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Estimation of emission mass from an eruption plume for the Aso volcano eruption, on October 8, 2016, using a fourdimensional variational method
Earth, Planets and Space volume 70, Article number: 202 (2018)
Abstract
When an explosive eruption, such as a Plinian eruption, occurs, in order to estimate ash fall around the volcano and for hazard mitigation, a numerical model is often used. Simulation by a numerical model needs emission mass from the eruption column including vertical profile and size distribution of ash particles. Hence, the accuracy of the emission mass from the eruption column is vital to estimate and forecast ash fall accurately. We developed a data assimilation system based on the fourdimensional variational method (4DVar) as an estimation method for emission mass from volcanic eruption columns as a function of altitude and ash particle size. This system includes a forward model which calculates volcanic ash forecast, and an observation operator, which are used for the calculation of misfit between observation and forecast. It also includes an adjoint model of the forward model which calculates the correction of emission mass from the misfit, and an algorithm to minimize the cost function as a measurement of optimization. In this system, observation and prior knowledge about emission mass from the volcanic eruption column, such as the Suzuki function, can be simultaneously treated with weight considering observation error and background error. Furthermore, this system has scalability for additional observations. That is to say, a variety of observations can be treated simultaneously, only if their observation operators which are an transformation from model parameters to observation value are developed. In this study, we applied this system to the October 8, 2016 Aso volcano eruption in Japan. After this eruption, ash fall observation (including lapilli) around Aso volcano was preformed, and operational weather radar captured the eruption cloud echo. Using both of these observations and the 4DVar system, we estimated emission mass from the eruption plume column as a function of altitude and particle size, and it led to ash fall simulation which was consistent with observations. In addition, the eruption mass which is the sum of emission mass from eruption column was estimated to be 1.32 × 10^{8} kg.
Introduction
When an eruption occurs, we need to know about eruption source parameters such as eruption mass, height of the eruption column and duration of the eruption to forecast next possible hazards for mitigation. Although ash fall from explosive eruptions does not always take human life directly, even a minor ash fall impacts human health, farming lifelines and aviation safety (Mannen 2014; Klawonn et al. 2012). In general, ash fall forecast is calculated by numerical models which need emission mass as an initial distribution (i.e. mass of ash and lapilli particle segregation from the eruption column) of volcanic ash including ash particle size and ash amount depending on altitude (e.g. Bonadonna et al. 2005), and the accuracy of ash fall simulation is particularly constrained by the difficulty of quantifying the ESPs (eruption source parameters) as initial distribution (e.g. Folch 2012). Some ash fall simulations use empirical and theoretical assumptions such as the Suzuki function for the initial distribution of volcanic ash. In the Suzuki function, the total size distribution of ash particles is assumed as lognormal distribution, and the emission mass for each particle size and each altitude is calculated by a simplified model of vertical velocity of an eruption column and terminal velocity (e.g. Shimbori et al. 2010; Folch 2012). However, the reliability of such distribution has yet to be verified in detail. For verification of empirical distribution and understanding of the nature of an eruption column, reconstruction of the ash particle segregation pattern along the eruption column from observation using an inversion method is key (Mannen 2014). There are a variety of inversion methods in diverse fields such as weather forecasts (e.g. Kalnay 2002). Of these, the fourdimensional variational method (4DVar) is one of the most sophisticated data assimilation methods and widely used in weather forecasts of meteorological organizations of various countries (e.g. WMO 2001). 4DVar is an optimization method based on Bayesian statistics. In 4DVar, prior information and observation can be simultaneously considered. In addition, in the 4DVar, because a variety of observations such as ash fall observation and radar observation are simultaneously possible, there is a possibility that diverse of information such as size distribution and the time series of emission can be extracted by combining them with model dynamics.
