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Retrospective analysis of uncertain eruption precursors at La Soufrière volcano, Guadeloupe, 1975–77: volcanic hazard assessment using a Bayesian Belief Network approach

Thea K Hincks1*, Jean-Christophe Komorowski2, Stephen R Sparks1 and Willy P Aspinall13

Author Affiliations

1 School of Earth Sciences, University of Bristol, Queen's Road, Bristol BS8 1RJ, UK

2 Institut de Physique du Globe de Paris, CNRS UMR 7154, PRES Sorbonne Paris-Cité, 1 rue Jussieu, 75238 Paris, Cedex 05, France

3 Aspinall & Associates, Cleveland House, High Street, Tisbury SP3 6HF, UK

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Journal of Applied Volcanology 2014, 3:3  doi:10.1186/2191-5040-3-3


The electronic version of this article is the complete one and can be found online at: http://www.appliedvolc.com/content/3/1/3


Received:24 July 2013
Revisions received:31 July 2013
Accepted:3 February 2014
Published:21 February 2014

© 2014 Hincks et al.; licensee Springer.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Background

Scientists monitoring active volcanoes are increasingly required to provide decision support to civil authorities during periods of unrest. As the extent and resolution of monitoring improves, the process of jointly interpreting multiple strands of indirect evidence becomes increasingly complex. Similarities with uncertainties in medical diagnosis suggest a formal evidence-based approach, whereby monitoring data are analysed synoptically to provide probabilistic hazard forecasts. A statistical tool to formalize such inferences is the Bayesian Belief Network (BBN). By explicitly representing conditional dependencies between the volcanological model and observations, BBNs use probability theory to treat uncertainties in a rational and auditable manner, as warranted by the strength of the scientific evidence. A retrospective analysis is given for the 1976 Guadeloupe crisis, using a BBN to provide inferential assessment of the state of the evolving magmatic system and probability of incipient eruption. Conditional dependencies are characterized quantitatively by structured expert elicitation.

Results

Analysis of the available monitoring data suggests that at the height of the crisis the probability of magmatic intrusion was high, in accordance with scientific thinking at the time. The corresponding probability of magmatic eruption was elevated in July and August 1976 and signs of precursory activity were justifiably cause for concern. However, collective uncertainty about the future course of the crisis was also substantial. Of all the possible scenarios, the most likely outcome evinced by interpretation of observations on 31 August 1976 was 'no eruption’ (mean probability 0.5); the chance of a magmatic eruption/blast had an estimated mean probability of ~0.4. There was therefore no evidential basis for asserting one scenario to be significantly more likely than another.

Conclusions

Our analysis adds objective probabilistic expression to the volcanological narrative at the time of the 1976 crisis, and demonstrates that a formal evidential case could have supported the authorities' concerns about public safety and decision to evacuate. Revisiting the episode highlights many challenges for modern, contemporary decision making under conditions of considerable uncertainty, and suggests the BBN is a suitable framework for marshalling multiple, uncertain observations, model results and interpretations. The formulation presented here can be developed as a tool for ongoing use in the volcano observatory.

Keywords:
Volcanic hazards; Multi-parameter monitoring; Bayesian inference; Uncertainty; Decision making; Expert judgement

Background

During a volcanic crisis, decisions typically have to be made with limited information and high uncertainty, on short time scales. The primary goal is to minimise loss and damage from any event, but social and economic loss resulting from false alarms and evacuations must also be considered (Woo 2008). Although it is not the responsibility of the scientist to call an evacuation or manage a crisis, there is an increasing requirement to assess risks and present scientific information and associated uncertainties in ways that enable public officials to make urgent evacuation decisions or other mitigation policy choices.

In the interests of safety for both exposed populations and scientists in the field, Aspinall et al. (2003) stress the need for a robust and defensible evidence-based approach to hazard and risk assessment. Although applied in medicine (Sackett et al. 1996), formal evidence-based decision-making is not common practice in volcanology (or indeed other areas of natural hazard and risk assessment). This paper aims to demonstrate the practicality and utility of this approach, particularly in situations with uncertain scientific information, limitations of understanding, and intrinsically unpredictable outcomes. The process of identifying and attempting to quantify sources of uncertainty (both epistemic and aleatory) can be very informative, from all key perspectives (scientists, risk assessors and stakeholders).

Scientific disagreements and political contretemps surrounded the 1976 volcanic crisis of La Soufrière volcano, Guadeloupe (Fiske, 1984; Feuillard et al. 1983)a. These controversies and subsequent costly evacuation highlighted the need for a more structured and transparent approach to the use of scientific advice in volcanic hazard assessment (Aspinall and Woo 1994). A key question that needed to be answered in 1976 was: “Is the current state of unrest magmatic in origin, and if so, what is the probability of an explosive volcanic eruption?”. An additional question was: “If an explosive eruption occurs, what is the probability that it occurs at the onset or very near the onset of the eruptive activity?” Indeed, the predominant working hypothesis - which influenced the scientific attitude of many and drove the political management of the crisis - was that if there were to be paroxysmal explosive activity then it was more likely to occur at the onset of an eruption, as in the tragic 1902 eruption of La Montagne Pelée, Martinique, rather than later.

These questions will be just as pressing if, in the near future, the signs of volcanic unrest manifested since 1992 in Guadeloupe (Komorowski et al. 2005; Villemant et al. 2005) were to escalate. The public clashes in 1976 left a legacy of loss of trust in scientists and authorities there. Any future crisis will need to be managed with care and a transparent and robust approach to information sharing if trust is to be fully restored. Villemant et al. (2005) remark that it remains very difficult to interpret monitoring data in terms of deep volcanic processes, for example to identify or differentiate between magmatic activity and purely hydrothermal activity. At the time of the 1976 crisis, knowledge of the style and magnitude of past eruptive activity of La Soufrière was limited, and multi-parameter monitoring data (e.g. seismicity, deformation, fluid geochemistry) inconclusive and, even jointly, insufficient to reliably inform hazard mitigation decisions.

Following the approach of Aspinall et al. (2003), we use a Bayesian Belief Network (BBN) to interpret jointly the observational evidence available in 1976, and make inferences about the key volcanic states and processes at the time. The output is a retrospective probabilistic forecast that communicates the perceived level of hazard and associated scientific uncertainty. BBNs have been widely applied in engineering and medical decision support systems (Spiegelhalter et al. 1993). The Bayesian methodology is also becoming increasingly used for decision support in natural hazard assessments, including flood risk, terrain analysis and water quality (see, e.g. Molina et al. 2005; Stassopoulou et al. 1998; Borsuk et al. 2003, respectively); and operational risk (Cowell et al. 2007; Neil et al. 2005).

The graphical nature of the BBN makes it an efficient and intuitive means for describing a complex, multi-faceted system. Causal relationships are easily visualised, and can be presented in a compact and easily communicated format (see Methods). Measures such as mutual information (the strength of the relationship between a pair of nodes – see Methods) and entropy can also be used to assess the relative evidential value of individual observations and identify where further data or research may improve the hazard forecast or help reduce uncertainty. All these features should aid communication between volcanologists, risk assessors and stakeholders - critical to the successful management of any volcanological unrest situation. Complementary approaches include logic or event trees (e.g. Newhall and Hoblitt 2002; Marzocchi et al. 2004; Marzocchi et al. 2008; Sobradelo and Martí 2010) - these are generally designed to capture a sequence of events and observations rather than describe the primary components and process interactions of the system; however, the basic probability calculus is largely the same.

Volcanic record of La Soufrière, Guadeloupe

La Soufrière is the most recently active part of the composite, andesitic La Grande Découverte-Soufrière volcanic (GDSV) complex, located in southern Basse-Terre, Guadeloupe (Figures 1 and 2). Comprehensive descriptions of the geological setting and volcanic activity (Komorowski et al. 2005; Boudon et al. 2008; Samper et al. 2009; Legendre et al. 2010; Legendre 2012) are summarised briefly here. Past activity has ranged from effusive fissure eruptions, explosive magmatic and phreatic episodes, and major sector collapse events (Boudon et al. 1987; Carlut et al. 2000; Komorowski et al. 2008a; Samper et al. 2007) (Table 1).

thumbnailFigure 1. Map of Guadeloupe. Location of La Soufrière volcano, Guadeloupe, in the Lesser Antilles island arc, eastern Caribbean.

thumbnailFigure 2. Schematic diagram of La Soufrière volcano. This illustration shows the types of process and phenomena identified at La Soufrière. The BBN in Figure 3 is a development of this simple conceptual volcano model (focussing on the observations available at the time of the crisis and associated volcanic processes) and aims to capture how the key elements of this system are linked.

