小样本岩土参数下土质边坡可靠度分析的条件概率法

岩土勘察取样数量有限是土工参数统计不确定性的重要来源,小样本条件下参数估计偏差向可靠度计算环节传递,导致分析结果呈现不确定性.针对小样本岩土参数X,围绕样本平均值(X-)构造具有不同随机偏离程度的一系列假想参数总体均值μ*,以μ*i的不同取值为发生条件P(Bi)、对应的可靠度计算结果Pfi为条件概率P(A|Bi),基于...     

The Kappa model of probability and higher-order rock sequences

In any depositional environment, the sequence of sediments follows specific high- and low-frequency patterns of rock occurrences or events. The occurrence of a rock in a spatial location is conditional to a prior rock event at a distant location. Subsequently, a third rock occurs between the two locations. This third event is conditional to both prior events and is driven by a third-order conditional probability P(C ∣ (A ∩ B)). Such probability has to be characterized beyond the classic conditional independence model, and this research has found that exact computation requires a third-order co-cumulant term. The co-cumulants provide the higher-order redundancy among multiple indicator variables. A Bayesian analysis has been performed with “known” numerical co-cumulants yielding a novel model of conditional probability that is called the “Kappa model.” This model was applied to three-point variables, and the concept has been extended for multiple events P(G ∣ A ∩ B ∩ C ∩ D... ∩ N), allowing the reproduction of complex transitions of rocks in sequence stratigraphy. The Kappa model and co-cumulants have been illustrated with simple numerical examples for clastic rock sequences. In addition, the co-cumulant has been used to discover an extension of the variogram called the indicator cumulogram. In this way, multiple prior events are no longer ignored for evaluating the probability of a posterior event with higher-order co-cumulant considerations.     

The Nu Expression for Probabilistic Data Integration

The general problem of data integration is expressed as that of combining probability distributions conditioned to each individual datum or data event into a posterior probability for the unknown conditioned jointly to all data. Any such combination of information requires taking into account data interaction for the specific event being assessed. The nu expression provides an exact analytical representation of such a combination. This representation allows a clear and useful separation of the two components of any data integration algorithm: individual data information content and data interaction, the latter being different from data dependence. Any estimation workflow that fails to address data interaction is not only suboptimal, but may result in severe bias. The nu expression reduces the possibly very complex joint data interaction to a single multiplicative correction parameter ν ₀, difficult to evaluate but whose exact analytical expression is given; availability of such an expression provides avenues for its determination or approximation. The case ν ₀=1 is more comprehensive than data conditional independence; it delivers a preliminary robust approximation in presence of actual data interaction. An experiment where the exact results are known allows the results of the ν ₀=1 approximation to be checked against the traditional estimators based on assumption of data independence.     

Simulation of Non-Gaussian Transmissivity Fields Honoring Piezometric Data and Integrating Soft and Secondary Information

The conditional probabilities (CP) method implements a new procedure for the generation of transmissivity fields conditional to piezometric head data capable to sample nonmulti-Gaussian random functions and to integrate soft and secondary information. The CP method combines the advantages of the self-calibrated (SC) method with probability fields to circumvent some of the drawbacks of the SC method—namely, its difficulty to integrate soft and secondary information or to generate non-Gaussian fields. The SC method is based on the perturbation of a seed transmissivity field already conditional to transmissivity and secondary data, with the perturbation being function of the transmissivity variogram. The CP method is also based on the perturbation of a seed field; however, the perturbation is made function of the full transmissivity bivariate distribution and of the correlation to the secondary data. The two methods are applied to a sample of an exhaustive non-Gaussian data set of natural origin to demonstrate the interest of using a simulation method that is capable to model the spatial patterns of transmissivity variability beyond the variogram. A comparison of the probabilistic predictions of convective transport derived from a Monte Carlo exercise using both methods demonstrates the superiority of the CP method when the underlying spatial variability is non-Gaussian.     

Indicator Simulation Accounting for Multiple-Point Statistics

Geostatistical simulation aims at reproducing the variability of the real underlying phenomena. When nonlinear features or large-range connectivity is present, the traditional variogram-based simulation approaches do not provide good reproduction of those features. Connectivity of high and low values is often critical for grades in a mineral deposit. Multiple-point statistics can help to characterize these features. The use of multiple-point statistics in geostatistical simulation was proposed more than 10 years ago, on the basis of the use of training images to extract the statistics. This paper proposes the use of multiple-point statistics extracted from actual data. A method is developed to simulate continuous variables. The indicator kriging probabilities used in sequential indicator simulation are modified by probabilities extracted from multiple-point configurations. The correction is done under the assumption of conditional independence. The practical implementation of the method is illustrated with data from a porphyry copper mine.     

