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1.
Ilnur Minniakhmetov Roussos Dimitrakopoulos Marcelo Godoy 《Mathematical Geosciences》2018,50(7):753-780
High-order sequential simulation techniques for complex non-Gaussian spatially distributed variables have been developed over the last few years. The high-order simulation approach does not require any transformation of initial data and makes no assumptions about any probability distribution function, while it introduces complex spatial relations to the simulated realizations via high-order spatial statistics. This paper presents a new extension where a conditional probability density function (cpdf) is approximated using Legendre-like orthogonal splines. The coefficients of spline approximation are estimated using high-order spatial statistics inferred from the available sample data, additionally complemented by a training image. The advantages of using orthogonal splines with respect to the previously used Legendre polynomials include their ability to better approximate a multidimensional probability density function, reproduce the high-order spatial statistics, and provide a generalization of high-order simulations using Legendre polynomials. The performance of the new method is first tested with a completely known image and compared to both the high-order simulation approach using Legendre polynomials and the conventional sequential Gaussian simulation method. Then, an application in a gold deposit demonstrates the advantages of the proposed method in terms of the reproduction of histograms, variograms, and high-order spatial statistics, including connectivity measures. The C++ course code of the high-order simulation implementation presented herein, along with an example demonstrating its utilization, are provided online as supplementary material. 相似文献
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Mineral deposits frequently contain several elements of interest that are spatially correlated and require the use of joint geostatistical simulation techniques in order to generate models preserving their spatial relationships. Although joint-simulation methods have long been available, they are impractical when it comes to more than three variables and mid to large size deposits. This paper presents the application of block-support simulation of a multi-element mineral deposit using minimum/maximum autocorrelation factors to facilitate the computationally efficient joint simulation of large, multivariable deposits. The algorithm utilized, termed dbmafsim, transforms point-scale spatial attributes of a mineral deposit into uncorrelated service variables leading to the generation of simulated realizations of block-scale models of the attributes of interest of a deposit. The dbmafsim algorithm is utilized at the Yandi iron ore deposit in Western Australia to simulate five cross-correlated elements, namely Fe, SiO2, Al2O3, P and LOI, that are all critical in defining the quality of iron ore being produced. The block-scale simulations reproduce the direct- and cross-variograms of the elements even though only the direct variograms of the service variables have to be modeled. The application shows the efficiency, excellent performance and practical contribution of the dbmafsim algorithm in simulating large multi-element deposits. 相似文献
4.
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
5.
Numerical representations of multivariate natural phenomena, including characteristics of mineral deposits, petroleum reservoirs
and geo-environmental attributes, need to consider and reproduce the spatial relationships between correlated attributes of
interest. There are, however, only a few methods that can practically jointly simulate large size multivariate fields. This
paper presents a method for the conditional simulation of a non-Gaussian vector random field directly on block support. The
method is derived from the group sequential simulation paradigm and the direct block simulation algorithm which leads to the
efficient joint simulation of large multivariate datasets jointly and directly on the block support. This method is a multistage
process. First, a vector random function is orthogonalized with minimum/maximum autocorrelation factors (MAF). Blocks are
then simulated by performing LU simulation on their discretized points, which are later back-rotated and averaged to yield
the block value. The internal points are then discarded and only the block value is stored in memory to be used for further
conditioning through a joint LU, resulting in the reduction of memory requirements. The method is termed direct block simulation
with MAF or DBMAFSIM. A proof of the concept using an exhaustive data set demonstrates the intricacies and the performance
of the proposed approach. 相似文献
6.
