Introduction

Infrasound observation at active volcanoes has played an important role in monitoring volcanic activity and understanding eruption dynamics (e.g., Vergniolle and Brandeis 1994; Ripepe et al. 2001; Ripepe and Marchetti 2002; Dabrowa et al. 2011; Fee et al. 2013). Volcanic infrasound signals vary in amplitude, duration, and peak frequency, which reflects the diversity of eruption styles and dynamics at the acoustic source (e.g., Johnson and Ripepe 2011; Fee and Matoza 2013; Marchetti et al. 2009). The characteristics of infrasound records accompanying intermittent Strombolian eruptions are well explained based on the model of vibration and burst of a bubble (Vergniolle and Brandeis 1996; Vergniolle and Caplan-Auerbach 2004; Vidal et al. 2006). Attempts at acoustic source quantification has also been made with acoustic power (Vergniolle et al. 2004) and by applying a monopole source assumption (Johnson et al. 2004). Previous works focusing on small-scale gas-rich (Strombolian) eruptions have demonstrated that infrasound waveform analysis can be used to estimate gas volumes, which give results comparable to those obtained by other observation methods (e.g., Oshima and Maekawa 2001; Dalton et al. 2010; Delle Donne et al. 2016). Vulcanian eruptions, which are more explosive than the Strombolian type, yield an infrasound “pulse signal” (e.g., Petersen et al. 2006; Yokoo et al. 2009; Firstov et al. 2013). Such a pressure pulse has also been explained by volume change of a monopole source assumed to be located at the vent (Johnson and Miller 2014). Instead, the source of infrasound signals accompanying stronger, steady emission events (such as Plinian eruptions) is usually modeled as a dipole (Vergniolle and Caplan-Auerbach 2006) or turbulent jet noise (Matoza et al. 2009). In case a monopole source is adopted, the inferred volume, Vinf, is considered to be equivalent to that of the mixture of hot volcanic particles and gases emerging from the vent that displace the atmosphere above the vent. For events having an eruption cloud with a height of the order of several kilometers, previous studies have reported that Vinf is considerably smaller than the video-derived volume, Vvideo (Johnson and Miller 2014; Yamada et al. 2017). A possible explanation of the cause of the discrepancy is the entrainment of surrounding air into the eruption cloud, which is not taken into account in the analyses based upon the monopole assumption. As a result, assuming that Vinf is the volume of the total eruption cloud appears to be inadequate. However, since the infrasound data can be obtained in real time as part of an operational monitoring system, a simple waveform analysis with the monopole assumption remains an effective and efficient method to approximately quantify the acoustic source in near-real time and immediately after the eruption. Therefore, it is important to understand how Vinf relates eruption cloud volume and its component parts.

An eruption cloud accompanying a short-lived eruption is usually modeled as a thermal (e.g., Turner 1962; Woods and Kienle 1994; Sparks et al. 1997). Terada and Ida (2007) revealed that the kinematic features of volcanic clouds accompanying short-lived eruptions can be explained by assuming that the cloud is an isolated spherical thermal. A simple method proposed by Terada and Ida (2007) allows initial buoyancy of the thermal, F0, to be estimated with the maximum cloud height and a vertical profile of ambient air density without considering thermodynamics. Here, F0 is given by the product of the volume of the thermal, Vb, gravity acceleration, and the difference in density between the thermal and the ambient air. The thermal is expected to obtain F0 when it has entrained enough of the surrounding air to ascend by buoyancy only. We focus on Vb at that moment of transition to buoyancy, which can be obtained with the value of F0. Investigating the relationship between Vb and Vinf can aid our understanding of the nature of Vinf and explain how Vinf can be linked to the dynamics of the eruption plume.

