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1.
The purpose of this paper is to extend the locally based prediction methodology of BayMar to a global one by modelling discrete spatial structures as Markov random fields. BayMar uses one-dimensional Markov-properties for estimating spatial correlation and Bayesian updating for locally integrating prior and additional information. The methodology of this paper introduces a new estimator of the field parameters based on the maximum likelihood technique for one-dimensional Markov chains. This makes the estimator straightforward to calculate also when there is a large amount of missing observations, which often is the case in geological applications. We make simulations (both unconditional and conditional on the observed data) and maximum a posteriori predictions (restorations) of the non-observed data using Markov chain Monte Carlo methods, in the restoration case by employing simulated annealing. The described method gives satisfactory predictions, while more work is needed in order to simulate, since it appears to have a tendency to overestimate strong spatial dependence. It provides an important development compared to the BayMar-methodology by facilitating global predictions and improved use of sparse data. 相似文献
2.
Seismic inverse modeling, which transforms appropriately processed geophysical data into the physical properties of the Earth, is an essential process for reservoir characterization. This paper proposes a work flow based on a Markov chain Monte Carlo method consistent with geology, well-logs, seismic data, and rock-physics information. It uses direct sampling as a multiple-point geostatistical method for generating realizations from the prior distribution, and Metropolis sampling with adaptive spatial resampling to perform an approximate sampling from the posterior distribution, conditioned to the geophysical data. Because it can assess important uncertainties, sampling is a more general approach than just finding the most likely model. However, since rejection sampling requires a large number of evaluations for generating the posterior distribution, it is inefficient and not suitable for reservoir modeling. Metropolis sampling is able to perform an equivalent sampling by forming a Markov chain. The iterative spatial resampling algorithm perturbs realizations of a spatially dependent variable, while preserving its spatial structure by conditioning to subset points. However, in most practical applications, when the subset conditioning points are selected at random, it can get stuck for a very long time in a non-optimal local minimum. In this paper it is demonstrated that adaptive subset sampling improves the efficiency of iterative spatial resampling. Depending on the acceptance/rejection criteria, it is possible to obtain a chain of geostatistical realizations aimed at characterizing the posterior distribution with Metropolis sampling. The validity and applicability of the proposed method are illustrated by results for seismic lithofacies inversion on the Stanford VI synthetic test sets. 相似文献
3.
Comparing the Gradual Deformation with the Probability Perturbation Method for Solving Inverse Problems 总被引：1，自引：0，他引：1
Jef Caers 《Mathematical Geology》2007,39(1):27-52
Inverse problems are ubiquitous in the Earth Sciences. Many such problems are ill-posed in the sense that multiple solutions
can be found that match the data to be inverted. To impose restrictions on these solutions, a prior distribution of the model
parameters is required. In a spatial context this prior model can be as simple as a Multi-Gaussian law with prior covariance
matrix, or could come in the form of a complex training image describing the prior statistics of the model parameters. In
this paper, two methods for generating inverse solutions constrained to such prior model are compared. The gradual deformation
method treats the problem of finding inverse solution as an optimization problem. Using a perturbation mechanism, the gradual
deformation method searches (optimizes) in the prior model space for those solutions that match the data to be inverted. The
perturbation mechanism guarantees that the prior model statistics are honored. However, it is shown with a simple example
that this perturbation method does not necessarily draw accurately samples from a given posterior distribution when the inverse
problem is framed within a Bayesian context. On the other hand, the probability perturbation method approaches the inverse
problem as a data integration problem. This method explicitly deals with the problem of combining prior probabilities with
pre-posterior probabilities derived from the data. It is shown that the sampling properties of the probability perturbation
method approach the accuracy of well-known Markov chain Monte Carlo samplers such as the rejection sampler. The paper uses
simple examples to illustrate the clear differences between these two methods 相似文献
4.
The spatial continuity of facies is one of the key factors controlling flow in reservoir models. Traditional pixel-based methods
such as truncated Gaussian random fields and indicator simulation are based on only two-point statistics, which is insufficient
to capture complex facies structures. Current methods for multi-point statistics either lack a consistent statistical model
specification or are too computer intensive to be applicable. We propose a Markov mesh model based on generalized linear models
for geological facies modeling. The approach defines a consistent statistical model that is facilitated by efficient estimation
of model parameters and generation of realizations. Our presentation includes a formulation of the general framework, model
specifications in two and three dimensions, and details on how the parameters can be estimated from a training image. We illustrate
the method using multiple training images, including binary and trinary images and simulations in two and three dimensions.
We also do a thorough comparison to the snesim approach. We find that the current model formulation is applicable for multiple training images and compares favorably to
the snesim approach in our test examples. The method is highly memory efficient. 相似文献
5.