High-order Statistics of Spatial Random Fields: Exploring Spatial Cumulants for Modeling Complex Non-Gaussian and Non-linear Phenomena 总被引：9，自引：7，他引：2
The spatial distributions of earth science and engineering phenomena under study are currently predicted from finite measurements
and second-order geostatistical models. The latter models can be limiting, as geological systems are highly complex, non-Gaussian,
and exhibit non-linear patterns of spatial connectivity. Non-linear and non-Gaussian high-order geostatistics based on spatial
connectivity measures, namely spatial cumulants, are proposed as a new alternative modeling framework for spatial data. This
framework has two parts. The first part is the definition, properties, and inference of spatial cumulants—including understanding
the interrelation of cumulant characteristics with the in-situ behavior of geological entities or processes, as examined in
this paper. The second part is the research on a random field model for simulation based on its high-order spatial cumulants.
Mathematical definitions of non-Gaussian spatial random functions and their high-order spatial statistics are presented herein,
stressing the notion of spatial cumulants. The calculation of spatial cumulants with spatial templates follows, including
anisotropic experimental cumulants. Several examples of two- and three-dimensional images, including a diamond bearing kimberlite
pipe from the Ekati Mine in Canada, are analyzed to assess the relations between cumulants and the spatial behavior of geological
processes. Spatial cumulants of orders three to five are shown to capture directional multiple-point periodicity, connectivity
including connectivity of extreme values, and spatial architecture. In addition, they provide substantial information on geometric
characteristics and anisotropy of geological patterns. It is further shown that effects of complex spatial patterns are seen
even if only subsets of all cumulant templates are computed. Compared to second-order statistics, cumulant maps are found
to include a wealth of additional information from underlying geological patterns. Further work seeks to integrate this information
in the predictive capabilities of a random field model. 相似文献
7.
Oscar Rondon 《Mathematical Geosciences》2012,44(4):469-504
Multivariate conditional simulation is used to assess the multivariate grade risk in mineral deposits. With the presence of several spatially correlated attributes, it is important to ensure that their joint simulation is carried out properly and that the observed spatial correlation is reproduced in the realizations. The method of minimum/maximum autocorrelation factors (MAF) is a well established and practical technique that can be used for this purpose. MAF offers tremendous advantages over standard full cosimulation, principal component analysis, and stepwise techniques. In what follows, a detailed review of the MAF technique, its applications, and examples are provided to guide the practitioner on its use. 相似文献
8.
To speed up multivariate geostatistical simulation it is common to transform the set of attributes into spatially uncorrelated factors that can be simulated independently. Spatial decorrelation methods are usually based on the diagonalisation of the variance/covariance and semivariogram matrices of the set of attributes for a chosen family of lag spacings. These matrices are symmetric and there are several efficient methods for the approximate joint diagonalisation of a family of symmetric matrices. One of these is the uniformly weighted exhaustive diagonalisation with Gauss iterations (U-WEDGE) method. In contrast to the method of minimum/maximum autocorrelation factors (MAF), where a two structure linear model of coregionalisation is assumed, U-WEDGE can be applied directly to the set of experimental semivariogram matrices without having to place restrictions on the number of structures in the linear model of coregionalisation, thus removing one of the restrictions placed on the subsequent modelling of the spatial structure of the factors. We use an iron-ore data set to illustrate the method and present a comparison between the simulated attributes obtained from U-WEDGE and MAF with the full co-simulation of the attributes. 相似文献
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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
10.
Modeling Combined Geological and Grade Uncertainty: Application of Multiple-Point Simulation at the Apensu Gold Deposit, Ghana 总被引：1，自引：1，他引：0
Traditionally within the mining industry, single models for both grade and geology of orebodies are created upon which all mine development decisions are based. These models provide a single interpretation of the extent and continuity of the mineralization envelope based on solids and sections interpreted from relatively widely spaced drilling. The inherent variable behavior of grade and geology cannot be understood from a single estimated resource model. To account for uncertainty in the geology and mineralization envelope, Newmont Mining Corporation uses multiple-point statistics (MPS), an emerging spatial simulation framework, which can be employed to generate multiple, geologically realistic, realizations of data representing attributes of mineral deposits that display complex non-linear features. MPS uses a conceptual model of the geology, termed a training image, to infer these high-order spatial relationships. A detailed application of the MPS algorithm at the structurally controlled Apensu gold deposit, Ghana, demonstrates the practical intricacies of the MPS framework and documents efficiency and effectiveness. Multiple realizations of the Apensu deposit allow for an assessment of the geologic and volumetric uncertainty, which is further combined with grade simulations to generate a more complete picture of the true uncertainty of the deposit. 相似文献
11.