The present study analyzes infrasound waveforms accompanying Vulcanian and phreatic eruptions at Aso and Shinmoedake volcanoes in Japan and Lokon-Empung volcano in Indonesia (Fig. 1) to estimate Vinf. We also refer to data for Vinf estimated in previous studies (Johnson and Miller 2014; Kim et al. 2015; Fee et al. 2017; Yamada et al. 2017). Estimation of F0 for each eruption follows the method of Terada and Ida (2007) which allow us to obtain Vb. We then examine the relationship between Vb and Vinf and discuss the dynamics of the eruption cloud behind the relationship.

Fig. 1
figure 1

a Map of the Pacific Ocean showing the location of Kyusyu Island (Japan) and the Minahasa Peninsula (Indonesia). Detail maps of b Kyusyu Island and c the Minahasa Peninsula, where red triangles show the location of volcanoes examined here. White diamonds denote the location of stations from which weather data were taken for the estimation of the buoyancy-derived volume, Vb. d, e Station networks at d Aso and e Shinmoedake (Kirishima volcano complex) volcanoes, where red triangles show the location of the active vents. White squares denote the stations of the NIED (ASHV, ASIV, ASTV, and ASNV at Aso and KRHV and KRMV at Shinmoedake) which has a microbarometer. The stations of the JMA (ASO2 at Aso and KIAM at Shinmoedake), that have infrasound microphones, are given by white pentagons. f Station map at Lokon-Empung volcano. A red triangle and a white square show the location of the active vent and the infrasound station (KKVO), respectively

Full size image

Following Sparks et al. (1997), here we use the term “plume” to describe the explosive ejection of material from the vent. We also use the term “eruption cloud” to mean a plume which has detached from the vent. The dynamics of a short-lived explosive eruption has two main phases: gas thrust and buoyancy (e.g., Wilson and Self 1980; Patrick 2007; Marchetti et al. 2009; Chojnicki et al. 2015). Patrick (2007) describes how the gas thrust phase consists of jets and starting plumes, and how the buoyancy phase can be classified into the following two parts: rooted thermals and discrete thermals. To be consistent with Terada and Ida (2007), who examined the discrete thermals, we focus on the discrete thermal for the buoyancy phase.

Estimation of the infrasound-derived volume, V inf

Dataset

Our dataset consists of 53 events, from which 22 events are available for waveform analysis (Aso, Shinmoedake, and Lokon-Empung). These include 31 events from previous studies (Johnson and Miller 2014; Kim et al. 2015; Fee et al. 2017; Yamada et al. 2017). A summary of these events is given in Table 1.

Table 1 Summary of infrasound dataset examined in the present study
Full size table

Aso volcano

Aso is an active volcano at the center of Kyusyu Island in Japan (Fig. 1b). Since December 2014, a number of Strombolian eruptions, continuous ash emission events, and phreatic eruptions have occurred (Yokoo and Miyabuchi 2015). Yamada et al. (2017) estimated Vinf of some phreatic eruptions in 2015 and early 2016. On 7 October 2016 (all dates and times are UTC), an explosive phreatic eruption occurred and its eruption cloud reached an altitude of 12 km (Sato et al. 2018). Infrasound signals accompanying the eruption were recorded by the observation networks of the National Institute of Earth Science and Disaster Resilience (NIED) and the Japan Metrological Agency (JMA) (Fig. 1d). The NIED operates four permanent monitoring stations at horizontal distances from the active crater ranging from 3.7 to 6.9 km. Each station has a microbarometer (AP-270, Setra) with a flat response at DC–100 Hz. The data is digitized with a sampling frequency of 1 Hz. Station ASO2 of the JMA has an infrasound microphone (ACO 3348, Aco) that has a flat response at 0.1–100 Hz with a sampling frequency of 100 Hz. Figure 2a shows microbarograms associated with the eruption. The amplitude of microbarograms is reduced at 1 km from the source assuming a geometric amplitude decrease (Johnson and Ripepe 2011). All waveforms in Fig. 2 are temporally co-located using the onset of the eruption signal at each station. Each microbarogram shares a prominent pulse at the onset and following a coherent time history. Since the sampling frequency of the microbarograms is 1 Hz, high-frequency components (> 0.5 Hz) are not recorded in the waveforms. Figure 2d shows the reduced (at 1 km from the source) pressure waveform recorded by the microphone of the JMA, a microphone that has a flat response for the high-frequency range. Although ASO2 is located at only 1.1 km away from the active vent, the maximum amplitude of the reduced pressure waveform at ASO2 is less than half of that of the reduced microbarograms. Therefore, we consider that the contribution of the high-frequency components is not significant for the eruption signal.

