Revisiting Multi-Gaussian Kriging with the Nataf Transformation or the Bayes’ Rule for the Estimation of Spatial Distributions

A multivariate probability transformation between random variables, known as the Nataf transformation, is shown to be the appropriate transformation for multi-Gaussian kriging. It assumes a diagonal Jacobian matrix for the transformation of the random variables between the original space and the Gaussian space. This allows writing the probability transformation between the local conditional probability density function in the original space and the local conditional Gaussian probability density function in the Gaussian space as a ratio equal to the ratio of their respective marginal distributions. Under stationarity, the marginal distribution in the original space is modeled from the data histogram. The stationary marginal standard Gaussian distribution is obtained from the normal scores of the data and the local conditional Gaussian distribution is modeled from the kriging mean and kriging variance of the normal scores of the data. The equality of ratios of distributions has the same form as the Bayes’ rule and the assumption of stationarity of the data histogram can be re-interpreted as the gathering of the prior distribution. Multi-Gaussian kriging can be re-interpreted as an updating of the data histogram by a Gaussian likelihood. The Bayes’ rule allows for an even more general interpretation of spatial estimation in terms of equality for the ratio of the conditional distribution over the marginal distribution in the original data uncertainty space with the same ratio for a model of uncertainty with a distribution that can be modeled using the mean and variance from direct kriging of the original data values. It is based on the principle of conservation of probability ratio and no transformation is required. The local conditional distribution has a variance that is data dependent. When used in sequential simulation mode, it reproduces histogram and variogram of the data, thus providing a new approach for direct simulation in the original value space.     

Sedimentology and genesis of the Cenozoic sediments of northwestern Himalayas (India)

The paper discusses at length the provenance and the environments of sedimentation of the Cenozoic sediments exposed in the northwestern sector of the Himalayas. The results are based on detailed sedimentological (including mineralogical, petrographical and petrochemical) investigations of more than 3,000 representative samples. It is concluded that the detritus of the Cenozoic formations was derived mainly from metamorphosed rocks exposed in the adjacent Himalayan regions. A comparatively smaller proportion of the sediments was contributed by acid plutonic, volcanic and sedimentary rocks. The detritus of the oldest unit of the Cenozoic sequence was deposited in shallow marine waters. The remaining formations represent the accumulations of nonmarine environments fluctuating in space and time.     

Representing Spatial Uncertainty Using Distances and Kernels

Assessing uncertainty of a spatial phenomenon requires the analysis of a large number of parameters which must be processed by a transfer function. To capture the possibly of a wide range of uncertainty in the transfer function response, a large set of geostatistical model realizations needs to be processed. Stochastic spatial simulation can rapidly provide multiple, equally probable realizations. However, since the transfer function is often computationally demanding, only a small number of models can be evaluated in practice, and are usually selected through a ranking procedure. Traditional ranking techniques for selection of probabilistic ranges of response (P10, P50 and P90) are highly dependent on the static property used. In this paper, we propose to parameterize the spatial uncertainty represented by a large set of geostatistical realizations through a distance function measuring “dissimilarity” between any two geostatistical realizations. The distance function allows a mapping of the space of uncertainty. The distance can be tailored to the particular problem. The multi-dimensional space of uncertainty can be modeled using kernel techniques, such as kernel principal component analysis (KPCA) or kernel clustering. These tools allow for the selection of a subset of representative realizations containing similar properties to the larger set. Without losing accuracy, decisions and strategies can then be performed applying a transfer function on the subset without the need to exhaustively evaluate each realization. This method is applied to a synthetic oil reservoir, where spatial uncertainty of channel facies is modeled through multiple realizations generated using a multi-point geostatistical algorithm and several training images.     

