Reconstructing eruptive source parameters from tephra deposit: a numerical study of mediumsized explosive eruptions at Etna volcano
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Abstract
Since the 1970s, multiple reconstruction techniques have been proposed and are currently used, to extrapolate and quantify eruptive parameters from sampled tephra fall deposit datasets. Atmospheric transport and deposition processes strongly control the spatial distribution of tephra deposit; therefore, a large uncertainty affects mass derived estimations especially for fall layer that are not well exposed. This paper has two main aims: the first is to analyse the sensitivity to the deposit sampling strategy of reconstruction techniques. The second is to assess whether there are differences between the modelled values for emitted mass and grainsize, versus values estimated from the deposits. We find significant differences and propose a new correction strategy. A numerical approach is demonstrated by simulating with a dispersal code a mild explosive event occurring at Mt. Etna on 24 November 2006. Eruptive parameters are reconstructed by an inversion information collected after the eruption. A full synthetic deposit is created by integrating the deposited mass computed by the model over the computational domain (i.e., an area of 7.5 × 10^{4} km ^{2}). A statistical analysis based on 2000 sampling tests of 50 sampling points shows a large variability, up to 50 % for all the reconstruction techniques. Moreover, for some test examples Power Law errors are larger than estimated uncertainty. A similar analysis, on simulated grainsize classes, shows how spatial sampling limitations strongly reduce the utility of available information on the total grain size distribution. For example, information on particles coarser than ϕ(−4) is completely lost when sampling at 1.5 km from the vent for all columns with heights less than 2000 m above the vent. To correct for this effect an optimal sampling strategy and a new reconstruction method are presented. A sensitivity study shows that our method can be extended to a wide range of eruptive scenarios including those in which aggregation processes are important. The new correction method allows an estimate of the deficiency for each simulated class in calculated mass deposited, providing reliable estimation of uncertainties in the reconstructed total (whole deposit) grainsize distribution.
Keywords
Tephra deposit Eruptive source parameters Dispersal model Grain size distribution TGSD Erupted mass Sampling strategyIntroduction
A primary objective in volcanology, critical for hazard assessment of active volcanoes, is the characterization of explosive volcanic eruptions and the quantification of their intensities. Accurately estimating eruptive source parameters (ESPs), such as mass eruption rate (MER), is necessary if we are to deepen our understanding of eruptive column and ash cloud dynamics. In addition, knowledge of the total (whole deposit) grain size distribution (TGSD) of the solid particles at the vent is needed if we are to better understand the mechanisms occurring within the conduit during an eruption. This is commonly viewed as comprising a primary fragmentation of magma during rapid decompression producing large fragments (Alidibirov and Dingwell 1996) and a secondary one producing fine ash (Kaminski and Jaupart 1998). These processes will result in solid particles of various sizes, ranging from meters to a few microns, with those of several centimeter and smaller (tephra) carried into the atmosphere and subsequently settled toward the ground under the action of atmospheric dynamics, aggregative processes, and gravity. Atmospheric transport processes, controlled by particle characteristics (size, density, and shape) and altitude of release from the column (depending on MER and atmospheric conditions), affect aerial distribution over timescales ranging from a few hours up to weeks. The area affected by ash fallout can extend over thousands of square kilometers around the vent (Sparks et al. 1997). Recently, Bursik et al. (2012), Degruyter and Bonadonna (2013), Woodhouse et al. (2013) and Mastin (2014) investigated wind effects on column dynamics and height, revealing the importance of taking wind into account in order to avoid strong underestimation of mass flowrates. For all the aforementioned reasons, uncertainty associated with the tephra dispersal process is quite large and an accurate estimation of total erupted mass and TGSD is a complex and still daunting task.
During a volcanic crisis, several direct measurements are a useful source of information about the ongoing activity. Remote sensing instruments provide estimates of column (Rose et al. 1995; Arason et al. 2011) and plume height (Vernier et al. 2013; Grainger et al. 2013), duration of the event (Johnson et al. 2004), gas and particle exit velocities (Dubosclard et al. 2004), and plume composition (Rose et al. 2000; Spinetti et al. 2013). However, satellite retrievals provide only columnintegrated information and are strongly sensitive to the presence of atmospheric clouds above the ash plumes and to assumptions about ash size distribution and ash composition (Wen and Rose 1994). Moreover, most of the band spectra used by remote sensing instruments are not very sensitive to the presence of particles larger than 32 microns, and consequently, they detect only a fraction of the erupted mass. Corradini et al. (2008) estimate satellite retrieved total mass uncertainty to be on the order of 40 % increasing up to 50 % when considering nonspherical particles (Kylling et al. 2014). For sun photometers coupled with Lidar measurements (Gasteiger et al. 2011) estimated a 50 % error in mass concentrations. Due to these limitations, the deposit is still one of the main products that must be analyzed in order to estimate the cumulative erupted mass (EM) and total grainsize distribution (TGSD) of an eruption.
Historically, the volume of solid erupted material has been estimated from discrete samplings of deposit thickness (Fisher 1964; Walker 1973). In recent decades, several techniques have been proposed to optimize the integration of field data and to obtain an estimate of the emitted material (Pyle 1989; Fierstein and Nathenson 1992). Bonadonna and Houghton (2005) and Bonadonna and Costa (2012) introduced other methods, commonly adopted by volcanologists during their field studies, for estimating volume and total grain size distribution. More, recently, Burden et al. (2013) and Engwell et al. (2013) used statistical methods to study the uncertainty in volume estimation (estimated to be between 1 and 10 % for small datasets), associated with uncertainties in tephra thickness measurements. Additional studies addressing estimation of uncertainty for erupted mass, based solely on field data, can be found in Andronico et al. (2014a), Klawonn et al. (2014b), Engwell et al. (2015), and Bonadonna et al. (2015).
In the past decade, several attempts have been made to integrate field data analysis with other approaches in order to better constrain the initial eruptive conditions. Gudmundsson et al. (2012) integrated ground measurements with satellite observations. Recently, Stevenson et al. (2015) integrated tephrochronology, dispersion modeling and satellite remote sensing to understand the discrepancy between tephra deposit and satellite infrared measurements.
The importance of accurately estimating eruptive source parameters (ESPs) is in part due to the growing use of dispersal codes for ash hazards assessment (Sparks et al. 1997; Textor et al. 2005; Folch 2012; Fagents et al. 2013). The reliability of the output from such tephra dispersal models depends strongly on the reliability and uncertainty of ESPs. In recent years, the application of inversion techniques to tephra dispersal models, based on advectiondiffusion sedimentation equations, has shown promise for determining EPSs (Connor and Connor 2006; Burden et al. 2011; Bursik et al. 2012; Pardini et al. 2016; Scollo et al. 2008; Bonasia et al. 2010; Volentik et al. 2010; Fontijn et al. 2011; Johnston et al. 2012; Klawonn et al. 2012; Magill et al. 2015).
