seismographic Networks (Leroy et al., 2024; Wilson et al., 2024). These stations are equipped with broadband seismometers and feature high-quality installations that enable the detection of robust signals within the sub-seismic frequency range (Davis & Berger, 2007; Freybourger et al., 1997; Lambotte et al., 2006; Martynov et al., 2020). Additionally, effective temperature insulation further enhances signal integrity (Doody et al., 2018). We retrieve continuous seismic data in time windows of 30 days with 20 days overlap for each broadband station. We meticulously exclude windows with any data gaps to circumvent the need for pre-processing such as gap filling or interpolation, which lead to potential data degradation during analysis. These data are then corrected for the instrument response and converted to acceleration and then bandpass filtered within the period range from 7 days to 2,000 s. Subsequently, each 30-day window, initially sampled at 1 sample per 1 s or 1 sample per 10 s, is resampled to 1 sample every 1,000 s.

In contrast to previous global data stacking approaches, which focused on superconducting gravimeters or the global position system (Ding et al., 2021), we calculate the fully normalized coherence between vertical component seismic signals recorded at each couple of stations (i, j), over an ensemble of time windows M (Bendat & Piersol, 2011) for frequencies (f):

$$C_{ij}^{T}(f) = \frac{1}{M} \frac{\sum_{t=1}^{M} S_{i}(f) \cdot S_{j}^{*}(f)}{\sum_{t=1}^{M} \sqrt{S_{i}^{2}(f) \cdot S_{j}^{2}(f)}} \quad (1)$$

The size of each time window T is 30 days, and the signals (S) are tapered in the time domain using a Hanning window. For each window T, the total number of windows M is variable and depends on the available data for a given time period. However, we ensure that M > 20 (600 days of data) for a reliable estimate of station-pair coherence Equation 1. With this analysis we derived approximately 4,000 cross-spectra for each 30-day window, resulting in over two million cross-spectra for the 14-year period.

To explore the global results we calculate the network average coherence over N station pairs, defined as

$$\text{Coherence}(f) = \frac{2}{N(N-1)} \sum_{i=1}^{N} \sum_{j=i+1}^{N} |C_{ij}^{T}(f)| \quad (2)$$

We estimate the uncertainty of this network averaging (Efron & Tibshirani, 1994) by resampling spectra over station pairs (i, j). This reveals that the coherence spectrum is not dominated by some station pairs and indicates the spatial stability of these observations (Figure 1b).

The coherence defined in Equation 1, while limiting the physical interpretability of the results, is particularly well-suited for identifying signals with constant phase. It is highly effective in reducing the impact of transient signals (e.g., spikes), instrumental response artifacts, and variable noise amplitude among various broadband instruments composing the global networks used (Leroy et al., 2024; Wilson et al., 2024).
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Observation of spatial evolution of gravity from seismometers: The real part of the coherence (black dots) as a function of longitudinal distance for periods of (a) 24 hr 50 m, (b) 24 hr, (c) 12 hr 25 m, (d) 12 hr, and (e) 8.3 hr. Cyan dots in panels (a, b) show the real part of the coherence for station pairs in different hemispheres. In each plot, the red lines represent the best-fitting cosine function to estimate the longitudinal wavelengths. (f) Frequency and longitudinal wavelength estimated from lunar (green) and solar (blue) peaks in coherence spectra. The green and blue continuous lines indicate the known solar and lunar days, respectively.</figcaption></figure>

### 3. Results

The resulting network average reveals a rich coherence spectrum characterized by several prominent peaks at physically interesting frequencies (Figure 1b). Around 1 cycle per day (cpd), we observe two distinct peaks: a weaker one at ~0.92 cpd, close to the period of the *O1* tide, and a larger peak at exactly 1 cpd, near the frequencies of the *P1*, *S1*, and *K1* tides (Hartmann & Wenzel, 1995). A similar pattern is seen around 2 cpd, with the largest peak at ~1.93 cpd, corresponding closely to the expected *M2* tide (Hartmann & Wenzel, 1995). The second peak occurs precisely at 2 cpd, near the *S2*, 2*SK2*, and *K2* tidal frequencies (Hartmann & Wenzel, 1995). Around the terdiurnal frequency, the largest peak at ~2.9 cpd corresponds to the *M3* tide, while a weaker peak of coherence is observed at 3 cpd, where the *SK3* and *SP3* tides are expected (Hartmann & Wenzel, 1995).

