Chapter 4 What is the Probability of Explosive Eruption at a Long-Dormant Volcano? Charles B. Connor, Alexander R. McBirney, Claudia Furlan

One of the most difficult problems we face in assessing volcanic hazards is that of evaluating the potential activity of volcanoes with little or no record of Holocene eruptions. Is there some minimum period of inactivity after which we can safely rule out a future eruption of large magnitude? Or, failing that, can we say how likely it is that such a volcano will return to activity within a particular span of time? Violent explosive eruptions are uncommon during the youthful stage of active growth. They are confined almost entirely to large mature volcanoes. Here, we use the global record of volcanic activity (Simkin and Siebert 1994) to evaluate the duration of repose intervals preceding such explosive volcanic eruptions. This analysis indicates that the hazard rate for explosive eruptions is not constant with time, but depends on the time since last eruption and that explosive eruptions may occur at volcanoes that have been quiescent for 10,000 years or more. The techniques we employ are common to a class of problems in survival analysis (Cox and Oakes 1984; Woo 1999), and can be applied to a variety of hazard problems on volcanoes (e.g., Hill et al. 1998; Connor et al. 2003; Calder et al. 2005). One of the major lessons of this type of analysis is that applied statistical methods can teach us much about the time scales of volcanic activity, and about the underlying physical mechanisms governing these rates. 1

4.1

Background

As the populations of volcanic regions become denser and more vulnerable to natural hazards, risks associated with volcanic eruptions also become greater. This means that even low probability events, such as the reactivation of a long-dormant volcano, must be evaluated quantitatively. When faced with this task, volcanologists may be called upon to estimate the probability of a major eruption from a mature volcano that has no record of eruptions during the last several thousand years (e.g., McBirney et al. 2003). How does one judge whether a volcano is extinct or merely dormant? How long must a volcano be inactive before it can be dismissed as extinct? However one addresses these questions, mistaken judgments can have serious consequences. An overly conservative decision may cause a much-needed project to be abandoned because of the exorbitant cost of needless safeguards, whereas the opposite choice confronts the correlation between the duration of repose periods and the violence of ensuing eruptions. We have no clear rules for defining volcanoes as extinct or dormant (Szak´acs 1994), and it may be futile to try to make such a categorical distinction. Most volcanologists seem to consider any Quaternary volcano potentially active, but some would also include Pliocene volcanoes. Smith and Luedke (1984), for example, concluded from their study of volcanism in the western United States that “any volcanic locus that has appeared within the last 5 Ma reflects a viable mantle source that has potential for volcanic eruption.” Given that perspective, our problem is not to decide whether a particular volcano is dormant or extinct but rather to estimate the probability that it will erupt in a given way within a specific interval of time. To that end, it may be useful to review some basic observations that have a bearing on the problem. Recent studies reveal two significant features of large, mature volcanoes. First, violent explosive eruptions rarely come from youthful, rapidly growing volcanoes. Instead, they are products of mature volcanoes that have prolonged periods of inactivity. Thus, the history of these volcanoes can be divided into two main periods: an early stage of active growth in which large-scale eruptions are rare, and a later one in which eruptions are less frequent but more explosive. Pyle (1998) has found that these two stages of activity are characteristic of many volcanoes that have had large explosive eruptions. Analyzing the data of Simkin and Siebert (1994) using a statistical method similar to that used to study the distribution of earthquakes of differing magnitudes (see Placios et al., this volume), Pyle found that large numbers of eruptions with few fatalities are statistically distinguishable from a much smaller number of large eruptions that caused 2

larger numbers of casualties. Thus the hazard rates and risks associated with volcanic activity are different in these two stages of activity. A second characteristic of eruptions from long-dormant volcanoes is that the strength of these eruptions, as measured by the volcanic explosivity index (VEI) of Newhall and Self (1982), appears to be related to the length of the preceding repose interval. Qualitatively, the largest eruptions have followed the longest intervals of quiet, and although the pattern is far from regular, all eruptions classified as VEI 6 or greater followed repose intervals of more than 100 years (Simkin and Siebert 1984, 1994). The processes responsible for the repose period and magnitude of eruptions during the mature stage of activity must be fundamentally different from those controlling behaviour during the stage of active growth, when long repose intervals are less likely. Although the physical processes underlying these relations are incompletely understood, it is possible to take a statistical approach and use the Holocene record to estimate the probability that an eruption of a given magnitude will occur after repose intervals of differing durations. Ideally, one can establish a probabilistic limit beyond which eruptions are so unlikely that they need no longer be considered hazardous. With the recent improvements of the record of global volcanism, we can now assess this problem in the context of the available data with more confidence than was previously possible.

