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 Ho's nonhomogeneous model and the Weibull renewal process seems to produce more plausible results.     

Bayesian Estimation of Earth’s Undiscovered Mineralogical Diversity Using Noninformative Priors

Recently, statistical distributions have been explored to provide estimates of the mineralogical diversity of Earth, and Earth-like planets. In this paper, a Bayesian approach is introduced to estimate Earth’s undiscovered mineralogical diversity. Samples are generated from a posterior distribution of the model parameters using Markov chain Monte Carlo simulations such that estimates and inference are directly obtained. It was previously shown that the mineral species frequency distribution conforms to a generalized inverse Gauss–Poisson (GIGP) large number of rare events model. Even though the model fit was good, the population size estimate obtained by using this model was found to be unreasonably low by mineralogists. In this paper, several zero-truncated, mixed Poisson distributions are fitted and compared, where the Poisson-lognormal distribution is found to provide the best fit. Subsequently, the population size estimates obtained by Bayesian methods are compared to the empirical Bayes estimates. Species accumulation curves are constructed and employed to estimate the population size as a function of sampling size. Finally, the relative abundances, and hence the occurrence probabilities of species in a random sample, are calculated numerically for all mineral species in Earth’s crust using the Poisson-lognormal distribution. These calculations are connected and compared to the calculations obtained in a previous paper using the GIGP model for which mineralogical criteria of an Earth-like planet were given.

     

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.     

Bayesian analysis of marked stress release models for time-dependent hazard assessment in the western Gulf of Corinth

We consider point processes defined on the space–time domain which model physical processes characterized qualitatively by the gradual increase over time in some energy until a threshold is reached, after which, an event causing the loss of energy occurs. The risk function will, therefore, increase piecewise with sudden drops in correspondence to each event. This kind of behaviour is described by Reid's theory of elastic rebound in the earthquake generating process where the quantity that is accumulated is the strain energy or stress due to the relative movement of tectonic plates. The complexity and the intrinsic randomness of the phenomenon call for probabilistic models; in particular the stochastic translation of Reid's theory is given by stress release models. In this article we use such models to assess the time-dependent seismic hazard of the seismogenic zone of the Corinthos Gulf. For each event we consider the occurrence time and the magnitude, which is modelled by a probability distribution depending on the stress level present in the region at any instant. Hence we are dealing here with a marked point process. We perform the Bayesian analysis of this model by applying the stochastic simulation methods based on the generation of Markov chains, the so called Markov chain Monte Carlo (MCMC) methods, which allow one to reconcile the model's complexity with the computational burden of the inferential procedure. Stress release and Poisson models are compared on the basis of the Bayes factor.     

Probabilistic landslide hazard assessment using homogeneous susceptible units (HSU) along a national highway corridor in the northern Himalayas, India

The increased socio-economic significance of landslides has resulted in the application of statistical methods to assess their hazard, particularly at medium scales. These models evaluate where, when and what size landslides are expected. The method presented in this study evaluates the landslide hazard on the basis of homogenous susceptible units (HSU). HSU are derived from a landslide susceptibility map that is a combination of landslide occurrences and geo-environmental factors, using an automated segmentation procedure. To divide the landslide susceptibility map into HSU, we apply a region-growing segmentation algorithm that results in segments with statistically independent spatial probability values. Independence is tested using Moran’s I and a weighted variance method. For each HSU, we obtain the landslide frequency from the multi-temporal data. Temporal and size probabilities are calculated using a Poisson model and an inverse-gamma model, respectively. The methodology is tested in a landslide-prone national highway corridor in the northern Himalayas, India. Our study demonstrates that HSU can replace the commonly used terrain mapping units for combining three probabilities for landslide hazard assessment. A quantitative estimate of landslide hazard is obtained as a joint probability of landslide size, of landslide temporal occurrence for each HSU for different time periods and for different sizes.     

Estimation of Geological Attributes from a Well Log: An Application of Hidden Markov Chains

Geophysical well logs used in petroleum exploration consist of measurements of physical properties (such as radioactivity, density, and acoustic velocity) that are digitally recorded at a fixed interval (typically half a foot) along the length of the exploratory well. The measurements are informative of the unobserved rock type alternations along the well, which is critical for the assessment of petroleum reservoirs. The well log data that are analyzed here are from a North Sea petroleum reservoir where two distinct strata have been identified from large scale seismic data. We apply a hidden Markov chain model to infer properties of the rock type alternations, separately for each stratum. The hidden Markov chain uses Dirichlet prior distributions for the Markov transition probabilities between rock types. The well log measurements, conditional on the unobserved rock types, are modeled using Gaussian distributions. Our analysis provides likelihood estimates of the parameters of the Dirichlet prior and the parameters of the measurement model. For fixed values of the parameter estimates we calculate the posterior distributions for the rock type transition probabilities, given the well log measurement data. We then propagate the model parameter uncertainty into the posterior distributions using resampling from the maximum likelihood model. The resulting distributions can be used to characterize the two reservoir strata and possible differences between them. We believe that our approach to modeling and analysis is novel and well suited to the problem. Our approach has elements in common with empirical Bayes methods in that unspecified parameters are estimated using marginal likelihoods. Additionally, we propagate the parameter uncertainty into the final posterior distributions.     