To experimentally confirm that these global signals truly represent tidally-induced accelerations, we analyze the spatial pattern in our data using an approach similar to spatial correlation methods (Aki, 1957). We analyze the in-phase part of the cross spectrum by plotting the real part of the coherence for each station pair against the interstation longitudinal distance, considering only station pairs with similar latitudes (i.e., within a range of 20° latitude). The spatial correlation for five peaks at frequencies from 1 to 3 cpd follows a sinusoidal structure along the longitudinal distance (Figure 2). This spatially-consistent structure indicates that nearby stations exhibit similar phases (real part of cross-spectrum close to 1), with the phase difference increasing with increasing longitudinal distance governed by the wavelength of a longitudinal sinusoidal wave at that frequency. We estimate the longitudinal wavelength for each frequency by fitting a cosine function to the data. This spatial analysis reveals that seismometers record coherent vertical acceleration with wavelengths of 360, 180, and 120° for frequencies around 1, 2, and 3 cpd, respectively. We use the longitudinal wavelength and frequency to obtain the rotation speeds of each resolved spatial structure, which are in agreement with the solar and lunar days (Figure 2f). Two peaks with similar spatial structures are observed at 24 hr 50 min and 24 hr, corresponding to the lunar and solar cycles, respectively (Figures 2a and 2b). A similar behavior is observed for the signals at 12 hr 25 min and 12 hr (Figures 2c and 2d). Additionally, we observe that the cross-spectra for station pairs crossing the equator follow an inverse cosine spatial dependence around 1 cpd (Figure 2). This behavior implies that two gravity
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Time evolution of coherence across 14 years for (a) 0.5–3.5 cpd and (b) 3.5–8 cpd. (c) Coherence plotted as a function of time for 4 cpd (red) and 6 cpd (green). The background colors highlight summer (red) and winter (blue) periods in the northern hemisphere. In panels (a, b) the red stars indicate the occurrence of earthquakes with magnitude larger than 8. The black line represents the background noise level estimated at 4.5 cpd.</figcaption></figure>

perturbations of opposite sign but with the same longitudinal wavelength are observed, as expected for the spatiotemporal evolution of the solid tide gravity potential for certain constituents (Agnew, 2010).

Although the detection of these primary tidal harmonics (*O1*, *S2*, 2SK2, K2, and less often *M3*) has been reported in previous studies using broadband seismological data (Davis & Berger, 2007; Freybourger et al., 1997; Lambotte et al., 2006; Martynov et al., 2020), our method offers a significant advance not only by detecting these fundamental harmonics but also by resolving the higher-order tidal harmonics with smaller amplitudes. For instance, at 4 cpd we observe a dominant peak at the expected frequency of 4 cpd, alongside the *M4* tide at 3.86 cpd. The detection of the *M4* tide implies an amplitude sensitivity of at least 20 nanogals (Boy et al., 2004), underscoring the enhanced sensitivity of our global stack to previously undetectable tidal signals that would not have been apparent in single-station seismological analyses (Davis & Berger, 2007; Freybourger et al., 1997; Lambotte et al., 2006; Martynov et al., 2020). This high sensitivity is further demonstrated by the detection of peaks corresponding to the *M6* and 2MS6 tidal components (Figure S1 in Supporting Information S1), where known amplitudes range from 1 to 2 nanogals (Boy et al., 2004; Merriam, 1995). Although most of these primary peaks align well with expected tidal constituents, we also observe deviations at higher frequencies (greater than 3 cpd), where the peaks no longer correspond to predicted periods for solid Earth tides (Figure 1). This behavior suggests that peaks at harmonics of 1 day for frequencies larger than 3 cpd are not associated with oceanic or solid Earth tides (Agnew, 2010; Hartmann & Wenzel, 1995), while they are reminiscent of acceleration peaks observed in superconducting gravimeters (Warburton & Goodkind, 1977), principally induced by Newtonian attraction and free air effect from atmospheric tides (Lindzen & Chapman, 1969).

