Applicability of statistical eruption analysis to the geological record of Villarrica and Lanín volcanoes, Southern Volcanic Zone, Chile

Journal of Volcanology and Geothermal Research 200 (2011) 99–115

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Journal of Volcanology and Geothermal Research

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Applicability of statistical eruption analysis to the geological record of Villarrica andLanín volcanoes, Southern Volcanic Zone, Chile

Heidi Wehrmann a,⁎, Yvonne Dzierma b

a SFB 574, IFM-GEOMAR, Wischhofstr. 1-3, 24148 Kiel, Germanyb SFB 574, University of Kiel, Otto-Hahn-Platz 1, 24118 Kiel, Germany

⁎ Corresponding author. Tel.: +49 431 600 2568; faxE-mail address: [email protected] (H. We

0377-0273/$ – see front matter © 2010 Elsevier B.V. Aldoi:10.1016/j.jvolgeores.2010.11.009

a b s t r a c t
a r t i c l e i n f o
Article history:Received 17 June 2010Accepted 3 November 2010Available online 12 November 2010

Keywords:eruption time seriesgeological recorddistribution functionsprobabilistic eruption forecastingvolcanic hazardsVillarrica VolcanoLanín Volcano

Standard stochastic failure-analysis techniques are applied to the geological eruption records of Villarrica andLanín volcanoes in the Chilean Southern Volcanic Zone. These statistical methods are used to estimate theprobabilities of eruptions large enough to leave a trace in the geological record. For Villarrica, the standardlifetime distributions predict a near 100% probability that such an eruption should have occurred between489 AD and today. If the VEI=3 eruptions observed in the historical record are large enough to be preservedin the geological record, the probability of such a future eruption of Villarrica is about 20% in the next100 years.In the case of Lanín, the applicability of the exponential, Weibull and log-logistic distributions to thegeological record is doubtful, since the volcano appears to have experienced alternating high- and low-activity regimes. Only the mixture-of-exponentials distribution can take into account the possible dormancydescending to final extinction, resulting in a probability of about 16% that Lanín will ever erupt again in thefuture.The contrasting behaviour of the eruption records of these two neighbouring volcanoes in terms of thenumber of eruptions and the length of the repose-time intervals serves to examine to what extent themethodcan be applied to geological eruption records. This study contributes to directing statistical eruption analysistowards time scales long enough to investigate large-magnitude eruptions, and it includes the possibility ofdormancy/extinction.

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1. Introduction

Intense volcanism occurs along the Western margin of SouthAmerica, caused by the subduction of the Nazca Plate and theAntarctic Plate beneath the South American Plate. Extending from33°S to 46°S, the Chilean Southern Volcanic Zone (SVZ) is one of thefour active arc segments contained in the Andes, entailing a potentialof destructive eruptions. In addition to three giant calderas (Diaman-te, Calabozos, and Caviahue) that covered large areas by the eruptionof voluminous pyroclastic flow deposits during the past 1.1 Ma(Hildreth et al., 1984; Muñoz and Stern, 1988; Stern, 2004; Sruoga etal., 2005), the SVZ comprises more than 60 large volcanic edifices, inaddition to numerous small eruptive centres. Most of these volcanoesare tagged as active or potentially active (Stern, 2004). Since morethan 70% of the Chilean population lives in this area of the SVZ,volcanic hazard assessment is essential to mitigate and preventdamage to people and property. The most effective volcanic hazardassessment is performed through multi-parameter surveillance of the

volcano, recording seismicity, degassing, deformation, etc. In manycases, this allows for an early alert with regard to impending eruptions(e.g., Scarpa and Tilling, 1996). In the SVZ, however, with a fewexceptions, most of the volcanoes are not continuously monitored(Stern, 2004; and information from Servicio Nacional de Geología yMinería (SERNAGEOMIN), Chile).

Although not developed to substitute for monitoring efforts,statistical analysis of eruption time series has advanced to becomean important component of hazard assessment (e.g., Mader et al.,2006). It provides the tool for probabilistic eruption forecasting, beingthe only approach at the numerous volcanoes where no permanentmonitoring data are available. In the ideal case of surveyed volcanoes,it complements the investigation of interacting geophysical, volcano-logical and geochemical processes. Moreover, it provides an approachto estimate the hazard posed by a volcano on a longer time scale,independently of its present state of activity.

Stochastic time-series modelling was first applied to volcanoeruption series by Wickman (1966), after which a number of studieshave been performed for volcanoes around the world (Hawaii (Klein,1982; Ho, 1990; Bebbington, 2008), New Zealand (Bebbington andLai, 1996a; Turner et al., 2008), Soufrière Hills (Connor et al., 2003),Italy (Ho, 1990; Marzocchi et al., 2004; Bebbington, 2007; Bebbington,

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100 H. Wehrmann, Y. Dzierma / Journal of Volcanology and Geothermal Research 200 (2011) 99–115

2008), Mexico (De la Cruz-Reyna, 1991; Mendoza-Rosas and De laCruz-Reyna, 2008, 2009), South America (Dzierma and Wehrmann,2010a,b), and others). These empirical approaches use methodsborrowed from the standard life distribution/failure analyses (see,e.g., Cox and Oakes, 1984; Marshall and Olkin, 2007).

Although the procedures have been successfully applied to thehistorical eruption record (usually extending a few centuries back) ofmany volcanoes, only few studies have included eruptions thatoccurred over the past thousands to millions of years (Ho, 1991;Nathenson et al., 2007; Mendoza-Rosas and De la Cruz-Reyna, 2008;Turner et al., 2008). This means that in most cases, the largest part ofthe known activity record has not been considered in the analysis. Ofthose eruptions dating further back into the past, the comparativelythin deposits of small eruptions are likely to be lost. Therefore, thegeological record can be expected to represent only the largereruptions, which occur less frequently, but would have a higherimpact on the surrounding regions.

Two volcanoes in the central SVZ are the focus of this study:Villarrica and Lanín. Volcanologically, with regard to volcanic hazardassessment, we aim at estimating the probabilistic likelihood that ageologically traceable eruption (as defined below) will occur within agiven amount of time in the future. Methodologically, we will exploreto what extent the standard statistical time-series analysis can beapplied to an eruption record spanning over the entire Holocene. Theanalysis of the eruption records of Villarrica and particularly of Lanínusing an established statistical procedure illustrates some of thepitfalls that may arise in the statistical analysis when applied toeruption records beyond historical documentation.

Fig. 1. Location map of Villarrica and Lanín volcanoes (large triangles). Several other largetriangles. The inset map shows the position of the region in South America.

2. Villarrica and Lanín Volcanoes

Villarrica Volcano is located at 39°25′14″S/71°56′23″W, at theintersection between the Northwest–Southeast trending Mocha–Villarrica Fault Zone and the large North–South trending Liquiñe–Ofqui Fault Zone (Fig. 1). It was built up since the Mid- to Late-Pleistocene by alternating effusive and highly-explosive activity withmultiple caldera collapses, resulting in a large complex stratovolcano.Presently, Villarrica is one of the most active volcanoes of SouthAmerica, situated in a populated region with local industry,agriculture and tourism. Its recent eruptions have mostly beensmall, yet they caused damage and loss in the surroundingcommunities by the occurrence of tephra fallout, lava flows, laketsunamis, and lahars (Lara and Clavero, 2004; Castruccio et al., 2010;Silva-Parejas et al., 2010). Villarrica Volcano has been the subject ofseveral recent studies focusing on quiescent gas release, eruptiveactivity, lahar debouching, lake sediments, remote sensing, etc.(Johnson et al., 2004; Mather et al., 2004; Witter et al., 2004; Lohmaret al., 2007;Moernaut et al., 2009; Castruccio et al., 2010; Silva-Parejaset al., 2010), it has also been investigated for its volcano–tectonicinteractions in the course of an SVZ-wide study by Cembrano and Lara(2009). It is continuously monitored by the Observatorio Volcanolo-gico de los Andes del Sur of the Servicio Nacional de Geología yMinería (Lara and Clavero, 2004; and current projects).