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
12.
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
13.
高阶统计量方法在地震信号分析中的应用 总被引：1，自引：0，他引：1
高阶统计量分析是近20年来国内外信号处理领域的一个前沿课题,广泛应用于所有需要考虑非高斯、非最小相位、有色噪声、非线性或循环平稳性的各类问题中。在地震信号处理方面,应用高阶统计可以消除高斯有色噪声的影响和提取与识别非最小相位子波;基于高阶统计特征的独立分量分析己成为信号处理领域的一个研究热点并应用于地震信号分析中。对高阶统计量的定义和性质以及高阶谱时频分析方法作了介绍,并以此为理论依据,进行了理论模型和实际地震数据的实验。 相似文献
14.
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
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.
Assessing spatial uncertainty in mapping soil erodibility factor using geostatistical stochastic simulation 总被引：1，自引：0，他引：1
Soil erosion is one of most widespread process of degradation. The erodibility of a soil is a measure of its susceptibility
to erosion and depends on many soil properties. Soil erodibility factor varies greatly over space and is commonly estimated
using the revised universal soil loss equation. Neglecting information about estimation uncertainty may lead to improper decision-making.
One geostatistical approach to spatial analysis is sequential Gaussian simulation, which draws alternative, equally probable,
joint realizations of a regionalised variable. Differences between the realizations provide a measure of spatial uncertainty
and allow us to carry out an error analysis. The objective of this paper was to assess the model output error of soil erodibility
resulting from the uncertainties in the input attributes (texture and organic matter). The study area covers about 30 km2 (Calabria, southern Italy). Topsoil samples were collected at 175 locations within the study area in 2006 and the main chemical
and physical soil properties were determined. As soil textural size fractions are compositional data, the additive-logratio
(alr) transformation was used to remove the non-negativity and constant-sum constraints on compositional variables. A Monte
Carlo analysis was performed, which consisted of drawing a large number (500) of identically distributed input attributes
from the multivariable joint probability distribution function. We incorporated spatial cross-correlation information through
joint sequential Gaussian simulation, because model inputs were spatially correlated. The erodibility model was then estimated
for each set of the 500 joint realisations of the input variables and the ensemble of the model outputs was used to infer
the erodibility probability distribution function. This approach has also allowed for delineating the areas characterised
by greater uncertainty and then to suggest efficient supplementary sampling strategies for further improving the precision
of K value predictions. 相似文献
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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. 相似文献
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Geotechnical engineering problems are characterized by many sources of uncertainty. Some of these sources are connected to the uncertainties of soil properties involved in the analysis. In this paper, a numerical procedure for a probabilistic analysis that considers the spatial variability of cross‐correlated soil properties is presented and applied to study the bearing capacity of spatially random soil with different autocorrelation distances in the vertical and horizontal directions. The approach integrates a commercial finite difference method and random field theory into the framework of a probabilistic analysis. Two‐dimensional cross‐correlated non‐Gaussian random fields are generated based on a Karhunen–Loève expansion in a manner consistent with a specified marginal distribution function, an autocorrelation function, and cross‐correlation coefficients. A Monte Carlo simulation is then used to determine the statistical response based on the random fields. A series of analyses was performed to study the effects of uncertainty due to the spatial heterogeneity on the bearing capacity of a rough strip footing. The simulations provide insight into the application of uncertainty treatment to geotechnical problems and show the importance of the spatial variability of soil properties with regard to the outcome of a probabilistic assessment. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献