Spatio-temporal drought risk mapping approach and its application in the drought-prone region of south-east Queensland,Australia

In view of the growing importance of stochastic earthquake modeling in disaster preparation, the present study introduces a new family of exponentiated Weibull distribution and examines its performance in earthquake interevent time analysis in a stationary point process. This three-parameter (one scale and two shapes) distribution not only covers the Weibull distribution, exponentiated exponential distribution, Burr-type X distribution, Rayleigh distribution, and exponential distribution as special sub-families, but also offers monotone and non-monotone hazard shapes. Here we first describe some of the exponentiated Weibull distribution properties, such as the survival rate, mode, median, and hazard rate. We then provide statistical inference and goodness-of-fit measures to examine the suitability of exponentiated Weibull model in comparison with other popular models, like exponential, gamma, lognormal, Weibull, and exponentiated exponential. Finally, we conduct real data analysis to assess the usefulness and flexibility of exponentiated Weibull distribution in the context of seismic interevent time modeling and associated applications. Results suggest that the exponentiated Weibull distribution has a comparable performance with other popular distributions of its nature. However, further investigations are necessary to confirm the importance and flexibility of exponentiated Weibull distribution in statistical seismology.     

A parametric Markov renewal model for predicting tropical cyclones in Bangladesh

In this paper, we consider a Markov renewal process (MRP) to model tropical cyclones occurred in Bangladesh during 1877–2009. The model takes into account both the occurrence history and some physical constraints to capture the main physical characteristics of the storm surge process. We assume that the sequence of cyclones constitutes a Markov chain, and sojourn times follow a Weibull distribution. The parameters of the Weibull MRP jointly with transition probabilities are estimated using the maximum likelihood method. The model shows a good fit with the real events, and probabilities of occurrence of different types of cyclones are calculated for various lengths of time interval using the model. Stationary probabilities and mean recurrence times are also calculated. A brief comparison with a Poisson model and a marked Poisson model has also been demonstrated.     

Robust Estimation for the Weibull Process Applied to Eruption Records

The Weibull process is a parsimoniously parameterized nonhomogeneous Poisson process with monotonic trend, which has been widely used in reliability applications. It has also been used in volcanology to model the process of eruption onsets for a volcano with waning or waxing activity, and thus produce hazard forecasts. However, particularly in the latter application, problems with missing or spurious data can strongly influence the parameter estimates, which are usually obtained by maximizing the log likelihood function, and hence the future hazard. We show how theory developed for robust estimation of a nonhomogeneous Poisson process can be implemented for the Weibull process. The flank eruptions of Mt. Etna, in Sicily, is one of the most complete and best studied records of volcanism. Nevertheless, a number of different catalogs exist. We show how these can be at least partially reconciled by robust estimation, and how the more dubious regions of the catalogs can be identified.     

Nonhomogeneous Poisson model for volcanic eruptions

A simple Poisson process is more specifically known as a homogeneous Poisson process since the rate was assumed independent of time t. The homogeneous Poisson model generally gives a good fit to many volcanoes for forecasting volcanic eruptions. If eruptions occur according to a homogeneous Poisson process, the repose times between consecutive eruptions are independent exponential variables with mean=1/. The exponential distribution is applicable when the eruptions occur at random and are not due to aging, etc. It is interesting to note that a general population of volcanoes can be related to a nonhomogeneous Poisson process with intensity factor(t). In this paper, specifically, we consider a more general Weibull distribution, WEI (, ), for volcanism. A Weibull process is appropriate for three types of volcanoes: increasing-eruption-rate (>1), decreasing-eruption-rate (<1), and constant-eruption-rate (=1). Statistical methods (parameter estimation, hypothesis testing, and prediction intervals) are provided to analyze the following five volcanoes: Also, Etna, Kilauea, St. Helens, and Yake-Dake. We conclude that the generalized model can be considered a goodness-of-fit test for a simple exponential model (a homogeneous Poisson model), and is preferable for practical use for some nonhomogeneous Poisson volcanoes with monotonic eruptive rates.     

