Global sensitivity analysis through polynomial chaos expansion of a basin-scale geochemical compaction model

We present a model-driven uncertainty quantification methodology based on sparse grid sampling techniques in the context of a generalized polynomial chaos expansion (GPCE) approximation of a basin-scale geochemical evolution scenario. The approach is illustrated through a one-dimensional example involving the process of quartz cementation in sandstones and the resulting effects on the dynamics of the vertical distribution of porosity, pressure, and temperature. The proposed theoretical framework and computational tools allow performing an efficient and accurate global sensitivity analysis (GSA) of the system states (i.e., porosity, temperature, pressure, and fluxes) in the presence of uncertain key mechanical and geochemical model parameters as well as boundary conditions. GSA is grounded on the use of the variance-based Sobol indices. These allow discriminating the relative weights of uncertain quantities on the global model variance and can be computed through the GPCE of the model response. Evaluation of the GPCE of the model response is performed through the implementation of a sparse grid approximation technique in the space of the selected uncertain quantities. GPCE is then be employed as a surrogate model of the system states to quantify uncertainty propagation through the model in terms of the probability distribution (and its statistical moments) of target system states.     

Combining meta-modeling and categorical indicators for global sensitivity analysis of long-running flow simulators with spatially dependent inputs

Variance-based global sensitivity analysis (GSA) is a powerful procedure for importance ranking of the uncertain input parameters of a given flow model. The application of GSA is made possible for long-running flow simulators (computation (CPU) time more than several hours) by relying on meta-modeling techniques. However, such flow models can involve one or several spatial inputs, for instance, the permeability field of a reservoir and of a caprock formation in the context of CO₂ geological storage. Studying the sensitivity to each of these spatial inputs motivated the present work. In this view, we propose a strategy which combines (1) a categorical indicator (i.e., a pointer variable taking discrete values) assigned to the set of stochastic realizations associated with each spatial input (spatial maps) and (2) meta-modeling techniques, which jointly handle continuous and categorical inputs. In a first application case, a costless-to-evaluate numerical multiphase flow model was used to estimate the sensitivity indices. Comparisons with results obtained using the meta-model showed good agreement using a two-to-three ratio of the number of learning samples to the number of spatial maps. On this basis, the strategy can be recommended for cases where the number of maps remains tractable (i.e., a few hundred), for example, for moderately complex geological settings, or where a set of such maps can be selected in a preliminary stage using ranking procedures. Finally, the strategy was applied to a more complex multiphase flow model (CPU time of a few hours) to analyze the sensitivity of CO₂ saturation and injection-induced pressure build-up to seven homogeneous rock properties and two spatial inputs.     

中国不同气候区河川径流对气候变化的敏感性

利用一个简单的月水量平衡模型,模拟了位于中国不同气候区的21个典型流域的径流量过程,采用假定的气候情景,分析了河川径流量对不同气候变化的敏感性。结果表明,所采用的月水量平衡模型能够较好地模拟不同气候区的月流量过程,21个典型流域的Nash-Sutcliffe模型效率系数大多超过65%,水量平衡误差也均控制在1%以内。黄河以北干旱半干旱地区的典型流域径流量对气温和降水变化的响应敏感,其次为华中、华南半湿润区和湿润区,西部高寒山区径流对气候变化的响应最弱。因此,中国适应气候变化的重点应集中在干旱半干旱地区。     

Conditioning Simulations of Gaussian Random Fields by Ordinary Kriging

Conditioning realizations of stationary Gaussian random fields to a set of data is traditionally based on simple kriging. In practice, this approach may be demanding as it does not account for the uncertainty in the spatial average of the random field. In this paper, an alternative model is presented, in which the Gaussian field is decomposed into a random mean, constant over space but variable over the realizations, and an independent residual. It is shown that, when the prior variance of the random mean is infinitely large (reflecting prior ignorance on the actual spatial average), the realizations of the Gaussian random field are made conditional by substituting ordinary kriging for simple kriging. The proposed approach can be extended to models with random drifts that are polynomials in the spatial coordinates, by using universal or intrinsic kriging for conditioning the realizations, and also to multivariate situations by using cokriging instead of kriging.     

