BET_VH: a probabilistic tool for long-term volcanic hazard assessment

In this paper, we illustrate a Bayesian Event Tree to estimate Volcanic Hazard (BET_VH). The procedure enables us to calculate the probability of any kind of long-term hazardous event for which we are interested, accounting for the intrinsic stochastic nature of volcanic eruptions and our limited knowledge regarding related processes. For the input, the code incorporates results from numerical models simulating the impact of hazardous volcanic phenomena on an area and data from the eruptive history. For the output, the code provides a wide and exhaustive set of spatiotemporal probabilities of different events; these probabilities are estimated by means of a Bayesian approach that allows all uncertainties to be properly accounted for. The code is able to deal with many eruptive settings simultaneously, weighting each with its own probability of occurrence. In a companion paper, we give a detailed example of application of this tool to the Campi Flegrei caldera, in order to estimate the hazard from tephra fall.     

Hazard and risk assessment in a complex multi-source volcanic area: the example of the Campania Region, Italy

In order to zone the territory of Campania Region (southern Italy) with regard to the hazard related to future explosive activity of Somma-Vesuvio, Campi Flegrei, and Ischia Island, we drew a multi-source hazard map for tephra and pyroclastic flows. This map, which merges the areas possibly endangered by the three volcanic sources, takes into account a large set of tephra fall and pyroclastic flow events that have occurred in the last 10 ka. In detail, for fall products at Campi Flegrei and Somma-Vesuvio we used the dispersal of past eruption products as deduced by field surveys and their recurrence over the whole area. For pyroclastic flows, the field data were integrated with VEI = 4 simulated events; about 100 simulations sourcing from different points of the area were performed, considering the different probability of vent opening. The spatial recurrence of products of both past eruptions and simulated events was used to assign a weight to the area endangered by the single volcanic sources. The sum of these weights in the areas exposed to the activity of two sources and/or to different kinds of products was used to draw a hazard map, which highlights the spatial trend and the extent of the single equivalent classes at a regional scale. A multi-source risk map was developed for the same areas as the graphic result of the product of volcanic hazard and exposure, assessed in detail from a dasymetric map. The resulting multi-source hazard and risk maps are essential tools for communication among scientists, local authorities, and the public, and may prove highly practical for long-term regional-scale mitigation planning.     

Probability hazard map for future vent opening at the Campi Flegrei caldera,Italy

The Campi Flegrei caldera is a restless structure affected by general subsidence and ongoing resurgence of its central part. The persistent activity of the system and the explosive character of the volcanism lead to a very high volcanic hazard that, combined with intense urbanization, corresponds to a very high volcanic risk. One of the largest sources of uncertainty in volcanic hazard/risk assessment for Campi Flegrei is the spatial location of the future volcanic activity. This paper presents and discusses a long-term probability hazard map for vent opening in case of renewal of volcanism at the Campi Flegrei caldera, which shows the spatial conditional probability for the next vent opening, given that an eruption occurs. The map has been constructed by building a Bayesian inference scheme merging prior information and past data. The method allows both aleatory and epistemic uncertainties to be evaluated. The probability map of vent opening shows that two areas of relatively high probability are present within the active portion of the caldera, with a probability approximately doubled with respect to the rest of the caldera. The map has an immediate use in evaluating the areas of the caldera prone to the highest volcanic hazard. Furthermore, it represents an important ingredient in addressing the more general problem of quantitative volcanic hazards assessment at the Campi Flegrei caldera.     

Tephra fallout hazard assessment at the Campi Flegrei caldera (Italy)

Tephra fallout associated with renewal of volcanism at the Campi Flegrei caldera is a serious threat to the Neapolitan area. In order to assess the hazards related with tephra loading, we have considered three different eruption scenarios representative of past activity: a high-magnitude event similar to the 4.1 ka Agnano-Monte Spina eruption, a medium-magnitude event, similar to the ∼3.8 ka Astroni 6 eruption, and a low-magnitude event similar to the Averno 2 eruption. The fallout deposits were reconstructed using the HAZMAP computational model, which is based on a semi-analytical solution of the two-dimensional advection–diffusion–sedimentation equation for volcanic tephra. The input parameters into the model, such as total erupted mass, eruption column height, and bulk grain-size and components distribution, were obtained by best-fitting field data. We carried out tens of thousands simulations using a statistical set of wind profiles, obtained from NOAA re-analysis. Probability maps, relative to the considered scenarios, were constructed for several tephra loads, such as 200, 300 and 400 kg/m². These provide a hazard assessment for roof collapses due to tephra loading that can be used for risk mitigation plans in the area.     