In this study, we developed a data assimilation system based on 4DVar for the estimation of emission mass from an eruption plume as a function of altitude and ash particle size, and we applied it to the case of the 2016 Aso volcano eruption in Japan. We obtained an initial distribution of ash particles which led to a forecast that was consistent with ash fall observation.
Data assimilation methodology
Overview of the 4DVar system for volcanic ash
We analyse emission mass from an eruption plume by 4DVar. This method is an optimization method based on Bayesian statistics (e.g. Kalnay 2002). The optimization is to find certain variables (called “analysis variables”) minimizing a cost function J, including a term of difference between forecast and the background, the difference between forecast and observation, and a penalty term. In this study, the cost function \( J \) is defined as follows
where \( \tilde{{\mathbf{x}}}\left( {t_{i} } \right) \) is the emission mass from the eruption column for a discretized diameter \( D_{n} \) and a discretized altitude \( z_{k} \) at a discretized time \( t_{i} \)
where \( {\mathbf{c}} \) is the emission mass of volcanic ash (g/m^{3}) and N is the index of eruption duration (i.e. \( t_{1} \) is the start time of the eruption, and \( t_{N} \) is the end time of the eruption). \( {\mathbf{x}}(t_{j} ) \) is the forecast value of ash concentration in the atmosphere and the ash fall amount (including lapilli) which are both functions of location and ash diameter. The forecast variable \( {\mathbf{x}}(t_{j} ) \) is calculated by a forward model \( \varvec{M}_{t} \) (atmospheric transport model) from emission mass \( \tilde{{\mathbf{x}}} \), i.e. \( {\mathbf{x}}(t_{j} ) = \varvec{M}_{{t_{j} }} (\tilde{{\mathbf{x}}}) \). \( \tilde{{\mathbf{x}}}_{\varvec{g}} \) is the first guess which is based on prior information. For example, emission mass \( \tilde{{\mathbf{x}}}_{\varvec{g}} \) for each altitude and each particle diameter is reproduced by plume height in Suzuki (1983). \( {\mathbf{B}} \) is the background error covariance which has characteristics of a model forecast error. \( \varvec{y}_{j} \) is defined as observation at time \( t_{j} (j = 1, \ldots ,K. \) \( t_{K} \) is the end time of the data assimilation window). \( \varvec{H} \) is the observation operator which is a transformation from the model variable \( {\mathbf{x}}(t_{j} ) \) to observation value \( \varvec{y}_{j} \). For ash fall amount (g/m^{2}), the observation operator is the sum of each size of ash fall amount with a diameter of 2 mm or less, and it includes interpolation to the location of ash fall observation from the model grid.
\( J_{\text{p}} \) is a penalty term which has the effect to add constraints to the optimization of the variational data assimilation system (Parrish and Derber 1992). In this case, the emission mass for any size particle and any altitude must be positive, i.e. we take the penalty term \( J_{\text{p}} \) as follows
where \( \mu \) is the penalty term factor. This penalty term has the effect of increasing cost function \( J \) for negative emission mass. A sufficiently large \( \mu \) restricts the optimization of the 4DVar system with positive emission mass.
Description of forward model
The 4DVar system includes a forward model \( \varvec{M}_{t} \) which calculates ash concentration and ash fall at the surface from emission mass \( \tilde{{\mathbf{x}}}(t_{i} ) \) from the eruption plume.
