Mark Bebbington

A flexible Weibull extension

We propose a new two-parameter ageing distribution which is a generalization of the Weibull and study its properties. It has a simple failure rate (hazard rate) function. With appropriate choice of parameter values, it is able to model various ageing classes of life distributions including IFR, IFRA and modified bathtub (MBT). The ranges of the two parameters are clearly demarcated to separate these classes. It thus provides an alternative to many existing life distributions. Details of parameter estimation are provided through a Weibull-type probability plot and maximum likelihood. We also derive explicit formulas for the turning points of the failure rate function in terms of its parameters. This, combined with the parameter estimation procedures, will allow empirical estimation of the turning points for real data sets, which provides useful information for reliability policies.

Probabilistic eruption forecasting at short and long time scales

Any effective volcanic risk mitigation strategy requires a scientific assessment of the future evolution of a volcanic system and its eruptive behavior. Some consider the onus should be on volcanologists to provide simple but emphatic deterministic forecasts. This traditional way of thinking, however, does not deal with the implications of inherent uncertainties, both aleatoric and epistemic, that are inevitably present in observations, monitoring data, and interpretation of any natural system. In contrast to deterministic predictions, probabilistic eruption forecasting attempts to quantify these inherent uncertainties utilizing all available information to the extent that it can be relied upon and is informative. As with many other natural hazards, probabilistic eruption forecasting is becoming established as the primary scientific basis for planning rational risk mitigation actions: at short-term (hours to weeks or months), it allows decision-makers to prioritize actions in a crisis; and at long-term (years to decades), it is the basic component for land use and emergency planning. Probabilistic eruption forecasting consists of estimating the probability of an eruption event and where it sits in a complex multidimensional time–space–magnitude framework. In this review, we discuss the key developments and features of models that have been used to address the problem.

On nonhomogeneous models for volcanic eruptions

Recently, a special nonhomogeneous Poisson process known as the Weibull process has been proposed by C-H. Ho for fitting historical volcanic eruptions. Revisiting this model, we learn that it possesses some undesirable features which make it an unsatisfactory tool in this context. We then consider the entire question of a nonstationary model in the light of availability and completeness of data. In our view, a nonstationary model is unnecessary and perhaps undesirable. We propose the Weibull renewal process as an alternative to the simple (homogeneous) Poisson process. For a renewal process the interevent times are independent and distributed identically with distribution function F where, in the Weibull renewal process, F has the Weibull distribution, which has the exponential as a special situation. Testing for a Weibull distribution can be achieved by testing for exponentiality of the data under a simple transformation. Another alternative considered is the lognormal distribution for F. Whereas the homogeneous Poisson process represents purely random (memoryless) occurrences, the lognormal distribution corresponds to periodic behavior and the Weibull distribution encompasses both periodicity and clustering, which aids us in characterizing the volcano. Data from the same volcanoes considered by Ho were analyzed again and we determined there is no reason to reject the hypothesis of Weibull interevent times although the lognormal interevent times were not supported. Prediction intervals for the next event are compared with Hos nonhomogeneous model and the Weibull renewal process seems to produce more plausible results.

Statistical analysis of New Zealand volcanic occurrence data

Abstract The Poisson process is considered to provide a good fit to many volcanoes for forecasting eruptions. However, several exceptions exist, usually where the intensity is non-stationary. The nonhomogeneous Poisson model (the Weibull process) caters for a monotonically varying intensity, but often results in physically unrealistic behaviour. A model which allows for more general behaviour than the Poisson process is the Weibull renewal model. This paper considers data from two New Zealand volcanoes, Mt. Ruapehu and Mt. Ngauruhoe, and fits the Poisson and Weibull renewal models. We conclude that a simple Poisson process fits Ngauruhoe very well. The behaviour of Mt. Ruapehu is considerably more complex, although quite reasonable forecasts can still be obtained from the renewal models. An interesting feature of our analysis is that there seems to be no correlation between the observed eruption sequences of these two closely neighbouring volcanoes.

