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Journal of Volcanology and Geothermal Research 193 (2010) 117–136

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Journal of Volcanology and Geothermal Research

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Modeling tephra dispersal in absence of wind: Insights from the climactic phase of the 2450 BP Plinian eruption of Pululagua volcano (Ecuador)

Alain C.M. Volentik a,⁎, Costanza Bonadonna b, Charles B. Connor a, Laura J. Connor a, Mauro Rosi c a Department of Geology, SCA 528, University of South Florida, 4202, E. Fowler Ave., Tampa FL 33620, USA b Section des Sciences de la Terre et de l'environnement, Université de Genève, Rue des Maraîchers 13, 1205 Genève, Switzerland c Dipartimento di Scienze della Terra, Università di Pisa, Via S. Maria 53, 56126 Pisa, Italy article info abstract

Article history: The determination of eruptive parameters is crucial in volcanology, not only to document past eruptions, but Received 18 July 2009 also for tephra fallout hazard assessments. In most tephra fallout studies, eruptive parameters have been Accepted 24 March 2010 determined either by empirical techniques or analytical models, but the uncertainty of such parameters is Available online 1 April 2010 usually not well described. We have applied both empirical and analytical models to characterize the climactic phase of the 2450 BP Plinian eruption of Pululagua (BF2 layer) and explore the variations in the Keywords: total erupted mass, column height and total grain size distribution. Both approaches yield comparable results Pululagua in the total mass of tephra erupted (4.5±1.5×1011 kg), while they show some discrepancies for the Plinian eruptions – – tephra fall deposits determination of the column height (36 20 km from empirical techniques and 30 20 km from analytical grain size analysis techniques). The total grain size distribution of the BF2 layers varies with the different techniques used for modeling the calculation and significantly affects the outputs of analytical models. Furthermore, the determination of inversion techniques the total grain size distribution depends strongly on the number and spatial distribution of the sample location. Inverting tephra fallout deposits on the total accumulation (or thickness) gives a good constraint on the total mass erupted but not on the column height. However, inverting on individual grain size classes better constrains the possible range of column heights (but cannot resolve particle release height). Results from the inversion on individual grain size classes show that large diffusion coefficients are necessary to model the BF2 layer and might be required to model proximal tephra deposits in order not to overestimate the total erupted mass. Additionally, we used a statistical method (smoothed bootstrap approach) to quantify the uncertainty in eruptive parameters such as column height and total erupted mass. Our uncertainty analysis yields a mean total erupted mass of 4.5±0.3×1011 kg and a mean column height of 30±3 km. Results from the uncertainty analysis compare well with results from other approaches. Finally, although the climactic phase of the 2450 BP Plinian eruption of Pululagua occurred in relatively calm atmospheric conditions, our results show that the dispersion of the BF2 layer was influenced by slight northeasterly wind conditions. © 2010 Elsevier B.V. All rights reserved.

1. Introduction in the field and can be used to infer eruption parameters such as column height, total mass of tephra erupted, total grain size Tephra dispersal models are important in volcanology, not only to distribution (TGSD), and wind direction and speed (Carey and Sparks, constrain physical processes leading to tephra transport and sedi- 1986; Pyle, 1989; Fierstein and Nathenson, 1992; Bonadonna and mentation following an explosive eruption at a given volcano (e.g. Houghton, 2005). These models are mainly based on curve-fitting Armienti et al., 1988; Bursik et al., 1992; Bonadonna and Phillips 2003; techniques of field observations and on simplified description of Costa et al., 2006), but also to assess tephra hazards that potentially tephra dispersal. Wind advection adds a level of complexity in the threaten populated areas (e.g. Connor et al., 2001; Bonadonna et al., study of tephra dispersion. Only three eruptions are known to have 2005a; Houghton et al., 2006; Macedonio et al., 2008) and critical occurred in approximately still atmospheric conditions, resulting in a facilities (e.g. Volentik et al., 2009). For past and unwitnessed circular-shaped dispersion of tephra around the vent: the ∼5000 BP eruptions, the thickness and/or accumulation of tephra is measured Fogo A eruption (Walker and Croasdale, 1971; Bursik et al., 1992), the 1210 BP eruption of Cotopaxi (layer 9 of Barberi et al., 1995) and the 2450 BP Plinian eruption of Pululagua, Ecuador (Papale and Rosi, 1993). We have chosen the latter to investigate sedimentation from ⁎ Corresponding author. New address: Department of Geology & Geophysics, School Plinian plumes and evaluate empirical and analytical models for the of Ocean & Earth Sciences & Technology, 1680 East-West Road, Honolulu, Hawaii 96822, USA. determination of crucial eruptive parameters (e.g. Pyle 1989, Bona- E-mail addresses: [email protected], [email protected] (A.C.M. Volentik). donna and Houghton 2005, Connor and Connor 2006) and analytical

0377-0273/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jvolgeores.2010.03.011 Author's personal copy

118 A.C.M. Volentik et al. / Journal of Volcanology and Geothermal Research 193 (2010) 117–136 models for the description of particle transport and deposition (e.g. White Ash deposit (WA), a thin ash bed that tops the Plinian sequence Bonadonna et al. 2005a; Connor et al., 2008). (Papale and Rosi, 1993). This ash bed (also defined as co-plinian ash There is uncertainty in the modeling of tephra deposits and the according to the general description of Fierstein and Hildreth (1992)) inference of eruptive parameters. Such uncertainties are usually not is thought to have originated from the slow settling of fines (b1 mm) well described in the literature. Therefore, we describe a Monte Carlo after cessation of the sustained Plinian column. Had a moderate wind approach, combined with a smoothed bootstrap method, to quantify field been present at the time of the eruption of the BF layer, the fine the uncertainty in the determination of total erupted mass and White Ash particles would have been advected downwind and would column height from inversion of field data. We investigate the have sedimented away from the vent. Papale and Rosi (1993) calc- variability of the results (i.e. column height and total erupted mass) in ulated a maximum column height of 36 km (based on the 3.2, 1.6 terms of the total accumulation observed and for each grain size. and 0.8 cm lithic isopleths) and 21 km (based on the 6.4 cm lithic isopleth), using the model of Carey and Sparks (1986) (thereafter 2. Geological setting and background referred as CS). The model of Wilson and Walker (1987) applied to the 4.9 and 6.4 cm lithic isopleths yielded a column height of 28 km (Papale Pululagua Volcano is part of the active Western Andean Volcanic and Rosi, 1993). Magma discharge rate was estimated to be 2×108 kg/s, Front of Ecuador (Hall et al., 2008) and is located 15 km north of Quito following both Sparks' (1986) and Wilson and Walker's (1987) models. (Fig. 1). Papale and Rosi (1993), Pallini (1996) and Andrade and Pallini (1996) revisited the BF deposit and subdivided it into additional Molina (2006) described the volcanic stratigraphy and evolution of layers compared to the study of Papale and Rosi (1993) and proposed a Pululagua. Pululagua is a 19 km2 dacitic caldera and is surrounded by volume for the BF of about 0.58 km3,basedonPyle's (1989) method. The ten older lava domes. The most recent volcanic activity at Pululagua eruption column height was estimated using CS and Pyle (1989) models started with the formation of old dacitic lava domes with their and yielded heights of 36 km and 28 km respectively. Thus, Pallini (1996) associated block-and-ash flow deposits, which are capped by an proposed a probable column height of 32 km, resulting from the average ubiquitous, well-developed palaeosoil. The 2450 BP Plinian sequence of these two estimates, and a magma discharge rate of 2×108 kg/s (based overlies this palaeosoil conformably. The explosive activity leading to on Sparks, 1986)and3×108 kg/s (based on Wilson and Walker, 1987). the formation of the irregularly shaped caldera occurred as a series of This whole BF tephra sequence is overlain by numerous pyroclastic volcanic eruptions during which ∼5–6km3 (DRE) of hornblende- density currents (PDCs) in the near-vent region intercalated within bearing dacitic magma was erupted. Papale and Rosi (1993) estimated other minor tephra fallout deposits (Papale and Rosi, 1993; Andrade that the main basal pumice fall (BF) deposit (Fig. 2) covers an area of and Molina, 2006). The latter tephra deposits show a global westward more than 2.2×104 km2 and has a volume of about 1.1 km3 (0.34 km3 dispersion (Papale and Rosi, 1993) compared to the BF sequence. DRE). The general stratigraphy of Pululagua deposits, as well as the circular isopach and isopleth maps, were presented by Papale and Rosi 3. New stratigraphy (1993) for the whole basal fallout deposit. The circular pattern of the isopach and isopleth maps indicates emplacement in relatively wind- We use Pallini's (1996) work to define a more detailed strati- free conditions, which is confirmed by an ubiquitous, normally graded graphic subdivision for the BF eruption (Fig. 2a–c): (i) a basal grey ash