Lanín Volcano, a large stratovolcano in a back-arc setting at 39°37′58″S/71°29′59″W(Fig. 1), has evolved sinceMid-Pleistocene by mainlyeffusive eruptions of bimodal compositions from basalt to trachydacite.No major caldera-forming events have been reconstructed, however, a

volcanic centres in the Chilean Southern Volcanic Zone are indicated by small open

101H. Wehrmann, Y. Dzierma / Journal of Volcanology and Geothermal Research 200 (2011) 99–115

series of pyroclastic deposits documents the occurrence of at leastmoderately explosive activity over the volcano's history (Lara, 2004;Lara et al., 2004). While no recent eruptions are known, there is noreason to believe it to be extinct, and indeed people living in thecommunities close to Lanín claim to feel occasional seismicity (thisactually caused some concern after the recent Maule earthquake on27.02.2010 (Iglesia Llanos, personal communication), although noactivity has yet been documented — and Lanín remained inactive evenafter the large 1960 Valdivia earthquake in spite of its immediategeographic proximity (Caselli, personal communication)). From a riskperspective, Lanín is rather remote from urban areas as compared withVillarrica. However, it is surroundedby several villages andwas recentlyproposed as a ski resort. Additionally, one of the main border crossingroads between Chile and Argentina passes directly at the foot of Lanín,exposing it to possible flank collapses and lahars, despite the fact thatfallout tephra, even of smaller eruptions, may spread out far beyond theimmediate surroundings (Lara, 2004).

Table 1Geological eruptions of (a) Villarrica and (b) Lanín Volcano as listed on the GVP webpage. Fresult of Silva-Parejas et al. (2010). The tephrostratigraphic age of 4940 years assigned to theseems to be simply an erroneous unit name assignment in the GVP list, the date is taken hecalibrated using calib (version 6.0.1) (Stuiver and Reimer, 1993), with the Southern hemispranges; of which a mean age was used here for the statistical analysis. The 1558 eruption fhistorical record hereafter has been omitted since it was analysed in Dzierma and Wehrmrecord. For the hypothesis that the integration of historical VEI=3 eruptions with the geol

Uncorrected (years BP) 1 σ calibrated results Corrected age (yr BC/AD) Dating

a. Villarica8600 7592:7571 7582 BC Radioc5700 4501:4452 4477 BC Radioc4940 3694:3680 3671 BC Tephro

3666:36444200 2863:2835 2746 BC Radioc

2817:28062759:27172710:26652645:2639

4100 2618:2609 2574 BC Radioc2597:25952582:25632534:2493

3930 2452:2442 2398 BC Tephro2440:24202405:23772350:2297

3510 1781 BC Radioc1878:18391828:17921785:17261725:1691

2600 792:756 685 BC Radioc684:669606:604

1800 253:263 281 AD Radioc277:329

1600 441:454 488 AD Radioc457:485532:556

1558 AD Histori

b. Lanin9240 BC±500 years Radioc6340 BC±200 years Radioc590 BC±200 years Radioc

220±200 BC Radioc80 BC±200 years Radioc90 AD±300 years Tephro

400 AD±150 years Radioc560 AD±150 years Radioc

3. Data sets

In this work, we distinguish between “historical” and “geological”eruptions. Historical eruptions are defined as eruptions known fromwritten or oral documentation, while geological eruptions refer tothose reconstructed from deposits of erupted material sufficientlyvoluminous and well-preserved to be related back to the eruptionevents— both effusive and explosive. Such eruptions will henceforthbe called eruptions of “geological magnitude”, although preservationis affected by variable environmental factors. Nevertheless, it cannotbe ruled out that additional eruptions may be identified as volcano-logical work proceeds, e.g., by investigation of lake sediment cores oradditional field work (compare, e.g., Sisson and Vallance, 2009).

The eruption records for Villarrica and Lanín are taken fromavailable literature (Moreno et al., 1994; Lara and Clavero, 2004; Laraet al., 2004; Naranjo and Moreno, 2005; Silva-Parejas et al., 2010); inaddition, compilations such as provided by the Smithsonian Global

or Villarrica Volcano, the age of the Pucón ignimbrite was updated to the recent datingEl Pirao lavas conflicts with the stratigraphic sequence in the literature; which howeverre to belong to the Antumalal lavas (marked by an asterisk in the table). The ages werehere correction of McCormac et al. (2004). Conversion results are given as one-sigmarom the historical record is included to mark the onset of the historical eruptions; theann (2010a,b) and it remains unclear how to correctly integrate it with the geologicalogical record is permissible, see Appendix C.

method Eruptive style Name of unit

arbon Explosive Voipir pyroclastic flowarbon Explosive Candelaria pyroclastic flowchronology Effusive *Antumalal lavas

arbon Explosive Scoria pyroclastic flow and fallout

arbon Explosive Afunalhue pyroclastic flow

chronology Effusive Zanjón Seco lavas

arbon Explosive Pucon ignimbrite

arbon Explosive Pedregoso pyroclastic surge

arbon Effusive Zanjon Seco pahoehoe basaltic lavas

arbon Explosive Cónquil pyroclastic flow

cal records Explosive

arbon (corrected) Explosivearbon (corrected) Explosivearbon (corrected) Explosivearbon (corrected) Explosive and effusivearbon (corrected) Explosivechronology Effusivearbon (corrected) Explosivearbon (corrected) Explosive

Fig. 2. Cumulative number of eruptions, as a function of time. (a) Villarrica volcano, alleruptions (geologic and historic). The slope of the graph changes markedly where thetransition from the geologic to the historic record occurs. (b) Villarrica volcano,geological eruptions only (onset of historical record is shown for clarity). (c) Lanínvolcano.


Volcanism Program website (GVP, http://www.volcano.si.edu) pro-vide a useful resource. However, the possibility remains that not alleruptions have been recorded. Within the applied statistical frame-work of point process analysis, any eruption occurrence is treatedhere as an “event”, not considering the eruption magnitude, style, ormagma production rates.

While Villarrica's historical eruption record shows frequentactivity with eruptions occurring every few years (statisticallyanalysed by Muñoz, 1983; Dzierma and Wehrmann, 2010a), wehere consider eruptions of geological magnitude spanning almost thepast 10,000 years. The numerically dated eruptions provided in theliterature and by the GVP are reported as uncalibrated 14C-years. Thoseages were converted here with the help of the software calib (version6.0.1) (Stuiver and Reimer, 1993), using the Southern hemispherecorrection of McCormac et al. (2004) and are expressed as years BC/AD(Tables 1, A1).Moreover, stratigraphicworkbyMorenoet al. (1994)andLara and Clavero (2004) evinces that more eruptions than those datedand listed on the GVP webpage occurred during the Holocene. Thevolcanic deposits intercalated between these registered eruptions areunderstood here to represent individual eruption events. As noindications for (a) extended time intervals such as the formation ofpalaeosols, or (b) very short time breaks such as gradual contactsbetween the volcanic deposits can be inferred from the stratigraphicsuccessions as given in the literature, the eruptions are temporarilyplaced between the dated ones by linear interpolation. We areconscious, however, that this introduces a piecewise artificial regularityinto the time series. The analysis of this extended and interpolatederuption record is performed in Appendix A, while in the main text wepresent theanalysis of the twelvedatederuptions. Lanínhas a geologicalrecord of eight documented eruptions in the Holocene, the last one in560 AD (Lara et al., 2004).

Following the usual convention (Klein, 1982), repose times aredefined as the time interval between the onset of successiveeruptions, thus neglecting the duration of each eruption. Someuncertainty is introduced because of the errors in the age dating ofthe deposits. The estimation of the error bounds for each individualeruption date is beyond the scope of this paper; nonetheless, this isless vital for the final results thanmay seem at first sight (more detailsare provided in Appendix B).

For most of these geological eruptions, the magnitude is notknown and no Volcanic Explosivity Indices (VEI, indicating mainly theerupted volume, besides several other physical eruption parameters(Newhall and Self, 1982; Simkin and Siebert, 1994)) have beenassigned.We will therefore use all geological eruptions without directVEI-filtering as would usually be done for historical data to eliminatesmaller eruptions. Also purely effusive eruptions are included. Thedata sets used are given in Table 1a,b. A first graphical impression canbe obtained by plotting the cumulative number of eruptions as afunction of time (Fig. 2a–c).

For Villarrica, a sharp increase in the slope of the plot occurs wherethe historical record begins, marking a transition towards anapparently higher observed eruption rate. Most of the historicaleruptions were classified as small eruptions (of VEI≤2), whileeruptions of VEIN2 have been extremely rare for Villarrica volcanoduring historical times (one eruption in 1948 of VEI=3, and one in1963 listed as “VEI 3(?)”). Such small eruptions, even if they tookplace, would be more likely to be lost from the geological recordbecause of erosional removal of the deposits. The geological recordtherefore seems to display larger eruptions at a lower eruptionfrequency. On the other hand, it can presently not be assessed whichand how many of the known historical eruptions will still begeologically traceable in the future. It is therefore complicated todecide how the geological and historical records could be combined;we present a tentative approach in Appendix C, for the main part ofthe paper we will arbitrarily neglect the historical record. In aprevious study the historical record has been used to estimate the

frequency and statistical probability of small eruptions (mainlyVEI=2) (Dzierma and Wehrmann, 2010a). Here, the statisticalanalysis of the geological record will serve to forecast possibly largereruptions.