A methodology for the evaluation of long-term volcanic risk from pyroclastic flows in Campi Flegrei (Italy)

The volcanological history of Campi Flegrei suggests that the most frequent eruptions are characterized by the emplacement of pyroclastic flow and surge deposits erupted from different vents scattered over a 150-km² caldera. The evaluation of volcanic risk in volcanic fields is complex because of the lack of a central vent. To approach this problem, we subdivided the entire area of Campi Flegrei into a regular grid and evaluated the relative spatial probability of opening of vents based on geological, geophysical and geochemical data. We evaluated the volcanic risk caused by pyroclastic flows based on the formula proposed by UNESCO (1972), R=H×V×Va, where H is the hazard, V is the vulnerability and Va is the value of the elements at risk. The product H×V was obtained by performing simulations of type eruptions centered in each cell of the grid. The simulation is based on the energy cone scheme proposed by Sheridan and Malin [J. Volcanol. Geotherm. Res. 17 (1983) 187–202], hypothesizing a column collapse height of 100 m for eruptions of VEI=3 and 300 m for eruptions of VEI=4 with a slope angle of 6°. Each simulation has been given the relative probability value associated with the corresponding cell. We made use of the GIS software ArcView 3.2 to evaluate the intersection between the energy cone and the topography. The superposition of the areas invaded by pyroclastic flows (124 simulations for VEI=3 and 37 for VEI=4) was used to obtain the relative hazard map of the area. The relative volcanic risk map is obtained by superimposing the urbanization maps.     

<Superscript>40</Superscript>Ar/<Superscript>39</Superscript>Ar dating of tuff vents in the Campi Flegrei caldera (southern Italy): toward a new chronostratigraphic reconstruction of the Holocene volcanic activity

The Campi Flegrei hosts numerous monogenetic vents inferred to be younger than the 15 ka Neapolitan Yellow Tuff. Sanidine crystals from the three young Campi Flegrei vents of Fondi di Baia, Bacoli and Nisida were dated using ⁴⁰Ar/³⁹Ar geochronology. These vents, together with several other young edifices, occur roughly along the inner border of the Campi Flegrei caldera, suggesting that the volcanic conduits are controlled by caldera-bounding faults. Plateau ages of ∼9.6 ka (Fondi di Baia), ∼8.6 ka (Bacoli) and ∼3.9 ka (Nisida) indicate eruptive activity during intervals previously interpreted as quiescent. A critical revision, involving calendar age correction of literature ¹⁴C data and available ⁴⁰Ar/³⁹Ar age data, is presented. A new reference chronostratigraphic framework for Holocene Phlegrean activity, which significantly differs from the previously adopted ones, is proposed. This has important implications for understanding the Campi Flegrei eruptive history and, ultimately, for the evaluation of related volcanic risk and hazard, for which the inferred history of its recent activity is generally taken into account.     

Long-term multi-hazard assessment for El Misti volcano (Peru)

We propose a long-term probabilistic multi-hazard assessment for El Misti Volcano, a composite cone located <20 km from Arequipa. The second largest Peruvian city is a rapidly expanding economic centre and is classified by UNESCO as World Heritage. We apply the Bayesian Event Tree code for Volcanic Hazard (BET_VH) to produce probabilistic hazard maps for the predominant volcanic phenomena that may affect c.900,000 people living around the volcano. The methodology accounts for the natural variability displayed by volcanoes in their eruptive behaviour, such as different types/sizes of eruptions and possible vent locations. For this purpose, we treat probabilistically several model runs for some of the main hazardous phenomena (lahars, pyroclastic density currents (PDCs), tephra fall and ballistic ejecta) and data from past eruptions at El Misti (tephra fall, PDCs and lahars) and at other volcanoes (PDCs). The hazard maps, although neglecting possible interactions among phenomena or cascade effects, have been produced with a homogeneous method and refer to a common time window of 1 year. The probability maps reveal that only the north and east suburbs of Arequipa are exposed to all volcanic threats except for ballistic ejecta, which are limited to the uninhabited but touristic summit cone. The probability for pyroclastic density currents reaching recently expanding urban areas and the city along ravines is around 0.05 %/year, similar to the probability obtained for roof-critical tephra loading during the rainy season. Lahars represent by far the most probable threat (around 10 %/year) because at least four radial drainage channels can convey them approximately 20 km away from the volcano across the entire city area in heavy rain episodes, even without eruption. The Río Chili Valley represents the major concern to city safety owing to the probable cascading effect of combined threats: PDCs and rockslides, dammed lake break-outs and subsequent lahars or floods. Although this study does not intend to replace the current El Misti hazard map, the quantitative results of this probabilistic multi-hazard assessment can be incorporated into a multi-risk analysis, to support decision makers in any future improvement of the current hazard evaluation, such as further land-use planning and possible emergency management.     