Fig. 1. Digital elevation model for the region of interest around Pululagua, with the three axes used in this study: 1, the ESE axis; 2, the SE axis and 3, the SW axis. Numbers refer to sample locations. Cities abbreviations are as follows, A: Atahualpa, C: Calacali, G: Guayllabamba, No: Nono, P: Perucho, SA: San Antonio, SJM: San Jose de Minas. The dark grey area represents the extent of Quito. Location of Pululagua volcano (P, black triangle) within Ecuador is shown in inset map (Q stands for Quito). Author's personal copy

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Fig. 2. Picture and detailed stratigraphy of the outcrop for three locations at various distances from the vent. (a) Proximal: PL40 located at about 4.5 km southeast from the inferred vent. (b) Medial: PL19 located at about 13 km east-southeast from the inferred vent. (c) Distal: PL24 located at about 21 km southeast from the inferred vent. We defined the inferred vent as being in the center of the caldera, in the current position of the central post-caldera domes. Note the White Ash sealing the top of the BF deposits. Author's personal copy

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(BGA), resulting from several discrete phreatomagmatic eruptions In proximal locations, the BF2 layer presents one or two thin characterizing the onset of explosive activity at Pululagua (Papale horizons of finer ash (Fig. 2a), that could represent a small pause in and Rosi, 1993); (ii) two early Plinian fallout deposits (BF1a and b), the eruption, resulting in the slow settling of fine particles, or small overlain by (iii) the main (climactic phase) fallout layer of the Plinian dilute PDCs, resulting from local ephemeral instabilities in the erup- eruption, the BF2 layer, that is the focus of this study; (iv) the BF3 tion column. These ash layers are not ubiquitous and thin quickly away Plinian fallout deposit, and (v) the White Ash (WA) fallout deposit. from the vent. Furthermore, in the same proximal areas, small density The BF2, BF3 and WA episodes are thought to have occurred in still current deposits (very small volume pyroclastic flows and/or surges) atmospheric conditions, whereas the BGA and BF1a and b events display are interbedded between the BF2 and BF3 layers (Fig. 2a), suggesting a NW dispersal axis (Pallini, 1996). These six tephra deposits are con- either another pause in the sustained phase of the Plinian eruption formable and are thought to have been deposited without any between the BF2 and BF3 layers or, again, some small instabilities of significant break in the explosive eruption (Papale and Rosi, 1993). the outer part of the volcanic plume. However, the sharp transition in Distinguishing features of these different fall layers are described below. grain size between the BF2 and BF3 layers in all locations argues in The BGA layer marks the beginning of the explosive eruption of favor of a pause in the Plinian phase of the eruption. Pululagua (Fig. 2a) (Papale and Rosi, 1993). The BGA is a fine-grained In the present study, we focus on the BF2 layer, which represents ash deposit, composed of multiple thin layers alternating coarse and the climactic phase of the 2450 BP Plinian eruption of Pululagua. We fine deposits that are the result of several phreatomagmatic pulses. measured thickness of the BF2 layers at 73 locations. BF2 thickness These are interpreted to characterize the vent-opening phase of the varies from about 40 cm in the most proximal location (i.e. about Plinian eruption. The thickness of BGA varies from a few millimeters 4.5 km from the inferred vent in the present caldera) to less than 1 cm up to almost 10 cm in the most proximal sections. in the most distal location we sampled (about 35 km from the inferred The Basal Fall (BF) deposit lies conformably on the BGA layer vent). Samples were collected at 53 locations for grain size analysis (Fig. 2), without any sign of break in the sedimentation process. The and for each location, we calculated the median clast diameter (Mdϕ) BF1a and BF1b layers are composed of white angular pumices and up and the graphical standard deviation (or sorting, σϕ) from Inman to 10% lithics by volume (Fig. 2a and b). The BF1a and b layers are (1952) (Fig. 3). separated by dilute PDCs in proximal areas (Fig. 2a) and by a thin bed of fine lapilli further away from the vent. Their axes of dispersion are 4. Empirical determination of eruptive parameters toward the NNW for BF1a and NW for BF1b (Pallini, 1996). The BF2 layer represents the climactic phase of the BF Plinian 4.1. Sample grain size and total grain size distribution eruption of Pululagua. This layer is the thickest and covers the widest area of all of the BF layers (Fig. 2). The transition between BF1b and 4.1.1. Grain size BF2 is marked by a sharp increase in grain size. Pumices are white in Samples were dry-sieved down to 4ϕ (63 μm) at 1ϕ intervals. color, angular and finely vesiculated. Accidental basement lithics are Proximal deposits were sieved in the field down to the −3ϕ mesh and usually highly oxidized and comprise up to 20% in volume of the the fraction finer than −3ϕ was first quartered to reduce the total deposit. volume of the remainder deposit, and then carried back to the lab for The transition between the BF2 and BF3 layers is represented by a further sieving. The ash fraction finer than 4ϕ (63 μm) was analyzed series of dilute PDCs in proximal locations (Fig. 2a). A pronounced for grain size characteristics down to 12ϕ with the Malvern decrease in grain size marks this transition in more distal sections. The PharmaVision 830 (PVS) automated optical device. Results from the nature of the pumice clasts in BF3 are the same as those of BF2 and PVS include different morphological parameters of the particles that BF1: white, angular and finely vesiculated pumices. Lithic fragments may be related to settling velocity, such as the maximum length, the are still present, but are less abundant than in the BF2 (5–10% in mean diameter, or the width of the particles. Therefore, the grain size volume). In the upper third part of the deposit, the BF3 displays a distribution can be recalculated using each morphological parameter. distinct increase in size of pumice clasts, which can be attributed to an We decided to constrain the grain size distribution of the fine ash by increase in the eruption intensity resulting in a higher column height. using the particle width parameter, as it gives the best result com- Unlike previous contacts, the contact between the BF3 layer and pared to hand-sieved data. The median grain size (Mdϕ) and sorting the WA layer is not sharp, but rather shows a gradation in grain size (σϕ) of the deposit vary from −4ϕ to 2.25ϕ and from 1.15ϕ to 3ϕ from the BF3 to the WA layers (Fig. 2a–c). We used the appearance of respectively, with the majority of the samples having a Mdϕ b0 and a the dominant white color from the fine white ash and the σϕ N2.0ϕ. Therefore, the BF2 layer is coarse grained and poorly sorted disappearance of large clasts (≥1 mm) to subdivide the BF3 and the (Cas and Wright, 1987), which can be attributed to the lack of sig- WA layers. The WA layer is normally graded and ubiquitously covers nificant wind during the eruption. As expected (Sparks et al., 1992) the the underlying lapilli fallout deposits of the BF eruption. Therefore, the Mdϕ and the σϕ decrease with distance from the vent (Fig. 3a and b), WA marks the end of the first Plinian phase of Pululagua. Where meaning that the overall grain size decreases away from the volcanic pristine, the WA thickness varies from 10 cm close to the vent to 6 cm vent and the sorting of the deposit improves. in distal locations. The area covered by the WA layer is greater than Fig. 4 shows the isomass maps of the individual grain size classes the area covered by BF layers, as the WA has been found as far as from −5ϕ down to 2ϕ. Sample locations showing grain size classes 63 km west of the caldera (Pallini, 1996) and even at the coastline larger than −5ϕ are not sufficient to trace isomass maps, while the 3ϕ ∼200 km west from the caldera (P. Mothes, personal communica- and 4ϕ grain size classes display such a low accumulation that tion). The presence of the WA on top of the other BF layers not only contouring the field data is not an objective task. Isomass maps for guarantees the integrity of the underlying units, but also gives a hint clast fractions −5ϕ to −2ϕ display a dispersion component toward about the atmospheric conditions at the time of the first Plinian the south and the west. In comparison, particles from −1ϕ to 1ϕ eruption of Pululagua. The occurrence of such a uniform fine-grained clearly show dispersion toward the west. The 2ϕ grain size class layer requires a nearly still atmospheric column, as fine particles have shows a more circular dispersal pattern. Therefore, when the deposit such low terminal velocities that they would have formed an asym- is broken down into individual grain size classes, the circular pattern metric deposit in windy conditions. observed in the isopach map (Fig. 5a) is not evident. The circularity These Plinian fallout deposits from Pululagua are overlain by a observed in the deposit thickness is the result of the differential sequence of PDCs, interlayered with additional minor Plinian fallout accumulation pattern of the different grain size classes. deposits from Pululagua (Papale and Rosi, 1993; Andrade and Molina, Fig. 5b shows the distribution of the Mdϕ around Pululagua, and 2006; Petriello, 2007). iso-Mdϕ lines tend to be roughly circular around the vent, although Author's personal copy