While the analysis of historical eruption time series usuallyconsiders the present day as the end date for the analysis, thesituation is not as clear for the geological data. In the example ofVillarrica, the last geologically dated eruption occurred in 489 AD, butstarting in 1558, historical eruptions are documented up to the

http://www.volcano.si.edu

image of Fig.�2


present day. If we therefore only consider geological eruptions andperform the analysis from the point of view of the year 2010, weimplicitly assume that all the historical eruptions were so small thatthey would not leave a trace in the geological record. This may not betrue, in particular it would be questionable to consider a volcano asinactive during a time when the historical record proves it erupted athigh frequency (though with generally small VEI). We wouldtherefore overestimate the last repose time. A mathematically morecorrect way to analyse the geological data set would therefore be toplace ourselves in the year 489 AD, i.e., in the year after the lastgeologically documented eruption, or the year 1557, before the firsthistorically documented eruption. Of these two, the first alternativewill be preferred because of the arbitrariness of the year 1557, whichis not geologically relevant but depends on the onset of the historicalrecord.

If the two historical VEI=3 eruptions were included in the recordtogether with the geological eruptions, the data set would still appearnearly uniform, passing both the stationarity test and independencetest (see below). Therefore, it might be permissible to combine thesetwo records; for comparison, this is shown in Appendix C. Whileposing the problem of stringency, this data set has the advantage thatit allows us to extend the analysis to the present day, based on the factthat no VEI=3 eruption has been observed between 1963 and 2010.

No historical eruptions of Lanín are documented. The first reposetimes appear to be longer than the average, resulting in an eruptionregime with a flatter slope at the beginning than at later times, whichmay be taken as an indication that the geological eruption record atthis time is not complete. If a record incompleteness is evinced, thestatistical analysis should be performed excluding this time interval.However, for Lanín the present inactivity gives some indications thatalternating high- and low-activity episodes may be the characteristicbehaviour for the volcano. Therefore, it may be more appropriate toeither average over the entire record or consider it piecewise, retainingthe very early eruptions with large repose times, which may sample alow-activity regime. The choice of the onset of the eruption recordwillbe discussed below after performing several statistical tests. Thischoice is evidently arbitrary to some degree: if we were in the year561 AD, we might consider only the high-activity phase of Lanín to berepresentative andwould exclude thefirst low-activity phase from theanalysis as possibly incomplete, resulting in a different analysis andpredicting an increased eruption likelihood.

4. Method

4.1. The Poisson process and the exponential distribution

The simplest and ideal representation of a stochastic sequence ofpoint events is in terms of a Poisson process, which obeys theconditions that eruptions

1. are rare (meaning no two events will occur simultaneously)2. are independent of each other (the probability of each eruption does

not depend on what occurred before the time interval considered)3. occur at constant probability λ=1/t0 (characteristic time t0)

(for a mathematical treatment of Poisson processes, see, e.g., Cox andLewis, 1966).

The cumulative number of eruptions as a function of time (Fig. 2a,b)can provide a first insight into the behaviour of an eruption time series.In such a representation, the possible existence of different eruptionregimesor episodeswithpiecewise constant eruption rate canbe tested.In the same way, if the cumulative number of eruptions increaseslinearly with time, this gives a first indication that the eruption timeseries may be consistent with Poisson behaviour.

The independence property (2) is tested by calculating the cor-relation coefficient R between successive repose times, visualized in acorrelation scatter plot (Cox and Lewis, 1966; Mendoza-Rosas and De

la Cruz-Reyna, 2008). The square of the correlation coefficient (thecoefficient of variation R²) corresponds to the fraction of the variationin the data that can be explained by the model.

The stationarity condition (3), meaning that the parameters of thedistribution functions are constant over time, can be checked by amoving average test (Klein, 1982; De la Cruz-Reyna, 1996; Mendoza-Rosas and De la Cruz-Reyna, 2008). This is implemented here bytaking the average of each three successive repose times and plottingit as a function of time. Taking the average will smooth over short-term statistical fluctuations, but can make a possible trend in reposetimes visible. A condition for this test is that the eruption record mustbe complete. Conversely, for a stationary time series, the test candetect whether or not the data set is indeed complete.

4.2. Repose-time distribution fits

For a Poisson process, the probability of k events in an interval oftime t is

P kð Þ = exp −λtð Þ −λtð Þkk!

ð1Þ

If the survival function of repose times S(t) is defined as theprobability that the repose time T is larger than t, this is given by theprobability of k=0 events in the time interval T:

Sexp tð Þ = exp −λtð Þ ð2Þ

The corresponding repose time distribution F(t) is the exponentialdistribution

Fexp tð Þ = 1−Sexp tð Þ = 1− exp −λtð Þ: ð3Þ

which is the probability that the observed repose time T is smallerthan or equal to t.

A generalisation of the exponential function leads to the Weibulldistribution (Ho, 1991; Bebbington and Lai, 1996a,b; Watt et al.,2007):

SWB tð Þ = exp − ktð Þdn o

ð4Þ

where the shape parameter d determines whether the hazard rateincreases (dN1) or decreases (db1) with time since the last eruption.For d=1, the Weibull distribution reverts into the original exponen-tial distribution. TheWeibull model as amodel of simple failure ceasesto be appropriate in cases with more than one dominant process, e.g.,in the case where several competing processes act in a way that someparameters increase the probability of an eruption, while other,contemporaneously counteracting parameters cause a decrease ineruption probability. Such a situation can be addressed with the log-logistic distribution (Pareto III distribution, Connor et al., 2003), whichphenomenologically includes the different influencing factors asshape parameter α and scale parameter x0:

Slog tð Þ = 11 + t=xoð Þα ð5Þ

Since we cannot a priori know which situation best describes thereal behaviour of the volcanoes, we will try to fit all three possiblemodels to the observed repose time distributions by the method ofleast squares implemented in standard software packages (e.g., R,Matlab or Origin; here, OriginPro 8.0 and Origin 7.0 were used).

To select the best model representing a data set, the correctedAkaike Information Criterion (AICc) is used, which balances the misfitof the model against the number of parameters necessary to obtainthe fit (Akaike, 1973; Bebbington, 2007; Turner et al., 2008). In the


special case of least squares fitting to the same data set of n points, theAICc is given by

AICc = n⋅ lnΣi y xið Þ−yið Þ2

n+ 2k +

2k k + 1ð Þn−k−1

ð6Þ

where k is the number of free parameters, and y(xi) are the points ofthe model y(x) used to fit the data points (xi, yi), i=1,...,n (Sigiura,1978; Burnham and Anderson, 1998). The AICc does not possess anabsolute scale, rather, it provides a relative comparison betweendifferent models.

To absolutely assess the quality of the fits, the Kolmogorov–Smirnov-test (KS-test) is used, which is a standard goodness-of-fittest appropriate also for small data sets.

The fitting of the repose times can be performed either with givenscale factor A=“the total observed number of repose times” or withvariable scale, with good reasons for either choice (Dzierma andWehrmann, 2010b). We implemented both alternatives and evaluat-ed the fits using the AICc. For the data sets presented in this study, inall cases the AICc is comparable or better where A is held fixed,indicating that the simplest models adequately describe these rathersmall data sets.

In the literature, it is tacitly implied that the fits of the distributionfunctions should be applied to the data set of repose times as groupedinto equal-size intervals. This standard approach is certainly justifiedas long as the range of possible repose times is small — e.g., thehistorical repose times of Villarrica, which are generally shorter than20 years and can easily be grouped into intervals of one-year scale(Dzierma and Wehrmann, 2010a). This becomes difficult for geolog-ical records, because the order of magnitude of the repose times isvery large, from several decades to some thousand years. Ifequidistant interval boundaries are forced, a large number of intervalshave no observed repose time associated with them. The effect of thisis that in the fit performed on the basis of this data set, these reposetimes will be systematically weighted higher. We therefore proposethat for unevenly distributed data points, the actual repose timevalues should be used in the fits without classifying them intointervals. To compare the results of both methods, the interval-basedanalysis for Lanín is presented in Appendix D.

