共查询到20条相似文献，搜索用时 31 毫秒
1.
Xavier Emery 《Mathematical Geosciences》2008,40(1):83-99
This paper presents random field models with Gaussian or gamma univariate distributions and isofactorial bivariate distributions,
constructed by composing two independent random fields: a directing function with stationary Gaussian increments and a stationary
coding process with bivariate Gaussian or gamma distributions. Two variations are proposed, by considering a multivariate
directing function and a coding process with a separable covariance, or by including drift components in the directing function.
Iterative algorithms based on the Gibbs sampler allow one to condition the realizations of the substitution random fields
to a set of data, while the inference of the model parameters relies on simple tools such as indicator variograms and variograms
of different orders. A case study in polluted soil management is presented, for which a gamma model is used to quantify the
risk that pollutant concentrations over remediation units exceed a given toxicity level. Unlike the multivariate Gaussian
model, the proposed gamma model accounts for an asymmetry in the spatial correlation of the indicator functions around the
median and for a spatial clustering of high pollutant concentrations. 相似文献
2.
Application of EM algorithms for seismic facices classification 总被引：1，自引：0，他引：1
Identification of the geological facies and their distribution from seismic and other available geological information is
important during the early stage of reservoir development (e.g. decision on initial well locations). Traditionally, this is
done by manually inspecting the signatures of the seismic attribute maps, which is very time-consuming. This paper proposes
an application of the Expectation-Maximization (EM) algorithm to automatically identify geological facies from seismic data.
While the properties within a certain geological facies are relatively homogeneous, the properties between geological facies
can be rather different. Assuming that noisy seismic data of a geological facies, which reflect rock properties, can be approximated
with a Gaussian distribution, the seismic data of a reservoir composed of several geological facies are samples from a Gaussian
mixture model. The mean of each Gaussian model represents the average value of the seismic data within each facies while the
variance gives the variation of the seismic data within a facies. The proportions in the Gaussian mixture model represent
the relative volumes of different facies in the reservoir. In this setting, the facies classification problem becomes a problem
of estimating the parameters defining the Gaussian mixture model. The EM algorithm has long been used to estimate Gaussian
mixture model parameters. As the standard EM algorithm does not consider spatial relationship among data, it can generate
spatially scattered seismic facies which is physically unrealistic. We improve the standard EM algorithm by adding a spatial
constraint to enhance spatial continuity of the estimated geological facies. By applying the EM algorithms to acoustic impedance
and Poisson’s ratio data for two synthetic examples, we are able to identify the facies distribution. 相似文献
3.
Positive definiteness is not enough 总被引：2，自引：0，他引：2
M. Armstrong 《Mathematical Geology》1992,24(1):135-143
Geostatisticians know that the mathematical functions chosen to represent spatial covariances and variograms must have the appropriate type of positive definiteness, but they may not realize that there are restrictions on the types of covariances and variograms that are compatible with particular distributions. This paper gives some examples showing that (1) the spherical model is not compatible with the multivariate lognormal distribution if the coefficient of variation is 2.0 or more (even in 1-D), and (2) the Gaussian covariance and several other models are not compatible with indicator random functions. As these examples concern quite different types of random functions, it is clear that there is a general problem of compatibility between spatial covariance models (or variograms) and a specified multivariate distribution. The problem arises with all distributions except the multivariate normal, and not just the two cited here. The need for a general theorem giving the necessary and sufficient conditions for a covariance or a variogram to be compatible with a particular distribution is stressed. 相似文献
4.
G. Bourgault 《Mathematical Geology》1997,29(3):315-334
Parametric geostatistical simulations such as LU decomposition and sequential algorithms do not need Gaussian distributions.
