共查询到20条相似文献，搜索用时 484 毫秒
1.
Conditional Simulation of Random Fields with Bivariate Gamma Isofactorial Distributions 总被引：5，自引：0，他引：5
Xavier Emery 《Mathematical Geology》2005,37(4):419-445
This work focuses on a random function model with gamma marginal and bivariate isofactorial distributions, which has been applied in mining geostatistics for estimating recoverable reserves by disjunctive kriging. The objective is to widen its use to conditional simulation and further its application to the modeling of continuous attributes in geosciences. First, the main properties of the bivariate gamma isofactorial distributions are analyzed, with emphasis in the destructuring of the extreme values, the presence of a proportional effect (higher variability in high-valued areas), and the asymmetry in the spatial correlation of the indicator variables with respect to the median threshold. Then, we provide examples of stationary random functions with such bivariate distributions, for which the shape parameter of the marginal distribution is half an integer. These are defined as the sum of squared independent Gaussian random fields. An iterative algorithm based on the Gibbs sampler is proposed to perform the simulation conditional to a set of existing data. Such ‘multivariate chi-square’ model generalizes the well-known multigaussian model and is more flexible, since it allows defining a shape parameter which controls the asymmetry of the marginal and bivariate distributions. 相似文献
2.
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
3.
On the Existence of Mosaic and Indicator Random Fields with Spherical,Circular, and Triangular Variograms 总被引：1，自引：1，他引：0
Xavier Emery 《Mathematical Geosciences》2010,42(8):969-984
This paper focuses on two specific families of stationary random fields (namely, mosaic and indicator random fields) and examines
the question of whether the variogram of a member of these two families could be spherical, circular, or triangular. It is
shown that there are such examples in one-dimensional spaces, while there are counterexamples in higher-dimensional spaces
for mosaic random fields, indicators of Boolean random sets and indicators of excursion sets of Gaussian random fields. Further
results concerning the spherical plus nugget and circular plus nugget variograms are provided. 相似文献
4.
Xavier Emery 《Mathematical Geology》2007,39(2):205-223
This work deals with a family of geostatistical models used in application fields concerned with a change of support, such
as mineral resources evaluation and polluted soil management. Three models are examined: the discrete Gaussian, Hermitian
and Laguerrian models, which rely on a transformation of the variable of interest (mineral grade or pollutant concentration)
defined on point and block supports into variables with Gaussian or gamma univariate distributions and isofactorial bivariate
distributions. The focus is given to the relationships between the transformed variables at both supports, and to the conditions
that these relationships imply on the model parameters. Additionally, guidelines are given for improving the variogram analysis
of the transformed variables and for validating the change-of-support model. 相似文献
5.
Non-stationary Geostatistical Modeling Based on Distance Weighted Statistics and Distributions 总被引：3，自引：0，他引：3
A common assumption in geostatistics is that the underlying joint distribution of possible values of a geological attribute at different locations is stationary within a homogeneous domain. This joint distribution is commonly modeled as multi-Gaussian, with correlations defined by a stationary covariance function. This results in attribute maps that fail to reproduce local changes in the mean, in the variance and, particularly, in the spatial continuity. The proposed alternative is to build local distributions, variograms, and correlograms. These are inferred by weighting the samples depending on their distance to selected locations. The local distributions are locally transformed into Gaussian distributions embedding information on the local histogram. The distance weighted experimental variograms and correlograms are able to adapt to local changes in the direction and range of spatial continuity. The automatically fitted local variogram models and the local Gaussian transformation parameters are used in spatial estimation algorithms assuming local stationarity. The resulting maps are rich in nonstationary spatial features. The proposed process implies a higher computational effort than traditional stationary techniques, but if data availability allows for a reliable inference of the local distributions and statistics, a higher accuracy of estimates can be achieved. 相似文献
6.
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
7.
Xavier Emery 《Mathematical Geology》2007,39(6):607-623
Conditioning realizations of stationary Gaussian random fields to a set of data is traditionally based on simple kriging.