Methods
Numerical model
We use the dispersal code VOLCALPUFF (Barsotti et al. 2008) to simulate the transport of volcanic ash cloud in a transient and 3D atmosphere and to compute a synthetic deposit. No other models have been tested here since the purpose of this study is analysing the sensitivity and uncertainty of reconstruction techniques instead of testing different models (for a review of dispersal model see Folch (2012)). VOLCALPUFF couples an Eulerian description of the initial plume rise phase, where plume theory equations are solved (Morton 1959; Bursik 2001), with a Lagrangian description of the transport of material leaving the eruptive column. The model calculates mass lost along the column for each class size as a function of settling velocities. Settling velocity is computed as a function of particle Reynolds number and depends on particle characteristics (dimension, density, shape) as well as atmospheric properties (air density and viscosity), according with the modification of the Wilson and Huang (1979) model, as presented in Pfeiffer et al. (2005). Particles of different sizes are released from the column as a series of Gaussian packets (puffs), which are transported and diffused by the wind during their fall toward the ground due to gravity. Tracking puff movements within the 3D domain, the code computes at each time step the amount of mass advected out of the domain, still suspended in the atmosphere and deposited on the ground. VOLCALPUFF has been tested and adopted to simulate volcanic ash dispersal and deposition at several volcanoes worldwide (Barsotti and Neri 2008; Barsotti et al. 2011; Spinetti et al. 2013; Barsotti et al. 2015).
For the application presented in this paper, we initialized VOLCALPUFF with meteorological input data produced by the nonhydrostatic code LAMI (Doms and Schättler 2002) with 7 km horizontal resolution and 23 vertical levels and refined down to one km horizontally and one hour in time by the processor CALMET (Scire et al. 2000).
Inversion modelling and bestfitting criterion
Reconstruction techniques
Tephra deposit isopach
The determination of the amount of erupted material is historically done starting from field measurements collected over the domain at discrete points (Walker 1973; Pyle 1989; Fierstein and Nathenson 1992). These data can be either measurement of deposit thickness for historical eruptions or of loading for more recent ones. Deposit information is generally completed by interpolating the pointwise measurements and by hand drawing isopachs. Klawonn et al. (2014b) analyzed the variability in handmade isopach, estimating a range of ±10 % for the associated uncertainty. Bursik and Sieh (2013) and Engwell et al. (2015) used a mathematical fitting function based on cubic splines in order to reduce the subjectivity in the process. Recently, Kawabata et al. (2015) proposed a different statistical method, based on fitting multiple ellipsoids. Here, to produce isopach maps in an objective and automatic way, we adopt the Natural Neighbor (NN) interpolation method introduced by Sibson (1981). The Natural Neighbor method is an exact interpolation technique based on Voronoi tessellation of a discrete set of spatial points. The method has been already applied to geophysical problems and performs well for irregularly distributed data (Watson and Phillip 1987; Sambridge et al. 1995). Moreover, NN provides a smooth approximation by a weighted average within the sample set of the interpolation point’s neighbor values. We used a NN algorithm distributed with the NCAR Graphics Library (Brown et al. 2012a).
Erupted mass
In the past, different methods have been proposed to estimate erupted volume from deposit data (Rose et al. 1973; Pyle 1989; Fierstein and Nathenson 1992; Sulpizio 2005; Bonadonna and Costa 2012), where the estimation process consists of several common steps. The first step is to draw a discrete number (N) of isopach (or isomass) contours, as described above. The second step is to express the thickness t _{ i } as a function of its corresponding isopach area squareroot x _{ i }. This is done with a curvefitting procedure, where the function t(x) is chosen from a fixed family, described by one or more parameter, by minimizing a residual function. Finally, the erupted volume is calculated by integrating the selected function. Recently, Burden et al. (2013) proposed an alternative method avoiding drawing isomass maps and using a statistical method on the sampling point measurements.
 1)
Exponential or Pyle’s method (Pyle 1989). This method assumes that thickness, calculated on circular isopachs, follows an exponential decay with distance from the vent. Fierstein and Nathenson (1992) and Bonadonna and Houghton (2005) generalized the method for elliptical isopachs and for accounting the breakinslope of some tephra deposits, by using various exponential segments. This method is now widely used even though it is very sensitive to the number of straight segments and their extremes. In order to reduce the arbitrariness of the process (in particular in the choice of the segments extremes), we define an automatic procedure, where we select also the breakinslope points within the minimization process involving the fitting.
 2)
Powerlaw method (Bonadonna and Houghton 2005). In their paper, the authors suggest a powerlaw best fit of field data to obtain the total erupted volume. This volume can be calculated as \( V=\int 2 t_{pl} x^{m+1} dx\), where t _{ p l } is the thickness associated to the unit area isomass. Unfortunately, this integral, when evaluated over the interval \( [0, +\infty ]\), is infinite. To avoid this problem such an interval is replaced by a smaller one [x _{0},x _{1}], where x _{0} and x _{1} represent respectively, the exclusion of the proximal area and the distal region from the domain. As already stated by Bonadonna and Costa (2013), this choice depends on the powerlaw exponent and strongly influences Powerlaw results . For this reason, in our statistical analysis, average values of these distances are fixed at 1 and 300 km (accordingly with the eruption size) with a range of variability equal to ±10 %.
 3)
Weibull method (Bonadonna and Costa 2012; 2013). The method is based on the assumption that the function x T(x) can be described by a Weibull distribution with three main parameters (λ,𝜃,n). λ represents the characteristic decay length scale of deposit thinning, 𝜃 represents a thickness scale and n is a shape parameter (dimensionless).
Reconstruction methods presented with explicit formulas for loading (t) as a function of corresponding isomass squareroot areas (x), and residual cost functions
Method  Formula t(x)  Residual function  Free parameters 

Exponential  \( t_{0} \exp (m x)\)  \( {\sum }_{i=1, N} (log(t(xi))log(t_{i}))^{2}\)  6 
Powerlaw  t _{ p } x _{ m }  \( {\sum }_{i=1, N} (log(t(xi))log(t_{i}))^{2}\)  2+2 arb 
Weibull 1  \( \theta (x/\lambda )^{n2} \exp ((x/\lambda )^{n})\)  \( {\sum }_{i=1, N} (t(x_{i})t_{i})^{2}\)  3 
Weibull 2  \( \theta (x/\lambda )^{n2} \exp ((x/\lambda )^{n})\)  \( {\sum }_{i=1, N} 1/{t_{i}^{2}} (t(x_{i})t_{i})^{2}\)  3 
Weibull 3  \( \theta (x/\lambda )^{n2} \exp ((x/\lambda )^{n}) \)  \( {\sum }_{i=1, N}1/t_{i} (t(x_{i})t_{i})^{2} \)  3 
In this work, model outcomes are expressed as loadings, so we apply all previous methods for the mass estimation by considering isomass contours instead of isopachs as frequently done in case of fresh deposit (Scollo et al. 2007; Andronico et al. 2008; Andronico et al. 2009).
