共查询到20条相似文献，搜索用时 15 毫秒
1.
Simulating Large Gaussian Random Vectors Subject to Inequality Constraints by Gibbs Sampling 总被引：4，自引：2，他引：2
The Gibbs sampler is an iterative algorithm used to simulate Gaussian random vectors subject to inequality constraints. This algorithm relies on the fact that the distribution of a vector component conditioned by the other components is Gaussian, the mean and variance of which are obtained by solving a kriging system. If the number of components is large, kriging is usually applied with a moving search neighborhood, but this practice can make the simulated vector not reproduce the target correlation matrix. To avoid these problems, variations of the Gibbs sampler are presented. The conditioning to inequality constraints on the vector components can be achieved by simulated annealing or by restricting the transition matrix of the iterative algorithm. Numerical experiments indicate that both approaches provide realizations that reproduce the correlation matrix of the Gaussian random vector, but some conditioning constraints may not be satisfied when using simulated annealing. On the contrary, the restriction of the transition matrix manages to satisfy all the constraints, although at the cost of a large number of iterations. 相似文献
2.
A new and simple method is proposed to obtain estimates of recovery functions: the Bi-Gaussian approach. Existing methods estimate recovery functions with conditional distributions where the conditioning set is all the data available. Here instead the simple kriging estimate of the Gaussian transform is proposed to be used. Results in the point recovery case are identical to the multi-Gaussian approach of Verly (1983, 1984), whereas in the non-point-support situation, an approximation is derived which saves computer time as compared to employing the strict multi-Gaussian hypothesis. Two examples compare favorably with the well-established disjunctive kriging method (discrete Gaussian model). 相似文献
3.
Most geostatistical studies consider multiple-related variables. These relationships often show complex features such as nonlinearity, heteroscedasticity, and mineralogical or other constraints. These features are not handled by the well-established Gaussian simulation techniques. Earth science variables are rarely Gaussian. Transformation or anamorphosis techniques make each variable univariate Gaussian, but do not enforce bivariate or higher order Gaussianity. The stepwise conditional transformation technique is proposed to transform multiple variables to be univariate Gaussian and multivariate Gaussian with no cross correlation. This makes it remarkably easy to simulate multiple variables with arbitrarily complex relationships: (1) transform the multiple variables, (2) perform independent Gaussian simulation on the transformed variables, and (3) back transform to the original variables. The back transformation enforces reproduction of the original complex features. The methodology and underlying assumptions are explained. Several petroleum and mining examples are used to show features of the transformation and implementation details. 相似文献
4.
Xavier Emery 《Mathematical Geosciences》2012,44(1):27-46
In oxide copper deposits, the acid soluble copper represents the fraction of total copper recoverable by heap leaching. Two
difficulties often complicate the joint modeling and simulation of total and soluble copper grades: the inequality constraint
linking both grade variables and the sampling design for soluble copper grade, which may be preferential and cause biases
in sample statistics. A methodology is presented in order to accurately estimate the total and soluble copper grade bivariate
distribution, based on an explicit modeling of the conditional distributions of soluble copper grade. Co-simulation is then
realized by converting the copper grades into Gaussian random fields, through stepwise conditional transformation, and by
fitting a coregionalization model while accounting for the preferential sampling design. The proposed approach is illustrated
through an application to an ore deposit located in northern Chile. 相似文献
5.
Kriging with Inequality Constraints 总被引：1，自引：0，他引：1
A Gaussian random field with an unknown linear trend for the mean is considered. Methods for obtaining the distribution of the trend coefficients given exact data and inequality constraints are established. Moreover, the conditional distribution for the random field at any location is calculated so that predictions using e.g. the expectation, the mode, or the median can be evaluated and prediction error estimates using quantiles or variance can be obtained. Conditional simulation techniques are also provided. 相似文献
6.
Some major challenges for geophysicists and structural geologists using three-dimensional boundary element method codes (3D-BEM)
are: (1) reducing the amount of memory required to solve large and dense systems and (2) incorporation of inequality constraints
such as traction inequality constraints (TIC) and displacement inequality constraints (DIC). The latter serves two purposes.
First, for example, inequality constraints can be used to simulate frictional slip (using TIC). Second, these constraints
can prevent element interpenetration while allowing opening mode (using DIC). We have developed a method that simultaneously
incorporates both types of functionality of the inequality constraints. We show that the use of an appropriate iterative solver
not only avoids the allocation of significant memory for solving the system (allowing very large model computation and simplifying
parallelization on multi-core processors), but also admits interesting features such as natural incorporation of TICs and
DICs. Compared to other techniques of contact management (e.g., Lagrange multipliers, penalty method, or complementarity problem),
this new simple methodology, which does not use any incremental trial-and-error procedures, brings more flexibility, while
making the system more stable and less subject to round-off errors without any computational overhead. We provide validations
and comparisons of the inequality constraints implementation using 2D analytical and numerical solutions. 相似文献
7.
