Since we have collected necessary information on the Lamb waves from the Tonga volcano, now we can calculate each of the three (Newtonian, free-air and inertial) terms of atmospheric loading that would have affected the gravity observed at Matsushiro. For the calculation, we use the data of atmospheric pressure recorded at Matsushiro as representative of the whole study area.

Zürn and Wielandt (2007) gives formulae for calculating the effects of atmospheric loading for several theoretical models. Here we adopt the “acoustic-gravity wave model” (Sect. 4.3) of that paper, because we deal with traveling plane waves. For a single angular frequency *ω*, the three equations (13), (15) and (16) of Zürn and Wielandt (2007) give the relevant Fourier components of Newtonian, free-air and inertial effects, respectively. Therefore, the necessary steps to be taken here will be (i) Fourier transform the time domain data of surface atmospheric pressure, (ii) multiply the Fourier components by the factors in the right-hand sides of the three equations and (iii) inverse Fourier transform the frequency domain data back to the time domain. These procedures are quite simple to implement. However, we found that applying this method to our data gives an unrealistically large value (larger than \(9\times {10}^{-9} \, {\text{m}}{\text{s}}^{-2}\)) for the Newtonian effect. The cause of this phenomenon is likely the assumption of the half-space adopted in the theory. Because the Newtonian attraction slowly decays with distance, a flat Earth model with infinitely wide spatial extension may lead to a serious overestimation of the total force. On the other hand, the free-air and the inertial effects are related with local displacement fields of the ground, which will be affected to a lesser degree by the half-space treatment compared with the Newtonian effect. Therefore, here we choose to adopt a less sophisticated grid-method for calculating the Newtonian effect, whereas the method of Zürn and Wielandt (2007) is applied to calculate the free-air and the inertial effects.

For calculating the Newtonian effect, we consider a square area of 100 km × 100 km size around Matsushiro (Figure 1). The whole area is divided by a 360 × 360 grid. The size of each cell is about 286 m, corresponding to the propagation length of the Lamb waves in 1 s (the sampling interval of the barometer data). Within one cell, atmospheric pressure is taken to be uniform. Each cell is further divided into 29 × 29 sub-cells, each having the size of about 10 m. Vertically, 10,000 layers with 5-m spacing are used. The total height of the atmosphere considered is 50 km. It is noted that the atmospheric Lamb waves are evanescent (see Fig. 5 of Arai et al. (2011)). Following equation (10) of Zürn and Wielandt (2007), we assume that density perturbations for the traveling waves depend on the height \(z\) as \(\mathrm{exp}\left(-z/H\gamma \right)\), where \(H\) is the scale height and \(\gamma\) is the specific heat ratio of the atmosphere. We assume that \(H=8\times {10}^{3} \, {\text{m}}\) and \(\gamma =1.4\). We ignore topography and curvature of the ground. For each epoch, Newtonian attraction due to density perturbations from all the sub-cells is summed up to give the total effect at station Matsushiro.

For calculating the free-air and the inertial effects, equations (15) and (16) of Zürn and Wielandt (2007) are applied. Converting the horizontal wavenumber into angular frequency *ω* in the two equations, the Fourier components for the free-air and the inertial effects are given by

$$\Delta {g}_{z}^{F}=-\left|\frac{\delta g}{\delta z}\right|\frac{1}{2\mu }\left[\frac{\lambda +2\mu }{\lambda +\mu }\right]\frac{c}{\omega }{p}_{\omega }$$

and

$$\Delta {g}_{z}^{I}=-\frac{1}{2\mu }\left[\frac{\lambda +2\mu }{\lambda +\mu }\right]c\omega {p}_{\omega } ,$$

respectively. \(\lambda\) and \(\mu\) are the Lamé constants and \(c\) is the wave velocity. \({p}_{\omega }\) is the Fourier component of temporal pressure changes for angular frequency \(\omega\). We assume that \(\lambda =\mu\), so that the factor \(\left(\lambda +2\mu \right)/\left(\lambda +\mu \right)\) is equal to 1.5. The vertical gravity gradient \(\left|\delta g/\delta z\right|\) is taken to be \(3.08\times {10}^{-6} {\text{s}}^{-2}\). Note that magnitudes of these two effects are inversely proportional to the value of rigidity \(\mu\). Because the Fourier component for the inertial effect is proportional to the angular frequency, it is necessary to suppress higher frequencies in the calculation. After some trials, we found that limiting the frequency band up to 0.0035 Hz gave a good result when compared with observed gravity residuals. The same cutoff is applied also to the Fourier component of the free-air effect.

