Statistical Analysis of the Frequency of Eruptions at Furnas Volcano, Saoä Miguel, Azores G

Journal of Volcanology and Geothermal Research 92Ž. 1999 31±38 www.elsevier.comrlocaterjvolgeores

Statistical analysis of the frequency of eruptions at Furnas Volcano, SaoÄ Miguel, Azores G. Jones a, D.K. Chester b, F. Shooshtarian c,) a Department of Computing and Mathematics, London Guildhall UniÕersity, London, EC3N 1JY, UK b Department of Geography, UniÕersity of LiÕerpool, LiÕerpool, L69 3BX, UK c Department of Electronics and Mathematics, UniÕersity of Luton, Luton, LU1 3JU, UK Accepted 20 April 1999

Abstract

Furnas Volcano is one of three major volcanic centres on the island of SaoÄ Miguel, AcËores. Both Furnas and Fogo have displayed violent explosive activity since the island was first occupied in the early 15th century AD. There is concern that future volcanic activity will not only cause major economic losses, but will also result in widespread mortality, and it is for these reasons that a major programme of hazard assessment has been undertaken on Furnas. The present study is part of this programme and involves both the general statistical modelling of the record of historic eruptions and, more specifically, develops a technique for determining the rate of volcanic eruptions Ž.l , an important parameter in the Poisson probability model. q 1999 Elsevier Science B.V. All rights reserved.

Keywords: frequency of eruptions; Furnas Volcano; SaoÄ Miguel, Azores

1. Introduction issue. and a chronology is provided in Table 1. These eruptions range from sub-plinianrphreato- Furnas volcano is one of three major active vol- magmatic events, such as 1630 AD, to larger plinian canic centres on the island of SaoÄ Miguel in the episodes Ð for example Furnas C. Even the smaller AcËores. The last eruption of Furnas in 1630 AD eruptions, if repeated, would lead to total devastation killed ;100 peopleŽ. Cole et al., 1995 . Queiroz et within the calderaŽ. Fig. 1 . In an assessment of the al.Ž. 1995 have demonstrated that there was also an volcanic hazard of Furnas, it is important to consider historic eruption ;1440 AD, during the period of the probability of a future eruption and MooreŽ. 1990 , the first settlement of the island by the Portuguese. on the basis of the number of eruptions over the last The eruptions of Furnas over the last 5000 years are 5000 years, showed that eruptions had taken place on well documentedŽ Booth et al., 1978, 1983; Cole et average once every 300 years and, with the last al., 1995, 1999-this issue; Guest et al., 1999-this eruption occurring almost 400 years ago, he consid- ered that there was at present a high probability of eruption. Eruptive mechanisms of volcanoes are not, how- ) Corresponding author. E-mail: [email protected] ever, sufficiently well understood to allow determin-

