The modeling of statistical distribution of the eruptive frequency provides basic information to quantitatively assess the volcanic hazard and constrain the physics of the eruptive process. Here, we discuss the statistics of the time series of lateral eruptions of the Nyiragongo volcano in the Virunga Volcanic Province, western branch of the East African Rift System. We examine eruption data with a volcanic explosivity index of at least 1 listed in the Global Volcanism Network Bulletins. After investigating the completeness, stationarity, and independence of the eruption time series, we employ five distribution models (Brownian passage time, gamma, log-logistic, lognormal, and Weibull) to fit the repose time. First, we identify a clear tendency for events to cluster in time. We hypothesize two clusters, the ‘pre-1927’ cluster related to the intracraterial and volcanic activity of the lava lake, and the ‘post-1977’ cluster mainly related to lateral eruptions (i.e. those potentially generating lava flows). Using the maximum likelihood estimations, we evaluate the model parameters with a 95% confidence interval. Next, we use the Akaike Information Criterion to determine the most suitable distribution and we perform the Bayesian Model Averaging approach to assess uncertainty issues in model selection process. The results suggest that the BPT distribution provides the best fit for data of lateral eruptions (post-1977). Then we estimate the time-dependent probability of the occurrence of a lateral eruption for the 50-year period between 2022 and 2072. The estimates reach 50.79%, 88.61%, 97.59%, 99.50%, and 99.89% for 2032, 2042, 2052, 2062, and 2072 years, respectively.
Volcanic activity often results in the formation of new volcanic vents. These new vents can create hazards in unexpected areas. Therefore, the probability of new vent formation should be assessed as part of volcanic hazard assessments. This paper describes our use of kernel density estimation (KDE) as a way to estimate the spatial density of future volcanic vents. The bivariate Gaussian kernel function is described step-by-step using pseudocode. Our computer code, written in PERL, is used to calculate the spatial density of existing vents and then create a contour map using GMT (Generic Mapping Tools). Application of this method and code relies on several assumptions about the definition of volcanic events, independence of events, the type of kernel function used, and the selection of kernel bandwidth. Three examples using the code are provided: (1) for volcanic vents located west of the city of Managua (Nicaragua), (2) for volcanic vents distributed within the Arsia Mons caldera (Mars), weighted by volume, and (3) for vents of the Lassen volcanic system (northern California), sub-divided by geochemistry.
Volcanic hazards assessments for critical facilities, such as nuclear power plants, often must consider the potential effects from phenomena that have very low likelihoods of occurrence. The International Atomic Energy Agency recently released a Technical Document that provides detailed guidance on conducting practical volcanic hazards assessments that include rare phenomena, using both deterministic and probabilistic approaches. This guidance develops a graded approach to the analysis, and provides a range of model complexities for assessing hazardous volcanic phenomena. Although this guidance focuses on the safety of nuclear installations, it is applicable to other sites that warrant consideration of low likelihood volcanic hazards that have potentially high consequences.
Long period (LP) earthquakes are common at active volcanoes, and are ubiquitous at persistently active andesitic and dacitic subduction zone volcanoes. They provide critical information regarding the state of volcanic unrest, and their occurrence rates are key data for eruption forecasting. LPs are commonly quasi-periodic or 'anti-clustered', unlike volcano-tectonic (VT) earthquakes, so the existing Poisson point process methods used to model occurrence rates of VT earthquakes are unlikely to be optimal for LP data. We evaluate the performance of candidate formulations for LP data, based on inhomogeneous point process models with four different inter-event time distributions: exponential (IP), Gamma (IG), inverse Gaussian (IIG), and Weibull (IW). We examine how well these models explain the observed data, and the quality of retrospective forecasts of eruption time. We use a Bayesian MCMC approach to fit the models. Goodness-of-fit is assessed using Quantile-Quantile and Kolmogorov-Smirnov methods, and benchmarking against results obtained from synthetic datasets. IG and IIG models were both found to fit the data well, with the IIG model slightly outperforming the IG model. Retrospective forecasting analysis shows that the IG model performs best, with the initial preference for the IIG model controlled by catalogue incompleteness late in the sequence. The IG model fits the data significantly better than the IP model, and simulations show it produces better forecasts for highly periodic data. Simulations also show that forecast precision increases with the degree of periodicity of the earthquake process using the IG model, and so should be better for LP earthquakes than VTs. These results provide a new framework for point process modelling of volcanic earthquake time series, and verification of alternative models.