4.3. Allowing for different eruption regimes: mixture of exponentials

If the eruption time series data strongly suggests that thevolcano alternates between different regimes, i.e. time intervalsover which the eruption rate can be taken to be approximatelyconstant, but with different values for the individual intervals, thebehaviour may be better described and predicted based on amixture-of-exponentials distribution (MOED) (Mendoza-Rosas andDe la Cruz-Reyna, 2008, 2009). An alternative interpretation of thisfeature is to assume different eruption episodes interspersed on amore “quiet” background state (Nathenson et al., 2007). Statisticalmethods exist that allow for a mathematically sound and objectivedefinition of regime boundaries in time series, which can involvethe application of hidden Markov models (Mulargia and Tinti, 1986;Mulargia et al., 1987; Bebbington, 2007). However, within the scopeof this work, we decided for the intuitive, visual definition ofregimes as used for MOED estimation and MOED-based Bayesiananalyses (De la Cruz-Reyna, 1996; Mendoza-Rosas and De la Cruz-Reyna, 2008, 2009), which is based on visual identification ofpiecewise linear regimes in a plot of cumulative number oferuptions vs. time (Fig. 2b). Since the high- and low-activityregimes in the record of Lanín are clearly marked, although basedon a rather sparse data set, the visual identification is evident. ForVillarrica, the application of regime-based methods does not appearto be necessary as changes in the activity regimes do not seem to beas clearly defined. The first repose time may correspond to a phase

of lower activity: indeed, later analysis suggests that the record maybe incomplete here and the first eruption should be neglected.

The MOED is calculated as the weighted sum of the distributionfunction for each individual regime:

FMOED tð Þ = ∑mi = 1wi 1− exp −λitð Þð Þ; ð7Þ

SMOED tð Þ = ∑mi = 1wi exp −λitð Þ ð8Þ

∑mi = 1wi = 1 ð9Þ

where m is the total number of identified regimes, λi is the eruptionrate (number of eruptions in regime i divided by duration of theregime), and wi are weighting factors given by

wi =Dt−Di

∑mi = 1 Dt−Dið Þ ð10Þ

with Di being the duration of each regime and Dt the total duration ofthe sampled interval of time. This weighting scheme is chosen so thatthe low-activity regimes are up-weighted to ensure that the overallresults are not dominated only by the highest-activity regime (De laCruz-Reyna, personal communication, 2010).

In the method introduced by Mendoza-Rosas and De la Cruz-Reyna (2008), the parameters of the MOED are read from the plot.Therefore, no fit of the parameters is performed here, and theagreement of theMOEDwith the observed datawill not be ideal sinceit has not been optimised. The advantage in this method is the easyand intuitive parameter determination, whereas it would be hard toobtain a unique stable result from fitting a sum of exponentialdistributions to the data. It is also directly insightful because thechanges between different eruption episodes are apparent in theplot.

4.4. Bayesian approach

When interpreted from a Bayesian point of view, what the MOEDdoes is to provide an estimate of the eruption rate used for theforecast, based on the length of time period each eruption rate hasbeen observed to occur in the past.

From the volcanological/physical point of view, the Bayesiananalysis assumes that the eruptions occur as a Poisson process, butwith unknown present-day eruption rate. Themost probable eruptionrate at the present time is estimated from the past eruption rates,weighted by the length of the time interval they were observed:

w�i =

Di

Dtð11Þ

This yields the prior expectation of the eruption rate

λ�prior = ∑iw

�i λi ð12Þ

as the weighted average over the observed eruption rates λi up tonow.

Making use of our posterior knowledge of the number of eruptionsthat have occurred recently, the posterior expectation for the present-day eruption rate can be calculated based on the Bayes' theorem that

P λi jyð Þ = P y jλið ÞP λið Þ∑m

i P y jλið ÞP λið Þ ð13Þ

Fig. 3. (a,b) Stationarity check for Villarrica (upper) and Lanín (lower), both for thecomplete geological eruption record.


Here P(λi|y) is the probability of each λi reviewed in the light of ourknowledge that no eruption has occurred until the present time. Informal terms, we write y= “no eruption at the present time”, P(λi)=wi

the prior (estimated “intrinsic”) probability of λi and P(y|λi) thelikelihood of observing y=“no eruption” given λi, calculated as theprobability of event “zero eruptions” in a Poisson distribution withparameter λi:

P 0;λið Þ = λiΔtð Þ0 exp −λiΔtð Þ0! = exp −λiΔtð Þ = exp −nið Þ ð14Þ

where ni is the number of eruptions in the ith regime (see De la Cruz-Reyna, 1996 for a full explanation and, e.g., Gelman et al., 2009, fordetails on the Bayesian approach).

The P(λi|y) can then be interpreted as the posterior weights in thecalculation of the posterior expected value of λ, given the knowledgeof the data:

λposterior = ∑iP λi jyð Þλi ð15Þ

which can then be used for the prediction of the eruption probabilitiesusing a “single” exponential distribution with the a-posteriori mostlikely eruption rate.

4.5. Probabilistic eruption forecast

The probability that an eruptionwill occur in the future time t aftera givenwaiting time (age) x is found from the residual life distribution

Sx tð Þ = S x + tð ÞS xð Þ ð16Þ

This function gives the probability of survival until time x+ t underthe condition of survival through the waiting time x.

The memoryless property of the exponential distribution meansthat Sx,exp(t)=Sexp(t), i.e., an exponential distribution is not affectedby wear or aging. For the other distributions, this is not the case,which makes the choice of the end time of the analysis x relevant.

For the purpose of hazard assessment, the probability ofoccurrence of at least one eruption in a specific time t in the futureis evaluated as (Marshall and Olkin, 2007)

P T≤s + t jT N sð Þ = 1−Sx tð Þ = 1−1−F s + tð Þ1−F sð Þ ð17Þ

where s is the time that elapsed since the last eruption.

5. Results

5.1. Correlation of successive eruptions and stationarity

The results for the independence and stationarity check are givenin Table 2. The correlation coefficients do not show significantcorrelation at the 5% level, neither for the total record nor if the firsttwo eruptions (exceptionally long repose times, possibly due to

Table 2Correlation coefficient R, coefficient of variation R2, and result from stationarity checkfor both volcanoes and different restrictions on the eruption record.

Volcano R R² Stationarity test

Villarrica (total) 0.24 0.06 1st point out of 1σ-envelopeFrom year 4477 BC −0.16 0.03 OkLanín (total) 0.31 0.10 1st point out of 1σ-envelopeFrom year 590 BC −0.59 0.35 OkBased on only 6 eruptions (4 samples)—no reliable statistic

record incompleteness) are excluded. We can therefore assumeindependence of successive repose times.

For both volcanoes and all time intervals, the stationarity check(when based on the 1σ-criterion) is satisfied except at the beginningof the record. However, while it is statistically possible that stochasticfluctuations cause some of the observed repose time averages to fallmore than 1σ distant from the average, it is improbable that thisshould happen exactly for the first repose time average as it does inthe case of Villarrica (Fig. 3a). We therefore exclude the first longrepose-time eruption and perform the following analysis for theeruption records starting in the year 4477 BC.With this restriction, thecumulative number of eruptions for Villarrica increases almostlinearly with time, which is what we would expect for a Poissonprocess.

In the case of Lanín, the long first repose-time average alsosuggests that the analysis should be based on the eruption recordstarting in the year 590 BC. This would certainly be the decision madeif the analysis was performed in the year 561 AD following the lasteruption. However, with hindsight, no eruption has been reported orreconstructed in the past 1450 years, which lends some credibility tothe longer repose times. It is possible that Lanín intrinsically switchesbetween high- and low-activity episodes (Fig. 2c). While this wouldcontradict the stationarity property, the inclusion of the low-activityregime in the fit may better reflect the real behaviour, which calls forthe application of techniques taking into account different regimes(mixture of exponentials and Bayesian method); therefore we willperform the analysis for the complete geological record for Lanín.

image of Fig.�3

Table 3Fit parameters for the repose time fits for Villarrica and Lanín, geological record starting in the years 4477 BC and 9240 BC, respectively. The K–S-significance value at the 5% level is0.432 for Villarrica and 0.457 for Lanín, respectively, so all fits pass this test.

Fit A t0/yr k/yr−1 d x0/yr α χ²/DoF R² K–S value AICc

Villarrica (from year 4477 BC)Exp 8.0±0.0 632.5±77.3 0.691 0.908 0.221 0.09Weibull 8.0±0.0 0.00157±0.0002 1.138±0.242 0.871 0.876 0.257 1.87Log–log 8.0±0.0 415.2±64.4 1.5285±0.319 0.900 0.896 0.243 2.61

Lanín (from year 9240 BC)Exp 7.0±0.0 380.3±52.1 0.449 0.925 0.269 −1.1Weibull 7.0±0.0 0.00307±0.0004 1.515±0.380 0.444 0.921 0.213 1.8Log–log 7.0±0.0 225.7±36.1 2.195±0.635 0.404 0.962 0.247 2.6


5.2. Repose time fits

All fit distributions (Table 3, Fig. 4a,b) pass the goodness-of-fit testat the 5% level. According to the AICc, the exponential distributionclearly gives the best fit. This is probably due to the aspect discussedabove: the data set is small (less than ten observed repose times),therefore including many free parameters would risk overfitting thedata.