Table 1. Eruptive chronology of La Soufrière in the last 9000 years, with calibrated radiocarbon dates (CE: common era and BCE: before common era) from Boudon et al. (1988); Komorowski et al. (2005; 2008); Boudon et al. (2007and 2008); Siebert and Simkin (2002–2011)

The GDSV complex consists of three major composite andesitic volcanoes formed over the last 445 ky: the Grande-Découverte volcano, the Carmichaël volcano, and the most recent La Soufrière volcano (Boudon et al. 1988; Komorowski et al. 2005; Samper et al. 2009). Although the Grande-Découverte phase (445 to 42 ka) was dominated by effusive activity it was interrupted by at least three major caldera-forming explosive eruptions approximately at 140, 108,and 42, ka . The 42,ka caldera eruption marks the end of the Grande-Découverte phase (Boudon et al. 1988; Komorowski et al. 2005). This was followed by effusive and explosive activity, forming a new edifice in the caldera. The Carmichaël phase (35 to 11.5 ka) was dominated by effusive to explosive dome-growth and ended with a sector collapse and associated laterally-directed explosions without a magmatic component. The Soufrière phase (last 9 ky) has been characterized by recurrent effusive to explosive dome-growth and up to eight sector collapse events (Komorowski et al. 2005; Boudon et al. 2007; Komorowski et al. 2008b) with major events about 8 and 3 ka (Boudon et al. 2007). Several of the sector-collapses were associated with lateral blasts (2–5 events) and a magmatic component (Boudon et al. 1987; Komorowski et al. 2008b).

At least three low to moderate (VEI 2–3) explosive eruptions occurred in the last 9000 years, and three or more much larger VEI 4–5 explosive eruptions in the period from 11.5 to 42.5 ka. The last significant explosive magmatic eruption (VEI 2–3, 1530 AD) produced a multi-stage sequence: a debris avalanche from sector collapse; pumice and scoria fallout from a short-lived subplinian convective column; column collapse scoria pyroclastic density currents, and an andesite lava dome that ended the eruption (Boudon et al. 2008; Komorowski et al. 2008a). The petrology and isotopic signature of the 1530 AD erupted products suggest a zoned magma chamber at a depth of about 6 km periodically fed by basaltic magma from depth (Touboul et al. 2007; Boudon et al. 2008).

Carbon-14 dating of lahar, pyroclastic surge and pumice fall deposits indicate that further activity occurred between 1530 AD and the arrival of European settlers in 1635 AD (Boudon et al. 2008; Legendre 2012). Historical activity has been phreatic, characterized by explosions, episodes of vigorous degassing and ash venting. Major phreatic eruptions occurred in 1797–98 and 1976–77, with minor events in 1690, 1812, 1836–1837 and 1956 (Boudon et al. 1988; Komorowski et al. 2005). Hazards associated with phreatic activity include vertical and laterally-directed blasts, ash fall, small-volume non-magmatic pyroclastic density currents, debris flows, acid degassing, potential contamination of the groundwater and aquifer. Major phreatic eruptions could trigger flank collapse and lateral non-magmatic blasts.

The 1975–77 episode of eruptive unrest at La Soufrière

Between 1956 and June 1975 the mean monthly number of recorded volcano-tectonic (VT) earthquakes was 15 and the mean monthly number of felt VT earthquakes was about 0.2 (Dorel and Feuillard 1980; IPGP 1956–2013). This 19-year average is considered the baseline level of VT activity. In contrast to previous phreatic eruptions of La Soufrière (including the last short-lived phreatic eruption of 19–27 October 1956: Jolivet, 1958), a one year period of steadily increasing volcanic seismicity was recorded and felt in Guadeloupe prior to the start of surface activity. This pre-eruptive seismicity was characterized by large numbers of recorded and felt events, and the occurrence of three distinct successive earthquake swarms of increasing released seismic energy. An initial swarm of 30 VTs, one of which was felt, occurred in July 1975. Monthly seismicity continued to increase in the following months and was characterized by two significant swarms of VTs in November to December 1975 (total of 296 VTs and 4 felt events) and in March to June 1976 (total of 2713 VTs and 59 felt events). Although by this stage the rapidly escalating seismicity was about 45 times greater than the baseline monthly rate there were no discernible changes in fumarolic activity (Dorel and Feuillard 1980; Feuillard et al. 1983). However, on 9 June 1976 a minor landslide occurred on the La Ty fault and new fractures were observed on the road at the base of the dome (Feuillard 2011). An explosion occurred at 08:55 local time on 8 July 1976, marking the onset of nine months of complex eruptive activity. The explosion reactivated the 1956 phreatic eruptive fracture and produced 60% by volume (0.6 × 106 m3) of the total ejected solids from the entire 1976–1977 crisis (Le Guern et al. 1980). Table 2 summarises the timeline of key eruptive phenomena, monitoring and phenomenological data (observables) and associated decisions by the scientific team and the public officials.

Table 2. Timeline of the volcanic crisis from July 1975 – June 1977

The 1975–1977 unrest and eruptive crisis lasted 22.5 months from 24 July 1975 until 15 June 1977, during which 16493 VTs were recorded of which 153 were felt (Dorel and Feuillard 1980; Feuillard et al. 1983; IPGP, 1956–2013; Feuillard 2011). A total of 26 explosions occurred in two phases over the 9 months from 8 July 1976–1 March 1977. The eruption can be divided into five main phases: a) a pre-eruptive unrest phase from 24 July 1975–7 July 1976 with steadily increasing and escalating seismicity (3189 VTs, 64 felt VTs); b) eruptive phase 1 from 8 July to 10 November 1976, the most intense in terms of explosive activity (17 explosions) and seismicity (11,649 VTs, 68 felt VTs); c) eruptive phase 2 from 11 November 1976 to 4 January 1977 (the least intense) with no explosions, frequent ash venting episodes, and a decrease in recorded seismicity (968 VTs; 9 felt VTs); d) eruptive phase 3 from 5 January to 1 March 1977, which showed a renewal of explosive activity (9 explosions, some amongst the largest of the entire crisis) and a decline in seismicity (475 VTs, 7 felt VTs); e) a post-eruptive unrest phase beginning 2 March during which seismicity returned to almost background levels by 15 June 1977, formally marking the end of the eruption. Excluding volcanic tremor, recorded seismicity was poorly correlated with eruptive phenomena.

The 1975–77 eruption produced an estimated 106 m3 of non-juvenile tephra deposited as very fine, thin ash up to 10–15 km west and south of the summit (Heiken et al. 1980; Le Guern et al. 1980). Ballistic blocks (a few kilogrammes to several tonnes) were ejected up to 1.6 km from the vent. The eruption was accompanied by morphological changes including opening of two new major sets of fractures (several hundred meters long, several decameters wide) in the dome, and the widening and deepening of historically old craters and fractures. Small-volume pyroclastic density currents from the explosive vents extended to about 1 km from the dome (Sheridan, 1980) before transforming into debris flows that formed from the resurgence of perched aquifers and reached up to 3.5 km to the east and south. The eruption was accompanied by low-temperature (100-200° C) degassing of H2O and minor quantities of CO2, H2S, SO2, as well as acid condensates (HCl, HF, Br) and a sustained renewal of fumarolic activity on and at the periphery of the dome.

Phreatic tephra consisted essentially of old hydrothermally-altered material from the dome and old pyroclastic fragments from the nearby Echelle scoria cone. Several authors (Marinelli 1976; Brousse et al. 1977; Heiken et al. 1980) reported that the phreatic products contained up to 10 % by weight of fresh unaltered vitreous andesitic fragments - these were later identified as material from the underlying 1530 AD pumice fallout deposits, although this was not understood at the time. This led to major scientific controversy as to whether there was evidence of juvenile material and therefore fresh magma in the conduit, compatible with the rapidly escalating and publicly obvious unrest. In the absence of conclusive monitoring data and independent evidence of a magmatic component, this single observation had a pivotal influence on the scientific management of the crisis, justifying a precautionary response by the authorities.