Fig. 2
figure 2

a Reduced microbarograms (at 1 km from the source) accompanying an eruption on 7 October 2016 (16:46) at ASHV, ASIV, ASTV, and ASNV, respectively. b Estimated time series of volumetric flow rate, q(t), at the monopole source. c Estimated time series of cumulative volume. d Reduced infrasound waveform (at 1 km from the source) at ASO2

Full size image

Assuming the observed pressure history ∆P(t) is excited by a volumetric flow rate of q(t) at a monopole source in a half-space, the relationship between ∆P(t) and q(t) can be expressed by

$$ \Delta P(t)=\frac{\rho_{\mathrm{atm}}}{2\pi r}\dot{q}\left(t-\frac{r}{c}\right), $$
(1)

where r is the distance from the source, ρatm is the density of atmosphere near the ground (1 kg/m3 is adopted for simplicity), and c is the sound velocity (Lighthill 1967). The relationship is valid under conditions of r ≫ λ/2π, and a ≪ λ/2π, where λ and a are the wavelength of the signal and a characteristic dimension of the source, respectively. In the case at Aso, the former condition seems to be valid: assuming a characteristic signal period of 10–20 s and sound velocity of 340 m/s yields λ/2π of 0.5–1.3 km. However, the latter condition for the source dimension can be contentious (Johnson and Lees 2010; Johnson and Miller 2014; Yamada et al. 2017). We discuss about the possibility of failure of the compact source approximation later in this paper. Assuming that the observed pressure change satisfies Eq. 1, single and double integrations of the observed waveform yield the time histories of q(t) and cumulative volume, respectively. To obtain a realistic time history by waveform integration, it is necessary to remove a linear trend from the observed waveform before integration and set an adequate integration time window. We use the method of Johnson and Miller (2014) to remove the trend from raw waveforms and to set the time window. Figure 2b, c shows the estimated time histories of q(t) and the cumulative volume as obtained by integrating the observed microbarograms. The determined length of the time window (10–98 s in Fig. 2b, c) is also comparable with the duration of the short period infrasound signal at ASO2. The value of the final cumulative volume at each station is distributed within a range of 0.7–1.2 × 109 m3 (Fig. 2c). At Sakurajima, Kim et al. (2015) demonstrated that the value of Vinf estimated by considering the effect of volcanic topography is about twice of the result which is obtained by considering a half-space medium. The contribution of signals from dipole and quadrupole sources (Woulff and McGetchin 1976) may cause the inhomogeneity of the infrasound amplitude at each station, since these sources induce an acoustic wavefield with directivity in the radiation pattern (Kim et al. 2012). The discrepancies between Vinf at each station may be caused by these factors. However, because we see no significant radiation pattern throughout the infrasound records, only the monopole source is assumed for the infrasound source. For simplicity, we define the average of all stations as a representative value of Vinf. We also attempted to estimate Vinf from the infrasound waveform recorded by the microphone of the JMA (Fig. 2d). However, this yields a bipolar-shaped time history of q(t). Since such a flux history is unrealistic for an eruption (Johnson and Miller 2014), the present study focuses on the pressure waveforms recorded by the microbarometers of the NIED only.