Grainsize distribution
Total grainsize distribution (TGSD) of tephrafall deposits is a crucial ESP for tephradispersal modelling and risk mitigation plans (Scollo et al. 2008; Folch 2012) and several techniques have been developed to reconstruct it starting from ground information, e.g., Murrow et al. (1980) and Bonadonna and Houghton (2005). Usually, grain size is expressed in ϕ scale where ϕ=−l o g _{2}(d/d _{0}), d is the particle diameter, and d_{0} is a reference diameter of 1 mm. For reconstructing the TGSD different techniques exist: weighted average, sectorization of the deposit, or isomass maps of individual ϕ classes. Nevertheless, the Voronoi method adopted in Bonadonna and Houghton (2005) for the TGSD reconstruction is widely used by the volcanological community. For this reason, we chose it in our work for a comparison with the emitted and deposited TGSD. Voronoi, or nearestneighbor, technique is a constant piecewise interpolation method built upon the Voronoi tessellation (Voronoi 1907). Given a set of points called seeds, the plane, over which they lay, is partitioned into convex cells, each one consisting of all those points closer to that seed than to any others.
Then, as proposed in Bonadonna and Houghton (2005), the grain size and loading of each sample point are assigned to the corresponding Voronoi cell and the TGSD is obtained with a massweighted (where the mass is defined as the product between the cell area and the cell loading) average of all the sampled values over the whole deposit. The main requirement to apply the Voronoi method is to fix the deposit extent, adding deposit zero values to the original dataset. This step is essential to prevent the external Voronoi cells from having unlimited area and thus infinite mass. Dependence of the reconstructed TGSD on the location of the zeros has been previously recognized and discussed in Bonadonna and Houghton (2005), Bonadonna et al. (2015), and Volentik et al. (2010).
Application to 24 November 2006 eruption at Mt. Etna
Mt. Etna is the most active volcano in Europe, and each year it shows a wide variability in its eruptive activity. It has been extensively studied, and a wide literature exists regarding past activities, eruptive styles and products (Branca and Del Carlo 2005; Andronico et al. 2005; Scollo et al. 2007; Andronico et al. 2008; Andronico et al. 2009; Andronico et al. 2014b). It is definitely an openpit laboratory of volcanology and for all these reasons we have chosen it as our test case. In this paper, VOLCALPUFF is used to simulate an eruption occurred on 24 November 2006, during which a 9hlong eruption was observed with a column height estimated between 2000 and 2500 m above the ground level (agl) (Andronico et al. 2009). At that time a quite persistent wind was blowing toward southeast with intensities up to 10 m/s at 3 km above the vent. Due to these meteorological conditions, the investigated domain, here fixed to 272 × 277 km ^{2}, is displaced southeast of the vent.
A sensitivity analysis of column height as function of Mass Eruption Rate (MER) has been carried out assuming a bimodal TGSD, obtained with a linear combination of two lognormal distributions with parameters μ _{1,2} and σ _{1,2} respectively ranging in the intervals [−3,5] and [0.5,6]. The initial distribution has been then partitioned in 23 classes equally spaced in the ϕ interval [ −5, 6]. Particles are assumed non spherical with a shapefactor of 0.43 within the range presented in Andronico et al. (2014b). Density is assumed varying with particles size according to Eychenne and Le Pennec (2012). Exit velocity and radius have been varied hourly.
Inversion result: simulated scenario
Computed deposit
Figure 4a shows the simulated tephra deposit calculated 24 h after the eruption beginning. A considerable part of the deposit interests the inland (96 %) while a small amount falls into the sea. Deposit crosswind extension mainly depends on wind temporal stability and direction during the eruptive event. Due to wind variability in direction with altitude and with time, the simulated eruption produces a wide deposit, which enlarges downwind the vent (Fig. 4a).
Sampling methodology
Here, we test the stability of the reconstruction techniques presented in “Methods” by applying the methods to different sampling datasets. The whole set of 250 × 255 integral loading values is sampled to create 2000 smaller datasets (”sampling tests”) of 50 points. For each sampling test, points are generated using a cylindrical coordinate system and the sampling procedure is performed using a uniform probability distribution along the major dispersal axis and a normal distribution for the angular coordinate. We calculate the major dispersal axis using simulated airborne concentrations and wind direction. Due to wind direction variability during the eruptive event, and therefore to the difficulty to find a major dispersal axis, a small random variability is considered in the angle for each sampling test. Tephra deposit sampling is often incomplete due to difficulties in sampling the very proximal or very distal regions. To reproduce this limitation, we exclude the area close to the vent from the sampling, but we include at least one point within 2 km from the vent in each test. We also sampled only inside the 10 ^{−2} kg/m ^{2} isomass contour (corresponding to 0.01 mm, assuming a constant deposit density of 1000 kg/m ^{3} (Andronico et al. 2014b)) represented by black points. This value is chosen as a threshold in agreement with studies on eruptions of similar size (Andronico et al. 2008; Bonadonna and Houghton 2005; Scollo et al. 2007) and objective limitations on measurements for historical deposits. In order to keep the analysis more general, we have included in our dataset also points collected over the sea. Figure 4b and c show two examples of sampling tests, respectively, TestA_1 and TestA_2, where the deposit mass is reconstructed using the Natural Neighbor (NN) interpolation method.
Results
In this section, mass and TGSD values, calculated using the “exact integration” (see “Erupted and deposited mass”) and classical reconstruction techniques are compared with the known values used as Scenario input.
Erupted and deposited mass
With gray dashed lines, we refer to the 10, 50, and 90 % of the total deposited mass and the corresponding loading which is, respectively, of 300, 90, and 0.8 kg/m ^{2}. This reveals how the domain size and the considered thickness threshold influence the deposited mass. More than 90 % of the emitted mass is deposited within an area of 3600 km ^{2}, and 10 % lies in an area smaller than 1.3 km ^{2}.
Comparison with reconstruction techniques
In this section, we test all the previously introduced reconstruction techniques (Exponential, Powerlaw, and Weibull) using the different sampling tests.
Deposited grain size distribution
In analogy with the previous section, here we first compare for each size the emitted mass (EM_{ i }) with the deposited (DM_{ i }) obtained from an exact integration of the simulated deposit for the 23 grain size classes. Afterwards, results are compared with the TGSD obtained using the commonly adopted Voronoi tessellation method, applied to the 2000 sampling tests.
As expected, the deposited mass estimation strongly depends also on the minimum loading (hereafter referred as threshold) considered for the integration: the higher the threshold, the larger the underestimation of deposited mass. Figure 7b presents for each simulated grainsize the deposited mass DM_{ i } when different thresholds are considered. Classes in the ϕ range interval [1,2] are underestimated of about 5–10 % considering a 0.01 kg/m ^{2} threshold (orange bars); these percentages rise to 30–60 % when 0.1 kg/m ^{2} threshold is considered (red bars in Fig. 7b).
As shown in Fig. 7a, for the investigated Scenario, a large amount of the emitted mass for classes ϕ>5 is advected out the domain (up to 90 %).
As we have already shown, the effect of the domain restriction drastically reduces the amount of fine classes deposited on such domain. This is well visible in Fig. 8c for classes finer than ϕ=5 where the peak is leaving the domain border. Therefore, the fine material found on the ground is mostly coming from the margin of the column. The cutoff effect due to a limited domain is not easy to overcome, in fact the area interested by the fallout of particle finer than ϕ=5 in the case of the described Scenario is larger than 10 ^{6} km ^{2}. However, the spatial distribution of small class is more homogeneous due to the larger diffusion. As a direct consequence, for a smaller class, the error committed using the interpolation techniques for reconstruct DM_{ i } is small.