4.2

The Record of Volcanic Activity

During the last 25 years, volcanologists at the Smithsonian Institution have compiled a list of all known eruptions that have occurred throughout the world during the last 10,000 years (Simkin et al. 1981; Simkin and Siebert 1984, 1994). Of the 7886 dated eruptions in the record, we extracted 267 eruptions that are classified as VEI 4 or greater and for which the preceding repose interval is known. The repose interval derived from the record is actually the number of years from the start of a previous eruption to the start of a subsequent eruption of magnitude VEI 4 or greater. Ideally, the repose interval should be measured from the end of the last preceding eruption until the beginning of the paroxysmal phase of the following one, but the historical and geologic records of eruptions are rarely adequate to support this refined definition. We find very few eruptions for which the end of the event is given. Nevertheless, only 16% of 1015 eruptions identified from the catalogue, VEI 3 or greater in magnitude, had durations exceeding one year and even in these cases the length of this period is an inconsequential fraction of the repose intervals being discussed. Moreover, only six of these eruptions that lasted more than one year involved caldera 3

collapse, a Plinian eruption, or both. In a few instances, the start of the large eruption did not immediately follow the repose interval, but was preceded by milder eruptions lasting several years. We have excluded 25 events in which eruptions of VEI 4 or greater occurred more than three years after the renewal of activity because in no way would these volcanoes be considered long dormant. Interestingly, most of these were eruptions of Vesuvius. Of course, this data set is only a sample of the eruptions that have occurred in the Holocene. The completeness and duration of the historical record varies regionally, and for much of the prehistoric Holocene, the record is scant. Some volcanoes have not been studied in sufficient detail to document all major eruptions. At some volcanoes, erosion rates may be so high that only large eruptions in the early Holocene were voluminous enough to leave deposits that are preserved, recognised, and dated. Additionally, many tephra layers for lesser eruptions have been identified stratigraphically, but not dated, and some dated tephra layers may actually represent series of eruptions separated by repose intervals. So we make the caveat that we have only a small sample of Holocene eruptions, and variable geologic preservation and reporting of eruptive activity may bias the sample. Nevertheless, we proceed because analysis of the sample (the currently known record of eruptions) holds the best promise of revealing the statistical structure of the overall pattern of Holocene volcanism. Simple statistics indicate that eruption magnitude does vary with the duration of the preceding repose interval (Table 4.1). Large eruptions are rare, and tend to occur after much longer repose intervals than smaller explosive eruptions. These observations, though interesting in a geological sense, are of little help in evaluating the probabilities of specific types of eruption from a given volcano during a given time interval. The latter requires a more detailed examination of the repose intervals preceding eruptions of different magnitudes. As indicated in Table 4.1, we subdivide the 267 eruptions into three categories: VEI 4, VEI 5, and VEI 6 - 7. VEI 6 and VEI 7 eruptions are grouped because of the small number of these events and because the frequent uncertainty associated with classifying these very large magnitude events by VEI. Note that the well-known Santorini eruption of 1628 BC is not included in the data set because the preceding eruption occurred greater than 10 ka. Volcanic eruptions larger than VEI 7 have occurred greater than 10 ka, but none are in the data set. Eruptions smaller than VEI 4 are not included in the analysis because of difficulty in determining the repose interval between these more frequent eruptions.

4

4.3

Analysis of the Repose Interval Data

We first show an empirical analysis of the repose intervals through the empirical survivor function, and then we compare it with the survivor functions of two possible models: the exponential and the log-logistic distributions. The survivor function S(t) is the complement of the distribution function and gives the probability that a measured value of the random variable T exceeds some value, t: ST (t) = P [T > t].

