Markov chain Monte Carlo (MCMC) sampling methods to determine optimal models,model resolution and model choice for Earth Science problems

We present an overview of Markov chain Monte Carlo, a sampling method for model inference and uncertainty quantification. We focus on the Bayesian approach to MCMC, which allows us to estimate the posterior distribution of model parameters, without needing to know the normalising constant in Bayes' theorem. Given an estimate of the posterior, we can then determine representative models (such as the expected model, and the maximum posterior probability model), the probability distributions for individual parameters, and the uncertainty about the predictions from these models. We also consider variable dimensional problems in which the number of model parameters is unknown and needs to be inferred. Such problems can be addressed with reversible jump (RJ) MCMC. This leads us to model choice, where we may want to discriminate between models or theories of differing complexity. For problems where the models are hierarchical (e.g. similar structure but with a different number of parameters), the Bayesian approach naturally selects the simpler models. More complex problems require an estimate of the normalising constant in Bayes' theorem (also known as the evidence) and this is difficult to do reliably for high dimensional problems. We illustrate the applications of RJMCMC with 3 examples from our earlier working involving modelling distributions of geochronological age data, inference of sea-level and sediment supply histories from 2D stratigraphic cross-sections, and identification of spatially discontinuous thermal histories from a suite of apatite fission track samples distributed in 3D.     

Statistical analysis of wave-induced extreme nonlinear load effects using time-domain simulations

A new hybrid method for the time-domain nonlinear simulation of the hydroelastic load effects and the peak-over-threshold (POT) method for the calculation of the short-term extreme responses are briefly described and applied to a flexible containership of the latest design. Statistical analysis has been carried out to study the sensitivity of the predicted extreme vertical bending moments and vertical shear forces to the changes in the threshold of the POT method, as well as the statistical uncertainty in the prediction due to the limited duration of the nonlinear simulation. It is recommended that 90%–95% quantile should be used as the threshold in the POT method and more than 100 h of time-domain simulation should be carried out in order to obtain satisfactory predictions of the short-term extreme nonlinear load effects.     

应用贝叶斯方法对有效信息较少的渔业资源进行资源评估-以印度洋长鳍金枪鱼的资源评估为例

众所周知,对有效信息较少的渔业资源进行资源评估面临很大的挑战,而贝叶斯方法在数据数量较少、质量较差的情况下能利用其它种群高质量的数据或已知的先验信息提高资源评估结果的可靠性。由于印度洋长鳍金枪鱼的数据质量较差而数据量有限,长鳍金枪鱼的资源评估结果存在很大的不确定性,为此,本文以印度洋长鳍金枪鱼的资源评估为例,以调查贝叶斯方法在有效信息较少的资源评估中的优势。本文根据不同的先验假设与捕捞数据系列,共构建了8个贝叶斯动态产量模型,以评估长鳍金枪鱼资源。结果表明:(1)分析参数的后验分布能提高捕捞数据系列选择与参数假设的合理性; (2) 利用种群统计学方法为内禀增长率(r)构建有信息先验,能提高资源评估结果的可靠性。与传统方法相比,当基于贝叶斯框架时,能将已知的知识表示为先验信息并能分析参数的后验分布,从而在数据较少或数据质量较差的情况下,能利用各种信息提高参数估计的合理性与资源评估的可靠性。因此,对数据量较少或数据质量较差情况下的渔业资源评估而言,贝叶斯方法非常有效,如本文所示的印度洋长鳍金枪鱼的资源评估。     