Construction of Binary Multi-grid Markov Random Field Prior Models from Training Images 总被引：1，自引：1，他引：0
Bayesian modeling requires the specification of prior and likelihood models. In reservoir characterization, it is common practice to estimate the prior from a training image. This paper considers a multi-grid approach for the construction of prior models for binary variables. On each grid level we adopt a Markov random field (MRF) conditioned on values in previous levels. Parameter estimation in MRFs is complicated by a computationally intractable normalizing constant. To cope with this problem, we generate a partially ordered Markov model (POMM) approximation to the MRF and use this in the model fitting procedure. Approximate unconditional simulation from the fitted model can easily be done by again adopting the POMM approximation to the fitted MRF. Approximate conditional simulation, for a given and easy to compute likelihood function, can also be performed either by the Metropolis–Hastings algorithm based on an approximation to the fitted MRF or by constructing a new POMM approximation to this approximate conditional distribution. The proposed methods are illustrated using three frequently used binary training images. 相似文献
6.
The Bayesian Maximum Entropy (BME) method of spatial analysis and mapping provides definite rules for incorporating prior information, hard and soft data into the mapping process. It has certain unique features that make it a loyal guardian of plausible reasoning under conditions of uncertainty. BME is a general approach that does not make any assumptions regarding the linearity of the estimator, the normality of the underlying probability laws, or the homogeneity of the spatial distribution. By capitalizing on various sources of information and data, BME introduces an epistemological framework that produces predictive maps that are more accurate and in many cases computationally more efficient than those derived by traditional techniques. In fact, kriging techniques can be derived as special cases of the BME approach, under restrictive assumptions regarding the prior information and the data available. BME is a more rigorous approach than indicator kriging for incorporating soft data. The BME formulation, in fact, applies in a spatial or a spatiotemporal domain and its extension to the case of block and vector random fields is straightforward. New theoretical results are presented and numerical examples are discussed, which use the BME approach to account for important sources of knowledge in a systematic manner. BME can be useful in practical situations in which prior information can be used to compensate for the limited amount of measurements available (e.g., preliminary or feasibility study levels) or soft data are available that can be combined with hard data to improve mapping significantly. BME may be then viewed as an effort towards the development of a more general framework of spatial/temporal analysis and mapping, which includes traditional geostatistics as its limiting case, and it also provides the means to derive novel results that could not be obtained by traditional geostatistics. 相似文献
7.
8.
长大深基坑施工空间效应研究 总被引：6，自引：0，他引：6
针对目前长大深基坑施工空间效应的研究主要集中在数值模拟方面,理论研究尚不多见的现状,提出了黏性土条件下长大深基坑施工空间效应的简化计算方法。引入等代内摩擦角的概念,将基坑坑周土层等代为无黏性均质体,应用土的塑性上限理论及相关联流动法则,采用极限平衡分析法,对长大深基坑拉裂-剪切和纯剪切两种三维破坏模式下的空间效应进行了具体研究,给出了相应的空间效应系数计算公式,并进行了算例分析。研究结果表明,该方法避免了数值模拟的低效率、高费用、长周期的缺陷,可手算完成;计算成果能直接得出基坑坑壁空间效应系数的分布特征及量值大小,可用于指导基坑支护结构、施工方案的设计以及信息化监测测点布置和断面选择。 相似文献
9.
Thomas Romary 《Computational Geosciences》2010,14(2):343-355
In history matching of lithofacies reservoir model, we attempt to find multiple realizations of lithofacies configuration
that are conditional to dynamic data and representative of the model uncertainty space. This problem can be formalized in
the Bayesian framework. Given a truncated Gaussian model as a prior and the dynamic data with its associated measurement error,
we want to sample from the conditional distribution of the facies given the data. A relevant way to generate conditioned realizations
is to use Markov chains Monte Carlo (MCMC). However, the dimensions of the model and the computational cost of each iteration
are two important pitfalls for the use of MCMC. Furthermore, classical MCMC algorithms mix slowly, that is, they will not
explore the whole support of the posterior in the time of the simulation. In this paper, we extend the methodology already
described in a previous work to the problem of history matching of a Gaussian-related lithofacies reservoir model. We first
show how to drastically reduce the dimension of the problem by using a truncated Karhunen-Loève expansion of the Gaussian
random field underlying the lithofacies model. Moreover, we propose an innovative criterion of the choice of the number of
components based on the connexity function. Then, we show how we improve the mixing properties of classical single MCMC, without
increasing the global computational cost, by the use of parallel interacting Markov chains. Applying the dimension reduction
and this innovative sampling method drastically lowers the number of iterations needed to sample efficiently from the posterior.
We show the encouraging results obtained when applying the methodology to a synthetic history-matching case. 相似文献
10.
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
11.