Successive Nonparametric Estimation of Conditional Distributions

Spatial characterization of non-Gaussian attributes in earth sciences and engineering commonly requires the estimation of their conditional distribution. The indicator and probability kriging approaches of current nonparametric geostatistics provide approximations for estimating conditional distributions. They do not, however, provide results similar to those in the cumbersome implementation of simultaneous cokriging of indicators. This paper presents a new formulation termed successive cokriging of indicators that avoids the classic simultaneous solution and related computational problems, while obtaining equivalent results to the impractical simultaneous solution of cokriging of indicators. A successive minimization of the estimation variance of probability estimates is performed, as additional data are successively included into the estimation process. In addition, the approach leads to an efficient nonparametric simulation algorithm for non-Gaussian random functions based on residual probabilities.     

Earthquake Hazard in Italy, 2001–2030

The study computes time-dependant earthquake probabilities on the basis of seismicity data mainly deriving from historic records. It provides a methodological approach useful for those countries where the scarcity of instrumental data and/or paleoseismological evidences requires that historical information shall be stressed. Thus, the conditional probability that damaging earthquakes (M ≥ 6) may occur in Italy in the next 30 years is shown, and the potential for the main worldwide known Italian cities with a cultural heritage is outlined. Earthquake probabilities are computed referring to the application of renewal processes, where the periodicity is analytically modelled by means of the Brownian Passage Time function; an estimate of the dispersion (i.e., uncertainty) introduced on probabilities is provided making use of Monte Carlo simulations. The computed probabilities refer to seismic source zones deriving from the spatial clustering of the historically documented seismicity. The computation of probabilities based on the interaction of earthquakes occurring in nearby zones, has been also attempted for a test area to explore the influence exerted by the stress transfer effect. The main findings of this study are that (1) seismic source zones in Southern Italy are the most prone to experience damaging earthquakes in the next 30-years, with conditional probabilities a large as 10%; and (2) the influence exerted by the earthquake interaction in increasing such probabilities, doesn’t seem to be relevant, because the mean recurrence times of large earthquakes (above the threshold magnitude of six chosen in this study) are in general much longer than the time shortening produced by the stress transfer.     

Methods for probabilistic assessments of geologic hazards

Although risk analysis today is considered to include three separate aspects (1) identifing sources of risk, (2) estimating probabilities quantitatively, and (3) evaluating consequences of risk, here only estimation of probabilities for natural geologic events, processes, and phenomena is addressed. Ideally, evaluation of potential future hazards includes an objective determination of probabilities that have been derived from past occurrences of identical events or components contributing to complex processes or phenomena. In practice, however, data which would permit objective estimation of those probabilities of interest may not be adequate, or may not even exist.Another problem that arises normally, regardless of the extent of data, is that risk assessments involve estimating extreme values. Probabilities are required for events that are the greatest or rarest because they commonly will have the greatest consequences; the largest, or rarest, events always fall in tails of frequency distributions. Rarely are extreme values accurately predictable even when an empirical frequency distribution is established well by data.In the absence of objective methods for estimating probabilities of natural events or processes, subjective probabilities for the hazard must be established through Bayesian methods, expert opinion, or Delphi methods. Alternative solutions may involve consequence analysis which may demonstrate that, although an event may occur, its consequences are sufficiently small that it safely may be ignored or by establishing bounds which may demonstrate that although probabilities are not known they cannot exceed a maximum value that is sufficiently small so that associated risk may be considered to be negligible.Uncertainty of every probability determination must be stated for each component of an event, process, or phenomenon. These uncertainties also must be propagated through the quantitative analysis so that a realistic estimate of total uncertainty can be associated with each final probability estimate for a geologic hazard.This paper was presented (by title) at Emerging Concepts, MGUS-87 Conference, Redwood City, California, 13–15 April 1987.     