Two models have been proposed to explain the 1975–77 eruption. The more widely-accepted model interprets the crisis as a still-born magmatic event (Feuillard et al. 1983). Seismic activity from a depth of about 6 km is compatible with the inferred depth of a magma reservoir, and observations of SO2 provided evidence for magmatic unrest. Analysis of Cl levels in thermal springs sampled in the 15 years following the crisis has attributed the cause to shallow magma intrusion (Villemant et al. 2005; Boichu et al. 2011), and the chlorine isotope signature of thermal spring waters is markedly magmatic (Li et al. 2012). Feuillard et al. (1983) proposed that abnormal heat flux, either caused by magma differentiation of the 1530 AD magma batch or magma chamber replenishment, led to formation and propagation of fractures towards the surface, enabling migration of magmatic gases into the hydrothermal system. The hydrothermal system acted as a buffer, dissipating heat and magmatic gases and inhibiting magmatic eruption. Recent modelling of the flux of noble gas isotopes (4He/3He; R/Ra) in hydrothermal fluids suggests that either new magma is being continuously injected in the 6 km deep magma chamber, with a fresh batch of magma emplaced between 1959 and 1962 (Ruzié et al. 2012) or that magmatic gases are being transferred into and through the shallow magma chamber. The attendant increase of heat flux, and migration of fluids into the locally sealed hydrothermal system led to progressive pressurization and the phreatic activity of 1976–77.

A second model, proposed by Zlotnicki et al. (1992), does not invoke physicochemical changes in the magma reservoir. Rather, it proposes that aquifers become isolated (sealed) by structural readjustments or deposition of impermeable clay minerals from hydrothermal activity. This sealing is assumed to limit convective heat transfer from depth, leading to pressurization. Phreatic eruption occurs when overpressures are sufficient to cause fracturing.

The 1976–1977 eruption engendered significant and recurrent disruption and risk to the population (Le Guern et al. 1980) largely due to: 1) frequently and strongly felt volcanic seismicity; 2) atmospheric contamination by acid gases (H2S, SO2) and fine corrosive volcanic dust rich in acid condensates and Ca-sulfate that sometimes also contained non-negligible quantities of silica polymorphs; 3) contamination of potable spring waters and water tanks due to soluble acid condensates (including halogens such as fluorine, chlorine, bromine) and other trace elements adsorbed on the surface of the erupted ash; 4) chemical and mechanical consequences of the contamination of crops and grazing land due to acid condensates and trace elements adsorbed on the ash, in particular fluorine.

The 6-month evacuation of around 73,000 people (with an estimated 2000 remaining) caused severe socio-economical difficulties for the population in southern Basse-Terre and the island as a whole, and had a profound and long-lasting influence on society. The cost of evacuation has been estimated at 60 % of the total annual per capita Gross Domestic Product (GDP) of Guadeloupe in 1976 (Lepointe, 1999; Blérald, 1986) or about 342 million USD using 1976 currency rates (Kokelaar 2002); this excludes the losses of uninsured and other personal assets including open-grazing livestock and farm animals. Hence, the crisis ranks amongst one of the most costly of the 20th century (Annen and Wagner 2003), although there was no loss of life.

A Bayesian Belief Network for Guadeloupe 1975–77

A BBN has been developed to describe the fundamental processes and interactions governing volcanic activity and the state of unrest at La Soufrière from 1975 to 1976 (Figure 3). This is an extension of the model developed in Hincks (2007), following re-analysis of the literature and reports about the 1976 crisis as part of the European Union EXPLORIS project (Explosive Eruption Risk and Decision Support for EU Populations Threatened by Volcanoes, Komorowski et al. 2004) and the 2009 Agence Nationale de la Recherche funded CASAVA project (Compréhension et Analyse des Scénarios, Aléas, et risques Volcaniques aux Antilles). This BBN is a simplified conceptual model of the volcanological system depicted in Figure 2, informed by understanding of the causal relationships between processes at depth and resulting surface manifestations of activity. As the aim here is to perform a retrospective analysis of the 1976 crisis, the model is based on what was then contemporary knowledge of the volcanic system and includes only observations available at the time. The basic structure could easily be adapted to build a model for renewed activity, incorporating additional nodes and arcs to reflect the increased scope, resolution and frequency of modern monitoring, and current understanding of the various interactions and time-scales involved (e.g. rates of degassing), informed by both observation and numerical simulation.

thumbnailFigure 3. Bayesian belief network diagram for La Soufrière. BBN showing the relationship between volcanic processes, states and observations available in 1976, used to infer future activity. Nodes represent both hidden (grey) and observable (blue) states. The arcs between nodes represent conditional dependencies (e.g. direct causal relationships or influence) and are characterized by conditional probability tables (CPTs). The arrows indicate the direction of influence e.g. venting tremor is believed to be a sign of perturbation of the hydrothermal system resulting from magma ascent. In this case, all conditional probability distributions (and associated uncertainties) were obtained by expert elicitation (for example, the probability of observing SO2 given that magma is ascending), the network structure being agreed by the group prior to elicitation. For data rich applications parameters can be estimated purely from data, or a combination of observations and prior knowledge (e.g. using Dirichlet priors or other parameter constraints: Niculescu et al. (2006). The basic difference between this and a logic or event tree representation is explained in the Methods section and Figure 4.

All nodes are discrete, with mutually exclusive and exhaustive states (Figure 3). Arcs between nodes indicate direct conditional dependencies (represented by conditional probability tables - CPTs), with the arrow showing the direction of influence. Observations (nodes shown in blue) can then be used to make inferences about unobservable or 'hidden’ states of the system (grey) and about the outcome of interest - whether or not an eruption will occur (the query node). This is a static network which, unlike an event tree, does not explicitly model time dependency, i.e. variables are evaluated only at a particular point in time. The probability of eruption and other unobservable states can, however, be evaluated at discrete time steps to give a changing 'hazard forecast'. Temporal associations between nodes can be incorporated in a dynamic BBN - see Future Developments.

In order to enumerate network parameters (conditional probabilities) and associated uncertainties for this analysis, we conducted a structured expert judgement elicitation over two days in November 2007 (see Methods, below). The group (the authors and colleagues, see acknowledgements) comprised seven volcanologists with expertise covering a range of relevant disciplines, some of whom were very familiar with the 1976 events. With the Cooke approach, a group of seven experts would be regarded as close to a minimum quorum, but here it is considered sufficient for deriving quantitative values for the BBN. Following an initial discussion of the network structure and related elicitation questions, the experts were asked to provide their personal 5, 50 and 95 percentile estimates of different volcano state probability values, taking into account uncertainties associated with the various processes, data and interpretations of evidence. After pooling these opinions into joint, group distributions (see Methods), we use the results to populate conditional probability tables (CPTs) for each node in the BBN. For example, one such question for node 1 was phrased (using the present tense) as follows: "What is the probability of ground deformation occurring due to a deep source in La Soufrière region, given magmatic intrusion has occurred at depth (5–10 km)?". The elicitation questionnaire and results are provided as supplementary material to this paper [see Additional files 1 and 2]. With respect to the elicitation, a key feature of the procedure is that it encourages experts to state, independently after group discussion, their true opinion; this limits direct peer influences and other biases (Aspinall 2006, p28).

Additional file 1. Expert elicitation questionnaire. The complete set of questions addressed by the expert group during the elicitation in Bristol, 28-29 November 2007.

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Additional file 2. Elicitation results. This document presents the individual experts responses to the elicitation questions, the resulting joint Decision Maker's estimates, and associated notes prepared by facilitator (W. Aspinall) immediately after elicitation responses were processed.

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To frame the elicitation questions we chose a time scale of three months to evaluate the likelihood of observing any given phenomena or sustained level of activity. Recognizing that uncertainty in hazards forecasts increases the further ahead one looks, this time period is a pragmatic choice, appropriate in terms of decision-making in a volcanic emergency. Moreover, during the pre-eruption unrest phase at La Soufrière from June 1975 to July 1976, earthquake swarms that lasted about 1 month were separated by periods of lower seismic activity on the order of 2–3 months (Dorel and Feuillard 1980; Feuillard et al. 1983). At the Soufrière Hills Volcano, Montserrat, a time scale of six months was chosen for hazard and risk assessment during the volcanic emergency (Sparks and Aspinall 2004), this period being long enough for authorities to make and implement mitigation plans but judged short enough by the science assessment team for making useful hazards forecasts. In practice, the forecast time period can be determined by the situation and other demands, with the network nodes and structure designed to accommodate multiple time periods if necessary.

Although observations available in 1975–77 only are used in our analysis, knowledge of volcanic processes and the assessment and interpretation of monitoring data have advanced since then. The group felt it was impossible to transport themselves back fully into the state of general volcanological knowledge and mindsets of the time so the elicitation process inevitably incorporates some elements of contextual advances in the science.