In practice, this approach may be demanding as it does not account for the uncertainty in the spatial average of the random
field. In this paper, an alternative model is presented, in which the Gaussian field is decomposed into a random mean, constant
over space but variable over the realizations, and an independent residual. It is shown that, when the prior variance of the
random mean is infinitely large (reflecting prior ignorance on the actual spatial average), the realizations of the Gaussian
random field are made conditional by substituting ordinary kriging for simple kriging. The proposed approach can be extended
to models with random drifts that are polynomials in the spatial coordinates, by using universal or intrinsic kriging for
conditioning the realizations, and also to multivariate situations by using cokriging instead of kriging. 相似文献
8.
Xavier Emery 《Mathematical Geology》2002,34(1):79-100
This paper presents a conditional simulation procedure that overcomes the limits of gaussian models and enables one to simulate regionalized variables with highly asymmetrical histograms or with partial or total connectivity of extreme values. The philosophy of the method is similar to that of sequential indicator technique, but it is more accurate because it is based on a complete bivariate model by means of an isofactorial law. The resulting simulations, which can be continuous or categorical, not only honor measured values at data points, but also reproduce the mono and bivariate laws of the random function associated to the regionalized variable, that is, every one or two-point statistic: histogram, variogram, indicator variograms. The sequential isofactorial method can also be adapted to conditional simulation of block values, without resorting to point–support simulations. 相似文献
9.
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
10.
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
11.
Geostatistical analysis of spatial random functions frequently uses sample variograms computed from increments of samples of a regionalized random variable. This paper addresses the theory of computing variograms not from increments but from spatial variances. The objective is to extract information about the point support space from the average or larger support data. The variance is understood as a parametric and second moment average feature of a population. However, it is well known that when the population is for a stationary random function, spatial variance within a region is a function of the size and geometry of the region and not a function of location. Spatial variance is conceptualized as an estimation variance between two physical regions or a region and itself. If such a spatial variance could be measured within several sizes of windows, such variances allow the computation of the sample variogram. The approach is extended to covariances between attributes that lead to the cross-variogram. The case of nonpoint sample support of the blocks or elements composing each window is also included. A numerical example illustrates the application of this conceptualization. 相似文献
12.
Eulogio Pardo-Igúzquiza 《Mathematical Geology》1998,30(1):95-108
In this paper, the maximum likelihood method for inferring the parameters of spatial covariances is examined. The advantages of the maximum likelihood estimation are discussed and it is shown that this method, derived assuming a multivariate Gaussian distribution for the data, gives a sound criterion of fitting covariance models irrespective of the multivariate distribution of the data. However, this distribution is impossible to verify in practice when only one realization of the random function is available. Then, the maximum entropy method is the only sound criterion of assigning probabilities in absence of information. Because the multivariate Gaussian distribution has the maximum entropy property for a fixed vector of means and covariance matrix, the multinormal distribution is the most logical choice as a default distribution for the experimental data. Nevertheless, it should be clear that the assumption of a multivariate Gaussian distribution is maintained only for the inference of spatial covariance parameters and not necessarily for other operations such as spatial interpolation, simulation or estimation of spatial distributions. Various results from simulations are presented to support the claim that the simultaneous use of maximum likelihood method and the classical nonparametric method of moments can considerably improve results in the estimation of geostatistical parameters. 相似文献
13.
Heuristic models for indicator variograms and their parameters (practical nugget effect and range) are proposed for a bivariate normal distribution with spherical correlogram. These models can be used conveniently as a preliminary check for bivariate normality. In the general non-Gaussian case, indicator variogram models for multiple threshold values must verify a certain number of order relations (inequalities) established directly from the properties of a general bivariate cumulative distribution function. An interesting, little-known maximum hole effect for indicator correlation is pointed out. 相似文献
14.
