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1.
Spatially distributed and varying natural phenomena encountered in geoscience and engineering problem solving are typically
incompatible with Gaussian models, exhibiting nonlinear spatial patterns and complex, multiple-point connectivity of extreme
values. Stochastic simulation of such phenomena is historically founded on second-order spatial statistical approaches, which
are limited in their capacity to model complex spatial uncertainty. The newer multiple-point (MP) simulation framework addresses
past limits by establishing the concept of a training image, and, arguably, has its own drawbacks. An alternative to current
MP approaches is founded upon new high-order measures of spatial complexity, termed “high-order spatial cumulants.” These
are combinations of moments of statistical parameters that characterize non-Gaussian random fields and can describe complex
spatial information. Stochastic simulation of complex spatial processes is developed based on high-order spatial cumulants
in the high-dimensional space of Legendre polynomials. Starting with discrete Legendre polynomials, a set of discrete orthogonal
cumulants is introduced as a tool to characterize spatial shapes. Weighted orthonormal Legendre polynomials define the so-called
Legendre cumulants that are high-order conditional spatial cumulants inferred from training images and are combined with available
sparse data sets. Advantages of the high-order sequential simulation approach developed herein include the absence of any
distribution-related assumptions and pre- or post-processing steps. The method is shown to generate realizations of complex
spatial patterns, reproduce bimodal data distributions, data variograms, and high-order spatial cumulants of the data. In
addition, it is shown that the available hard data dominate the simulation process and have a definitive effect on the simulated
realizations, whereas the training images are only used to fill in high-order relations that cannot be inferred from data.
Compared to the MP framework, the proposed approach is data-driven and consistently reconstructs the lower-order spatial complexity
in the data used, in addition to high order. 相似文献
2.
3.
Joint geostatistical simulation techniques are used to quantify uncertainty for spatially correlated attributes, including mineral deposits, petroleum reservoirs, hydrogeological horizons, environmental contaminants. Existing joint simulation methods consider only second-order spatial statistics and Gaussian processes. Motivated by the presence of relatively large datasets for multiple correlated variables that typically are available from mineral deposits and the effects of complex spatial connectivity between grades on the subsequent use of simulated realizations, this paper presents a new approach for the joint high-order simulation of spatially correlated random fields. First, a vector random function is orthogonalized with a new decorrelation algorithm into independent factors using the so-termed diagonal domination condition of high-order cumulants. Each of the factors is then simulated independently using a high-order univariate simulation method on the basis of high-order spatial cumulants and Legendre polynomials. Finally, attributes of interest are reconstructed through the back-transformation of the simulated factors. In contrast to state-of-the-art methods, the decorrelation step of the proposed approach not only considers the covariance matrix, but also high-order statistics to obtain independent non-Gaussian factors. The intricacies of the application of the proposed method are shown with a dataset from a multi-element iron ore deposit. The application shows the reproduction of high-order spatial statistics of available data by the jointly simulated attributes. 相似文献
4.
Mathematical Geosciences - A training image free, high-order sequential simulation method is proposed herein, which is based on the efficient inference of high-order spatial statistics from the... 相似文献
5.
Validation Techniques for Geological Patterns Simulations Based on Variogram and Multiple-Point Statistics 总被引：2，自引：2，他引：0
Traditional simulation methods that are based on some form of kriging are not sensitive to the presence of strings of connectivity
of low or high values. They are particularly inappropriate in many earth sciences applications, where the geological structures
to be simulated are curvilinear. In such cases, techniques allowing the reproduction of multiple-point statistics are required.
The aim of this paper is to point out the advantages of integrating such multiple-statistics in a model in order to allow
shape reproduction, as well as heterogeneity structures, of complex geological patterns to emerge. A comparison between a
traditional variogram-based simulation algorithm, such as the sequential indicator simulation, and a multiple-point statistics
algorithm (e.g., the single normal equation simulation) is presented. In particular, it is shown that the spatial distribution
of limestone with meandering channels in Lecce, Italy is better reproduced by using the latter algorithm. The strengths of
this study are, first, the use of a training image that is not a fluvial system and, more importantly, the quantitative comparison
between the two algorithms. The paper focuses on different metrics that facilitate the comparison of the methods used for
limestone spatial distribution simulation: both objective measures of similarity of facies realizations and high-order spatial
cumulants based on different third- and fourth-order spatial templates are considered. 相似文献
6.