Shinmoedake volcano

Shinmoedake volcano is a part of the Kirishima volcanic complex in the south of Kyusyu Island (Nagaoka and Okuno 2011). It had a series of eruptions comprising three sub-Plinian eruptions, the formation of a lava dome, several Vulcanian eruptions, and continuous ash emission events in 2011 (Nakada et al. 2013). We here analyze infrasound signals accompanying nine Vulcanian eruption events that occurred during the period 28 January to 18 February 2011. In the case of continuous ash emission events, Yamada et al. (2017) estimated Vinf focusing on the very long period components of the infrasound signal with microbarograms recorded by the observation network of the NIED (Fig. 1e). Each station of the NIED at Shinmoedake is equipped with the same model of microbarometer with the same sampling frequency that is installed at stations on Aso. Station KIAM of the JMA also has the same microphone with a sampling frequency of 100 Hz as that at ASO2. As a representative event, reduced microbarograms accompanying a Vulcanian eruption on 31 January 2011 are shown in Fig. 3a. Both microbarograms at KRHV and KRMV show a pressure pulse with a large amplitude at the onset, which is a characteristic of an infrasound signal accompanying a Vulcanian eruption (Johnson 2003). Figure 3b, c represents the estimated time histories of the q(t) and the cumulative volume at the source obtained with the same method described above. As above, we adopt the microbarograms only to estimate Vinf at Shinmoedake.

Fig. 3
figure 3

a Reduced microbarograms (at 1 km from the source) accompanying a Vulcanian eruption on 31 January 2011 (22:54) at KRHV and KRMV. b Estimated time series of volumetric flow rate, q(t), at the monopole source. c Estimated time series of cumulative volume. d Reduced infrasound waveform (at 1 km from the source) at KIAM

Full size image

Lokon-Empung

Lokon-Empung volcano is an active volcano on the Minahasa Peninsula of Sulawesi island in Indonesia (Fig. 1c). During the period of September 2012 to September 2013, infrasound signals accompanying 56 Vulcanian eruption and continuous ash emission events were recorded (Yamada et al. 2016). The infrasound signals were recorded at KKVO (Fig. 1f) with a sampling frequency of 100 Hz by an infrasound microphone (SI102, Hakusan) having a flat response at 0.05–1500 Hz. We examine the signals accompanying 12 of Vulcanian eruption events which have a sufficiently high signal-to-noise ratio. Figure 4a shows the infrasound waveform associated with a Vulcanian eruption on 5 October 2012. Figure 4b, c represents the time histories of q(t) and the cumulative volume estimated by the method described above. It must be considered that the infrasound waveforms at Lokon-Empung were recorded by the microphone that has a narrower flat response range than the microbarometers at Aso and Shinmoedake. The instrumental response of the microphone (SI102) is not provided analytically (Yamada et al. 2017). Therefore, no corrections are completed for infrasound waveforms at Lokon-Empung.

Fig. 4
figure 4

a Raw infrasound waveform accompanying a Vulcanian eruption on 5 October 2012 (13:37) at KKVO. b Estimated time series of volumetric flow rate, q(t), at the monopole source. c Estimated time series of cumulative volume

Full size image

Sakurajima and Kuchinoerabujima

The eruption activity of Sakurajima volcano in Japan (Fig. 2b) has been characterized by numerous repeating Vulcanian eruptions since 1955 (Iguchi et al. 2013). Previous studies observed acoustic signals accompanying Vulcanian eruptions at Sakurajima (e.g., Iguchi and Ishihara 1990; Morrissey et al. 2008; Yokoo et al. 2013), and some of them have estimated Vinf by waveform analysis (Johnson and Miller 2014; Kim et al. 2015; Fee et al. 2017). The Vinf data of Kim et al. (2015) and Fee et al. (2017) was estimated by taking the effect of the volcanic topography into account.

Kuchinoerabujima is a volcanic island located at offshore of Kyusyu Island, Japan (Fig. 2b). Infrasound signals accompanying phreatic eruptions in 2014 and 2015 were recorded by two microbarometers of the NIED on the island, and Vinf for these eruptions were estimated using the microbarograms (Yamada et al. 2017). We here compile Vinf reported by these previous studies to examine the relation between Vb and Vinf.