http://www.sciencedirect.com/science/article/pii/S1674987115000389

A very fast simulated annealing(VFSA) global optimization is used to interpret residual gravity anomaly.Since,VFSA optimization yields a large number of best-fitted models in a vast model space;the nature of uncertainty in the interpretation is also examined simultaneously in the present study.The results of VFSA optimization reveal that various parameters show a number of equivalent solutions when shape of the target body is not known and shape factor 'q' is also optimized together with other model parameters.The study reveals that amplitude coefficient k is strongly dependent on shape factor.This shows that there is a multi-model type uncertainty between these two model parameters derived from the analysis of cross-plots.However,the appraised values of shape factor from various VFSA runs clearly indicate whether the subsurface structure is sphere,horizontal or vertical cylinder type structure.Accordingly,the exact shape factor(1.5 for sphere,1.0 for horizontal cylinder and 0.5 for vertical cylinder)is fixed and optimization process is repeated.After fixing the shape factor,analysis of uncertainty and cross-plots shows a well-defined uni-model characteristic.The mean model computed after fixing the shape factor gives the utmost consistent results.Inversion of noise-free and noisy synthetic data as well as field data demonstrates the efficacy of the approach.     

A Functional Data Analysis Approach to Surrogate Modeling in Reservoir and Geomechanics Uncertainty Quantification

Uncertainty quantification for geomechanical and reservoir predictions is in general a computationally intensive problem, especially if a direct Monte Carlo approach with large numbers of full-physics simulations is used. A common solution to this problem, well-known for the fluid flow simulations, is the adoption of surrogate modeling approximating the physical behavior with respect to variations in uncertain parameters. The objective of this work is the quantification of such uncertainty both within geomechanical predictions and fluid-flow predictions using a specific surrogate modeling technique, which is based on a functional approach. The methodology realizes an approximation of full-physics simulated outputs that are varying in time and space when uncertainty parameters are changed, particularly important for the prediction of uncertainty in vertical displacement resulting from geomechanical modeling. The developed methodology has been applied both to a subsidence uncertainty quantification example and to a real reservoir forecast risk assessment. The surrogate quality obtained with these applications confirms that the proposed method makes it possible to perform reliable time–space varying dependent risk assessment with a low computational cost, provided the uncertainty space is low-dimensional.     

基岩表面地脉动谱比研究

目前,场地影响评价方法可以分为理论法和经验法。地脉动水平分量与竖向分量谱比法属经验法。这种方法将地脉动水平分量与竖向分量谱比的最大值对应的频率和幅值分别看作是观测场地的卓越频率和放大因子。虽然该法在国内外很多工程实例中得到了理想的结果。但由于这种方法建立过程中应用了许多假设,所以很多地震工程学家对这种方法持怀疑态度。本文讨论了基岩地脉动水平分量与竖向分量谱比为1这一假定的合理性。并采用泊松弹性半空间模型,应用概率分析方法考虑地脉动的随机性,推导了地下体波斜入射弹性半空间自由表面形成地脉动的水平分量与竖向分量谱比表达式。通过计算论证体波斜入射泊松弹性半空间形成的地脉动水平分量与竖向分量谱比特性。结果表明：当地脉动主要成分为体波时,基岩表面地脉动水平分量与竖向分量谱比接近于1;当地脉动主要成分为面波时,基岩表面地脉动水平分量与竖向分量谱比介于0.54～0.79之间。对泊松基岩,该比值是0.68。     

Statistical inverse analysis based on genetic algorithm and principal component analysis: Applications to excavation problems and pressuremeter tests

This study concerns the identification of constitutive models from geotechnical measurements by inverse analysis. Soil parameters are identified from measured horizontal displacements of sheet pile walls and from a measured pressuremeter curve. An optimization method based on a genetic algorithm (GA) and a principal component analysis (PCA), developed and tested on synthetic data in a previous paper, is applied. These applications show that the conclusions deduced from synthetic problems can be extrapolated to real problems. The GA is a robust optimization method that is able to deal with the non‐uniqueness of the solution in identifying a set of solutions for a given uncertainty on the measurements. This set is then characterized by a PCA that gives a first‐order approximation of the solution as an ellipsoid. When the solution set is not too curved in the research space, this ellipsoid characterizes the soil properties considering the measured data and the tolerate margins for the response of the numerical model. Besides, optimizations from different measurements provide solution sets with a common area in the research space. This intersection gives a more relevant and accurate identification of parameters. Finally, we show that these identified parameters permit to reproduce geotechnical measurements not used in the identification process. Copyright © 2009 John Wiley & Sons, Ltd.     