Methods

Bayesian Networks

The basic concept behind the treatment of uncertainty in Bayesian Belief Networks is conditional probability (Jensen and Graven-Nielsen 2007). A BBN is a directed acyclic graph and comprises a set of variables (nodes) with either continuous or discrete state-sets, together with a set of directed links or arcs representing the direction of causality or influence between the nodes (see basic example in Figure 4a). Nodes can represent observable or hidden states of the system. A link between two nodes is described by a conditional probability distribution (CPD), with multiple state distributions entered on a Conditional Probability Table (CPT). There are similarities with the logic or decision tree (a BBN can have a tree structure, or be approximated by a tree); however, the BBN captures conditional independence and in some cases can represent the system more compactly. The tree (Figure 4b) has a hierarchical as opposed to a network structure, and represents a sequence of events occurring in time culminating in the outcome of interest. In a BBN, decision or logic tree, node states are mutually exclusive and exhaustive, but this is not necessarily the case in an event tree (e.g. see Marzocchi et al. 2006).

thumbnailFigure 4. Comparing a Bayesian Network and logic or event tree. (a) This is an elementary BBN representing a very simple volcanological model to infer the probability of eruption resulting from magmatic unrest. Unrest (shaded grey) cannot be observed directly, and must be inferred from observations (e.g. seismicity and SO2 emission, shaded blue). The query node (shaded pink) is the outcome of interest - eruption. For a generic Bayesian Network B = {X,Y,U}; X represents the set of unobserved or hidden states, Y the set of observable states, and U the set of arcs or directed links between nodes. Arrows indicate direction of causality or influence. For the simple case where nodes are assigned discrete states, node relationships are described by conditional probability tables (CPTs). States must be mutually exclusive and exhaustive. (b) presents an equivalent logic tree representation of the basic BBN in (a). For simplicity two alternative states are shown for each node.

In this study, OpenPNLb (an open source C++ library initially developed by Intel) was used for the computational BBN. A separate “link connection strength” package, an add-on to OpenPNL by Ebert-Uphoff (2007), was used to calculate mutual information (described below). There are numerous alternative software tools with graphical user interfaces which make BBN inference and analysis accessible. Many also provide API (Application Programming Interface) functionality enabling custom code to be written for automated inference and data input/output. Examples include GeNIe and SMILE (Decision Systems Laboratory 2013), Netica (Norsys 2013) and Uninet (Cooke et al. 2007).

In the static BBN presented here, data from the 1976 episode were limited and it was necessary to use expert judgement to fully describe the stochastic and scientific uncertainties. The ability of the volcanologists to quantify uncertainty was therefore critical to the reliability of the model, and we used a structured elicitation procedure (described below) to obtain collective, quantified uncertainty distributions.

BBN network structure

The network structure and node states for the Guadeloupe BBN are shown in Figure 3. There are three different node types: 'hidden’ or latent state nodes; 'observable’ nodes, and the all-important eruption outcome 'query’ node. The state of the hidden nodes (unobservable volcanic processes) and eruptive outcome can then be estimated from the input observations using Bayesian inference (see Aspinall et al. 2003). The following section describes each of the nodes in more detail. Additional file 1 documents the elicitation questions used to populate the conditional probability tables for each node in the BBN.

Hidden nodes

These represent unobservable volcanic states and processes that can only be inferred from observations.

0 Magmatic intrusion at depth. Depth is defined as 5 km or greater, in the region of the magma chamber. Node 0 has no parents, therefore experts were asked to estimate 5, 50 and 95 percentile values to characterize a probability distribution that new magma was being intruded in the absence of any volcanological evidence – i.e. the baseline probability of intrusion taking place at La Soufrière at any point in time (see Additional file 1 – elicitation questions). This is a simple binary node – the state is either true or false.

1 Deep source ground deformation. This is defined as a source at 5 km or greater depth, giving rise to wide field surface deformation. It is anticipated that only magmatic processes could result in such deformation. Node states: true/false.

2 Perturbation of the hydrothermal system due to magmatic processes at depth. The network considers the effects on the hydrothermal system due to magmatic processes at depth (typically 5–10 km), and magma ascent (above 5 km depth, node 4 below) separately. Identification of either effect can be complicated by exogenic forcing (tectonic and meteorological). Node states: true/false.

3 Magma ascent. Ascent is defined as any movement of magma above the magma chamber (around 5 km depth) toward the near-surface. Magma need not necessarily reach the surface and could stall at shallow depth. This network node represents the probability of ascent over a three month period, given magmatic intrusion is occurring at depth. If there is no magma intrusion then the probability of hydrothermal perturbations due to magmatic processes at depth is assumed to be zero (and therefore not elicited). States: true/false.

4 Perturbation of the hydrothermal system due to magma ascent. As for node 2, effects can be complicated by exogenic forcing. If there is no magma ascent then the probability of hydrothermal perturbations due to magma ascent is zero (therefore not elicited). States: true/false.

Observable nodes

Figure 5 (lower panel) shows the time series of observational evidence input to the BBN, replicated from 1976 records and narrative accounts from Guadeloupe. As this is intended as an indicative example rather than a complete chronology of observations, these inputs span the whole crisis but focus on key dates when significant observations were made, according to contemporary reports. Computation time is very short, a few seconds, so a modern day BBN could be updated frequently and coupled with real time monitoring data streams to give near continuous hazard estimates. The following summary describes the chosen thresholds for different levels of activity for each of the observables, with some (modern) commentary on the available data, processes and reliability of evidence. The commentary reflects discussions and decisions among the expert group, referenced where possible. Volcano-tectonic earthquakes are denoted as VT.

5 Number of VT earthquakes. The baseline rate for Guadeloupe was assumed to be approximately 12 VT earthquakes per month (Dorel and Feuillard, 1980; Feuillard et al. 1983). Recent access to more detailed reports (IPGP, 1956–2013) gives a revised baseline rate of 15 recorded VTs/month and 0.2 felt VTs/month (as quoted in the description of eruptive unrest earlier in the manuscript); this, however, has little effect on the original findings. Three states are defined: low ( ≤ 24/month, up to double the baseline rate), moderate (between 24 and 120/month) and high (greater than 120/month, or in excess of 10 times the baseline).

6 Number of felt VT earthquakes. Felt earthquakes are typically VT events with a duration magnitude (Md) of approximately 2 or greater. This node has three states: low (≤ 1 felt earthquake per year), moderate (1-10/year) and high (>10/year). The baseline rate from historic records for the period from 1956 through 1975 assessed in contemporary reports (Smithsonian Institution 1976; 1977) was assumed to be approximately 0.6/year (although note that as for node 5, IPGP (1956–2013) gives a revised baseline of 0.2 felt VTs/month). There are good data on felt earthquakes associated with previous episodes of unrest (Jolivet 1958; Dorel and Feuillard 1980), and although various factors influence detection and reporting (population distribution, survival of records etc.) this non-instrumental observable is assumed to be an informative parameter, with a relatively consistent historical detection threshold. Felt VTs can also be used as a proxy for seismic energy release. However, felt VTs could be non-magmatic in origin, e.g. due to activation of faults related to disturbance of the hydrothermal system. It is possible that magmatic intrusion may occur but not be accompanied by detected or felt VTsc.

7 Total seismicity. This node represents the observation of deep seismicity, and the likelihood of it being caused by magmatic intrusion. A high level of activity is defined as more than 24 recorded VT signals and more than 1 felt earthquake per month. Non-magmatic triggers could include tectonic activity associated with the regional fault system and high fluid pressures and stresses in the hydrothermal system.

8 Borehole-type tiltmeter observations. Following vigorous ash venting on the 30 August 1976, scientists from the Los Alamos National Laboratory deployed four novel borehole-type tiltmeters on the southwest flank of La Soufrière, at distances of 0.8 to 8 km from the vent in temporary surface installations. Measurements for the 22–23 September, 27 September and 2 October 1976 are given in West et al. (1976). In 1976 this was new, unproven, experimental technology. The instruments were not fixed in boreholes and hence the data are considered unreliable, with non-volcanic causes of tilt (e.g. ground surface instability, wind, heavy rainfall). For this node, a high tilt rate is defined as >10 μrad/day.

9 Dry tilt data. In 1976 dry tilt was considered to be an accurate and tested method, likely to detect widespread deformation. In August 1976, R. Fiske and K. Kinoshita set up four dry tilt stations on the SW flank of La Soufrière. Data for the period 29 August - 30 September are presented in Smithsonian Institution (1976) reports. The report states that tilt measurements from 1–16 October lacked coherence, likely due to effects of rainfall and evaporation at the stations. Smithsonian Institution (1977) reports "ground deformation measurements … (pendulum, borehole, dry tilt) have indicated no significant changes in shape". We therefore assume no evidence of deformation from dry or borehole-type tilt for 30/11/76 and 1/3/77. High dry tilt is defined as >10 μrad/day.

