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1.
 Being a non-linear method based on a rigorous formalism and an efficient processing of various information sources, the Bayesian maximum entropy (BME) approach has proven to be a very powerful method in the context of continuous spatial random fields, providing much more satisfactory estimates than those obtained from traditional linear geostatistics (i.e., the various kriging techniques). This paper aims at presenting an extension of the BME formalism in the context of categorical spatial random fields. In the first part of the paper, the indicator kriging and cokriging methods are briefly presented and discussed. A special emphasis is put on their inherent limitations, both from the theoretical and practical point of view. The second part aims at presenting the theoretical developments of the BME approach for the case of categorical variables. The three-stage procedure is explained and the formulations for obtaining prior joint distributions and computing posterior conditional distributions are given for various typical cases. The last part of the paper consists in a simulation study for assessing the performance of BME over the traditional indicator (co)kriging techniques. The results of these simulations highlight the theoretical limitations of the indicator approach (negative probability estimates, probability distributions that do not sum up to one, etc.) as well as the much better performance of the BME approach. Estimates are very close to the theoretical conditional probabilities, that can be computed according to the stated simulation hypotheses.  相似文献   

2.
Categorical data play an important role in a wide variety of spatial applications, while modeling and predicting this type of statistical variable has proved to be complex in many cases. Among other possible approaches, the Bayesian maximum entropy methodology has been developed and advocated for this goal and has been successfully applied in various spatial prediction problems. This approach aims at building a multivariate probability table from bivariate probability functions used as constraints that need to be fulfilled, in order to compute a posterior conditional distribution that accounts for hard or soft information sources. In this paper, our goal is to generalize further the theoretical results in order to account for a much wider type of information source, such as probability inequalities. We first show how the maximum entropy principle can be implemented efficiently using a linear iterative approximation based on a minimum norm criterion, where the minimum norm solution is obtained at each step from simple matrix operations that converges to the requested maximum entropy solution. Based on this result, we show then how the maximum entropy problem can be related to the more general minimum divergence problem, which might involve equality and inequality constraints and which can be solved based on iterated minimum norm solutions. This allows us to account for a much larger panel of information types, where more qualitative information, such as probability inequalities can be used. When combined with a Bayesian data fusion approach, this approach deals with the case of potentially conflicting information that is available. Although the theoretical results presented in this paper can be applied to any study (spatial or non-spatial) involving categorical data in general, the results are illustrated in a spatial context where the goal is to predict at best the occurrence of cultivated land in Ethiopia based on crowdsourced information. The results emphasize the benefit of the methodology, which integrates conflicting information and provides a spatially exhaustive map of these occurrence classes over the whole country.  相似文献   

3.
Using auxiliary information to improve the prediction accuracy of soil properties in a physically meaningful and technically efficient manner has been widely recognized in pedometrics. In this paper, we explored a novel technique to effectively integrate sampling data and auxiliary environmental information, including continuous and categorical variables, within the framework of the Bayesian maximum entropy (BME) theory. Soil samples and observed auxiliary variables were combined to generate probability distributions of the predicted soil variable at unsampled points. These probability distributions served as soft data of the BME theory at the unsampled locations, and, together with the hard data (sample points) were used in spatial BME prediction. To gain practical insight, the proposed approach was implemented in a real-world case study involving a dataset of soil total nitrogen (TN) contents in the Shayang County of the Hubei Province (China). Five terrain indices, soil types, and soil texture were used as auxiliary variables to generate soft data. Spatial distribution of soil total nitrogen was predicted by BME, regression kriging (RK) with auxiliary variables, and ordinary kriging (OK). The results of the prediction techniques were compared in terms of the Pearson correlation coefficient (r), mean error (ME), and root mean squared error (RMSE). These results showed that the BME predictions were less biased and more accurate than those of the kriging techniques. In sum, the present work extended the BME approach to implement certain kinds of auxiliary information in a rigorous and efficient manner. Our findings showed that the BME prediction technique involving the transformation of variables into soft data can improve prediction accuracy considerably, compared to other techniques currently in use, like RK and OK.  相似文献   