Conditional Latin Hypercube Simulation of (Log)Gaussian Random Fields

In earth and environmental sciences applications, uncertainty analysis regarding the outputs of models whose parameters are spatially varying (or spatially distributed) is often performed in a Monte Carlo framework. In this context, alternative realizations of the spatial distribution of model inputs, typically conditioned to reproduce attribute values at locations where measurements are obtained, are generated via geostatistical simulation using simple random (SR) sampling. The environmental model under consideration is then evaluated using each of these realizations as a plausible input, in order to construct a distribution of plausible model outputs for uncertainty analysis purposes. In hydrogeological investigations, for example, conditional simulations of saturated hydraulic conductivity are used as input to physically-based simulators of flow and transport to evaluate the associated uncertainty in the spatial distribution of solute concentration. Realistic uncertainty analysis via SR sampling, however, requires a large number of simulated attribute realizations for the model inputs in order to yield a representative distribution of model outputs; this often hinders the application of uncertainty analysis due to the computational expense of evaluating complex environmental models. Stratified sampling methods, including variants of Latin hypercube sampling, constitute more efficient sampling aternatives, often resulting in a more representative distribution of model outputs (e.g., solute concentration) with fewer model input realizations (e.g., hydraulic conductivity), thus reducing the computational cost of uncertainty analysis. The application of stratified and Latin hypercube sampling in a geostatistical simulation context, however, is not widespread, and, apart from a few exceptions, has been limited to the unconditional simulation case. This paper proposes methodological modifications for adopting existing methods for stratified sampling (including Latin hypercube sampling), employed to date in an unconditional geostatistical simulation context, for the purpose of efficient conditional simulation of Gaussian random fields. The proposed conditional simulation methods are compared to traditional geostatistical simulation, based on SR sampling, in the context of a hydrogeological flow and transport model via a synthetic case study. The results indicate that stratified sampling methods (including Latin hypercube sampling) are more efficient than SR, overall reproducing to a similar extent statistics of the conductivity (and subsequently concentration) fields, yet with smaller sampling variability. These findings suggest that the proposed efficient conditional sampling methods could contribute to the wider application of uncertainty analysis in spatially distributed environmental models using geostatistical simulation.     

基于MODFLOW参数不确定性的地下水水流数值模拟方法

考虑到模型不确定性引起的地下水数值模拟不确定性对模拟过程的影响,在简要介绍含水层水文地质参数变异性研究进展和地质统计学的基础上,基于常用的确定性地下水流数值模拟软件MODFLOW开发了MODFLOW-Gslib软件,相较于传统的数值模拟方法,将地质统计学与数值模拟结合的方法能够模拟非均质含水层中的参数变异性问题。将MODFLOW-Gslib软件运用于模拟实例中,选择常见的不确定性因素进行模拟,并对其模拟产生的数据进行统计分析,结果表明,软件转化后的参数符合水文地质参数不确定性的相关特征;与原模拟结果进行对比,该软件能够更加真实地刻画含水层参数变异性特征。     

Decision-theoretic sensitivity analysis for reservoir development under uncertainty using multilevel quasi-Monte Carlo methods

At various stages of petroleum reservoir development, we encounter a large degree of geological uncertainty under which a rational decision has to be made. In order to identify which parameter or group of parameters significantly affects the output of a decision model, we investigate decision-theoretic sensitivity analysis and its computational issues in this paper. In particular, we employ the so-called expected value of partial perfect information (EVPPI) as a sensitivity index and apply multilevel Monte Carlo (MLMC) methods to efficient estimation of EVPPI. In a recent paper by Giles and Goda, an antithetic MLMC estimator for EVPPI is proposed and its variance analysis is conducted under some assumptions on a decision model. In this paper, for an improvement on the performance of the MLMC estimator, we incorporate randomized quasi-Monte Carlo methods within the inner sampling, which results in an multilevel quasi-Monte Carlo (MLQMC) estimator. We apply both the antithetic MLMC and MLQMC estimators to a simple waterflooding decision problem under uncertainty on absolute permeability and relative permeability curves. Through numerical experiments, we compare the performances of the MLMC and MLQMC estimators and confirm a significant advantage of the MLQMC estimator.     