The Long Valley volcanic region is an active volcanic area situated at the eastern base of the Sierra Nevada escarpment and dominated by a 32 km wide resurgent caldera created ~760 ka. Eruptions after 180 ka have been localized at Mammoth Mountain on the western rim of the caldera and along the Mono-Inyo Craters volcanic chain stretching about 45 km northward. Three different probability models have been developed and then combined in a logic tree to estimate the long-term spatial probability of vent opening. These models include (i) an anisotropic kernel density estimator based on past vent locations, (ii) a new Bayesian model for coupling new vents with pre-existing faults, and (iii) a uniformly distributed probability map. The model combination procedure relies on Bayesian model averaging. This doubly stochastic framework enables us to incorporate some of the main sources of epistemic uncertainty about the interpretation of the volcanic system, thereby exploring their effect. Our vent-opening probability maps show two higher likelihood regions for new vent opening, one around Mammoth Mountain to the south, and the other along the Mono-Inyo Craters to the north. The spatial vent opening probability, conditional on an eruption, is estimated as ~64% in the northern region and ~36% in the southern region, with an uncertainty of about ±20%. These findings provide a rational basis for the hazard mapping of a potential eruption in the Long Valley volcanic region, suggesting that the hazard associated with Mammoth Mountain volcanism should be fully evaluated.
In volcanology, the sparsity of datasets for individual volcanoes is an important problem, which, in many cases, compromises our ability to make robust judgments about future volcanic hazards. In this contribution we develop a method for using hierarchical Bayesian analysis of global datasets to combine information across different volcanoes and to thereby improve our knowledge at individual volcanoes. The method is applied to the assessment of mobility metrics for pyroclastic density currents in order to better constrain input parameters and their related uncertainties for forward modeling. Mitigation of risk associated with such flows depends upon accurate forecasting of possible inundation areas, often using empirical models that rely on mobility metrics measured from the deposits of past flows, or on the application of computational models, several of which take mobility metrics, either directly or indirectly, as input parameters. We use hierarchical Bayesian modeling to leverage the global record of mobility metrics from the FlowDat database, leading to considerable improvement in the assessment of flow mobility where the data for a particular volcano is sparse. We estimate the uncertainties involved and demonstrate how they are improved through this approach. The method has broad applicability across other areas of volcanology where relationships established from broader datasets can be used to better constrain more specific, sparser, datasets. Employing such methods allows us to use, rather than shy away from, limited datasets, and allows for transparency with regard to uncertainties, enabling more accountable decision-making.
Based on rapid analyses done in near real-time and in support of scientific advice, I present two cases of prima facie evidence for an earth tide triggering eect on earthquake activity during two volcanic eruption episodes: the first, in relation to large magnitude earthquakes, occurred during August2014 in the 2014 2015 Bar﷿arbunga-Holuhruan, Iceland, eruption; and the second concerns the timing of VT strings at the Soufriere Hills Volcano, Montserrat. In both cases, statistical testing of the hypothesis that the timings of the seismic events of interest and vertical earth tide phase were not correlated produced probabilities at or below a 5% significance level, suggesting rejection of the hypothesis can be argued. This note outlines the method used for computing earth tide amplitudes at the two volcanoes and the basis and results of the Schuster test for relative phase timings; some comments on the volcanological contexts are added. These findings suggest that further, more formal analysis may be warranted.
Six modules are presented that are designed for use in an undergraduate physical volcanology course. Each module uses a four-step mathematical problem-solving approach following Polya's How to Solve It, to guide students to: (i) understand the problem, (ii) devise a plan for solving the problem, (iii) implement the plan, and (iv) reflect on their solution. The modules cover topics in estimation of volumes of debris flow and tephra fallout deposits, lava effusion rate, calculation of melt density and viscosity, and development of dynamic models of bubble ascent in magma. Core quantitative issues, such as linear regression and error propagation, are introduced using these modules. The modules are included as supplementary material in PowerPoint and Portable Document Format (PDF). Cumulatively, the modules are designed to help volcanology students construct their own schemas, through which they can master abstract concepts in volcanology and in the geosciences generally. Physical volcanology can play a central role in the Earth Sciences curriculum by promoting quantitative problem solving using modules such as these.
Aeromagnetic surveys over the Amargosa Desert, Nevada, have revealed the presence of several magnetic anomalies that have been interpreted to be caused by buried volcanoes; many of these anomalies have been confirmed by drilling. We present data collected from a high-resolution, ground-based magnetic survey over Anomaly B, the largest of these anomalies, that reveal details about a buried crater and its associated lava flow, not observed in the aeromagnetic surveys. These details provide insight into the nature of the eruption and volume of this buried volcano. Results from non-linear inversion demarcate a crater with a diameter of approximately 700 m and a base approximately 150 m below the ground surface. Coupled with well log data, the inversion results suggest a total volume for the Anomaly B crater area and associated lava flows of approximately 1.0 ± 0.4 km3, based on an estimated lava flow field area of 24 km2 and a lava thickness of 42 ± 15 m. A workflow is presented for processing such large ground-based magnetic data sets with attendant GPS data, filtering these data and constructing maps and models using the provided PERL scripts.
In estimating hazard from a currently quiescent volcano, the most basic quantity of interest is the likelihood of an eruption in some defined time horizon. Starting with the dichotomy of stationarity (where the average future level of activity is equal to the average level of past activity) or non-stationarity, we outline several classes of stochastic models that can be used to forecast future onsets. Renewal models, including the simple Poisson process and mixtures, are compared with models that incorporate volumes of past eruptions, and models that include a trend in the activity level. The mathematical formulations are supplemented by Matlab programs that fit the models using maximum likelihood. Tests are provided for whether a particular model is consistent with the data, and for identifying the best model from those considered. The philosophy behind assumptions and the limitations of each class of models are discussed, and suggestions for further exploration are given. The models are illustrated on a data set of VEI > 1 eruptions from Mt Ruapehu (New Zealand) since 1860.