Comparison with the Voronoi technique
On average Voronoi reconstructs well the deposited mass over the considered area (orange line) but can present artificial mode on the reconstructed TGSD due to the piecewise approximation. In fact, different classes, having different settling velocities, will generate peaks in the deposit at different distances from the vent (Fig. 8c). Indeed, if a sampling point is close to a peak of a specific class, this maximum value of the deposit will be extended over the entire Voronoi cell. This could have the final effect of overestimating the reconstructed class mass as visible in Fig. 9a for ϕ(3). Due to diffusion, finer particles (ϕ>2) present a more uniform distribution at the ground and a smaller maximum load. For this reason, it is easier to sample points with a loading close to the maximum value and also a piecewise constant method, as Voronoi, well approximates the deposited mass. Conversely, for coarser particles, the area with a loading higher than 50 % of the peak is smaller (see Fig. 8c) and the probability to have a sample within this area is smaller, resulting in a larger variability in the results. It is also worth to note that the modes observed in Fig. 9b, associated with the samples and the reconstruction technique, are in the relative amount of mass and not in the absolute value, and thus they can also be due to an underestimation of the other classes, as visible in Fig. 9a.
For the classes corresponding to the mode, a larger gap is present between the 5^{ t h } and the 95^{ t h } percentiles (lightgray area). A large variability in the results obtained for the different sampling tests is also indicated by the large standard deviation. Conversely, for finer particles (ϕ>3), a better accordance between values obtained with the exact integration and values reconstructed with the Voronoi technique is found. For these classes, also the standard deviation and the gap between the 5^{ t h } percentile and the 95^{ t h } percentile are small, and thus the reconstruction is more stable with respect to the choice of sampling points. Consequently, Voronoi seems reproducing well the deposited mass if we restrict our analysis to ϕ(−2), as visible in Fig. 8c.
Discussion
The analysis presented in the previous sections confirms that inferring quantitative data from the deposits can be very difficult and a large uncertainty affects the results. Estimated values can be far from representing a full picture of the initial eruptive condition at the vent. On the one hand, the reconstruction of the total mass and the grain size with an exact integration over the considered computational domain (area 7.5 × 10^{2} km ^{2}) shows that, for finer particles, a large amount of the emitted mass is not found at the ground (see Fig. 5b). Precisely, 78 % of particles with ϕ>4 is leaving the domain. On the other hand, information on coarser particles is partially lost when very proximal area is excluded (Fig. 7a).
A combination of these two effects, here obtained considering a loading larger than 0.01 kg/m ^{2} (roughly corresponding to 0.01 mm) and excluding an area within about 2 km from the vent, causes an underestimation of the deposited mass up to about 70 %. In this case, the two constraints on the integration domain adopted to figure out the TGSD can reflect a reasonable minimum value sampled in the field and the objective difficulties in reaching areas very close to the vent.
The underestimation reflects also in an almost unimodal reconstructued TGSD, where the tails are both depleted with respect to the initial ones. We remark that these results are not an effect of the procedure, since no extrapolation techniques are used. The mass underestimation only depends of the choice of the integration domain, although quite extended in this study.
Results are obtained for a ϕ range spanning from −5 to 6, but the same investigations, considering finer particles, would produce similar outcomes. Similar results, on modifying the sampling region by excluding a proximal area, have been found on Andronico et al. (2014a) whereas the mass lost was estimated to be 22 % of the EM.
Deposit information is further degraded when inferred from a finite number of sampling values by using reconstruction methods. In general, despite the large number of samples considered for each sampling test (50 points are considered a very well sampled deposit), the mass values obtained with Exponential, Powerlaw and Weibull methods show a large variability and, for a significant percentage of tests, an underestimation of the erupted mass (Fig. 4d).
On average, Exponential and Weibull1 fitting methods seem to produce more stable results than the Power Law and other Weibull functions providing a better estimate of erupted mass. Numerical integral provides a smaller standard deviation within the 2000 sampling tests performed.
When the Voronoi technique is used, the TGSD presents fictitious modes in the resulting distribution (Fig. 9) and a large variability affects the results depending on the choice of the sampling points. This is mostly due to a problem of the deposit exposure and not on the particular choice of the averaging strategy (here Voronoi).
Best sampling strategy
Based on our study on the column model, columns height is weakly dependent the emitted TGSD. Consequently, from Fig. 8c, we can conclude that for each class the peak distance from the vent is minimally depending on the emitted percentage. Similar results have been found using analytical model for monodisperse plume released from high altitude (Tirabassi et al. 2009). Because the peak distance, for a fixed column height, is not depending on the emitted mass but only on the atmospheric condition, we can easily estimate it. This would help to find the best sampling region to reconstruct the eruptive scenario reducing the number of sampled points (see also Spanu et al. 2015; Costa et al. 2016). In fact, using Fig. 8c, we can obtain for each class the distance from the vent where to sample to measure the loading peaks. For example, to correctly estimate the deposited mass ϕ(1), we should sample at least one point at 3 km from the vent. This suggests an optimal sampling strategy where we collect points along the main dispersal axis with a growing distance from the vent as shown with the black line in Fig 8c. For coarser classes, only a very small part of the emitted material is deposited in the considered domain (see Fig. 7). When we consider the number of particles for unit area, instead of the mass, we found less than one particle per square meter already at 2 km away from the vent with ϕ<(−4). Thus, the probability to sample coarse particles at these distances is really small and this produces a large variability in the reconstruction of the emitted mass. One solution to reduce this uncertainty is to collect more points on the same region to make the statistic more robust. This will also increase the probability of sampling a peak. As we can see in Fig. 8c, the peak isomass area is a function of the particles size. For coarse classes (ϕ<−2), the peak is concentrated in a small area consequently it is more difficult to locate.
Correction factors
This method also allows an estimation of the emitted mass by summing the results obtained for all the classes (see Fig. 11d, h). For a sampling distance within 1 and 100 km from the vent (Fig. 11a), the deposited mass DM_{ t o t } only represents the 70 % of the emitted one, whereas the interval provided by our method correspond to 90 and 112 % of the emitted mass. For the second example (Fig. 11e), where the sampling points are collected within 2 and 100 km from the vent, the deposited mass represents only the 30 % of the emitted, whereas our method gives an interval corresponding to 82 and 120 % of the real value. Despite the effect of excluding a proximal area region is smaller for higher columns, for a 6000 m agl column only 10 % of ϕ(−5) is still depositing outside the proximal region (see Fig. 10b). In the auxiliary material, we provide a table with more examples of correction inverse factor for different column height.
The same technique can be adopted to estimate information losses over a generic domain, for example, when the median region or areas above the sea are missing. However, when applying the method we need to be careful in considering the uncertainty associated with the reconstruction techniques applied to samplings dataset (in our examples NN) because this error will affect directly our estimate interval. We remark that approximation errors are decreasing with the number of the collected sample, whereas the bias between the emitted and deposited mass only depends on the considered domain.