The model \( \varvec{M}_{t} \) is a Euler model using finite difference methods to calculate time evolution of volcanic ash concentration using dynamical processes such as advection and gravity settling. For time discretization, the Euler scheme (e.g. Kalnay 2002) is used. For space discretization, we adopt a finitevolume method (FVM) which calculates the time evolution of volcanic ash concentration from flux of the grid boundaries. The forecast variables are volcanic ash concentration and ash fall. The volcanic ash concentration is defined as a function of each grid for each ash particle diameter, i.e. the number of forecast variables in one grid is the number of particle diameters. Ash fall forecast at the surface is also a forecast variable. For the advection process, we adopted thirdorder upwind discretization (Wicker and Skamarock 2002) and a flux limiter function (Koren 1993). The meteorological field, including wind, temperature and pressure, used in the advection process and gravitational process with interpolation linearly by space and time is the Meso Analysis. The Meso Analysis is a 3 + 1dimensional (space and time) meteorological field, such as wind and temperature calculated by the Japan Meteorological Agency (JMA 2013) with approximately 5km resolution horizontally for daily weather forecasts and meteorological hazard mitigation. The spatial coordinates are uniform for latitude, longitude and altitude. Gravitational settling is vertically downward advection calculated from terminal velocity which depends on the diameter of ash particles based on Shimbori et al. (2010). In the calculation of terminal velocity, particle density is based on Wilson and Huang (1979). In addition, diameter D of ash particles is defined as the mean of diameter of the major axis \( a_{1} \) and short diameter \( a_{2} = a_{3} \) with a shape factor \( \frac{{a_{2} }}{{a_{1} }} = \frac{1}{3} \) (Shimbori 2015). In this study, ash particle diameter \( D \) is defined by \( D = \frac{1}{3}(a_{1} + a_{2} + a_{3} ) \).
Optimization algorithm
The 4DVar system calculates an optimized value (called the “analysis value”) from a first guess by minimizing the cost function. For minimizing the cost function, we use an iteration method using the gradient of the cost function \( {\varvec{\nabla }}_{\text{i}} J = \partial J/\partial \tilde{{\mathbf{x}}}(t_{i} ) \) as follows
where \( {\mathbf{M}}^{{\mathbf{T}}} \) is the adjoint of the forward model \( \varvec{M}_{t} \), as follows
Similarly, \( {\mathbf{H}}^{\text{T}} \) is adjoint of the observation operator \( \varvec{H} \), as follows
The adjoint model \( {\mathbf{M}}^{{\mathbf{T}}} \) calculates the sensitivity of cost function \( J \) for emission mass \( \tilde{{\mathbf{x}}}(t_{1} ), \ldots ,\tilde{{\mathbf{x}}}(t_{N} ) \) from misfits between observation and forecast \( {\mathbf{H}}^{{\mathbf{T}}} \varvec{R}^{  1} [\varvec{H}({\mathbf{x}}(t_{j} ))  \varvec{y}_{j} ] \) in a model space at time \( t_{j} \), including observation error as weight. Once we obtain a gradient of cost function \( {\varvec{\nabla }}_{\text{i}} J \), the iteration method can be performed to minimize the cost function. In this study, the limitedmemory quasiNewton (LBFGS) algorithm is used as the iteration minimizing algorithm (Liu and Nocedal 1989). In Fig. 1, the flow of the overall system is shown.
Application to the 2016 Aso volcano eruption
We adopted the 4DVar system for the 01:46 (JST) October 8, 2016 eruption of Aso volcano in Kyushu, Japan, which is an active volcano that JMA monitors with various instruments such as seismometers, tiltmeters and remote cameras. Most recently, several ash emissions (2003–2005) and volcanic gases and ash emissions (2014–2016) occurred (Miyabuchi et al. 2008; Marumoto et al. 2017). According to Ishii et al. (2018), in the October 2016 eruption, SO_{2} emitted from the eruption reached an altitude of 13–14 km, and the ash and lapilli fall spread several kilometres, mainly to the northeast by the ambient wind. Furthermore, lapilli which fell approximately 4.5 km from the vent broke window panes, and lapilli which fell approximately 6.5 km away broke more than 1500 solar panels.
Observation used in the 4DVar system
In this study, the input into the 4DVar system is ash fall (including lapilli) and radar observations.