Assessment of <Emphasis Type="Italic">pre-crisis</Emphasis> and <Emphasis Type="Italic">syn-crisis</Emphasis> seismic hazard at Campi Flegrei and Mt. Vesuvius volcanoes,Campania, southern Italy

In this study, we address the issue of short-term to medium-term probabilistic seismic hazard analysis for two volcanic areas, Campi Flegrei caldera and Mt. Vesuvius in the Campania region of southern Italy. Two different phases of the volcanic activity are considered. The first, which we term the pre-crisis phase, concerns the present quiescent state of the volcanoes that is characterized by low-to-moderate seismicity. The second phase, syn-crisis, concerns the unrest phase that can potentially lead to eruption. For the Campi Flegrei case study, we analyzed the pattern of seismicity during the 1982–1984 ground uplift episode (bradyseism). For Mt. Vesuvius, two different time-evolutionary models for seismicity were adopted, corresponding to different ways in which the volcano might erupt. We performed a site-specific analysis, linked with the hazard map, to investigate the effects of input parameters, in terms of source geometry, mean activity rate, periods of data collection, and return periods, for the syn-crisis phase. The analysis in the present study of the pre-crisis phase allowed a comparison of the results of probabilistic seismic hazard analysis for the two study areas with those provided in the Italian national hazard map. For the Mt. Vesuvius area in particular, the results show that the hazard can be greater than that reported in the national hazard map when information at a local scale is used. For the syn-crisis phase, the main result is that the data recorded during the early months of the unrest phase are substantially representative of the seismic hazard during the whole duration of the crisis.     

Vertical ground movements in the Campi Flegrei caldera as a chaotic dynamic phenomenon

The ground level in the Campi Flegrei caldera has never been stationary in the last 2,000 years. Historical data, and a nearly continuous tide-gauge record 20 years long, show that uplift and sinking have taken place on a variety of different time scales. In addition, the Campi Flegrei volcanic system appears to be sensitive to weak external forces such as tidal forces. We infer from these elements that the Campi Flegrei system is far from thermodynamic equilibrium, and suggest that its dynamics may be chaotic. We analyze the short-term variations of the ground level, and find that they can be described in a low-dimensional phase space. The dynamics of the Campi Flegrei system seems to have been phase-locked with tidal forces in the period following the 1970–1972 climax, and to have undergone a transition to chaos in some moment that preceded the presently continuing sinking phase.     

A reappraisal of shear wave splitting parameters from Italian active volcanic areas through a semiautomatic algorithm

Shear wave splitting parameters represent a useful tool to detail the stress changes occurring in volcanic environments before impending eruptions. In the present paper, we display the parameter estimates obtained through implementation of a semiautomatic algorithm applied to all useful datasets of the following Italian active volcanic areas: Mt. Vesuvius, Campi Flegrei, and Mt. Etna. Most of these datasets have been the object of several studies (Bianco et al., Annali di Geofisica, XXXXIX 2:429–443, 1996, J Volcanol Geotherm Res 82:199–218, 1998a, Geophys Res Lett 25(10):1545–1548, 1998b, Phys Chem Earth 24:977–983, 1999, J Volcanol Geotherm Res 133:229–246, 2004, Geophys J Int 167(2):959–967, 2006; Del Pezzo et al., Bull Seismol Soc Am 94(2):439–452, 2004). Applying the semiautomatic algorithm, we confirmed the results obtained in previous studies, so we do not discuss in much detail each of our findings but give a general overview of the anisotropic features of the investigated Italian volcanoes. In order to make a comparison among the different volcanic areas, we present our results in terms of the main direction of the fast polarization (φ) and percentage of shear wave anisotropy (ξ).     

