### Modified probabilistic estimation method

In order to investigate the height of the eruption plume from its radar echo, the author developed a modified probabilistic estimation (MPE) method, an improved version of the PE method (Sato et al. 2018).

The path of radio waves in the spherically stratified atmosphere is expressed as (e.g., Hartree et al. 1946; Battan 1973):

$$\frac{{d}^{2}h}{{ds}^{2}}-\left(\frac{2}{R+h}+\frac{1}{n}\frac{dn}{dh}\right){\left(\frac{dh}{ds}\right)}^{2}-{\left(\frac{R+h}{R}\right)}^{2}\left(\frac{1}{R+h}+\frac{1}{n}\frac{dn}{dh}\right)=0,$$

(1)

where \(h\) is the altitude above the surface of the Earth, \(s\) is the horizontal distance, \(R\) is the radius of the Earth, and \(n\) is the refractive index of the atmosphere.

Assuming \({\left(dh/ds\right)}^{2}\ll 1\), \(n\approx 1\) and \(h\ll R\), Eq. (1) is simplified to:

$$\frac{{d}^{2}h}{{ds}^{2}}=\frac{1}{R}+\frac{dn}{dh}.$$

(2)

If the elevation angle \({\theta }_{e}\) of the beam path is small, it can be assumed that \(dh/ds\approx {\theta }_{e}\), and Eq. (2) becomes:

$$\frac{d{\theta }_{\mathrm{e}}}{ds}=\frac{1}{R}+\frac{dn}{dh}=\frac{1}{{k}_{\mathrm{e}}R},$$

(3)

where \({k}_{\mathrm{e}}\) is the equivalent radius ratio (e.g., Doviak and Zrnic 1993; Sato et al. 2018).

Whereas the PE method assumes the 4/3 *R* equivalent earth radius model, the MPE method incorporates the ellipticity of the Earth by using Euler’s (1767) radius of curvature:

$${R}_{\varphi }^{\alpha }=\frac{{M}_{\varphi }{N}_{\varphi }}{{N}_{\varphi }{\mathrm{cos}}^{2}\alpha +{M}_{\varphi }{\mathrm{sin}}^{2}\alpha }.$$

(4)

Here, \({M}_{\varphi }\) is the radius of curvature of the meridian at latitude \(\varphi\), \({N}_{\varphi }\) is the radius of curvature of the prime vertical, and \(\alpha\) is the azimuth of the target relative to the radar. Using \({R}_{\varphi }^{\alpha }\) obtained for each radar, Eq. (3) becomes:

$$\frac{1}{{k}_{\mathrm{e}}R}=\frac{1}{{R}_{\varphi }^{\alpha }}+\frac{dn}{dh}.$$

(5)

We obtained \(dn/dh\) from the mean field of the lower atmosphere (< 5 km altitude) between the radar and the target by using the initial value of the JMA Mesoscale Model (MSM) at 0:00 UTC on January 23, 2018 as the atmospheric field. Since the MSM does not explicitly predict the refractive index of the atmosphere, it was calculated as:

$$N=\left(n-1\right)\times {10}^{6}=7.76\times {10}^{-5}\frac{p}{T}+3.75\times {10}^{5}\frac{e}{{T}^{2}},$$

(6)

where \(N\) is refractivity, \(p\) is atmospheric pressure, \(T\) is air temperature, and \(e\) is vapor pressure (e.g., Bean and Dutton 1968). Saturated water vapor pressure \({e}_{s}\) was approximated by using the expression used by the World Meteorological Organization (e.g., Mizuno 2000):

$${e}_{s}=\mathrm{exp}\left[19.482-4303.4/\left({T}_{\mathrm{c}}+243.5\right)\right],$$

(7)

where \({T}_{\mathrm{c}}\) is the atmospheric temperature in degrees Celsius. In the MPE method, these formulas are used to obtain the equivalent earth radius ratio for each radar.