In the October 8, 2016 eruption, ash and lapilli fall spread to the northeast area of Aso volcano. Ash fall observations were performed from a few hours after the eruption by JMAMOT (JMA Mobile Observation Team). However, in this experiment, it is assumed that observed ash fall was sedimented within 2 h after the eruption (i.e. 01:47–03:47 JST). Although we have no time series of ash fall observation, accounting for wind from aerological observation, etc., on the day, this assumption is sufficiently realistic. In addition to JMAMOT field observations, JMA directly confirmed if ash had fallen or not by asking public offices (such as police stations) in the wider region. In this information, unquantitative values, such as “Ash fell but the amount is unknown” and “Lapilli is on the ground but the amount is unknown”, are included. Therefore, for this result of asking public offices, reports of “No Ash Fall” were used as ash fall 0 (g/m^{2}) [including lapilli 0 (g/m^{2})], but unquantitative values such as “Ash fell but the amount is unknown” were not used in this experiment. We assumed that observation of “Ash Fall Only” includes “No Lapilli”. Besides, the eruption plume was captured as radar echoes by JMA operational weather radars, and the top height of the plume was estimated about 12,000 m ± 687 m above sea level (Sato et al. 2018). In this study, radar observation which is input into the 4DVar system assumes that there is no ash and lapilli above 13 km above sea level (in this study, radar data do not have a quantitative parameter in the eruption column).
Simulation settings and first guess
In this simulation, the spatial coordinate is a horizontal grid spacing of 0.01 degrees and a vertical grid spacing of 1 km. The area of calculation is 32.75–33.55 degrees for latitude and 130.95–131.75 degrees for longitude, and the surface—16 km for altitude. The time step is 1 s. The forecast time is 2 h from eruption (i.e. time of ash fall observation). Forecast variables are ash concentration of \( \phi \)scale − 4 to + 4 discretized to 20 bins. Each variable has density which is based on Wilson and Huang (1979) (Table 1). The emission mass profile is based on the emission source model Suzuki (1983). In this study, as a centre diameter \( d_{m} \) and standard deviation \( \sigma \) of lognormal distribution, we used \( d_{m} = 0.25 \) mm, \( \sigma = 1.0 \) based on Shimbori (2015). The start time of the eruption and duration of the eruption are 01:46 October 8, 2016 (JST) and 180 s from seismometer observation (Ishii et al. 2018). The vent is located at 32.8836N, 131.0969E. Total mass emission (i.e. eruption mass) for ash fall deposit (including lapillisize crusts) is assumed as 1.8 × 10^{8} kg from field survey (Miyabuchi et al. 2017). In this study, the meteorological field is the Meso Analysis of initial time 00, 03 and 06 JST to be interpolated linearly by the time and space. The background error covariance \( {\mathbf{B}} \) assumes Gaussian statistics between distance and correlation for vertical distribution, and between distance and correlation for size distribution. The element of \( {\mathbf{B}} \) is as follows.
where \( z_{i} ,z_{j} \) are altitudes which are discretized and \( \phi_{n} ,\phi_{m} \) are ash particle diameters which are discretized.
The dump coefficient of Gaussian \( \sigma_{z} \) and \( \sigma_{\phi } \) is, respectively, 500 m and \( 1\phi \), and \( B_{0} \) is 10 g/m^{3}. It is assumed that the observation error covariance is diagonal. Error of ash fall observation is 10% of the measured value, and error of radar observation is 10^{−3} g/m^{3}.
As mentioned above, the penalty term is used to avoid negative emission mass. The parameter \( \mu \) in the penalty term needs to be sufficiently large. However, a too large value leads that a penalty term is too dominant in the cost function, while a too small value leads negative mass emission. As a result of trying some values, we chose \( \mu \) = 20,000 (m^{6}/g^{2}).
Simulation results
Figure 2 shows that the ash fall simulation results with three hundred 4DVar iteration steps, updating the first guess every 20 steps, are shown in Fig. 2. The ash fall forecast with the 4DVar system is closer to observation than that with no 4DVar. Especially, the main axis of ash fall moved southward which is more consistent with observation, and in the west and south of the main axis of ash fall, we can see an improvement of ash fall forecast. That is, in the result of no assimilation, the ash fall is overestimated clearly around the northwest and south ash fall area of the main axis; however, with 4DVar this is improved. As expected, these improvements were due to ash fall observations.