(4.1)

If we do not assume a distribution for the repose intervals T , we cannot evaluate the survivor function using parametric statistics. Nevertheless, we can still calculate the empirical survivor function from the observed distribution of repose intervals. The empirical survivor function is used as a descriptive instrument and, in the next two subsections, as a visual tool for the goodness of fit of the model adopted. For each category of volcanic eruptions, we put the observations in rank order so that t1 ≤ t2 ≤ . . . ≤ tN , where N is the total number of observations. Then, we define the empirical survivor function, at repose interval ti , as follows: ˆ i ) = N − i i = 1, . . . , N. (4.2) S(t N Figure 4.1 shows the empirical survivor function in a particular logit scale, for the three categories of volcanic eruptions (VEI 4, VEI 5, VEI 6 - 7). We calculate the empirical logit of the empirical survivor function (L) ! ˆ i ) + 0.5/N S(t d S(t ˆ i )) = log (4.3) L = logit( ˆ i ) + 0.5/N 1 − S(t where 0.5/N (an adjustment with respect to the standard logit function) ˆ i ) = 1. We adopt the “logit” scale avoids numerical problems when S(t simply because the plot (Figure 4.1) allows better discrimination between the three categories of eruptions. It can be seen (Figure 4.1) that the three categories of data have markedly different empirical survivor functions. Also note that the 104 a limit on the data set is likely to truncate the distributions, especially for the rare large magnitude (VEI 6 - 7) eruptions. For smaller explosive eruptions (e.g., VEI 4) the 104 a limit appears to have less effect. The 104 a limit is imposed by the sampling (the eruptions in the catalogue are documented to 104 a) and is not likely to be a feature of the underlying unknown true distribution. Given this important caveat, our goal is to model these three distributions and use these models to assess their utility for forecasting future volcanism at long-dormant volcanoes. 5

4.3.1

Using the Exponential Distribution

As the first step, we model the repose intervals as independent realizations from an exponential distribution Ti ∼ Exp(1/µ) i = 1, . . . , N,

(4.4)

where N is the number of observations and µ the mean of the repose intervals, for each category of volcanic eruptions. This distribution is often used to model volcanic hazards because it naturally describes the duration between events in a Poisson process. In this context, events are the volcanic eruptions and durations are the repose intervals. Although the exponential distribution is easy and widely used, it requires that the process generating the data satisfy the following strong characteristic P (T ≥ t + s|T ≥ s) = P (T ≥ t) t, s > 0 (4.5) that is, the fact that an eruption has not happened yet, tells us nothing about how much more time will elapse before it does happen. For an exponential distribution, with mean µ, the probability density function is   1 ti fTi (ti ) = exp − , ti > 0. (4.6) µ µ the survivor function is (combining Equations 4.1 and 4.6): STi (ti ) = exp(−ti /µ) i = 1, . . . , N,

(4.7)

and the hazard function is hTi (ti ) =

fTi (ti ) 1 = STi (ti ) µ

i = 1, . . . , N.

(4.8)

The hazard function for the exponential distribution is constant over time and it implies that volcanoes do not become extinct: clearly not a geologically useful feature of the distribution! For each category of eruptions, we estimated µ via maximum likelihood: PN ti µ ˆ = i=1 (4.9) N where ti is the i-th observed repose interval. As a consequence of the properties of the maximum likelihood estimator, the standard error of µ ˆ is approximately: µ ˆ (4.10) st. err. (ˆ µ) = √ . N 6