Characterization of complex geological features and patterns remains one of the most challenging tasks in geostatistics. Multiple point statistics (MPS) simulation offers an alternative to accomplish this aim by going beyond classical two-point statistics. Reproduction of features in the final realizations is achieved by borrowing high-order spatial statistics from a training image. Most MPS algorithms use one training image at a time chosen by the geomodeler. This paper proposes the use of multiple training images simultaneously for spatial modeling through a scheme of data integration for conditional probabilities known as a linear opinion pool. The training images (TIs) are based on the available information and not on conceptual geological models; one image comes from modeling the categories by a deterministic approach and another comes from the application of conventional sequential indicator simulation. The first is too continuous and the second too random. The mixing of TIs requires weights for each of them. A methodology for calibrating the weights based on the available drillholes is proposed. A measure of multipoint entropy along the drillholes is matched by the combination of the two TIs. The proposed methodology reproduces geologic features from both TIs with the correct amount of continuity and variability. There is no need for a conceptual training image from another modeling technique; the data-driven TIs permit a robust inference of spatial structure from reasonably spaced drillhole data. 相似文献
7.
Conditional Simulation with Patterns 总被引：17，自引：0，他引：17
An entirely new approach to stochastic simulation is proposed through the direct simulation of patterns. Unlike pixel-based
(single grid cells) or object-based stochastic simulation, pattern-based simulation simulates by pasting patterns directly
onto the simulation grid. A pattern is a multi-pixel configuration identifying a meaningful entity (a puzzle piece) of the
underlying spatial continuity. The methodology relies on the use of a training image from which the pattern set (database)
is extracted. The use of training images is not new. The concept of a training image is extensively used in simulating Markov
random fields or for sequentially simulating structures using multiple-point statistics. Both these approaches rely on extracting
statistics from the training image, then reproducing these statistics in multiple stochastic realizations, at the same time
conditioning to any available data. The proposed approach does not rely, explicitly, on either a statistical or probabilistic
methodology. Instead, a sequential simulation method is proposed that borrows heavily from the pattern recognition literature
and simulates by pasting at each visited location along a random path a pattern that is compatible with the available local
data and any previously simulated patterns. This paper discusses the various implementation details to accomplish this idea.
Several 2D illustrative as well as realistic and complex 3D examples are presented to showcase the versatility of the proposed
algorithm. 相似文献
8.
Luis Manuel de Vries Jesus Carrera Oriol Falivene Oscar Gratacós Luit Jan Slooten 《Mathematical Geosciences》2009,41(1):29-42
Simulation of flow and solute transport through aquifers or oil reservoirs requires a precise representation of subsurface
heterogeneity that can be achieved by stochastic simulation approaches. Traditional geostatistical methods based on variograms,
such as truncated Gaussian simulation or sequential indicator simulation, may fail to generate the complex, curvilinear, continuous
and interconnected facies distributions that are often encountered in real geological media, due to their reliance on two-point
statistics. Multiple Point Geostatistics (MPG) overcomes this constraint by using more complex point configurations whose
statistics are retrieved from training images. Obtaining representative statistics requires stationary training images, but
geological understanding often suggests a priori facies variability patterns. This research aims at extending MPG to non-stationary
facies distributions. The proposed method subdivides the training images into different areas. The statistics for each area
are stored in separate frequency search trees. Several training images are used to ensure that the obtained statistics are
representative. The facies probability distribution for each cell during simulation is calculated by weighting the probabilities
from the frequency trees. The method is tested on two different object-based training image sets. Results show that non-stationary
training images can be used to generate suitable non-stationary facies distributions. 相似文献
9.