Aggregation
Since we obtained our results using a numerical study, they are clearly dependent on model assumptions. However, we tested the dependency on settling velocity, factor shape and particles density, and results are poorly sensitive to those parameters.
Conclusion

Deposited mass was a function of considered domain and exact integration allowed to quantify the underestimation with respect to the emitted mass. When the deposited mass was integrated over a distance of 1 and 100 km from the vent only 30 % of the emitted mass was found on the ground.

Even by using a large and welldistributed dataset of sampled points (50 points) over the modeled domain, large errors can affect the results. Reconstruction techniques, performed over 2000 different sampling tests, showed a gap in the mass values up to an order of magnitude between the 5^{ t h } and the 95^{ t h } percentile. Besides, estimated confidence intervals were not representative of committed errors.

Large standard deviation values and large relative errors affected reconstructed TGSD obtained in case of poorly exposed deposits. In particular, the emitted fractions of coarse and fine particles were generally underestimated underrating the hazard associated with volcanic airborne particulate. Furthermore, when the fractions obtained with the Voronoi technique were rescaled with the total emitted mass, median classes were always overestimated in mass.

We proposed a new method to reconstruct the emitted TGSD and the emitted mass starting from single classes measurements and column height observations. We used a sensitivity study on eruptive parameters to generalize the analysis so far depending on the eruption scale and assumed TGSD.

Column height showed to be weakly dependent on the TGSD emitted. Consequently, for a fixed column height, for each class the distance from the vent of the maximum loading is independent from the emitted percentage. This provides a useful criterion to sample a deposit in order to reduce the uncertainty in the TGSD.
Finally, this work showed how, in support of fieldbased studies, numerical studies represent a useful tool to assess ESP uncertainty.
Notes
Acknowledgments
This work presents results achieved in the PhD work of the first author (A.S.), carried out at Scuola Normale Superiore and Istituto Nazionale di Geofisica e Vulcanologia. The activity has been partially funded by the Italian Presidenza del Consiglio dei Ministri Dipartimento della Protezione Civile (Project V1). Further, we should mention ARPA_SIMCINECA for the LAMI code runs. The manuscript benefited from fruitful discussions with Dr. A. Neri, Dr. S. Engwell and Prof. A.B. Clarke. We wish to thank the Editor Prof. C. Bonadonna, M. Pyle, and two other anonymous reviewers for their constructive comments that greatly improved the quality of the manuscript.
Supplementary material
References
 Alidibirov M, Dingwell DB (1996) Magma fragmentation by rapid decompression. Nature 380:146–148. doi: http://dx.doi.org/10.1038/380146a0 CrossRefGoogle Scholar
 Andronico D, Branca S, Calvari S, Burton MR, Caltabiano T, Corsaro RA, Del Carlo P, Garfí G., Lodato L, Miraglia L, Muré F., Neri M, Pecora E, Pompilio M, Salerno G, Spampinato L (2005) A multidisciplinary study of the 200203 Etna eruption: insights into a complex plumbing system. Bull Volcanol 67:314–330. doi: 10.1007/s0044500403728
 Andronico D, Scollo S, Caruso S, Cristaldi A (2008) The 200203 Etna explosive activity: tephra dispersal and features of the deposits. J Geophys Res:113:B4 B04:209. doi: 10.1029/2007JB005126
 Andronico D, Scollo S, Cristaldi A, Ferrari F (2009) Monitoring ash emission episodes at Mt. Etna: the 16 November 2006 case study. J Volcanol Geotherm Res 180:123–134. doi: 10.1016/j.jvolgeores.2008.10.019 CrossRefGoogle Scholar
 Andronico D, Scollo S, Cristaldi A, Lo Castro MD (2014a) Representivity of incompletely sampled fall deposits in estimating eruption source parameters: a test using the 1213 January 2011 lava fountain deposit from Mt Etna volcano, Italy. Bull Volcanol 76(10):1–14. doi: 10.1007/s0044501408613
 Andronico D, Scollo S, Lo Castro MD, Cristaldi A, Lodato L, Taddeucci J (2014b) Eruption dynamics and tephra dispersal from the 24 November 2006 paroxysm at SouthEast Crater Mt Etna, Italy. J Volcanol Geotherm Res. doi: 10.1016/j.jvolgeores.2014.01.009
 Arason P, Petersen GN, Bjornsson H (2011) Observations of the altitude of the volcanic plume during the eruption of Eyjafjallajökull, AprilMay 2010. Earth Syst Sci Data 3(1):9–17. doi: 10.5194/essd392011 CrossRefGoogle Scholar
 Barsotti S, Bignami C, Buongiorno MF, Corradini S, Doumaz F, Guerrieri L, Merucci L, Musacchio L, Nannipieri L, Neri A, Piscini A, Silvestri M, Spanu A, Spinetti C, Stramondo C, Wegmuller U (2011) SAFER Response to Eyjafjallajökull and Merapi Volcanic Eruptions. In: Commission E. (ed) Let’s embrace space’ Space Research achievements under the 7th Framework Programme. DG Enterprise and Industry. http://hdl.handle.net/2122/7646, pp –
 Barsotti S, Neri A (2008) The VOLCALPUFF model for atmospheric ash dispersal: 2 Application to the weak Mount Etna plume of July 2001. J Geophys Res 113. doi: 10.1029/2006JB004624
 Barsotti S, Neri A, Bertagnini A, Cioni R, Mulas M, Mundula S (2015) Dynamics and tephra dispersal of violent Strombolian eruptions at Vesuvius: insights from field data, wind reconstruction and numerical simulation of the 1906 event. Bull Volcanol 58:77 7. doi: 10.1007/s0044501509396 Google Scholar
 Barsotti S, Neri A, Scire J (2008) The VOLCALPUFF model for atmospheric ash dispersal: 1 Approach and physical formulation. J Geophys Res. 113:B3 and B03:208 10.1029/2006JB004623
 Biass S, Bagheri G, Aeberhard WH, Bonadonna C (2013) TError: towards a better quantification of the uncertainty propagated during the characterization of tephra deposits. Stat Volcanol 1:1–27. doi: 10.5038/2163338X.1.2 CrossRefGoogle Scholar