Table 1 Eruption chronology, SaoÄ Miguel, AcËoresŽ. based on Queiroz et al. and other sources G. Jones et al.rJournal of Volcanology and Geothermal Research 92() 1999 31±38 33 Ž. Fig. 1. Furnas Volcano: general location map from Chester et al., 1995 . 34 G. Jones et al.rJournal of Volcanology and Geothermal Research 92() 1999 31±38 istic predictions of future activity to be made with `without a memory' of previous events and, hence, confidence. On Furnas and many other volcanoes are termed Simple Poissonian Volcanoes. prediction must rely on statistical modelling using It should be noted that events can have: eruption records. Such models have allowed many 1. A random distributionŽ not to be confused with important features of future eruptions to be pre- the term `random variable'. . dicted. These have included: the frequencies; types; 2. A regular distribution. spatial patterns and probabilities of eruptions, all of 3. A contagious distribution. which have been incorporated into risk assessments 4. A random distribution, implying that events occur ŽWickman, 1966a,b,c,d,e; Klein, 1982, 1984; `at random'. Newhall, 1982, 1984; Scandone, 1983; Mulargia et A Poisson distribution provides a suitable model al., 1985; Condit et al., 1989; Cruz-Reyna, 1991; Ho for such behaviour and it is the hypothesis of ran- et al., 1991. . domness that is usually the first to be tested. A In this paper we propose a statistical approach regular distribution suggests that events occur at which is applicable equally to Furnas and to other fixed, regular intervals and a contagious distribution volcanoes with similarly deficient data sets. In par- implies that there is a clustering of events. In this ticular, we estimate the unknown parameters which paper, therefore, a Poisson model will be fitted to the are required to calculate the probability of eruptions. eruptions of Furnas. Before doing so it should be Taking into account the limited availability of pre- noted that Ho et al.Ž. 1991 has introduced the con- cise data on the past eruptions of Furnas, it is our cept of a Nonhomogeneous Poisson model, that is a contention that estimates made from our model are model in which the occurrence of an event is depen- meaningful, both statistically and geologically. dent on the elapsed time since the previous event, but Babbington and LaiŽ. 1996 argue that such a model can possess certain undesirable features. 2. The data The Poisson process is used to describe a wide variety of stochastic phenomena that share certain Data are presented in Table 1 and dates of erup- characteristics and in which some `happening' Ð or tion are referred to two time axes. The first, labelled event Ð takes place sporadically over time, in a RADIOCARBON, is scaled from 5000 years BP to manner which is commonly believed to be `random.' present, but note that radiocarbon dates have an If events in a Poisson process occur at a mean rate arbitrary reference point of 0 BPs1950 AD. . The of l per unit timeŽ. 1 year, 105 year, etc. , then the second scale is chronological, having a range from expected numberŽ. long-run average of occurrences 3000 BC to 2000 AD. Both scales use intervals of in an interval of time in t units is lt Že.g., see Ho et 200 years. The nature of these data are such that al., 1991. . Using Table 1, a volcanic eruption is values are estimated when they do not coincide with defined as such an event. either of the axesŽ. Table 1 . The rate of occurrence of volcanic eruptions, l is a critical parameter in the probability calculation. Various statistical methods for calculating l are 3. Choice of model examined to show how the record of volcanism on Furnas may be used to estimate values of l and so Application of statistical methods to volcanic demonstrate the limitations involved in calculating eruptions began with the pioneering research of l. WickmanŽ. 1966a; b; c; d; e; 1976 , who discussed the applicability of various Poisson models to volcanic events. His research involved the calculation of eruption recurrence rates for a number of volcanoes 4. Methodology with different styles of activity. Wickman observed that for this class of volcano age specific eruption By electing to employ a Poisson model means rates are not time dependent. Such volcanoes are that volcanic eruptions occur at random, according to G. Jones et al.rJournal of Volcanology and Geothermal Research 92() 1999 31±38 35