We next analyze the time evolution of the observed peaks by performing a moving average over time windows spanning 5 months. This analysis reveals that the observed spectral structure (Figure 1) is stable over time, for peaks up to 11 cpd (Figure 3). The resolved spectral peaks maintain nearly constant coherence throughout the 14-year period and show no significant changes in association with major earthquakes (Figure 3). Even the
24 hr harmonics at frequencies larger than 3 cpd remain relatively stable and do not respond to transient stress perturbations, which could potentially lead to harmonics generation through non-linear response (Agnew, 1981).

## 4. Discussion

The results of our global data analysis reveal a continuous set of signals that persistently appear throughout the analysis period. These signals exhibit frequencies corresponding to solid Earth tides (Hartmann & Wenzel, 1995) and integer fractions of 1 day. The spectral characteristics and temporal consistency of these observations suggest that a nearly constant source is responsible for exciting these signals. The diurnal, semidiurnal, and terdiurnal peaks align with the main tidal harmonics (Figure 1b) and exhibit the typical spatial patterns associated with solid Earth tides (Figure 2). These findings corroborate previous studies that reported a strong sensitivity of broadband seismic instruments to the strongest tidal harmonics (Davis & Berger, 2007; Freybourger et al., 1997; Lambotte et al., 2006; Martynov et al., 2020).

In addition, our global stacked-data approach reveals tidal peaks at 4 and 6 cpd, which are notoriously difficult to identify in single-station analyses due to the self-noise level of most seismometers (Freybourger et al., 1997; Lambotte et al., 2006; Rosat et al., 2015). Despite the coherence stacking method not resolving the physical units of the peaks, we can compare our coherence spectrum with previous seismological and gravimetric studies to estimate the sensitivity achieved through our stacking approach. For instance, Freybourger et al. (1997) reported that broadband seismometers could hardly detect quaterdiurnal tides (M4) with an amplitude of approximately 20 nanogals, whereas these peaks clearly emerge in our stacked coherence. Additionally, we resolve peaks corresponding to the M6 and 2MS6 non-linear tides, which are typically only observed with superconducting gravimeters with a sensitivity of ~1 nanogal (Boy et al., 2004).

This comparison illustrates the power of stacking data from a large number of seismic stations, providing a complementary data set with nanogal sensitivity, which can be employed for future studies of gravity perturbations from tides or geodynamic origins. Moreover, the extension of our method to other data sets (e.g., superconducting gravimeters) might help to mitigate the effect of transient signals, unknown or uncertain instrument responses, and even to combine different types of instruments to build new global stacks in search of elusive geodynamic signals (Crossley et al., 1992; Rosat et al., 2003).

In addition to the well-known linear and non-linear Earth tides, we observed a series of peaks at harmonics of 24 hr, whose frequencies do not align with solid Earth tidal constituents (Figure 1b) and are consistently observed throughout the full analysis period (Figure 3). These peaks resemble those observed in pressure-uncorrected gravimetric data (Warburton & Goodkind, 1977), with their origin primarily associated with the Newtonian attraction induced by moving masses above the instruments and free air effect. The discrete nature of these peaks is a result of their excitation by atmospheric tides (Hedlin et al., 2018; Lindzen & Chapman, 1969). Despite a full characterization of their origin being beyond the scope of this work, and potentially requiring a more detailed comparison with barometric data, we propose that these peaks (Figure 1a) have an atmospheric origin, and their strength reflects the time evolution (Figures 3a and 3b) of the Newtonian attraction and free air effect induced by the processes in the atmosphere as resolved by superconducting gravimeters (Hinderer et al., 2014).