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pretation that at least part of the BF (i.e. the BF2, BF3 and WA layers) eruption happened in still atmospheric conditions, because if a signif- icant wind field had been present at the time of the eruption, the iso- Mdϕ lines would have been more strongly distorted by the wind field. In many tephra fall studies (e.g. Pyle, 1989; Sparks et al., 1992; Papale and Rosi, 1993), Mdϕ is correlated with distance from the vent along the dispersal axis, but might not show a clear correlation when all the sample locations are plotted versus distance from the vent (Rose et al., 2008). We investigate such relationships and focus our analysis along three axes around the volcano (ESE axis, SE axis and SW axis, see Fig. 1). As for the BF2, Fig. 3a and b shows that both the Mdϕ and σϕ decrease with distance from the vent, and follows a power-law thinning trend. However, the rate at which Mdϕ and σϕ decrease with distance from the vent is not the same along the different axes investigated. The decrease in Mdϕ and σϕ is faster along the ESE axis (axis 1 in red) than along the SE axis (axis 2 in blue) and the SW axis (axis 3 in green). The scatter of the Mdϕ vs. distance from the vent might be the result of non-uniform sedimentation due to atmospheric diffusion or wind interaction. The difference in the horizontal position of the plume corner along the different axes (see discussion below) might be another explanation for the scatter of the Mdϕ vs. distance from the vent. While Mdϕ and σϕ do not correlate in many fallout deposits (Walker, 1971; Houghton et al., 2004, Rose et al., 2008), the BF2 layer shows a relatively good linear correlation between these two parameters (R2 =0.79–0.83) (Fig. 3c). The absence of wind may be a factor in improving the correlation, as the wind field tends to improve the sorting of the deposit regardless of the particle size.

4.1.2. Total grain size distribution A crucial eruptive parameter for the modeling of tephra fall deposits is given by the total grain size distribution (TGSD). The TGSD is an important parameter used to (i) constrain tephra sedimentation models (e.g. Bursik et al., 1992; Bonadonna and Phillips, 2003), (ii) infer fragmentation and eruption processes (e.g. Kaminski and Jaupart, 1998), (iii) assess tephra hazards for population vulnerability (e.g. Connor et al., 2001; Bonadonna et al., 2005a), critical facilities vulnerability (e.g. Volentik et al., 2009) and aviation safety (e.g. Durant et al., 2009; Mastin et al., 2009; Rose and Durant, 2009), and (iv) evaluate human health hazards due to the settling of fine particles in populated areas (e.g. Horwell and Baxter, 2006). Several ways to estimate the TGSD have been proposed (Walker, 1981; Carey and Sigurdsson, 1982; Bonadonna and Houghton, 2005; Rose et al., 2008). The most recent one, the Voronoi tessellation, a spatial analysis method developed by Bonadonna and Houghton (2005), can be defined as the partitioning of the plane (e.g. the tephra blanket) such that, for any set of distinct data points, the cell associated with a particular data point contains all spatial locations closer to that point than to any other. In this study, we have used the following ap- proaches: (i) simple unweighted average of grain size analysis from all the available locations (Technique A, Walker, 1981), (ii) mass- weighted average of grain size analysis from all the available locations (Technique B, Walker, 1981), (iii) isopach weighted (Technique C, Fig. 3. Grain size characteristics for the BF2 layer. (a) Plot of the median diameter (Mdϕ) Rose et al., 2008) and (iv) the Voronoi tessellation (Technique D, σ vs. distance from the vent. (b) Plot of the standard deviation ( ϕ) vs. distance from the Bonadonna and Houghton, 2005). We applied these four approaches vent. (c) Plot of σϕ vs. Mdϕ. Red dots are for data from axis 1 (ESE), blue dots are for data from axis 2 (SE), green dots are for data from axis 3 (SW), and black dots are for the rest to two sets of samples: (1) the current set of samples (set 1) available of the data set. Solid lines are regressions through the data from the different axes from our field study and (2) another set, in which we have added (color scheme same as before, see Fig. 1). The pyroclastic flow and fall fields are defined virtual sample locations (set 2) in areas where the deposit is now after Walker (1971). lacking, based on the assumption of circularity of the deposit and on the current samples collected in the field. The Voronoi tessellation approach requires the definition of a “zero accumulation limit”, which they show a dispersion toward the southwest, similar to the pattern is not available for past eruptions. We used a circular “zero accumu- displayed by the isopach map (Fig. 5a). The −2ϕ and −1ϕ contours lation line” with variable radius (35, 40, 50, 100 and 200 km, see Fig. 6 also display a distortion toward the southwest. The iso-Mdϕ lines are and Table 1) to investigate the sensitivity of the technique to the typically more sensitive to wind dispersal than isopachs, as noted by position of the “zero accumulation line”. Rose et al. (2008) for the 1974 Fuego (Guatemala) sub-Plinian tephra The results of the different techniques used to calculate the TGSD fall deposit. This characteristic is another factor leading to the inter- of the BF2 layer on the two datasets are presented in Fig. 6 and Table 1. Author's personal copy

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The Voronoi tessellation (technique D) clearly shows a greater mulation line”, needed to constrain and calculate the TGSD applying consistency in the Mdϕ values from the two different sets of data the Voronoi tessellation (Bonadonna and Houghton, 2005), as shown points used to calculate the TGSD compared to Techniques A and B. by the only slight variations in the grain size parameters from the Fig. 6g and h and Table 1 show that the Voronoi technique is more Voronoi technique using a circular “zero accumulation line” at 35, 40, sensitive to the distribution of sample locations than to the actual 50, 100 and 200 km from the inferred vent for the BF2 eruption. The position of the “zero accumulation line”. Techniques A, C and D seem main difference between the different “zero accumulation line” to show approximately the same pattern of TGSD (Fig. 6), while results lies in the amount of fine ash present in the deposit, an Technique B displays a coarser, wide bell-shaped distribution com- amount increasing with an increasing distance from the vent of the pared to the other techniques. Therefore, the Voronoi tessellation is “zero accumulation line”, therefore giving a higher weight to the fine the least sensitive technique to the number of data points available. portion of the deposit. Technique A, and to a lesser extent Technique B, is very sensitive to the distribution of the sample locations, as shown by the large 4.2. Erupted volume difference in the Mdϕ (−0.55 for data set 1 and 0 for data set 2 in technique A). Another important observation is that the Voronoi Statistical models are widely used to estimate eruption volumes technique is not that sensitive to the position of the “zero accu- from comparatively sparse data. The volume of tephra emitted during

Fig. 4. Isomass maps for each individual grain size class from −5ϕ to 4ϕ of the BF2 layer. See Fig. 1 for abbreviations. Individual location accumulations and contours are shown. Values are in kg m−2. Dashed lines where isomass contours are extrapolated. Author's personal copy

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Fig. 4 (continued). an explosive volcanic eruption can be inferred using curve-fitting We have also applied the model of Bonadonna and Houghton methods on a semi-logarithmic plot of thickness vs. the square root (2005) that consists of fitting field data using a power-law curve on a of the area enclosed by the isopach of a given thickness. Following semi-log plot of thickness vs. square root of the isopach areas. As for the model of Pyle (1989), an exponential trend line is fit through the exponential model, we have applied the power-law method to measurements of the BF2 deposit thicknesses plotted against the both the data sets, with and without the 1 and 2 cm isopach areas square- root of the area (Fig. 7). Volume is calculated using two (Fig. 7c and d, respectively). An important step in the application of different exponential curves. The first (Fig. 7a) does not take into the power-law method is the choice of the outer limit of integration, account the 1 and 2 cm isopach areas, because these areas are not i.e. the maximum distance from the vent reached by the deposit well-constrained by field observations (Fig. 5a). The total volume is (i.e. thickness=0). A small tephra layer from the 2450BP eruption about 0.3 km3 (R2 =0.99). The second exponential curve (Fig. 7b) of Pululagua has been identified along the Pacific coast of Ecuador takes into account the whole suite of field observations and thus (P. Monthes, personal communication) some 200–250 km away from includes the 1 cm and 2 cm isopach areas. In this case, the correlation Pululagua. Using 200 km or 250 km as the outer integration limit, the coefficient is lower (R2 =0.95), but the volume is also about 0.3 km3. total volume, excluding the 1 cm and 2 cm isopach areas, is 0.8– In both cases, these results indicate the BF2 layer was a VEI 4 eruption 1.0 km3, while it is reduced to 0.4–0.5 km3 when the 1 cm and 2 cm (Newhall and Self, 1982). isopach areas are included. These volumes estimates also indicate a Author's personal copy