Estimation of the buoyancy-derived volume V b

Following Terada and Ida (2007), we estimate F0 and consequently Vb for the explosive events reviewed in the previous section. Assuming that an eruption cloud is as an isolated thermal ascending vertically with self-similarity, the radius of the thermal r spreads linearly with the height of the thermal center from the ground zc (Woods and Kienle 1994), as

$$ r=k{z}_{\mathrm{c}}, $$
(2)

where k is an empirical constant (Scorer 1957). Using Eq. 2, the rate of volume change of the thermal is given by

$$ \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{4}{3}\pi {r}^3\right)=4\pi {r}^2k{w}_{\mathrm{c}}, $$
(3)

where wc = dzc/dt is vertical velocity of the thermal center (Terada and Ida 2007). Equation 3 represents the volume of entrained gas through in a unit area of the thermal surface. Therefore, constant k corresponds to the entrainment constant (Morton et al. 1956). For simplicity, the present study adopts k as 0.25, which is the value obtained from laboratory experiments for a discrete thermal (Scorer 1957). The total buoyancy F of a spherical thermal is expressed as (Morton et al. 1956; Turner 1962)

$$ F=\frac{4}{3}\uppi {r}^3g\left({\rho}_{\mathrm{air}}-{\rho}_{\mathrm{th}}\right), $$
(4)

where g is the gravitational acceleration, ρair is the density of ambient air, and ρth is the mean density of the thermal. Using Eqs. 2 and 4, and the mass conservation of the thermal with the entrainment hypothesis (Morton et al. 1956), one obtains the equation (Terada and Ida 2007):

$$ \frac{\mathrm{d}F}{\mathrm{d}{z}_{\mathrm{c}}}=\frac{4\pi }{3}g{k}^3{z}_{\mathrm{c}}^3\frac{\mathrm{d}{\rho}_{\mathrm{air}}}{\mathrm{d}{z}_{\mathrm{c}}}. $$
(5)

Integrating Eq. 5 with respect to z and adopting a vertical profile for the ambient air density, we obtain the vertical profile of F for an arbitrary initial value of F0 at zc = 0. As an example of the estimation of F0, Fig. 5 shows the case of the eruption at Aso on 7 October 2016 (Fig. 2). Vertical profiles of the ambient air density observed at Fukuoka and Kagoshima meteorological stations of the JMA (Fig. 1b) are shown in Fig. 5a. The present study adopts the weather (pressure and temperature) data distributed by the University of Wyoming (2017). The equation of state for ambient air (p = ρairRT) is used to derive the ambient air density, where p is the pressure, R is the gas constant (287.05 J/kg/K), and T is the temperature of the corresponding altitude. The solid line in Fig. 5a is a quadratic least square fit for data at both Fukuoka and Kagoshima stations. With the regression of the ambient air density, vertical profiles of F for values of F0 of 108–1011 are calculated in Fig. 5b. When the thermal reaches the neutral buoyancy height, the value of F drops sharply to 0. Figure 5c represents the relationship between the neutral buoyancy height of the thermal and the corresponding value of F0. The height of the top of the thermal z is given by z = zc + r. In the case of the eruption at Aso, the eruption cloud reached a height of 10.7 km above the crater rim. Thus, we obtain F0 as 5.0 × 1010 for this eruption (Fig. 5c).