10 Increased or decreased pressure, acid-rich fumarolic activity. Magmatic processes (degassing, interaction with groundwater etc.) can perturb the hydrothermal system and increase fumarolic activity. The presence of highly water-soluble halogen species (e.g. acids such as HCl, HBr) in thermal springs can be explained by magma intrusion and degassing that interacts with the hydrothermal system (Boichu et al. 2011estimated a volume of 0.01 - 0.52 km3 of intruded magma). Halogen acids are stable in water and largely unaffected by cooling and decompression (Villemant et al. 2005). Fumarolic activity, Cl content of thermal springs and seismicity slowly declined in the 10–15 years following the crisis. The slow decline could be associated with reducing supply of magmatic fluids. Alternatively changes in porosity and permeability, along with self-sealing of the host rock and conduit, could reduce rates of surface degassing without the deep source flux of magmatic gases necessarily changing. The hydrothermal system could also be affected by tectonic and weather-related processes, although this was considered by the expert group to be low probability.

Temperatures of fumaroles and hot springs had been measured since 1956 (Zlotnicki et al. 1992). Limited gas chromatography data existed, but problems with sampling (reaction of gas species in the sample vial during and after sampling) make these unreliable measurements (Feuillard et al. 1983). Given this uncertainty, and that regular sampling of fumaroles did not begin until after the crisis in 1979, this node has simply two states (true/false) to capture any reported increase in fumarolic activity in the absence of reliable geochemical data. Between 1970 and 1976 the only active fumaroles were at the base of the dome, and in 1976 fumarolic activity developed rapidly on new and re-opened fractures on the summit, flanks and base of the dome. Heiken et al. (1980) report near-continuous background fumarolic activity during the period August-October 1976. Activity decreased rapidly towards the end of the crisis, ceasing first at the summit, then at the periphery in 1977, with the north fault fumarole disappearing in May 1977 (Zlotnicki et al. 1992). For purposes of this analysis we therefore assume elevated fumarolic activity was present from mid 1976, returning to a low level by 30 November 1976.

11 SO2 present in gas/steam emissions. This node represents the probability of detecting SO2, given magma ascent (or otherwise). SO2 is regarded as diagnostic of shallow depth magmatic origin (Villemant et al. 2005), however, chemical and other processes (decompression, cooling, distance from source, chemical reactions etc.) can affect concentration, speciation and the time lag from release to detection.

In assessing SO2 there is need to account for false positive and false negative observations. For example, the potential for SO2 emissions due to deep magmatic unrest, but no magma ascent above 5 km. The meteoric system can also act as a filter or sink for chemical species that might otherwise signify magmatic activity (e.g. scrubbing of SO2 or halogen species). La Soufrière has an extensive hydrothermal system (Villemant et al. 2005; Zlotnicki et al. 2006) and scrubbing due to tropical rainfall will increase the chance of a false negative observation. Conditions at the time adversely affected the quality and frequency of observations, and as a result there are no published SO2 data from the time of the crisis (only pH). The pH is not diagnostic if anions (SO42- and SO32) are not analysed to rule out the contribution of acid halogens (e.g. HCl) to the acidity of fumarolic condensates. Moreover false positive secondary SO2 can be formed at fumaroles due to rapid oxidation of H2S by the atmosphere or bacteria.

12 Petrological evidence - observation of juvenile material. This is defined as observation of abundant, unaltered fresh juvenile glass. On 12 August 1976 some scientists identified pumice in ejecta from steam explosions, and this was presented as strong evidence for fresh juvenile material. However, this sample may have been recycled material from the previous magmatic eruption in 1530 AD or even misidentified clay minerals. At the time juvenile glassy material was typically assumed to be vesicular, but recent research suggests otherwise. For example, non-vesicular, microlite-rich glassy fragments of the Mt St Helens cryptodome have been identified (post hoc) by Cashman and Hoblitt (2004) as a magmatic precursor. Here the experts were required to evaluate the probability of a false negative result or misclassification. Fresh glass may be present but not identified in the sample, magma may be ascending but fresh material may not be present at the surface, and old products remobilised by phreatic activity could be misidentified as fresh material.

13 Presence of continuous venting tremor. Continuous tremor can be generated by gas escape or fluid flow, and episodes of tremor can be associated with phreatic explosions (e.g. Barberi et al. 1992; Young et al. 1998; Nakada et al. 1999). It is possible (although much less likely) that non-magmatic processes such as changes in groundwater circulation or atmospheric pressure could perturb a hydrothermal system and generate low-level seismic tremor (e.g. Aspinall et al. 1976).

14 Seismic hypocenters ascending. This is defined as the emergence of an average of 24 or more VT earthquakes above 5 km depth (above the magma chamber) over three months. Such events could be caused by magma ascent (movement of magma, rock fracturing, gas escape) or alternatively hydrothermal processes driving movement of fluids, gas or steam, or fracturing. By considering a 3-month period of activity significantly above the baseline, the aim is to make it easier to differentiate between ascending magmatic and static non-magmatic drivers, as it was considered unlikely that shallow seismicity would persist at such elevated and escalating levels without an element of upward-moving magmatic activity. Discussion of Hirn and Michel’s (1979) post-crisis analysis of seismic hypocentres during the elicitation may have influenced the experts here – highlighting the difficulty of performing a reanalysis some 40 years on.

thumbnailFigure 5. La Soufrière Volcano observations and eruption probabilities for July 1975 to March 1977. Upper three panels: time variations in BBN probability estimates for: (a) a magmatic eruption or magmatic blast; (b) a phreatic eruption, or (c) no eruption, given observation states shown in the lower part of the figure. The unbroken black line denotes the expected (mean) probability estimate and the dashed line the median, as determined by Monte Carlo re-sampling of BBN input distributions; the shaded bands show the corresponding 5–95 percentile ranges, indicating the uncertainty in the forecast probability. Lower ten panels: sequence of observation states input to the BBN to estimate probability of eruption (see text).

Query node

15 Eruption within 3 months The outcome (in any subsequent three-month period) is defined as the probability of: (a) no eruption; (b) a phreatic eruption; (c) a magmatic eruption, or (d) a magmatic blast.

Expert elicitation

A structured expert elicitation is a formalized method for assimilating group judgements in a robust and reproducible way. In a crisis, a carefully targeted elicitation can be completed in an hour or so, including expert calibration, and the findings processed within a further hour, as happened many times in Montserrat in 1995–96 (e.g. Aspinall et al. 2002, pp82-83).

In the present exercise, individual expert uncertainty judgements were combined with equal weights using the “Classical Model” formulation (Cooke 1991; Aspinall and Cooke, 2013) and its implementation in the EXCALIBUR software package (Cooke et al. 2000). The elicitation questions [see Additional file 1] were structured to obtain enough information to enumerate all CPTs, and hence fully characterize the BBN. To quantify uncertainty in each parameter in an elemental distribution form, the volcanologists were asked to provide lower and upper tail quantile markers, corresponding to 5% percentile and 95% percentile values, together with a median estimate to locate central tendency. Because of the small size of the group and the variety of specialisms involved, the Classical Model performance-based differential weighting option was considered not to be appropriate; instead, equal weights combinations of experts’ uncertainty distributions were computed. For each variable, this took the form of a 'joint’ estimate of the relevant 5, 50 and 95 percentile values, expressing the spread of uncertainty for each BBN node item; these quantiles were used to fit and parameterize standard statistical distributionsd (see section below “BBN parameterization and evaluation”).

Elicitation responses

The outcomes of the elicitation exercise are provided in an additional file [see Additional file 2]. Most of the responses to questions demonstrate clearly (even four decades after the event ) that considerable uncertainty would still attend nearly every aspect involved in assessing the internal state of La Soufrière volcano from observations, and that the diagnostic power , in terms of eruption forecasting, is generally weak. This would have been even more so in 1976. However, this approach enables expressions of scientific uncertainty in more objective terms than possible at the time.

By and large, elicitation responses are reasonably coherent across the group, but the preponderance of wide credible intervals tends to mask any systematic differences between experts. Some of the results are informative with regard to evidential value of particular types of observations. Here we consider three specific target items (out of the 62 elicited, in total) to illustrate some generic issues.

Figure 6 shows the results of the elicitation relating to a question on the probability of observing/detecting ground deformation from a deep source in La Soufrière region. Two versions of this question, involving different conditional stipulations, were elicited: the first given there is no magmatic intrusion/unrest at depth, and the second given there is magmatic intrusion or unrest at depth. In this case, most experts judged that widespread ground deformation would accompany either intrusion or magmatic unrest. Five experts showed a high degree of confidence in the positive link, whereas the other two experts, whilst agreeing that this link was very likely, also gave a very wide uncertainty range. In contrast, all but one expert placed the likelihood of having either intrusion or magmatic unrest but no surface deformation as low. The same two experts again gave much wider uncertainties to this question than their other colleagues.

thumbnailFigure 6. Example expert range graph. Plot showing the individual experts responses to a question on the probability of observing/detecting ground deformation from a deep source in La Soufrière region.