Multiple-point statistics (MPS) provides a flexible grid-based approach for simulating complex geologic patterns that contain high-order statistical information represented by a conceptual prior geologic model known as a training image (TI). While MPS is quite powerful for describing complex geologic facies connectivity, conditioning the simulation results on flow measurements that have a nonlinear and complex relation with the facies distribution is quite challenging. Here, an adaptive flow-conditioning method is proposed that uses a flow-data feedback mechanism to simulate facies models from a prior TI. The adaptive conditioning is implemented as a stochastic optimization algorithm that involves an initial exploration stage to find the promising regions of the search space, followed by a more focused search of the identified regions in the second stage. To guide the search strategy, a facies probability map that summarizes the common features of the accepted models in previous iterations is constructed to provide conditioning information about facies occurrence in each grid block. The constructed facies probability map is then incorporated as soft data into the single normal equation simulation (snesim) algorithm to generate a new candidate solution for the next iteration. As the optimization iterations progress, the initial facies probability map is gradually updated using the most recently accepted iterate. This conditioning process can be interpreted as a stochastic optimization algorithm with memory where the new models are proposed based on the history of the successful past iterations. The application of this adaptive conditioning approach is extended to the case where multiple training images are proposed as alternative geologic scenarios. The advantages and limitations of the proposed adaptive conditioning scheme are discussed and numerical experiments from fluvial channel formations are used to compare its performance with non-adaptive conditioning techniques. 相似文献
10.
在有限数据条件下,可靠度敏感性分析是研究各种不确定性因素对边坡失稳概率影响规律的重要途径。基于直接蒙特卡洛模拟和概率密度加权分析方法提出了一种高效边坡稳定可靠度敏感性分析方法。所提出的方法通过随机场表征岩土体参数的空间变异性,并采用局部平均理论建立岩土体参数的缩维概率密度函数,用于概率密度加权分析中高效、准确地计算不同敏感性分析方案对应的边坡失稳概率。最后,通过一个工程案例--詹姆斯湾堤坝说明了所提出方法的有效性和准确性。结果表明：在敏感性分析过程中,所提出的方法只需要执行一次直接蒙特卡洛模拟,避免了针对不同敏感性分析方案重新产生随机样本和执行边坡稳定分析,节约了大量的计算时间和计算资源,显著提高了基于蒙特卡洛模拟的敏感性分析计算效率;在概率密度加权分析中采用岩土体参数的缩维概率密度函数能够准确地计算边坡失稳概率,避免了有偏估计,使概率密度加权分析方法适用于考虑空间变异性条件下的边坡稳定可靠度敏感性分析问题。 相似文献
11.
Geostatistical Seismic Inversion with Direct Sequential Simulation and Co-simulation with Multi-local Distribution Functions 总被引：1，自引：0，他引：1
Rúben Nunes Amílcar Soares Leonardo Azevedo Pedro Pereira 《Mathematical Geosciences》2017,49(5):583-601
Stochastic sequential simulation is a common modelling technique used in Earth sciences and an integral part of iterative geostatistical seismic inversion methodologies. Traditional stochastic sequential simulation techniques based on bi-point statistics assume, for the entire study area, stationarity of the spatial continuity pattern and a single probability distribution function, as revealed by a single variogram model and inferred from the available experimental data, respectively. In this paper, the traditional direct sequential simulation algorithm is extended to handle non-stationary natural phenomena. The proposed stochastic sequential simulation algorithm can take into consideration multiple regionalized spatial continuity patterns and probability distribution functions, depending on the spatial location of the grid node to be simulated. This work shows the application and discusses the benefits of the proposed stochastic sequential simulation as part of an iterative geostatistical seismic inversion methodology in two distinct geological environments in which non-stationarity behaviour can be assessed by the simultaneous interpretation of the available well-log and seismic reflection data. The results show that the elastic models generated by the proposed stochastic sequential simulation are able to reproduce simultaneously the regional and global variogram models and target distribution functions relative to the average volume of each sub-region. When used as part of a geostatistical seismic inversion procedure, the retrieved inverse models are more geologically realistic, since they incorporate the knowledge of the subsurface geology as provided, for example, by seismic and well-log data interpretation. 相似文献
12.
E. H. Timothy Whitten 《Mathematical Geology》1970,2(2):141-152
The advantages of using orthogonal rather than nonorthogonal polynomials for trend-surface analysis are discussed briefly.
A method for calculating orthogonal polynomial trend surfaces of any degree on the basis of irregularly spaced data is described.
The method is illustrated with subsurface data for the elevation on top of the Devonian Dundee Limestone, Michigan. 相似文献
13.