To support this point, we investigate the coherence values at the 4 and 6 cpd peaks through time (Figure 3c). The influence of solid and ocean tides is limited at these frequencies, and the coherence variations reflect the strength of the resolved signals over the array. We observe coherence peaks to be more pronounced at the end of the northern hemisphere winter season for both frequencies (Figure 3c). This time evolution is remarkably similar to the observation of atmospheric tidal strength at 4 and 6 cpd, which is mostly controlled by sudden stratospheric warming (SSW) events at the end of winter in the northern hemisphere (He et al., 2020, 2024). A similar time evolution was also observed in the strength of pressure signals observed with ground barometric data by Hedlin et al. (2018). As SSW events are expected to be more prominent in the northern hemisphere (He et al., 2020, 2024), we further analyze the strength of coherence for 6 cpd by considering stations separately in the northern and southern hemispheres, corroborating the hypothesis that these peaks are significantly correlated with the behavior of atmospheric dynamics (Figure S2 in Supporting Information S1).
The pressure variation at the free surface of the Earth, induced by atmospheric tides, ranges from a few to a few tens of Pascal (Hedlin et al., 2018; Ray & Ponte, 2003; Warburton & Goodkind, 1977). Assuming that these perturbations induce gravity changes primarily through Newtonian attraction (Zürn & Wielandt, 2007), we can use a theoretical admittance of $-4.27 \times 10^{-11} \text{ m/s}^2/\text{Pa}$ to estimate gravity perturbations of $4 \times 10^{-11} \text{ m/s}^2$ and $2 \times 10^{-9} \text{ m/s}^2$ for pressure variations of 1 Pa and 50 Pa, respectively. Using these estimates, we perform a synthetic test to assess the ability of our method to resolve such small gravity perturbations. We simulate the same amount of data used in our study for global seismic stations (Figure 1a). The synthetic data consist of two sinusoidal signals with periods of 4 and 6 hr and amplitudes ranging from 5 to 50 Pa, converted into gravity variations using the admittance for Newtonian attraction (Zürn & Wielandt, 2007). We then add random noise with an amplitude varying as $1/f$ and taking values typical of the noise level for broadband stations at sub-tidal frequencies ($1 \times 10^{-6} \text{ m/s}^2/\sqrt{Hz}$; Freybourger et al. (1997), Rosat et al. (2015)). Despite the noise being 10–1,000 times larger than the signal, our test reveals that signals induced by reasonable pressure variations can be retrieved by stacking the data using Equations 1 and 2 (Figure S3 in Supporting Information S1).

This test is not without several approximations. First, the admittance for seismometers (Lambotte et al., 2006; Rosat et al., 2015) has been observed to differ from the theoretical value assuming only the contribution of Newtonian attraction (Zürn & Wielandt, 2007). Second, the amplitude of atmospheric tides is spatially variable, nor constant, as we imposed in our simple model. Nevertheless, this numerical test illustrates the efficiency of our stacking method applied to seismological networks in resolving tiny acceleration signals.

## 5. Conclusion

Applying our stacking method to global seismological data at sub-seismic frequencies enables the observation of the spectral and spatial evolution of the main tidal harmonics across daily to terdiurnal frequencies. This approach not only enhances the resolution of previously observed tidal signals in seismological data, but also uncovers non-linear oceanic tides with amplitudes as small as 1 nanogal—previously undetected with seismological data. The improved resolution, achieved by stacking data from a vast global seismic network, effectively turns the network into a single, extensive instrument, enabling complementary observations of subtle gravity perturbations.