Risk assessment for the Yucca Mountain high-level nuclear waste repository site: Estimation of volcanic disruption

In this article, we model the volcanism near the proposed nuclear waste repository at Yucca Mountain, Nevada, U.S.A. by estimating the instantaneous recurrence rate using a nonhomogeneous Poisson process with Weibull intensity and by using a homogeneous Poisson process to predict future eruptions. We then quantify the probability that any single eruption is disruptive in terms of a (prior) probability distribution, since not every eruption would result in disruption of the repository. Bayesian analysis is performed to evaluate the volcanic risk. Based on the Quaternary data, a 90% confidence interval for the instantaneous recurrence rate near the Yucca Mountain site is (1.85×10^–6/yr, 1.26×10^–5/yr). Also, using-these confidence bounds, the corresponding 90% confidence interval for the risk (probability of at least one disruptive eruption) for an isolation time of 10⁴ years is (1.0×10^–3, 6.7×10^–3), if it is assumed that the intensity remains constant during the projected time frame.     

Periodicity in namurian sediments of derbyshire,england (a quantitative sedimentary analysis)

Five facies associations are present in the basin-filling Namurian sediments (Grindslow Shales and Kinderscout Grit) of the southern Central Pennine Basin of England. Visual study of sequences fails to disclose any pattern in the occurrence of facies. A simple mathematical statistical analysis shows that patterns do occur and approximate models can be proposed. The homogeneous siltstones (facies A) occur roughly in accord with a Poisson process model; the striped siltstones (facies B) of the sequence deviate from a renewal process model in many ways and may be most nearly approximated by some process showing trend in the rate of occurrence of events; the sharp-based sandstones (facies C) seem to occur in agreement with some type of renewal process, but not a Poissonian variety. The intervals between the sediment-building periods of these facies categories may be renewal processes of some kind, although all deviate from a simple Poisson model. Owing to the limitations of the data available only tentative conclusions can be drawn.     

基于统计理论的青海河南县地区5种草本植物根系力学特性研究

为合理定量评价草本植物根径、抗拉力和抗拉强度等指标分布特征,本研究选取生长于青海河南县地区的黄花棘豆(Oxytropis ochrocephala)、早熟禾(Poa annua)、紫花针茅(Stipa purpurea)、青藏苔草(Carex moorcroftii)和矮嵩草(Kobresia humilis)5种草本植物为研究对象,通过室内单根拉伸试验对上述草本单根抗拉力、根径和抗拉强度进行了测定;在此基础上,利用正态分布、伽马分布、泊松分布、瑞利分布以及威布尔分布等统计模型对上述指标的分布进行了定性拟合分析,最后采用柯尔莫哥洛夫-斯米洛夫检验对该指标进行了定量检验。结果表明:5种统计模型中,泊松分布对3项指标值分布的描述性相对较差,威布尔和伽马分布则对上述3项指标的分布描述较好,剩余分布对指标描述的适用性则介于两者之间。此外,所有指标均不服从泊松分布,根径均服从正态分布、伽马分布和威布尔分布,个别植物服从瑞利分布,抗拉力和抗拉强度均服从伽马分布和威布尔分布,个别植物服从瑞利分布和正态分布;除紫花针茅根径最优分布为威布尔分布外,其余植物根径最优分布均为伽马分布,除矮嵩草抗拉力服从威布尔分布外,剩余植物抗拉力最优分布均为伽马分布,紫花针茅根系抗拉强度最优分布为正态分布、青藏苔草和矮嵩草为伽马分布、早熟禾与黄花棘豆最优分布均为威布尔分布。该项研究结果对于实现合理定量评价草本植物根系根径、抗拉力和抗拉强度性能具有重要的理论参考价值。     

Measuring multifractality in seismic sequences

We investigated the multifractal structure of the interevent times between successive earthquakes that occurred in Umbria-Marche, which is one of the most seismically active areas of central Italy. We used the Multifractal Detrended Fluctuation Analysis (MF-DFA), which permits detection of multifractality in nonstationary series. Analyzing the time evolution of the multifractal behaviour of the seismic sequence, a loss of multifractality during the aftershocks is revealed.     

