For harmonic atmospheric forcing, the surface boundary condition can be written as 3P(z=0,t)=P0cos⁡ωt, and the corresponding analytical solution for Eq. (1) is 4P(z,t)=P0e-zω/2Dcos⁡ωt-zω/2D where ω is the frequency of the atmospheric fluctuations (Hanks and Woodruff, 1958). Equation () shows that atmospheric tides generate a depth-dependent oscillatory pore-gas pressure field characterized by exponential amplitude decay and a progressive phase lag with depth, both controlled by the hydraulic diffusivity. These cyclic pore-gas pressure changes can imprint on seismic observations as subdaily modulations of the relative seismic velocity changes (e.g., Kramer et al., 2025; Wang et al., 2025; Mao et al., 2019).

In particular, for unsaturated and dry conditions in a regime where the relative motion between gas and solid phases is negligible, Conte et al. (2009) express the shear- and compressional-wave velocities as 5Vs2=G(1-ϕ)ρS+ϕρg, and 6VP2=2G1-νSK/1-2νSK+Kg/ϕ(1-ϕ)ρS+ϕρg, where G is the shear modulus, φ is the porosity, νSK is Poisson's ratio of the soil skeleton, ρg and ρS are the densities of gas and solid phases, and Kg denotes the bulk modulus of the gas filling the soil pores. Crucially, only VP depends directly on pore-gas pressure through the pressure-dependent bulk modulus (Kg∝∂P∂V). Therefore, the pore-gas pressure oscillations predicted by Eq. () provide a pressure-controlled mechanism for generating subdaily changes in compressional-wave velocity within the vadose zone.

Beyond the subdaily response described above, sustained diffuse degassing can progressively modify the background state of the vadose zone by increasing gas content and mean pore-gas pressure. Under a gas-injection scenario, increasing mean pore-gas pressure reduces the effective gas compressibility (cg), thereby decreasing (ct) and increasing the hydraulic diffusivity (D). Consequently, atmospheric pressure oscillations are expected to penetrate deeper and attenuate less strongly, producing larger pore-gas pressure amplitudes at depth and modifying the phase lag relative to the surface forcing (Eq. 4). Progressive gas accumulation should therefore strengthen the subsurface imprint of atmospheric tides by enhancing pressure transmission through the vadose zone.

Long-term gas injection can also affect seismic velocities through changes in pore-gas density. An input of additional gas into the pore space increases ρg, slightly raising the overall bulk density ((1-ϕ)ρS+ϕρg). As implied by Eq. (), this leads to a slight decrease in VS if the shear modulus (G) remains unchanged. In contrast, the long-term evolution of VP reflects the combined influence of density and the pressure-dependent gas bulk modulus (Kg) (Eq. 6), such that evolving pore-gas conditions are expected to modulate more strongly VP than VS (e.g., Sánchez-Pastor et al., 2023). Accordingly, increasing gas accumulation is expected to manifest as a modest VS decrease, together with an enhanced and more persistent subdaily modulation of VP driven by atmospheric tides.

Importantly, the seismic observable used in this study (RWE) is sensitive to near-surface velocity variability but does not uniquely separate the relative contributions of VP and VS nor does it determine the sign of the underlying perturbation without independent constraints (e.g., direct velocity measurements). The analysis of barometric pumping reduces this ambiguity because it introduces an externally imposed forcing with a constrained physical effect: in the unsaturated vadose-zone regime considered here, subdaily pressure oscillations primarily modulate the pore-gas bulk modulus rather than the shear modulus of the solid skeleton. Therefore, although our methodology does not explicitly distinguish VP from VS, examining the long-term evolution of the subdaily atmospheric-tide response provides a more constrained basis for interpretation of the observed RWE variations in terms of pressure-driven, gas-controlled changes in the vadose zone.

For this reason, the Results section focuses on the long-term evolution of the subdaily atmospheric-tide response. We examine temporal changes in its amplitude, persistence, and phase and evaluate whether they are compatible with progressive gas injection and a more efficient transmission of pressure oscillations into the vadose zone. In this framework, RWE variability is interpreted primarily in terms of pressure-driven, gas-controlled changes in the shallow subsurface.