It is shown that variogram model reproduction is obtained when Uniform or Dipole distributions are used instead of Gaussian
distributions for drawing i. i.d. random values in LU simulation, or for modeling the local conditional probability distributions
in sequential simulation. Both algorithms yield simulated values with a marginal normal distribution no matter if Gaussian,
Uniform, or Dipole distributions are used. The range of simulated values decreases as the entropy of the probability distribution
decreases. Using Gaussian distributions provides a larger range of simulated normal score values than using Uniform or Dipole
distributions. This feature has a negligible effect for reproduction of the normal scores variogram model but have a larger
impact on the reproduction of the original values variogram. The Uniform or Dipole distributions also produce lesser fluctuations
among the variograms of the simulated realizations. 相似文献
5.
Xavier Emery 《Mathematical Geology》2005,37(2):163-181
The multigaussian model is used in mining geostatistics to simulate the spatial distribution of grades or to estimate the recoverable reserves of an ore deposit. Checking the suitability of such model to the available data often constitutes a critical step of the geostatistical study. In general, the marginal distribution is not a problem because the data can be transformed to normal scores, so the check is usually restricted to the bivariate distributions. In this work, several tests for diagnosing the two-point normality of a set of Gaussian data are reviewed and commented. An additional criterion is proposed, based on the comparison between the usual variogram and the variograms of lower order: the latter are defined as half the mean absolute increments of the attribute raised to a power between 0 and 2. This criterion is then extended to other bivariate models, namely the bigamma, Hermitian and Laguerrian models. The concepts are illustrated on two real data-sets. Finally, some conditions to ensure the internal consistency of the variogram under a given model are given. 相似文献
6.
Hou Weisheng Cui Chanjie Yang Liang Yang Qiaochu Clarke Keith 《Mathematical Geosciences》2019,51(1):29-51
In each step of geological modeling, errors have an impact on measurements and workflow processes and, so, have consequences that challenge accurate three-dimensional geological modeling. In the context of classical error theory, for now, only spatial positional error is considered, acknowledging that temporal, attribute, and ontological errors—and many others—are part of the complete error budget. Existing methods usually assumed that a single error distribution (Gaussian) exists across all kinds of spatial data. Yet, across, and even within, different kinds of raw data (such as borehole logs, user-defined geological sections, and geological maps), different types of positional error distributions may exist. Most statistical methods make a priori assumptions about error distributions that impact their explanatory power. Consequently, analyzing errors in multi-source and conflated data for geological modeling remains a grand challenge in geological modeling. In this study, a novel approach is presented regarding the analysis of one-dimensional multiple errors in the raw data used for model geological structures. The analysis is based on the relationship between spatial error distributions and different geological attributes. By assuming that the contact points of a geological subsurface are decided by the geological attributes related to both sides of the subsurface, this assumption means that the spatial error of geological contacts can be transferred into specific probabilities of all the related geological attributes at each three-dimensional point, which is termed the “geological attribute probability”. Both a normal distribution and a continuous uniform distribution were transferred into geological attribute probabilities, allowing different kinds of spatial error distributions to be summed directly after the transformation. On cross-points with multiple raw data with errors that follow different kinds of distributions, an entropy-based weight was given to each type of data to calculate the final probabilities. The weighting value at each point in space is decided by the related geological attribute probabilities. In a test application that accounted for the best estimates of geological contacts, the experimental results showed the following: (1) for line segments, the band shape of geological attribute probabilities matched that of existing error models; and (2) the geological attribute probabilities directly show the error distribution and are an effective way of describing multiple error distributions among the input data.
相似文献7.
The majority of geostatistical estimation and simulation algorithms rely on a covariance model as the sole characteristic of the spatial distribution of the attribute under study. The limitation to a single covariance implicitly calls for a multivariate Gaussian model for either the attribute itself or for its normal scores transform. The Gaussian model could be justified on the basis that it is both analytically simple and it is a maximum entropy model, i.e., a model that minimizes unwarranted structural properties. As a consequence, the Gaussian model also maximizes spatial disorder (beyond the imposed covariance) which can cause flow simulation results performed on multiple stochastic images to be very similar; thus, the space of response uncertainty could be too narrow entailing a misleading sense of safety. The ability of the sole covariance to adequately describe spatial distributions for flow studies, and the assumption that maximum spatial disorder amounts to either no additional information or a safe prior hypothesis are questioned. This paper attempts to clarify the link between entropy and spatial disorder and to provide, through a detailed case study, an appreciation for the impact of entropy of prior random function models on the resulting response distributions. 相似文献
8.