The loghyperbolic: An alternative to the lognormal for modeling oil field size distribution

The lognormal distribution is widely used to represent the distribution of deposit or reserve size of oil and gas fields. The purpose of this paper has been to investigate the potential usefulness of the loghyperbolic distribution as an alternative to the lognormal distribution. This hypothesis is tested using a set of data from the Denver basin. The results indicate that the loghyperbolic distribution shows a better fit to the empirical data than the lognormal distribution.     

Distribution of hydrocarbon accumulations in a basin: a mathematical model for the West Siberian petroleum province

The distribution of hydrocarbon accumulations in a basin is modeled as a nonstationary Poisson field of points with the average density of accumulations as a function of distance from the basin margin. The model, in which this distance is a unique parameter to define the intensity function, is suitable, in a first approximation, for Monte-Carlo simulation of the real pattern of accumulations. The Poisson random field of points is described with a power function, where the power is a fractal dimension used as an integral numerical parameter of the distribution.     

Estimation of the waiting time distributions of earthquakes

Whether the earthquake occurrences follow a Poisson process model is a widely debated issue. The Poisson process model has great conceptual appeal and those who rejected it under pressure of empirical evidence have tried to restore it by trying to identify main events and suppressing foreshocks and aftershocks. The approach here is to estimate the density functions for the waiting times of the future earthquakes. For this purpose, the notion of Gram-Charlier series which is a standard method for the estimation of density functions has been extended based on the orthogonality properties of certain polynomials such as Laguerre and Legendre. It is argued that it is best to estimate density functions in the context of a particular null hypothesis. Using the results of estimation a simple test has been designed to establish that earthquakes do not occur as independent events, thus violating one of the postulates of a Poisson process model. Both methodological and utilitarian aspects are dealt with.     

Stage Predictions of Landslide and Subsidence from an Once-Through Cycle

In this paper both processes of landslide and subsidence are consid-ered to be limited systems.Each of these systems in nature might be re-garded as an organism.Generally their lifespan must develop with com-mon ecological characteristics,including several evolutional stages,such asinitiation,growth,maturation,decline and death.Among these stages,maturation is emphasized so as to find the occurring or thriving date ofboth systems.An once-through cycle of both landslide and subsidence isestablished and is accurately predicted by a developed,mathematic model of thePoisson cycle.The Weibull distribution is cited for a landslide example.Both fundamentals are discussed.Stage predictions of landslide and subsid-ence are performed for several examples.Back analysis of landslides thathave already happened are studied with the same model.And when com-pared with results from the biological mathematic model and with practicalresults,it is found that they correspond.Stage prediction of subsidences isalso researched     

Lognormality in the observed size distribution of oil and gas pools as a consequence of sampling bias

Economic filtration has been offered as an explanation of the observed lognormality in the size distribution of discovered oil and gas deposits. The result leads to the conclusion that one cannot impute the shape of the underlying parent distribution from the observed discoveries size distribution. The fact that the largest pools tend to be discovered early in the exploration history of an area of interest suggests the existence of an inherent sampling bias in the discovery process. The bias is influenced by the levels of geologic knowledge and technological sophistication. Furthermore, the existence of the bias leads to lognormality in the observed discoveries size distribution of oil and gas pools. A discovery process model explicitly incorporating the notion of sampling bias was applied to a series of Weibull parent frequency size distributions. The selected parent distributions are of a class suggested in the literature as more reflective of nature's size distribution and have empirical support. The distribution of discoveries resulting from the application of the model to the chosen parent size distributions were tested for lognormality using a chi-squared test. Lognormality was found to be an acceptable model of the discoveries size distribution over a wide range of resource exhaustion measures. When combined with the notion of economic filtration, sampling bias leads to the conclusion that one should not expect the lognormal distribution to accurately represent the underlying parent size distribution of oil and gas deposits.     