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Fig. 5. (a) Isopach map for the BF2 layer of the 2450BP Plinian eruption of Pululagua Volcano. Individual location thicknesses and contours are shown. Values are in centimeters. Note the circular shape of the isopachs. Dashed lines where isopach contours are extrapolated. (b) Map showing the distribution of the Mdϕ values of the BF2 layer. Individual location Mdϕ and contours are shown. Values are in Phi unit. Note that the circular shape is less pronounced than for the isopach map, and that the distribution of Mdϕ is slightly toward the west. Dashed lines where iso-Mdϕ contours are extrapolated. See Fig. 1 for abbreviations.

VEI 4 eruption. Bonadonna and Costa (in press) have shown that dent diffusion and particle density, a stratified atmosphere, particle volume calculations for deposits characterized by a power-law diffusion time within the rising plume, and settling velocities that exponent b2 (i.e. widely spread deposits) are sensitive to the choice include particle Reynolds number variations along the particle fall of the outer integration limit. In fact, the volume of the BF2 layer (i.e. Bonadonna et al., 1998; Bonadonna and Phillips, 2003). Modeled varies between 0.5 km3 and 1.8 km3 for an outer integration limit particles are assumed to be spherical, vertical atmospheric diffusion between 100 and 500 km, respectively and a power-law exponent negligible and horizontal atmospheric diffusion uniform and isotropic. of 1.2 (field data without the 1 and 2 cm isopach areas). The volume However, by using only forward modeling of tephra dispersal, the of the BF2 layer varies between 0.3 km3 and 0.6 km3 for an outer highly dimensional space of possible eruption input parameters integration limit between 100 and 500 km, respectively and a power- cannot possibly be fully investigated due to the great number of initial law exponent of 1.7 (field data with the 1 and 2 cm isopach areas). input parameters. Therefore it is unlikely that the best set of eruption The lower volume estimate yielded by the exponential curve- parameters can be found using a forward modeling approach. By fitting method compared to that from the power-law fitting technique using an inversion technique, it is possible to find a set of eruption applied to an outer integration limit of 500 km is probably due to the parameters (especially column height, total mass of tephra erupted, absence of both the very proximal and the distal part of the BF2 total grain size distribution and wind condition) that best reproduce deposit in the geological record. In fact, the exponential model can the observed tephra accumulation on the ground at each sample underestimate the total erupted volume unless four segments can location. be identified in the semi-log plots vs. distance from the vent plots Connor and Connor (2006) proposed a technique to better under- (Bonadonna and Houghton 2005). stand eruption dynamics by inverting tephra fallout. This inversion In conclusion, we consider a total volume of tephra ejected during technique searches for the optimal set of eruptive parameters that the climactic phase of the eruption (layer BF2) of Pululagua of about best explain variation in the field data (Fig. 8 and Table 2) using the 0.5±0.15 km3 (from the application of the power-law method to the downhill simplex algorithm. The goal is to discover a set of eruptive complete field data set and considering the outer limit of integration parameters that minimizes the error between the measured and between 100 and 500 km from the vent). The bulk density of the calculated tephra accumulation at each field point. The Root Mean deposit (920±80 kg/m3) has been measured in the field and was Square Error (RMSE) represents a criterion of goodness-of-fit be- close to 1000 kg/m3, yielding a total mass of about 5±1.5×1011 kg. tween the calculated and observed tephra deposit, following: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5. Analytical determination of eruptive parameters N ðÞMc −Mo 2 RMSE = ∑ a a Mo We use the Tephra2 semi-analytical forward model (Bonadonna et a =1 a al., 2005a; Connor and Connor 2006; Connor et al., 2008)to investigate the dispersion and sedimentation of the BF2 layer. The where N is the number of field observations, Moa is the observed mass numerical simulation of tephra accumulation is based on an analytical per unit area at location a and Mca is the calculated mass per unit area solution to the advection–diffusion equation and calculates the total at location a. mass per unit area M (kg m−2) of tephra accumulated at a given As tephra deposits contain more information than deposit location on the ground with the coordinates (x,y), which is one of the thickness alone, namely the grain size distribution of the deposit, quantities of greatest interest in tephra sedimentation models and in we decided to run the inversion technique also on grain size data at tephra hazard assessments. The model allows for grainsize-depen- each location. Basically, we divided the total tephra accumulation for a Author's personal copy

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Fig. 6. Results of total grain size distributions (TGSD) calculations using different techniques. (a) Technique A with the two sets of data points; (b) Technique B with the two sets of data points; (c) Technique C; (d) Technique D with the first set of data points, with a zero accumulation line at 35, 40, 50 and 100 km; (e) Technique D with the second set of data points, with a zero accumulation line at 35, 40, 50 and 100 km; (f) From the inversion on grain size, including and excluding the −7ϕ size fraction; (g) Technique D with a zero accumulation line at 100 km for the two data sets; (h) Technique D with a zero accumulation line at 100 and 200 km with the first data set. given location into an accumulation of particles for each grain size 5.1. Erupted mass class (at 1ϕ interval) according to the grain size distribution obtained by sieving the tephra deposit at each location. We then inverted those Mass, and hence volume of the eruption, can be estimated from the data to find the best-fit eruption parameters that would reproduce inversion and compared with volume estimated based on the curve- best the observed accumulation by grain size on the ground. fitting methods described earlier. All non-linear inversion methods Author's personal copy

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Table 1 Compilation of grain size characteristics for the different technique of TGSD calculations. 1980 MSH stands for the 1980 eruption of Mount St. Helens. Data on 1980 MSH are from Durant et al. (2009).

Technique Data set MdϕσϕSkG Ash Fine ash (wt.%) (wt.%)

TechniqueA Set 1 −0.55 2.43 −0.2 57.35 1.92 Unweighted Set 2 0 2.5 −0.24 64.56 3.33 Technique B Mass- Set 1 −1.85 2.78 −0.05 40.05 0.99 Weighted Set 2 −2.15 2.95 −0.03 36.81 0.94 Technique C isopach- Set 1 −1.1 2.68 −0.16 49.33 1.68 weighted TechniqueD Set 1 −0.84 2.46 −0.14 52.53 1.22 Voronoi—35 km Set 2 −0.72 2.75 −0.18 53.71 1.96 Technique D Set 1 −0.71 2.41 −0.15 54.64 1.27 Voronoi—40 km Set 2 −0.6 2.79 −0.18 55.24 2.31 Technique D Set 1 −0.51 2.32 −0.16 57.96 1.37 Voronoi—50 km Set 2 −0.36 2.82 −0.21 57.89 2.92 Technique D Set 1 −0.01 2.09 −0.19 66.48 1.68 Voronoi—l00 km Set 2 0.84 2.78 −0.36 69.78 5.66 Technique D Set 1 0.25 1.92 −0.18 71.93 1.98 Voronoi—200 km Set 2 1.75 1.92 −0.37 84.57 9.06 Inversion on grain All data 0.82 2.31 −0.41 75.46 n/a Fig. 8. Comparison between the observed tephra accumulation on the ground and the Size data Without −7ϕ 0.95 1.93 −0.34 79.53 n/a calculated tephra accumulation from one of the best-fit inversion results (see dashed 1980 MSH 4.8 2.5 ∼0.21 ∼95 ~57 black circle in Fig. 9) on the BF2 thickness at each locality. The diagonal black line represents the optimal case, when the model equals the actual accumulation. can be sensitive to local minima. To avoid this issue, we plot a two- result of the inversion investigation of the column height-mass space dimensional space represented by the column height vs. total mass is presented in Fig. 9. (Fig. 9). We invert the measured BF2 tephra deposit by using different Our investigation of this two-dimensional space shows that there ranges of input parameters (Table 2). The total mass was incremented is a non-unique solution in terms of column height and total mass by 0.2 log of the mass (from 10.6 to 12.4 log of the mass), and the erupted for the BF2 layer from the inversion of total tephra accu- column height was incremented by 2 km, from 8 km to 40 km. The mulation on the ground at each location. Indeed, the column height