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.     

地质条件不确定性下TBM管片结构失事概率计算

不良地质条件是影响TBM施工隧洞管片结构安全的重要因素。综合考虑围岩地质条件和衬砌结构的不确定性,提出了一种定量分析TBM管片结构失事概率的新方法。在基于Markov过程估计隧洞沿程地质岩性变化概率的基础上,建立了隧洞任意位置处管片选型不匹配的概率模型;考虑围岩和管片参数的不确定性,采用随机有限元方法计算某一类型管片在不同围岩下的失事概率;由此,采用全概率公式,可计算隧洞沿程任意位置处管片结构的失事概率。结合实际工程,针对施工期工况,确定了该隧洞管片沿程的失事概率、最大失事概率及其所对应的位置等,为管片选型、优化设计及TBM施工期的风险防范提供了依据。     

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。     

Using Multiple Grids in Markov Mesh Facies Modeling

A multigrid Markov mesh model for geological facies is formulated by defining a hierarchy of nested grids and defining a Markov mesh model for each of these grids. The facies probabilities in the Markov mesh models are formulated as generalized linear models that combine functions of the grid values in a sequential neighborhood. The parameters in the generalized linear model for each grid are estimated from the training image. During simulation, the coarse patterns are first laid out, and by simulating increasingly finer grids we are able to recreate patterns at different scales. The method is applied to several tests cases and results are compared to the training image and the results of a commercially available snesim algorithm. In each test case, simulation results are compared qualitatively by visual inspection, and quantitatively by using volume fractions, and an upscaled permeability tensor. When compared to the training image, the method produces results that only have a few percent deviation from the values of the training image. When compared with the snesim algorithm the results in general have the same quality. The largest computational cost in the multigrid Markov mesh is the estimation of model parameters from the training image. This is of comparable CPU time to that of creating one snesim realization. The simulation of one realization is typically ten times faster than the estimation.     

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.     

挡土结构物随机地震响应分析及动力可靠度研究

本文提供了一种简单的确定性数值方法,来分析在平稳随机地震荷载作用下的结构随机地震响应及动力可靠度。该方法基于有限元动力分析软件,以单位加速度脉冲函数作为地震荷载的输入,当计算出结构的脉冲响应函数后,再运用傅立叶变换得到随机激励和结构响应之间的传递函数,由此来计算结构的均方根响应和峰值响应。基于此方法,分析了挡土结构物在平稳随机地震荷载作用下的位移、弯矩、基底水平合力、基底竖向合力以及沿墙高的土压力极值的随机地震响应及动力可靠度。从分析结果可以看出:用Kanai谱模型的计算值比欧进萍谱模型的计算值更趋保守,而把响应过程当作马尔可夫过程似比泊松过程更精确。     

Prediction of earthquake hazard by hidden Markov model (around Bilecik,NW Turkey)

Earthquakes are one of the most important natural hazards to be evaluated carefully in engineering projects, due to the severely damaging effects on human-life and human-made structures. The hazard of an earthquake is defined by several approaches and consequently earthquake parameters such as peak ground acceleration occurring on the focused area can be determined. In an earthquake prone area, the identification of the seismicity patterns is an important task to assess the seismic activities and evaluate the risk of damage and loss along with an earthquake occurrence. As a powerful and flexible framework to characterize the temporal seismicity changes and reveal unexpected patterns, Poisson hidden Markov model provides a better understanding of the nature of earthquakes. In this paper, Poisson hidden Markov model is used to predict the earthquake hazard in Bilecik (NW Turkey) as a result of its important geographic location. Bilecik is in close proximity to the North Anatolian Fault Zone and situated between Ankara and Istanbul, the two biggest cites of Turkey. Consequently, there are major highways, railroads and many engineering structures are being constructed in this area. The annual frequencies of earthquakes occurred within a radius of 100 km area centered on Bilecik, from January 1900 to December 2012, with magnitudes (M) at least 4.0 are modeled by using Poisson-HMM. The hazards for the next 35 years from 2013 to 2047 around the area are obtained from the model by forecasting the annual frequencies of M ≥ 4 earthquakes.     

Probabilistic tsunami hazard assessment for Messina Strait Area (Sicily, Italy)

The general modular Bayesian procedure is applied to provide a probabilistic tsunami hazard assessment (PTHA) for the Messina Strait Area (MSA), Italy. This is the first study in an Italian area where the potential tsunamigenic events caused by both submarine seismic sources (SSSs) and submarine mass failures (SMFs) are examined in a probabilistic assessment. The SSSs are localized on active faults in MSA as indicated by the instrumental data of the catalogue of the Italian seismicity; the SMFs are spatially identified using their propensity to failure in the Ionian and Tyrrhenian Seas on the basis of mean slope and mean depth, and using marine geology background knowledge. In both cases the associated probability of occurrence is provided. The run-ups were calculated at key sites that are main cities and/or important sites along the Eastern Sicily and the Southern Calabria coasts where tsunami events were recorded in the past. The posterior probability distribution combines the prior probability and the likelihood calculated in the MSA. The prior probability is based on the physical model of the tsunami process, and the likelihood is based on the historical data collected by the historical catalogues, background knowledge, and marine geological information. The posterior SSSs and SMFs tsunami probabilities are comparable and are combined to produce a final probability for a full PTHA in MSA.     