It should be mentioned that while the fits with variable Aconsistently give worse AICc results, also they would pass the KS-test. However, in the case of Lanín, the Weibull function with variableA cannot be fit stably — this is probably due to the large range inrepose times, where the very long repose times (N1000 years) arehard to reconcile with the smaller repose times (100–400 years), and

Fig. 4. (a,b) Repose time fits for Villarrica (a) and Lanín (b).

the shape parameter of the Weibull function retrieved from the fitdepends very much on the starting values and in some cases does notconverge.

5.3. Fitting interval-based data

If the fits are performed for data grouped into intervals, forVillarrica the changes are slight — most of the fit parameters agreewithin the given uncertainty, and neither the choice of the best modelnor the overall shape of the plots are changed noticeably. For Lanín,however, a different picture emerges (Appendix D). Here the fitparameters found for different-sized intervals (100 years vs.1000 years) neither agree with each other nor with the fits basedon the ungrouped data. In particular, the shape parameters do notconsistently always give increasing or decreasing hazard rates, butchange from one to the other— this means not even the gross shape ofthe curves can be retrieved stably from the interval data, andpredictions as to waxing/waning volcanic activity may be mistaken.

Since the number of data points that enter the fit now depends onthe number of intervals and not on the number of actually observedrepose times, the AICc values can no longer really assess the quality ofthe models, because for the apparently large data sets the morecomplicated models (log–log andWeibull as opposed to exponential)perform better. This would lead us to prefer the log-logistic model forLanín, assuming that competing processes are at work, whereas in factthere is no reason to believe so.

Finally, if the scale A is varied in the fits, this does not usually resultin significantly different fit parameters in the “normal case” (based onthe point data); it does, however, for the interval-based fits for Lanín.This is also due to the fact that many “empty” intervals contribute tothe fit, overweighting the long repose times.

In summary, for uniform data sets like the Villarrica record, forwhich all repose times are of the same order of magnitude, theclassification of the data into a sensible number of intervals ispermissible without difficulties. This applies particularly to thestudies of the generally much denser historical data sets (Dziermaand Wehrmann, 2010a). For scarce data, and a wide range of reposetimes, the classification into intervals can in the worst case result insevere distortions of the fits and hence both the interpretation of thecharacteristic volcano behaviour (increasing/decreasing hazard rateetc.) and the predictions of future eruption probabilities.

Table 4MOED regimes and parameters for Lanín volcano.

Regime Start year End year Nr. erupt. Rate(λi/yr–1)

Weight(wi)

1 −9240 −590 2 2.31e−4 0.11552 −590 560 6 0.00522 0.44893 560 2009 0 0.0 0.4356

image of Fig.�4

Fig. 5. (a,b) Probability of at least one eruption as a function of time of (a) Villarrica,starting in the year 489 AD, and (b) Lanín, starting in 2010 (b).

Table 6aProbability of at least one eruption of geological magnitude of Villarrica Volcano withinthe next T years, in %, as starting from 489 AD.

T Exp. Weibull Log–log

0 0.00 0.00 0.001 0.16 0.08 0.022 0.32 0.16 0.045 0.79 0.43 0.14


For an interval-based analysis of the eruption time series data, wepropose that the intervals should satisfy the following criteria:

• The number of intervals should be approximately equal to thenumber of the observed repose times. This makes sure that the AICc

can sensibly be applied to identify overfitting.

• The interval spacing should be large enough so that there is no (orhardly any) intervals corresponding to repose times that have notbeen observed (“empty intervals” forming the horizontal linesegments in Fig. D1 for 100-year intervals). This avoids over-weighting of parts of the data set, particularly at long repose times.

• The interval spacing should be small enough to distinguish betweendifferent repose times (avoiding high steps like the first value inFig. D1 for 1000-year intervals). This allows for a good resolution ofthe “fine structure” and avoids underweighting short repose times.

In many historical and some geological data sets, the choice ofintervals according to these rules is straightforward. In cases such asLanín, this may be problematic, particularly since both very long andrather short repose times occur in the record, which may indicatevarying eruptive regimes. In such cases, a logarithmic grouping ofintervals might be envisaged — this may be worthwhile furtherinvestigation. At this point, however, we prefer again the pragmaticmethod of avoiding the possible problems from interval classificationand using the ungrouped point data for the fits.

5.4. Lanín — regime-based analysis methods

While the shortened Villarrica eruption record can be taken torepresent a single eruption regime of constant volcano behaviour,Lanín appears to change between regimes of high vs. low activity.Since no eruption has occurred for about 1450 years, the last regimehas been defined as one of quiescence, with an eruption rate of zero.Although it seems bizarre to assign a regime on the basis of no data,this is mathematically sound and the best representation of the stateof volcanic activity we are currently witnessing.

Since the MOED (Table 4) is not obtained from a fitting procedurebut simply “read off” from the data, no AICc is calculated. Thegoodness-of-fit is assessed with the KS-test, giving a difference of0.466, which just fails. We must, however, take into account that thisis based on data read off from the plot without any kind ofoptimisation. Intrinsically, since theMOED averages over the differentregimes, it is accompanied by deviations from the data at any giventime (in any given regime). It is not the aim of a perfect fit that theMOED pursues, but to give a good estimate of the average behaviourwhich will serve to predict the present behaviour of the volcano.

The Bayesian approach is followed based on the same regimesidentified for the MOED (Table 5), giving a posterior eruption rate ofλposterior=1.03e− 4 yr−1, which will be used in the followingprediction of the statistical eruption probabilities.

6. Probabilistic eruption forecast and discussion

The statistical eruption probabilities were calculated for Villarricaas in the year 489 AD and for Lanín as at the beginning of 2010 (Fig. 5a,b, Tables 6a, b).

Table 5Calculation of posterior eruption rate for Lanín Volcano using the Bayesian approach.λposterior=1.03e−4 yr−1.

Erupt rateλi(yr−1)

DurationΔt(yr)

PriorP(λi)

LikelihoodP(y|λi)

PosteriorP(λi|y)

2.31e−4 8650 0.769 0.135 0.4450.00522 1150 0.102 2.48e−3 1.09e−40.0 1449 0.129 1 0.554

6.1. Villarrica

Seen from the year 489 AD, the probability of at least one eruptionof geologicalmagnitude up to 2010 is between88% and 94%dependingon the fit function (Fig. 5a). This may be an indication that it wouldhave been justified to include VEI=3 historical eruptions in thegeological record. A repetition of the analysis including the two

10 1.57 0.92 0.3820 3.11 1.98 1.0250 7.60 5.44 3.89

100 14.62 11.52 10.32200 27.11 23.55 24.80500 54.64 53.25 57.12

1000 79.42 81.21 79.331521 90.97 93.24 87.932000 95.77 97.47 91.715000 99.96 100.00 97.82

10,000 100.00 100.00 99.23

image of Fig.�5

Table 6bProbability of at least one eruption of Lanín Volcano within the next T years, in %,starting from 2010.

T Exp Weibull Log–log MOED Bayes

0 0.00 0.00 0.00 0.0 0.01 0.26 1.00 0.13 0.0 0.02 0.52 1.98 0.27 0.0 0.05 1.31 4.89 0.66 0.0 0.1

10 2.59 9.55 1.32 0.0 0.120 5.12 18.22 2.61 0.1 0.250 12.32 39.68 6.35 0.2 0.5

100 23.12 63.93 12.11 0.4 1.0200 40.89 87.43 22.16 0.7 2.0500 73.14 99.56 43.81 1.8 5.0

1000 92.79 100.00 64.11 3.3 9.82000 99.48 100.00 81.74 5.9 18.65000 100.00 100.00 94.72 11.0 40.2

10,000 100.00 100.00 98.31 14.4 64.3


historical VEI=3 eruptions (Appendix C) reveals that the time seriesfrom this combination can be considered as stationary and complete.This would mean that the predicted geological magnitude eruptionhad already occurred. For a cautious interpretation, this might point tothe minimal magnitude a geological eruption would need to have inorder to become documented in the record.