Gilles Bourgault 《Mathematical Geosciences》2014,46(7):841-868
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. 相似文献
9.
The majority of geostatistical estimation and simulation algorithms rely on a covariance model as the sole characteristic of the spatial distribution of the attribute under study. The limitation to a single covariance implicitly calls for a multivariate Gaussian model for either the attribute itself or for its normal scores transform. The Gaussian model could be justified on the basis that it is both analytically simple and it is a maximum entropy model, i.e., a model that minimizes unwarranted structural properties. As a consequence, the Gaussian model also maximizes spatial disorder (beyond the imposed covariance) which can cause flow simulation results performed on multiple stochastic images to be very similar; thus, the space of response uncertainty could be too narrow entailing a misleading sense of safety. The ability of the sole covariance to adequately describe spatial distributions for flow studies, and the assumption that maximum spatial disorder amounts to either no additional information or a safe prior hypothesis are questioned. This paper attempts to clarify the link between entropy and spatial disorder and to provide, through a detailed case study, an appreciation for the impact of entropy of prior random function models on the resulting response distributions. 相似文献
10.
Spatial data analytics provides new opportunities for automated detection of anomalous data for data quality control and subsurface segmentation to reduce uncertainty in spatial models. Solely data-driven anomaly detection methods do not fully integrate spatial concepts such as spatial continuity and data sparsity. Also, data-driven anomaly detection methods are challenged in integrating critical geoscience and engineering expertise knowledge. The proposed spatial anomaly detection method is based on the semivariogram spatial continuity model derived from sparsely sampled well data and geological interpretations. The method calculates the lag joint cumulative probability for each matched pair of spatial data, given their lag vector and the semivariogram under the assumption of bivariate Gaussian distribution. For each combination of paired spatial data, the associated head and tail Gaussian standardized values of a pair of spatial data are mapped to the joint probability density function informed from the lag vector and semivariogram. The paired data are classified as anomalous if the associated head and tail Gaussian standardized values fall within a low probability zone. The anomaly decision threshold can be decided based on a loss function quantifying the cost of overestimation or underestimation. The proposed spatial correlation anomaly detection method is able to integrate domain expertise knowledge through trend and correlogram models with sparse spatial data to identify anomalous samples, region, segmentation boundaries, or facies transition zones. This is a useful automation tool for identifying samples in big spatial data on which to focus professional attention.
相似文献11.
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
12.
《Mathematical Geology》1997,29(6):779-799
Generalized cross-covariances describe the linear relationships between spatial variables observed at different locations.
They are invariant under translation of the locations for any intrinsic processes, they determine the cokriging predictors
without additional assumptions and they are unique up to linear functions. If the model is stationary, that is if the variograms
are bounded, they correspond to the stationary cross-covariances. Under some symmetry condition they are equal to minus the
usual cross-variogram. We present a method to estimate these generalized cross-covariances from data observed at arbitrary
sampling locations. In particular we do not require that all variables are observed at the same points. For fitting a linear
coregionalization model we combine this new method with a standard algorithm which ensures positive definite coregionalization
matrices. We study the behavior of the method both by computing variances exactly and by simulating from various models. 相似文献
13.