These results reflect the widespread view that surface deformation can be equated with intrusion of new magma, magma chamber replenishment or some internal magmatic event, such as convective over-turn, any one of which processes will increase pressure. The minority two experts felt that it was possible to envisage a deep magma system perturbed in some way that would not be necessarily accompanied by surface deformation. In an ideal exercise, these findings would be debated and all experts then re-elicited, having discussed the reasons why colleagues came to different conclusions. The principle here is not to enforce consensus but to be sure that all the pertinent evidence and arguments have been presented, and equally well-understood by the group.

Target Item 11 (Q12 on questionnaire [see Additional file 2]) concerns false positives for SO2 detection. Some colleagues considered that there is no diagnostic power in this information, but others suggest there is a small chance of it having evidential worth. The 1976 crisis occurred before the availability of high quality SO2 measurements from ground instruments and from satellite remote sensing. SO2 observations (now routine) would likely now be given more weight. However, experience in Montserrat indicates that SO2 data commonly remain enigmatic or ambiguous; for example, a decrease in SO2 might mean either a decline in deep activity and hence a diminution of eruption likelihood, or that gas has become trapped enhancing the prospect of an imminent explosion. As with most observables, SO2 has more evidential worth when analyzed in conjunction with changes in other observables.

With Target Item 36 (Q26 row 4 on table, Additional file 2), four experts seem convinced that there is a high probability of seeing shallow VTs when the hydrothermal system is perturbed by deep magma action but magma is not ascending, while the other three are much more cautious and gave far lower probability values. With respect to Target Items 51 and 52 (Q27 two rows on table, Additional file 2), two Experts (2 and 3) diverge from colleagues in estimating 'No eruption’ or 'Phreatic eruption’ probabilities when the hydrothermal system is perturbed by deep magmatic action but magma is not ascending. These are examples of “two schools of thought”, where further discussion might either resolve the issue to everyone’s satisfaction (i.e. some consensus is reached) or fail to resolve it so that, collectively, uncertainty is consequently high. In this case, the issue remained unresolved within the present exercise.

These examples illustrate ways in which important uncertainties in our retrospective re-appraisal of the 1975–76 crisis were brought out by adopting a structured elicitation approach.

BBN parametrization and evaluation

For the BBN calculations, we transformed the expert group quantile marker values into a standard statistical functional distribution. Two alternative distributions were considered: Dirichlet and Generalized Trapezoidal; and results compared to find which provided the best fit to the elicited joint percentile estimates [see Additional file 3]. Best fits were determined using SolvOpt (Kuntsevich and Kappel 1997), a solver for non-smooth optimization. The solver minimizes a least-squares function: the square of the difference between the solution at each of the three percentiles and the elicited values for the percentiles. Constraints on the distribution parameters are imposed by a penalty function (e.g. setting a large but finite penalty coefficient outside the required bounds of the function). Figure 7 shows the best-fit Dirichlet and Generalized Trapezoidal distributions for nodes 5 (recorded VT rate), 6 (Felt VT rate) and 15 (eruption) alongside the corresponding expert group quantiles. The Dirichlet distribution (i.e. the multivariate Beta) was thus chosen to characterize uncertainty distributions in the final network.

thumbnailFigure 7. Comparison of the best-fit Dirichlet and Generalized Trapezoidal distributions with expert group quantiles. Comparison of the best-fit Dirichlet and Generalized Trapezoidal distributions for nodes 5 (recorded VT rate), 6 (Felt VT rate) and 15 (eruption), shown alongside the corresponding expert group quantiles. Plot shows conditional probability distributions obtained by fitting Dirichlet distributions (red) and Generalized Trapezoidal distributions (green) to the joint Decision Maker’s (DM) 5, 50 and 95th percentile estimates (black; the joint DM being the optimised combination of the individual expert’s estimates). The Dirichlet distribution gave the overall best-fit and was chosen for the final network.

Additional file 3. Fitting distributions to expert group quantiles. A description of the procedure used to find the best fit distribution for the joint DM quantile marker values.

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In determining the distribution parameters, some logical constraints have been applied. For example, based on past activity the probability of a magmatic blast P(blast)at La Soufrière can be expected to be lower than the probability of a magmatic eruption without a blast P(magmatic eruption). However, if the uncertainty spread in the estimate for P(blast) is much greater than that of P(magmatic eruption), and the experts gave a broader distribution, this aspect may not be captured properly by fitting a Dirichlet distribution in the standard way with least squares (with all three percentiles weighted equally). To avoid this, we used weights of 1, 1.5, 1 for the 5, 50 and 95%ile respectively – that is, giving a higher weighting to the median value, and imposed the further condition that P(blast) < P(magmatic eruption). This approach more closely reproduced the groups’ 'best estimate’ values, and creates a logically consistent overall outcome.

Code has been written to perform random sampling from the best-fit Beta (for binary nodes) and Dirichlet distributions for each node using the Scythe Statistical Library (Pemstein et al. 2007). Given the relevant distribution parameters, a set of random samples is returned, corresponding to the probability of each possible outcome (with the total summing to 1). These samples populate the conditional probability tables (CPTs) for each node. As there is no real basis for enforcing correlations between the elicited probability density functions, each distribution was sampled independently. Each run results in a fully-defined CPT for each node, completely parametrizing the network. At this stage we calculate various measures, such as Mutual Information and entropy (described below and in supplementary material [see Additional file 4]), which characterize the strength of node relationships and overall uncertainty. The network is updated with the time series of observational evidence (bottom panel in Figure 5), and computes the probability of each unobservable state or outcome (e.g. magmatic eruption), at each time step. The sampling procedure is repeated for a total of 10,000 runs to build output distributions for P(eruption), Mutual Information etc., the goal being to propagate uncertainty from the experts’ initial estimates through to the ultimate query node.

Additional file 4. Fitting distributions to expert group quantiles. A description of the procedure used to find the best fit distribution for the joint DM quantile marker values.

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Mutual Information

To investigate the strength of the relationship between any particular node on the BBN and the query node (eruption probability), we compute Mutual Information (MI). The MI of nodes X and Y, MI(X,Y), is the reduction in uncertainty (entropy) in Y by knowing X [see Additional file 4]. In a Bayesian Network, higher entropy means the node states are more randomly distributed, therefore more uncertain (see Bedford and Cooke 2001). Zero entropy means the state is known exactly. We can also express the strength of the relationship between nodes as Mutual Information percentage (MI%), the percentage reduction in entropy of node Y given information about X. Zero MI implies conditional independence – node X does not give any information about the state of node Y. MI (and MI%) can be computed for any pair of nodes, regardless of whether they are directly connected. This can be used to assess how individual states and processes might affect hazard outcomes, and as a consequence, impact on risk. Computing this measure for all observables makes it possible to identify objectively which parameters provide the greatest information about future activity. As the BBN presented here has been developed using expert opinion (rather than observational data), MI is a measure of the perceived strength of connection between nodes, and the perceived value of the various observables in forecasting eruptive activity.

Setting the Guadeloupe BBN in context

The network (Figure 3) was used to infer the various hidden states (e.g. magmatic unrest or ascent) and calculate an evolving probability of eruption using the sequence of observations made between July 1975 and March 1977 (see Additional file 5 and Figure 5). Before discussing the observation-based probability estimates, we set the scheme in context by looking first at implied recurrence rates for a simple “reference scenario”. In this “no signs” scenario (denoted 'no activity' in the table in Additional file 5), the assumed situation is that monitoring and observational data are continuously available, but no abnormal activity is detected in any variable. This scenario yields a median probability of magmatic blast or magmatic eruption of the order 10-11 for an eruption within a three month interval (i.e. with no unrest evident in the three months prior to eruption), with an upper 95%ile probability of order 10-6. Thus the group assessed the chance of a totally “out-of-the-blue” magmatic eruption (unaccompanied by any precursory signs) within three months to be negligible. The median probability of a phreatic eruption per three month period under the same conditions (i.e. no prior unrest) is much higher, at 0.003 (corresponding to a recurrence rate of approximately one in 80 years), while the expected (mean) probability is 0.01 (1%, a recurrence rate of one in 25 years). Uncertainties are large, and differences between mean and median estimates signal strong skew in the distributions.

Additional file 5. Table showing Marginal distributions for probability of eruption, calculated for the baseline case (no activity) and given observational evidence from the period 1975-1977.