We present TError, a Matlab package designed to quantify systematically the uncertainty associated with the characterization of tephra deposits, in which the most commonly used methods to quantify eruption source parameters are implemented. Inputs of the code are a range of field-based, model-based and empirical parameters (i.e., clast diameter, crosswind and downwind ranges, thickness measurement, area of isopach contours, bulk deposit density, empirical constants and wind speed), for which the user defines an uncertainty and an associated distribution. The TError package contains two main functions. The first function deterministically varies one input parameter at a time and quantifies the sensitivity of each Eruption Source Parameter (ESP; i.e., plume height, erupted volume, mass eruption rate) to the variability of input parameters. The second function propagates input parameters as stochastic distributions of noise through all ESPs. The resulting distributions can then be used to express the uncertainty of physical parameters of explosive eruptions in a systematic way. For both functions, comprehensive reports and sets of figures assist the user in the interpretation of the results. As an example, the TError package was applied to Layer 5 of Cotopaxi volcano. Using the median, the 2nd percentile and the 98th percentile as central value, lower bound and upper bound respectively, a new quantification of the ESP suggests a plume height of $30 +/- 1 km a.s.l, a mass eruption rate of 1.8 (-0.2, +0.3) x 108 kg s-1 and a tephra volume between 0.23 (-0.04, +0.13) and 0.43 (-0.06, 0.08 km3, depending on the empirical model used.
Statistics in Volcanology is a comprehensive guide to modern statistical methods applied in volcanology written by today's leading authorities. The volume aims to show how the statistical analysis of complex volcanological data sets, including time series, and numerical models of volcanic processes can improve our ability to forecast volcanic eruptions. Specific topics include the use of expert elicitation and Bayesian methods in eruption forecasting, statistical models of temporal and spatial patterns of volcanic activity, analysis of time series in volcano seismology, probabilistic hazard assessment, and assessment of numerical models using robust statistical methods. Also provided are comprehensive overviews of volcanic phenomena, and a full glossary of both volcanological and statistical terms.Statistics in Volcanology is essential reading for advanced undergraduates, graduate students, and research scientists interested in this multidisciplinary field.
The literature on geophysical time series analysis is so extensive that to review even one topic, such as volcanic tremor series, is a major task (e.g. Konstantinou and Schlindwein 2002). The purpose of this chapter is not to attempt such a review but rather to outline some new tools for nonstationary and nonlinear time series analysis that have been developed and used successfully in other areas of the environmental sciences and appear to have good potential for application in a volcanological or wider geophysical context. These stochastic methods of time series analysis have the advantage that they all exploit powerful recursive (sequential updating) methods of estimation that facilitate the analysis of data generated by nonstationary and nonlinear systems (e.g. Young 1984). This Chapter starts by reviewing briefly some of the model-based methods of time series analysis, signal extraction and forecasting that have appeared in the statistical and time series analysis literature over many years and then proceeds to describe in more detail one approach that has attracted considerable interest over the past two decades. This is based on the concept of an ‘unobserved component’ model and it exploits recursive estimation for the purposes of estimating time variable parameters in nonstationary systems. It is shown that such an approach allows for various, practically useful procedures in time series analysis: signal extraction; interpolation over gaps in time series; and forecasting or backcasting. It then goes on to outline the basic aspects of input-output time series modelling, considering both discrete-time and continuous-time ‘transfer function’ models that are
Statistics in Volcanology is a comprehensive guide to modern statistical methods applied in volcanology written by today's leading authorities. The volume aims to show how the statistical analysis of complex volcanological data sets, including time series, and numerical models of volcanic processes can improve our ability to forecast volcanic eruptions. Specific topics include the use of expert elicitation and Bayesian methods in eruption forecasting, statistical models of temporal and spatial patterns of volcanic activity, analysis of time series in volcano seismology, probabilistic hazard assessment, and assessment of numerical models using robust statistical methods. Also provided are comprehensive overviews of volcanic phenomena, and a full glossary of both volcanological and statistical terms.Statistics in Volcanology is essential reading for advanced undergraduates, graduate students, and research scientists interested in this multidisciplinary field.
Statistics in Volcanology is a comprehensive guide to modern statistical methods applied in volcanology written by today's leading authorities. The volume aims to show how the statistical analysis of complex volcanological data sets, including time series, and numerical models of volcanic processes can improve our ability to forecast volcanic eruptions. Specific topics include the use of expert elicitation and Bayesian methods in eruption forecasting, statistical models of temporal and spatial patterns of volcanic activity, analysis of time series in volcano seismology, probabilistic hazard assessment, and assessment of numerical models using robust statistical methods. Also provided are comprehensive overviews of volcanic phenomena, and a full glossary of both volcanological and statistical terms.Statistics in Volcanology is essential reading for advanced undergraduates, graduate students, and research scientists interested in this multidisciplinary field.