On Selection of Analog Volcanoes

Estimating the occurrence probability of volcanic eruptions with VEI ??3 is challenging in several aspects, including data scarcity. A?suggested approach has been to use a simple model, where eruptions are assumed to follow a Poisson process, augmenting the data used to estimate the eruption onset rate with that from several analog volcanoes. In this model the eruption onset rate is a random variable that follows a gamma distribution, the parameters of which are estimated by an empirical Bayes analysis. The selection of analog volcanoes is an important step that needs to be explicitly considered in this model, as we show that the analysis is not always feasible due to the required over-dispersion in the resulting negative binomial distribution for the numbers of eruptions. We propose a modification to the method which allows for both over-dispersed and under-dispersed data, and permits analog volcanoes to be chosen on other grounds than mathematical tractability.     

Irreducible water distribution in sandstone rock: Two phase flow simulations in CT-based pore network

A simple cutoff approach based on the capillary bundle model has become an industrial standard method to quickly obtain the irreducible water saturation from NMR T₂ distribution. However this approach is not always valid. To overcome the shortcoming of the capillary bundle model that ignores pore-pore connectivity, we have conducted a two-phase flow simulation in a CT-based pore network. The CT -based pore network is a representation of a real rock pore structure that is described by a binary X-ray tomographic data set. Simulation on such network mimics a process of the porous plate measurement. The generated capillary curve is quite reasonable. The oil accession distributions at different water saturations plotted as a function of pore size provide an insight for the immiscible displacement process in the real rock pore structure. Water is trapped not only in dead ends but also in the well-connected pores due to a pore level by-pass mechanism. At the capillary end point pressure, a plot of the trapped water distribution as a function of pore size has the same lognormal distribution as the pore size distribution, which is much different from what the simple capillary bundle model suggests.     

Micromechanical Model for Simulating the Fracture Process of Rock

Summary A micromechanical model is proposed to study the deformation and failure process of rock based on knowledge of heterogeneity of rock at the mesoscopic level. In this numerical model, the heterogeneity of rock at the mesoscopic level is considered by assuming the material properties in rock conform to the Weibull distribution. Elastic damage mechanics is used to describe the constitutive law of meso-level elements, the finite element method is employed as the basic stress analysis tool and the maximum tensile strain criterion as well as the Mohr-Coulomb criterion is utilized as the damage threshold. A simple method, similar to a smeared crack model, is used for tracing the crack propagation process and interaction of multiple cracks. Based on this model, a numerical simulation program named Rock Failure Process Analysis Code (RFPA) is developed. The influence of parameters that include the Weibull distribution parameters, constitutive parameters of meso-level elements and number of elements in the numerical model, are discussed in detail. It is shown that the homogeneity index is the most important factor to simulate material failure with this model. This model is able to capture the complete mechanical responses of rock, which includes the crack patterns associated with different loading stages and loading conditions, localization of deformation, stress redistribution and failure process. The numerical simulation of rock specimens under a variety of static loading conditions is presented, and the results compare well with experimental results.     

Turbidite bed thickness distributions: methods and pitfalls of analysis and modelling