Fig. 5
figure 5

a Vertical profiles of ambient air density at Fukuoka and Kagoshima stations on 7 October 2016 (12:00). A quadratic regression fit is given by the solid line. b Vertical profiles of the calculated buoyancy of the thermal F with the initial value, F0, of 108–1011 at zc = 0. c Neutral buoyancy height reached for a given buoyancy F0

Full size image

We use the weather data from both Fukuoka and Kagoshima stations for the events at Aso, Shinmoedake, Sakurajima, and Kuchinoerabujima volcanoes, and the data at Manado station for the events at Lokon-Empung (Fig. 1c) to estimate F0. The data for maximum cloud height used in the analysis are summarized in Table 2. To obtain Vb (\( \frac{4}{3}\pi {r}^3 \) of Eq. 4) from the value of F0, it is necessary to assume a difference in density between the thermal and the ambient air (ρair − ρth). Following Wilson and Self (1980), Woods and Kienle (1994), and Yamamoto et al. (2008), we set a value of 0.3 kg/m3 as a characteristic density difference with an error range of ± 0.2 kg/m3.

Table 2 Date, Vinf, Vb, F0, the maximum eruption cloud height from the ground (zmax), elevation of the crater rim (z0), and date of weather data for each event
Full size table

Results

Figure 6 shows the estimated relationship between Vb and Vinf for 53 events analyzed here. A correlation coefficient of 0.81 is obtained between Vb and Vinf for all events. The ratio of Vb/Vinf ranges from 0.012 to 180, with an average of 27. The ratio of Vb/Vinf is almost always greater than 1 (51 events) and most commonly within the range of 3–30 (33 events). In Fig. 6, we define the linear regression function between Vb and Vinf of 16. Because the thermal volume increases linearly with altitude, we hypothesize that the relation Vb/Vinf also follows the linear function. Although some events deviate from this regression, the relationship between Vb and Vinf of most analyzed events roughly follows the ratio expressed by the linear regression. Table 2 summarizes the values of Vinf, Vb, and F0 for each event.

Fig. 6
figure 6

Relationship between the infrasound-derived volume, Vinf, and the buoyancy-derived volume, Vb. The events represented by diamonds express the relation between Vinf and Vb. These events have a horizontal error bar defined by uncertainty in Vinf at each station. Inverted triangles are events for which only Vb is estimated here, but for which the value of Vinf is taken from other studies (Johnson and Miller 2014; Kim et al. 2015; Fee et al. 2017; Yamada et al. 2017). Vertical error bars for all events correspond to uncertainty in Vb based on the value of (ρair − ρth) which has an error range of ± 0.2 kg/m3. Some events at Sakurajima (as outlined by the red box) have Vinf estimated considering the effect of topography (Kim et al. 2015; Fee et al. 2017). The solid regression line shows that Vb/Vinf = 16. Dashed lines represent the Vb/Vinf ratio of 1, 10, and 100

Full size image

Discussion

Figure 6 shows that the estimated Vinf is generally 3 to 30 times smaller than Vb. This is consistent with the comparison between Vinf and Vvideo by Johnson and Miller (2014) and Yamada et al. (2017). As we have shown in Figs. 2, 3, and 4, the analyzed infrasound waveforms exhibit a prominent compressional phase at the onset. From a perspective of the time history of observed overpressure, the estimated Vinf is likely to relate to the onset of plume emission which is driven by gas thrust, i.e., jets (e.g., Crapper 1977; Sparks et al. 1997; Patrick 2007). This interpretation is supported by the velocity profile with height commonly recorded for ascending plumes: rapid deacceleration in the gas thrust phase, followed by steady velocity in the buoyancy phase (e.g., Wilson and Self 1980; Patrick et al. 2007; Marchetti et al. 2009; Delle Donne and Ripepe 2012). Our interpretation is also supported by following field observation results: infrasound energy reflects the driving force of the gas thrust phase (Marchetti et al. 2009), and the duration of the gas thrust phase is highly correlated with that of infrasound signal (Delle Donne et al. 2016).