the Poisson process. Then the number, X, of occur- independence exists, it follows that x12, x ,...,xn rences per unit time has a probability PXŽ.sx s are observations of a random sample of size n. The eyll xrx!, xs0, 1, 2, . . . and the time between distribution from which the sample arises is the eruptions has a negative exponential distribution population. The observed sample values, x12, x , sl yl t whose probability density function is ftŽ. e, ..., xn, are used to determine information about the t, l)0. Observe that l is a parameter to be esti- unknown populationŽ. or distribution ''. Assuming mated from the data. that x12, x , ..., xn represent a random sample from Mathematically, the assumptions for a Poisson a Poisson population with parameter l, the likeli- distribution are as follows: hood function is: 1. the probability of an event occurring in a small yl nne l x i interval of time Ž.t,tqdt is ltqo Ž.dt ; l s l s LŽ.ŁŁfx Ži ; . ! 2. the probability of two or more events occurring in is1 is1 xi a small interval of time Ž.Ž.t,tdt is o dt ; n 3. occurrences in non-overlapping intervals are inde- Ý xi s pendent of one another. L l seyn ll i 1 2 Ž. n The term oŽ.dt denotes that terms such as dt ! Ł xi and higher powers are present, but are so small that is1 they can be ignored. The model indicates that the Many statistical procedures employ values for the length, dt of the time interval is such that at most population parameters that `best' explain the ob- one event can occur. Of course, l is assumed to served data. One meaning of `best' is to select the remain constant throughout, since this is the rate of parameter values that maximise the likelihood func- occurrence of volcanic eruptions. tion. This technique is called maximum likelihood estimation, and the maximising parameter values are called maximum likelihood estimates, also denoted 5. The analysis MLE, or lÃ. Note that any value of l that maximises LŽ.l will also maximises that log-likelihood, ln L Ž.l . Let R denote `the number of eruptions per unit Thus, for computational convenience, the alternate time'. Then R follows a Poisson distribution with form of the maximum likelihood equation, will often l l parameter , where is the rate of volcanic erup- be used, and the log-likelihood for a random sample s tions. This means that the probability of R r erup- from a Poisson distribution is: tions in unit time is given by: nn s s yl rr s l sy lq ly PRŽ.r e l r! for r 0,1,2... ln LŽ. n ÝŁxiiln x !. s ž/is1 i 1 Also, the time between eruptions follows an exponential distribution whose probability density func- The maximum likelihood equation is therefore: n tionŽ. pdf is given by: d xi ln LŽ.l synqs0, yl t llÝ ftŽ.sle , l,t)0. d is1 x To obtain an estimator for l the method of Maxi- n i which has the solution lÃ sÝ s sx. This is in- mum Likelihood is used. Ho et al.Ž. 1991, p. 50 , i 1 n argue that, ``in dealing with distributions, repeating a deed a maximum because the second derivative: 2 n random experiment several times to obtain informa- d xi Ž. ln LŽ.l sy , tion about the unknown parameter s is useful. The ll22Ý d is1 collection of resulting observations, denoted x12, x , ..., xn is a sample from the associated distribution. is negative when evaluated at x. Often these observations are collected so that they Let this estimation technique be demonstrated. are independent of each other. That is, one observa- Let X denote the number of volcanic eruptions for a tion must not influence the others. If this type of 105 -year period for the NTSŽ. Nevada Test Site 36 G. Jones et al.rJournal of Volcanology and Geothermal Research 92() 1999 31±38 region from this assumed Poisson process. Then X Turning once more to the data relevant to the follows a Poisson distribution with average recur- Furnas research problem and taking the origin on the rence rate m, with mslts10 5l. If we wish to chronological scaleŽ. Table 1 to be 1000 BC, then: estimate l for the Quaternary using the Poisson Ž.a Three events occur during a period of 870 count data for the NTS region, the successive num- yearsŽ. AqBqC. ber of eruptions from the last 16 consecutive inter- Ž.b Three events take place over a 730 year period vals of length 105 yearsŽ 16=10 56s1.6=10 s Ž.DqEqF. Quaternary period. must be estimated. The number Ž.c One event takes place after 360 years Ž G . . of observed eruptions per interval are denoted as x1, Ž.d One event occurs after 265 years Ž H . . x216, ..., x . Thus, these 16 values represent a Ž.e One event occurs after 35 years Ž. I . sample of size 16 from a Poisson random variable Ž.f One event occurs after 170 years Ž. J . with average recurrence rate m. Estimating the mean Ž.g No events have happened for a period of 365 of the Poisson variable from these count data gives: years. Given that LŽ.l;t ,t ,...,t is the joint probabil- 16 12 n xi msxs Ý , ity of the observations we may write: is1 16 LŽ.l;t1 ,...,tn and y l 33y l s Ž.Ž.1ye1870 ye 730 m 16 y l y l y l y l y l Ã xi =l 360 l 265 l 35 l 170 365 lÃ ss e e e e e . 5 Ý = 6 10 is1 Ž.16 10 Taking natural logarithms we have: s y This shows that the estimated annual recurrence ln L 4lnl 1195l rate, lÃ, is the average number of eruptions during q3lnŽ.Ž. 1yey870l q3ln 1yey730l . the observation periodŽ. in years . Based on this Differentiating and rearranging terms we have: estimation technique, lÃ can be defined as: dlnŽ.L 43ey870l Ž.870 E sy1195q lÃ s Ž.1 dll 1yey870l T 3ey710Ž.710 4 where Estotal number of eruption during the ob- qsy1195 y y710l l servation period, and Tsobservation period. 1 e l 5610 2130 Note that for the estimation of in this model, an qq 1870l 710l individual observation xi is not required. However, e y1ey1 the distribution of x 's can provide information for i dlnŽ.L model selection, for checking the adequacy of the s0 model and for parameter estimation in general. dl The above description indicates that Eq.Ž. 1 is the Equate this to zero and solve numerically, using `best' estimator arising out of the application of Program 1 shown below, to give: ls0.00403075. method of MLE. However, although we also use MLE, our method in estimating l is different from 10 Ts0 Ho et al.Ž. 1991 and we believe it provides a better 20 T1s300 estimate of the parameter l. We, therefore, estimate 30 WHILE ABSŽ. T1yT )0.0000001 l as follows: 40 TsT1 s y r r y First, the likelihood function LŽ.l;t12,t ,...,tn is 50 T1 298.75 1402.5 ŽŽEXP 1870)1 T ..1 set up, the natural logarithm of the equation is taken y532.5rŽŽEXP 710)1rT ..y1 and then differentiated with respect to l. The result- 60 PRINT TABŽ. 1 ;1rT;TAB Ž 20 . ;1rT1 ing expression is made equal to zero and the equa- 70 NEXT tion is solved using an appropriate method. 80 END G. Jones et al.rJournal of Volcanology and Geothermal Research 92() 1999 31±38 37