Effect of surge uncertainty on probabilistically computed dune erosion

To assess the flood protection capacity of dunes in The Netherlands, a semi-probabilistic dune-erosion prediction method is currently in use in which uncertainties in input parameters of an empirical dune erosion model were taken into account, with the exception of the uncertainty in the extreme surge distribution. Previous research has shown that the surge is by far the most influential parameter affecting erosion in the currently used erosion model, which is due both to the influence of the surge level itself and to the conditional dependence of the wave height and period on the surge level in the probabilistic model used for the assessment. Furthermore, the distribution of extreme surge levels has been shown to contain large statistical uncertainty. The inclusion of uncertainty in input variables into probabilistic models results in more extreme events (in this case erosion) for the same exceedance probability, largely due to the incorporation of higher values of the input variables. The goal of the research described in this paper was to determine the impact of the inclusion of uncertainty in the extreme surge distribution on the estimate of critical erosion (erosion associated with an exceedance frequency of 10^− 5 per year). The uncertainty in the surge distributions was estimated and parameterized, and was incorporated into the probabilistic model. A reduction in uncertainty was subsequently imposed to estimate what value a reduction in uncertainty can offer, in terms of the impact on critical erosion. The probabilistic technique first-order reliability method (FORM) was applied to determine the relative contribution of the uncertainty in the surge distribution (as well as the remaining stochastic variables) to the critical erosion. The impact of the inclusion of uncertainty in the surge distribution on the critical retreat distance was found to be substantial with increases ranging from 34% to 93% of the original estimate at five locations along the Dutch coast. The reduced uncertainty showed a more subtle impact, with increases in critical retreat distance ranging from 10% to 26% of the original estimate. The relative importance analysis showed that the uncertainty in the surge distribution has a strong influence, with the relative importance ranging from 10% to 23% for an exceedance frequency of 10^− 5 per year.     

A Bayesian method for the estimation of return values of wave heights

Bayesian statistics offer a novel means of estimating return values of wave heights and hence of establishing design criteria for offshore structures. The Bayesian method has significant advantages over the classical method since it enables all types of uncertainty (physical, parameter, distribution) associated with the design wave prediction to be handled in a consistent manner in the same analysis.The basic principles of the Bayesian method for drawing inferences are outlined step-by-step. It is shown how Bayesian estimators of return values for wave heights are established by taking an expectation over all parameters and contending distributions. When the Bayesian procedure is applied to large data sets, such as wave data sets, computational difficulties could be encountered, making a “remedial” procedure necessary. However, the Bayesian procedure has been used successfully with wave data sets from the northern North Sea. Furthermore, the associated remedial procedure is such that the program can be made suitable for many existing computers, e.g. desk computers.     

Failure Statistics Analysis Based on Bayesian Theory: A Study of FPSO Internal Turret Leakage

The load and corrosion caused by the harsh marine environment lead to the severe degradation of offshore equipment and to their compromised security and reliability. In the quantitative risk analysis, the failure models are difficult to establish through traditional statistical methods. Hence, the calculation of the occurrence probability of small sample events is often met with great uncertainty. In this study, the Bayesian statistical method is implemented to analyze the oil and gas leakages of FPSO internal turret, which is a typical small sample risk but could lead to severe losses.According to the corresponding failure mechanism, two Bayesian statistical models using the Weibull distribution and logarithmic normal distribution as the population distribution are established, and the posterior distribution of the corresponding parameters is calculated. The optimal Bayesian statistical model is determined according to the Bayesian information criterion and Akaike criterion. On the basis of the determined optimal model, the corresponding reliability index is solved to provide basic data for the subsequent risk assessments of FPSO systems.     

Automated threshold selection methods for extreme wave analysis

The study of the extreme values of a variable such as wave height is very important in flood risk assessment and coastal design. Often values above a sufficiently large threshold can be modelled using the Generalized Pareto Distribution, the parameters of which are estimated using maximum likelihood. There are several popular empirical techniques for choosing a suitable threshold, but these require the subjective interpretation of plots by the user.In this paper we present a pragmatic automated, simple and computationally inexpensive threshold selection method based on the distribution of the difference of parameter estimates when the threshold is changed, and apply it to a published rainfall and a new wave height data set. We assess the effect of the uncertainty associated with our threshold selection technique on return level estimation by using the bootstrap procedure. We illustrate the effectiveness of our methodology by a simulation study and compare it with the approach used in the JOINSEA software. In addition, we present an extension that allows the threshold selected to depend on the value of a covariate such as the cosine of wave direction.     