 Bonadonna C, Biass S, Costa A (2015) Physical characterization of explosive volcanic eruptions based on tephra deposits: propagation of uncertainties and sensitivity analysis. J Volcanol Geotherm Res 296:80–100. doi: 10.1016/j.jvolgeores.2015.03.009 CrossRefGoogle Scholar
 Bonadonna C, Costa A (2012) Estimating the volume of tephra deposits: a new simple strategy. Geology 40:415–418. doi: 10.1130/G32769.1 CrossRefGoogle Scholar
 Bonadonna C, Costa A (2013) Plume height, volume and classification of volcanic eruptions based on the Weibull function. Bull Volcanol. doi: 10.1007/s0044501307421
 Bonadonna C, Houghton BF (2005) Total grainsize distribution and volume of tephra fall deposits. Bull Volcanol 67:441–456. doi: 10.1007/s0044500403862 10.1007/s0044500403862 CrossRefGoogle Scholar
 Bonadonna C, Scollo S, Cioni R, Pioli L, Pistolesi M (2016) Determination of the largest clasts of tephra deposits for the characterization of explosive volcanic eruptions: report of the IAVCEI Commission on Tephra Hazard Modelling. https://vhub.org/resources/870
 Bonasia R, Macedonio G, Costa A, Mele D, Sulpizio R (2010) Numerical inversion and analysis of tephra fallout deposits from the 472 AD subPlinian eruption at Vesuvius (Italy) through a new bestfit procedure. J Volcanol Geotherm Res 189(3–4):238–246. doi: 10.1016/j.jvolgeores.2009.11.009 CrossRefGoogle Scholar
 Branca S, Del Carlo P (2005) Types of eruptions of Etna volcano AD 16702003: implications for shortterm eruptive behaviour. Bull Volcanol 67(8):732–742. doi: 10.1007/s004450050412z CrossRefGoogle Scholar
 Brown D, Brownrigg R, Haley M, Huang W (2012a) NCAR Command Language (NCL). doi: 10.5065/D6WD3XH5
 Brown RJ, Bonadonna C, Durant AJ (2012b) A review of volcanic ash aggregation. Phys. Chem. Earth 4546:65–78. doi: 10.1016/j.pce.2011.11.001 10.1016/j.pce.2011.11.001
 Burden RE, Chen L, C PJ (2013) A statistical method for determining the volume of volcanic fall deposits. Bull Volcanol 75(6):1–10. doi: 10.1007/s0044501307074 CrossRefGoogle Scholar
 Burden RE, Phillips JC, Hincks TK (2011) Estimating volcanic plume heights from depositional clast size. J Geophys Res 116. doi: 10.1029/2011JB008548
 Bursik M (2001) Effect of wind on the rise height of volcanic plumes. Geophys Res Lett 18:3621–3624. doi: 10.1029/2001GL013393 CrossRefGoogle Scholar
 Bursik M, Jones M, Carn S, Dean K, Patra A, Pavolonis M, Pitman EB, Singh T, Singla P, Webley P, Bjornsson H, Ripepe M (2012) Estimation and propagation of volcanic source parameter uncertainty in an ash transport and dispersal model: application to the Eyjafjallajökull plume of 1416 April 2010. Bull Volcanol 74:2321–2338. doi: 10.1007/s0044501206652 CrossRefGoogle Scholar
 Bursik M, Sieh K (2013) Digital database of the Holocene tephras of the MonoInyo Craters, California. US Geological Survey Data. http://pubs.usgs.gov/ds/758/
 Carey SN, Sigurdsson H (1982) Influence of particle aggregation on deposition of distal tephra from the May 18, 1980, eruption of Mount St. Helens Volcano. J Geophys Res 87:7061–7072. doi: 10.1029/JB087iB08p07061 CrossRefGoogle Scholar
 Connor LJ, Connor CB (2006) Inversion is the key to dispersion: understanding eruption dynamics by inverting tephra fallout. Stat Volcanol 1:231–242Google Scholar
 Cornell W, Carey SN, Sigurdsson H (1983) Computer simulation of transport and deposition of the Campanian Y5 ash. J Volcanol Geotherm Res 17:89–109. doi: 10.1016/03770273(83)90063X 10.1016/03770273(83)90063X CrossRefGoogle Scholar
 Corradini S, Spinetti C, Carboni E, Tirelli C, Buongiorno MF (2008) Mt Etna tropospheric ash retrieval and sensitivity analysis using moderate resolution imaging spectroradiometer measurements. J Appl Remote Sens 2(1):023550. doi: 10.1117/1.3046674 CrossRefGoogle Scholar
 Costa A, Pioli L, Bonadonna C (2016) Assessing tephra total grainsize distribution: insights from field data analysis. Earth Planet Sci Lett 443:90–107. doi: 10.1016/j.epsl.2016.02.040 CrossRefGoogle Scholar
 Daggitt ML, Mather TA, Pyle DM, Page S (2014) AshCalca new tool for the comparison of the exponential, powerlaw and Weibull models of tephra deposition. J Appl Volcanol 3. doi: 10.1186/2191504037
 Degruyter W, Bonadonna C (2013) Impact of wind on the condition for column collapse of volcanic plumes. Earth Planet Sci Lett 377378:218–226. doi: 10.1016/j.epsl.2013.06.041 CrossRefGoogle Scholar
 Doms G, Schättler U. (2002) A Description of the Nonhydrostatic Regional Model LM, Part I: Dynamics and Numerics. Deutscher WetterdienstGoogle Scholar
 Dubosclard G, Donnadieu F, Allard P, Cordesses R, Hervier C, Coltelli M, Privitera E, Kornprobst J (2004) Doppler radar sounding of volcanic eruption dynamics at Mount Etna. Bull Volcanol 66:443–456. doi: 10.1007/s0044500303248 CrossRefGoogle Scholar
 Engwell SL, Aspinall WP, Sparks RSJ (2015) An objective method for the production of isopach maps and implications for the estimation of tephra deposit volumes and their uncertainties. Bull Volcanol 77(7). doi: 10.1007/s004450150942y
 Engwell SL, Sparks RSJ, Aspinall WP (2013) Quantifying uncertainties in the measurement of tephra fall thickness. J of Applied Volcanology. doi: 10.1186/2191504025
 Eychenne J, Le Pennec JL (2012) Sigmoidal particle density distribution in a subplinian scoria fall deposit. Bull Volcanol 74(10):2243–2249. doi: 10.1007/s0044501206714 CrossRefGoogle Scholar
 Fagents SA, Gregg TKP, Lopes RMC (2013) Modeling Volcanic Processes the Physics and Mathematics of Volcanism. Cambridge University Press, CambridgeCrossRefGoogle Scholar
 Fierstein J, Nathenson M (1992) Another look at the calculation of fallout tephra volumes. Bull Volcanol 54:156–167. doi: 10.1007/BF00278005 10.1007/BF00278005 CrossRefGoogle Scholar
 Fisher RV (1964) Maximum size, median diameter and sorting of tephra. J Geophys Res 69:341–355. doi: 10.1029/JZ069i002p00341 10.1029/JZ069i002p00341 CrossRefGoogle Scholar