Fig. 7. Semi-logarithmic plots of log of thickness (cm) against the square root of the area enclosed by an isopach map contour for the BF2 tephra deposit. (a) Field data are fitted according to the exponential decay proposed by Pyle (1989), excluding the 1 and 2 cm isopachs. (b) Field data are fitted according to the exponential decay proposed by Pyle (1989), including the 1 and 2 cm isopachs. (c) Field data are fitted using the power-law technique proposed by Bonadonna and Houghton (2005), excluding the 1 and 2 cm isopachs. (d) Field data are fitted using the power-law technique proposed by Bonadonna and Houghton (2005), including the 1 and 2 cm isopachs. Author's personal copy

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Table 2 Example of input parameter ranges for the inversion and resulting output from the inversion. This solution is represented in Fig. 9 by the dashed black circle.

Modeled parameters Input Output Units

Minimum Maximum

Maximum column height 26,000 28,000 27,101 m Total mass ejected 1.5849×l011 2.5119×1011 2.50249×1011 kg Log (Total mass ejected) 11.2 11.4 11.4 Dimensionless Mean particle size (Mdϕ) −2.0 2.0 −0.2 ϕ Standard deviation of particle size (σϕ) 1.0 3.0 2.0 ϕ Diffusion coefficient 0.1 100,000 92066.1 m2s−1 vs. log(mass) space shows an area of possible eruption parameters 5.2. Column height that describe the BF2 tephra deposit equally well. However, our model shows that the total mass of the eruption is relatively well- Inverting tephra accumulation did not uniquely constrain the constrained between 2.5×1011 kg and 4.0×1011 kg, corresponding column height, as column heights ranging from 8 km to 40 km give to about 0.25–0.4 km3, assuming a bulk density of the deposit of solutions that equally reproduce the observed deposit (Fig. 9). In 1000 kg/m3. Conversely, the column height shows a wide range of contrast, the inversion on individual grain size classes proves possible solutions, from a height as low as 8 km and up to about especially useful for constraining the column height. By inverting on 40 km. Scollo et al. (2008) also shown that the erupted mass can be grain size, that is the mass per unit area of individual size classes, better calibrated than the plume height through the application of the ambiguity is removed that stems from uncertainty in grain size, and model Tephra2. hence particle fall velocity. By adding together all of the simulations by grain size, it is then A first set of results from the inversion on grain size is shown in possible to reconstruct the total mass of tephra erupted that deposited Fig. 10, where we compared the calculated accumulation versus the the BF2 layer. The total mass calculated by the inversion on grain size observed accumulation for two different grain sizes (−3ϕ and 0ϕ). for the BF2 layer is 5.0×1011 kg, which corresponds to a volume of Fig. 10 suggests that our inversion model based on grain size data does a 0.5 km3 (assuming a bulk density for the BF2 deposit of 1000 kg m−3), reasonably good job in reproducing the observed deposit. In fact all the which agrees well with the total mass predicted by the inversion on points lie relatively close to the ideal model represented by the thick line the total tephra accumulation at each location (0.25–0.4 km3) and with a slope of 1.0. Note that, although we do not present all the results narrows the range of total volume estimated with curve-fitting here, the results for other grain size classes (from −7ϕ down to b4ϕ) techniques (0.3–1.0 km3). show the same good fit between modeled and observed data.

Fig. 9. 2-D space of input parameters for the inversion on the BF2 thickness. Plot showing the column height vs. log(mass). The black dots represent solutions with a RMSEN100. The dashed black circle represents the solution to the inversion showed in Fig. 8 and with the input and output parameters presented in Table 2. Author's personal copy

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Fig. 10. Comparison between the observed tephra accumulation on the ground and the calculated tephra accumulation from the improved-inversion results on the BF2 grain size at each locality. Results shown are for two different grain sizes at each location: (a) the −3ϕ grain size fraction and (b) the 0ϕ grain size fraction.

The column heights obtained from the inversion on the different Using the inversion model by grain size, it is possible to reconstruct grain size classes are presented in Table 3, along with the diffusion the grain size distribution at each sample site and to compare these coefficients found from the inversions on grain size to model the with the observed grain size distributions (Fig. 11). The modeled grain sedimentation of each phi class with Tephra2. Excluding the−7ϕ,4ϕ size distribution at the different localities shown in Fig. 11 mimics the andN4ϕ size classes, we recorded no significant variation in column observed grain size characteristic of the actual BF2 layer. However, the heights (24 km to 30 km, Table 3) as a function of grain size, −5ϕ and especially the −6ϕ and −7ϕ for the 2 more proximal indicating no difference in the release heights for the different particle localities are not well modeled by our inversion approach on grain sizes. The column height obtained for the −7ϕ class is 20 km a.s.l., and size data. This may be a consequence of the sparseness of big particles therefore could represent a lower release height for bigger particles. in the deposit (especially for the −7ϕ fraction) or the sedimentation But the model for the −7ϕ (and to a smaller extent, for the −6ϕ as from the plume margins, which is not well described in the Tephra2 well) is not well-constrained, because of the low number of locations model. where −7ϕ particles sedimented (only 2 locations in the most proximal areas). Furthermore, these clasts might have fallen off early 5.3. Total grain size distribution from the margin of the volcanic plumes. In contrast to coarser particle models, the column heights for the 4ϕ and N4ϕ particle sizes show a The inversion of grain size data also yields an estimate of TGSD of low column height (about 10 km a.s.l.). We suspect this difference is the BF2 layer (Fig. 6f), and it can be compared to the field-based attributable to the overall proximity of all our locations (all closer than TGSD's (Fig. 6a–e) discussed previously. Although the TGSD obtained 35 km from the vent). Therefore we are missing much of the fine from the inversion of grain size seems to underestimate the amount fraction of the deposit, which is likely to have been sedimented of coarse particles compared to the other averaging techniques con- further away from Pululagua (volcanic ash from the 2450 BP eruption sidered (from simple unweighted average to the Voronoi tessellation; of Pululagua has been identified on the coast, Patty Mothes, personal Table 1), the results are in reasonable agreement. The coarser Mdϕ communication). predicted by the other averaging techniques is probably due to the lack of distal locations in our grain size data set, which creates a bias of the TGSD toward coarser particles. The inversion of grain size is less affected by this bias because it is based on a physical model of tephra Table 3 dispersion and sedimentation. Results from the inversion on grain size and the uncertainty analysis. GS stands for

Grain Size. Ht for Column Height. DC for Diffusion Coefficient and FTT for Fall Time Threshold. 5.4. Uncertainty analysis

GS Ht Mass DC FTT Uncertainty Ht Uncertainty Mass Uncertainties exist in the determination of eruptive parameters (phi) (m) (kg) (m2 s−1) (s) (m) (kg) such as the total mass of erupted tephra or the column height, and 10 −7 19,498 2.57E+10 4002.6 311.8 24,630±4750 2.78±1.91×10 are usually not addressed in the literature. The uncertainty on these −6 27,976 3.23E+09 27260.1 5262.8 28,500±1850 4.36±0.03×109 parameters can be due to (i) the model used to infer eruptive pa- −5 26,878 1.05E+10 75193.0 333.7 27,700±2550 9.89±1.64×109 fi −4 25,782 1.14E+10 53585.0 9514.2 29,250±1400 1.75±0.l0×l010 rameters and (ii) the original data set (i.e. sample distribution, eld −3 27,792 1.73E+10 46045.2 1687.7 29,410±1380 2.38±0.07×1010 observations, erosion of the deposit). Scollo et al. (2008) have already −2 29,894 2.15E+10 27992.5 2185.8 29,870±230 2.65±0.04×1010 performed a systematic sensitivity analysis of the model Tephra2 − 10 1 29,872 3.34E+10 36003.2 3532.1 29,710 ±452 4.19±0.4×10 and discussed the associated uncertainties concluding that the total 0 29,747 4.92E+10 16204.9 9921.4 29,770±290 5.00±0.2×1010 1 28,113 9.12E+10 33354.0 2200.0 28,180±930 9.11±0.6×1010 erupted mass can be well calibrated because it affects the model out- 2 24,000 1.36E+11 95984.0 2026.1 21,336±90 1.08±0.01×1011 put but not through the interaction with other parameters, whereas 3 25,732 9.37E+10 29661.3 1039.1 20,850±760 5.07±0.6×1010 parameters such as plume height and diffusion coefficient cannot 4 10,185 3.74E−09 7291.3 3592.8 n/a n/a be well calibrated as they affect the model output but through the b − 4 10,008 9.60E 10 23517.9 659.1 n/a n/a interaction with other parameters. In order to explore further and Author's personal copy