Geostatistical analyses require an estimation of the covariance structure of a random field and its parameters jointly from
noisy data. Whereas in some cases (as in that of a Matérn variogram) a range of structural models can be captured with one
or a few parameters, in many other cases it is necessary to consider a discrete set of structural model alternatives, such
as drifts and variograms. Ranking these alternatives and identifying the best among them has traditionally been done with
the aid of information theoretic or Bayesian model selection criteria. There is an ongoing debate in the literature about
the relative merits of these various criteria. We contribute to this discussion by using synthetic data to compare the abilities
of two common Bayesian criteria, BIC and KIC, to discriminate between alternative models of drift as a function of sample size when drift and variogram parameters are
unknown. Adopting the results of Markov Chain Monte Carlo simulations as reference we confirm that KIC reduces asymptotically to BIC and provides consistently more reliable indications of model quality than does BIC for samples of all sizes. Practical considerations often cause analysts to replace the observed Fisher information matrix
entering into KIC with its expected value. Our results show that this causes the performance of KIC to deteriorate with diminishing sample size. These results are equally valid for one and multiple realizations of uncertain
data entering into our analysis. Bayesian theory indicates that, in the case of statistically independent and identically
distributed data, posterior model probabilities become asymptotically insensitive to prior probabilities as sample size increases.
We do not find this to be the case when working with samples taken from an autocorrelated random field. 相似文献
12.
A Methodology for Establishing a Data Reliability Measure for Value of Spatial Information Problems 总被引：2，自引：0，他引：2
We propose a value of information (VOI) methodology for spatial Earth problems. VOI is a tool to determine whether purchasing
a new information source would improve a decision-makers’ chances of taking the optimal action. A prior uncertainty assessment
of key geologic parameters and a reliability of the data to resolve them are necessary to make a VOI assessment. Both of these
elements are challenging to obtain, as this assessment is made before the information is acquired. We present a flexible prior
geologic uncertainty modeling scheme that allows for the inclusion of many types of spatial parameter. Next, we describe how
to obtain a physics-based reliability measure by simulating the geophysical measurement on the generated prior models and
interpreting the simulated data. Repeating this simulation and interpretation for all datasets, a frequency table can be obtained
that describes how many times a correct or false interpretation was made by comparing them to their respective original model.
This frequency table is the reliability measure and allows a more realistic VOI calculation. An example VOI calculation is
demonstrated for a spatial decision related to aquifer recharge where two geophysical techniques are considered for their
ability to resolve channel orientations. As necessitated by spatial problems, this methodology preserves the structure, influence
and dependence of spatial variables through the prior geological modeling and the explicit geophysical simulation and interpretations. 相似文献
13.
Weidong Li 《Mathematical Geology》2007,39(3):321-335
Multi-dimensional Markov chain conditional simulation (or interpolation) models have potential for predicting and simulating
categorical variables more accurately from sample data because they can incorporate interclass relationships. This paper introduces
a Markov chain random field (MCRF) theory for building one to multi-dimensional Markov chain models for conditional simulation
(or interpolation). A MCRF is defined as a single spatial Markov chain that moves (or jumps) in a space, with its conditional
probability distribution at each location entirely depending on its nearest known neighbors in different directions. A general
solution for conditional probability distribution of a random variable in a MCRF is derived explicitly based on the Bayes’
theorem and conditional independence assumption. One to multi-dimensional Markov chain models for prediction and conditional
simulation of categorical variables can be drawn from the general solution and MCRF-based multi-dimensional Markov chain models
are nonlinear. 相似文献
14.
Weidong Li 《Mathematical Geosciences》2007,39(3):321-335
Multi-dimensional Markov chain conditional simulation (or interpolation) models have potential for predicting and simulating categorical variables more accurately from sample data because they can incorporate interclass relationships. This paper introduces a Markov chain random field (MCRF) theory for building one to multi-dimensional Markov chain models for conditional simulation (or interpolation). A MCRF is defined as a single spatial Markov chain that moves (or jumps) in a space, with its conditional probability distribution at each location entirely depending on its nearest known neighbors in different directions. A general solution for conditional probability distribution of a random variable in a MCRF is derived explicitly based on the Bayes’ theorem and conditional independence assumption. One to multi-dimensional Markov chain models for prediction and conditional simulation of categorical variables can be drawn from the general solution and MCRF-based multi-dimensional Markov chain models are nonlinear. 相似文献
15.
A Bayesian/maximum-entropy view to the spatial estimation problem 总被引：12，自引：0，他引：12
George Christakos 《Mathematical Geology》1990,22(7):763-777
The purpose of this paper is to stress the importance of a Bayesian/maximum-entropy view toward the spatial estimation problem. According to this view, the estimation equations emerge through a process that balances two requirements: High prior information about the spatial variability and high posterior probability about the estimated map. The first requirement uses a variety of sources of prior information and involves the maximization of an entropy function. The second requirement leads to the maximization of a so-called Bayes function. Certain fundamental results and attractive features of the proposed approach in the context of the random field theory are discussed, and a systematic spatial estimation scheme is presented. The latter satisfies a variety of useful properties beyond those implied by the traditional stochastic estimation methods. 相似文献
16.