时频分析技术在储层分析中的应用

时频分析是对地震信号进行分解,可以在不同频带上显示不同级别的地质现象。采用连续小渡变换方法,对地震资料进行储层时频分析和分频属性提取,时频主极值频率曲线与伽马曲线有较好对应关系,尤其是其低频趋势线。对A区块周边的两口井井旁道进行了时频分析,揭示出该区15Hz低频数据体与储层及含油气相关性较好。针对这一特征,对该区三维地震体进行了运算及属性提取,描述其储层展布。     

基于现场实测数据的半潜式平台运动响应分析和极值预测研究

浮式平台承受风浪流等多种海洋环境载荷作用,呈现出复杂的运动学响应状态.通过对"南海挑战号"半潜式平台的实测六自由度响应数据进行分析,采用广义极值分布建立六自由度响应的概率密度和分布模型,并通过K-S(Kolmogorov-Smirnov)检验验证了分布模型的合理性,进而开展了对该平台多年一遇重现期的六自由度响应极值预测研究.通过与平台的初始设计指标进行对比,发现平台的横荡、纵荡等五个自由度表现良好,而垂荡的响应极值超出设计指标,在现场作业中应予以注意研究成果对平台的安全作业具有辅助指导意义,可将预测结果作为极端恶劣海况下,人员提前撤离的辅助决策支持.通过更新平台的监测数据进行极值分析和预测研究可评估平台的性能变化行为.     

Blind marine seismic deconvolution using statistical MCMC methods

In order to improve the resolution of seismic images, a blind deconvolution of seismic traces is necessary, since the source wavelet is not known and cannot be considered as a stationary signal. The reflectivity sequence is modeled as a Gaussian mixture, depending on three parameters (high and low reflector variances and reflector density), on the wavelet impulse response, and on the observation noise variance. These parameters are unknown and must be estimated from the recorded trace, which is the reflectivity convolved with the wavelet, plus noise. Two methods are compared in this paper for the parameter estimation. Since we are considering an incomplete data problem, we first consider maximum likelihood estimation by means of a stochastic expectation maximization (SEM) method. Alternatively, proper prior distributions can be specified for all unknown quantities. Then, a Bayesian strategy is applied, based on a Monte Carlo Markov Chain (MCMC) method. Having estimated the parameters, one can proceed to the deconvolution. A maximum posterior mode (MPM) criterion is optimized by means of an MCMC method. The deconvolution capability of these procedures is checked first on synthetic signals and then on the seismic data of the IFREMER ESSR4 campaign, where the wavelet duration blurs the reflectivity, and on the SMAVH high-resolution marine seismic data.     

Quadrature-based approach for the efficient evaluation of surge hazard

The Joint Probability Method (JPM) has been used for hurricane surge frequency analysis for over three decades, and remains the method of choice owing to the limitations of more direct historical methods. However, use of the JPM approach in conjunction with the modern generation of complex high-resolution numerical models (used to describe winds, waves, and surge) has become highly inefficient, owing to the large number of costly storm simulations that are typically required. This paper describes a new approach to the selection of the storm simulation set that permits reduction of the JPM computational effort by about an order of magnitude (compared to a more conventional approach) while maintaining good accuracy. The method uses an integration scheme called Bayesian or Gaussian-process quadrature (together with conventional integration methods) to evaluate the multi-dimensional joint probability integral over the space of storm parameters (pressure, radius, speed, heading, and any others found to be important) as a weighted summation over a relatively small set of optimally selected nodes (synthetic storms). Examples of an application of the method are shown, drawn from the recent post-Katrina study of coastal Mississippi.     

A simulation study on the uncertainty of environmental contours due to sampling variability for different estimation methods

Environmental contours are often used in design of marine structures to identify extreme environmental conditions that may give rise to extreme loads and responses. Recently, attention has been given to the fact that different methods exist for establishing such contours, and that in some cases significant differences may be obtained from the various methods.In this study, another aspect of the uncertainty related to the calculation of environmental contours is addressed, namely the uncertainty due to sampling variability when environmental contours are constructed based on metocean data of finite sample size. The uncertainty of environmental contours for the joint distribution of significant wave height and wave period for different sample sizes (10, 25 and 100 years of data) are investigated considering different underlying datasets and for different estimation methods for the joint distribution. Both cases where samples are drawn from a known joint distribution of wave height and periods and cases where samples are drawn from a real hindcast dataset and fitted to the joint distribution are considered. The uncertainty of the estimated contours is quantified and discussed in light of differences that can be anticipated from the different methods used to calculate the contours. Moreover, the potential bias from assuming different estimation methods is illustrated.     

