Identifying volcanic regimes using Hidden Markov Models

Abstract. The Canary Islands are an active volcanic region densely populated and visited by several millions of tourists every year. Nearly twenty eruptions have been reported through written chronicles in the last 600 yr, suggesting that the probability of a new eruption in the near future is far from zero. This shows the importance of assessing and monitoring the volcanic hazard of the region in order to reduce and manage its potential volcanic risk, and ultimately contribute to the design of appropriate preparedness plans. Hence, the probabilistic analysis of the volcanic eruption time series for the Canary Islands is an essential step for the assessment of volcanic hazard and risk in the area. Such a series describes complex processes involving different types of eruptions over different time scales. Here we propose a statistical method for calculating the probabilities of future eruptions which is most appropriate given the nature of the documented historical eruptive data. We first characterize the eruptions by their magnitudes, and then carry out a preliminary analysis of the data to establish the requirements for the statistical method. Past studies in eruptive time series used conventional statistics and treated the series as an homogeneous process. In this paper, we will use a method that accounts for the time-dependence of the series and includes rare or extreme events, in the form of few data of large eruptions, since these data require special methods of analysis. Hence, we will use a statistical method from extreme value theory. In particular, we will apply a non-homogeneous Correspondence to: R. Sobradelo (rsobradelo@ija.csic.es) Poisson process to the historical eruptive data of the Canary Islands to estimate the probability of having at least one volcanic event of a magnitude greater than one in the upcoming years. This is done in three steps: First, we analyze the historical eruptive series to assess independence and homogeneity of the process. Second, we perform a Weibull analysis of the distribution of repose time between successive eruptions. Third, we analyze the non-homogeneous Poisson process with a generalized Pareto distribution as the intensity function.

Section: Introductionmentioning

confidence: 99%

Volcanic hazard assessment for the Canary Islands (Spain) using extreme value theory

Sobradelo

Martı́

Mendoza-Rosas³

et al. 2011

“…Some of the earliest, (Wickman, 1965(Wickman, , 1976Reyment, 1969;Klein, 1982) employed stochastic principles to analyze eruption patterns. Further studies included transition probabilities of Markov chains (Carta et al, 1981;Aspinall et al, 2006;Bebbington, 2007), Bayesian analysis of volcanic activity (Ho, 1990;Solow, 2001;Newhall and Hoblitt, 2002;Ho et al, 2006;Marzocchi et al, 2008), homogeneous and non-homogeneous Poisson process applied to volcanic series (De la Cruz-Reyna, 1991;426 A. T. Mendoza-Rosas and S. De la Cruz-Reyna: A simple and precise assessment of the volcanic hazard 1996a, b), geostatistical hazard-estimation methods (Jaquet et al, 2000;Jaquet and Carniel, 2006), a mixture of Weibull distributions (Turner et al, 2008) and non-homogeneous statistics to link geological and historical eruption time series (Mendoza-Rosas and De la ). An exhaustive list of the available literature on this subject is beyond the scope of this paper, and the above references only attempt to illustrate the diversity of methods that have been applied to the volcanic eruption sequences.…”

Section: Introductionmentioning

confidence: 99%

A mixture of exponentials distribution for a simple and precise assessment of the volcanic hazard

Mendoza-Rosas¹,

Cruz‐Reyna²

2009

Abstract. The assessment of volcanic hazard is the first step for disaster mitigation. The distribution of repose periods between eruptions provides important information about the probability of new eruptions occurring within given time intervals. The quality of the probability estimate, i.e., of the hazard assessment, depends on the capacity of the chosen statistical model to describe the actual distribution of the repose times. In this work, we use a mixture of exponentials distribution, namely the sum of exponential distributions characterized by the different eruption occurrence rates that may be recognized inspecting the cumulative number of eruptions with time in specific VEI (Volcanic Explosivity Index) categories. The most striking property of an exponential mixture density is that the shape of the density function is flexible in a way similar to the frequently used Weibull distribution, matching long-tailed distributions and allowing clustering and time dependence of the eruption sequence, with distribution parameters that can be readily obtained from the observed occurrence rates. Thus, the mixture of exponentials turns out to be more precise and much easier to apply than the Weibull distribution. We recommended the use of a mixture of exponentials distribution when regimes with well-defined eruption rates can be identified in the cumulative series of events. As an example, we apply the mixture of exponential distributions to the repose-time sequences between explosive eruptions of the Colima and Popocatépetl volcanoes, México, and compare the results obtained with the Weibull and other distributions.

“…Although the MOED probabilities also are the averages of a renewal process, they result to be more sensitive to the existence of previous regimes than the Poisson estimates. It must be remarked that this does not mean that the MOED estimates are "regime variables" as those obtained by Bebbington (2007), who uses Hidden Markov (HM) models to identify regime changes, and thus estimate the probability of being in a regime. The renewal processes are series of events in which the times between events are independently and identically distributed (Cox and Lewis, 1966) as is the case of the models used here.…”

Section: Discussionmentioning

confidence: 99%

“…The Akaike Information Criterion penalises the misfit and the number of parameters used in the tested distributions (Akaike, 1973;Bebbington, 2007;Turner et al, 2008;Dzierma and Wehrmann, 2010).…”

Section: A T Mendoza-rosas and S De La Cruz-reyna: El Chichón Volcmentioning

confidence: 99%

Hazard estimates for El Chichón volcano, Chiapas, México: a statistical approach for complex eruptive histories

Mendoza-Rosas¹,

Cruz‐Reyna²

2010

Abstract. The El Chichón volcano (Chiapas, México) most recent eruption occurred in 1982 causing the worst volcanic disaster in the recorded history of Mexico. Prior to the eruption, El Chichón volcano was not considered a very hazardous volcano, a perception mostly caused by the low eruption rate of the past eruptions. The correct assessment of volcanic hazard is the first step to prevent a disaster. In this paper, we analyze two periods of the reported eruptive history of El Chichón volcano during the Holocene, searching for the eruption rates of different VEI magnitude categories and testing their time dependence. One period accounting the eruptions of the last 3707 years before the last eruption (BLE) is assumed to be complete, with no missing relevant events. More scarce information of a period extending to 7772 years BLE is then added. We then apply the Non-Homogeneous Generalized Pareto-Poisson Process (NHGPPP), and the Mixture of Exponentials Distribution (MOED) methods to estimate the volcanic hazard of El Chichón considering both periods. The results are compared with the probabilities obtained from the homogeneous Poisson and Weibull distributions. In this case the MOED and the Weibull distribution are rather insensitive to the inclusion of the extended period. In contrast, the NHGPPP is strongly influenced by the extended period.