Beyond the primary tidal harmonics, our approach discerns peaks at integer fractions of 1 day, which were previously difficult to detect in seismometer data. These observations suggest that seismological instruments are sensitive to atmospheric dynamics, potentially capturing signals from atmospheric tides. A more thorough analysis is needed to further quantify the contribution of atmospheric effects and to eventually separate or remove these unwanted signals in the pursuit of detecting novel signals related to geodynamic processes.

The diverse spectrum of signals that emerge from the seismic data typically only appear in gravimeter data. However, unlike gravimeter measurements, our global seismic stack provides a much broader spatial and continuous temporal resolution, offering enhanced insights into the sources of these signals. In our analysis, we demonstrate the versatility of the data provided by seismological networks, highlighting the potential to track temporal and spatial tidal variations via seismic signals. This capability can be particularly valuable for addressing concerns related to climate dynamics, where distinct patterns in atmospheric mass redistribution above 3 cpd may hold key clues (He et al., 2020, 2024). This study highlights the power of using the global seismic network to observe gravity perturbations at nanogal sensitivity, and it lays the groundwork for further investigations into the potential for seismological data to capture not only tidal effects but also other geophysical signals, including those related to atmospheric dynamics.

## Data Availability Statement

The data used in this study are publicly available from the Global Seismograph Network (GSN—IRIS/USGS) (Albuquerque Seismological Laboratory/USGS, 1988), the Global Seismograph Network—IRIS/IDA (Scripps Institution of Oceanography, 1986), and GEOSCOPE, the French global network of broadband seismic stations (Institut de Physique du Globe de Paris (IPGP) & Ecole et Observatoire des Sciences de la Terre de Strasbourg (EOST), 1982).
Acknowledgments

This work has benefited from discussions
with Jean Paul Montagner, Duncan
Agnew, Rudolf Widmer-Schnidrig,
Severine Rosat, Martin Vallet, Jacopo
Boaga, Andy Armstrong, Kasper van
Wijk, and Alexander Stott. We especially
acknowledge the significant suggestions
from Adam Ringler.

References

Agnew, D. C. (1981). Nonlinearity in rock: Evidence from earth tides. *Journal of Geophysical Research*, **86**(B5), 3969–3978. https://doi.org/10.1029/jb086ib05p03969

Agnew, D. C. (2010). 6-Earth tides. *Treatise Geophys*, 3, 163. Retrieved from https://books.google.co.in/books?hl=en&lr=&id=SIHNCgAAQBAJ&oi=fnd&pg=PA163&dq=6-earth+tides&ots=TKo9DoMHgq&sig=wVHMunJQEEfo_wyzAy0xX6A6EcM&redir esc=y#v=onepage&q=6-earth%20tides&f=false

Aki, K. (1957). Space and time spectra of stationary stochastic waves, with special reference to microtremors. *Bulletin of the Earthquake Research Institute*, 35, 415–456. Retrieved from https://oceanrep geomar.de/id/eprint/43280/1/Aki.pdf

Albuquerque Seismological Laboratory/USGS. (1988). Global seismograph network (GSN - IRIS/USGS) [Dataset]. https://doi.org/10.7914/SN/IU

Bendat, J. S., & Piersol, A. G. (2011). *Random data: Analysis and measurement procedures*. John Wiley & Sons.

Boy, J.-P., Gegout, P., & Hinderer, J. (2002). Reduction of surface gravity data from global atmospheric pressure loading. *Geophysical Journal International*, 149(2), 534–545. https://doi.org/10.1046/j.1365-246x.2002.01667.x

Boy, J.-P., Llubes, M., Ray, R., Hinderer, J., Florsch, N., Rosat, S., et al. (2004). Non-linear oceanic tides observed by superconducting gravimeters in europe. *Journal of Geodynamics*, 38(3–5), 391–405. https://doi.org/10.1016/j.jog.2004.07.017