For the observational data of gravity to be compared with theoretical calculations, we apply tidal correction using the parameters determined from more than ten years of gravity observations at Matsushiro. We do not apply the usual atmospheric correction with an admittance (\(-3.3\times {10}^{-11} \, {\text{m}}{\text{s}}^{-2}{ \, {\text{Pa}}}^{-1}\) in the case of Matsushiro). The residual gravity time series are then lowpass-filtered. Again, after some trials, we found that choosing a cutoff frequency of 0.0025 Hz gave a reasonable result. The difference in the optimal cutoff frequencies for pressure and gravity is likely due to the different frequency (\(f\)) dependence of the power spectral densities of these data at the millihertz band (\(\sim {f}^{-1}\) for pressure and \(\sim {f}^{0}\) for gravity).

Figure 4b shows the calculation results of the three terms of atmospheric loading, using the atmospheric pressure data at Matsushiro shown in Figure 4a. The attraction term (blue) looks like a smoothed version of the atmospheric pressure. Because we assume that the atmospheric disturbances take the form of plane waves traveling in one direction, spatially integrating attractions from the cells over the study area is similar in effect to applying a lowpass filter to the time domain data. When the change in atmospheric pressure takes its maximum value of 174 Pa at 11:37 (UTC), the gravity decrease from the attraction term reaches \(7.3\times {10}^{-9} \, {\text{m}}{\text{s}}^{-2}\). If we simply calculate the ratio between these numbers, we obtain \(-4.2\times {10}^{-11} \, {\text{m}}{\text{s}}^{-2}{ \, {\text{Pa}}}^{-1}\) as an apparent admittance. The reason why this is larger than usually seen comes from the fact that in the case of Lamb waves the atmospheric admittance due to Newtonian attraction is theoretically about 1.4 times as large (in the absolute sense) as that for the atmosphere in static equilibrium (Zürn and Wielandt 2007). During the initial positive pressure pulse of the Lamb waves (from 11:18 to 11:46), the negative changes in gravity due to the Newtonian effect are partly cancelled by positive changes due to the free-air (green) and the inertial (red) effects. The rigidity \(\mu\) is taken to be 40 GPa. The free-air effect is temporally smoother than the inertial effect, because the latter is derived by differentiating the former twice with respect to time for each Fourier component. The sum of the three terms (magenta) is shown in Figure 4c. It reproduces well the observed gravity changes (cyan). The calculation reproduces also the latter parts of the wave train (from 11:50 to 12:30), indicating that our model also fits those later phases of atmospheric waves.

The good agreement between observations and theoretical calculations proves that the theory of Zürn and Wielandt (2007) is correct. In particular, incorporating the inertial effect of atmospheric loading which is often neglected is the key feature to understand the observed gravity signals. In other words, this result may be regarded as a clear identification of the inertial effect of atmospheric loading by surface gravity observations. If we subtract the Newtonian and the free-air terms from the observed gravity, we can isolate the inertial term. Figure 4d shows the inertial effect thus extracted, to be compared with the calculated one shown in Figure 4b. This is the “signal” of the inertial effect of atmospheric loading, identified by precise gravity observations with a superconducting gravimeter. This identification was made possible because the atmospheric disturbances were (1) energetic enough and (2) spatially coherent. Although this is not the first report of the signal of this kind (Zürn and Meurers 2009), the Tonga event has provided a rare opportunity for studying such a phenomenon with sufficient signal-to-noise ratio.

The rigidity \(\mu\) is an undetermined parameter in the theoretical calculations. After some trials, we found that the observation is well explained when we choose \(\mu\) = 40 GPa, not 30 GPa or 50 GPa. In other words, a rough estimate of rigidity has also been obtained from our analysis (Wang and Tanimoto 2020). The theory we rely upon in this study was developed on the basis of an elastic half-space with homogeneous elastic properties. Therefore, it is not easy to specify the effective depth to which the present estimate of rigidity is sensitive. Considering that the wavelength of the Lamb waves used in our analysis ranges from a few tens to a few hundreds of kilometers, the estimated rigidity may represent the property in the upper part of the Earth with similar vertical scales. Indeed, rigidity of 40 GPa corresponds to the depth of 15–20 km according to PREM (Dziewonski and Anderson 1981).