Results shown in the previous sections may be unreliable if low amplitude EXs were not correctly counted due to large noise amplitudes. For example, by removing the five lowest-amplitude bins from the Sakurajima EXs plotted on the double-logarithmic scale, correlation coefficients increase from 0.892 to 0974 for 1963–1999 data and from 0.948 to 0.996 for 2008–2011 data. However, EXs are some of the largest earthquakes occurring at Sakurajima and the signal amplitudes are well beyond the noise level. Hence, SVRC correctly detects EXs without interference from noise signals. At Semeru, EXs were selected by setting a high threshold level (10–40 times the noise level), and the magnitude–frequency distributions show a convex shape even in the high amplitude range (see Fig. 4). Similar convex shapes are recognized in full amplitude ranges of Stromboli volcano. These results suggest that low detection levels of EXs in the low amplitude range are not the main cause of the convex shape of the magnitude–frequency distributions, or of the low correlations for the power-law distributions.

We have fitted a straight line on the double-logarithmic scale magnitude–frequency distribution at each volcano. However, as Tanaka et al. (1972) explained in an investigation of data from Akita-Komagatake, Japan, the magnitude–frequency distribution at volcanoes such as Semeru, Stromboli, and Tokachi may be fitted by two straight lines on the double-logarithmic scale. For example, we can see inflections in the magnitude–frequency distribution at about −4.8 on the horizontal axis for Stromboli data, at about −2.8 and −3.1 for Semeru vulcanian and gas burst data, respectively, and −4.7 for Tokachi volcano. Fitting two straight lines to the data implies that the magnitude distribution follows two different power laws; we cannot statistically reject this possibility. Also, we are not able to objectively distinguish it from the exponential distribution, although the power laws are limited in narrow ranges of <1 order. If the distributions comprise two power laws, the inflection point may be related to a scale change in the magma system to generate explosions.

EXs are often explained by a single force acting at the shallow part of a volcano (e.g., Kanamori et al. 1984; Ohminato et al. 2006) or by pressure release of a magma chamber (Uhira and Takeo 1994). The amplitude of the single force is proportional to the product of the cross-sectional area of the vent and the pressure stored before explosion. The amplitude is related to the time derivative of the seismic moment that is defined by the volume of the chamber. These theoretical predictions indicate that the amplitudes of these sources distribute exponentially. Since EXs at each volcano repeatedly occurred from the same vent or crater, we infer that the characteristic scales are related to the size of conduit or chamber, and fluctuations of excess pressure stored before explosion may change for each explosion.

We have represented the observed exponential distributions by \(\log_{10} N = N_{0} \exp ( - a/a_{0} ),\) where *N* is the number of events at the maximum amplitude of *a*, *N*
_{0} is the constant related to the total number of events. The parameter *a*
_{0} is measured from the slope of the semilogarithmic graphs. To compare the characteristic scales of EXs at different volcanoes, we use the reduced displacement *D*
_{R} that is an indication of the strength of the seismic source (Aki and Koyanagi 1981), although *D*
_{R} is generally used to evaluate volcanic tremor sources. EXs generally excite large amplitude Rayleigh waves from a shallow depth in the conduit (Kanamori and Given 1983; Tameguri et al. 2002). Hence, we have assume that the dominant waves are surface waves and determine the reduced displacement, *D*
_{R}, for the surface wave, which is expressed as \(D_{\text{R}} = A_{0} \sqrt {\lambda r/2}\) where *r* is the epicentral distance, *A*
_{0} the maximum amplitude in displacement, and *λ* the wavelength. We assume a homogenous structure with a surface wave velocity of 1000 m/s, and the wavelength of the surface waves is calculated from the dominant frequencies *f*. When the observed maximum amplitude is in velocity, we obtain the displacement by using the equation \(A_{0} = a_{0} /2\pi f.\) Table 1 compares the reduced displacements for different EXs. Stromboli data at 0.05–0.2 Hz are excluded from the analysis, because the long-period waves are near-field and not surface waves. We also evaluate the seismic magnitude *M* using Tsuboi’s formula (Tsuboi 1954), where \(M = 1.73\log r + \log A_{0} - 0.83.\) Seismic magnitudes range from about −0.2 to 2. From the reduced displacements and seismic magnitudes obtained, the largest EXs are those of vulcanian eruptions at Sakurajima, followed by Tokachi, Semeru, Lokon and Suwanosejima volcanoes. Reduced displacement and seismic magnitude for the 1–8 Hz Stromboli EXs are much smaller than the others. If the excess pressure before explosion is almost constant, as suggested by Nishimura and Hamaguchi (1993) and Nishimura (1998), which shows a scaling relationship in the amplitude of single force and source duration time, the differences in reduced displacements and seismic magnitudes at these volcanoes may be attributed to vent radius. The differences could be examined in more detail if site amplification factors and attenuation structure, or more detailed waveform analyses using seismic waves, are incorporated in the analyses.