Prediction and assimilation of surf-zone processes using a Bayesian network: Part II: Inverse models

A Bayesian network model has been developed to simulate a relatively simple problem of wave propagation in the surf zone (detailed in Part I). Here, we demonstrate that this Bayesian model can provide both inverse modeling and data-assimilation solutions for predicting offshore wave heights and depth estimates given limited wave-height and depth information from an onshore location. The inverse method is extended to allow data assimilation using observational inputs that are not compatible with deterministic solutions of the problem. These inputs include sand bar positions (instead of bathymetry) and estimates of the intensity of wave breaking (instead of wave-height observations). Our results indicate that wave breaking information is essential to reduce prediction errors. In many practical situations, this information could be provided from a shore-based observer or from remote-sensing systems. We show that various combinations of the assimilated inputs significantly reduce the uncertainty in the estimates of water depths and wave heights in the model domain. Application of the Bayesian network model to new field data demonstrated significant predictive skill (R² = 0.7) for the inverse estimate of a month-long time series of offshore wave heights. The Bayesian inverse results include uncertainty estimates that were shown to be most accurate when given uncertainty in the inputs (e.g., depth and tuning parameters). Furthermore, the inverse modeling was extended to directly estimate tuning parameters associated with the underlying wave-process model. The inverse estimates of the model parameters not only showed an offshore wave height dependence consistent with results of previous studies but the uncertainty estimates of the tuning parameters also explain previously reported variations in the model parameters.     

Long-term response analysis of FPSO mooring systems

The design of mooring systems for floating production units usually considers extreme environmental conditions as a primary design parameter. However, in the case of FPSO (Floating, Production, Storage and Offloading) units, the worst response for the mooring system may be associated with other sea state conditions due to the fact that its extreme response may be associated with a resonant period instead of an extreme wave height. The best way to deal with this problem is by performing long-term analysis in order to obtain extreme response estimates. This procedure is computationally very demanding, since many short-term environmental conditions, and their associated stochastic nonlinear time domain numerical simulations of the mooring lines, are required to obtain such estimates. A simplified approach for the long-term analysis is the environmental contour-line design approach. In this paper a Monte Carlo-based integration procedure combined with an interpolation scheme to obtain the parameters of the short-term response distribution is employed to hasten the long-term analysis. Numerical simulations are carried out for an FPSO at three different locations considering a North Sea joint probability distribution for the environmental parameters. The long-term analysis results are compared against those obtained using extreme environmental conditions and environmental contour-line methodology. These results represent the characteristic load effect for the design of mooring systems of floating units using the reliability analysis for mooring line. The results show that the long-term results are usually more critical than those obtained with the other approaches and even different mooring lines can be identified as the critical ones.     

Non-Gaussian gas hydrate grade simulation at the Mallik site,Mackenzie Delta,Canada

For the past decades, gas hydrate reservoirs have beneficiated from an increasing attention in the academic and industrial worlds. As a result, there is a growing need to develop specific and comprehensive gas hydrate reservoir characterization methods. This study explores the use of a stochastic Bayesian algorithm to integrate well-logs and 3D acoustic impedance in order to estimate gas hydrate grades (product of saturation and total porosity) over a representative volume of the Mallik gas hydrate field, located in the Mackenzie Delta, Northwest Territories of Canada. First, collocated log data from boreholes Mallik 5L-38 and 2L-38 are used to estimate the statistical relationship between acoustic impedance and gas hydrate grades. Second, conventional stochastic Bayesian simulation is applied to generate multiple gas hydrate grade 3D fields integrating log data and lateral variability of 3D acoustic impedance. These equiprobable scenarios permit to quantify the uncertainty over the estimation, and identify zones where this uncertainty is greater. Contrary to conventional stochastic reservoir modeling workflows, the proposed method allows integrating non Gaussian and non linear distributions. This permits to handle bimodal distributions without using complex stochastic transforms. The results present gas hydrate grade values that are in accordance with well-log data. The relatively low standard deviation calculated at each pixel using all realizations suggests that gas hydrate grades is well explained by acoustic impedance and log data.     