New Zealand and UK Holocene flooding demonstrates interhemispheric climate asynchrony

The timing and controls of interhemispheric Holocene climate change have remained poorly understood, primarily because of the absence of well-dated and continuous climatic records in terrestrial environments. Here we report a new probability-based meta-analysis of 1185 14 C dates from fluvial sedimentary sequences in New Zealand and the UK, which provides a robust means of identifying centennial- and multicentennial-length episodes of Holocene river flooding. Statistical analysis shows that prior to large-scale human impact, which began at ca. 1000 cal. yr B.P., the incidence of extreme floods in New Zealand and the UK has been largely asynchronous during the Holocene. Major periods of flooding are controlled by large-scale shifts in atmospheric circulation, which alter the frequency of extreme precipitation events. Our novel synthesis demonstrates that short-term climate change, of sufficient magnitude to modify flooding regimes, was out of phase in the temperate maritime regions of the Northern and Southern Hemispheres during much of the Holocene. This supports recent evidence from both glacial and marine records that Holocene climate changes may have been antiphased between the polar regions and that this could have been related to variation in the strength of deep water formation.

A linked stress release model for historical Japanese earthquakes: coupling among major seismic regions

A linked stress release model is proposed for the analysis of spatial interaction of earthquake occurrences through stress transfer within a large area of the Earth’s crust. As an example, the model is used for statistical analysis for the Japanese historical earthquakes in central Japan and offshore in the Nankai and Sagami troughs with magnitude M ≥ 6.5 during the period from 1400 to 1997. This area is divided into four smaller regions of roughly comparable size and activity. Based on the Akaike information criterion (AIC), the results demonstrate the existence of coupling between certain of the regions. With the evidence that the crust may lie in a near-critical state, this has implications for the possible triggering of earthquakes at long distances from the origin event. In particular, we find evidence for the dependence of Nankai trough events on the Chubu/Kinki triangle region, whose events are themselves dependent on the the Fossa Magna/Sagami trough. Evidence for the validity of the model includes simulation results indicating that the model had a higher forecast hazard post-1991 for an event in the Chubu/Kinki triangle region than did models not incorporating regional coupling.

On the Statistical Design of Geometric Control Charts

Abstract We address several theoretic issues involved in the design of geometric control charts. Primary among these is the question of setting probability limits, based on notions of “unbiasedness” and “conditional power”. A nearly ARL-unbiased design is defined by setting the in-control ARL as near as possible, given the discrete nature of the geometric distribution, to the peak of the ARL curve. An optimal design criterion, based on minimizing the sum of out-of-control ARLs for upward and downward shift is also suggested. These are shown to produce improved conditional power compared with the usual design.

Modelling Deceleration in Senescent Mortality

Mortality deceleration is the observed but yet to be understood phenomenon that the increase in the late-life death rate slows down after a certain species-related advanced age. Various definitions of onsets of mortality deceleration are examined. A new distribution based on the Strehler-Mildvan theory of aging takes on the required shapes. The application is done on mortality data from the 1892 cohort of Swedish women and on Mediterranean fruit flies.

Relating magma composition to eruption variability at andesitic volcanoes: A case study from Mount Taranaki, New Zealand

Acquiring accurate eruption records and understanding the volcanic processes behind eruption periodicity are important in the development of realistic hazard assessments and volcanic emergency planning. Here, we use a detailed study of the Holocene ( 0.5 km 3 ) eruptions appear to be statistically predictable because they tend to occur just prior to a period of repose, and they erupt the most strongly evolved magmas. The fundamental properties of magma-volcano systems identified here offer a paradigm for constraining the time scales and nature of magmatic processes, in addition to providing a foundation for more robust probabilistic time-varying hazard forecasts.

Models for Temporal Volcanic Hazard

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.