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.     

Bayesian model selection for complex geological structures using polynomial chaos proxy

Different interpretation of sedimentary environments lead to “scenario uncertainty” where the prior reservoir model has a high level of discrete uncertainty. In a real field application, the scenario uncertainty has a considerable effect on flow response uncertainty and makes the uncertainty quantification problem highly nonlinear. We use clustering methods to address the scenario uncertainty. Our approach to cluster analysis is based on the posterior probabilities of models, known as “Bayesian model selection.” Accordingly, we integrate overall possible parameters in each scenario with respect to their corresponding priors to give the measure of how well a model is supported by observations. We propose a cluster-based reduced terms polynomial chaos proxy to efficiently estimate the posterior probability density function under each cluster and calculate the posterior probability of each model. We demonstrate that the convergence rate of the reduced terms polynomial chaos proxy is significantly improved under each cluster comparing to the non-clustered case. We apply the proposed cluster-based polynomial chaos proxy framework to study the plausibility of three training images based on different geological interpretation of the second layer of synthetic Stanford VI reservoir. We demonstrate that the proposed workflow can be efficiently used to calculate the posterior probability of each scenario and also sample from the posterior facies models within each scenario.     

“证据权”法在西南“三江”南段矿产资源评价中的应用

证据权法本身是一种离散的多元统计方法。阐述了矿物资源潜力区的概念,简述证据权法的基本原理。以西南“三江”南段为例,首先在地质、地球物理、地球化学以及遥感地质等致矿信息提取的基础上,运用证据权法定量分析各类致矿信息(控矿因素)与矿产资源空间分布的关系;进而选择最佳控矿因素作为资源评价变量,据其变量计算每个单元的后验成矿概率;最终根据后验成矿概率定量圈定和评价找矿靶区。实践证明:在GIS环境下“,证据权”与找矿信息结合能够有效区分矿化有利地段和不利地段,从而达到定量快速圈定和评价找矿靶区之目的。整个过程是借助于MOPAS3.0评价系统实现的。     

基于贝叶斯方法的船坞双排钢板桩围堰整体稳定性可靠度分析

双排钢板桩围堰的整体稳定性具有较大的不确定性,同样安全系数情况下对应的失稳概率可能不同。为了更准确地分析双排钢板桩围堰的整体稳定性,降低稳定性分析中的不确定性,采用贝叶斯方法对船坞双排钢板桩围堰的整体稳定性进行可靠度分析。首先通过统计数据获取土体参数的先验分布,然后基于实测数据采用贝叶斯方法更新参数以得到后验分布,最后根据参数的后验分布采用一次二阶矩计算围堰结构的可靠度。贝叶斯方法从理论的角度解决了已有工程经验和实际案例数据两方面信息有效综合的问题,能在更接近实际情况的前提下进行可靠度分析。     

Bridging the ensemble Kalman filter and particle filters: the adaptive Gaussian mixture filter

The nonlinear filtering problem occurs in many scientific areas. Sequential Monte Carlo solutions with the correct asymptotic behavior such as particle filters exist, but they are computationally too expensive when working with high-dimensional systems. The ensemble Kalman filter (EnKF) is a more robust method that has shown promising results with a small sample size, but the samples are not guaranteed to come from the true posterior distribution. By approximating the model error with a Gaussian distribution, one may represent the posterior distribution as a sum of Gaussian kernels. The resulting Gaussian mixture filter has the advantage of both a local Kalman type correction and the weighting/resampling step of a particle filter. The Gaussian mixture approximation relies on a bandwidth parameter which often has to be kept quite large in order to avoid a weight collapse in high dimensions. As a result, the Kalman correction is too large to capture highly non-Gaussian posterior distributions. In this paper, we have extended the Gaussian mixture filter (Hoteit et al., Mon Weather Rev 136:317–334, 2008) and also made the connection to particle filters more transparent. In particular, we introduce a tuning parameter for the importance weights. In the last part of the paper, we have performed a simulation experiment with the Lorenz40 model where our method has been compared to the EnKF and a full implementation of a particle filter. The results clearly indicate that the new method has advantages compared to the standard EnKF.     

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.     

华北克拉通沉积变质型铁矿床的特征与预测评价模型

华北克拉通是我国面积最大、时代最古老的陆块,地质构造演化历史复杂,也是我国重要的铁矿资源分布区.本文在系统总结该区域成矿规律的基础上,利用GIS平台提取了多元成矿有利信息(变质建造、控矿构造、矿致磁异常、氧化铁异常、自然重砂信息).在此基础上,建立了沉积变质型铁矿床的找矿预测模型,并采用MRAS系统中的证据权模块进行矿产资源定量预测.按照后验概率的大小划分了两级区域有利成矿远景区,分别是冀东、辽东、五台-吕梁三个A级成矿远景区和大青山-色尔能腾山、鲁山-舞阳、鲁西三个B级成矿远景区.这些远景区与已知矿床(点)重叠或比邻,显示远景区具有较好的成矿条件和较大的成矿潜力,应在今后的找矿勘查工作中予以充分重视.     