The 44-year Mediterranean HIPOCAS wind database: A useful tool to analyse offshore extreme wind events from a long-term regional perspective

Nowadays, many efforts are leading to use the high potential offshore wind energy resources. A detailed assessment of the offshore wind resources arises as a first-rate requirement. Most of such assessment is based on extreme offshore wind atlas generated mainly from global reanalysis and satellite data. Both sort of data show certain shortcomings related, among others, to coarse spatial resolution and time inhomogeneity issues, respectively. This snag seems to be crucial over areas such as the Mediterranean Basin, which is characterized by a complex land–sea distribution and a significant orography. The HIPOCAS Mediterranean long-term (1958–2001) wind database comes to overcome the aforementioned reanalysis shortcoming and provides a Mediterranean wind data set useful to perform extreme wind analysis. This contribution also deals with a statistical extreme wind analysis over the whole Mediterranean offshore areas. Extreme return periods and levels are obtained from annual maxima using a number of distributions. Additionally, an alternative regional statistical method based on regional L-moment statistics is also proposed. The regional technique is applied to reduce uncertainty and allows a higher number of measurements to be included in the analysis, using data from a homogeneous region instead of from a single location. The herein performed extreme wind analysis provides a detailed assessment of high wind offshore areas over the Mediterranean and constitutes a subject of great interest for evaluation of wind resources.     

海洋重现期波高统一化计算方法

重现期波高是港口海岸及海洋工程设计中不可回避的一个重要设计参数,尤其对深水海港、海上平台、海底油气管道、沿海核电站等重大涉海工程设计具有巨大的经济价值和深远的社会效益。但是,现有重现期波高推算缺乏统一的计算方法,导致计算结果相差悬殊。研究重现期波高的统一化计算方法,分析重现期波高计算中存在的各种不确定因素,提出减少这些不确定因素的新方法,建立误差小、应用方便、方法统一的重现期波高计算方法。基于澳大利亚悉尼站的长期连续观测波浪数据,研究发现:广义帕累托函数(generalized Pareto distribution III,GPD-III)和威布尔(Weibull)是重现期波高计算的最佳候选极值分布函数,新推导的函数形状参数计算公式较好提高重现期波高的计算精度,极值波高数据的分析方法和样本大小是影响重现期波高计算精确度的两个重要因素,短期波浪资料和年极值法可能高估重现期波高值。逐个风暴的极值波高数据分析法及最佳候选极值分布函数GPD-III和Weibull建议应用于涉海工程设计的重现期波高推算。     

Extreme value prediction via a quantile function model

Methods for estimating extreme loads are used in design as well as risk assessment. Regression using maximum likelihood or least squares estimation is widely used in a univariate analysis but these methods favour solutions that fit observations in an average sense. Here we describe a new technique for estimating extremes using a quantile function model. A quantile of a distribution is most commonly termed a ‘return level’ in flood risk analysis. The quantile function of a random variable is the inverse function of its distribution function. Quantile function models are different from the conventional regression models, because a quantile function model estimates the quantiles of a variable conditional on some other variables, while a regression model studies the conditional mean of a variable. So quantile function models allow us to study the whole conditional distribution of a variable via its quantile function, whereas conventional regression models represent the average behaviour of a variable.Little work can be found in the literature about prediction from a quantile function model. This paper proposes a prediction method for quantile function models. We also compare different types of statistical models using sea level observations from Venice. Our study shows that quantile function models can be used to estimate directly the relationships between sea condition variables, and also to predict critical quantiles of a sea condition variable conditional on others. Our results show that the proposed quantile function model and the developed prediction method have the potential to be very useful in practice.     

Prediction of visibility in the Arctic based on dynamic Bayesian network analysis

With the accelerated warming of the world, the safety and use of Arctic passages is receiving more attention.Predicting visibility in the Arctic has been a hot topic in recent years because of navigation risks and opening of ice-free northern passages. Numerical weather prediction and statistical prediction are two methods for predicting visibility. As microphysical parameterization schemes for visibility are so sophisticated, visibility prediction using numerical weather prediction models inclu...     

基于浮标资料的中国东部海域最大波高特征分析

利用原国家海洋局2010—2015年的浮标资料,计算渤、黄、东海有效波高和最大波高的线性关系,并通过1992—2011年共20 a的数值模拟有效波高资料计算中国东部海域各月的2.5 m、4 m、6 m以上最大波高频率和最大波高月极值分布。结果发现:中国东部海域由北至南,最大波高与有效波高的比值逐渐增大;最大波高频率和最大波高月极值空间分布均由渤海、黄海至东海逐渐增大,最大波高频率的极值12月最大,4或5月最小,最大波高月极值9月最大,4月最小。其时空分布表明:受不同天气系统影响,夏秋季台风较多,容易出现极值较大的最大波高;秋冬季冷空气较强,虽然最大波高极值相对较小,但大浪持续时间长、频率大、影响范围广。