Improvements in Grade Tonnage Curve Prediction via Sequential Gaussian Fractal Simulation 总被引：1，自引：0，他引：1
An application of sequential Gaussian fractal conditional simulation is presented, using actual sparse, irregularly spaced open cut gold data, to predict the grade tonnage curve associated with a single mining bench. It is shown that this constitutes an improvement over the prediction of the grade tonnage curves obtained via standard sequential Gaussian simulation and median indicator simulation. The fractal conditional simulation method uses the model of the covariance of increments of fractional Brownian motion together with the fractal co-dimension derived from a truncated power semivariogram model to create conditional simulations from irregularly spaced data. The grades and tonnages above several discrete cut offs are calculated for each of one hundred simulations, and the mean and variance of the grade and tonnage values for each cut off are computed to give an average grade tonnage curve with associated confidence limits. 相似文献
8.
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
9.
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
10.
Histograms of observations from spatial phenomena are often found to be more heavy-tailed than Gaussian distributions, which
makes the Gaussian random field model unsuited. A T-distributed random field model with heavy-tailed marginal probability density functions is defined. The model is a generalization
of the familiar Student-T distribution, and it may be given a Bayesian interpretation. The increased variability appears cross-realizations, contrary
to in-realizations, since all realizations are Gaussian-like with varying variance between realizations. The T-distributed random field model is analytically tractable and the conditional model is developed, which provides algorithms
for conditional simulation and prediction, so-called T-kriging. The model compares favourably with most previously defined random field models. The Gaussian random field model
appears as a special, limiting case of the T-distributed random field model. The model is particularly useful whenever multiple, sparsely sampled realizations of the
random field are available, and is clearly favourable to the Gaussian model in this case. The properties of the T-distributed random field model is demonstrated on well log observations from the Gullfaks field in the North Sea. The predictions
correspond to traditional kriging predictions, while the associated prediction variances are more representative, as they
are layer specific and include uncertainty caused by using variance estimates. 相似文献
11.
This paper proposes an augmented Lagrangian method for production optimization in which the cost function to be maximized
is defined as an augmented Lagrangian function consisting of the net present value (NPV) and all the equality and inequality
constraints except the bound constraints. The bound constraints are dealt with using a trust-region gradient projection method.
The paper also presents a way to eliminate the need to convert the inequality constraints to equality constraints with slack
variables in the augmented Lagrangian function, which greatly reduces the size of the optimization problem when the number
of inequality constraints is large. The proposed method is tested in the context of closed-loop reservoir management benchmark
problem based on the Brugge reservoir setup by TNO. In the test, we used the ensemble Kalman filter (EnKF) with covariance
localization for data assimilation. Production optimization is done on the updated ensemble mean model from EnKF. The production
optimization resulted in a substantial increase in the NPV for the expected reservoir life compared to the base case with
reactive control. 相似文献
12.
An important issue of using the multiple-point (MP) statistical approach for reservoir modeling concerns the integration of
auxiliary constraints derived, for instance, from seismic information. There exist two methods in the literature for these
non-stationary MP simulations. One is based on an analytical approximation (the “τ-model”) of the conditional probabilities that involve auxiliary data. The degree of approximation with this method depends
on the parameter τ, whose inference is difficult in practice. The other method is based on the inference of these conditional probabilities
directly from training images. This method classifies the auxiliary data into a few classes. This classification is in general
arbitrary and therefore inconvenient in practice, especially in the case of continuous auxiliary constraints. In this paper,
we propose an alternative method for performing non-stationary MP simulations. This method accounts for the data support in
the modeling procedure and allows, in particular, continuous auxiliary data to be integrated into MP simulations. This method
avoids the major limitations of the previous methods, namely the use of an approximate analytical model and the reduction
of the auxiliary data into a limited number of classes. This method can be easily implemented in the existing MP simulation
codes. Numerical tests show good performance of this method both in reproducing the geometrical features of the training image
and in honouring the auxiliary data. 相似文献
13.
State estimation in high dimensional systems remains a challenging part of real time analysis. The ensemble Kalman filter addresses this challenge by using Gaussian approximations constructed from a number of samples. This method has been a large success in many applications. Unfortunately, for some cases, Gaussian approximations are no longer valid, and the filter does not work so well. In this paper, we use the idea of the ensemble Kalman filter together with the more theoretically valid particle filter. We outline a Gaussian mixture approach based on shrinking the predicted samples to overcome sample degeneracy, while maintaining non-Gaussian nature. A tuning parameter determines the degree of shrinkage. The computational cost is similar to the ensemble Kalman filter. We compare several filtering methods on three different cases: a target tracking model, the Lorenz 40 model, and a reservoir simulation example conditional on seismic and electromagnetic data. 相似文献
14.
Michael Stein 《Mathematical Geology》1987,19(5):387-405
Suppose a multi-Gaussian process is observed at some set of sites, and we wish to obtain the conditional block grade distribution given some observations. We show that this conditional distribution is approximately Gaussian under certain conditions. In particular, given a single observation from a continuous multi-Gaussian process, the conditional distribution under a small change of support is approximately Gaussian unless, roughly speaking, the observed process is twice differentiable and the observation site is at the center of mass of the support region. A Gaussian approximation for the conditional prediction error of the total ore in a fixed region is considered also, although an example demonstrates that a naive analysis can give incorrect limiting conditional means. 相似文献
15.