 Folch A (2012) A review of tephra transport and dispersal models: evolution, current status and future perspectives. J Volcanol Geotherm Res 235236:96–115. doi: 10.1016/j.jvolgeores.2012.05.020 CrossRefGoogle Scholar
 Fontijn K, Ernst GGJ, Bonadonna C, Elburg MA, Mbede E, Jacobs P (2011) The ∼4ka Rungwe Pumice (SouthWestern Tanzania): a windstill Plinian eruption. Bull Volcanol 73 (9):1353–1368. doi: 10.1007/s0044501104868 CrossRefGoogle Scholar
 Gasteiger J, Groß S., Freudenthaler V, Wiegner M (2011) Volcanic ash from Iceland over Munich: mass concentration retrieved from groundbased remote sensing measurements. Atm Chem and Phys 11(5):2209–2223. doi: 10.5194/acp1122092011 CrossRefGoogle Scholar
 Girault F, Carazzo G, Tait S, Ferrucci F, Kaminski E (2014) The effect of total grainsize distribution on the dynamics of turbulent volcanic plumes. Earth Planet Sci Lett 394:124–134. doi: 10.1016/j.epsl.2014.03.021 CrossRefGoogle Scholar
 Grainger RG, Peters DM, Thomas GE, Smith AJA, Siddans R, Carboni E, Dudhia A (2013) Measuring volcanic plume and ash properties from space. Geol Soc Lond Spec Publ 380(1):293–320. doi: 10.1144/SP380.7 CrossRefGoogle Scholar
 Gudmundsson MT, Thordarson T, Hoskuldsson A, Larsen G, jornsson H, Prata FJ, Oddsson B, Magnusson E, Hognadottir T, Petersen N, Hayward CL, Stevenson JA, Jónsdóttir I (2012) Ash generation and distribution from the AprilMay 2010 eruption of Eyjafjallajökull, Iceland Scientific Report, 2, 572. doi: 10.1038/srep00572
 Hartmann WK (1969) Terrestrial, lunar, and interplanetary rock fragmentation. Icarus 10(2):201–213CrossRefGoogle Scholar
 Johnson JB, Aster RC, Kyle PR (2004) Volcanic eruptions observed with infrasound. Geophys Res Lett 31. doi: 10.1029/2004GL020020 10.1029/2004GL020020
 Johnston EN, Phillips JC, Bonadonna C, Watson IM (2012) Reconstructing the tephra dispersal pattern from the Bronze Age eruption of Santorini using an advectiondiffusion model. Bull Volcanol 74(6):1485–1507. doi: 10.1007/s004450120609x CrossRefGoogle Scholar
 Kaminski E, Jaupart C (1998) The size distribution of pyroclasts and the fragmentation sequence in explosive volcanic eruptions. J Geophys Res: Solid Earth 103(B12):29759–29779. doi: 10.1029/98JB02795 CrossRefGoogle Scholar
 Kawabata E, Bebbington MS, Cronin SJ, Wang T (2013) Modeling thickness variability in tephra deposition. Bull Volcanol 75(8):1–14. doi: 10.1007/s004450130738x CrossRefGoogle Scholar
 Kawabata E, Cronin SJ, Bebbington MS, Moufti MRH, ElMasry N, Wang T (2015) Identifying multiple eruption phases from a compound tephra blanket: an example of the ad1256 almadinah eruption, saudi arabia. Bull Volcanol 77(1):1–13. doi: 10.1007/s004450140890y CrossRefGoogle Scholar
 Klawonn M, Frazer LN, Wolfe CJ, Houghton BF, Rosenberg MD (2014a) Constraining particle sizedependent plume sedimentation from the 17 June 1996 eruption of Ruapehu Volcano, New Zealand, using geophysical inversions. J Geophys Res: Solid Earth 119(3):1749–1763. doi: 10.1002/2013JB010387
 Klawonn M, Houghton BF, Swanson DA, Fagents SA, Wessel P, Wolfe CJ (2014b) From field data to volumes: Constraining uncertainties in pyroclastic eruption parameters. Bull Volcanol 76(7):1–16. doi: 10.1007/s0044501408391
 Klawonn M, Wolfe CJ, Frazer LN, Houghton BF (2012) Novel inversion approach to constrain plume sedimentation form the tephra deposit data: Application to the 17 June 1996 eruption of Ruapehu volcano, New Zealand. J Geophys Res 117. doi: 10.1029/2011JB008767
 Kueppers U, Perugini D, Dingwell DB (2006) Explosive energy during volcanic eruptions from fractal analysis of pyroclasts. Earth Planet Sci Lett 248(34):800–807 . doi: 10.1016/j.epsl.2006.06.033 CrossRefGoogle Scholar
 Kylling A, Kahnert M, Lindqvist H, Nousiainen T (2014) Volcanic ash infrared signature: porous nonspherical ash particle shapes compared to homogeneous spherical ash particles. Atmos Meas Tech 7:919–929. doi: 10.5194/amt7919201 CrossRefGoogle Scholar
 Magill C, Mannen K, Connor LJ, Bonadonna C, Connor CB (2015) Simulating a multiphase tephra fall event: inversion modelling for the 1707 Hoei eruption of Mount Fuji, Japan. Bull Volcanol 77(9):1–18. doi: 10.1007/s0044501509672 CrossRefGoogle Scholar
 Mastin LG (2014) Testing the accuracy of a 1D volcanic plume model in estimating mass eruption rate: VOLCANIC PLUME MODELS ERUPTION RATE. J Geophys ResAtmos 119(5):2474–2495. doi: 10.1002/2013JD020604 CrossRefGoogle Scholar
 McKay MD, Beckman RJ, Conover WJ (1979) Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code. Technometrics; (United States) 21(2):239–245. doi: 10.1080/00401706.1979.10489755 Google Scholar
 de’ Michieli Vitturi M, Neri A, Barsotti S (2015) Plumemom 1.0: a new integral model of volcanic plumes based on the method of moments. Geosci Model Dev 8(8):2447–2463. doi: 10.5194/gmd824472015 CrossRefGoogle Scholar
 Morton BR (1959) Forced plumes. J Fluid Mech 5(151):16. doi: 10.1017/S002211205900012X Google Scholar
 Murrow PJ, Rose WI, Self S (1980) Determination of the total grain size distribution in a Vulcanian eruption column and its implications to stratospheric aerosol perturbation. Geophys Res Lett 7:893–896. doi: 10.1029/GL007i011p00893 CrossRefGoogle Scholar
 Pardini F, Spanu A, de’ Michieli Vitturi M, Salvetti MV, Neri A (2016) Grain size distribution uncertainty quantification in volcanic ash dispersal and deposition from weak plumes. J Geophys Res: Solid Earth 121(2):538–557. doi: 10.1002/2015JB012536 CrossRefGoogle Scholar
 Perugini D, Kueppers U (2012) Fractal analysis of experimentally generated pyroclasts: A tool for volcanic hazard assessment. Acta Geophysica 60(3):682–698. doi: 10.2478/s1160001200197 CrossRefGoogle Scholar
 Pfeiffer T, Costa A, Macedonio G (2005) A model for the numerical simulation of tephra fall deposits. J Volcanol Geotherm Res 140. doi: 10.1016/j.jvolgeores.2004.09.001
 Pyle DM (1989) The thickness, volume and grainsize of tephra fall deposits. Bull Volcanol 51:1–15. doi: 10.1007/BF01086757 CrossRefGoogle Scholar