为了分析将子域解析元素法应用于煤矿地下水流场模拟的可行性,并探究如何提高此方法的模拟精度,首先推导出了强度非线性变化的高阶线汇的复势表达式,分析了其流量势与流函数的空间分布特征,在此基础上应用python语言构建了基于子域解析元素法的煤矿地下水流场模型并应用于求解某煤矿放水试验后水位分布问题.模拟结果显示,模拟水位与观测孔水位偏差绝对值范围为1.36~5.27 m,模型外边界（实际定水头边界）上的水位接近实际值（900 m）,且通过模型外边界（实际隔水边界）的流量近似为零.对模拟原理及模拟结果的分析表明,基于子域解析元素法的煤矿地下水流场模型在全域上满足质量守恒及达西流梯度场,在全域内任意一点的水位可通过该点所处的子域所对应的流量势函数求得,因此应用子域解析元素法进行煤矿地下水流场模拟是可行的,而且将代表模型边界的非线性强度线汇剖分为更短的长度可进一步提高模拟精度. 相似文献
14.
15.
In order to determine to what extent a spatial random field can be characterized by its low-order distributions, we consider
four models (specifically, random spatial tessellations) with exactly the same univariate and bivariate distributions and
we compare the statistics associated with various multiple-point configurations and the responses to specific transfer functions.
The three- and four-point statistics are found to be the same or experimentally hardly distinguishable because of ergodic
fluctuations, whereas change of support and flow simulation produce very different outcomes. This example indicates that low-order
distributions may not discriminate between contending random field models, that simulation algorithms based on such distributions
may not reproduce the spatial properties of a given model or training image, and that the inference of high-order distribution
may require very large training images. 相似文献
16.
Digital simulation of multivariate two- and three-dimensional stochastic processes with a spectral turning bands method 总被引：3，自引：0，他引：3
Aristotelis Mantoglou 《Mathematical Geology》1987,19(2):129-149
The space domain version of the turning bands method can simulate multidimensional stochastic processes (random fields) having particular forms of covariance functions. To alleviate this limitation a spectral representation of the turning bands method in the two-dimensional case has shown that the spectral approach allows simulation of isotropic two-dimensional processes having any covariance or spectral density function. The present paper extends the spectral turning bands method (STBM) even further for simulation of much more general classes of multidimensional stochastic processes. Particular extensions include: (i) simulation of three-dimensional processes using STBM, (ii) simulation of anisotropic two- or three-dimensional stochastic processes, (iii) simulation of multivariate stochastic processes, and (iv) simulation of spatial averaged (integrated) processes. The turning bands method transforms the multidimensional simulation problem into a sum of a series of one-dimensional simulations. Explicit and simple expressions relating the cross-spectral density functions of the one-dimensional processes to the cross-spectral density function of the multidimensional process are derived. Using such expressions the one-dimensional processes can be simulated using a simple one-dimensional spectral method. Examples illustrating that the spectral turning bands method preserves the theoretical statistics are presented. The spectral turning bands method is inexpensive in terms of computer time compared to other multidimensional simulation methods. In fact, the cost of the turning bands method grows as the square root or the cubic root of the number of points simulated in the discretized random field, in the two- or three-dimensional case, respectively, whereas the cost of other multidimensional methods grows linearly with the number of simulated points. The spectral turning bands method currently is being used in hydrologic applications. This method is also applicable to other fields where multidimensional simulations are needed, e.g., mining, oil reservoir modeling, geophysics, remote sensing, etc. 相似文献
17.
Kernel Principal Component Analysis for Efficient,Differentiable Parameterization of Multipoint Geostatistics 总被引：6，自引：5，他引：1
This paper describes a novel approach for creating an efficient, general, and differentiable parameterization of large-scale
non-Gaussian, non-stationary random fields (represented by multipoint geostatistics) that is capable of reproducing complex
geological structures such as channels. Such parameterizations are appropriate for use with gradient-based algorithms applied
to, for example, history-matching or uncertainty propagation. It is known that the standard Karhunen–Loeve (K–L) expansion,
also called linear principal component analysis or PCA, can be used as a differentiable parameterization of input random fields
defining the geological model. The standard K–L model is, however, limited in two respects. It requires an eigen-decomposition
of the covariance matrix of the random field, which is prohibitively expensive for large models. In addition, it preserves
only the two-point statistics of a random field, which is insufficient for reproducing complex structures.
