Frequency spectra of the TEC oscillations
The amplitudes of the TEC oscillation (Fig. 3) vary in time, possibly by the interference of oscillations in multiple frequency peaks. Long continuous TEC records enabled by QZSS are suitable for studying their frequency spectra. Figure 5a shows the time series of slant TEC of J07 observed at three stations (0603 on Hahajima, 2007 and P217 on Chichijima) in 5:00–9:30 UT. A positive pulse at ~ 8:50 UT observed at 2007 and P217 would possibly be a sporadic-E irregularity (e.g., Maeda and Heki 2015) irrelevant to the volcanic eruption. We select the 4-h data 5:20–9:20 and estimated their frequency components using the Blackman–Tukey method (Fig. 5b).
They show four frequency peaks at about 3.7, 4.4, 4.8, and 5.4 mHz, with the 4.8 mHz peak weaker than the other three. The power shows a sharp drop for frequencies lower than 3.7 mHz possibly because this frequency is close to the acoustic cut-off of the atmospheric filter (Blanc 1985). These frequencies correspond to periods of about 270, 227, 208, and 185 s, respectively. The first two are the atmospheric resonance frequencies detected by seismometers after the 1991 eruption of the Pinatubo volcano (e.g., Kanamori and Mori 1992). They also coincide with the two modes with abnormally large amplitudes in the background free oscillation of the Earth (Nishida et al. 2000). The higher two frequencies are also overtones of the atmospheric resonant oscillation (Watada and Kanamori 2010).
These frequencies would depend on local atmospheric structures such as the mesopause altitudes. Precise identification of these peaks is important to investigate differences in atmospheric structures. We emphasize the benefit of long arcs enabled by QZSS because such a study would have been difficult with short arcs of conventional GNSS like GPS.
Temporal decay of the TEC oscillations
Next, we analyze how such TEC oscillation decayed in time. Figure 6a compares time series in two sequential 3-h periods (5:20–8:20 and 8:20–11:20) and two spectrograms made using the initial 2 hours of these time windows. Strong atmospheric mode peaks in the earlier time disappear in the later time (Fig. 6b). Figure 6c shows gradual decay of the three peak frequency intensities. Because the time corresponds to local afternoon (5:20 UT is 14:20 in the local time), background TEC also decays. However, the decay of the oscillation exceeds that in the background TEC calculated with a global ionospheric map (GIM) (Mannucci et al. 1998) suggesting that the atmospheric resonance would have decayed substantially within 3 h (Fig. 6c). We consider this decay reflects that of the excitation source (the Plinian eruption itself) rather than the diminishing of the oscillation governed by its quality factor.
Comparison of 3 different combinations of L1, L2, and L5
The QZSS satellites transmit microwave signals in three different frequencies L1 (~ 1.575 GHz), L2 (~ 1.228 GHz), and L5 (~ 1.176 GHz). So far, we have been combining L1 and L2 phases to calculate TEC, but the three frequencies allow us to compare three different combinations (L1–L2, L1–L5, and L2–L5) for TEC. Let fh and fl be the higher and lower frequencies to be combined, then we multiply their phase differences (expressed in lengths) with the factor fh2 fl2/(fh2−fl2) to obtain TEC. This factor becomes smaller (TEC data become less noisy) if the two frequencies are more different. The actual values of this factor are 7.76, 9.52, and 42.08 for the L1–L5, L1–L2, and L2–L5 combinations, i.e., the L1–L5/L2–L5 combinations would have the smallest/largest noises.