 Rose WI, Bluth GJS, Ernst GGJ (2000) Integrating retrievals of volcanic cloud characteristics from satellite remote sensors: a summary, vol 385. Phil. Trans. R. Soc. Lond. A. doi: 10.1098/rsta.2000.0605
 Rose WI, Bonis S, Stoiber RE, Keller M, Bickford T (1973) Studies of volcanic ash from two recent Central American eruptions. Bull Volcanol 37(3):338–364. doi: 10.1007/BF02597633 CrossRefGoogle Scholar
 Rose WI, Kostinski A, Kelley L (1995) Realtime CBand and radar observations of 1992 eruption clouds form Crater Peak, Mount Spurr, Alaska. US Geol Surv Bull 2139:19–26Google Scholar
 Sambridge M, Braun J, McQueen H (1995) Geophysical parameterization and interpolation of irregular data using natural neighbours. Geophys J Int 122:837–857. doi: 10.1111/j.1365246X.1995.tb06841.x CrossRefGoogle Scholar
 Scire J, Robe F, Fernau M, Yamartino R (2000) A users guide for the CALMET Meteorological Model. Earth Tech. http://www.asg.src.com/calpuff/download/download.htm
 Scollo S, Del Carlo P, Coltelli M (2007) Tephra fallout of 2001 Etna flank eruption: Analysis of the deposit and plume dispersion. J Volcanol Geotherm Res 160(1):147–164. doi: 10.1016/j.jvolgeores.2006.09.007 10.1016/j.jvolgeores.2006.09.007 CrossRefGoogle Scholar
 Scollo S, Tarantola S, Bonadonna C, Coltelli M, Saltelli A (2008) Sensitivity analysis and uncertainty estimation for tephra dispersal models. J Geophys Res 113. doi: 10.1029/2006JB004864
 Sibson R (1981) A Brief Description of Natural Neighbor Interpolation, vol 2. Wiley, ChichesterGoogle Scholar
 Spanu A, de’ Michieli Vitturi M, Barsotti S (2015) Reconstructing eruptive source parameters from tephra deposit: a numerical approach for mediumsized explosive eruptions. arXiv:1509.00386
 Sparks RSJ, Bursik MI, Carey SN, Gilbert JS, Glaze LS, Siggurdsson H, Woods AW (1997) Volcanic Plumes. Wiley, New York. http://eprints.lancs.ac.uk/id/eprint/53491 Google Scholar
 Spinetti C, Barsotti S, Neri A, Buongiorno MF, Doumaz F, Nannipieri L (2013) Investigation of the complex dynamics and structure of the 2010 Eyjafjallajökull volcanic ash cloud using multispectral images and numerical simulations. J Geophys Res: Atmospheres 118:4729–4747. doi: 10.1002/jgrd.50328 CrossRefGoogle Scholar
 Stevenson JA, Millington SC, Beckett FM, Swindles GT, Thordarson T (2015) Big grains go far: understanding the discrepancy between tephrochronology and satellite infrared measurements of volcanic ash. Atmos Meas Tech 8(5):2069–2091. doi: 10.5194/amt820692015 CrossRefGoogle Scholar
 Sulpizio R (2005) Three empirical methods for the calculation of distal volume of tephrafall deposits. J Volcanol Geotherm Res 145:3–4. doi: 10.1016/j.jvolgeores.2005.03.001 CrossRefGoogle Scholar
 Sulpizio R, Folch A, Costa A, Scaini C, Dellino P (2012) Hazard assessment of farrange volcanic ash dispersal from a violent Strombolian eruption at SommaVesuvius volcano, Naples, Italy: implications on civil aviation. Bull Volcanol 74(9):2205–2218. doi: 10.1007/s0044501206563 CrossRefGoogle Scholar
 Taddeucci J, Scarlato P, Montanaro C, Cimarelli C, Del Bello E, Freda CD (2011) Aggregationdominated ash settling from the Eyjafjallajökull volcanic cloud illuminated by field and laboratory highspeed imaging. Geology 39:891–894. doi: 10.1130/G32016.1 CrossRefGoogle Scholar
 Textor C, Graf HF, Herzog M, Oberhuber JM, Rose WI, Ernst GGJ (2006) Volcanic particle aggregation in explosive eruption columns. Part I: Parameterization of the microphysics of hydrometeors and ash. J Volcanol Geotherm Res 150:359–377. doi: 10.1016/j.jvolgeores.2005.09.007 CrossRefGoogle Scholar
 Textor C, Graf HF, Longo A, Neri A, Esposti Ongaro T, Papale P, Timmreck C, Ernst GJ (2005) Numerical simulation of explosive volcanic eruptions from the conduit flow to global atmospheric scales. Annals Geophys 48(4):5. http://hdl.handle.net/2122/942 http://hdl.handle.net/2122/942 Google Scholar
 Tirabassi T, Tiesi A, Buske D, Vilhena MT, Moreira DM (2009) Some characteristics of a plume from a point source based on analytical solution of the twodimensional advectiondiffusion equation. Atmos Environ 43 (13):2221–2227. doi: 10.1016/j.atmosenv.2009.01.020 10.1016/j.atmosenv.2009.01.020 CrossRefGoogle Scholar
 Turcotte DL (1986) Fractals and fragmentation. J Geophys Res 91(B2):1921–1926CrossRefGoogle Scholar
 Vernier JP, Fairlie TD, Murray JJ, Tupper A, Trepte C, Winker D, Pelon J, Garnier A, Jumelet J, Pavolonis M, Omar AH, Powell KA (2013) An advanced system to monitor the 3d structure of diffuse volcanic ash clouds. J Appl Meteorol Climatol 52(9):2125–2138. doi: 10.1175/JAMCD120279.1 CrossRefGoogle Scholar
 Volentik ACM, Bonadonna C, Connor CB, Connor LJ, Rosi M (2010) Modeling tephra dispersal in absence of wind: insights from the climactic phase of the 2450 BP Plinian eruption of Pululagua volcano (Ecuador). J Volcanol Geotherm Res 193(12):117–136. doi: 10.1016/j.jvolgeores.2010.03.011 CrossRefGoogle Scholar
 Voronoi G (1907) Nouvelles applications des paramètres continus à la théorie des formes quadratiques. J reine angew Math 133:97–178Google Scholar
 Walker GPL (1973) Explosive volcanic eruptions—a new classification scheme. Geol Rundsch 62:431–446. doi: 10.1007/BF01840108 CrossRefGoogle Scholar
 Watson DF, Phillip G (1987) Neighborhoodbased interpolation. Geobyte 2(2):12–16Google Scholar
 Wen S, Rose WI (1994) Retrieval of sizes and total masses of particles in volcanic clouds using AVHRR bands 4 and 5. J Geophys Res: Atmospheres 99(D3):5421–5431. doi: 10.1029/93JD03340 CrossRefGoogle Scholar
 Wilson L, Huang TC (1979) The influence of shape on the atmospheric settling velocity of volcanic ash particles. Earth Planet Sci Lett 44:311–324. doi: 10.1016/0012821X(79)901791 CrossRefGoogle Scholar
 Woodhouse MJ, Hogg AJ, Phillips JC, Sparks RSJ (2013) Interaction between volcanic plumes and wind during the 2010 Eyjafjallajökull eruption, Iceland. J Geophys Res 118:92–109. doi: 10.1029/2012JB009592 CrossRefGoogle Scholar
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