基于统计理论方法的水文模型参数敏感性分析

参数敏感性分析是模型不确定性量化的重要环节,有助于有效识别关键参数,减少参数的不确定性影响,进而提高参数优化效率。利用Morris筛选方法定性识别相对重要参数,耦合方差分解的Sobol方法和统计理论的响应曲面模型构建一种新的定量敏感性分析方法——RSMSobol方法。以长江支流沿渡河流域的日降雨径流过程模拟为例,系统分析4种不同目标函数响应条件下新安江模型的参数敏感性。结果表明Morris方法和RSMSobol方法的集成应用极大地提高了全局敏感性分析的效率,Morris定性筛选结果为定量评估减少了模型参数维数,采用代理模型技术的RSMSobol方法减少了模型的计算消耗。     

Measures of Parameter Uncertainty in Geostatistical Estimation and Geostatistical Optimal Design

Studies of site exploration, data assimilation, or geostatistical inversion measure parameter uncertainty in order to assess the optimality of a suggested scheme. This study reviews and discusses measures for parameter uncertainty in spatial estimation. Most measures originate from alphabetic criteria in optimal design and were transferred to geostatistical estimation. Further rather intuitive measures can be found in the geostatistical literature, and some new measures will be suggested in this study. It is shown how these measures relate to the optimality alphabet and to relative entropy. Issues of physical and statistical significance are addressed whenever they arise. Computational feasibility and efficient ways to evaluate the above measures are discussed in this paper, and an illustrative synthetic case study is provided. A major conclusion is that the mean estimation variance and the averaged conditional integral scale are a powerful duo for characterizing conditional parameter uncertainty, with direct correspondence to the well-understood optimality alphabet. This study is based on cokriging generalized to uncertain mean and trends because it is the most general representative of linear spatial estimation within the Bayesian framework. Generalization to kriging and quasi-linear schemes is straightforward. Options for application to non-Gaussian and non-linear problems are discussed.     

Pilot Block Method Methodology to Calibrate Stochastic Permeability Fields to Dynamic Data

In the present paper, a new geostatistical parameterization technique is introduced for solving inverse problems, either in groundwater hydrology or petroleum engineering. The purpose of this is to characterize permeability at the field scale from the available dynamic data, that is, data depending on fluid displacements. Thus, a permeability model is built, which yields numerical flow answers similar to the data collected. This problem is often defined as an objective function to be minimized. We are especially focused on the possibility to locally change the permeability model, so as to further reduce the objective function. This concern is of interest when dealing with 4D-seismic data. The calibration phase consists of selecting sub-domains or pilot blocks and of varying their log-permeability averages. The permeability model is then constrained to these fictitious block-data through simple cokriging. In addition, we estimate the prior probability density function relative to the pilot block values and incorporate this prior information into the objective function. Therefore, variations in block values are governed by the optimizer while accounting for nearby point and block-data. Pilot block based optimizations provide permeability models respecting point-data at their locations, spatial variability models inferred from point-data and dynamic data in a least squares sense. A synthetic example is presented to demonstrate the applicability of the proposed matching methodology.     

Combining Areal and Point Data in Geostatistical Interpolation: Applications to Soil Science and Medical Geography

A common issue in spatial interpolation is the combination of data measured over different spatial supports. For example, information available for mapping disease risk typically includes point data (e.g. patients’ and controls’ residence) and aggregated data (e.g. socio-demographic and economic attributes recorded at the census track level). Similarly, soil measurements at discrete locations in the field are often supplemented with choropleth maps (e.g. soil or geological maps) that model the spatial distribution of soil attributes as the juxtaposition of polygons (areas) with constant values. This paper presents a general formulation of kriging that allows the combination of both point and areal data through the use of area-to-area, area-to-point, and point-to-point covariances in the kriging system. The procedure is illustrated using two data sets: (1) geological map and heavy metal concentrations recorded in the topsoil of the Swiss Jura, and (2) incidence rates of late-stage breast cancer diagnosis per census tract and location of patient residences for three counties in Michigan. In the second case, the kriging system includes an error variance term derived according to the binomial distribution to account for varying degree of reliability of incidence rates depending on the total number of cases recorded in those tracts. Except under the binomial kriging framework, area-and-point (AAP) kriging ensures the coherence of the prediction so that the average of interpolated values within each mapping unit is equal to the original areal datum. The relationships between binomial kriging, Poisson kriging, and indicator kriging are discussed under different scenarios for the population size and spatial support. Sensitivity analysis demonstrates the smaller smoothing and greater prediction accuracy of the new procedure over ordinary and traditional residual kriging based on the assumption that the local mean is constant within each mapping unit.     