Because a jet barely entrains the surrounding air, where Patrick (2007) found k = 0.06 in the jet region of plumes at Stromboli, and Bombrun et al. (2018) estimated k = 0.178 at Santiaguito, the bulk density of the jet is expected to be almost the same as that of the particle-gas mixture. From the literature, the possible range of the bulk density of the jet is 2–20 kg/m3 (e.g., Sparks and Wilson 1976; Sparks et al. 1997; Formenti et al. 2003). Fee et al. (2017) demonstrated that total eruption mass inferred from Vinf and the bulk density of the flow near the vent (2–5 kg/m3) shows good agreement with that estimated from ground-based ash sampling. Therefore, we consider that Vinf represents the initial volume of the eruption plume driven by momentum near the vent (Fig. 7b). Since the present study focuses on short-lived eruptions, emission duration of the mixture of particles and gas from the vent is transient for each event. Hence, Vinf may be almost equivalent to the total emitted volume before entraining surrounding air. As it ascends from the vent, the plume gradually loses its momentum by gas thrust (Patrick et al. 2007). At the same time, the plume entrains the surrounding air and acquires buoyancy (e.g., Wilson and Self 1980; Patrick 2007; Marchetti et al. 2009). The estimated Vb can be regarded as the volume at the moment when the thermal has entrained a sufficient amount of the surrounding air to ascend by buoyancy only (Fig. 7c). Previous observations and theoretical considerations have reported the bulk density of the ascending thermal as 0.6–1.1 kg/m3 (e.g., Wilson and Self 1980; Woods and Kienle 1994; Yamamoto et al. 2008). The difference between the bulk density of both eruption plume regimes yields 1.8–32 times of volume expansion as the plume dynamic transitions from a jet to a discrete thermal. This rate shows good agreement with the range of Vb/Vinf ratio of most events (3.0–30) and the regression of Vb/Vinf in Fig. 6 which is 16. Therefore, the ratio of Vb/Vinf can be explained by the rate of volume change of the eruption plume that occurs between the gas thrust and buoyant regimes.

Fig. 7
figure 7

Schematic illustration of the plume volume inferred in the present study. a Before an eruption, where the atmosphere, denoted by the blue dome, corresponds to the volume that will be displaced by the emerging jet. b At the onset of the eruption, the plume is driven by gas thrust where the jet displaces the atmosphere and induces an infrasound pulse observed at the surrounding stations. Vinf corresponds to the volume of the jet. c The eruption cloud entrains surrounding air and obtains buoyancy. Vb corresponds to the volume of the plume at that moment

Full size image

In Fig. 6, we set error ranges for Vinf considering the discrepancy between the final values of the cumulative volume for each station, and for Vb considering the uncertainty of the difference in density between the thermal and the surrounding air. However, the errors derived from other factors should also be considered. The maximum eruption cloud height is an important parameter to consider in our estimate of Vb. The present study adopts data for the maximum eruption cloud height both inferred from weather radar and ground-based visual observation (Table 2). The maximum cloud height of some events at Shinmoedake was estimated by both methods: weather radar (Shimbori et al. 2013) and visual observation (Kato and Yamasato 2013). For example, for an event on 2 January 2011 (06:53), the weather radar detected the maximum eruption cloud height as 4.6 km above the crater (Shimbori et al. 2013). On the other hand, the visual observation from the ground estimated the maximum cloud height as 3.0 km above the crater (Kato and Yamasato 2013). If we adopt the maximum height data by the ground-based visual observation, this yields a value of Vb that is 4.8 times smaller than that estimated by the weather radar data. Such an error derived from the differences in estimation of cloud height is included in our results. The maximum cloud height is usually affected by several factors, such as the wind (Bursik 2001) and the total grain-size distribution (Girault et al. 2014). For example, the event at Kuchinoerabujima volcano in 2014 (Vinf = 5.5 × 107 m3, Vb = 6.8 × 105 m3) deviates considerably from the trend of most of the events represented by the regression line in Fig. 6. Iguchi and Nakamichi (2015) reported that strong wind from a nearby typhoon affected the maximum plume height of this eruption. The value of F0 and Vb may also be overestimated if the eruption plume overshoots the neutral buoyancy height (Holasek and Self 1995). Another critical parameter for estimation of Vb is the entrainment constant, k. It is known that this value can be affected by the interaction between an eruption plume and wind in the atmosphere (Suzuki and Koyaguchi 2015). Although there are theoretical models that consider the effect of wind on entrainment (Bursik 2001), quantification of the wind effect is still challenging even for recent numerical simulation models (Costa et al. 2016). Jessop and Jellinek (2014) reported that vent geometry modifies the shape of eddies on the edge of the plume and, thus, can affect entrainment of the ambient air. Although most previous studies have demonstrated that k can be considered as a constant approximately, Carazzo et al. (2008) pointed out that a better fit for experimental data is obtained setting k as a function of buoyancy. Terada and Ida (2007) determined k from video image analyses and reported that k varies in the range of 0.22–0.59. Bombrun et al. (2018) inferred k of a rooted thermal at Santiaguito as 0.356 from an analysis of thermal video images. For example, if we adopt k as 0.36, the average value from Terada and Ida (2007), Vb of the event at Aso (Fig. 2) becomes 1.1 × 1011 m3, i.e., 2.2 times greater than that obtained by setting k at 0.25. The linear regression function for the Vb/Vinf ratio of all events with k as 0.36 becomes 34, where it is also similar to the rate of volume change expected from the transition of the eruption plume regime.