Dae S. Young 《Mathematical Geology》1987,19(6):481-501
Indicator kriging (IK) is extended to analyze three-dimensional random unit vectors and evaluate the local probability distribution of rock joint orientations in geological formations. The pole vector representing joint orientations is regionalized and projected on a plane normal to the mean attitude of the joint family and centered at the mean. A two-dimensional cutoff system is developed to define the indicator variable, and corresponding indicator variograms and indicator kriging. The cutoff system defines probability regions similar to those of a bivariate distribution, concentric rings sliced into radial sectors. A case study made on an open pit mine proved positively the efficiency of IK and encourages its applications to localized probabilistic structural modeling for geotechnical or geohydrological analysis and oil and gas reservoir analysis. 相似文献
15.
Multivariable spatial prediction 总被引：1，自引：0，他引：1
For spatial prediction, it has been usual to predict one variable at a time, with the predictor using data from the same type of variable (kriging) or using additional data from auxiliary variables (cokriging). Optimal predictors can be expressed in terms of covariance functions or variograms. In earth science applications, it is often desirable to predict the joint spatial abundance of variables. A review of cokriging shows that a new cross-variogram allows optimal prediction without any symmetry condition on the covariance function. A bivariate model shows that cokriging with previously used cross-variograms can result in inferior prediction. The simultaneous spatial prediction of several variables, based on the new cross-variogram, is then developed. Multivariable spatial prediction yields the mean-squared prediction error matrix, and so allows the construction of multivariate prediction regions. Relationships between cross-variograms, between single-variable and multivariable spatial prediction, and between generalized least squares estimation and spatial prediction are also given. 相似文献
16.
Gonzalo Sapriza Azuri Jorge Jódar Jesús Carrera Hoshin V. Gupta 《Mathematical Geosciences》2013,45(5):621-645
A model for generating daily spatial correlated rainfall fields suitable for evaluating the impacts of climate change on water resources is presented. The model, termed Stochastic Rainfall Generating Process, is designed to incorporate two major nonstationarities: changes in the frequencies of different precipitation generating mechanisms (frontal and convective), and spatial nonstationarities caused by interactions of mesoscale atmospheric patterns with topography (orographic effects). These nonstationarities are approximated as discrete sets of the time-stationary Stochastic Rainfall Generating Process, each of which represents the different spatial patterns of rainfall (including its variation with topography) associated with different atmospheric circulation patterns and times of the year (seasons). Each discrete Stochastic Rainfall Generating Process generates daily correlated rainfall fields as the product of two random fields. First, the amount of rainfall is generated by a transformed Gaussian process applying sequential Gaussian simulation. Second, the delimitation of rain and no-rain areas (intermittence process) is defined by a binary random function simulated by sequential indicator simulations. To explore its applicability, the model is tested in the Upper Guadiana Basin in Spain. The result suggests that the model provides accurate reproduction of the major spatiotemporal features of rainfall needed for hydrological modeling and water resource evaluations. The results were significantly improved by incorporating spatial drift related to orographic precipitation into the model. 相似文献
17.
Flat variograms often are interpreted as representing a lack of spatial autocorrelation. Recent research in earthquake engineering shows that nearby field noise can substantially mask a prominent spatial autocorrelation and result in what appears to be a purely random spatial process. A careful selection of threshold in assigning an indicator function can yield an indicator variogram which reveals underlying spatial autocorrelation. Although this application involves use of seismic data, the results are relevant to geostatistical applications in general. 相似文献
18.
In this paper, entropy is presented as an alternative measure to characterize the bivariate distribution of a stationary spatial process. This non-parametric estimator attempts to quantify the concept of spatial ordering, and it provides a measure of how Gaussian the experimental bivariate distribution is. The concept of entropy is explained and the classical definition presented, along with some important results. In particular, the reader is reminded that, for a known mean and covariance, the bivariate Gaussian distribution maximizes entropy. A relative entropy estimator is introduced in order to measure departure of an experimental bivariate distribution from the bivariate Gaussian. Two case studies are presented as examples. 相似文献
19.
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
20.
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