Flood hazard assessment in the Kujukuri Plain of Chiba Prefecture,Japan, based on GIS and multicriteria decision analysis

A simple flood hazard assessment based on GIS and multicriteria decision analysis was presented, and the sensitivity analysis was applied to evaluate the uncertainty of input factors. The location chosen for the study is the Kujukuri Plain, Chiba Prefecture, Japan. The model incorporates six factors: river system, elevation, depression area, ratio of impermeable area, detention ponds, and precipitation. A hazard map for the year 2004, as an example, was obtained. The method of analytic hierarchy process was applied to calculate the weighting values of each factor. The hazard map was compared with the actual flood area, and good coincidence was found between them. The relative importance and uncertainty of the six input factors and weights were evaluated by using the global sensitivity analysis, i.e., extended FAST method, and the results showed a robust behavior of the model. The flood hazard assessment method presented here is meaningful for the flood management and environment protection in the area under the similar condition as this study.     

Three-dimensional finite element analysis of spatially variable PVD improved ground

A stochastic approach that investigates the effects of soil spatial variability on stabilisation of soft clay via prefabricated vertical drains (PVDs) is presented and discussed. The approach integrates the local average subdivision of random field theory with the Monte Carlo finite element (FE) technique. A special feature of the current study is the investigation of impact of spatial variability of soil permeability and volume compressibility in the smear zone as compared to that of the undisturbed zone, in conjunction with uncoupled three-dimensional FE analysis. A sensitivity analysis is also performed to identify the random variable that has the major contribution to the uncertainty of the degree of consolidation achieved via PVDs. The results of this study indicate that the spatial variability of soil properties has a significant impact on soil consolidation by PVDs; however, the spatial variability of soil properties in the smear zone has a dominating impact on soil consolidation by PVDs over that of the undisturbed zone. It is also found that soil volume compressibility has insignificant contribution to the degree of consolidation estimated by uncoupled stochastic analysis.     

Integrating Large-Scale Soft Data by Simulated Annealing and Probability Constraints

Interpretation of geophysical data or other indirect measurements provides large-scale soft secondary data for modeling hard primary data variables. Calibration allows such soft data to be expressed as prior probability distributions of nonlinear block averages of the primary variable; poorer quality soft data leads to prior distributions with large variance, better quality soft data leads to prior distributions with low variance. Another important feature of most soft data is that the quality is spatially variable; soft data may be very good in some areas while poorer in other areas. The main aim of this paper is to propose a new method of integrating such soft data, which is large-scale and has locally variable precision. The technique of simulated annealing is used to construct stochastic realizations that reflect the uncertainty in the soft data. This is done by constraining the cumulative probability values of the block average values to follow a specified distribution. These probability values are determined by the local soft prior distribution and a nonlinear average of the small-scale simulated values within the block, which are all known. For each realization to accurately capture the information contained in the soft data distributions, we show that the probability values should be uniformly distributed between 0 and 1. An objective function is then proposed for a simulated annealing based approach to enforce this uniform probability constraint. The theoretical justification of this approach is discussed, implementation details are considered, and an example is presented.     

气候变化下贵州省水资源安全空间评价及敏感性分析

探讨气候变化下水资源安全的时空演变规律,对喀斯特地区水资源安全的保障有着重要意义。文章采用GA-BP神经网络模型,研究了贵州省水资源安全的空间分异特征,并分析其对气候要素变化的敏感性。结果表明:（1）研究区水资源安全有较强的空间异质性。2001－2015年,黔南的水资源安全一直是全省最差的地区,贵阳的水资源安全改善最为明显,变化幅度最小的是安顺;（2）当变动率相同时,年平均降雨量的变动对水资源安全的影响最大,其增加10%时水资源安全指数上升0.95%,单位地表水资源量变动的影响其次,单位地下水资源量变动的影响最小;（3）对年平均降雨量变化最为敏感的地区是遵义、毕节、六盘水和黔西南。研究结果可为贵州省水资源的调控和开发提供参考。      

Exploring job centers by accessibility using fuzzy set approach: the case study of the Columbus MSA

When comparing accessibility, the interpretation of results is complex because of lack of standard or universal norm. This uncertainty issue of the distinction from the lack of standard can be solved using the multi-level approach of fuzzy set: universal, relative, and absolute index. Since a fuzzy set approach deals with the vagueness and indiscernibility of accessibility index, the proposed approach suggests a better solution to classify the index than a crisp set or even a single-level fuzzy set approach. In this study, we evaluate job accessibility of locations in the Columbus MSA in Ohio, USA for 18 worker groups. The uncertain distinction between strong/weak, rich/poor, and higher/lower accessibility is improved by the multi-level approach. Moreover, this study attempts to enhance our understanding of spatial structure of job accessibility disaggregated by occupation type and gender.     