Program 1 the Poisson probability model. The method is based on the number of eruptions occurring over a defined From this we can see that: period of observation. A recently published analysis PTŽ.)5 s0.980 of global volcanic activity arguesŽ. Ho et al., 1991 that the number of eruptions occurring per unit time, PT)10 s0.960 Ž. also follows a Poisson distribution. By using the PTŽ.)25 s0.904 method of maximum likelihood, an estimated value PTŽ.)50 s0.817 for l has been derived. Further, a 95% interval estimate for l has been estimated and this may be ) s PTŽ.75 0.739 used to provide interval estimates of probabilities of PTŽ.)100 s0.668 eruption over a range of times from 5 to 100 years. Thus, the probability that no eruption will occur within the next 5 years is 0.980; that is, there is a 2% References chance of an eruption occurring. The estimates above of the probability of no Booth, B., Walker, G.P.L., Croasdale, R., 1978. A quantitative eruption occurring prior to a given elapsed time are study of five thousand years of volcanism on SaoÄ Miguel, based on the point estimate of l, namely 0.00403075. Azores. Philos. Trans. R. Soc. London 288A, 271±319. We suggest that estimates based on a 95% confi- Booth, B., Croasdale, R., Walker, G.P.L., 1983. Volcanic hazard Ž. l on SaoÄ Miguel, Azores. In: Tazieff, H., Sabroux, J.-C. Eds. , dence interval for would be more useful. It is well Forecasting Volcanic Events. Elsevier, Amsterdam, pp. 99± known that if a random variable, X, has a negative 109. exponential distribution with parameter l then 2nxl Chester, D.K., Dibben, C., Coutinho, R., 1995. Report on the is distributed as a x 2 random variable on 2n de- evacuation of the Furnas District, SaoÄ Miguel, Azores. Open r grees of freedom. Thus, with xs320 and ns10 File Report 4, University College London Commission for l the European Union. we can set up a 95% confidence interval for . With Cole, P.D., Queiroz, G., Wallenstein, N., Gaspar, J. L, Duncan, 20 degrees of freedom, the lower 2.5% point is 9.591 A.M., Guest, J.E., 1995. An historic subplinianr and the upper 2.5% point is 34.170. These values phreatomagmatic eruption: the 1630 AD eruption of Furnas lead to the 95% interval 0.00148467 - l - Volcano, SaoÄ Miguel, Azores. J. Volcanol. Geotherm. Res. 69, 0.00528974. This enables us to refine our estimates 117±135. Cole, P.D., Guest, J.E., Queiroz, G., Wallenstein, N., Pacheco, J., to give: Gaspar, J.L, Ferreira, T., Duncan, A.M., 1999. Styles of PTŽ.)5 s0.973 volcanism and volcanic hazards on Furnas volcano, SaoÄ Miguel, Azores. J. Volcanol. Geotherm. Res., 39±53. PTŽ.)10 s0.947 Condit, C.D., Crumpler, L.S., Aubele, J.C., Elstow, W.E., 1989. ) s Patterns of volcanism along the southern margin of the Col- PTŽ.25 0.868 orado plateau: the Springeville Field. J. Geophys. Res. 94, PTŽ.)50 s0.736 1082±1975. Cruz-Reyna, S.De la., 1991. Poisson-distributed patterns of explo- PTŽ.)75 s0.603 sive eruptive activity. Bull. Volcanol. 54, 57±67. PT)100 s0.471 Guest, J.E., Gaspar, J.L., Cole, P.D., Queiroz, G., Duncan, A.M., Ž. Wallenstein, N., Ferreira, T., Pacheco, J-M., 1999. The vol- Thus, the revised probability that no eruption canic geology of Furnas volcano, SaoÄ Miguel, Azores. J. occurring within the next 5 years is 0.973; that is, Volcanol. Geotherm. Res. 92, 1±29. Ho, C.H., Smith, E.I., Feuerbach, D. L, Naumann, T.R., 1991. there is a 3% chance of an eruption occurring. Eruptive probability calculation for the Yucca Mountain site, USA: statistical estimation of recurrence rates. Bull. Volcanol. 54, 50±56. 6. Conclusions Klein, F.W., 1982. Patterns of historical eruptions at Hawaiian volcanoes. J. Volcanol. Geotherm. Res. 12, 1±35. Klein, F.W., 1984. Eruption forecasting at Kilauea volcano, The problem discussed in this paper involves the Hawaii. J. Geophys. Res. 89, 3059±3073. derivation of a technique for estimating the rate of Moore, R.B., 1990. Volcanic geology and eruption frequency, SaoÄ eruptions by means of Ž.l , an important parameter in Miguel, Azores. Bull. Volcanol. 52, 602±614. 38 G. Jones et al.rJournal of Volcanology and Geothermal Research 92() 1999 31±38