The estimate of µ and the relative standard error, for each category of eruptions, can be found in Table 4.2. Note that the standard errors are proportional to the estimated mean of the repose intervals and inversely proportional to the number of observations and for these reasons they grow with the VEI: larger eruptions are associated with longer repose intervals and a smaller sample size. Figures 4.2, 4.3 and 4.4 show estimates of the survivor functions obtained substituting the relative maximum likelihood estimate in Equation 4.7 for µ. We observe that exponential distribution models are poor descriptors of the observed distributions of repose intervals. It seems that the process generating the repose intervals of our dataset does not properly satisfy Equation 4.5 and does not have a constant hazard function (Equation 4.8). Thus, we should try an alternative distribution that provides some predictive information for time to next event based on time since the previous event. In the preceding analysis we adopted the principle of pooling all information across volcanoes. This assumption is strictly valid only if all the volcanoes are homogeneous in their behaviour. It is possible that the exponential distribution does not adequately describe the behaviour of repose intervals because the mean repose interval preceding eruptions of a given magnitude is different at different volcanoes. Naturally, it would be more informative to make assessments that are volcano-specific, but from a statistical perspective this creates difficulties due to the scarcity of data.

4.3.2

Using the Log-logistic Distribution

As the exponential distribution is unsuccessful, we consider alternative distributions. In particular, we need a model that handles the large standard deviation in our observations, which suggests the observed distribution arises from a set of complex processes, and a hazard function that is not constant. The log-logistic distribution is an alternative probability density function that can be applied to volcano repose data (Connor et al. 2003). This distribution arises when competing geological processes affect repose intervals on different time scales. For example, the magma source region may undergo a comparatively long term change in the rate of magma production that will alter repose intervals in a given volcanic system. In contrast, processes operating within the crust, such as magma differentiation and crystallization in a magma chamber, may influence repose intervals on comparatively short time scales. If such mechanisms are operating in magmatic systems, as they must, then log-logistic distributions of repose intervals may arise. 7

The probability density, survivor, and hazard functions for the loglogistic distribution with parameters α and β, that is Ti ∼ log-logi(α, β), are tiβ−1 αβ , [1 + ti αβ ]2

(4.11)

1 , 1 + (αti )β

(4.12)

fTi (ti ) = β STi (ti ) =

hTi (ti ) = β

tiβ−1 αβ 1 + (αti )β

,

(4.13)

where α, β > 0. If β > 1 the hazard function for the log-logistic distribution first increases and then decreases toward zero, while if β ≤ 1 the hazard function decays monotonically toward zero. We estimate α and β via maximum likelihood. Since an analytic formulation of the estimator is not available we use a numerical optimisation. The estimates of both the parameters and the relative standard errors are shown in Table 4.3. The 95% confidence intervals for β are approximately ˆ βˆ + 1.96 × st. err. (β)), ˆ (βˆ − 1.96 × st. err. (β), that is, (0.68, 0.85) for VEI 4, (0.65, 1.1) for VEI 5 and (1.26, 2.89) for VEI 6 - 7. Note that for VEI 4 the set of plausible values for β, at 95% level, is smaller than 1, for VEI 6 - 7 is larger than 1, while for VEI 5 includes values smaller and larger than 1. As a consequence, for VEI 4 only monotonically decreasing hazard functions are plausible, for VEI 6 - 7 only increasing and then decreasing hazard functions, while for VEI 5 both types of hazard functions are plausible. It seems that the shape of the hazard function changes gradually with the VEI, since βˆ increases with it. Figures 4.2, 4.3 and 4.4, for each category of eruptions, show the maximum likelihood estimates of the survivor functions of the repose interval. For repose intervals preceding VEI 4 and VEI 6 - 7 eruptions, the log-logistic model fits the observations well (see Figure 4.2 and Figure 4.4). The distribution of repose intervals preceding VEI 5 eruptions is also reasonably well modelled using the log-logistic distribution (see Figure 4.3). Note that in this case the model produces higher exceedance probabilities for repose intervals greater than approximately 1000 years. This misfit may result from truncation of our observations to those eruptions that have occurred in the last 10 ka. Estimated probabilities for repose intervals greater than t = 100 a, t = 1000 a, t = 10000 a, that is STi (t) (Equation 4.1), are shown in Table 4.4 for eruptions of various magnitudes. The results indicate that some volcanoes 8