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Fig. 11. Reconstruction of the modeled grain size distribution (accumulation in kg m−2) in red and comparison with the actual grain size distribution from field data, in black, for localities from proximal to medial, along axis 1 (ESE axis, see Fig. 1). We also showed the total modeled accumulation (mod.), calculated by summing the modeled accumulation for each grain size at a given location and the observed total accumulation (obs.) at the same location.

assess these uncertainties, we used a modified version of a smoothed times on the total accumulation and 50 times for each grain size class bootstrap approach developed by Press et al. (1992), which in turned (from −6ϕ to 3ϕ) following a Monte Carlo approach. We did not was based on the bootstrap methods proposed by Efron and perform the analysis for the −7ϕ,4ϕ and b4ϕ grain sizes because of Tibshirani (1991). We assumed that the original data set contains a the lack of control in the original inversion on grain size (see above). representative distribution of sample locations and the accumulation Our analysis yields a mean total erupted mass of 4.5±0.3×1011 kg and measurements at each site are true values. Furthermore, we also a mean column height of 30±3 km (with a range between 20 km and assumed that the Tephra2 model is able to accurately reproduce the 33 km). Results for individual grain size classes are presented in observed accumulation on the ground. Then, we randomly selected a Table 3 (uncertainties given at one standard deviation from the mean). location within a square kilometer around each original sample Results from the uncertainty analysis compare well with the results location (the pseudo set of points) and calculated the predicted from the inversion on original field data, except for the 2ϕ and 3ϕ accumulation for each new location using the forward solution of the fractions, for which the mean column height calculated by the Tephra2 model (using eruptive parameters drawn from the inversion smoothed bootstrap method yields lower estimates in the column on single grain classes). We applied the inversion technique using this heights. new set of data points to calculate a new set of eruptive parameters that reproduce the pseudo set of points. Our approach follows the 6. Mass discharge rate and eruption duration bootstrap theory as we are re-sampling the original set of sample locations to derive a new one and it is smoothed because the new Papale and Rosi (1993) proposed a magma discharge rate of accumulation value at each point is calculated from the forward 2×108 kg s−1, using the Sparks (1986) model, for the BF layer based solution of the Tephra2 model rather than sampled from a subset of on the maximum grain size data collected on the coarsest part of the the original sample locations. This approach has been repeated 100 BF deposit, which we defined in this paper as being the BF2 layer. By Author's personal copy

130 A.C.M. Volentik et al. / Journal of Volcanology and Geothermal Research 193 (2010) 117–136 using this magma discharge rate and the total mass calculated from (applying the equation described in Bonadonna and Phillips 2003), the empirical methods discussed above, we obtained an eruption whereas a corner of 10 km would result in a maximum plume height duration of about 37±13 min for the BF2 layer, a relatively short- of 42 km, which is not realistic, and therefore indicates a shift due lived eruption. some wind component. We used three different models to calculate the magma discharge rate (MDR) of the BF2 layer (Sparks, 1986; Wilson and Walker, 1987; 8. Discussion Sparks et al., 1997), using a column height range of 24–30 km. From the MDR, we then calculated the eruption duration based on the 8.1. Statistical vs. analytical determination of eruptive parameters mass resulting from the inversion analysis (2.5–5.0×1011 kg). Sparks' (1986) model for a tropical atmosphere yields a MDR of 6.2± The total erupted mass obtained using the exponential fitting 3.8×107 kg s−1, resulting in an eruption duration of 194±153 min, model of Pyle (1989) (i.e. 3×1011 kg) and the power-law fitting assuming a magma eruption temperature of 1000 °C and a magma model of Bonadonna and Houghton (2005) (i.e. 5±1.5×1011 kg) density of 2400 kg m−3. Calculations made using Wilson and Walker's compare well with the total erupted mass obtained from our ana- (1987) model yield a MDR of 1.8±0.8×108 kg s−1 and an eruption lytical analyses using the Tephra2 model (i.e. 4.5±0.3×1011 kg). This duration of 50±34 min. Finally, values of MDR and eruption duration value is smaller than the one proposed by Papale and Rosi (1993) of resulting from the application of Sparks et al. (1997) model are 1.2± 1.1 km3 (equivalent to 1.1×1012 kg with a deposit bulk density of 0.5×108 kg s−1 and 72±47 min, assuming a magma density of 1000 kg/m3) and the one proposed by Pallini (1996) of 0.5 km3 2400 kg m−3. The values of MDR and eruption duration calculated (equivalent to 5×1011 kg with a deposit bulk density of 1000 kg/m3). here are smaller and larger respectively compared to the values These discrepancies are due to the fact that Papale and Rosi (1993) calculated by Papale and Rosi (1993), because the column height studied the BF deposit as a whole (including the BF1 layers that show used to derive the MDR is higher (32 km compared to our 24–30 km a NE dispersion, BF3 and WA) and Pallini (1996) included both BF2 range), resulting in higher MDR and shorter eruption durations. and BF3 layers in his calculation, while we focused only on the climactic phase of the eruption (BF2 layer). It is also worth men- 7. Plume dynamics: corner position tioning that Pallini's (1996) isopach map for the BF2 + BF3 layers together shows a slight dispersion toward the west. We use grain size data along the three axes mentioned previously As noted from the total grain size distribution analysis, the BF2 (Fig. 1) to investigate the variation of accumulation of individual grain layer is lacking the fine part of the grain size distribution. The missing size classes with distance from the vent. Instead of using the accu- fine particles might have settled with the successive layers (BF3 and/ mulation per unit area, we followed the approach of Bursik et al. (1992) or WA) or might have been blown downwind by the slight wind and calculated the accumulation per unit distance (kg/m) at any given conditions at the time of the eruption of the BF2 layer. Since the fine locality by multiplying the mass accumulated by unit area by the particles are missing, the total erupted mass calculated represents a perimeter length of the circular isopach contour that passes through minimum estimate of the mass of the BF2 layer. this locality (Figs. 12–14). A fifth-order polynomial has been fitted to Column height determination by Papale and Rosi (1993) and field data to calculate the horizontal distance of the maximum in Pallini (1996) yielded heights ranging from 21 km up to 36 km, with a accumulation on the ground of each individual grain size classes. median value of 32 km. Inverting on the total accumulation at each From Figs. 12–14, we can observe that the accumulation of sample location did not improve the solution in the column height particles from 64 to 32 mm decreases away from the vent, for the determination, as column heights ranging from 8 km to 40 km would three axes of interest in this study. This pattern reflects particles reproduce the observed accumulation on the ground (Fig. 9). How- falling out of the plume margins. The transition from a sedimentation ever, inverting on individual grain size classes narrows the range from the plume margins to a sedimentation from the umbrella cloud of solutions for the column height to 27±3 km, which compares occurs for particles between 32 mm and 16 mm, for the ESE and SW relatively well with the results from empirical methods used by axes (Figs. 12 and 14), and for the 16 mm particles for the SE (Fig. 13). Papale and Rosi (1993) and Pallini (1996). Bursik et al. (1992) found a Clasts ranging between 8 mm and 2 mm grain size classes display first discrepancy between the height calculated from grain size data and an increase in accumulation per unit distance, reach a maximum and the height from CS (21 km and 27 km, respectively) for the Fogo A then decrease away from the vent. According to Bursik et al. (1992), eruption, a discrepancy similar to the one between the results from the maximum is associated with the plume corner and can be the CS and inversion technique on grain size from this study. It is also determined using the fifth-order polynomial function fitted through worth noting that the CS approach on the −6ϕ fraction yields a field data: 7 km for the ESE axis (Fig. 12) and 10 km for the SE and SW column height of 21 km, close to the one resulting from the inversion axes (Figs. 13 and 14). This is in the same range of distances found for on the −7ϕ fraction (20 km, see Table 3). Column heights calculated the Fogo A eruption (6.8–8.0 km) characterized by a column height of from the model of CS with the −6ϕ clasts yield an underestimate of 21–27 km (Bursik et al., 1992). For particles smaller than 2 mm, a the column height because these clasts fall from the plume margins, secondary maximum in accumulation can be identified down to the and this is true too for the coarser clasts as well, such as the −7ϕ 125 and 63 μm fraction, especially for the ESE and SE axes (Figs. 12 modeled with the inversion on grain size data. and 13), while it is not clearly defined along the SW axis Fig. 13). This The results drawn from the uncertainty analysis using a smoothed secondary maximum is located about 17 km from the vent for both bootstrap method shows that the uncertainty in the determination of axes. Bursik et al. (1992) also showed a secondary maximum for the total erupted mass (4.5±0.3×1011 kg) and column height (30± particles sizes of 1 and 0.5 mm, also located around 17 km from 3 km) is in the range found from the inversion on total accumulation the vent. As for the SW axis, there is no clearly defined secondary (2.5–4×1011 kg) and individual grain size classes (27±3 km). How- maximum, probably because it may be located beyond our area of ever, the results for the 2ϕ and 3ϕ fractions show a lower estimate of observation. This result combined with the different positions of the the column height resulting from the uncertainty model compared to plume corner along the different axes lead us to think that a south- the results from the inversion on these grain size classes. This might westward dispersion of the deposit has occurred, shifting the position be due to the lack of more distal deposit which would have improved of the plume corner on the ground from 7 km to 10 km from the the spatial resolution of our data set, resulting in a more robust ESE axis to the SW axis and shifting the position of the secondary correlation with the inversion on grain size. Furthermore this ap- maximum outside the range of observation for the SW axis. In ad- proach shows that inverting tephra deposits on grain size rather than dition, a corner of 7 km results in a maximum column height of 29 km on total accumulation gives relatively good answers (with Author's personal copy