Turbidite bed thickness distributions are often interpreted in terms of power laws, even when there are significant departures from a single straight line on a log–log exceedence probability plot. Alternatively, these distributions have been described by a lognormal mixture model. Statistical methods used to analyse and distinguish the two models (power law and lognormal mixture) are presented here. In addition, the shortcomings of some frequently applied techniques are discussed, using a new data set from the Tarcău Sandstone of the East Carpathians, Romania, and published data from the Marnoso‐Arenacea Formation of Italy. Log–log exceedence plots and least squares fitting by themselves are inappropriate tools for the analysis of bed thickness distributions; they must be accompanied by the assessment of other types of diagrams (cumulative probability, histogram of log‐transformed values, q–q plots) and the use of a measure of goodness‐of‐fit other than R², such as the chi‐square or the Kolmogorov–Smirnov statistics. When interpreting data that do not follow a single straight line on a log–log exceedence plot, it is important to take into account that ‘segmented’ power laws are not simple mixtures of power law populations with arbitrary parameters. Although a simple model of flow confinement does result in segmented plots at the centre of a basin, the segmented shape of the exceedence curve breaks down as the sampling location moves away from the basin centre. The lognormal mixture model is a sedimentologically intuitive alternative to the power law distribution. The expectation–maximization algorithm can be used to estimate the parameters and thus to model lognormal bed thickness mixtures. Taking into account these observations, the bed thickness data from the Tarcău Sandstone are best described by a lognormal mixture model with two components. Compared with the Marnoso‐Arenacea Formation, in which bed thicknesses of thin beds have a larger variability than thicknesses of the thicker beds, the thinner‐bedded population of the Tarcău Sandstone has a lower variability than the thicker‐bedded population. Such differences might reflect contrasting depositional settings, such as the difference between channel levées and basin plains.     

Comparison of Two Probability Distributions Used to Model Sizes of Undiscovered Oil and Gas Accumulations: Does the Tail Wag the Assessment?

Undiscovered oil and gas assessments are commonly reported as aggregate estimates of hydrocarbon volumes. Potential commercial value and discovery costs are, however, determined by accumulation size, so engineers, economists, decision makers, and sometimes policy analysts are most interested in projected discovery sizes. The lognormal and Pareto distributions have been used to model exploration target sizes. This note contrasts the outcomes of applying these alternative distributions to the play level assessments of the U.S. Geological Survey's 1995 National Oil and Gas Assessment. Using the same numbers of undiscovered accumulations and the same minimum, medium, and maximum size estimates, substitution of the shifted truncated lognormal distribution for the shifted truncated Pareto distribution reduced assessed undiscovered oil by 16% and gas by 15%. Nearly all of the volume differences resulted because the lognormal had fewer larger fields relative to the Pareto. The lognormal also resulted in a smaller number of small fields relative to the Pareto. For the Permian Basin case study presented here, reserve addition costs were 20% higher with the lognormal size assumption.     

Joint return probability analysis of wind speed and rainfall intensity in typhoon-affected sea area

Strong wind and rainfall induced by extreme meteorological processes such as typhoons have a serious impact on the safety of bridges and offshore engineering structures. A new bivariate compound extreme value distribution is proposed to describe the probability dependency structure of annual extreme wind speed and concomitant process maximum rainfall intensity in typhoon-affected area. This probability model takes full account of the case that there may be no rainfall in a typhoon process. A case study based on the observation data of typhoon maximum wind speed and maximum rainfall intensity in Shanghai is conducted to testify the efficiency of the model. Weibull distributions with two parameters are applied to fit respective probability margins, and the joint probability distribution is constructed by Gumbel–Hougaard copula. The fitting results and K–S tests show that these models describe the original data well. The joint return periods are calculated by Poisson bivariate compound extreme value distribution we have proposed. They indicate that typhoons with no rain have smaller joint return periods, and wind speed is the main factor which impacts the change of the joint return periods.     

Normal and lognormal data distribution in geochemistry: death of a myth. Consequences for the statistical treatment of geochemical and environmental data

All variables of several large data sets from regional geochemical and environmental surveys were tested for a normal or lognormal data distribution. As a general rule, almost all variables (up to more than 50 analysed chemical elements per data set) show neither a normal or a lognormal data distribution. Even when different transformation methods are used more than 70 % of all variables in every single data set do not approach a normal distribution. Distributions are usually skewed, have outliers and originate from more than one process. When dealing with regional geochemical or environmental data normal and/or lognormal distributions are an exception and not the rule. This observation has serious consequences for the further statistical treatment of geochemical and environmental data. The most widely used statistical methods are all based on the assumption that the studied data show a normal or lognormal distribution. Neglecting that geochemcial and environmental data show neither a normal or lognormal distribution will lead to biased or faulty results when such techniques are used. Received: 21 June 1999 · Accepted: 14 August 1999