are likely to have repose intervals greater than t = 10000 a, although we have little confidence in the maximum likelihood estimates of α and β for this duration. Figure 4.5 shows the logarithm of the hazard function (see Equation 4.13), for each category of eruptions, and Table 4.5 presents the hazard function for some values of repose intervals. As βˆ < 1 for both VEI 4 and VEI 5 data sets, the hazard functions decrease monotonically. That is, eruptions are most likely immediately following preceding activity, after which the hazard gradually decreases with time. After approximately 1000 a, hazard rates for VEI 4 and VEI 5 eruptions are roughly the same (see Table 4.5). As βˆ > 1 for VEI 6 - 7 the hazard curve first increases and then decreases with a unique maximum at approximately 1000 a. Judging from our limited data set of VEI 6 - 7 eruptions, it is clear that the probability of these large eruptions after short repose intervals, say less than 100 a, is very low (see Table 4.5). In general, the hazard rate for VEI 6 - 7 eruptions is much less than VEI 4 and VEI 5 eruptions, but after 1000 a repose, the hazard rates become comparable. Of course, we cannot have great confidence in the model parameters estimated from analysis of our repose interval data set because of truncation of the data set at 10 ka. Nevertheless, based on the model we can say that it appears such volcanoes remain hazardous, that is capable of future volcanic eruptions of large magnitude, even after 10 ka.

4.4

Concluding Remarks

The question remains, what is the probability that a long-dormant volcano will erupt explosively? We cannot answer this question explicitly because we do not have a record including all volcanoes, rather only a record of some Holocene eruptions. Nevertheless, the record of Holocene eruptive activity does provide insight. With this record we have prepared models of the probable duration of repose intervals, given that the volcano will erupt explosively again. The exponential distribution is a poor description of the observed distribution of repose intervals, while the log-logistic distribution fits the data rather well. The model based on the log-logistic distribution suggests that, given the resolution of the global record, it is incorrect to assume that volcanoes that have not been active during the Holocene are incapable to future eruptions, simply on the basis of this period of repose. The fact that such special distributions work at all for modelling repose interval data suggests that improved data resolution and improved models of magma transport may yield rich rewards in improved long-term volcanic hazard forecasts. As more data are gathered on the world’s volcanoes, it is 9

very likely the specific parameter estimates will need to be revised. As we use such data sets in increasingly sophisticated ways, it is equally important to assess possible bias in the geologic record. For the more frequent, comparatively less explosive eruptions (VEI 4 and VEI 5), the identification of a monotonically decreasing hazard function suggests that hazards are not necessarily reduced in the years immediately following an eruption. Resources invested monitoring recently active volcanoes are well spent. Nevertheless, repose intervals prior to large explosive eruptions are commonly longer than 100 a, and in the case of very large eruptions longer than 1000 a. Certainly a volcano should not be considered extinct, simply because it has no record of activity in the last 10 ka. Although we cannot be confident in the probability estimates for t > 10 ka, the analysis does suggest that some volcanoes are likely to have repose intervals greater than 10 ka. It is clear that our need to refine these estimates is ever increasing. For example, annual hazard rates of 1 × 10−6 are considered significant for some facilities, such as nuclear power plants and high-level radioactive waste disposal sites (IAEA 1997; McBirney and Godoy 2003; Apted et al. 2004). Hazard rates at volcanoes that have been quiescent for 10 ka appear to exceed such thresholds by approximately two orders of magnitude. Our analysis illustrates the need to expand and refine the global database of recent volcanism.

Further Reading The book by Cox and Oakes (1984) provides a clear and concise introduction to analysis of survival functions and the exponential and log-logistic distributions. Woo (1999) also provides detailed discussion about using probability distributions in analysis of natural hazards. Any work on volcano repose intervals should begin with thorough reading of the introductory material found in Volcanoes of the World by Simkin and Seibert (1994).

Acknowledgments This paper could not have been completed without the gracious support of Tom Simkin of the Smithsonian Institution. Careful review and comments by Stuart Coles and Lee Siebert greatly improved the manuscript. Claudia Furlan is grateful for financial support from the University of Padova (Italy) grant CPDA037217: “Methods for the analysis of extreme sea levels and for coastal erosion”. Attendance of CBC at the “Statistics in Volcanology” workshop was sponsored as part of the Environmental Mathematics and Statistics Programme funded jointly by NERC/EPSRC, UK.