Markov Chain Random Fields for Estimation of?Categorical Variables

Multi-dimensional Markov chain conditional simulation (or interpolation) models have potential for predicting and simulating categorical variables more accurately from sample data because they can incorporate interclass relationships. This paper introduces a Markov chain random field (MCRF) theory for building one to multi-dimensional Markov chain models for conditional simulation (or interpolation). A MCRF is defined as a single spatial Markov chain that moves (or jumps) in a space, with its conditional probability distribution at each location entirely depending on its nearest known neighbors in different directions. A general solution for conditional probability distribution of a random variable in a MCRF is derived explicitly based on the Bayes’ theorem and conditional independence assumption. One to multi-dimensional Markov chain models for prediction and conditional simulation of categorical variables can be drawn from the general solution and MCRF-based multi-dimensional Markov chain models are nonlinear.     

四川罗家青杠岭崩塌风险的定量评价研究

我国多山,崩塌灾害频繁发生,相应的风险评价也得到了越来越多的关注。由于崩塌发生和运移过程的高度不确定性以及历史数据的不完备,往往很难进行相应的定量风险评价。四川罗家青杠岭的崩塌现场非常典型,而且现场资料较全、历史数据较多并且明确,是开展崩塌风险定量研究的很好实例。通过现场工程地质调查、测绘和统计分析,确定了历史崩塌的物源区、堆积区、最大运移距离、年发生概率以及坡体上的4块典型危岩体A-D。基于历史崩塌堆积区的块石统计特征以及物源区危岩体失稳启动位置的不确定性,利用二维Rockfall模拟软件对所在坡面的恢复系数及摩擦系数进行了反演。在此基础上,对危岩体A-D失稳后的运动特征进行了随机性数值模拟和统计分析,从而确定了崩塌的到达概率。基于崩塌发生概率、到达概率、承灾体时空分布概率和易损性的乘积,作者对罗家青杠岭崩塌进行了定量风险评价。评价结果表明,危岩体A和D的风险值处于不可接受的风险区间,块石B和C的风险值处于警告的风险区间,严重威胁着坡脚附近居民的生命财产安全,有必要采取相应的防灾减灾措施。     

基于条件期望的全概率公式及其应用

基于条件期望和随机事件A的示性函数IA，推导了离散型和连续型随机变量的全概率公式，要解决诸如离散型和连续型等一类随机问题时，若采用传统的全概率公式来解，势必带来很大的困难。但是，运用本文所研究的全概率公式。结果令人满意，具有较强的实用性，从而拓展了全概率公式的应用范围和解题思路，值得推广应用，同时该公式在可靠性理论研究中也有重要应用。     

Probabilistic stability evaluation for wildlife-damaged earth dams: a Bayesian approach

ABSTRACT

There is a growing body of evidence confirming the detrimental effects of invasive wildlife activities on the performance and – ultimately – the safety of earth dams and levee systems. Modelling cavities and burrows dug by animals in earth structures is rather cumbersome due to their geometrical complexity and randomness. This study proposes the use of the probabilistic approach to estimate the risk associated with the presence of wildlife chambers in an arbitrary dam section. Two dimensional limit equilibrium slope stability analyses were conducted for a homogeneous silty sand dam. The animal chambers were modelled as idealised highly permeable circles with a near-zero strength. Uncertainty in the dam material was introduced using the Monte-Carlo technique to randomise the input parameters. The conditional probability of failure, P(F|C), was spatially evaluated by manual perturbations of chamber locations within the dam section. Using the Bayesian approach, the section was accordingly mapped into three stability zones: “safe”, “marginally safe” and “likely to fail” based on P(F|C) values associated with the chamber location. For this very complex problem, the probabilistic handling seems to provide a mathematically normalised basis for risk assessment in lieu of the classical belief-based single-value factor of safety (FoS).     

Approximate local confidence intervals under change of support

Approximate local confidence intervals are constructed from uncertainty models in the form of the conditional distribution of the random variable Z given values of variables [Z_i, i=1,...,n]. When the support of the variable Z is any support other than that of the data, the conditional distributions require a change of support correction. This paper investigates the effect of change of support on the approximate local confidence intervals constructed by cumulative indicator kriging, class indicator kriging, and probability kriging under a variety of conditions. The conditions are generated by three simulated deposits with grade distributions of successively higher degree of skewness; a point support and two different block supports are considered. The paper also compares the confidence intervals obtained from these methods using the most used measures of confidence interval effectiveness.     