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Fig. 12. Mass accumulation per unit distance (kg m−1) of particles in grain size classes from 64 mm to 63 μm(−6ϕ to 4ϕ, respectively) for the ESE axis defined in Fig. 1. comparatively lower uncertainty) in terms of both total erupted mass tracking model (ATHAM-LPM). Their analysis is based on the isopleth and eruption column height, because the settling velocity of particles proposed by Papale and Rosi (1993), which in turn was complied on the is better constrained. BF2 layer as mentioned earlier. The column heights calculated by Kobs et Interestingly, Kobs et al. (in preparation) found a column height of al. (in preparation), although lower than our estimated values, compare 21–24 km for the BF eruption of Pululagua using a Lagrangian particle relatively well with the column heights proposed in our study. Author's personal copy

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Fig. 13. Mass accumulation per unit distance (kg m−1) of particles in grain size classes from 64 mm to 63 μm(−6ϕ to 4ϕ, respectively) for the SE axis defined in Fig. 1.

8.2. Plume dynamics combined with the relatively short duration of the eruption, could lead to the conclusion that the eruptive column was well-mixed, Contrary to our expectations, the inversion on grain size does not contradicting the “envelope model” of CS (although it might also show any difference in particle release heights (except maybe for the represent an artifact or the resolution limit of the model), and that the −7ϕ, with reservations about the validity of the inversion on this explosive eruption was more a transient rather than a sustained data) for the different classes of grain size (Table 3). This observation, event, or better, a short-lived sustained event. However, the variation Author's personal copy

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Fig. 14. Mass accumulation per unit distance (kg m−1) of particles in grain size classes from 64 mm to 63 μm(−6ϕ to 4ϕ, respectively) for the SW axis defined in Fig. 1. in accumulation per unit distance of each individual grain size with sedimentation from the umbrella cloud occurs in between the distance from the vent for the three axes studied in this paper 32 mm and 16 mm fraction (−5ϕ and −4ϕ, respectively) and clasts (Figs. 12–14) resolves this problem. In fact, Figs. 12–14 show that of the 8 mm (−3ϕ) fraction and smaller are clearly sedimenting from particles from the 64 mm and 32 mm (−6ϕ and −5ϕ, respectively) the umbrella cloud. Therefore, the lack of consistent variations in fractions are falling from the plume margins and therefore do not column height for individual grain size from the inversion cannot be reach the top of the eruptive column and the umbrella cloud. The linked to plume characteristics, probably because the variations transition from a sedimentation from the plume margins to a observed are outside the resolution limits of the model. Author's personal copy

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The secondary maximum located at ∼17 km from the vent (Figs. 12 map, the isomass maps for individual grain sizes and the inversion on and 13) observed in the accumulation per unit distance for particles grain size show a general dispersion toward the southwest, indicating smaller than 2 mm (1ϕ) was already observed but not discussed by that a light NE wind was present at the time of the eruption of the BF2 Bursik et al. (1992) and could be attributed to (i) convective instabilities, layer. The variations of Mdϕ and σϕ with distance from the vent for the (ii) aggregation (although not observed in the deposit), (iii) preferential different axes investigated in this study (see Fig. 3)showthattheMdϕ fallout of crystals (also considering that BF is crystal rich as observed by and σϕ decrease faster toward the ESE than toward the SW. Therefore, Papale and Rosi (1993)) or (iv) hydrometeor formation in the cloud clasts of a given size are transported further away from the vent along (Durant et al., 2009). However, convective instabilities, aggregation and the SW axis than along the ESE axis, probably as a consequence of a hydrometer formation should also result in polymodal grain size slight wind transport. This observation is emphasized by the shift in the distribution which is not observed at this distance from the vent, position of the plume corner from 7 km to 10 km for the ESE and SW although a slight secondary population of fine ash (up to 4 wt.% at axes respectively (see Figs. 12–14). A corner of 7 km results in a 17 km from the vent) with a mode around 6ϕ seems to appear. The maximum column height of 29 km (in good agreement with other secondary maximum could also be explained by a change in the results in this study), whereas a corner of 10 km would result in a sedimentation regime, from particles mostly falling in the turbulent maximum plume height of 42 km, which is significantly larger than our regime to particles falling mostly in the intermediate regime. results described above, and therefore indicates a shift probably due to the wind. Connor et al. (2006) have demonstrated that the circularity of 8.3. Diffusion coefficient tephra deposits can be reproduced in windy conditions (with actual wind data, implying wind shear) for Cotopaxi volcano (Ecuador). Diffusion coefficients used in semi-analytical tephra dispersal models, However, we think that the 2450 BP eruption of Pululagua occurred in such as Tephra2, ASHFALL and HAZMAP (Hurst and Turner, 1999; very calm atmospheric conditions, because of the presence of the WA, Bonadonna et al., 2005a; Macedonio et al., 2005), are not true atmospheric the position of the plume corner and the modeling of the grain size diffusion coefficients (typically dependent on the scale of the phenom- data proposed in this study. Results from the inversion of both total enon and range between 1 and 6000 m2 s−1), but represent empirical accumulation and grain size show wind directions mainly toward the parameters describing complex plume and atmospheric processes not south or the west with speed ≤10 m s−1. captured in the physical model (e.g. gravitational spreading). This is The overall circularity of the deposit may also be the result of confirmed by the discrepancies between diffusion coefficients normally differential dispersion of individual grain size classes in different used in Tephra2 (b10 m2 s−1) and diffusion coefficients obtained in directions, as shown by Fig. 4. Then, the wind might have died out ASHFALL and HAZMAP (2000–6000 m2 s−1) related to a different after the sedimentation of the BF2 (and possibly BF3) layer to allow description of plume dynamics in the model (Bonadonna et al. 2005a). the slow settling of the fine particle composing the White Ash layer. An important result from the inversion on each grain size of BF2 However, the more irregular isomass contours shown by the coarsest identifies large values for diffusion coefficient (values N15,000 m2 s−1,see particles (−5ϕ and −4ϕ; Fig. 4) could indicate an asymmetrical Table 3), which are higher than values typically used in advection– sedimentation from the plume margins towards the SE in agreement diffusion models). We suggest that large values of the diffusion coefficient with a possible model for proximal sedimentation described by are necessary to describe the gravitational spreading in a no-wind Houghton et al. (2004). These irregular isomass contours could reflect condition. In particular, smaller values of the diffusion coefficient cannot a mixing of volcanic clasts falling from different release heights, describe the accumulation of coarse particles away from the vent as therefore sedimenting simultaneously from different transport observed on the field (Fig. 11). Large values of diffusion coefficients regimes (the plume margins and the umbrella cloud). Such an asym- (30,000–90,000 m2 s−1)werealsofoundbyBonadonna et al. (2005b) in metrical sedimentation is also confirmed by the shift of plume corner the description of the Ruapehu 1996 plume and of the associated deposit. shown by grain size data (Figs. 12–14). Atmospheric diffusion in Tephra2 is described by two different laws for particle with fall time smaller or larger than the fall-time threshold (FTT in Table 3). If the total particle fall time is smaller than FTT, the 9. Conclusions diffusion is linear and depends on the diffusion coefficient (Fickian law). Otherwise, diffusion follows a power-law relationship (Suzuki 1983)and Our study of the climactic phase (BF2) of the 2450 BP Plinian does not depend on the diffusion coefficient. As a result, values of eruption of Pululagua volcano based on field data (both thickness and diffusion coefficient in Tephra2 do not affect the dispersal of fine particles. grain size data), empirical techniques, and analytical modeling shows Inversion on individual grain sizes show that larger particles (N1ϕ)have that: relatively high FTT, and therefore will diffuse mainly following Fick's law, since the total fall time of particles is likely to be smaller or similar to the (1) Results from empirical and analytical models for the determi- FTT. The diffusion of these particles is therefore strongly dependent on nation of total erupted mass of BF2 layer are in good agreement, the value of the diffusion coefficient. For smaller particles, the FTT is likely with the empirical mass being 3×1011 kg (exponential fit) and to be smaller than their total fall time, and therefore will experience a 5±1.5×1011 kg (power-law fit) and the analytical mass being shift in diffusion law during fall (from linear to power-law). of 4.5±0.3×1011 kg (inversion on observed mass/area). Another important observation made during the inversion mod- (2) Inverting tephra fallout deposit on the total accumulation (or eling was that it was possible to model the BF2 deposit with a smaller thickness) gives a good constraint on the total mass erupted diffusion coefficient but a higher total erupted mass. Therefore, using a but not on the column height. By inverting on individual grain small value for the diffusion coefficient when inversion on tephra- size classes, the possible range in column heights that can thickness data is applied only on proximal and medial locations might reproduce the deposit is better constrained. overestimate the total mass of the deposit. In contrast, modeling on (3) The plume had a maximum height of 36 km, 28 km, and 27± individual grain size classes will help avoid this possible issue. 3 km when determined using the model of Carey and Sparks (1986), Pyle (1989), and from the inversion on individual grain 8.4. Wind or no wind? size data, respectively. (4) We found that the position of the plume corner shifted from 7 The circularity of the deposit along with the occurrence of the White to 10 km (from the east to the southwest of the vent) and this Ash layer strongly suggests nearly still atmospheric conditions at the shift could be due both to a wind effect and to an asymmetrical time of the eruption. Nevertheless, the isopach map, the Mdϕ contour sedimentation from the plume margins. Author's personal copy