Statistically coherent approach involving log-ratio transformation of geochemical data enabled tephra correlations of two late Pleistocene tephra from the eastern Adriatic shelf

Tephrostratigraphic correlations commonly rely on geochemical composition supported by additional constraints (e.g., multiple stratigraphically ordered tephra, geochronological-stratigraphical constraints, and isotopic determinations), which provide key clues to restrict the number of possible candidates and disambiguate the correlation of a specific tephra among compositionally similar volcanic sources/tephra. However, such additional data may not be available or acquirable, leaving the geochemical data as sole, but challenging viable approach. In this study, two geochronologically poorly constrained late Pleistocene tephra from the eastern Adriatic –from a sand profile on Mljet Island (M-53/2) and from a marine sediment core from Pirovac Bay (PROS 721)– were correlated to known eruptions using only geochemical data (major and trace elements of glass shards), which were treated using both log ratio transformed and raw data. After the statistical treatment of the geochemical data using bivariate plots, linear discrimination analyses and selbal algorithm, the tephra M-53/2 and PROS 721 were suitably correlated with the widespread tephra generated during the Campi Flegrei eruptions of Massereia del Monte (Y-3 marine tephra, 29.0 ± 0.8 ka) and Neapolitan Yellow Tuff (14.5 ± 0.4 ka), respectively. This study showed that the correlation was hardly tenable when using the raw data, as opposed to compositional approach, which yielded satisfactory results. As a consequence, the distribution of Massereia del Monte/Y-3 tephra extended far toward the northeast, while a better chronological model, for reconstructing the paleoenvironmental changes at the Pirovac Bay location and the Holocene sea-level dynamics, could be obtained.     

Structure and evolution of the Bay of Pozzuoli (Italy) using marine seismic reflection data: implications for collapse of the Campi Flegrei caldera

Digital marine seismic reflection data acquired in 1973 in the Bay of Pozzuoli, and recently reprocessed, were used to study the volcanological evolution of the marine sector of Campi Flegrei Caldera during the last 37 ka. In order to gain more information, interpretation also involved estimation of the "pseudo-velocity" and the "pseudo-density" from the resistivity logs of two onshore deep exploration wells. The main results are: (1) discovery of ancient pre-18 ka and post-37 ka submarine and mainly effusive volcanic activity, along coeval emission centers located at the edges of Campi Flegrei Caldera; (2) confirmation that the caldera collapse in the marine sector of Campi Flegrei seems strongly controlled by regional NE–SW and NW–SE structural discontinuities; (3) the finding of at least two episodes of collapse in the bay; and (4) identification of a post-18 ka volcanic deflation phase that has caused about 150–200 m of subsidence in the central sector of the Bay of Pozzuoli in the last 18 ka.Editorial responsibilty: T. Druitt     

龙岗金龙顶子火山空降碎屑物数值模拟及概率性灾害评估

空降碎屑物为爆炸式火山喷发产生的一种重要的灾害类型,数值模拟已成为一个快速有效地确定火山灰扩散和沉积范围的方法。本文根据改进的Suzuki(1983)二维扩散模型,编写了基于Windows环境下的火山灰扩散程序。通过对前人资料的分析,模拟了龙岗火山群中最新火山喷发——金龙顶子火山喷发产生的空降碎屑物扩散范围,与实测结果具有很好的一致性,证实了模型的可靠性和参数的合理性。根据该区10年的风参数,模拟了7021次不同风参数时金龙顶子火山灰的扩散范围,以此制作了火山灰沉积厚度超过1cm和0.5cm时的概率性空降碎屑灾害区划图。本文的研究可为龙岗火山区火山危险性分析和灾害预警与对策提供重要的科学依据。     