Markov Chain Random Fields for Estimation of Categorical Variables

Multi-dimensional Markov chain conditional simulation (or interpolation) models have potential for predicting and simulating categorical variables more accurately from sample data because they can incorporate interclass relationships. This paper introduces a Markov chain random field (MCRF) theory for building one to multi-dimensional Markov chain models for conditional simulation (or interpolation). A MCRF is defined as a single spatial Markov chain that moves (or jumps) in a space, with its conditional probability distribution at each location entirely depending on its nearest known neighbors in different directions. A general solution for conditional probability distribution of a random variable in a MCRF is derived explicitly based on the Bayes’ theorem and conditional independence assumption. One to multi-dimensional Markov chain models for prediction and conditional simulation of categorical variables can be drawn from the general solution and MCRF-based multi-dimensional Markov chain models are nonlinear.     

全新世北半球典型冷事件的模拟研究

全新世冷事件期间的气候格局及其成因是过去气候变化研究的热点问题.利用基于通用气候系统模式开展的TraCE-21ka气候模拟试验资料,在定义和提取典型冷事件的基础上,分析了全强迫试验模拟的全新世北半球多次冷事件的规模及冷事件发生时温度与降水的空间特征,并结合全强迫试验中使用的4个外强迫序列(淡水注入、轨道强迫、大气温室气体、大陆冰盖)及其对应的单因子敏感性试验,初步探讨了部分典型冷事件的成因.结果表明: TraCE-21ka模拟的冷事件年份与重建/集成序列的冷事件年份对应较好,模式较好地模拟出了全新世北半球的冷事件;全新世期间,北半球共发生了 10 次典型冷事件( 9. 7 ka B. P.、 8. 3 ka B. P.、 7. 3 ka B. P.、6. 2 ka B. P.、 5. 2 ka B. P.、 4. 2 ka B. P.、 3. 4 ka B. P.、 2. 1 ka B. P.、 1. 0 ka B. P.和 0. 2 ka B. P.);每次冷事件发生时,北半球大范围降温和变干,温度变化呈现明显的纬度地带性差异,中高纬地区降温最显著,低纬10°N附近降水减少最显著;在 8. 3 ka B. P.、 7. 3 ka B. P.、 6. 2 ka B. P.、 5. 2 ka B. P.、 4. 2 ka B. P.、 3. 4 ka B. P.、 2. 1 ka B. P.和1. 0 ka B. P.共8次冷事件中,北半球温度和降水的空间变化较为相似,北大西洋经圈翻转流( Atlantic Meridional Overturning Circulation,简称 AMOC)变弱导致了冷事件,格陵兰岛南部的北大西洋海域降温和变干尤为显著;9. 7 ka B. P.和3. 4 ka B. P.的冷事件可能与轨道强迫有关,淡水注入造成了8. 3 ka B. P.和7. 3 ka B. P.的冷事件, 0. 2 ka B. P.冷事件可能与大气温室气体波动有关.地球系统内部变率对于冷事件的发生可能也有一定影响.     

Mixture of populations

A simple example simulating a mixture of two normal populations results in some important observations, nonnormality and nonsymmetry of the mixture conditional pdf, nonlinearity of the conditional mean as a function of the conditioning data, heteroscedasticity of the conditional variance and its nonmonotonicity as a function of distance of the unknown to the conditioning data. A comparison of the mixture statistics with those predicted by traditional models ignoring the mixture reveals the inadequacy and inappropriateness of these traditional approaches. A mixture of two multivariate normal populations is illustrated through the analytical expressions of its conditional distribution and moments.     

Decision support system based on the history of flood and flash flood events in Romania

A lightweight decision support system is presented, oriented also to statistics, useful for assisting weather forecasters and other parties interested in hazard assessment associated with extreme weather. The system can be used in enhancing the warning procedures, ahead of a flood or a flash flood whose probability of occurrence is based on the history of such events in a particular region. A software application has been built that integrates meteorological data with Geographical Information Systems procedures, in a unified informational aggregate. This system stores various types of data related to flood and flash flood events, so it is able to provide the user with any piece of information related to a documented event. It also catalogues any information that users provide it with, to further document a past, or an ongoing event. The system can be used to raise awareness of forecasters over a particular context, before a possibly hazardous situation, and it can also offer automatic warnings and suggestions to those interested in disaster mitigation.     