Xavier Emery 《Mathematical Geology》2006,38(7):801-819
Multigaussian kriging aims at estimating the local distributions of regionalized variables and functions of these variables
(transfer or recovery functions) at unsampled locations. In this paper, we focus on the evaluation of the recoverable reserves
in an ore deposit accounting for a change of support and information effect caused by ore/waste misclassifications. Two approaches
are proposed: the multigaussian model with Monte Carlo integration and the discrete Gaussian model. The latter is simpler
to use but requires stronger hypotheses than the former. In each model, ordinary multigaussian kriging gives unbiased estimates
of the recoverable reserves that do not utilize the mean value of the normal score data.
The concepts are illustrated through a case study on a copper deposit which shows that local estimates of the metal content
based on ordinary multigaussian kriging are close to the optimal conditional expectation when the data are abundant and are
not dominated by the global mean when the data are scarce. The two proposed approaches (Monte Carlo integration and discrete
Gaussian model) lead to similar results when compared to two other geostatistical methods: service variables and ordinary
indicator kriging, which show strong deviations from conditional expectation. 相似文献
16.
Soil contamination by heavy metals and organic pollutants around industrial premises is a problem in many countries around
the world. Delineating zones where pollutants exceed tolerable levels is a necessity for successfully mitigating related health
risks. Predictions of pollutants are usually required for blocks because remediation or regulatory decisions are imposed for
entire parcels. Parcel areas typically exceed the observation support, but are smaller than the survey domain. Mapping soil
pollution therefore involves a local change of support. The goal of this work is to find a simple, robust, and precise method
for predicting block means (linear predictions) and threshold exceedance by block means (nonlinear predictions) from data
observed at points that show a spatial trend. By simulations, we compared the performance of universal block kriging (UK),
Gaussian conditional simulations (CS), constrained (CK), and covariance-matching constrained kriging (CMCK), for linear and
nonlinear local change of support prediction problems. We considered Gaussian and positively skewed spatial processes with
a nonstationary mean function and various scenarios for the autocorrelated error. The linear predictions were assessed by
bias and mean square prediction error and the nonlinear predictions by bias and Peirce skill scores. 相似文献
17.
A Bayesian linear inversion methodology based on Gaussian mixture models and its application to geophysical inverse problems are presented in this paper. The proposed inverse method is based on a Bayesian approach under the assumptions of a Gaussian mixture random field for the prior model and a Gaussian linear likelihood function. The model for the latent discrete variable is defined to be a stationary first-order Markov chain. In this approach, a recursive exact solution to an approximation of the posterior distribution of the inverse problem is proposed. A Markov chain Monte Carlo algorithm can be used to efficiently simulate realizations from the correct posterior model. Two inversion studies based on real well log data are presented, and the main results are the posterior distributions of the reservoir properties of interest, the corresponding predictions and prediction intervals, and a set of conditional realizations. The first application is a seismic inversion study for the prediction of lithological facies, P- and S-impedance, where an improvement of 30% in the root-mean-square error of the predictions compared to the traditional Gaussian inversion is obtained. The second application is a rock physics inversion study for the prediction of lithological facies, porosity, and clay volume, where predictions slightly improve compared to the Gaussian inversion approach. 相似文献
18.
J. A. Vargas-Guzmán 《Mathematical Geology》2004,36(3):307-322
This paper introduces geostatistical approaches (i.e., kriging estimation and simulation) for a group of non-Gaussian random fields that are power algebraic transformations of Gaussian and lognormal random fields. These are power random fields (PRFs) that allow the construction of stochastic polynomial series. They were derived from the exponential random field, which is expressed as Taylor series expansion with PRF terms. The equations developed from computation of moments for conditional random variables allow the correction of Gaussian kriging estimates for the non-Gaussian space. The introduced PRF geostatistics shall provide tools for integration of data that requires simple algebraic transformations, such as regression polynomials that are commonly encountered in the practical applications of estimation. The approach also allows for simulations drawn from skewed distributions. 相似文献
19.
Hypo-elastic relations are often adopted to simulate the recoverable non-linear behaviour of soils within elasto-plastic constitutive models. In reality, they are unable to reproduce the elastic, ie, recoverable, response of materials; hence, they introduce severe inconsistencies in models based on the decomposition of the total strain tensor into its recoverable and permanent parts. Hyper-elasticity should then be used. However, existing models developed within this framework do not satisfy a number of fundamental theoretical requirements. A new hyper-elastic model is proposed, which is rigourously formulated by integrating some of the main relations which emerge from experimental results. The model satisfies all theoretical requirements and also possesses features that are fundamental for its numerical integration. The model can be considered as the correct hyper-elastic version of the classical hypo-elastic constitutive relation adopted in models based on the Critical State framework, such as the Modified Cam-Clay, with a constant Poisson's ratio. 相似文献
20.
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