The interpretation of subdaily pressure oscillations in terms of hydraulic diffusivity D, and therefore in terms of subsurface variability, requires isolating the contribution of atmospheric forcing from other potential drivers. As indicated by Eq. (), both amplitude attenuation and phase shift depend on the hydraulic diffusivity D. An increase in D reduces attenuation and decreases the phase lag with depth, while a decrease in D has the opposite effect. However, when multiple natural forcings act simultaneously and induce pressure oscillations in the subsurface (e.g., atmospheric tides, solid Earth tides, and aquifer fluctuations), changes in the observed amplitude or phase of the induced velocity response may not exclusively reflect variations in diffusivity but can also result from the superposition of different forcings (e.g., Kuang et al., 2013).

To minimize this ambiguity, we concentrate on atmospheric tidal components that are spectrally well separated from other periodic forcings. In La Palma, the most robust strategy to confidently isolate the atmospheric contribution is to focus on the 8 h atmospheric tide. This component is clearly expressed in the atmospheric pressure spectrum, whereas the corresponding 8 h oceanic, solid Earth, and temperature tidal constituents are nearly absent (Fig. 2). This spectral separation makes the 8 h band a clean subdaily diagnostic of barometric forcing. In La Palma, the 8 h barometric tide has an amplitude more than an order of magnitude smaller than the dominant 12 h component (Fig. 2), with a peak-to-peak variation of 3.5 Pa. In contrast, the 8 h ocean tide (M3) is nearly three orders of magnitude weaker than the principal 12 h ocean tide (M2) (Dirección General de Costas, 2004), further reinforcing its suitability as a barometric indicator in this setting.

The applicability of the 8 h barometric tide in other regions is expected to depend on its spatial variability. While the 24 and 12 h atmospheric tides exhibit relatively consistent large-scale patterns (e.g., McMillan et al., 2019), the 8 h component is less well constrained and can vary significantly with longitude and regional atmospheric conditions (e.g., Moudden and Forbes, 2013), as it may arise from the superposition of multiple tidal components. Therefore, although the 8 h signal provides a useful diagnostic in La Palma due to its spectral isolation, its detectability and amplitude may differ substantially elsewhere, and its applicability should be evaluated on a site-by-site basis.

To investigate the link between RWE and known tidal forcings, we calculated the magnitude-squared coherence (MSC) between RWE and potential external drivers: atmospheric pressure and temperature time series derived from the ERA5 reanalysis database, and vertical solid Earth tides computed with the Python package pysolid (Yunjun et al., 2022). MSC quantifies, in the frequency domain, the linear correlation between two time series through their cross-spectra, and helps identify phase and amplitude synchronizations. It is defined as: 7γxy2(f)=|Pxy(f)|2Pxx(f)⋅Pyy(f) Where Pxx(f) and Pyy(f) are the power spectral densities (PSD) of the two time series, and Pxy(f) is their cross-spectral density. We estimated MSC using Welch PSD method (Welch, 1967), as implemented in the MATLAB function pwelch. The RWE and driver series were aligned to a common 2 h interval, segmented into overlapping windows of 60 d, tapered with a Hanning window, and averaged in the frequency domain.

The MSC spectra shown in Fig. 4c for the RWE function at 48 Hz compared to the three different tidal forcings indicate that the observed spectral peaks in RWE (Fig. 4a and b) are statistically significant and highly correlated with the external drivers. However, statistically significant coherence does not necessarily imply a causal relationship. For instance, in the MSC between RWE at 48 Hz and the vertical displacement of solid Earth tides, the most prominent peak appears at 2 cpd. In contrast, at 1 cpd where the solid Earth tide spectrum exhibits its strongest peak (Fig. 2d), no clear correspondence with RWE is observed. This discrepancy can be explained by the fact that both barometric and solid Earth tides share the 2 cpd harmonic. In this case, coherence does not reflect a direct causal link between solid Earth tides and RWE, but rather an indirect relationship mediated by the common periodicity present in barometric pressure oscillations. We infer that barometric pressure is the dominant driver at 2 cpd, as only the MSC between atmospheric pressure and ellipticity displays a clear correlation at 1 cpd and across all harmonics (Fig. 4c), as also reported by Kramer et al. (2023).

Figure 4d presents the MSC maps for all frequencies of the RWE function at station EHIG with respect to the three external drivers considered. Consistent with the spectrograms (Fig. 4a and b), the E₁ and E₂ components exhibit the highest coherences with well-defined spectral peaks. In contrast, the targeted terdiurnal component (E₃) is only discernible at the highest frequencies (>45 Hz) in the coherence map with atmospheric pressure. These results motivate restricting the long-term analysis of the 3 cpd behavior to the 40–50 Hz frequency band.