Probabilistic model for eruptions and associated flood events in the Katla caldera,Iceland

Eruptions in the subglacial Katla caldera, South Iceland, release catastrophic jokulhlaups (meltwater floods). The ice surface topography divides the caldera into three drainage sectors (Ko, So and En sectors) that drain onto Myrdalssandur, Solheimasandur and Markarfljot plains, respectively. In historical times, floods from the Ko sector have been dominant, with only two recorded So events. Geological records indicate that floods from the En sector occur every 500–800 years. A probabilistic model for an eruption is formulated in general terms by a stochastic parameter that simulates a series giving the time interval in years between two consecutive events. The model also contains a Markovian matrix that controls the location of the event and thereby what watercourse is hit by the flood. A record of Katla eruptions since the 8th and the 9th century a.d., and geological information of volcanogenic floods towards the west over the last 8,000 years is used to calibrate the model. The model is then used to find the probabilities for floods from the three sectors: Ko, So and En. The simulations predict that the most probable eruption interval for the En sector and the So sector is several times smaller than the average time interval, implying infrequent periods of high activity in these sectors. A correlation is found between the magnitude of eruptions and the following time intervals. Using the statistical approach and considering this magnitude–time interval correlation, the probability of an eruption in Katla volcano is considered to be 20% within the next 10 years. This compares to a probability of 93% if only a simple average is considered. These probabilities do not take account of long-term eruption precursors and should therefore be regarded as minimum values.     

辽宁本溪高分辨率石笋氧同位素记录的东亚夏季风“2.8 ka”事件

本研究利用辽宁本溪庙洞石笋MD12,通过230Th定年和高分辨率氧同位素分析,并集成庙洞另一支石笋MD11序列,重建了3.04~2.60 ka B.P.时期辽东地区夏季风(降水)变化历史。该石笋氧同位素序列记录了2.8 ka事件的详细过程,事件的核心阶段起止时间约为2.76~2.66 ka B.P.,中心点位于2.68 ka B.P.,稍微晚于西南石笋氧同位素记录(约2.71 ka B.P.),但是在事件的开始时间上,庙洞石笋氧同位素记录比西南石笋氧同位素记录晚了约100年。因此,庙洞石笋氧同位素记录与西南石笋氧同位素记录中显示的缓慢减弱-快速增强模式不同,2.8 ka事件在辽东石笋氧同位素记录中表现为快速减弱-突然增强的变化模式,与太阳活动代用指标一致变化,支持太阳活动是2.8 ka弱季风事件驱动因子的观点。庙洞石笋氧同位素快速响应太阳活动的变化特征,说明其动力学机制可能是太阳活动减弱导致北半球高纬度地区温度降低,然后通过大气"遥相关"作用影响东亚夏季风。     

Estimation of Probabilities of Three Kinds of Petrologic Hypotheses with Bayes Theorem

Physical-chemical explanations of the causes of variations in rock suites are evaluated by comparing predicted to measured compositions. Consistent data turn an explanation into a viable hypothesis. Predicted and measured values seldom are equal, creating problems of defining consistency and quantifying confidence in the hypthesis. Bayes theorem leads to methods for testing alternative hypotheses. Information available prior to data collection provides estimates of prior probabilities for competing hypotheses. After consideration of new data, Bayes theorem updates the probabilities for the hypotheses being correct, returning posterior probabilities. Bayes factors, B, are a means of expressing Bayes theorem if there are two hypotheses, H ₀ and H ₁. For fixed values of the prior probabilities, B > 1 implies an increased posterior probability for H ₀ over its prior probability, whereas B < 1 implies an increased posterior probability for H ₁ over its prior probability. Three common problems are: (1) comparing variances in sets of data with known analytical uncertainties, (2) comparing mean values of two datasets with known analytical uncertainties, and (3) determining whether a data point falls on a predicted trend. The probability is better than 0.9934 that lava flows of the 1968 eruption of Kilauea Volcano, Hawaii, are from a single magma batch. The probability is 0.99 that lava flows from two outcrops near Mount Edziza, British Columbia, are from different magma batches, suggesting that the two outcrops can be the same age only by an unlikely coincidence. Bayes factors for hypotheses relating lava flows from Volcano Mountain, Yukon Territory, by crystal fractionation support the hypothesis for one flow but the factor for another flow is so small it practically guarantees the fractionation hypothesis is wrong. Probabilities for petrologic hypotheses cannot become large with a single line of evidence; several data points or datasets are required for high probabilities.