In this work, kernel PCA is applied to address the limitations associated with the standard K–L expansion. Although widely
used in machine learning applications, it does not appear to have found any application for geological model parameterization.
With kernel PCA, an eigen-decomposition of a small matrix called the kernel matrix is performed instead of the full covariance
matrix. The method is much more efficient than the standard K–L procedure. Through use of higher order polynomial kernels,
which implicitly define a high-dimensionality feature space, kernel PCA further enables the preservation of high-order statistics
of the random field, instead of just two-point statistics as in the K–L method. The kernel PCA eigen-decomposition proceeds
using a set of realizations created by geostatistical simulation (honoring two-point or multipoint statistics) rather than
the analytical covariance function. We demonstrate that kernel PCA is capable of generating differentiable parameterizations
that reproduce the essential features of complex geological structures represented by multipoint geostatistics. The kernel
PCA representation is then applied to history match a water flooding problem. This example demonstrates that kernel PCA can
be used with gradient-based history matching to provide models that match production history while maintaining multipoint
geostatistics consistent with the underlying training image. 相似文献
18.
A hierarchical finite element is presented for the geometrically nonlinear free and forced vibration of a non-uniform Timoshenko beam resting on a two-parameter foundation. Legendre orthogonal polynomials are used as enriching shape functions to avoid the shear-locking problem. With the enriching degrees of freedom, the accuracy of the computed results and the computational efficiency are greatly improved. The arc-length iterative method is used to solve the nonlinear eigenvalue equation. The computed results of linear and nonlinear vibration analyses show that the convergence of the proposed element is very fast with respect to the number of Legendre orthogonal polynomials used. Since the elastic foundation and the axial load applied at both ends of the beam affect the ratios of linear frequencies associated with the internal resonance, they influence the nonlinear vibration characteristics of the beam. The axial tensile stress of the beam in nonlinear vibration is investigated in this paper, and attention should be paid to the geometrically nonlinear vibration resulting in considerably large axial tensile stress in the beam. 相似文献
19.
Assessing the risk of soil vulnerability to wind erosion through conditional simulation of soil water content in Sistan plain, Iran 总被引：1，自引：0，他引：1
Surface soil water content (SWC) is one of the key factors controlling wind erosion in Sistan plain, southeast of Iran. Knowledge of the spatial variability of surface SWC is then important to identify high-risk areas over the region. Sequential Gaussian simulation (SGSIM) is used to produce a series of equiprobable models of SWC spatial distribution across the study area. The simulated realizations are used to model the uncertainty attached to the surface SWC estimates through producing a probability map of not exceeding a specified critical threshold when soil becomes vulnerable to wind erosion. The results show that SGSIM is a suitable approach for modelling SWC uncertainty, generating realistic representations of the spatial distribution of SWC that honour the sample data and reproduce the sample semivariogram model. The uncertainty model obtained using SGSIM is compared with the model achieved through sequential indicator simulation (SISIM). According to accuracy plots, goodness statistics and probability interval width plots, SGSIM performs better for modelling local uncertainty than SISIM. Sequential simulation provided a probabilistic approach to assess the risk that SWC does not exceed a critical threshold that might cause soil vulnerability to wind erosion. The resulted risk map can be used in decision-making to delineate “vulnerable” areas where a treatment is needed. 相似文献
20.
Methods reported in the literature for rock fracture simulations include approaches based on stochastic geometry, multiple-point
statistics and a combination of geostatistics for fracture density and object-based modelling for fracture geometries. The
advantages and disadvantages of each of these approaches are discussed with examples. By way of review, the authors begin
with the geostatistical indicator simulation method, based on the truncated–Gaussian algorithm; this is followed by multiple-point
statistical simulation and then the stochastic geometry approach, which is based on marked point process simulation. A new
approach, based on pluriGaussian structural simulation, is then introduced. The new approach incorporates in the simulation
the spatial correlation between different sets of fractures, which in general, is very difficult, if not impossible, to accomplish
in the three methods reviewed. Each simulation method is summarised together with detailed simulation procedures for each.
A published two-dimensional fracture dataset is used as a means of assessing the performance of each simulation method and
of demonstrating the concepts discussed in the text. 相似文献