It is well known that the magnitude–frequency distribution of the volcanic explosivity index (VEI, Newhall and Self 1982), which is determined from the volume of ejecta or column height, is expressed by a power law for all volcanoes around the world (Simkin 1993). Recently, Nakada (2015) examined Sakurajima and Izu-Oshima, Japan, and Merapi in Indonesia and indicated that the magnitude–frequency distributions of VEI in the range of 2–6 obey a power law, even for individual volcanoes. Power-law distributions represent self-similar properties in the eruption system and suggest there is no characteristic scale, even in an individual volcano. This is different from the exponential distribution found for EXs and may be attributed to different scaling relations of eruption mechanisms. However, we are not able to discuss the inconsistency in the magnitude–frequency distributions between EXs and VEIs, because the amount of ejecta associated with each EX we analyzed is often too small to evaluate its VEI. Even for large vulcanian eruptions from Minami-dake at Sakurajima, column heights reach about 5000 m and their VEI is equivalent to 2. The EX magnitude–frequency distributions reflect the statistical characteristics of mechanical properties and may not be directly related to the amount of ejecta that determines the VEI. Further investigation on the volume of ejecta associated with vulcanian eruptions, for example, using ash collectors (e.g., Tajima et al. 2013), is necessary.

We are not contending that the magnitude–frequency distributions of eruption earthquakes always show exponential or two power-law relationships. Suwanosejima shows a better fit to the power law, and Minakami (1960) indicated that EXs and B-type earthquakes that may be related to small explosions at Sakurajima and Asama volcanoes, Japan, follow the Ishimoto–Iida’s formula. It is also noted that our observation periods are limited, and there must be temporal changes in the magnitude–frequency distributions. Figure 6 shows an example of temporal changes in the magnitude–frequency distributions of Sakurajima. Most of the plots show exponential distributions, but the slopes of the graphs change with time: *a*
_{0} values representing the slopes are larger in 1984–1987 than in the other periods, which may reflect changes in eruptive activity and magma system at Sakurajima. We also divide the data at Suwanosejima into two periods, January–February and March–June, but we do not see significant temporal changes in the magnitude–frequency distributions. The origins of temporal changes in the magnitude–frequency distribution at each volcano may be clarified by carefully examining relationships with eruptive activities. It is worthwhile mentioning that the amplitude ranges of most EXs are limited to around one order of magnitude for each volcano. Also, magnitude–frequency distributions may deviate from an exponential form at low and high amplitude ranges, as shown, for example, in the data for amplitudes of <0.05 mm/s at Sakurajima (Fig. 6). These observations need to be considered to understand the overall characteristics of magnitude–frequency distributions of volcanic explosions.

Finally, we discuss the possibility of airwave contamination of EX seismograms. At Sakurajima, maximum amplitudes are measured in the seismograms before the arrival of airwaves. We see ground-coupled airwaves in the seismograms at Semeru and Suwanosejima, but these signals are generally smaller than the maximum amplitude of ground motion. At Lokon, ground-coupled waves are not well recognized, probably because a hill located between the station and crater obstructs the propagation of airwaves. Hence, we simply measured the maximum amplitudes in the EX seismograms for these three volcanoes. There is no detailed description in Okada et al. (1990) on the amplitude measurements at Tokachi-dake. However, according to Ichihara et al. (2012), the amplitude of ground-coupled airwaves is estimated to be <about 20 % of that of maximum ground motion, based on observation at a nearby station. Hence, the data in Okada et al. (1990) can be reliably used for the examination of EX magnitude–frequency distributions. We analyzed very low-frequency seismic waves (0.05–0.2 Hz) only for Stromboli, as EXs at the other volcanoes are dominated by short-period waves around 1 Hz, and no significant low-frequency waves are recognized. Also, since impulsive airwaves are generated from vulcanian eruptions, low-pass-filtered seismograms may represent ground-coupled airwaves. Hence, we have not analyzed low-frequency bands for the other volcanoes.