Figure 4 shows the seismic trace and the corresponding spectrogram for two stations located at different epicentral distances, POHA (Pohakuloa, Hawaii), at 4955 km, and PAB (San Pablo, central Spain) at 17735 km. In both cases, the arrival of the seismic waves generated by the volcanic explosion is shown with blue dashed lines. Red dashed lines show the Lamb waves, which reach POHA around 8:30 and PAB around 20:00. The most prominent feature of the spectrograms is the high energy detected around 3 mHz, lasting 10–12 hours.
This low-frequency, long-duration signal appears clearly in a seismic section similar to the one presented in Fig. 1, but now filtering between 2.5 and 6.5 mHz (Fig. 5a) shows a seismic section similar to the one presented in Fig. 1, but now filtering between 2.5 and 6.5 mHz. The first observations of this kind of signals date back to the early 1990s, following the eruption of Mount Pinatubo in 1991 10,11. The signals were described as a bichromatic, with two spectral peaks at 3.7 y. 4.3 mHz. Inspecting previous data, 11 identified similar signals produced by the 1982 El Chinchón eruption, with spectral peaks located in this case at 3.7 and 5.1 mHz. No further evidence of low-frequency seismic signals observed worldwide since then has been reported, but Dautermann et al. (2009) reported signals with a maximum amplitude of around 4 mHz, after the collapse of the explosive lava dome. de Soufrière Hills (Montserrat, Lesser Antilles) in July 2003 km on borehole dilatometers measuring volumetric strain on Montserrat Island.
To better explore the properties of these signals, we have calculated the spectra of each seismic station using a 24 hours interval. The spectra obtained for a selection of sites, evidences the presence of clearly defined amplitude peaks between 3 and 6 mHz (Fig. 6). As the frequency and relative amplitude of these peaks are rather constant, we have stacked the results obtained from the 95 available stations to produce the stacked spectra shown as a blue line. The regular spacing of the peaks suggest that they may be related to the spheroidal normal modes of the Earth that form the so-called “Earth’s Hum”. To verify this point we have added to the figure the eigenfrequencies of the Earth, calculated from the PREM Earth reference model 13, as reproduced in 14. As clearly shown, the dominant frequency is 3.7 mHz and corresponds to the oS29 spheroidal mode. A second group of spectral peaks with high energy is seen around 4.65 mHz. Moving to higher frequencies, a third group can be identified around 5.4 mHz and a last one around 6.0 mHz. It is interesting to note that for each of these groups, the modes around the dominant one also have significant energy.
Kanamori and Mori (1992) and Widmer and Zürn (1992) interpreted the two spectral peaks observed in their data as resulting from two atmospheric waves, the low frequency one being a gravity waves and the other a pressure wave.Widmer and Zürn 1992) pointed out a feedback regime between the atmosphere and the volcano and attributes the difference in the high frequency value between El Chinchón and Mount Pinatubo (5.1 vs 4.4 mHz) to temperature changes in the atmosphere affecting the pressure wave. Following a different approach,15 considered the solid Earth and the atmosphere as a single system and calculated the theoretical normal modes, concluding that the 3.7 and 4.4 mHz nodes are the ones with the highest energy in the atmosphere. Using these theoretical modes, Lognonné (2009) built synthetic seismograms that fit the signals from Mount Pinatubo and showed, as in our dataset, a significant increase in amplitude near the antipodal point. The Mount Pinatubo model found reasonable amplitudes only for a source located at 24–28 km of altitude. Although it will be necessary to elaborate a specific synthetic model for the case of the Hunga-Tonga eruption, we would like to indicate that the twin waveforms separated by about 200 s observed for the events of 04:00 and 5:30 could be explained by a source located in this height range and a wave reflection at the top of the stratosphere.
The spectra stack constructed using 95 stations distributed around the world shows that the signal cannot be described as bichromatic, as there is a clearly dominant mode group around 3.7 mHz, but also three other mode groups with relatively high amplitudes, with maximums at 4.5, 5.3 and 6.0 mHz. For all four groups, the main mode has peaks of lower amplitude on both sides and that all of them coincide with the eigenfrequencies of the PREM Earth reference model. Thus, we favor an origin related to a coupling of the atmospheric system-solid Earth, in which the large atmospheric explosion resulted in the excitation of the normal modes of the Earth between 2 and 6 mHz, the so-called “Earth's Hum”, with higher amplitudes at frequencies where atmospheric waves carry more energy and can be better transferred to the ground.
As seen in Fig. 5b, at stations located at large epicentral distances, the amplitude of the low-frequency signal decreases smoothly with time, only to increase again between 09:00 and 10:00 UTC. The timing of this increase coincides with the arrival of surface seismic waves generated by the 08:30 volcanic explosion (see section 2), which suggests that this event may have increased the level of normal mode excitation, a hypothesis that should be confirmed by future modelling.