The Neapolitan Yellow Tuff — A large volume multiphase eruption from Campi Flegrei,Southern Italy

the Neapolitan Yellow Tuff (NYT) (12 ka BP) is considered to be the product of a single eruption. Two different members (A and B) have been identified and can be correlated around the whole of Campi Flegrei. Member A is made up of at least 6 fall units including both ash and lapilli horizons. The basal stratified ash unit (A1) is interpreted to be a phreatoplinian fall deposit, since it shows a widespread dispersal (>1000 km²) and a constant thickness over considerable topography. The absence of many lapilli fall units in proximal and medial areas testifies to the erosive power of the intervening pyroclastic surges. The overlying member B was formed by many pyroclastic flows, radially distributed around Campi Flegrei, that varied widely in their eruptive and emplacement mechanisms. In some of the most proximal exposures coarse scoria and lithic-rich deposits, sometimes welded, have been identified at the base of member B. Isopach and isopleth maps of fall-units, combined with the distribution of the coarse proximal facies, indicate that the eruptive vent was located in the NE area of Campi Flegrei. It is considered that the NYT eruption produced collapse of a caldera approximately 10 km diameter within Campi Flegrei. The caldera rim, located by geological and borehole evidence, is now largely buried by the products of more recent eruptions. Initiation of caldera collapse may have been contemporaneous with the start of the second phase (member B). It is suggested that there was a single vent throughout the eruption rather than the development of multiple or ring vents. Chemical data indicate that different levels of a zoned trachyte-phonolite magma chamber were tapped during the eruption. The minimum volume of the NYT is calculated to be about 50 km³ (DRE), of which 35 km³ (70%) occurs within the caldera.     

The late Pleistocene pyroclastic deposits of the Campanian Plain: New insights into the explosive activity of Neapolitan volcanoes

The lithological and compositional characteristics of eighteen different pyroclastic deposits of Campanian origin, dated between 125 cal ky BP and 22 cal ky BP, were described. The pyroclastic deposits were correlated among different outcrops mainly located on the Apennine slopes that border the southern Campanian Plain. They were grouped in two main stratigraphic and chronologic intervals of regional significance: a) between Pomici di Base (22.03 cal ky BP; Somma–Vesuvius) and Campanian Ignimbrite (39 cal ky BP; Campi Flegrei) eruptions; and b) older than Campanian Ignimbrite eruption. Three new ¹⁴C AMS datings support the proposed correlations. Six eruptions were attributed to the Pomici di Base-Campanian Ignimbrite stratigraphic interval, while twelve eruptions are older than Campanian Ignimbrite. Of the studied deposits two originated from Ischia island, five are related to Campi Flegrei, and three to Somma–Vesuvius. Two eruptions have an uncertain correlation with Somma–Vesuvius or Campi Flegrei, while six eruptions remain of uncertain source. Minimum volumes of five eruptions were assessed, ranging between 0.5 km³ and 4 km³. Two of the studied deposits were correlated with Y-3 and X-5 tephra layers, which are widely dispersed in the central Mediterranean area. The new stratigraphic and chronologic data provide an upgraded chrono-stratigraphy for the explosive activity of Neapolitan volcanoes in the period between 125 and 22 cal ky BP.     

Evaluating fracture patterns within a resurgent caldera: Campi Flegrei, Italy

Understanding deformation of active calderas allows their dynamics to be defined and their hazard mitigated. The Campi Flegrei resurgent caldera (Italy) is one of the most active and hazardous volcanoes in the world, characterized by post-collapse resurgence, eruptions, ground deformation, and seismicity. An original structural analysis provides an overview of the main fracture zones. NW-SE and NE-SW fractures (normal or transtensive faults and extensional fractures) predominate along the rim and within the caldera, suggesting a regional control, both during and after the collapses. While the NE-SW fractures are ubiquitous in the deposits of the last ∼37 ka, NW-SE fractures predominate in the last 4.5 ka, during resurgence. The most recently (<4.5 ka) strained area lies in the caldera center (Solfatara area), where the faults, with an overall ∼ENE-WSW extension direction, appear to be associated with the bending due to resurgence. Solfatara lies immediately to the east of the most uplifted part of the caldera (Pozzuoli area), where domes form and culminate both on the long-term (resurgence, accompanied by volcanic activity) and short-term deformation (1982–1984 bradyseism, accompanied by seismic and hydrothermal activity). Similar volcano-tectonic behavior characterizes the short- and long-term uplifts, and only the intensity of the tectonic and volcanic activity varies, being related to varying amounts of uplift. Seismicity and hydrothermal manifestations occur during the bradyseisms, with moderate uplift, while surface faulting and eruptions occur during resurgence, with higher uplift. The features observed at Campi Flegrei are found at other major calderas, suggesting consistent behavior of large magmatic systems.     