Figure 7a compares slant TEC time series obtained with the same station–satellite pair (2007 – J07) using the three different phase combinations. Indeed, the L2–L5 combination shows the largest noise, and the L1–L5 combination shows a slightly smaller noise than the conventional L1–L2 combination. The total standard deviation (σtot) would be composed of real ionospheric TEC changes (σion), caused, e.g., by small-scale electron density irregularities (scintillations), and phase reading errors within GNSS receivers (σrx). The latter would be enhanced by the frequency factor F as discussed above while the former would remain the same for any frequency combinations. Then, their variances would have the following relationship:
$$ \sigma_{{\mathop {{\text{tot}}}\nolimits^{2} }} \; = \;\sigma_{{\mathop {{\text{ion}}}\nolimits^{2} }} \; + \;F^{2} \sigma_{{\mathop {{\text{rx}}}\nolimits^{2} }} \qquad F\; \equiv \;f_{{\mathop {\text{h}}\nolimits^{2} }} \;f_{{\mathop 1\nolimits^{2} }} \;/\;(f_{{\mathop {\text{h}}\nolimits^{{2}} }} - \;f_{{\mathop 1\nolimits^{2} }} ).\; $$
(1)
Figure 7b compares the total standard deviation σtot as a function of the factor F. We used (1) as the observation equation and estimated the two quantities σion and σrx for three different periods by the least-squares method. Figure 7c indicates that the receiver noise σrx remains constant for the three periods while ionospheric scintillation σion decays rapidly as time elapses. This provides an additional support for the decay of atmospheric modes within 3 hours (Fig. 6c).
TEC oscillation amplitudes
Cahyadi et al. (2020) compared the TEC oscillation amplitudes relative to the background vertical TEC from three eruption cases and suggested that they might be proportional to the mass eruption rate (MER). The motions of electrons in the ionospheric F region are constrained in the direction of the ambient geomagnetic fields. This causes the directivity of ionospheric disturbances. They appear strongly on the equator side of the volcano, i.e., stronger disturbances emerge on the southern side in northern hemisphere (e.g., Heki 2006; Kundu et al. 2021), and northern side in southern hemisphere (Nakashima et al. 2016).
Figure 8a shows how such north–south asymmetry occurs using numerical simulation of upward propagation of atmospheric acoustic waves following Kundu et al. (2021). There, hypothetical line-of-sight with elevation angle 45º from two points assumed 270 km due south (Ps) and due north (Pn) are given with white lines. In Fig. 8b, c we compare slant TEC changes as viewed from the points Pn and Ps. Their amplitude differs by a factor of ~ 4, and the amplitude observed at 0603 using J07, calculated assuming real azimuth and elevation of J07 from 0603 (red curve in Fig. 8b), is similar to the point Pn case. We discuss waves close to the acoustic cut-off frequency, and a simple raytracing (Kundu et al. 2021) involves errors by neglecting gravitational restoring forces. Nevertheless, this factor ~ 4 is supported by the real north–south asymmetry observed in the 2003 Tokachi-oki earthquake (Heki and Ping 2005).
The current 0603-J07 slant TEC peak-to-peak amplitude of ~ 0.23 TECU becomes ~ 0.19 TECU in vertical TEC by multiplying with the cosine of the incidence angle of line-of-sight (~ 33º) with the F region ionosphere. This corresponds to ~ 0.76% of the background vertical TEC, ~ 25 TECU according to GIM.
This relative TEC oscillation amplitude in the 2021 Fukutoku-Okanoba eruption is comparable to those associated with the 2015 Calbuco and 2010 Merapi eruptions (Cahyadi et al. 2020). However, we would have observed ~ 4 times as strong oscillation if we had a GNSS station to the south of the volcano (Fig. 8). If we consider a factor 4 difference, the TEC oscillation amplitude of the 2021 Fukutoku-Okanoba eruption may reach ~ 3% of the background vertical TEC, which exceeds the value for the 2014 Kelud eruption (Nakashima et al. 2016; Cahyadi et al. 2020). Then, the MER at the peak time (~ 5:20 UT) of the TEC oscillation may have been as large as 5 × 107 kg s−1. This is consistent with the total amount of ejecta in this eruption inferred as 3–10 × 1011 kg (GSJ 2021).