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Bibliography [1] Apted, M., Berryman, K., Chapman, N., Cloos, M., Connor, C., Kitayama, K., Sparks, S., & Tsuchi, H. 2004. Locating a radioactive waste repository in the ring of fire. Eos, Transactions of the American Geophysical Union, 85, 465, 471. [2] Calder, E.S., Cortes, J.A., Palma, J.L., & R. Luckett 2005. Probabilistic analysis of rockfall frequencies during an andesite dome eruption: The Soufri`ere Hills volcano, Montserrat. Geophysical Research Letters, 32, L16309, doi:10.1029/2005GRL023594. [3] Connor, C.B., Sparks, R.S.J., Mason, R.M., Bonadonna, C. & Young, S.R. 2003. Exploring links between physical and probabilistic models of volcanic eruptions: The Soufri`ere Hills Volcano, Montserrat. Geophysical Research Letters, 30, 1701, doi:10.1029/2003GL017384. [4] Cox, D.R. & Oakes, D. 1984. Analysis of Survival Data. Chapmanand Hall, London. [5] Druitt, T.H., Edwards, L., Mellors, R.M., Pyle, D.M., Sparks, R.S.J., Lanphere, M., Davies, M. & Barriero, B. 2002. Santorini Volcano. Geological Society, London, Memoirs. [6] Friedrich, W.L. 1999. Fire in the Sea: The Santorini Volcano, Natural History and the Legend of Atlantis. Cambridge University Press. [7] Hill, B.E., Connor, C.B., Jarzemba, M.S. & La Femina, P.C. 1998. 1995 eruptions of Cerro Negro, Nicaragua and risk assessment for future eruptions. Geological Society of America Bulletin, 110, 1231-1241. [8] International Atomic Energy Agency 1997. Volcanoes and associated topics in relation to nuclear power plant siting. Provisional Safety Standard Series 1. [9] McBirney, A.R. & Godoy, A. 2003. Notes on the IAEA Guidelines for assessing volcanic hazards at nuclear facilities. Journal of Volcanology and Geothermal Research, 126, 1-9. 11

[10] McBirney, A.R., Serva, L., Guerra, M. & Connor, C.B. 2003. Volcanic and seismic hazards at a proposed nuclear power plant site in central Java. Journal of Volcanology and Geothermal Research, 126, 11-30. [11] Newhall, C.G., & Self, S. 1982. The volcanic explosive index (VEI): an estimate of explosive magnitude for historical volcanism. Journal of Geophysical Research, 87, 1231-1238. [12] Pyle, D.M. 1998. Forecasting sizes and repose times of future extreme volcanic events. Geology, 26, 367-370. [13] Simkin, T., Siebert, L., McClelland, L., Bridge, D., Newhall, C. & Latter, J.H. 1981. Volcanoes of the World: A Regional Directory, Gazetteer, and Chronology of Volcanism During the Last 10,000 Years. Hutchinson Ross, Stroudsburg, PA. [14] Simkin, T. & Siebert, L. 1984. Explosive Eruptions in Space and Time: Durations, Intervals, and a Comparison of the Worlds Active Volcanic Belts. In Boyd, R. F. (ed) Explosive Volcanism: Inception, Evolution, and Hazards. National Academy Press, Washington, D. C., 110-121. [15] Simkin, T. & Siebert, L. 1994. Volcanoes of the World, 2nd Edition. Geoscience Press, Tucson. [16] Smith, R.L. & Luedke, R.G. 1984. Potentially active volcanic lineaments and loci in Western conterminous United States. In: Boyd, R.F. et al. (eds) Explosive Volcanism. National Academy Press, Washington, DC, 47-66. [17] Szak´acs, A. 1994. Redefining active volcanoes: a discussion. Bulletin of Volcanology, 56, 321-325. [18] Woo, G. 1999. The Mathematics of Natural Catastrophes. Imperial College Press, London.