孤南洼陷古近系沙三段中亚段可容空间转换系统研究

传统的层序地层学认为,在盆地演化过程中,海(湖)平面或可容空间的变化是统一的。然而根据对可容空间变化的分析研究发现,由于受到基底的差异沉降、沉积物的沉积以及绝对湖平面的变化三种因素的控制,断陷盆地内可容空间在时间和空间上的变化是非统一性的。根据可容空间的变化划分出的可容空间增加带、减小带以及转换带所组成的体系及其在平面和垂向上的空间展布称之为可容空间转换系统。文中首先从理论模型的角度分析了断陷盆地可容空间转换系统的构成要素及其相互关系,论述了可容空间转换系统的组成和划分方案,并以孤南洼陷沙三中亚段为研究对象,划分了一个三级可容空间转换系统并细分为7个四级可容空间转换系统。最后从石油地质学理论以及可容空间转换带处准层序垂向发育特征两方面阐述了转换带与油气成藏之间的关系,并结合典型油气藏实例解剖,认为可容空间转换带控制了有利的成藏组合的发育位置。     

陆相断陷盆地层序类型及隐蔽圈闭预测

对中国东部古近系和新近系陆相断陷盆地层序地层的研究结果表明,陆相断陷盆地存在着同生断裂坡折带、弯折带2种基本的坡折带类型.当水平面下降到坡折带以下时,构成一个完整的水平面升降旋回,可形成包括低水位、湖扩展、湖萎缩3个体系域的层序(三分体系域);当水平面未下降到坡折带以下时,构成一个不完整的水平面升降旋回,可形成包括湖扩展、湖萎缩2个体系域的层序(二分体系域).这2种层序在纵向上有规律的叠置组合就构成了盆地层序充填序列.水平面上升期是寻找充填作用形成的水下扇(低水位扇)、小型(扇)三角洲(低水位楔)隐蔽圈闭的有利层序地层部位;而水平面下降期则是寻找(扇)三角洲前缘滑塌作用形成的浊积扇隐蔽圈闭的有利层序地层部位.     

Uncertainty of soil erosion modelling using open source high resolution and aggregated DEMs

Digital Elevation Model (DEM) is one of the important parameters for soil erosion assessment. Notable uncertainties are observed in this study while using three high resolution open source DEMs. The Revised Universal Soil Loss Equation (RUSLE) model has been applied to analysis the assessment of soil erosion uncertainty using open source DEMs (SRTM, ASTER and CARTOSAT) and their increasing grid space (pixel size) from the actual. The study area is a part of the Narmada river basin in Madhya Pradesh state, which is located in the central part of India and the area covered 20,558 km². The actual resolution of DEMs is 30 m and their increasing grid spaces are taken as 90, 150, 210, 270 and 330 m for this study. Vertical accuracy of DEMs has been assessed using actual heights of the sample points that have been taken considering planimetric survey based map (toposheet). Elevations of DEMs are converted to the same vertical datum from WGS 84 to MSL (Mean Sea Level), before the accuracy assessment and modelling. Results indicate that the accuracy of the SRTM DEM with the RMSE of 13.31, 14.51, and 18.19 m in 30, 150 and 330 m resolution respectively, is better than the ASTER and the CARTOSAT DEMs. When the grid space of the DEMs increases, the accuracy of the elevation and calculated soil erosion decreases. This study presents a potential uncertainty introduced by open source high resolution DEMs in the accuracy of the soil erosion assessment models. The research provides an analysis of errors in selecting DEMs using the original and increased grid space for soil erosion modelling.     

Vertical Carbon Flux of Biogenic Matter in a Coastal Area of the Aegean Sea: The Importance of Appendicularians