The distribution functions and eruption probabilities retrievedfrom this approach agree relatively well with those found before(Tables C1, C2). If we therefore assume that the last historicallydocumented VEI=3 eruption in the year 1963 was indeed ofgeological magnitude, we can calculate the future eruption probabil-ities as seen from today (2010). If this is done, the probability that aneruption of similar size occurs in the next 100 years is about 13–17%(Table C2).

In the case of Villarrica, the comparatively good quality of the data –and particularly the fact that only one eruptive regime had to beconsidered – makes the fitting procedure very stable with respect tothe eruption forecast: the probabilities predicted by the three differentdistribution functions agree closely.

6.2. Lanín

Despite the long recent inactivity of Lanín of 1450 years, thesurvival functions based on the fits give fairly high probabilities forfuture eruptions, and very large variations between the individualfunctions (Table 6b, Fig. 5b). The Weibull distribution, which isincreasing hazard rate for Lanín, results in an extremely highprobability of future eruptions, exceeding 60% within the coming100 years. Since the likelihood of an eruption within this time span ismuch lower for the exponential and log-logistic distribution (about20% and 10%, respectively), and also in the light of the regime-basedmethods, we believe this is an overestimate.

Nevertheless, all the fit distributions result in a probability of aneruption within the coming 1000 years of more than 50%, which mayseem surprising from common association since no eruption isdocumented during the last 1450 years. This impression is muchmore adequately reflected in the Bayesian approach, which consis-tently results in much lower eruption probabilities of about 1% in thenext 100, and about 10% in the next 1000 years.

However, all these distributions suffer from the problem that theycannot take the possibility into account that a volcano may be extinct.All of them average over the past eruption rates in a way that onesingle past eruption will cause the outcome to be non-zero andtherefore always predict the volcano to be currently active. Mathe-matically, the future eruption probability for non-zero parameters

will always tend to the limit of 100%, resulting from the fact that thesurvival function S in Eq. (17) tends to zero.

This also applies to the MOED, as long as all eruption regimes havenon-zero eruption rate λi:

limt→∞

P T ≤ s + t jT N sð Þ = 1− limt→∞

∑wi exp −λisð Þ exp −λitð Þ∑wi exp −λisð Þ

= 1−0 = 1ð18Þ

since limt→∞

exp(−λit)=0 ifλi≠0.However, if there is at least one zero-eruption rate regime (λj), the term exp (−λjt) becomes 0 and theabove limit is unequal to 1:

limt→∞

P T≤s + t jT N sð Þ = 1−wj exp −λjs

� �

∑wi exp −λisð Þ= 1−const:≠b1

ð19Þ

Therefore, the MOED is the only one of the methods used here thatallows the eruption probability of Lanín to never reach 100%, nomatter how far we extrapolate into the future. Here, we find aprobability of about 16% for the very far future, meaning that theprobability of Lanín ever erupting again is about 16% according to theMOED approach. Accordingly, the probability of witnessing aneruption in the nearer future is much lower for the MOED, with lessthan 5% probability for an eruption in the next 1000 years.

It remains open whether Lanín's recent inactivity may or may notbe representative for its overall behaviour and state of activity.Depending on how strong this information is weighted in themodelling, very different conclusions are drawn regarding the presenthazard. If the present regime is still ongoing, we would indeed bejustified to just extrapolate this last regime into the future, meaningthat we could assume Lanín to be extinct. On a geologic time scale,however, 1450 years of dormancy cannot be classified as long, and itcannot be ruled out that a new regime is not just beginning. In settingssuch as the one explored in this study, where the large-scale drivingforce of the eruptions – here the subduction of the Nazca Platebeneath the South American Plate – remains active, due care shouldbe exerted before declaring a volcano to be extinct.

7. Applicability of the method and evaluation of the results

Although statistical hazard assessment has mostly been appliedto historical eruption records, there is no a priori reason why thetechnique should not be applicable to geological records. However,several complications occur due to the variability in the availableinformation on past eruptions that are only reconstructed fromgeologic evidence. An important aspect to ensure comparability ofthe eruptions within one time series is that the eruptions are ofsimilar dimensions. The erupted volume has to be constrained fromthe deposit by detailed field mapping, corrected for variabledegrees of non-preservation, which is a function of the climaticconditions with regard to erosion and weathering, as well as ofremoval or sealing by subsequent eruptions. Since such studiesrequire considerable effort, for many past eruptions the volume ofthe deposit has not been determined. Consequently, no VEI hasbeen assigned – thus impeding to filter the eruption record by anumerically expressed magnitude – but as the “completeness” ofthe record defines itself under the same conditions for eachvolcano, eruptions of similar size experience a similar risk of beingentirely removed or weathered to absolute indiscernibility.Therefore, from a certain volume threshold upwards, there is agood probability that large eruptions of the recent geologic past areequally recorded, although the method applied here only focuseson the occurrence of an eruption as an event, without distinguish-ing between large and very large eruptions.


Difficulties may arise during the analysis in response to the dataset quality. Some of those we pointed out at the example of Lanín:

• Short eruption records (i.e., fewknowneruptions),which increase therisk of overfitting the data with too complicated models— in a worstcase of exaggerated fitting this may verge on futility, when themodelbecomes detached from what it is physically supposed to express.

• Much temporal variability in the record: the uncertainty in theprediction is severely increased as soon as it becomes necessary toassume several regimes of different eruption rates. Even though thedefinition of the regimes itself may be obvious, and slight variationsin the definition have a negligible influence on the outcome, thedifferent approaches treat regimes very differently. While theexponential, Weibull and log-logistic distributions are notmeant forapplication to regimes and therefore cannot adequately treat thisphenomenon, the Bayesian and MOED approaches are bothdesigned particularly for the application to changing regimes, butstill give very different results.

• The former problem is particularly aggravated if the present regimeis one of inactivity. In this case, only theMOED can take into accountthe basic possibility that the volcanomay never erupt again. Even incases in which the model predicts this to be likely, however,categorising a volcano as extinct has to be done hesitatively, asmany volcanoes have been known to reawake after long periods ofdormancy (e.g., Connor et al., 2006).

• Independent of the current state of activity or dormancy, afundamental decision has to be made on how to define the timereference for theanalysis. Nogenerally valid advice canbe issuedhere,every volcano's eruption record needs to be assessed individually.

In the case of Lanín, the eruption record is both short and fairlyvariable in terms of the repose time lengths, evoking the problemsmentioned above when performing the statistical analysis. However,within this wider frame of variability, the method can be appliedreasonably well to Lanín. We provide the uncertainty-marked resultshere to demarcate the range of probabilities for expected eruptions atLanín. At the present time, since no geophysical/geochemicalmonitoring data are available for Lanín (e.g., seismicity, degassing orthe like), the statistical analysis is our onlyway of assessing the hazardposed by this volcano.

In the case of “good records” such as the one of Villarrica – with asubstantial amount of eruptions that are not too variably spaced forthe part of the record considered – the method performs well andproduces stable, comparable results for the different fit functions. Theonly major issue in the course of implementing the analysis was aboutsetting the time reference for the end point of the considered eruptionrecord. When analysing volcanoes that have been recently active andthe geological record has to be combined with a historical record, thequestion of how to combine the two records needs to be assessed.

Overall, given the amount of possible problems that must beaddressed when performing this kind of analysis, the applicability ofthe statistical method to geological eruption series is surprisinglygood even where the geological record is far from perfect. Still, it isimportant to keep in mind the number of difficulties that may ariseduring the analysis, of which this study could only give an exemplaryinsight. In spite of these drawbacks, the assessment of the hazardposed by large eruptions almost inevitably has to rely on thegeological record, since these eruptions are so rare that the historicalrecord alone does not generally include enough data to allow forstatistical analysis. Therefore, the analysis of the geological recordcomplements the forecast derived from the historical record byincluding also the very high-hazard, large eruptions, that devastat-ingly may interrupt the sequence of the “usual” smaller eruptions (of

VEI=1–2 at Villarrica), thus allowing us to characterise the volcanoon both time scales at both eruption dimensions.

8. Conclusions

The implementation of standard failure-analysis statistical tech-niques to the geological eruption record of the very active volcanoVillarrica and its presently dormant neighbour Lanín in the ChileanSouthern Volcanic Zone allows us to estimate the probabilities offuture eruptionswithmagnitudes large enough to leave amark in thegeological record.

For Villarrica, the geological record shows nearly Poissonianbehaviour and appears to have operated with approximatelyconstant eruption rate over the past 6000 years. The time-seriesanalysis predicts that the probability of at least one eruption ofgeological magnitude occurring in the time postdating the lasteruption of the geological record until today is close to 100%. Thissuggests that either the VEI=3 eruptions in 1948 and 1963 weresufficiently strong to meet this prediction, or that the volcano is(over-)due to produce a major eruption. If it is assumed that theeruption of geological magnitude has already taken place, theprobability of another such eruption occurring in the next100 years is about 20%.