4.
The Bayesian maximum entropy (BME) method can be used to predict the value of a spatial random field at an unsampled location given precise (hard) and imprecise (soft) data. It has mainly been used when the data are non-skewed. When the data are skewed, the method has been used by transforming the data (usually through the logarithmic transform) in order to remove the skew. The BME method is applied for the transformed variable, and the resulting posterior distribution transformed back to give a prediction of the primary variable. In this paper, we show how the implementation of the BME method that avoids the use of a transform, by including the logarithmic statistical moments in the general knowledge base, gives more appropriate results, as expected from the maximum entropy principle. We use a simple illustration to show this approach giving more intuitive results, and use simulations to compare the approaches in terms of the prediction errors. The simulations show that the BME method with the logarithmic moments in the general knowledge base reduces the errors, and we conclude that this approach is more suitable to incorporate soft data in a spatial analysis for lognormal data.  相似文献   

5.
Bayesian data fusion in a spatial prediction context: a general formulation   总被引：1，自引：1，他引：1  
In spite of the exponential growth in the amount of data that one may expect to provide greater modeling and predictions opportunities, the number and diversity of sources over which this information is fragmented is growing at an even faster rate. As a consequence, there is real need for methods that aim at reconciling them inside an epistemically sound theoretical framework. In a statistical spatial prediction framework, classical methods are based on a multivariate approach of the problem, at the price of strong modeling hypotheses. Though new avenues have been recently opened by focusing on the integration of uncertain data sources, to the best of our knowledges there have been no systematic attemps to explicitly account for information redundancy through a data fusion procedure. Starting from the simple concept of measurement errors, this paper proposes an approach for integrating multiple information processing as a part of the prediction process itself through a Bayesian approach. A general formulation is first proposed for deriving the prediction distribution of a continuous variable of interest at unsampled locations using on more or less uncertain (soft) information at neighboring locations. The case of multiple information is then considered, with a Bayesian solution to the problem of fusing multiple information that are provided as separate conditional probability distributions. Well-known methods and results are derived as limit cases. The convenient hypothesis of conditional independence is discussed by the light of information theory and maximum entropy principle, and a methodology is suggested for the optimal selection of the most informative subset of information, if needed. Based on a synthetic case study, an application of the methodology is presented and discussed.  相似文献   

6.
Inverse distance interpolation for facies modeling   总被引：1，自引：0，他引：1  
Inverse distance weighted interpolation is a robust and widely used estimation technique. In practical applications, inverse distance interpolation is oftentimes favored over kriging-based techniques when there is a problem of making meaningful estimates of the field spatial structure. Nowadays application of inverse distance interpolation is limited to continuous random variable modeling. There is a need to extend the approach to categorical/discrete random variables. In this paper we propose such an extension using indicator formalism. The applicability of inverse distance interpolation for categorical modeling is then illustrated using Total’s Joslyn Lease facies data.  相似文献   

7.
With rapid advances of geospatial technologies, the amount of spatial data has been increasing exponentially over the past few decades. Usually collected by diverse source providers, the available spatial data tend to be fragmented by a large variety of data heterogeneities, which highlights the need of sound methods capable of efficiently fusing the diverse and incompatible spatial information. Within the context of spatial prediction of categorical variables, this paper describes a statistical framework for integrating and drawing inferences from a collection of spatially correlated variables while accounting for data heterogeneities and complex spatial dependencies. In this framework, we discuss the spatial prediction of categorical variables in the paradigm of latent random fields, and represent each spatial variable via spatial covariance functions, which define two-point similarities or dependencies of spatially correlated variables. The representation of spatial covariance functions derived from different spatial variables is independent of heterogeneous characteristics and can be combined in a straightforward fashion. Therefore it provides a unified and flexible representation of heterogeneous spatial variables in spatial analysis while accounting for complex spatial dependencies. We show that in the spatial prediction of categorical variables, the sought-after class occurrence probability at a target location can be formulated as a multinomial logistic function of spatial covariances of spatial variables between the target and sampled locations. Group least absolute shrinkage and selection operator is adopted for parameter estimation, which prevents the model from over-fitting, and simultaneously selects an optimal subset of important information (variables). Synthetic and real case studies are provided to illustrate the introduced concepts, and showcase the advantages of the proposed statistical framework.  相似文献   