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During the elicitation exercise it was not feasible to weigh up such end-member recurrence rate uncertainties in detail, not least because there is no real basis on which to calibrate them against geological and historical records. If major discrepancies were suspected, perhaps due to members of the group inflating their rate estimates to reflect their uncertainty in the absence of substantive evidence, this would invite potentially intractable debate. Faute de mieux, the rate estimates are reproduced here to illustrate how even extremely unlikely, exceptional scenarios can be accommodated in a BBN, albeit with gross uncertainties; the numbers are not definitive.

Results and discussion

Inferring eruption probabilities using 1975–1977 observations

Figure 5 shows the sequence of estimated probabilities over the following three months for: (a) a magmatic eruption or blast, (b) phreatic eruption, or (c) no eruption, with associated uncertainties in these estimates. The solid black lines show the expected (mean) probability estimates, the broken lines indicate the medians (50th percentile), with the shaded areas outlining the 5–95 percentile ranges. The data are presented in Additional file 5.

From July 1975 until early July 1976, the likelihood of a magmatic eruption was only very marginally elevated above the assumed base-rate: at the 95%ile level this probability would have been less than 0.01 (i.e. only a 5% chance it would be greater), while the corresponding 95%ile probability for a phreatic explosion was 0.07. This very limited increase in probability at the start of July 1976 indicates the influence of false positive and true negative rates for signs associated with magmatic eruption, including VT activity. If post-hoc this appears surprisingly low it serves to illustrate the challenge of setting up a CPT with appropriate diagnostic powers for the various precursors when these are poorly known a priori, and can also reflect the fact that some observations can run counter to others. Perhaps even more pertinently, it illustrates potential pitfalls with hindsight interpretation of evidence: some scientists present in 1976 were quite convinced a magmatic eruption was not going to happen.

From 9 July 1976 onwards, the perceived median probability of a magmatic eruption or magmatic blast within three months increased to 0.3, the probability of a phreatic eruption also increased (to around 0.1) [see Additional file 5] and the most likely outcome was still “no eruption”. The occurrence of the explosion on 8 July 1976 did not modulate future event probabilities as these were solely conditional on observational data trends. In other words, the likelihood of another explosion in the following three months did not change in the wake of one that had just happened - incorporating this additional dependence would improve the BBN model.

The probability of a magmatic eruption fell back marginally at 13 August 1976, but rose again to peak at the end of the month when fresh glass was reported as having been detected in a microscope sample and all volcanic other phenomena were observed as present or elevated (see lower panels, Figure 5). Magmatic eruption probability remained elevated until the end of November 1976. Thereafter the states of a number of observables dropped (see Figure 5, lower panels) and the probability of eruption fell noticeably. However, it still remained above the pre-July level, and stayed there through until March 1977 (at which time accessible data became incomplete and our analysis was curtailed). From early July 1976, the likelihood of a phreatic explosion remained elevated at a roughly constant level, reducing slightly through March 1977. Whilst uncertainty in magmatic and phreatic eruption probability was higher after 29 August 1976 than before 9 July [see Additional file 5] , lower bound estimates returned close to pre-July levels, with a corresponding shift up in the upper bound likelihood of no eruption.

The corresponding marginal distributions for probability of magmatic intrusion and ascent are given in Additional file 6. The median probability of magmatic unrest with no precursory activity is estimated to be of the order 10-8 (mean 3 × 10-4), which can be compared to the 'baseline' elicited probability of magmatic unrest (the CPT for node 0) - a median value of 0.08, or 1 episode every 3.1 years. In considering this estimate, one hypothesis is that exceptionally elevated VT swarm activity is evidence of magma intrusion rather than abnormal non-magmatic hydrothermal perturbation. If we deem ~15 VTs/ month to represent the baseline seismic activity related to a steadily degassing magma reservoir, observing a VT rate well above this might be taken as evidence of a switch due to upward magma intrusion from the shallow chamber. Applying this logic to the period before 1975, we find six potential instances of possible intrusion between 1956 and 1975 (including that at the beginning of the 1975–77 crisis, in July 1975). Six intrusions in 20 years implies one every 3.3 years, largely in line with the elicited joint median value of 1 event per 3.1 years. There is, however, large uncertainty in the experts' joint probability estimate – the corresponding 5 and 95 percentile estimates are 7.6 × 10-5 (1 event per ~3300 years) and 0.81 (just over 3 episodes / year).

Additional file 6. Table showing Marginal distributions for probability of magmatic intrusion and ascent, calculated for the baseline case (no activity) and given observational evidence from the period 1975-1977.

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This hypothesis presumes that all phreatic eruptions are still-born failed magmatic eruptions, and that magma intruded towards the surface heated the hydrothermal system, pressurized it, opened fractures and generated VTs in competent rock, but then stalled at shallow depth. More recent occurrences of such "intrusive swarms", if that is what they are, suggest almost continuous intrusive activity from 1992, ramping up to 1997–98 when sudden and sustained chlorine degassing appeared and peaked. This is compatible with a phase of magma intrusion into the dome system, with declining degassing at its periphery but much increased gas and heat flux at the summit. An earlier, notable pulse of "intrusive swarms" occurred in 1962–1968; magma intrusion at that time is inferred by Ruzié et al. (2012), based on Noble gas isotope systematics and other data.

The question of whether all pre-1976 historical phreatic eruptions were magmatic in origin is moot. An example of an alternative hypothesis (not considered in our study) is that gas fluxes in magma reservoirs may be decoupled and independent of magma fluxes. Thus episodes of unrest record the transfer of gas to higher levels unaccompanied by magma. The great value of a structured probabilistic elicitation approach is that it helps raise scientific questions of this kind, and encourages development of causal conceptual models that can be tested with observations and theoretical models. The preferred conceptual model can then be used to structure a BBN, dynamically trained on the available observations (and other information) and then used to evaluate scenario probabilities for new unrest situations.

BBN mutual information

The BBN formulation allows analytic testing of node state conditionalities and uncertainty in the network. For instance, entropy is a measure of unpredictability in a system (Bedford and Cooke 2001), with higher entropy indicating the node states are more randomly distributed, and therefore more uncertain. Here we restrict our investigation to the strength of the relationship between individual nodes in the BBN and the query node, and compute corresponding Mutual Information percentages MI% for each pair (Ebert-Uphoff 2007; see Methods and Additional file 4). MI% is a measure of the reduction of uncertainty in the target node (eruption) due to knowledge of another node; either an observational node or an inferred latent (hidden) node. This allows us to identify which monitoring parameters provide the greatest information about future activity.

Figure 8 summarizes MI% results: black diamonds denote the mean MI % estimate, with boxes denoting the median (centreline) and first and third quantiles; whiskers on the plot depict 5th and 95th percentiles. In terms of eruption probability, the four most important nodes (0, 2, 3 and 4 – see above for definitions) are all latent factors concerning aspects of magma condition or behaviour at depth. These cannot be observed directly and can only be inferred from related observations. Deep source deformation (node 1) is a latent node with moderate potential to influence eruption probability.

thumbnailFigure 8. Mutual Information. Mutual Information (MI) is a measure of the strength of the relationship between two nodes. By computing the MI for eruptive outcome with each observable, it is possible to identify which parameters provide the greatest information about future activity. The black diamonds denote the mean MI % estimate. Boxes denote the median (centreline), first and third quantiles. The whiskers on the plot denote the 5th and 95th percentiles. The data are provided in Additional file 7.

Additional file 7. Mutual Information percentage (MI%) calculated for the eruption node. Here we evaluate the perceived influence of each individual node in the network (both hidden and observable states) on the probability of eruption within 3 months.

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In 1976, the most informative observational nodes in relation to eruption potential are inferred to be seismicity (i.e. nodes 5, 6, 13 and 14), with shallow seismicity above 5 km depth identified as having the strongest indicative power. In the BBN, shallow seismicity is closely associated with inferences about magma ascent, and therefore acts to influence the query node through its links to those latent nodes (e.g. nodes 3 and 4). The 1976 tilt measurements (nodes 8 and 9) and measurements of SO2 (node 11) were of little significance for assessing eruption likelihood. These findings might be less applicable in current times with modern techniques, better instruments, and advances in data interpretation and physical models.

Identification of fresh glass (node 12) can be a valuable petrological indicator, but turned out to be a false positive here. Properly, the presence or absence of fresh glass is not a single binary yes/no parameter, and its evidential worth should be considered in conjunction with other observables. As Cashman and Hoblitt (2004) have shown, petrological evidence from early erupted products can be highly diagnostic. However, in an escalating eruption, obtaining samples can be challenging and risky, and dedicated resources are needed to perform fast petrological analysis. Rapid interpretation with a simple binocular microscope requires experience, care and awareness of context. Here the BBN formulation is indispensable – the implications should be modulated by the reliability of the observation. In the escalating sequence at Merapi 2010, for instance, evidence for vesicular material in the very earliest tephra was found in material erupted three days before the dome was first seen and then again in material before the paroxysmal dome explosion on 5 November , however the analysis was not performed until some months after the events unfolded (Komorowski et al. 2013). The example of fresh glass highlights the vital importance of characterizing fully both the diagnostic sensitivity and specificity (e.g. Sackett et al. 1996) when weighing strands of scientific evidence.