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.     

Integrating production data under uncertainty by parallel interacting Markov chains on a reduced dimensional space

In oil industry and subsurface hydrology, geostatistical models are often used to represent the porosity or the permeability field. In history matching of a geostatistical reservoir model, we attempt to find multiple realizations that are conditional to dynamic data and representative of the model uncertainty space. A relevant way to simulate the conditioned realizations is by generating Monte Carlo Markov chains (MCMC). The huge dimensions (number of parameters) of the model and the computational cost of each iteration are two important pitfalls for the use of MCMC. In practice, we have to stop the chain far before it has browsed the whole support of the posterior probability density function. Furthermore, as the relationship between the production data and the random field is highly nonlinear, the posterior can be strongly multimodal and the chain may stay stuck in one of the modes. In this work, we propose a methodology to enhance the sampling properties of classical single MCMC in history matching. We first show how to reduce the dimension of the problem by using a truncated Karhunen–Loève expansion of the random field of interest and assess the number of components to be kept. Then, we show how we can improve the mixing properties of MCMC, without increasing the global computational cost, by using parallel interacting Markov Chains. Finally, we show the encouraging results obtained when applying the method to a synthetic history matching case.     

估计迹长概率分布函数的新方法及其应用

本文提出一种估计迹长概率分布函数的新方法。该方法的优点是不需要有节理在测窗上出露长度的数据，有效的减少了野外工作量     

An iterative version of the adaptive Gaussian mixture filter

The adaptive Gaussian mixture filter (AGM) was introduced as a robust filter technique for large-scale applications and an alternative to the well-known ensemble Kalman filter (EnKF). It consists of two analysis steps, one linear update and one weighting/resampling step. The bias of AGM is determined by two parameters, one adaptive weight parameter (forcing the weights to be more uniform to avoid filter collapse) and one predetermined bandwidth parameter which decides the size of the linear update. It has been shown that if the adaptive parameter approaches one and the bandwidth parameter decreases, as an increasing function of the sample size, the filter can achieve asymptotic optimality. For large-scale applications with a limited sample size, the filter solution may be far from optimal as the adaptive parameter gets close to zero depending on how well the samples from the prior distribution match the data. The bandwidth parameter must often be selected significantly different from zero in order to make large enough linear updates to match the data, at the expense of bias in the estimates. In the iterative AGM we introduce here, we take advantage of the fact that the history matching problem is usually estimation of parameters and initial conditions. If the prior distribution of initial conditions and parameters is close to the posterior distribution, it is possible to match the historical data with a small bandwidth parameter and an adaptive weight parameter that gets close to one. Hence, the bias of the filter solution is small. In order to obtain this scenario, we iteratively run the AGM throughout the data history with a very small bandwidth to create a new prior distribution from the updated samples after each iteration. After a few iterations, nearly all samples from the previous iteration match the data, and the above scenario is achieved. A simple toy problem shows that it is possible to reconstruct the true posterior distribution using the iterative version of the AGM. Then a 2D synthetic reservoir is revisited to demonstrate the potential of the new method on large-scale problems.     

估计迹长概率分布函数的新方法及其应用

本文提出一种估计迹长概率分布函数的新方法。该方法的优点是不需要有节理在测窗上出露长度的数据, 有效的减少了野外工作量。     

地基沉降修正系数的Bayes概率推断

通过分析常规方法在沉降修正系数的选取中具有的定值性和随意性,引入建立在过去信息和现在样本信息之上的Bayes理论,结合某客运专线红黏土路基工程,提出用后验分布得到修正系数的取值范围。实例研究表明,用以往经验综合样本信息,估计修正系数的先验概率在某一区间上服从均匀分布。由现场载荷试验实测沉降量与理论计算沉降量分析所得的修正系数,将现场量测的沉降变形信息与先验信息结合起来,利用Bayes统计理论,由小样本试验数据推算得到修正系数的后验概率服从正态分布。对后验分布所得参数进行区间估计,得到该区域红黏土地基沉降修正系数的取值优化区间为 [1.0, 1.7],分析了不同荷载作用条件下沉降修正系数的概率分布模型。     