The assumption of the compact source adopted in the estimation of Vinf may not be adequate for some events. The waveform analysis of the event at Aso examined in Fig. 2 yields Vinf as 9.5 × 108 m3. Assuming that the eruption cloud is a sphere, the radius of the sphere is 0.61 km. For infrasound signal with periods of 10–20 s and a sound velocity of 340 m/s, the source dimension is no longer compact relative to the value of λ/2π of 0.5–1.3 km. This issue is a challenging topic for the study of volcanic infrasound and may be solved with detailed analysis techniques, such as the superposition of multiple monopole sources discussed by Johnson and Lees (2010). The present study compiles Vinf of some events that are estimated by considering the effect of volcanic topography, which affects the estimation of Vinf on the order of two times reported by Kim et al. (2015). However, we see no significant difference between topography-considered and half-space-considered events in our result in Fig. 6. Since we focus on the macroscopic scale over several orders of magnitude, it seems that the effect of volcanic topography is not influential in our results. However, accounting for topography in the infrasound analysis will be necessary for accurate estimation of Vinf and for examining plume dynamics at a more detailed scale.

Although our results contain unresolved errors, we consider that they are representative of the macroscopic dynamics of an eruption plume emitted during a short-lived eruption: the rate of volume change of the plume as it transitions from gas thrust-driven to buoyancy-driven regimes appears valid. The estimated relationship between Vinf and Vb can also be of value in constraining the volume of plume with infrasound data. The maximum plume height can also be constrained with our method if the vertical profile of the ambient air density is available.

Conclusions

We have here examined the relationship between infrasound-waveform-derived eruption plume volume, Vinf, and the buoyancy-derived plume volume, Vb, obtained by considering plume as a discrete thermal. The infrasound waveforms accompanying short-lived (Vulcanian and phreatic) eruptions at Aso, Shinmoedake, and Lokon-Empung volcanoes were analyzed to estimate Vinf. We also referred to values of Vinf reported by previous studies at several other volcanoes. The estimation of Vb followed the method of Terada and Ida (2007) which uses the maximum eruption cloud height and a vertical profile of the ambient air density. Estimated relationships of Vb/Vinf for most events are in the range of 3–30, and we obtain a ratio of Vb to Vinf of 16. This ratio can be explained by the rate of volume change of the eruption plume as it transitions from a jet regime to that of a discrete thermal. The relationship can be a valuable index to estimate the eruption cloud volume with infrasound data. Since infrasound data can be obtained in real time at appropriately configured monitoring systems, the Vb/Vinf relation can be useful in assessing the magnitude of the eruption immediately after the onset of the event. Real-time provision of Vb and F0 inferred from Vinf will also be useful in forecasting ash fall (Bonadonna et al. 2005).