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VEI 4 5 6-7

N mean (a) median (a) σ (a) skewness 194 448.6 51.5 1204.4 4.7 56 905.9 361.5 1280.8 2.1 17 1490 1072 1081.2 0.7

Table 4.1: Descriptive statistics of repose intervals preceding explosive volcanic eruptions. VEI: volcano explosivity index; N: number of eruptions; σ: standard deviation. Note that the mean and median repose interval increases for eruptions of increasing intensity.

VEI 4 5 6-7

µ ˆ st. err. (ˆ µ) 448.63 31.96 905.83 121.06 1490.23 361.43

Table 4.2: Estimate of µ and relative standard error using the exponential distribution, for each category of eruptions (see Equations 4.9 and 4.10 ).

VEI 4 5 6-7

α ˆ st. err. (α) ˆ 0.019 0.003 0.003 0.001 0.001 0.0002

ˆ βˆ st. err. (β) 0.765 0.045 0.829 0.092 2.075 0.414

Table 4.3: Estimates of α and β and relative standard errors using the log-logistic distribution, for each category of eruptions.

VEI 4 5 6-7

t = 100 a t = 1000 a t = 10000 a 0.379 0.095 0.018 0.711 0.269 0.052 0.994 0.575 0.011

Table 4.4: Probabilities the repose interval will exceed t years (survivor functions), given that a VEI 4 - 7 eruption eventually occurs. Probability models are based on the log-logistic models.

13

VEI 4 5 6-7

t = 1a t = 10 a t = 100 a t = 1000 a t = 10000 a 0.035 0.017 0.0047 0.0007 0.00007 0.007 0.0047 0.0024 0.0006 0.00008 0.0000009 0.00001 0.0001 0.0009 0.0002

Table 4.5: Hazard rates for volcanoes that will eventually experience VEI 4 7 eruptions, using the log-logistic hazard function with the estimate of α and β of Table 4.3 in Equation 4.13. Hazard rates are greatest for VEI 4 and VEI 5 eruptions in the years following volcanic eruptions. In contrast, hazard rates are greatest for VEI 6 - 7 eruptions after approximately 1000 year repose.

14

logit of estimated function, L

4 2 0 -2

VEI=4 VEI=5 VEI=6−7

-4 -6 1

10

100

1000

10000

Repose Interval (a)

Figure 4.1: Empirical logit of the empirical survivor function of the repose intervals (L), preceding eruptions in our data set, as a function of repose interval.

1.0

Survivor function

0.8 0.6 0.4 0.2 0.0 1

10

100

1000

10000

Repose Interval (a) Preceding VEI 4 Eruptions

Figure 4.2: Empirical survivor function for observed repose intervals preceding VEI 4 eruptions (black circles), compared with the survivor function estimated with the exponential distribution (broken line) and with the loglogistic distribution (solid black line). In contrast to the log-logistic model, the exponential distribution is a poor fit to these data.

15

Survivor function

1.0 0.8 0.6 0.4 0.2 0.0 2

5

20 50

200

1000

5000

Repose Interval (a) Preceding VEI 5 Eruptions

Figure 4.3: Empirical survivor function for observed repose intervals preceding VEI 5 eruptions (black circles), compared with the survivor function estimated with the Exponential distribution (broken line) and with the log-logistic distribution (solid black line). Note that the model predicts higher exceedance probabilities than observed at repose intervals greater than approximately 1000 years, which may result from truncation of the observations at 10 ka.

Survivor function

0.8

0.6

0.4

0.2

1000

2000

4000

Repose Interval (yr) Preceding VEI 6−7 Eruptions

Figure 4.4: Empirical survivor function for observed repose intervals preceding VEI 6 - 7 eruptions (black circles), compared with the survivor function estimated with the exponential distribution (broken line) and with the loglogistic distribution (solid black line).

16

−2 −4

log(h)

−6 −8 −10 −12 −14 1

10

100

1000

10000

Repose Interval (a)

Figure 4.5: Hazard functions on a logarithmic scale, log(h) = log(hTi (ti )), for VEI 4 (broken line), VEI 5 (dotted line) and VEI 6 - 7 (solid line) eruptions, calculated using the log-logistic hazard function, using the estimate of α and β of Table 4.3 in Equation 4.13.

17