Crossley, D. J., Hinderer, J., & Riccardi, U. (2013). The measurement of surface gravity. *Reports on Progress in Physics*, 76(4), 046101. https://doi.org/10.1088/0034-4885/76/4/046101

Crossley, D. J., Rochester, M. G., & Peng, Z. (1992). Slichter modes and love numbers. *Geophysical Research Letters*, 19(16), 1679–1682. https://doi.org/10.1029/92gl01574

Davis, P., & Berger, J. (2007). Calibration of the global seismographic network using tides. *Seismological Research Letters*, 78(4), 454–459. https://doi.org/10.1785/gssrl.78.4.454

Ding, H., Xu, X., Pan, Y., & Li, M. (2021). Array processing techniques for analyzing global geophysical observation networks: Comprehensive analysis and typical applications. *Earth-Science Reviews*, 221, 103807. https://doi.org/10.1016/j.earscirev.2021.103807

Doody, C., Ringler, A. T., Anthony, R. E., Wilson, D. C., Holland, A. A., Hutt, C. R., & Sandoval, L. D. (2018). Effects of thermal variability on broadband seismometers: Controlled experiments, observations, and implications. *Bulletin of the Seismological Society of America*, 108(1), 493–502. https://doi.org/10.1785/0120170233

Efron, B., & Tibshirani, R. J. (1994). *An introduction to the bootstrap*. Chapman and Hall/CRC.

Forbriger, T., Zürn, W., & Widmer-Schnidrig, R. (2021). Challenges and perspectives for lowering the vertical-component long-period detection level. *Seismological Research Letters*, 92(4), 2498–2512. https://doi.org/10.1785/0220200399

Freybourger, M., Hinderer, J., & Trampert, J. (1997). Comparative study of superconducting gravimeters and broadband seismometers sts-1/z in seismic and subseismic frequency bands. *Physics of the Earth and Planetary Interiors*, 101(3–4), 203–217. https://doi.org/10.1016/s0031-9201(97)00003-4

Hartmann, T., & Wenzel, H.-G. (1995). The hw95 tidal potential catalogue. *Geophysical Research Letters*, 22(24), 3553–3556. https://doi.org/10.1029/95gl03324

He, M., Forbes, J. M., Chau, J. L., Li, G., Wan, W., & Korotyshkin, D. V. (2020). High-order solar migrating tides quench at SSW onsets. *Geophysical Research Letters*, 47(6), e2019GL086778. https://doi.org/10.1029/2019gl086778

He, M., Forbes, J. M., Jacobi, C., Li, G., Liu, L., Stober, G., & Wang, C. (2024). Observational verification of high-order solar tidal harmonics in the Earth's atmosphere. *Geophysical Research Letters*, 51(8), e2024GL108439. https://doi.org/10.1029/2024gl108439

Hedlin, M., de Groot-Hedlin, C., Forbes, J., & Drob, D. (2018). Solar terminator waves in surface pressure observations. *Geophysical Research Letters*, 45(10), 5213–5219. https://doi.org/10.1029/2018gl078528

Hinderer, J., Hector, B., Boy, J.-P., Riccardi, U., Rosat, S., Calvo, M., & Littel, F. (2014). A search for atmospheric effects on gravity at different time and space scales. *Journal of Geodynamics*, 80, 50–57. https://doi.org/10.1016/j.jog.2014.02.001

Institut de Physique du Globe de Paris (IPGP) & Ecole et Observatoire des Sciences de la Terre de Strasbourg (EOST). (1982). Geoscope, French global network of broad band seismic stations [Dataset]. https://doi.org/10.18715/GEOSCOPE.G

Lambotte, S., Rivera, L., & Hinderer, J. (2006). Vertical and horizontal seismometric observations of tides. *Journal of Geodynamics*, 41(1–3), 39–58. https://doi.org/10.1016/j.jog.2005.08.021