The emission mass of the first guess and analysis, and corrections by the 4DVar system (called “analysis increment” or “increment”) are shown in Fig. 3. The overestimation of ash fall, which is forecast from the first guess to be compared to the ash fall observations in the west area, decreases emission mass of fine ash from low altitude (surface—5000 m, \( 0 \le \phi \le 3 \)). Besides, the difference between ash fall observations (with no ash) and ash fall forecast with no 4DVar (Fig. 2a) in the south area of the main axis decreases emission mass at high altitude and coarse ash (surface—10,000 m, \(  \,4 \le \phi \le  \,1 \)). In addition, there are positive increment areas at “7000–12,000 m of altitude around \( 0 \le \phi \le 1 \)”. This positive increment is due to underestimations of forecast with no 4DVar around the main axis from observation (around the area of ash fall observation 109 g/m^{2}, 183 g/m^{2}, see Fig. 2a).
Figure 4 is a plot between observation and forecast for ash fall. It shows that the ash fall forecast calculated from the emission mass with the 4DVar system is better than that with no 4DVar. Therefore, this improvement of forecast suggests that ash emission mass from the eruption column was improved by the 4DVar system and observation. However, for some ash fall observations with large amounts (> 500 g/m^{2}, i.e. observation close to the vent), the ash fall forecast with 4Dvar was worse than that from the first guess (small window in Fig. 4). This implies that phenomena close to the vent, including complex transport and depositional processes with a huge variation of ash fall locally, cannot be treated properly in this system.
Discussion
Generally, because ash particles from each altitude and in each size fall at a variety of locations, an appropriate spatial arrangement of ash fall observation is needed for the accurate estimation of emission mass. Actually, in this case, inaccuracy from a lack of ash fall observation appeared in the estimation of emission mass by the 4DVar system. In Fig. 3, there is no correction for high altitude and fine ash (above 6000 m, \( 2 \le \phi \le 4 \)). Typically, fine ash is blown a long distance. Especially, fine ash particles emitted from high altitude reach several hundreds to several thousands of kilometres away. Therefore, in order to correct fine ash emission from high altitude, ash observations in distant locations are needed.
However, in this study, we only have ash fall observations in the region of Aso volcano. (Observation points are shown in Fig. 2.) For this reason, fine ash emission from high altitude of the first guess could not be corrected by ash fall observations. That is, the analysis value is almost the value of the first guess in the region for high altitude and fine ash. Except for large eruptions, we cannot obtain quantitative ash fall observations several hundreds to several thousands of kilometres away because often the ash fall amount is too small to be measured accurately and to be represented locally because of dispersion over a long time, that is, more than 1 day. In this case, ash particles could only be identified by microscopic analysis at the Okayama Meteorological Observatory, over 300 km from Aso volcano (done by Dr. Yaguchi, who reported it). Therefore, when the ash fall amount is too small, it is difficult to obtain accurate and local representative observation values, so it is not expected that a 4DVar system with ash fall observation can correct emission mass of fine ash from high altitude. For example, although the eruption mass which is the sum of emission mass from the eruption column could be estimated to be 1.32 × 10^{8} kg, it should be noted that this value includes some errors associated with a lack of observation such as this case. For a solution to this problem of fine particles from high altitude, other observations such as volcanic ash mass loading from satellite observation (e.g. Bessho et al. 2016) are necessary. The volcanic ash mass loading from satellite observation provides the amount of fine ash in the atmosphere before fall and a few hours after eruption. In addition, in this study, ash fall observation includes only the amount of ash (and lapilli) and does not include size distribution or a time series of ash fall. For example, additional observation including these, such as PARSIVEL (e.g. Iriyama 2018), could be expected to improve the accuracy of estimation. In the 4DVar system, it is necessary to develop an observation operator \( \varvec{H}({\mathbf{x}}) \), its tangent linear \( \frac{{\partial \varvec{H}}}{{\partial {\mathbf{x}}}} \) and its adjoint \( {\mathbf{H}}^{{\mathbf{T}}} \) to introduce new observations. The 4DVar system can treat a variety of observations simultaneously by the development of the observation operators. This scalability for additional observation is the most significant point for the 4DVar system.
