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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
3.
We consider point processes defined on the space–time domain which model physical processes characterized qualitatively by the gradual increase over time in some energy until a threshold is reached, after which, an event causing the loss of energy occurs. The risk function will, therefore, increase piecewise with sudden drops in correspondence to each event. This kind of behaviour is described by Reid's theory of elastic rebound in the earthquake generating process where the quantity that is accumulated is the strain energy or stress due to the relative movement of tectonic plates. The complexity and the intrinsic randomness of the phenomenon call for probabilistic models; in particular the stochastic translation of Reid's theory is given by stress release models. In this article we use such models to assess the time-dependent seismic hazard of the seismogenic zone of the Corinthos Gulf. For each event we consider the occurrence time and the magnitude, which is modelled by a probability distribution depending on the stress level present in the region at any instant. Hence we are dealing here with a marked point process. We perform the Bayesian analysis of this model by applying the stochastic simulation methods based on the generation of Markov chains, the so called Markov chain Monte Carlo (MCMC) methods, which allow one to reconcile the model's complexity with the computational burden of the inferential procedure. Stress release and Poisson models are compared on the basis of the Bayes factor. 相似文献
4.
Constraining stochastic models of reservoir properties such as porosity and permeability can be formulated as an optimization problem. While an optimization based on random search methods preserves the spatial variability of the stochastic model, it is prohibitively computer intensive. In contrast, gradient search methods may be very efficient but it does not preserve the spatial variability of the stochastic model. The gradual deformation method allows for modifying a reservoir model (i.e., realization of the stochastic model) from a small number of parameters while preserving its spatial variability. It can be considered as a first step towards the merger of random and gradient search methods. The gradual deformation method yields chains of reservoir models that can be investigated successively to identify an optimal reservoir model. The investigation of each chain is based on gradient computations, but the building of chains of reservoir models is random. In this paper, we propose an algorithm that further improves the efficiency of the gradual deformation method. Contrary to the previous gradual deformation method, we also use gradient information to build chains of reservoir models. The idea is to combine the initial reservoir model or the previously optimized reservoir model with a compound reservoir model. This compound model is a linear combination of a set of independent reservoir models. The combination coefficients are calculated so that the search direction from the initial model is as close as possible to the gradient search direction. This new gradual deformation scheme allows us for reducing the number of optimization parameters while selecting an optimal search direction. The numerical example compares the performance of the new gradual deformation scheme with that of the traditional one. 相似文献
5.
Victor N. Podkovyrov 《Mathematical Geology》1987,19(7):697-717
Grain sequences of Precambrian rapakivi granites of the Vyborg and Salmi Massifs have been compared with the stochastic model for ideal granite. These sequences show that classical rapakivi granites correspond to metasomatically weakly altered granites with a simple loss of Markov transitions from quartz and plagioclase. Observed parameters of the model indirectly indicate rapakivi magma had small volatile content and large viscosity which is also characteristic of many other Precambrian granites. 相似文献
6.
We present a method for fitting trishear models to surface profile data, by restoring bedding dip data and inverting for model parameters using a Markov chain Monte Carlo method. Trishear is a widely-used kinematic model for fault-propagation folds. It lacks an analytic solution, but a variety of data inversion techniques can be used to fit trishear models to data. Where the geometry of an entire folded bed is known, models can be tested by restoring the bed to its pre-folding orientation. When data include bedding attitudes, however, previous approaches have relied on computationally-intensive forward modeling. This paper presents an equation for the rate of change of dip in the trishear zone, which can be used to restore dips directly to their pre-folding values. The resulting error can be used to calculate a probability for each model, which allows solution by Markov chain Monte Carlo methods and inversion of datasets that combine dips and contact locations. These methods are tested using synthetic and real datasets. Results are used to approximate multimodal probability density functions and to estimate uncertainty in model parameters. The relative value of dips and contacts in constraining parameters and the effects of uncertainty in the data are investigated. 相似文献
7.
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
8.
岩土力学参数空间变异性的集合卡尔曼滤波估值 总被引：3，自引：1，他引：2
岩土参数具有结构性和随机性的空间变异特征,该特征导致岩土参数具有不确定性。以地质统计学作为岩土参数空间变异性分析的理论基础,将分布于研究区的岩土参数视为区域化变量,变异函数既描述了岩土参数整体的空间结构性变化,又描述了其局部的随机性变化,用变异函数理论模型作为描述岩土参数空间变异规律的数学模型。引入集合卡尔曼滤波（EnKF）分析方法,利用时空分布的观测数据,对岩土参数空间变异性进行估值。数值算例表明,EnKF能够有效地融合观测数据,较好地提供岩土参数空间变异性的估值。 相似文献
9.
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
10.