Leroy, N., Vallée, M., Zigone, D., Romanowicz, B., Stutzmann, E., Maggi, A., et al. (2024). Geoscope network: 40 yr of global broadband seismic data. *Seismological Research Letters*, 95(3), 1495–1517. https://doi.org/10.1785/0220230176

Lindzen, R. S., & Chapman, S. (1969). Atmospheric tides. *Space Science Reviews*, 10(1), 3–188. https://doi.org/10.1007/bf00171584

Martynov, V. G., Astiz, L., Kilb, D., & Vernon, F. L. (2020). The M₂ tidal tilt results from USArray seismic data from the western United States. *Bulletin of the Seismological Society of America*, 110(6), 3196–3210. https://doi.org/10.1785/0120190314

Merriam, J. (1995). Non-linear tides observed with the superconducting gravimeter. *Geophysical Journal International*, 123(2), 529–540. https://doi.org/10.1111/j.1365-246x.1995.tb06869.x

Nishida, K. (2013). Earth's background free oscillations. *Annual Review of Earth and Planetary Sciences*, 41(1), 719–740. https://doi.org/10.1146/annurev-earth-050212-124020

Rademacher, H. (2023). A brief history of broadband seismometry—part i. *Summary of the Bulletin of the International Seismological Centre*, 57(II), 28–46. https://doi.org/10.31905/5kbo4882

Ray, R., & Ponte, R. (2003). Barometric tides from ECMWF operational analyses. *Annales Geophysicae*, 21(8), 1897–1910. https://doi.org/10.5194/angeo-21-1897-2003

Rosat, S., Calvo, M., Hinderer, J., Riccardi, U., Arnoso, J., & Zürn, W. (2015). Comparison of the performances of different spring and superconducting gravimeters and STS-2 seismometer at the gravimetric observatory of Strasbourg, France. *Studia Geophysica et Geodaetica*, 59(1), 58–82. https://doi.org/10.1007/s11200-014-0830-5

Rosat, S., Hinderer, J., Crossley, D., & Rivera, L. (2003). The search for the slighter mode: Comparison of noise levels of superconducting gravimeters and investigation of a stacking method. *Physics of the Earth and Planetary Interiors*, 140(1–3), 183–202. https://doi.org/10.1016/j.pepi.2003.07.010

Scripps Institution of Oceanography. (1986). Global seismograph network - IRIS/IDA [Dataset]. https://doi.org/10.7914/SN/II

Warburton, R. J., & Goodkind, J. M. (1977). The influence of barometric-pressure variations on gravity. *Geophysical Journal International*, 48(3), 281–292. https://doi.org/10.1111/j.1365-246x.1977.tb03672.x

Wielandt, E., & Streckeisen, G. (1982). The leaf-spring seismometer: Design and performance. *Bulletin of the Seismological Society of America*, 72(6A), 2349–2367. https://doi.org/10.1785/BSSA07206A2349
Wilson, D. C., Hutt, C. R., Gee, L. S., Ringler, A. T., & Anthony, R. E. (2024). Global seismic networks operated by the US Geological Survey. *Seismological Research Letters*, 95(3), 1578–1590. https://doi.org/10.1785/0220230178

Zürn, W., Exß, J., Steffen, H., Kroner, C., Jahr, T., & Westerhaus, M. (2007). On reduction of long-period horizontal seismic noise using local barometric pressure. *Geophysical Journal International*, 171(2), 780–796. https://doi.org/10.1111/j.1365-246x.2007.03553.x

Zürn, W., & Widmer, R. (1995). On noise reduction in vertical seismic records below 2 MHz using local barometric pressure. *Geophysical Research Letters*, 22(24), 3537–3540. https://doi.org/10.1029/95gl03369

Zürn, W., & Wielandt, E. (2007). On the minimum of vertical seismic noise near 3 MHz. *Geophysical Journal International*, 168(2), 647–658. https://doi.org/10.1111/j.1365-246x.2006.03189.x