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
17.
Updating Markov chain models using the ensemble Kalman filter 总被引：1，自引：0，他引：1
18.
A Fixed-Path Markov Chain Algorithm for Conditional Simulation of Discrete Spatial Variables 总被引：4，自引：0，他引：4
Weidong Li 《Mathematical Geology》2007,39(2):159-176
The Markov chain random field (MCRF) theory provided the theoretical foundation for a nonlinear Markov chain geostatistics.
In a MCRF, the single Markov chain is also called a “spatial Markov chain” (SMC). This paper introduces an efficient fixed-path SMC algorithm for conditional simulation of discrete spatial variables
(i.e., multinomial classes) on point samples with incorporation of interclass dependencies. The algorithm considers four nearest
known neighbors in orthogonal directions. Transiograms are estimated from samples and are model-fitted to provide parameter
input to the simulation algorithm. Results from a simulation example show that this efficient method can effectively capture
the spatial patterns of the target variable and fairly generate all classes. Because of the incorporation of interclass dependencies
in the simulation algorithm, simulated realizations are relatively imitative of each other in patterns. Large-scale patterns
are well produced in realizations. Spatial uncertainty is visualized as occurrence probability maps, and transition zones
between classes are demonstrated by maximum occurrence probability maps. Transiogram analysis shows that the algorithm can
reproduce the spatial structure of multinomial classes described by transiograms with some ergodic fluctuations. A special
characteristic of the method is that when simulation is conditioned on a number of sample points, simulated transiograms have
the tendency to follow the experimental ones, which implies that conditioning sample data play a crucial role in determining
spatial patterns of multinomial classes. The efficient algorithm may provide a powerful tool for large-scale structure simulation
and spatial uncertainty analysis of discrete spatial variables. 相似文献
19.
We present a general framework that enables decision-making when a threshold in a process is about to be exceeded (an event).
Measurements are combined with prior information to update the probability of such an event. This prior information is derived
from the results of an ensemble of model realisations that span the uncertainty present in the model before any measurements
are collected; only probability updates need to be calculated, which makes the procedure very fast once the basic ensemble
of realisations has been set up. The procedure is demonstrated with an example where gas field production is restricted to
a maximum amount of subsidence. Starting with 100 realisations spanning the prior uncertainty of the process, the measurements
collected during monitoring bolster some of the realisations and expose others as irrelevant. In this procedure, more data
will mean a sharper determination of the posterior probability. We show the use of two different types of limits, a maximum
allowed value of subsidence and a maximum allowed value of subsidence rate for all measurement points at all times. These
limits have been applied in real world cases. The framework is general and is able to deal with other types of limits in just
the same way. It can also be used to optimise monitoring strategies by assessing the effect of the number, position and timing
of the measurement points. Furthermore, in such a synthetic study, the prior realisations do not need to be updated; spanning
the range of uncertainty with appropriate prior models is sufficient. 相似文献
20.
Conditioning complex subsurface flow models on nonlinear data is complicated by the need to preserve the expected geological connectivity patterns to maintain solution plausibility. Generative adversarial networks (GANs) have recently been proposed as a promising approach for low-dimensional representation of complex high-dimensional images. The method has also been adopted for low-rank parameterization of complex geologic models to facilitate uncertainty quantification workflows. A difficulty in adopting these methods for subsurface flow modeling is the complexity associated with nonlinear flow data conditioning. While conditional GAN (CGAN) can condition simulated images on labels, application to subsurface problems requires efficient conditioning workflows for nonlinear data, which is far more complex. We present two approaches for generating flow-conditioned models with complex spatial patterns using GAN. The first method is through conditional GAN, whereby a production response label is used as an auxiliary input during the training stage of GAN. The production label is derived from clustering of the flow responses of the prior model realizations (i.e., training data). The underlying assumption of this approach is that GAN can learn the association between the spatial features corresponding to the production responses within each cluster. An alternative method is to use a subset of samples from the training data that are within a certain distance from the observed flow responses and use them as training data within GAN to generate new model realizations. In this case, GAN is not required to learn the nonlinear relation between production responses and spatial patterns. Instead, it is tasked to learn the patterns in the selected realizations that provide a close match to the observed data. The conditional low-dimensional parameterization for complex geologic models with diverse spatial features (i.e., when multiple geologic scenarios are plausible) performed by GAN allows for exploring the spatial variability in the conditional realizations, which can be critical for decision-making. We present and discuss the important properties of GAN for data conditioning using several examples with increasing complexity.
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