In this paper, given an estimate of the bearing capacity of the soil, by treating settlement at a given load as a random variable and the evolution of settlement of footing on cohesionless soil with the increasing load as a stochastic process, a tri-level homogeneous Markov chain (TLHMC) model is proposed for prediction of settlement. Comparison of the predicted mean and bounds on settlements, obtained using TLHMC, with the respective field values obtained from literature shows that the stochastic evolution can be modelled using TLHMC with a correlation coefficient of 0.90. A methodology for reliability-based design of footings is also presented and its use is demonstrated through a numerical example. 相似文献
11.
岩土参数具有结构性和随机性的空间变异特征，该特征导致岩土参数具有不确定性。以地质统计学作为岩土参数空间变异性分析的理论基础，将分布于研究区的岩土参数视为区域化变量，变异函数既描述了岩土参数整体的空间结构性变化，又描述了其局部的随机性变化，用变异函数理论模型作为描述岩土参数空间变异规律的数学模型。引入集合卡尔曼滤波（EnKF）分析方法，利用时空分布的观测数据，对岩土参数空间变异性进行估值。数值算例表明，EnKF能够有效地融合观测数据，较好地提供岩土参数空间变异性的估值。 相似文献
12.
Integrating production data under uncertainty by parallel interacting Markov chains on a reduced dimensional space 总被引：2，自引：0，他引：2
Thomas Romary 《Computational Geosciences》2009,13(1):103-122
In oil industry and subsurface hydrology, geostatistical models are often used to represent the porosity or the permeability
field. In history matching of a geostatistical reservoir model, we attempt to find multiple realizations that are conditional
to dynamic data and representative of the model uncertainty space. A relevant way to simulate the conditioned realizations
is by generating Monte Carlo Markov chains (MCMC). The huge dimensions (number of parameters) of the model and the computational
cost of each iteration are two important pitfalls for the use of MCMC. In practice, we have to stop the chain far before it
has browsed the whole support of the posterior probability density function. Furthermore, as the relationship between the
production data and the random field is highly nonlinear, the posterior can be strongly multimodal and the chain may stay
stuck in one of the modes. In this work, we propose a methodology to enhance the sampling properties of classical single MCMC
in history matching. We first show how to reduce the dimension of the problem by using a truncated Karhunen–Loève expansion
of the random field of interest and assess the number of components to be kept. Then, we show how we can improve the mixing
properties of MCMC, without increasing the global computational cost, by using parallel interacting Markov Chains. Finally,
we show the encouraging results obtained when applying the method to a synthetic history matching case. 相似文献
13.
Multivariate Spatial Modeling for Geostatistical Data Using Convolved Covariance Functions 总被引：1，自引：0，他引：1
Soil pollution data collection typically studies multivariate measurements at sampling locations, e.g., lead, zinc, copper
or cadmium levels. With increased collection of such multivariate geostatistical spatial data, there arises the need for flexible
explanatory stochastic models. Here, we propose a general constructive approach for building suitable models based upon convolution
of covariance functions. We begin with a general theorem which asserts that, under weak conditions, cross convolution of covariance
functions provides a valid cross covariance function. We also obtain a result on dependence induced by such convolution. Since,
in general, convolution does not provide closed-form integration, we discuss efficient computation.
We then suggest introducing such specification through a Gaussian process to model multivariate spatial random effects within
a hierarchical model. We note that modeling spatial random effects in this way is parsimonious relative to say, the linear
model of coregionalization. Through a limited simulation, we informally demonstrate that performance for these two specifications
appears to be indistinguishable, encouraging the parsimonious choice. Finally, we use the convolved covariance model to analyze
a trivariate pollution dataset from California. 相似文献
14.
The CMC (coupled Markov chain) model, which is based on the extension of Markov chains in two-dimensions, is used in the reduction
of uncertainty in geological structures when conditioned (i.e., honours the data and their location) on a number of boreholes.
The model has been applied to an unconsolidated aquifer deposit located in the central Rhine-Meuse delta (the Gorkum study
area) in the Netherlands. A comparison is also made between the CMC and the SIS (sequential indicator simulation) model, which
is based on Kriging and co-Kriging theories on the same deposit. The results show the potential applicability of the CMC model
in reducing the uncertainty in geological configurations when a sufficient number of boreholes is available. Reproduction
of the global geological features requires relatively few boreholes (in this case study, nine boreholes with 30-m spacing
over a distance of 240 m). However, reproduction of the proportion of each state requires a relatively large number of boreholes
(in this case study 31 boreholes with 8-m spacing over a distance of 240 m). It has been shown that variograms can be deceptive
in modeling the spatial pattern and that they reflect only part of the complete spatial structure in the field. The use of
transition probabilities via the CMC model provides a better alternative approach, because it uses multiple point information.
Amro M. M. Elfeki on leave from Department of Irrigation and Hydraulics, Faculty of Engineering, Mansoura University, Mansoura,
Egypt 相似文献
15.
D. N. Ivanov 《Mathematical Geology》1987,19(7):655-666
Petrographic data from the Iultin Massif (Chuckotka) were analyzed for crystallization and postmagmatic transformation using a Markov stochastic model. The goal was to ascertain whether this model can be used to divide the massif into distinctive portions. A comparison was made between the stochastic approach and conventional geological methods for understanding the historical development of granitic massifs. 相似文献
16.