Scott Painter 《Mathematical Geology》1998,30(2):163-179
Stochastic simulations of subsurface heterogeneity require accurate statistical models for spatial fluctuations. Incremental values in subsurface properties were shown previously to be approximated accurately by Levy distributions in the center and in the start of the tails of the distribution. New simulation methods utilizing these observations have been developed. Multivariate Levy distributions are used to model the multipoint joint probability density. Explicit bounds on the simulated variables prevent nonphysical extreme values and introduce a cutoff in the tails of the distribution of increments. Long-range spatial dependence is introduced through off-diagonal terms in the Levy association matrix, which is decomposed to yield a maximum likelihood type estimate at unobserved locations. This procedure reduces to a known interpolation formula developed for Gaussian fractal fields in the situation of two control points. The conditional density is not univariate Levy and is not available in closed form, but can be constructed numerically. Sequential simulation algorithms utilizing the numerically constructed conditional density successfully reproduce the desired statistical properties in simulations. 相似文献
14.
The problem to predict a direction, axis, or orientation (rotation) from corresponding geocoded data is discussed and a general solution by virtue of embedding a sphere/hemisphere in a real vector space is presented. Its explicit justification in terms of mathematical assumptions concerning stationarity/homogeneity and isotropy is included. The data are modelled by a stationary random field, and the spatial correlation is represented by modified multivariate variograms and covariance functions. Various types of isotropy assumptions concerning invariance under translation/rotation of the data locations, the measurements, or a combination of both, can be distinguished and lead to different simplifications of the general cross-covariance function. Beyond spatial prediction a measure of confidence in the estimates is provided. 相似文献
15.
Local and Spatial Joint Frequency Uncertainty and its Application to Rock Mass Characterisation 总被引：1，自引：1，他引：0
Stability is a key issue in any mining or tunnelling activity. Joint frequency constitutes an important input into stability
analyses. Three techniques are used herein to quantify the local and spatial joint frequency uncertainty, or possible joint
frequencies given joint frequency data, at unsampled locations. Rock quality designation is estimated from the predicted joint
frequencies. The first method is based on kriging with subsequent Poisson sampling. The second method transforms the data
to near-Gaussian variables and uses the turning band method to generate a range of possible joint frequencies. The third method
assumes that the data are Poisson distributed and models the log-intensity of these data with a spatially smooth Gaussian
prior distribution. Intensities are obtained and Poisson variables are generated to examine the expected joint frequency and
associated variability. The joint frequency data is from an iron ore in the northern part of Norway. The methods are tested
at unsampled locations and validated at sampled locations. All three methods perform quite well when predicting sampled points.
The probability that the joint frequency exceeds 5 joints per metre is also estimated to illustrate a more realistic utilisation.
The obtained probability map highlights zones in the ore where stability problems have occurred. It is therefore concluded
that the methods work and that more emphasis should have been placed on these kinds of analyses when the mine was planned.
By using simulation instead of estimation, it is possible to obtain a clear picture of possible joint frequency values or
ranges, i.e. the uncertainty. 相似文献
16.
利用地质统计学方法模拟岩石裂隙网络的三维空间分布——以云南个旧高松矿田为例 总被引：1，自引：0，他引：1
裂隙在地学的诸多领域中均具有重要意义,其空间分布可以使用地质统计学方法进行模拟,同时考虑裂隙的方向(走向和倾角)。利用序贯高斯模拟方法可以估计裂隙密度的空间分布,并根据裂隙密度数值随机产生裂隙位置的空间分布。裂隙方向被划分成若干(非)均等的方向组,将裂隙方向归属到其所属方向组,表示为由若干二值变量组成的指示形式,0和1分别代表该裂隙方向不属于和属于该组。为了便于计算,减少方向指示变量的成分数目,使用主成分分析法求出方向指示变量的主成分,用普通克里格法估计各主成分的空间分布。把估计结果反演为原始的指示形式,并找出其中数值最大的方向组且将其赋值为1。按照对应方向组内裂隙方向的累积密度函数,随机产生裂隙的方向。根据估计结果,将符合一定距离和角度标准的裂隙元连接为一个裂隙面,从而形成裂隙网络。根据在云南个旧锡矿高松矿田白云岩中进行裂隙网络模拟的应用,可见该方法由于组合了序贯高斯模拟法、普通克里格法和主成分分析法,可以较好地对岩石裂隙位置和方向进行合理的模拟。 相似文献
17.