The Campania-Campi Flegrei area: a contribution to discern the best structural model from gravity interpretation

Structural interpretations of negative gravity anomalies at Campi Flegrei and Campania are reviewed. The interpretations are not uniquely determined, as different models correspond to equally good anomaly fits.These models are examined using a new method based on the interpretation of the apparent density map computed from the gravity anomaly map.As regards the Campi Flegrei anomaly, a funnel-shaped caldera 3D model is readily obtained. This is more appropriate than previous interpretations, as not only anomaly but also apparent densities are fitted.As about the regional structures, the used approach suggests that the more pertinent model is a large magmatic intrusion, 9 to 14 km deep.A final scheme is proposed linking this intrusion to the funnel-shaped caldera as the main regional reservoir.     

Volcanic CO2 flux measurement at Campi Flegrei by tunable diode laser absorption spectroscopy

Near-infrared room temperature tunable diode lasers (TDL) have recently found increased usage in atmospheric chemistry and air monitoring research, but applications in volcanology are still limited to a few examples. Here, we explored the potential of a commercial infrared laser unit (GasFinder 2.0 from Boreal Laser Ltd) for measurement of volcanic CO₂ mixing ratios, and ultimately for estimating the volcanic CO₂ flux. Our field tests were conducted at Campi Flegrei near Pozzuoli, Southern Italy, where the GasFinder was used during three campaigns in October 2012, January 2013 and May 2013 to repeatedly measure the path-integrated mixing ratios of CO₂ along cross sections of the atmospheric plumes of two major fumarolic fields (Solfatara and Pisciarelli). By using a tomographic post-processing routine, we resolved, for each of the two fields, the contour maps of CO₂ mixing ratios in the atmosphere, from the integration of which (and after multiplication by the plumes’ transport speeds) the CO₂ fluxes were finally obtained. We evaluate a total CO₂ output from the Campi Flegrei fumaroles of ～490 Mg/day, in line with independent estimates based on in situ (Multi-GAS) observations. We conclude that TDL technique may enable CO₂ flux quantification at other volcanoes worldwide.     

Numerical simulation of pyroclastic density currents on Campi Flegrei topography: a tool for statistical hazard estimation.

We describe a numerical simulation of both concentrated and dilute gravity-driven pyroclastic flows on a digital topographic model of the Campi Flegrei volcanic field. Families of numerical flows are generated by sampling a multi-dimensional matrix of vent coordinates, flow properties and dynamical parameters within a wide range of values. Hazard maps are constructed from the data base of simulated flows, using a mixed deterministic–statistical approach. The set of probable vents covers the area of recent eruptions. Results show the key role of topography in controlling the flow dispersion. The maximum hazard appears to be the NE sector of the caldera. Flows in the eastern sector, including the city of Naples, are shown to be efficiently hindered by the Posillipo and Camaldoli hills at the caldera borders, thus reducing the hazard. The results represent the first physically based estimate of hazard from pyroclastic flows in this densely populated area, and can be used for civil defence purposes.     

Evaluating suitability of a tephra dispersal model as part of a risk assessment framework

In volcanic risk assessment it is necessary to determine the appropriate level of sophistication for a given predictive model within the contexts of multiple sources of uncertainty and coupling between models. A component of volcanic risk assessment for the proposed radioactive waste repository at Yucca Mountain (Nevada, USA) involves prediction of dispersal of contaminated tephra during violent Strombolian eruptions and the subsequent transport of that tephra toward a hypothetical individual via surface processes. We test the suitability of a simplified model for volcanic plume transport and fallout tephra deposition (ASHPLUME) coupled to a surface sediment-transport model (FAR) that calculates the redistribution of tephra, and in light of inherent uncertainties in the system. The study focuses on two simplifying assumptions in the ASHPLUME model: 1) constant eruptive column height and 2) constant wind speed and direction during an eruption. Variations in tephra dispersal resulting from unsteady column height and wind conditions produced variations up to a factor of two in the concentration of tephra in sediment transported to the control population. However, the effects of watershed geometry and terrain, which control local remobilization of tephra, overprint sensitivities to eruption parameters. Because the combination of models used here shows limited sensitivity to the actual details of ash fall, a simple fall model suffices to estimate tephra mass delivered to the hypothetical individual.