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(5) Total grain size calculations for the BF2 layer are strongly Bonadonna, C., Houghton, B.F., 2005. Total grain-size distribution and volume of tephra- fall deposits. Bulletin of Volcanology 67 (5), 441–456. dependant on the number and distribution of sample locations Bonadonna, C. and Costa, A., in press. Modeling of tephra sedimentation from volcanic (as already observed by Bonadonna and Houghton, 2005) and plumes. In: S.A. Fagents, T.K.P. Gregg and R.M.C. Lopes (Editors), Modeling Volcanic less dependent on the position of the “zero accumulation line”. Processes: The Physics and Mathematics of Volcanism. “ Bonadonna, C., Ernst, G.G.J., Sparks, R.S.J., 1998. Thickness variations and volume The TGSD resulting from the Voronoi technique with a zero estimates of tephra fall deposits: the importance of particle Reynolds number. accumulation line” at 100 km and 200 km and from the Journal of Volcanology and Geothermal Research 81 (3–4), 173–187. inversion on individual grain size data are in good agreement. Bonadonna, C., Connor, C.B., Houghton, B.F., Connor, L., Byrne, M., Laing, A., Hincks, T.K., (6) Based on the inversion on single grain size classes, large values 2005a. Probabilistic modeling of tephra dispersal: hazard assessment of a multi- phase rhyolitic eruption at Tarawera, New Zealand. Journal of Geophysical of the diffusion coefficient are necessary to model the BF2 layer. Research 110 (B03203). doi:10.1029/2003JB002896. Although smaller values of the diffusion coefficient can model Bonadonna, C., Phillips, J.C., Houghton, B.F., 2005b. Modeling tephra fall from a Ruapehu the deposit relatively well by inverting on thickness (or weak plume eruption. Journal of Geophysical Research 110 (B08209). doi:10.1029/ 2004JB003515. accumulation) only, the resulting total mass erupted will be Bursik, M.I., Sparks, R.S.J., Gilbert, J.S., Carey, S.N., 1992. Sedimentation of tephra by an overestimate of the true mass of the deposit. This conclusion volcanic plumes: I, Theory and its comparison with a study of the Fogo A plinian has to be acknowledged in the modeling of relatively proximal deposit, Sao Miguel (Azores). Bulletin of Volcanology 54 (4), 329–344. fl – Carey, S., Sigurdsson, H., 1982. In uence of particle aggregation on deposition of distal deposits by using advection diffusion models. We believe that tephra from the May 18, 1980, eruption of Mount St. Helens volcano. Journal of such high values are necessary to describe the gravitational Geophysical Research 87 (B8), 7061–7072. spreading of the plume in a no-wind condition. Carey, S., Sparks, R.S.J., 1986. Quantitative models of the fallout and dispersal of tephra from volcanic eruption columns. Bulletin of Volcanology 48, 109–125. (7) The inversion on individual grain size classes cannot resolve Cas, R.A.F., Wright, J.V., 1987. Volcanic Successions, Modern and Ancient: A Geological the difference in particle release heights, while the approach Approach to Processes, Products, and Successions. Springer. 544 pp. proposed by Bursik et al. (1992) shows that the transition from Connor, L.J., Connor, C.B., 2006. Inversion is the key to dispersion: understanding eruption dynamics by inverting tephra fallout. In: Mader, H.M., Cole, S.G., Connor, C.B., plume-margin and umbrella-cloud sedimentation occurs for Connor, L.J. (Eds.), Statistics in Volcanology. Special Publications of IAVCEI. Geological particles between 32 mm and 16 mm, whereas particles Society, London, pp. 231–242. ≤8 mm mostly fell from the umbrella cloud. Connor, C.B., Hill, B.E., Winfrey, B., Franklin, N.M., La Femina, P.C., 2001. Estimation of – (8) Our uncertainty analysis confirmed that inversion on individual volcanic hazards from tephra fallout. Natural Hazards Review 2 (1), 33 42. Connor, L.J., Ruiz, G., Volentik, A., Byrne, M., Mittal, A., Bonadonna, C., Connor, C.B., 2006. grain size classes rather than total accumulation gives better Probabilistic forecasts of tephra dispersion: application in Ecuador, Cities on estimates of the column height, with relatively low uncertainty Volcanoes 4, Quito (Ecuador). . on the calculated values. Therefore, integrating grain size data Connor, L.J., Connor, C.B., Bonadonna, C., 2008. Forecasting tephra dispersion using Tephra2. http://www.cas.usf.edu/~cconnor/vg@usf/tephra.html. to the total tephra accumulation in modeling tephra deposits Costa, A., Macedonio, G., Folch, A., 2006. A three-dimensional Eulerian model for transport should be considered in future studies. and deposition of volcanic ashes. Earth and Planetary Science Letters 241 (3–4), 634–647. (9) The climactic phase of the 2450 BP Plinian eruption of Pululagua Durant, A.J., Rose, W.I., Sarna-Wojcicki, A.M., Carey, S., Volentik, A.C.M., 2009. Hydrometeor-enhanced tephra sedimentation: constraints from the 18 May 1980 occurred in relatively calm atmospheric conditions, as demon- eruption of Mount St. Helens (USA). Journal of Geophysical Research 114 (B03204). strated by the occurrence and ubiquity of the WA layer. doi:10.1029/2008JB005756. Efron, B., Tibshirani, R., 1991. Statistical data analysis in the computer age. Science 253, 390–395. 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