This work focuses on the direct measurement of the vertical flux of appendicularian houses in order to assess their importance as a component of vertical carbon flux in coastal areas. For this purpose, arrays of cylindrical sediment traps were deployed for 5 to 8 days at two depths in a coastal area of the northern Aegean Sea (inner Thermaikos Gulf) during spring. The data support the contention that resuspension was minimal. Fecal pellet (FP) production and grazing experiments with the dominant copepods (Acartia clausi) were conducted to provide additional information on the potential FP contribution to the total carbon flux. The magnitude of the vertical flux of particulate organic carbon (POC) ranged between 310 and 724 mg C m^?2 day^?1. The proportion of phytoplankton carbon in the POC vertical flux was up to 45 %. The contribution of zooplankton FPs to the total carbon never exceeded 5 %. On the contrary, appendicularian houses were an essential component of the biogenic carbon flux contributing up to 55.3 % of the total vertical carbon flux. Consequently, both phytoplankton and appendicularian houses contributed equally to the biogenic carbon flux exceeding 80 % of the total sinking POC. Taking into account the sinking speed of the particles and the environment in the area, all this carbon probably reaches the seafloor, thus indicating a strong pelagic–benthic coupling.     

氧化钕中杂质氧化镍含量的测定及测量结果的不确定度分析

采用电感耦合等离子体发射光谱法(ICP-OES)法测定氧化钕中杂质氧化镍的含量,得到氧化镍含量为0.002 7%,分析了检测过程中的各个影响因素,并对测量结果的可靠性以相对扩展不确定度表示(U_rel=12%,k=2)。相对不确定度分量计算结果表明,标准曲线拟合时引入的相对不确定度分量最大,u_rel(c₂)=5.639%,其次为标准溶液配制过程中引入的相对不确定度分量u_rel(c₁)=0.879%。本文可为相关检测人员的结果不确定度分析提供参考。     

Automatic Histogram and Variogram Reproduction in Simulated Annealing Simulation

Simulated annealing simulation (SAS) is a flexible tool for generating stochastic simulations conditioned to data at various scales and precision. However, a number of important drawbacks exists: (1) SAS requires considerable CPU time, (2) it requires experience in setting the so-called cooling schedule to obtain convergence, and (3) its space of uncertainty is not well understood. In this paper, I propose a new simulated annealing method that guarantees histogram and variogram reproduction through the implementation of a novel perturbation mechanism. The novel perturbation method is based on the Metropolis–Hastings sampler for a Markov-type random field. This new annealing method allows removal of the histogram and variogram from the objective function in the simulated annealing procedure. Furthermore, for the proposed method one finds that (1) the uncertainty space can be quantified, (2) the convergence properties can be linked to convergence of stationary Markov chains, and (3) the convergence speed is improved over traditional simulated annealing.     

Modelling Short‐Scale Variability and Uncertainty During Mineral Resource Estimation Using a Novel Fuzzy Estimation Technique

For mineral resource assessment, techniques based on fuzzy logic are attractive because they are capable of incorporating uncertainty associated with measured variables and can also quantify the uncertainty of the estimated grade, tonnage etc. The fuzzy grade estimation model is independent of the distribution of data, avoiding assumptions and constraints made during advanced geostatistical simulation, e.g., the turning bands method. Initially, fuzzy modelling classifies the data using all the component variables in the data set. We adopt a novel approach by taking into account the spatial irregularity of mineralisation patterns using the Gustafson–Kessel classification algorithm. The uncertainty at the point of estimation was derived through antecedent memberships in the input space (i.e., spatial coordinates) and transformed onto the output space (i.e., grades) through consequent membership at the point of estimation. Rather than probabilistic confidence intervals, this uncertainty was expressed in terms of fuzzy memberships, which indicated the occurrence of mixtures of different mineralogical phases at the point of estimation. Data from different sources (other than grades) could also be utilised during estimation. Application of the proposed technique on a real data set gave results that were comparable to those obtained from a turning bands simulation.     

床面附近泥沙运动的分析

基于水、沙两相的分相测量试验结果,分析了床面附近泥沙颗粒的脉动和力学特性,指出床面附近的泥沙运动有着特殊的力学机制:颗粒相具有较强的非湍流脉动,其产生的脉动应力对颗粒的运动起着重要作用。论述了沙粒在水流中从推移运动到扬起悬浮的物理过程,讨论了过去一些理论中存在的问题和不足,概括了泥沙颗粒在水流中从床面扬起的基本模式,运用两相流理论分析了沙粒在水流中扬起的动力学机理。根据颗粒运动的垂向动量平衡原理,对泥沙颗粒的垂向浓度分布规律作了新的分析解释。证实了除浓度梯度之外,颗粒相的垂向脉动强度梯度也是泥沙扩散的重要扩散势,进一步揭示了悬移质浓度垂线分布存在两种类型的内在机理。     