三维构造恢复中原位归属问题的方法技术

针对目前复杂构造三维恢复过程中地质体产生空间位置变化的不确定性,采用三维原位恢复方法,以巴楚断隆S1 构造为例,通过对比S1 构造恢复过程中所选取的标准区域与样本区域在构造变形前后的空间位置变化,分析构造变化过程中S1 构造整体模型的位置变化规律; 并据此将构造复原后的模型整体变化至初始位置,消除S1 构造因去掉石炭系顶面断距带来西南方向的位移,也消除了因去掉石炭系褶皱所带来西南方向的位移。研究表明,运用三维原位恢复技术,能够确保地质模型在构造复原前后空间位置不变,进而保证研究主体在构造恢复过程中空间位置的相对稳定,为分析构造变形带来的影响奠定空间比较基础。     

Review: Current and emerging methods for catchment-scale modelling of recharge and evapotranspiration from shallow groundwater

A review is provided of the current and emerging methods for modelling catchment-scale recharge and evapotranspiration (ET) in shallow groundwater systems. With increasing availability of data, such as remotely sensed reflectance and land-surface temperature data, it is now possible to model groundwater recharge and ET with more physically realistic complexity and greater levels of confidence. The conceptual representation of recharge and ET in groundwater models is critical in areas with shallow groundwater. The depth dependence of recharge and vegetation water-use feedback requires additional calibration to fluxes as well as heads. Explicit definition of gross recharge vs. net recharge, and groundwater ET vs. unsaturated zone ET, in preparing model inputs and reporting model results is necessary to avoid double accounting in the water balance. Methods for modelling recharge and ET include (1) use of simple surface boundary conditions for groundwater flow models, (2) coupling saturated groundwater models with one-dimensional unsaturated-zone models, and (3) more complex fully-coupled surface-unsaturated-saturated conceptualisations. Model emulation provides a means for including complex model behaviours with lower computational effort. A precise ET surface input is essential for accurate model outputs, and the model conceptualisation depends on the spatial and temporal scales under investigation. Using remote sensing information for recharge and ET inputs in model calibration or in model–data fusion is an area for future research development. Improved use of uncertainty analysis to provide probability bounds for groundwater model outputs, understanding model sensitivity and parameter dependence, and guidance for further field-data acquisition are also areas for future research.     

气候自然变异在气候变化对水资源影响 评价中的贡献分析——Ⅰ.基准期的模型与方法

基于气候变化影响的水资源评价对水资源规划和管理具有重要意义,随着全球气候变化影响的加剧,这一研究显得越来越紧迫。在目前的气候变化研究中,很少考虑气候自然波动的影响(气候自然变异),常将所有的变化单独归因于气候变化的影响,这在气候变化的影响评价中可能导致错误的理解与判断。气候自然变异分析由于缺乏超长系列的数据资料而长期被人为避开。针对这一问题,本研究提出模型方法体系,通过历史基准期的长系列模拟来分析气候自然变异的影响。选取常用的1961~1990年水文系列作为基准期,提出一种基于拉丁超立方体抽样技术的季节分段抽样模拟方法,实现对气候自然变异的模拟。应用水文模型TOPMODEL对基准期的径流系列进行模拟,基于不确定性分析GLUE方法对基准期内水文模型参数不确定性进行分析,并探讨了气候自然变异的影响。研究结果表明,在气候变化影响评价中,气候自然变异的影响不可忽略,应在气候变化的影响中加以区分和界定。     

基于线框单元体的三维闭合地质块体构建方法

为能充分利用空间数据模型中的拓扑关系来实现三维闭合块体的构建,引入了表示简单块体轮廓的线框单元体概念来组织三维模型间各要素之间的拓扑关系;并以此为基础给出了以方向边和方向三角形作为基本识别单元进行三维简单形体的自动识别方法。应用实例表明：利用该方法不仅可以准确构建复杂地质块体,还可以准确描述块体内各要素的拓扑关系,再现局部特殊地质现象,如小断层（悬面）等。为三维地质模型在地震正演模拟、射线追踪等分析应用方面提供了基础。