Having established that the terdiurnal component (E₃) is restricted to the 40–50 Hz band, we next examine its long-term variability over 2007–2023. This analysis provides the basis for linking the 3 cpd oscillations to diffuse gas injections and to their imprint on the long-term subdaily velocity response in the shallow vadose zone. To do so, we assume that the coupling between atmospheric pressure and RWE variations can be described as a linear time invariant system at that period. Under this assumption, the system response can be characterized by the complex transfer function 8G(f)=Pxy(f)Pxx(f) The magnitude |G(f)| quantifies the gain of the RWE response relative to the input, while its phase expresses the frequency-dependent delay. As in the coherence analysis, all spectra were estimated with the Welch method using Hanning tapers.

Figure 5 shows the temporal evolution of the gain, phase lag, and MSC at the terdiurnal frequency (3 cpd) for the EHIG station from 2007 to 2023. These results were obtained using 15 d sliding windows with a 2 d step, and the resulting time series were smoothed with a 10 d moving median. In each window, the 3 cpd component was isolated by averaging the cross- and auto-spectra within a narrow frequency band centered at 3 cpd. The top panel displays the gain, deemed as well as the amplitude ratio between the RWE response and atmospheric pressure variations. The middle panel shows the corresponding phase lag expressed in hours, with positive values indicating that RWE is delayed with respect to pressure. The bottom panel presents the MSC, which quantifies the statistical stability of the coupling at this frequency.

The comparison between gain variations and CO2 effluxes is challenging for several reasons. First, the two observables represent different physical scales: fluxes quantify the amount of gas released into the atmosphere, whereas RWE variations are most sensitive to changes within the upper meters of the subsurface. Specifically, in the 40–50 Hz band and using a reference velocity model constrained by tomographic studies of La Palma (e.g., Serrano et al., 2023; Cabrera-Pérez et al., 2023b), the RWE kernel is confined to depths shallower than ∼5 m, with maximum sensitivity at ∼2 m. A temporal delay between surface fluxes and subsurface RWE variations is therefore expected. Second, the CO2 efflux dataset represents a spatial average over the entire volcanic edifice, while our RWE measurement reflects a point-specific response. Under a diffuse degassing regime, it is reasonable that individual sites exhibit their own local trends. Third, the temporal resolutions are markedly different: RWE gain was estimated every two days, whereas CO2 effluxes are available as annual averages.

Despite these differences in scale, resolution, and sensitivity, both curves display broadly consistent long-term trends, suggesting a moderate agreement between the two observations (see Fig. A2 in Appendix). As reasoned in Sect. 3.1, an increase in gas concentration reduces the effective gas compressibility and enhances the hydraulic diffusivity. In the transfer function analysis, this process is expected to manifest as an increase in gain, reflecting the more efficient transmission of pressure oscillations into the subsurface.

Between 2015 and 2017 we observe a marked increase in coherence, accompanied by a rise in gain that could indicate a stronger degassing episode in this area (Fig. 5). Unfortunately, no direct records of gas emissions are available for this period to validate this interpretation. Over the entire study period, the phase lag displays a clear seasonal variability. Soil moisture and precipitation are known to influence gas transport by attenuating barometric pumping (Forde et al., 2019). Consistently, comparison of the phase lag results with the rainfall pattern for La Palma suggests a possible influence of precipitation. The seasonal oscillations in phase show a moderate alignment with the island's annual precipitation cycle, which may reflect transient reductions in diffusivity caused by water infiltration into the vadose zone. This observation departs from our initial assumption of a purely gas-filled vadose zone and underscores the role of hydrological processes in modulating the pressure response.

No sustained trend in gain, coherence, or phase that could be interpreted as a precursor signal to the 2021 eruption is observed at this station, suggesting that the eruptive episode did not correspond to the most intense period of diffuse degassing within its long-term record. Only a slight increase in coherence during the eruption episode and in early 2021 is remarkable, together with a disruption of the purely annual phase seasonality observed in mid-2020 and mid-2021, when the phase lag tended to approach zero (Anomalies marked in Fig. 5). As these phase lag perturbations are not accompanied by a corresponding increase in gain, we interpret them as changes in the hydrological state of the vadose zone, most likely related to variations in soil moisture or permeability of the soil, rather than as evidence of enhanced gas injections.