Stochastic simulation model for tropical cyclone tracks,with special emphasis on landfall behavior

We consider a spatial stochastic model for the simulation of tropical cyclone tracks, which has recently been introduced. Cyclone tracks are represented as labeled polygonal lines, which are described by the movement directions, translational speeds, and wind speeds of the cyclones in regular 6-h intervals. In the present paper, we compare return levels for wind speeds of historically observed cyclone tracks with those generated by the simulator, where a mismatch is shown for most of the considered coastal regions. To adjust this discrepancy, we develop a stochastic algorithm for acceptance and rejection of simulated cyclone tracks with landfall. It is based on the fact that the locations, translational speeds, and wind speeds of cyclones at landfall constitute three-dimensional Poisson point processes, which are a basic model type in stochastic geometry. Due to that, a well-known thinning property of Poisson processes can be applied. This means that to each simulated cyclone, an acceptance probability is assigned, which is higher for cyclones with suitable landfall characteristics and lower for implausible ones. More intuitively, the algorithm comprises the simulation of a more comprehensive cyclone event set than needed and the random selection of those tracks that best match historical observations at landfall. A particular advantage of our algorithm is its applicability to multiple landfalls, i.e., to cyclones that successively make landfall at two geographically distinct coastlines, which is the most relevant case in applications. It turns out that the extended simulator provides a much better accordance between landfall characteristics of historical and simulated cyclone tracks.     

沉积模拟的基本数学模型及其在第四纪研究中的应用

内容提要本文扼要介绍了沉积模拟的基本数学模型,其中包括十种随机模拟数学模型和十种确定模拟数学模型,阐明其简要原理和应用范围,并讨论它们在第四纪研究中的应用现状和前景。本文涉及的应用范围主要为:第四纪沉积环境、第四纪沉积物特征和成因类型。最后讨论了地质过程数学模拟的特点和意义、第四纪沉积过程数学模拟对提高第四纪地质学研究定量化水平的作用,以及进一步开展工作的方向。     

Impact of horizontal resolution on prediction of tropical cyclones over Bay of Bengal using a regional weather prediction model

The present study is carried out to examine the performance of a regional atmospheric model in forecasting tropical cyclones over the Bay of Bengal and its sensitivity to horizontal resolution. Two cyclones, which formed over the Bay of Bengal during the years 1995 and 1997, are simulated using a regional weather prediction model with two horizontal resolutions of 165 km and 55 km. The model is found to perform reasonably well towards simulation of the storms. The structure, intensity and track of the cyclones are found to be better simulated by finer resolution of the model as compared to the coarse resolution. Rainfall amount and its distribution are also found to be sensitive to the model horizontal resolution. Other important fields, viz., vertical velocity, horizontal divergence and horizontal moisture flux are also found to be sensitive to model horizontal resolution and are better simulated by the model with finer horizontal grids.     

The fast dynamo in interstellar turbulence

The α effect and coefficient of eddy diffusivity are calculated for the magnetic field in a random flow with recovery. Such a flow loses its memory abruptly at random times that form a Poisson flow of events. Interstellar turbulence sustained by supernova outbursts is one physical realization of such a flow. The growth rates and configurations of large-scale galactic magnetic fields for this situation are close to those predicted by simple galactic dynamo models. At the same time, the model of a flow with recovery makes it possible to trace the role of the effective “forgetting” of correlations. The presence of this forgetting distinguishes interstellar turbulence from other types of random flows.     

Wind erosion occurrence probabilities maps in the watershed of the Ningxia–Inner Mongolia reach of the Yellow River,China

The watershed of the Ningxia–Inner Mongolia reach of the Yellow River suffers serious wind erosion hazards and the areas with high wind erosion probabilities need to be identified to help in the building of the correct wind-sand blown hazard protection systems. In this study, the Integrated Wind-Erosion Modelling System model and Normalized Difference Vegetation Index (NDVI) data set were used to identify the distributions of threshold wind speeds and wind erosion occurrence probabilities. Through field observations, the relationships among NDVI, vegetation cover, frontal area (lateral cover), roughness length, and threshold friction velocity were obtained. Then, using these relationships, the spatial distributions of threshold wind speeds for wind erosion at a height of 10 m for the different months were mapped. The results show that the threshold wind speed ranged from 7.91 to 35.7 m/s. Based on the threshold wind speed distributions, the wind erosion occurrence probabilities of different months were calculated according to the current wind speed. The results show that the distributions of wind erosion occurrence probabilities and threshold wind speeds were related to each other. The resulting maps of threshold wind speeds and wind erosion occurrence probabilities would help environmental and agricultural researchers in determining some strategies for mitigating or adapting from wind erosion hazards.     

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.