The wide uncertainty spreads shown on Figure 8 (see also Additional file 7) illustrate clearly just how weak, individually, these various indicators were for eruption forecasting on Guadeloupe in 1976. Calculation of Mutual Information allows parameters that have poor diagnostic value to be identified, potentially enabling resources to be focussed on more diagnostic observations and allowing simplification of the BBN by removing links which emerge as uninformative.

Discussion and conclusions

The findings of this exploratory Bayesian Belief Network analysis of the 1976 Guadeloupe crisis lend objective support to the retrospective view that the authorities’ concern for public safety and decision to evacuate were rational and defensible, given all the major scientific uncertainties that existed at the time. It is clear that even now, despite 40 years of intervening advances in research, the evidence of the time engenders considerable concern as well as uncertainty. Highly definite views, expressed at the time by some scientists, are hard to justify given the large uncertainties in the evidence, and the official precautionary decision to evacuate, given the potential consequences of an explosive magmatic eruption, is not weakened by this hindcasting analysis. The BBN analysis also helps highlight some of the challenges of providing science-based decision support under conditions of considerable uncertainty that will emerge again in a future volcanic crisis, on Guadeloupe or elsewhere.

In the present case, some of our retrospective probability estimates may be unduly or conservatively high (for example, the relatively high probability of magmatic blast as compared to other, lower intensity outcomes). In part, this may be due to the restricted scope and rather perfunctory nature of the elicitation; a more comprehensive exercise, with more experts, would be desirable. However, the BBN calculations illustrate the substantial uncertainties that are typically associated with interpreting incomplete observational information or monitoring data of limited quality. Volcanology has moved on since 1976, of course, and many monitoring techniques have improved immensely. This said, the basic evidential principles outlined here remain the same, and demonstrate how crucial it is, when resources are limited, to focus monitoring efforts on those parameters which maximise strength of inference about key hidden conditions and latent factors, such as magma ascent.

The information that goes into building the BBN model can offer considerable help in expressing and communicating scientific uncertainties and their sources, and in elucidating how forecast outcomes are sensitive to different assumptions and relationships. For example, the mutual information measure of conditional dependency between two variables is an elegant way of identifying and quantifying dependencies between elements in a system that may not be immediately obvious, especially in a complex situation involving a model with numerous inputs, nodes, and interactions.

Assessing and presenting uncertainty to policy- and decision-makers has emerged as an important topic in all branches of hazard and risk analysis, not least for the situation where we are confronted by escalating volcanic unrest (for a review of approaches to volcanic hazard assessment, see Marzocchi and Bebbington 2012). Due to the intrinsic process complexities and limitations in our understanding (present in all natural hazards domains), volcano forecasting assessments are inevitably imprecise and a sense of vagueness can be communicated through the use of confidence ranges.

Recent work by Dieckmann et al. (2010) suggests that decision makers are not necessarily 'ambiguity averse’ in a forecast context, and presenting ranges of probability can have distinct advantages as a way to communicate probability and diagnostic confidence. The study by Dieckmann et al. (in a military intelligence context) indicated a smaller ambiguity effect in decisions that were taken when a narrative forecast was accompanied by a probability range as opposed to the same narrative with simply a point value. However, in one of their tests, it was thought that the point estimate was more useful for decision making at low probabilities. Dieckmann et al. also found that when evaluating a forecast in hindsight, their decision makers tended to report lower levels of blame and higher levels of source credibility for forecasts that reported uncertainty ranges as compared to single value point assessments. How these findings might map across to volcanic hazard forecasts and decision support situations requires further research.

Future developments

This retrospective analysis is necessarily simple in scope, reflecting the limited information available during the 1976 crisis. This said, the basic framework can easily be developed to reflect current scientific understanding and to incorporate a much wider range of monitoring data and observations. The diversification of new volcano monitoring techniques is such that a formulation for weighing and pooling multiple strands of observational evidence is becoming indispensable, especially if an audit trail is required to track science-based inputs to decisions under rapidly-changing conditions. Such a network can be utilised as an automated tool for real-time use in the volcano observatory, using streams of monitoring data to generate and continuously update probabilistic hazard forecasts.

The network presented here is a static BBN that weighs the evidence in discrete and unconnected time steps. With sufficient knowledge of the volcanic system and with comprehensive, repeated observational data, more sophisticated Dynamic Bayesian Networks (DBNs) can be constructed to model temporal relationships between nodes (Pearl 2000; Murphy 2002; Jensen and Graven-Nielsen 2007).

Whereas a static BBN describes the state of a system without using information about its prior history, a dynamic BBN can incorporate crucial information about system evolution in which the state of the volcano at any time is dependent on any number of past states. The order of such a model is the length of history, or 'memory’ of the processes concerned. Dynamic BBN nodes can be tied over many time-slices to represent higher order processes, as appropriate.

One volcanological example of a DBN is the Hidden Multi-state Markov Model (HMM) of Aspinall et al. (2006). The HMM is a simple case of a DBN with a single discrete hidden node (e.g. see Rabiner 1989; Jensen 1996). The HMM model is based on the assumption that multi-parameter monitoring data can be jointly evaluated to infer time to eruption, and hence used to inform hazard alert levels. For data-rich applications, network parameters and even the network structure itself can be estimated from data, using learning algorithms (Murphy 2002). Such an extension is presented in Hincks et al. (2006), which applies learning algorithms to parameterize a DBN for forecasting dome collapse on Montserrat using multiple time series of monitoring data. Determining network structure from time-evolving data is an advanced technique which would be especially salient for observational volcanology in an unrest crisis; this theme will be developed elsewhere.

Endnotes

aOne of us, WPA, was present at various times in Guadeloupe in 1976, participating in monitoring activities. Another, JCK, was Director of the Guadeloupe Volcano Observatory (IPGP) from 1997 to 2001.

bOpenPNL is available at http://sourceforge.net/projects/openpnl/ webcite and https://github.com/crishoj/OpenPNL webcite.

cAs, for example, in two eruptions of the nearby Soufriere of St. Vincent volcano in 1971 and 1979 (Aspinall et al. 1973; Shepherd et al. 1979). In these instances, there was almost no (recorded) precursory seismicity: in 1971 the nearest seismometer on St Vincent was 30 km from the volcano, while in 1979 there were seismometers on the volcano but only low-level instrumental tremor was detected and then only in the 12 hours before the first explosion.

dBoth Dirichlet and Generalised Trapezoidal distributions were compared, the Dirichlet distribution gave a better fit to the joint DM quantiles and was used in the final analysis.

Abbreviations

BBN: Bayesian Belief Network; CPT: Conditional probability table; DBN: Dynamic Bayesian Network; DM: Decision maker (elicitation group pooling); GDSV: La Grande Découverte-Soufrière volcanic complex; HMM: Hidden Multi-state Markov Model; VT: Volcano-tectonic earthquake; VEI: Volcanic Explosivitiy Index.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The BBN structure was developed through discussions between the authors and all four contributed to writing the paper. R.S.J.S., W.P.A. and J-C.K. advised on the key volcanic processes and participated in the expert judgement elicitation. T.K.H. wrote the code and performed the the BBN analysis and network parameter fitting, and drafted the diagrams (with the exception of Figures 5 and 6, produced by W.P.A.). W.P.A. ran EXCALIBUR for processing elicitation responses, analysed the results of the expert elicitation; he also provided direct knowledge about monitoring and events during the 1976 Guadeloupe crisis. J-C.K. provided background information on the eruptive history, the chronology of the crisis (Tables 1 and 2), and monitoring data for La Soufrière. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank A. Rust, J. Gottsmann and Y. Legendre for participating in the expert elicitation and their contributions to discussions. This work was supported by VOLDIES (Advanced European Research Council grant to RSJ Sparks). EU Project EXPLORIS (EVR1-2001-00047) and Aspinall & Associates partly funded early stages of this work. Support was provided by the Institut National des Sciences de l’Univers (INSU-CNRS) and the CASAVA Project (ANR-09-RISK-02, J.C. Komorowski). JCK is grateful for valuable and insightful discussions with F. Beauducel. We gratefully acknowledge the original intellectual stimulation and advice of Dr Gordon Woo, who encouraged us to undertake this study.

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