In the case of Lanín, the applicability of the exponential, Weibulland log-logistic distributions to the geological record is doubtful, sincethe volcano appears to have experienced alternating high- and low-activity regimes. Although the non-observation of activity during thepast 1450 years may be interpreted as a sign of extinction, such aperiod is not exceptionally long in a volcano's lifetime and might beonly a repose period between regimes of higher activity. Only theMOED can take into account the possible dormancy descending tofinal extinction, resulting in a probability of about 16% that thevolcano will ever erupt again in the future.

Despite some methodological limitations as discussed, the analysisperformed here substantiates the applicability of the statisticaltechnique even to geological eruption records of limited comprehen-siveness. If careful attention is paid to thedecisions that have to bemadeduring theprocedureaswepointedout,misunderstandingof the resultscan be avoided and the analysis will serve as a valuable component ofvolcanic hazard assessment. Dealing with eruption records on geologictime scales is accompanied by the possibility that the probabilisticeruption forecast may reach a tangible likelihood at a time in the futuremuch beyond a human lifetime. While this may seem to range in aremote sphere of theoretical imagination for some volcanoes, it issuesan even more emphatic alert for those volcanoes that have not shownany eruptions of geological magnitude in the extended recent past. Thestatistical analysis may reveal that a volcano is overdue to erupt; and asthe geological record documents the rather large-magnitude eruptions,any predicted future event in such series is likely to be of destructivenature geographically not limited to the immediate vicinity of thevolcano. While active volcano monitoring with regard to hazardassessment is performed by surveillance of deformation, seismicactivity, degassing, temperature, and magma composition, statisticalanalyses as the ones presented here contribute a parametric descriptionof the volcano's chronological character, thus allowing us to integratethe physico-chemical behaviour of the volcano with the temporallikelihood of future eruptions.

Acknowledgements

We strongly thank David Pyle and Manny Nathenson for verydetailed reviews and helpful comments that greatly improved themanuscript. This publication is contribution no. 193 of theSonderforschungsbereich 574 “Volatiles and Fluids in SubductionZones” at Kiel University, funded by the German ResearchFoundation.


Appendix A. Using the complete known geological recordfor Villarrica

As stratigraphic data have shown that more eruptions occurred atVillarrica volcano during the Holocene than those that have beennumerically dated and are presented in the GVP, we here process theextended geological record which includes the non-dated eruptionsintercalated between the dated deposits. The ages for the non-datederuptions were interpolated to identical time intervals between theknown eruption dates, which should be taken as an approximation,although it introduces an artificial regularity into the time series(Table A1).

The cumulative number of eruptions (Fig. A1) shows a relativelyuniform increase. However, since this uniformity is in part producedby the interpolated ages, this cannot be taken as a proof ofstationarity. Conversely, the interpolation has an adverse effect onthe stationarity test, which can also not be interpreted as convincingevidence. In themoving average test (Fig. A2), the variation in 5-pointrepose time averages is larger than one standard deviation around themean. Even though a relaxed condition of two standard deviations ofvariation around themeanwould include all 5-point averages, still thefact that all repose times prior to ca. 2500 BC fall above the mean andall later repose times below themean points to non-stationarity in therecord. In the same way, the deviation of the last five 5-point averagesfrom the mean (all more than one standard deviation away from themean) is inconsistent with stationarity. Since the non-dated eruptionsbetween two known events were interpolated to occur at equidistant

Fig. A1. Cumulative number of eruptions for Villarrica volcano, extended geologicalrecord. While the frequency of eruptions increases somewhat in the younger part of therecord, the overall behaviour can still be well approximated by a linear increase in thenumber of eruptions with time, suggesting that the record may be near-stationary(dashed green line).

Fig. A2. Stationarity test for Villarrica Volcano, extended geological record.

Fig. A3. Scatter plots showing the result of the independence test for the extendedgeological record of Villarrica volcano. The plots show the observed repose times as afunction of the preceding repose times. Left: result if the independence test isperformed indiscriminately in the “standard” way (linear regression shown by redline). Right: test after deleting the artificially produced diagonal elements.

time intervals, the scatter of the single repose times was artificiallyreduced. Therefore, the scatter of each five successive repose timesaround the mean (used to estimate the mean of the 5-point standarddeviations) is preconditioned to be very small, and cannot be taken asrepresentative for the overall variation in repose times. Due to thiseffect, the stationarity test cannot be applied to this data set in thetypical way, and we do not consider its result for the further analysis.As stationarity was confirmed for the dated eruptions, we assume italso holds true for the extended record.

Fig. A4. Fits to the repose time distribution function for Villarrica volcano, extendedgeological record.

Fig. A5. Probability of at least one eruption of Villarrica, starting in the year 489 AD, andusing the extended geological record.


The applicability of the independence test in this scenario is equallybiased. The equidistant interpolation produces a suite of equal reposetimeswhich are plotted along the diagonal (dashed green line in Fig. A3a)

Table A1Extended geological eruption record of Villarrica Volcano, with datingmethod, taken from thal. (2010). Uncalibrated radiocarbon ages were converted using calib (version 6.0.1) (Stuiverare taken from the correction data set of Stuiver et al. (1993) as they impinged at the rangeevents, were placed between the under- and overlying dated deposits at regular time intervastatistical analysis, these dates shall not be taken as real eruption age constraints. The temp

Uncorrected year (BP) 1 σ calibrated results Mean of converted years

13,850 12,712 BC13,800 12,645 BC

11,841 BC11,038 BC10,235 BC9,432 BC

10,600 8,629 BC8,280 BC7,930 BC

8,600 7592:7571 7,582 BC6,804 BC6,028 BC5,252 BC

5,700 4501:4452 4,477 BC3,611 BC

4,200 2863:2835 2,746 BC2817:28062759:27172710:26652645:2639

4,100 2618:2609 2,574 BC2597:25952582:25632534:2493

2,177 BC3,510 1878:1839 1,781 BC

1828:17921785:17261725:1691

1,415 BC1,049 BC

2,600 792:756 685 BC684:669606:604

491 BC298 BC105 BC88 AD

1,800 253:263 281 AD277:329

1,600 441:454 488 AD457:485532:556

or close to it (by rounding the repose times to discrete values), thusleading to a seemingly significant correlation. If the diagonal is deletedfrom the scatter plot, no significant correlation is found, but it isquestionable if this test is valid in this special case. Since the extendedgeological record canbe taken to bemore complete than the dated record,we now assume stationarity and completeness as a working hypothesis.

Comparing the eruption probabilities (Table A3, Fig. A5) from theextended geological record with those from the shorter recordwithout interpolation, we find:

e sanlimlso

D

TRIIIIRIIRIIIRIR

R

IR

IIR

IIIIR

R

Years in the future

tratigraphic logs of Moreno et al. (1994d Reimer, 1993), with southern hemispit of the software. Non-dated deposit

—we emphasise that this is a tephrochral interpolation also leads to a piecew

ating method Eruptive styl

ephrochronology Explosiveadiocarbon Explosiventerpolated Explosiventerpolated Explosiventerpolated Effusiventerpolated Explosiveadiocarbon Explosiventerpolated Effusiventerpolated Effusiveadiocarbon Explosiventerpolated Effusiventerpolated Explosiventerpolated Explosiveadiocarbon Explosiventerpolated Effusiveadiocarbon Explosive

adiocarbon Explosive

nterpolated Effusiveadiocarbon Explosive

nterpolated Explosiventerpolated Explosiveadiocarbon Explosive

nterpolated Explosiventerpolated Effusiventerpolated Explosiventerpolated Explosiveadiocarbon Effusive

adiocarbon Explosive

Eruption probability (%) derived from

Long record

), Lara and Clavero (2004), andhere correction; values for thes, understood to represent indironological approximation onlyise artificial regularity in the ti

e Name of unit

Cudico pyroclastic flowLican ignimbritePucura pyroclastic flowScoria fallout depositPucura lavasScoria pyroclastic flowTurbio pyroclastic flowBasaltic lavasBasaltic lavasVoipir pyroclastic flowEl Pirao lavasScoria pyroclastic flowChépica pyroclastic flowCandelaria pyroclastic fl

Antumalal lavasScoria pyroclastic flow

Afunalhue pyroclastic fl

Zanjón Seco lavasPucon ignimbrite

Chaimilla 1Chaimilla 2Pedregoso pyroclastic s

Refugio scoria falloutMolco basaltic lavasSki scoria pyroclastic fl

Los Nevados pyroclastiZanjón Seco pahoehoe

Cónquil pyroclastic flow

Short record

5
0–1 0–1 10 0–2 0–2
100
5–16 10–15 1000 83–91 79–81
Formost intermediate times (up to century scales), the predictionsof the long record agree very well with those from the short record; attimes far in the future, the long record gives slightly higher eruptionprobabilities, probably due to the fact that large repose times are

Silva-Parejas etfirst three datesvidual eruptionto facilitate theme series.

and fallout

ow

and fallout

ow

urge

owc flowbasaltic lavas

Table A2Fit parameters for the repose time fits for the extended geological record of Villarrica. The KS-significance value at the 5% level is 0.328, so all fits pass the KS-test. Based on the AICc,the exponential, Weibull and log-logistic distributions all perform comparatively.