An additional and particularly noteworthy finding is the presence of a clear semiannual cycle in the gain, consistently observed throughout the study period (Fig. 6b). When examining the gain at the three analyzed stations, semiannual cycles are also apparent (Figs. 5–7). At EHIG, for example, during the low CO2 emission period (2007–2010) the cycle is clearly expressed (Fig. 5), and at TBT a semiannual modulation emerges with peaks in March and September and minima in July and December (Fig. 7b), a seasonal pattern that is synchronized across the three stations. Although environmental variables such as volumetric soil water content, wind, or even atmospheric pressure can also exhibit semiannual variability, none of them display a pattern as regular and well defined as the solid Earth tides. The latter reproduce the cycles observed at the three stations, supporting our hypothesis that the detected semiannual modulation reflects the influence of solid tidal forcing.

This semiannual modulation can be physically explained by the response of the fractured medium to solid Earth tides. At the peaks of the vertical tidal component, fractures are expected to open, leading to an increase in the effective permeability of the medium. As described by Eqs. ()–(), an increase in permeability directly translates into higher diffusivity and thus a more efficient propagation of the barometric pressure wave. Consequently, earth tidal peaks are expressed as gain maxima, in agreement with the observed cycles at the three stations (Figs. 5a, 6b and 7b). This result highlights not only the potential of our approach to monitor subsurface gas accumulation, but also its ability to detect permeability changes driven by tidal forcing. In settings where degassing or significant gas fluxes from the critical zone are not expected, analyzing the 3 cpd cycle and its response to solid Earth tides provides a novel tool to probe short term variations in subsurface permeability. We therefore propose that tidal modulation of barometric pumping can serve as a powerful diagnostic for permeability changes, with applications extending well beyond volcanic environments.

Although the semiannual modulation provides a common background signal at all stations, its expression depends on the degassing regime. At EHIG, for example, the cycle is most clearly observed during periods of low CO2 emission and becomes less evident when degassing intensifies (Fig. 5a). TBT, however, shows an additional feature: a marked increase in coherence together with a stabilization of the lag prior to the eruption (Fig. 7), pointing to localized changes in permeability or gas flux that were not equally expressed elsewhere. Notably, this anomaly would not be apparent if one considered gain alone, since the initial increase in coherence coincided with a decrease in the vertical solid Earth tide, corresponding to a scenario of fracture compression. Under such conditions, even if gas injection occurred, the simultaneous closure of fractures would dampen the RWE response. While we lack direct measurements of gas flux to validate this observation, we interpret the increase in coherence and stabilization of the lag at TBT as being consistent with subsurface gas accumulation and/or changes in vadose zone permeability. These processes may also have favored anomalous efflux to the atmosphere about six months before the eruption.

This distinctive behavior at TBT can be explained by its structural and hydrogeological setting. The station is located within a highly fractured zone affected by abundant vertical dike intrusions and intersected by water wells and galleries, which together enhance vertical permeability and pressure transmission. Such structural discontinuities likely provide preferential pathways for gas ascent from the underlying hydrothermal system, amplifying the coupling between barometric pumping and subsurface gas accumulation. In contrast, stations located in less fractured domains or outside the main hydrothermal influence, such as Taburiente Caldera (Fig. 1), lack this level of connectivity and therefore do not record the same precursor signal. To further verify that the observed anomaly at TBT reflects atmospheric forcing rather than random fluctuations or other mechanisms, we examined the frequency dependence of the coherence (γ2) and the standard deviation of the lag (σLag) around the terdiurnal cycle (Fig. 8a). The results show that the enhanced coherence and reduced lag variability preceding the eruption are consistently observed between 2.9 and 3.1 cpd, with the strongest signal at 3.0 cpd. This narrowband expression supports the interpretation that the anomaly is linked to barometric pumping and thus constitutes a genuine precursor signal.

A closely similar behavior is also observed at EXILP, where the standard deviation of the lag exhibits two marked reductions, including a drop towards nearly zero immediately prior to the eruption (Fig. 8b). When lag variability is examined as a function of period, both TBT and EXILP independently show a narrowing and stabilization of the 3 cpd response in the pre-eruptive interval. This coherent pattern at two stations with different geological settings strengthens our interpretation of the signal as a precursor, consistent with a pre-eruptive increase in shallow gas accumulation and/or permeability changes that enhanced barometric pressure coupling near the future vent area.