Latent Compositional Factors in The Llobregat River Basin (Spain) Hydrogeochemistry

The study of hydrogeochemical data sets frequently calls for statistical dimension reducing techniques. It is well known that hydrochemical parameters are compositions and, for this type of data, the direct application of classical statistical methods based on the correlation matrix yield spurious results. But new results on compositional data analysis have identified the sampling space, the simplex, with an Euclidean space, a fact that allows us to define a simplicial factor analysis strategy, thus overcoming the problem. For illustration, we use samples from the Llobregat River and its tributaries (NE Spain). Three unobservable or latent factorial components are extracted, which are identified with pristine waters, potash-mining influence and urban sewage influence. These three factorial components or compositional factors are plotted in a factorial ternary diagram, which reflects the relative influence of each one of these factors on each observation.     

微机整谱拟合方法在矿物红外光谱多组分定性和定量分析中的应用

本文提出了一种利用微机整谱拟合技术对多组分物相的红外光谱进行定性鉴定和定量分析的方法,阐述了该方法的原理和计算方法。通过与其它红外光谱定性和定量分析方法的对比,该方法不需选择特征峰,不需制备更多的标样。每种物相只需一个含量(重量)已知的标样即可分析。它能够一次性地对多种组分物相做出鉴定并提供其中每种组分的含量。该方法完全实现了计算机程序化,减少了很多人为的因素影响,提高了分析精度,并达到了方便实用的要求。     

贵州中三叠世盘县动物群产出地层剖面样品元素地球化学数据因子分析及其环境演化意义

应用地质统计学方法研究剖面沉积物化学组分的含量变化规律和相互关系,以了解剖面沉积物及其化学组分的影响因素和剖面地层的古环境演化。从R型和Q型因子分析结果得出,剖面沉积物及其化学组分含量主要受陆源物质影响,其次受火山源物质的影响,成岩作用及生物源物质也发挥了一定的作用。剖面的古环境演化以陆源物质、火山源物质和生物源物质混合沉积为主,其间至少经历了7次火山喷发。短期频繁的火山喷发及大量的火山喷发物可能严重影响该时期剖面位置的爬行动物及其他生物的生活环境,这可能是它们在这段时间大量死亡的重要原因之一。     

A New Nonparametric Discriminant Analysis Algorithm Accounting for Bounded Data Errors

In a statistical pattern recognition context, discriminant analysis is designed to classify, when possible, objects into predefined categories. Because this method requires precise input data, uncertainties cannot be propagated in the classifying process. In real case studies, this could lead to drastic misinterpretations of objects. A new nonparametric algorithm based on interval arithmetic has thus been developed to propagate interval-form data. They consist in calculating interval conditional probability density functions and interval posterior probabilities. Objects are eventually assigned to a subset of classes, consistent with the data and their uncertainties. The classifying model is thus less precise, but more realistic than the standard one, which we prove on a real case study.     

Analysis of bimodal orientation data

Statistical models underlying the analysis of orientation data commonly assume a unimodal symmetric population, such as the circular normal distribution. If the sample distribution is skewed or bimodal, standard procedures usually produce misleading results. Where such situations occur, a mixture of two or more circular normal distributions may be used as the population model. The parameters describing each mode and the mixing proportion may be estimated by the method of maximum likelihood using numerical techniques. This approach is applied to a distinctly bimodal set of cross-bedding data from the Mississippian Salem Limestone of central Indiana.     

P-wave velocities of polymineralic rocks: comparison of theory and experiment and test of elastic mixture rules

We compare the P-wave velocities (Vp) of 696 dry samples measured at pressures up to 0.6–1.0 GPa with values calculated from the volume fraction and room pressure elastic constants of each constituent mineral using 16 averaging methods. The exceptional large number of samples covers almost all common lithologic types of igneous and metamorphic rocks. The calculated Vp data are in good agreement with laboratory values measured at about 300 MPa, even though elastic constants of only 22 common minerals are used in the computation. The mean Vp of a polymineralic rock is exclusively controlled by the volume fractions of its constituent minerals while grain shape and crystallographic preferred orientations, anisotropy and other perturbations have minimum effects. The mean Vp can be fairly well predicted as long as a relevant mixture rule is used and the volume fraction is accurately determined for each mineral. However, none of the mixture rules can simultaneously produce the best fit to measured P-wave velocities for all the lithologic types. One method may work well for one lithology but poor for other lithology. Even for a given lithology, an averaging method may yield good agreement at moderate pressure but poor agreement at high pressure. Applications of an inappropriate mixture rule will potentially cause the misinterpretation of the crust and mantle Vp data in terms of mineralogical compositions and structures.