Exp 27±0 567.2±37.3 8.295 0.877 0.217 20.6Weibull 27±0 0.00184±0.00008 1.4156±0.119 5.502 0.916 0.179 19.5Log–log 27±0 394.3±16.8 2.097±0.184 5.152 0.927 0.175 19.7

Table A3Probability of at least one eruption of a geological magnitude of Villarrica Volcanowithin the next T years, in %, as starting from 489 AD, based on the extended geologicalrecord.


0 0.00 0.00 0.001 0.18 0.02 0.002 0.35 0.05 0.005 0.88 0.16 0.02

10 1.75 0.39 0.0520 3.46 0.98 0.2150 8.44 3.44 1.35

100 16.16 8.81 5.44200 29.72 21.69 19.58500 58.59 58.98 62.29

1000 82.85 90.69 87.582000 97.06 99.82 96.795000 99.99 100.00 99.52

10,000 100.00 100.00 99.89

Fig. C1. Cumulative number of eruptions for the short geological record of Villarrica,starting in the year 4477 BC, which includes the two documented historical VEI=3eruptions. The eruption rate is approximately constant, giving rise to a near-linear increasein thenumber of eruptionswith time. Thedata set passesboth the stationarity test at the 1-sigma level and the independence test (correlation coefficient R=−0.1, plots not shown).


under-represented in the extended and interpolated eruption record.For hazard assessment purposes, however, where timescales up tocenturies are the most relevant, the good agreement builds ourconfidence in the representativeness of the eruption records used.

Appendix B. Influence of dating errors in the analysis

The dating of the geologically recorded eruptions involves uncer-tainties both in the age measurements themselves and ambiguity inseveral sections of the calibration curve for the radiocarbon method.This problem is common in volcanology; sometimes statigraphicallysuccessive deposits are dated with uncertainties large enough so thatthe error bounds overlap. In such cases, the stratigraphic considerationsare more decisive for establishing the temporal eruption sequenceprovided that no overturn has occurred (which is a reasonablepresumption for unconsolidated volcanic material); while less impor-tance is attributed to the dating uncertainties.

For the statistical analysis, processing the error bars of individualnumerical eruption dates is not straightforward when attempting toconfine the presentation to elemental statistical techniques only.However, the effect of age uncertainties is not as severe as may at firstglance seem when thinking of the propagation of errors in thecalculation of the repose times (as the difference in time of occurrenceof successive eruptions) and the distribution fits based on theresulting data set. This can most readily be illustrated by taking alook at the exponential distribution. Let us assume that the “real”volcano behaviour is perfectly described by an exponential distribu-tion with eruption rate Λ. Our observations are one random sample ofthis underlying “true” distribution. We calculate the sample eruptionrate λ as an estimator of Λ. However, if several samples could bedrawn (if another time interval of observations were available or ifanother volcano obeying the same distribution function existed), asomewhat different result λi would emerge for each sample i.Considering λi to be unbiased estimators of the “real” eruption rateΛ, the expectation E(λ) would give the real value of the underlyingparameter Λ. Therefore, what we are observing in this scenario is justone possible realisation of sampling from the exponential distribu-

tion; in a repeat experiment, the dates of each eruption would varywith respect to what is observed. Given this statistical variationcaused by the random process, the dating errors of the eruptions donot bias the results as strongly as might be the case for a deterministicprocess. Since the uncertainties are not systematic in a way to shift allthe repose times uniquely towards shorter or longer repose times, theoverall distribution of the repose times is still approximately retained.Indeed, if one eruption actually occurred earlier than the assumeddate, this would decrease the preceding repose time and increase thesucceeding repose time.

Indeed, in the simple case of the exponential distribution, theeruption rate could be approximated simply by dividing the totalnumber of observed eruptions by the total observed time. This iscertainly a rough estimate; although it is unbiased, this will not yieldas efficient an approximation to the real Λ as fitting the datadistribution; but the expectation value E(λ) would still be Λ. Aslong as we assume completeness, i.e., that there is no error in thenumber of eruptions, the estimated value of Λ based on this methodwould not depend on any uncertainties in the intermediate eruptiondates, and include an uncertainty only because of the first eruptionand last eruption date. If we estimate the error in Λ introduced by this,using, e.g., 10% uncertainty of the total observed time, the uncertaintyin eruption rate would also be approximately 10% (making use ofGaussian error propagation).

In the case of theWeibull and log-logistic distributions, the shape ofthe fit distributions dependsmore strongly on the distribution of shortas compared to long repose times. Still, as long as no systematic shifttowards shorter or longer repose times is introduced by the datinguncertainties, the main properties of the distributions should not bechanged. While being aware of the difficulties of the statisticalapproach where dating errors are as large as they generally are forgeological data sets, we believe the approach remains justified.

Appendix C. Combining the historical VEI=3 eruptions with theVillarrica geological record

Table C1Fit parameters for the repose time fits for the geological record of Villarrica, starting in the year 4477 BC, which includes the two documented historical VEI=3 eruptions. The KS-significance value at the 5% level is 0.391, so all fits pass the KS-test. Based on the AICc, the exponential, Weibull and log-logistic distributions all perform similarly well, of which theexponential function provides a slightly worse model for the data.


Exp 10±0 712.3±45.9 0.641 0.915 0.248 0.98Weibull 10±0 0.00133±0.0002 0.7511±0.129 0.581 0.917 0.265 5.01Log –log 10±0 434.8±56.4 1.009±0.174 0.574 0.933 0.282 6.12

Table C2Probability of at least one future eruption of Villarrica, regarded from the year 2010, based on the restricted geological record from 4477 BC on, which includes the two documentedhistorical VEI=3 eruptions. Again, for the relatively near future (up to the centuries scale), the predictions agree well with those calculated without the inclusion of the VEI=3historical eruptions, with deviations of only a few percent. For larger time scales, lower eruption probabilities are predicted.


0 0.00 0.00 0.001 0.14 0.20 0.212 0.28 0.40 0.415 0.70 0.98 1.02

10 1.39 1.92 2.0220 2.77 3.73 3.9650 6.78 8.62 9.36

100 13.10 15.54 17.15200 24.48 26.56 29.34500 50.44 48.46 51.07

1000 75.43 68.58 67.722000 93.97 86.43 80.845000 99.91 98.27 91.40

10,000 100.00 99.90 95.53


Appendix D. Interval-based fitting — the case of Lanín

Fig. D1. Interval-based fits for Lanín (a: 100-year intervals and b: 1000-year intervals).

Table D1Fit parameters for the repose time fits for Lanín, for repose times grouped into intervals of 50 years. The K–S-significance value at the 5% level is 0.177 for the 100-year intervals and0.486 for the 1000-year intervals. Note that the AICc values for the 100-year intervals for Lanín cannot be compared with the 1000-year intervals, since the number of intervalsentering the fits is different.

Fit A t0/yr k/yr−1 d x0/yr α χ²/DoF R² KS–value AICc

Lanín (100-year intervals)Exp 7.0±0.0 453.70±53.8 0.551 0.895 0.042 33.2Weibull 7.0±0.0 0.00225±0.0003 1.145±0.307 0.664 0.845 0.039 39.3log–log 7.0±0.0 307.2±33.4 1.682±0.362 0.423 0.924 0.039 23.8

Lanín (1000-year intervals)Exp 7.0±0.0 1260.5±231.9 0.510 0.904 0.548 1.2Weibull 7.0±0.0 0.00132±0.0006 0.439±0.142 0.191 0.731 0.677 −1.5log–log 7.0±0.0 413.4±224.2 0.846±0.271 0.174 0.973 0.677 −0.6


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Documents

Applicability of statistical eruption analysis to the geological record of Villarrica and Lanín volcanoes, Southern Volcanic Zone, Chile