It also should be noted that the emission mass estimated by the 4DVar system includes also errors associated with uncertainty of the numerical model. In this case, even if ash fall observation is used, ash fall observation close to the vent could not be reproduced by the 4DVar system because of model uncertainties such as complex transport and depositional processes with a huge variation of ash fall locally. For the solution to this problem, the model uncertainties will need to be reduced. One solution may be a highresolution model including complex processes such as entrainment of ambient air into eruption clouds by turbulent mixing (e.g. Suzuki et al. 2005).
Conclusion and future work
In this study, in order to obtain emission mass from the eruption column for each altitude and each size of ash particles, we developed a data assimilation system based on 4DVar. This system was applied to the Aso volcano eruption at 01:46 on October 8, 2016, and observation for the data assimilation system is the ash fall (including lapilli) and meteorological radar. Except for fine ash from high altitude, the first guess was corrected by ash fall observation in the data assimilation system. Using the corrected first guess (i.e. analysis), the numerical model led to an ash fall forecast which is more consistent with ash fall observations than the first guess.
However, the 4DVar data assimilation system did not correct emission mass of fine ash from high altitude because the ash fall observation in the region of the volcano does not have sensitivity for fine ash from high altitude, as it does not fall in the region of the volcano. In order to correct fine ash from high altitude, it is necessary to have other observations such as mass loading from satellites.
In this study, we developed a novel data assimilation system for volcanic ash. There are many unknown parameters such as \( \sigma_{z} \), \( \sigma_{\phi } \) and \( B_{0} \) background error covariance \( {\mathbf{B}} \), error of ash fall observation in the observation error covariance \( {\mathbf{R}} \), and the penalty term factor \( \mu \). We could not check these parameters sufficiently. In addition, ash fall close to vent also could not be reproduced by this system. The solution to this problem may be the use of highresolution models including complex processes. From the above, a lack of understanding remains in the interpretation of the emission mass estimated by the 4DVar system. Therefore, it is should be noted that eruption mass 1.32 × 10^{8} kg estimated by 4DVar system includes some errors. Nevertheless, the ash emission mass estimated by the 4DVar system led to an ash fall forecast which is much more consistent with ash fall observation than that with no 4DVar, except for the area close to the vent.
Recently, quantitative retrieval techniques using remote sensing have been developed. For example, Marzano et al. (2013) developed quantitative retrieval techniques inside eruption columns based on weather radar. Comparison between the 4DVar estimate and radar observation for an eruption column may reveal disadvantages and validate/improve each scheme.
Abbreviations
 JMA:

Japan Meteorological Agency
 4DVar:

fourdimensional variational method
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Authors’ contributions
KI developed the data assimilation system and applied it to the October 8, 2016 Aso volcano eruption in Japan. The author read and approved the final manuscript.
Acknowledgements
We thank the two anonymous reviewers for their useful comments and suggestions.
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Ishii, K. Estimation of emission mass from an eruption plume for the Aso volcano eruption, on October 8, 2016, using a fourdimensional variational method. Earth Planets Space 70, 202 (2018). https://doi.org/10.1186/s4062301809648
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DOI: https://doi.org/10.1186/s4062301809648
Keywords
 Data assimilation
 Fourdimensional variational method
 4DVar
 Atmospheric transport model
 Aso volcano