关于频率电磁测深几个问题的探讨(六)——频率电磁测深的电磁场分布与观测参量

分析了水平电偶源与垂直磁偶源的电磁场在半空间大地中的分布特点,指出观测各场分量的合适地面部位。为了定性解释,一般将场分量计算为单分量视电阻率或比值视电阻率;做定量解释时,可直接采用场强进行反演,以减小换算误差。基于当前电偶源与磁偶源的发射功率和电磁场衰减规律,在实际应用中,水平电偶源可探测更大深度(<3 000 m),垂直磁偶源探测浅部(<500 m)。      

Poroelastic model for pile–soil interaction in a half‐space porous medium due to seismic waves

In this paper, frequency domain dynamic response of a pile embedded in a half‐space porous medium and subjected to P, SV seismic waves is investigated. According to the fictitious pile methodology, the problem is decomposed into an extended poroelastic half‐space and a fictitious pile. The extended porous half‐space is described by Biot's theory, while the fictitious pile is treated as a bar and a beam and described by the conventional 1‐D structure vibration theory. Using the Hankel transformation method, the fundamental solutions for a half‐space porous medium subjected to a vertical or a horizontal circular patch load are established. Based on the obtained fundamental solutions and free wave fields, the second kind of Fredholm integral equations describing the vertical and the horizontal interaction between the pile and the poroelastic half‐space are established. Solution of the integral equations yields the dynamic response of the pile to plane P, SV waves. Numerical results show the parameters of the porous medium, the pile and incident waves have direct influences on the dynamic response of the pile–half‐space system. Significant differences between conventional single‐phase elastic model and the poroelastic model for the surrounding medium of the pile are found. Copyright © 2007 John Wiley & Sons, Ltd.     

Marine palaeoseismology from very high resolution seismic imaging: the Gondola Fault Zone (Adriatic foreland)

We present a marine palaeoseismology analysis of a dense network of very high resolution seismic profiles along the Gondola Fault Zone (GFZ), a right-lateral, E–W-striking, active fault system in the Adriatic foreland. This case-study aims to show how time and space variations in the activity of a dominantly right-lateral fault system can be assessed using the vertical component of slip. The GFZ has been investigated for a length of 50 km. It includes two parallel subvertical fault sets and two main anticlines. The late Middle Pleistocene to Holocene vertical component of displacement along the fault is bell-shaped, suggesting that in the long-term the fault zone acts as a single, kinematically coherent structure. Slip rates are 0–0.18 mm a⁻¹ and vary temporally on individual segments. This variability is consistent with a model in which individual fault segments rupture independently during earthquakes with magnitudes up to 6.4 and 1.3–1.8 ka recurrence intervals.     

高速公路工程中粉喷桩复合地基最大侧向变形研究

分析了粉喷桩复合地基路堤填筑过程中最大侧向变形发展的特性,确定了坡脚处最大侧向变形与路中沉降呈双直线的关系,坡脚处最大侧向变形在大部分填筑过程中与路中沉降呈 0.115的关系。对比分析了粉喷桩复合地基和天然地基侧向变形发展规律的差异,并基于复合地基理论分析,认为导致它们差异变化的原因是粉喷桩复合地基桩-土应力分担比的变化。     

Reliability-based semi-analytical solution for ground improvement by PVDs incorporating inherent (spatial) variability of soil

The design of soil consolidation via prefabricated vertical drains (PVDs) has been traditionally carried out deterministically and thus can be misleading due to the ignorance of the uncertainty associated with the inherent (spatial) variation of soil properties. To treat such uncertainty in the design process of soil consolidation by PVDs, stochastic approaches that combine the finite element method with the Monte Carlo technique (FEMC) have been usually used. However, such approaches are complex, computationally intensive and time consuming. In this paper, a simpler reliability-based semi-analytical (RBSA) method is proposed as an alternative tool to the complex FEMC approach for soil consolidation by PVDs, considering soil spatial variability. The RBSA method is found to give similar results to those obtained from the FEMC approach and can thus be used with confidence in practice.