P. Juan J. Mateu M.M. Jordan J. Mataix-Solera I. Meléndez-Pastor J. Navarro-Pedreño 《Journal of Geochemical Exploration》2011
The problem of estimating and predicting spatial distribution of a spatial stochastic process, observed at irregular locations in space, is considered in this paper. Environmental variables usually show spatial dependencies among observations, with lead one to use geostatistical methods to model the spatial distributions of those observations. This is particularly important in the study of soil properties and their spatial variability. In this study geostatistical techniques were used to describe the spatial dependence and to quantify the scale and intensity of spatial variations of soil properties, which provide the essential spatial information for local estimation. In this contribution, we propose a spatial Gaussian linear mixed model that involves (a) a non-parametric term for accounting deterministic trend due to exogenous variables and (b) a parametric component for defining the purely spatial random variation due possibly to latent spatial processes. We focus here on the analysis of the relationship between soil electrical conductivity and Na content to identify spatial variations of soil salinity. This analysis can be useful for agricultural and environmental land management. 相似文献
18.
针对高斯混合模型(GMM)在空间聚类中由于忽视目标对象之间的空间关联性而导致的高误判率等问题,本文提出了一种顾及梯度的高斯混合模型:GMM-G,并将其应用在三维属性场的空间聚类中。GMM-G用反映标量场最大属性变化方向的梯度因子来定义邻域规则,设定梯度正交平面所通过的邻域体元更倾向于与中心体元归属于相同或相近的类别;并据此设计了符合归一性和空间连续性的空间邻域信息函数,来定义中心体元属于各类别的具有空间领域规则约束的后验概率。通过对由蒙特卡洛随机抽样构建的实验场的空间聚类结果进行对比表明,相对GMM方法,GMM-G具有更优的聚类精度及效率。最后,把GMM-G方法用于红透山铜矿区可控源音频大地电磁法(CSAMT)三维视电阻率场的空间聚类,得到了与已知岩性划分具有较高匹配度的分类结果,该方法可为物性属性场的岩性划分及地质推断提供相关的依据和参考。 相似文献
19.
Ute Christina Herzfeld 《Mathematical Geology》1989,21(7):693-713
During the German Antarctic Expedition VI (leg 3, December 1987 to March 1988), bathymetric surveys were made in the Weddell Sea by the SEABEAM sonar system. For the first time geostatistical methods were applied in the SEABEAM-postprocessing. The investigations of variography that were necessary prior to the cartographical-geomorphological evaluation shed new light on classical geostatistical concerns. SEABEAM data provide a good example of a mean square, differentiable regionalized variable, where data are sampled over a two-dimensional support due to the technique of the sonar device. By deregularizations of the sample variograms, spatial continuity can be shown to be a property of seafloor depth as well as a point variable. The results are discussed in a sedimentological context. As an application of the regional variogram analyses, large-scale kriged bathymetric maps are presented. 相似文献
20.
储层物性受诸多地质因素的影响，经常表现出强烈的非均质性。这种非均质性在一定的空间尺度上往往具有明显的结构特点。本文以青海尕斯库勒油田E3^1油藏为例，讨论了地质统计学方法在油田储层特性空间结构特征分析与预测中的应用。采用变异函数定量描述了孔隙度和渗透率的空间分布结构特征，并在此基础上利用克立格法进行了最成插值预测。结果表明，研究区储层物性（孔隙度和渗透率）具有显著的空间结构性特点，变程一般在800－2000m之间；不同小层的储层物性具有不同的空间结构方向性。这种特征主要沉积相带空间展布的影响，各小层孔隙度和渗透率的实验半变异函数均可用具有块金效应的球状模型来拟合并进行预测。作为验证，本文还采用“多重趋势面”预测模型对储层的孔隙度和渗透率进行了预测分析。 相似文献