INTRODUCTIoNImagetextureanalysisisanimportantpartofresearchin-tocomputervision.Forobjectidentificationandunderstand-ing,textureanalysisisthebasisoftheresearchwork.Ingeneral,texturecanberegardedasakindofstructurethatconsistsofmanytextureelementsorpatternswhicharemoreorlesssimilar,i.e.,primitivesthatformtexturesandspatialdependenceorinteractionbetweentheprimitives.Ex-tensiveresearchhasbeendoneintextureanalysis,andrefer-ence(Haralick,l979)hasmadeathoroughsurvey.Amongthesetextureanalysismetho… 相似文献
17.
Gradual Deformation and Iterative Calibration of Gaussian-Related Stochastic Models 总被引：11，自引：0，他引：11
Lin Y. Hu 《Mathematical Geology》2000,32(1):87-108
This paper describes a new method for gradually deforming realizations of Gaussian-related stochastic models while preserving their spatial variability. This method consists in building a stochastic process whose state space is the ensemble of the realizations of a spatial stochastic model. In particular, a stochastic process, built by combining independent Gaussian random functions, is proposed to perform the gradual deformation of realizations. Then, the gradual deformation algorithm is coupled with an optimization algorithm to calibrate realizations of stochastic models to nonlinear data. The method is applied to calibrate a continuous and a discrete synthetic permeability fields to well-test pressure data. The examples illustrate the efficiency of the proposed method. Furthermore, we present some extensions of this method (multidimensional gradual deformation, gradual deformation with respect to structural parameters, and local gradual deformation) that are useful in practice. Although the method described in this paper is operational only in the Gaussian framework (e.g., lognormal model, truncated Gaussian model, etc.), the idea of gradually deforming realizations through a stochastic process remains general and therefore promising even for calibrating non-Gaussian models. 相似文献
18.
M. Panzeri E. L. Della Rossa L. Dovera M. Riva A. Guadagnini 《Computational Geosciences》2016,20(3):637-653
We present a methodology that allows conditioning the spatial distribution of geological and petrophysical properties of reservoir model realizations on available production data. The approach is fully consistent with modern concepts depicting natural reservoirs as composite media where the distribution of both lithological units (or facies) and associated attributes are modeled as stochastic processes of space. We represent the uncertain spatial distribution of the facies through a Markov mesh (MM) model, which allows describing complex and detailed facies geometries in a rigorous Bayesian framework. The latter is then embedded within a history matching workflow based on an iterative form of the ensemble Kalman filter (EnKF). We test the proposed methodology by way of a synthetic study characterized by the presence of two distinct facies. We analyze the accuracy and computational efficiency of our algorithm and its ability with respect to the standard EnKF to properly estimate model parameters and assess future reservoir production. We show the feasibility of integrating MM in a data assimilation scheme. Our methodology is conducive to a set of updated model realizations characterized by a realistic spatial distribution of facies and their log permeabilities. Model realizations updated through our proposed algorithm correctly capture the production dynamics. 相似文献
19.
Yalchin Efendiev Akhil Datta-Gupta Xianlin Ma Bani Mallick 《Mathematical Geosciences》2008,40(2):213-232
In this paper, the Markov Chain Monte Carlo (MCMC) approach is used for sampling of the permeability field conditioned on
the dynamic data. The novelty of the approach consists of using an approximation of the dynamic data based on streamline computations.
The simulations using the streamline approach allows us to obtain analytical approximations in the small neighborhood of the
previously computed dynamic data. Using this approximation, we employ a two-stage MCMC approach. In the first stage, the approximation
of the dynamic data is used to modify the instrumental proposal distribution. The obtained chain correctly samples from the
posterior distribution; the modified Markov chain converges to a steady state corresponding to the posterior distribution.
Moreover, this approximation increases the acceptance rate, and reduces the computational time required for MCMC sampling.
Numerical results are presented. 相似文献
20.
Towards Stochastic Time-Varying Geological Modeling 总被引：3，自引：1，他引：2
Guillaume Caumon 《Mathematical Geosciences》2010,42(5):555-569
The modeling of subsurface geometry and properties is a key element to understand Earth processes and manage natural hazards
and resources. In this paper, we suggest this field should evolve beyond pure data fitting approaches by integrating geological
concepts to constrain interpretations or test their consistency. This process necessarily calls for adding the time dimension
to 3D modeling, both at the geological and human time scales. Also, instead of striving for one single best model, it is appropriate
to generate several possible subsurface models in order to convey a quantitative sense of uncertainty. Depending on the modeling
objective (e.g., quantification of natural resources, production forecast), this population of models can be ranked. Inverse
theory then provides a framework to validate (or rather invalidate) models which are not compatible with certain types of
observations. We review recent methods to better achieve both stochastic and time-varying geomodeling and advocate that the
application of inversion should rely not only on random field models, but also on geological concepts and parameters. 相似文献