Next, we derived the probabilistic estimation equations used in the MPE method. In the PE method, the radar beam height that captures the plume is calculated by:

$$h=\left({k}_{e}R+{h}_{0}\right)\left[\frac{\mathrm{cos}{\theta }_{e}}{\mathrm{cos}\left\{{\theta }_{e}+s/\left({k}_{e}R\right)\right\}}-1\right]+{h}_{0},$$

(8)

where \({h}_{0}\) is the antenna height. The JMA weather radars perform 28 scans per 10 min over a range of elevation angles from about 0 to 25 degrees. Furthermore, for the low elevation angles used in this study (about 0–5°), most of the angles are scanned twice, and as a result, the time resolution is about 5 min. Whether the target echo was caused by volcanic plumes or not is judged from the continuity between elevation angles.

One of the improvements from the PE method is a geoid correction. Whereas the PE method does not consider the geoid height for height estimations, the MPE method includes a geoid correction. Specifically, because the antenna height is usually given relative to the geoid, the geoid height at the location of the radar \({h}_{G,\mathrm{radar}}\) is added to convert the antenna height to an altitude relative to the Earth ellipsoid. Then, after deriving the altitude of the target, the geoid height at the location of the target \({h}_{G,\mathrm{target}}\) is subtracted to convert the target height to an altitude relative to the geoid. For the geoid height, GSIGEO2011 by the Geospatial Information Authority of Japan (2016) was used. By this correction, Eq. (8) becomes:

$$h=\left({k}_{\mathrm{e}}R+{h}_{0}+{h}_{G,\mathrm{radar}}\right)\left[\frac{\mathrm{cos}{\theta }_{e}}{\mathrm{cos}\left\{{\theta }_{e}+s/\left({k}_{e}R\right)\right\}}-1\right]+{h}_{0}+{h}_{G,\mathrm{radar}}-{h}_{G,\mathrm{target}}.$$

(9)

The obtained heights are treated as altitudes above sea level (m ASL).

The probability density function of the heights obtained by each radar is calculated as:

$${f}_{i}\left({h}_{j}\right)=\frac{1}{\sqrt{2\pi }{\sigma }_{i}}\mathrm{exp}\left\{-\frac{{\left({h}_{j}-{\mu }_{i}\right)}^{2}}{2{\sigma }_{i}^{2}}\right\}.$$

(10)

Here, \({h}_{j}\left(=j\Delta h\right)\) is the discretized height above sea level (*j* = 0,1,2…, ∞), and \({\mu }_{i}\) and \({\sigma }_{i}\) are the median height and standard deviation of the *i*th radar, respectively. In addition, we assume that \({\mu }_{i}={H}_{\mathrm{c}}\), \({\sigma }_{i}=\beta \left({H}_{\mathrm{u}}-{H}_{l}\right)/2\) and \(\beta =1\). \({H}_{\mathrm{c}}\) is the altitude at the center of the beam, \({H}_{u}\) is the altitude at the top of the beam, and \({H}_{\mathrm{l}}\) is the altitude at the bottom of the beam. Finally, the composite probability density combining all radars is obtained as:

$${f}_{\mathrm{composite}}\left({h}_{j}\right)=\frac{{\prod }_{i=1}^{n}{f}_{i}\left({h}_{j}\right)}{{\sum }_{j=0}^{\infty }\left\{{\prod }_{i=1}^{n}{f}_{i}\left({h}_{j}\right)\right\}\Delta h}.$$

(11)

### Erupted mass estimation methods

One of the author’s goals is to quickly estimate the scale of the eruption, that is, the erupted mass, using the eruption plume height estimated by weather radars. We used four empirical methods in Costa et al. (2016) to estimate the total mass of ejecta by the 2018 Kusatsu-Shirane eruption. For simplicity, each model is reproduced as closely as possible from the literature, and the variables in this section are independent of those used in the preceding section. We briefly describe each method and the references therein which provide more detailed information.

The method of Carazzo et al. (2014; hereafter ‘C14’) estimates the mass eruption rate \(\dot{M } \left[\mathrm{kg}/\mathrm{s}\right]\) of weak mid-latitude plume es:

$${\mathrm{ln}}\left(\dot{M}\right)=\mathrm{ln}\left({b}_{1}{H}^{{n}_{1}}\right)+cWH,$$

(12)

where \(H\) is the maximum observed plume height [km], \(W=83.66 \mathrm{m}/\mathrm{s}\) is the wind velocity at the tropopause height, and \({n}_{1}=4.06\), \({b}_{1}=63.22\) and \(c=0.0025\) are fitting parameters.