Mulargia, F., Tinti, S., Boschi, E., 1985. A statistical analysis of Wickman, F.E., 1966a. Repose period patterns of volcanoes: I. flank eruptions on Etna Volcano. J. Volcanol. Geotherm. Res. Volcanic eruptions regarded as random phenomena. Ark. Min- 23, 263±272. eral. Geol. 4, 291±301. Newhall, C.G., 1982. A method for estimating intermediate and Wickman, F.E., 1966b. Repose period patterns of volcanoes: II. long-term risks from volcanic activity, with an example from Eruption histories of some East Indian volcanoes. Ark. Min- Mount St. Helens. Open File Report, United States Geological eral. Geol. 4, 303±317. Survey, Washington, DC, 82±396. Wickman, F.E., 1966c. Repose period patterns of volcanoes: III. Newhall, C.G., 1984. Semiquantitative assessment of changing Eruption histories of some Japanese volcanoes. Ark. Mineral. volcanic risk at Mount St. Helens, Washington. Open File Geol. 4, 319±335. Report, United States Geological Survey, Washington, DC, Wickman, F.E., 1966d. Repose period patterns of volcanoes: IV. 84±272. Eruption histories of some selected volcanoes. Ark. Mineral. Queiroz, G., Gaspar, J.L., Cole, P.D., Guest, J.E., Wallenstein, N., Geol. 4, 337±350. Duncan, A.M., Pacheo, J., 1995. ErupcËoesÄÃ vulcanicas no Vale Wickman, F.E., 1966e. Repose period patterns of volcanoes: V. das FurnasŽ. Inla de S. Miguel. AcËores na primera metade do General discussion and a tentative stochastic model. Ark. SecobÂ XV. AcËoreana 8, 159±165. Mineral. Geol. 4, 351±367. Scandone, R. 1983. Problems related with the evaluation of Wickman, F.E., 1976. Markov models of repose-period patterns of volcanic risk. In: Tazieff, H., Sabroux, J.C.Ž. Eds. , Forecasting volcanoes. In: Merriam, D.F.Ž. Ed. , Random Processes in Volcanic Events, Developments in Volcanology, 1. Elsevier, Geology. Springer, Berlin, pp. 135±161. Amsterdam, pp. 57±69.