The method of Degruyter and Bonadonna (2012; hereafter ‘DB12’) calculates \(\dot{M}\) as:

$$\dot{M}=\pi \frac{{\rho }_{a0}}{{g}^{\mathrm{^{\prime}}}}\left(\frac{{2}^{5/2}{\alpha }^{2}{\overline{N} }^{3}}{{z}_{l}^{4}}{H}^{4}+\frac{{\beta }^{2}{\overline{N} }^{2}\overline{v}}{6}{H }^{3}\right),$$

(13)

where \({\rho }_{a0}=1.105\mathrm{kg}/{\mathrm{m}}^{3}\) is the density of the atmosphere, \({g}^{^{\prime}}=41.289\mathrm{ m}/{\mathrm{s}}^{2}\) is the reduced gravity at the plume source, \(\alpha =0.1\) is the radial entrainment coefficient, \(\overline{N }=0.0134 {\mathrm{s}}^{-1}\) is the average buoyancy, \({z}_{l}=2.8\) is the maximum non-dimensional height of Morton et al. (1956), \(\beta =0.5\) is the wind entrainment coefficient, and \(\overline{v }=31.3935\mathrm{m}/\mathrm{s}\) is the average wind velocity over the plume height.

Mastin et al. (2009; hereinafter ‘M09’) estimated \(\dot{M}\) as:

$$\dot{M}={\rho }_{m}{\left(H/2\right)}^{4.15},$$

(14)

where \({\rho }_{m}=2500 \mathrm{kg}/{\mathrm{m}}^{3}\) is the density of magma. M09 estimated the volumetric flow rate (dense-rock equivalent) \(\dot{V} \left[{\mathrm{m}}^{3}/s\right]\) at first, and converted it to \(\dot{M}\) with this magma density.

Finally, the method of Woodhouse et al. (2013, 2016; hereafter ‘W16’) uses:

$$\dot{M}=0.35{\alpha }^{2}{f\left({W}_{\mathrm{s}}\right)}^{4}\frac{{\rho }_{a0}}{{g}^{\mathrm{^{\prime}}}}{N}^{3}{H}^{4},$$

(15)

$$f\left({W}_{\mathrm{s}}\right)=\frac{1+1.4266{W}_{\mathrm{s}}+0.3527{W}_{\mathrm{s}}^{2}}{1+1.373{W}_{\mathrm{s}}},$$

(16)

$${W}_{\mathrm{s}}=1.44\dot{\gamma }/N,$$

(17)

and,

$${g}^{\mathrm{^{\prime}}}=g\left[\frac{\left({C}_{\mathrm{v}}{n}_{0}+{C}_{\mathrm{s}}\left(1-{n}_{0}\right)\right){\theta }_{0}-{C}_{\mathrm{a}}{\theta }_{\mathrm{a}0}}{{C}_{\mathrm{a}}{\theta }_{\mathrm{a}0}}\right].$$

(18)

Here, \({W}_{\mathrm{s}}\) is a dimensionless wind strength, \(\alpha =0.09\), \({\rho }_{\mathrm{a}0}=1.104\mathrm{kg}/{\mathrm{m}}^{3}\), \({\theta }_{0}=1273 \mathrm{K}\) and \({\theta }_{\mathrm{a}0}=268.8\mathrm{ K}\) are the source temperature and the atmospheric temperature, respectively, \(N=0.014 {\mathrm{s}}^{-1}\), \(\dot{\gamma }=0.007 {\mathrm{s}}^{-1}\) is the shear rate of the atmospheric wind, \({n}_{0}=0.03\) is the mass fraction of gas at the vent, \({C}_{\mathrm{v}}=1810\mathrm{ J}/\left(\mathrm{kg\,K}\right)\), \({C}_{\mathrm{s}}=1100\mathrm{ J}/\left(\mathrm{kg\,K}\right)\), and \({C}_{\mathrm{a}}=1000\mathrm{ J}/\left(\mathrm{kg\,K}\right)\) are the specific heat capacity at constant pressure of water vapor, solid pyroclasts, and dry air, respectively.