8.
Application of the BME approach to soil texture mapping   总被引：3，自引：1，他引：3  
In order to derive accurate space/time maps of soil properties, soil scientists need tools that combine the usually scarce hard data sets with the more easily accessible soft data sets. In the field of modern geostatistics, the Bayesian maximum entropy (BME) approach provides new and powerful means for incorporating various forms of physical knowledge (including hard and soft data, soil classification charts, land cover data from satellite pictures, and digital elevation models) into the space/time mapping process. BME produces the complete probability distribution at each estimation point, thus allowing the calculation of elaborate statistics (even when the distribution is not Gaussian). It also offers a more rigorous and systematic method than kriging for integrating uncertain information into space/time mapping. In this work, BME is used to estimate the three textural fractions involved in a texture map. The first case study focuses on the estimation of the clay fraction, whereas the second one considers the three textural fractions (sand, silt and clay) simultaneously. The BME maps obtained are informative (important soil characteristics are identified, natural variations are well reproduced, etc.). Furthermore, in both case studies, the estimates obtained by BME were more accurate than the simple kriging (SK) estimates, thus offering a better picture of soil reality. In the multivariate case, classification error rate analysis in terms of BME performs considerably better than in terms of kriging. Analysis in terms of BME can offer valuable information to be used in sampling design, in optimizing the hard to soft data ratio, etc.  相似文献   

9.
Assimilation of fuzzy data by the BME method   总被引：1，自引：1，他引：0  
Modern spatiotemporal geostatistics provides a powerful framework for generation of predictive maps over a spatiotemporal domain by accounting for general knowledge to define a space of plausible events and then restricting this space of plausible events to be consistent with available site-specific knowledge. The Bayesian maximum entropy (BME) method is one of the most widely used modern geostatistics methods. BME results from assigning probabilities of plausible events based on general knowledge through information maximization and then applying operational Bayesian conditionalization that can explicitly assimilate stochastic representations of various uncertain (soft) data bases. The paper demonstrates that fuzzy data sets can be indirectly assimilated by BME through a two-step process: (a) reinterpretation of the fuzzy data as probabilistic through a generalized defuzzification procedure, and (b) efficient assimilation of the probabilistic results of generalized defuzzification by the BME method. A numerical demonstration involves site-specific probabilistic results obtained from the generalized defuzzification of a simulated fuzzy data set and general knowledge that includes the spatial mean trend and correlation structure models. The parameters of these models can be inferred from the hard data equivalent values of the probabilistic results. Accordingly, details of inference based on probabilistic soft data are also considered.  相似文献   

10.
The mapping of saline soils is the first task before any reclamation effort. Reclamation is based on the knowledge of soil salinity in space and how it evolves with time. Soil salinity is traditionally determined by soil sampling and laboratory analysis. Recently, it became possible to complement these hard data with soft secondary data made available using field sensors like electrode probes. In this study, we had two data sets. The first includes measurements of field salinity (ECa) at 413 locations and 19 time instants. The second, which is a subset of the first (13 to 20 locations), contains, in addition to ECa, salinity determined in the laboratory (EC2.5). Based on a procedure of cross-validation, we compared the prediction performance in the space-time domain of 3 methods: kriging using either only hard data (HK) or hard and mid interval soft data (HMIK), and Bayesian maximum entropy (BME) using probabilistic soft data. We found that BME was less biased, more accurate and giving estimates, which were better correlated with the observed values than the two kriging techniques. In addition, BME allowed one to delineate with better detail saline from non-saline areas.  相似文献   

11.
We propose a spectral turning-bands approach for the simulation of second-order stationary vector Gaussian random fields. The approach improves existing spectral methods through coupling with importance sampling techniques. A notable insight is that one can simulate any vector random field whose direct and cross-covariance functions are continuous and absolutely integrable, provided that one knows the analytical expression of their spectral densities, without the need for these spectral densities to have a bounded support. The simulation algorithm is computationally faster than circulant-embedding techniques, lends itself to parallel computing and has a low memory storage requirement. Numerical examples with varied spatial correlation structures are presented to demonstrate the accuracy and versatility of the proposal.  相似文献   

12.
The spatial distribution of residual light non-aqueous phase liquid (LNAPL) is an important factor in reactive solute transport modeling studies. There is great uncertainty associated with both the areal limits of LNAPL source zones and smaller scale variability within the areal limits. A statistical approach is proposed to construct a probabilistic model for the spatial distribution of residual NAPL and it is applied to a site characterized by ultra-violet-induced-cone-penetration testing (CPT–UVIF). The uncertainty in areal limits is explicitly addressed by a novel distance function (DF) approach. In modeling the small-scale variability within the areal limits, the CPT–UVIF data are used as primary source of information, while soil texture and distance to water table are treated as secondary data. Two widely used geostatistical techniques are applied for the data integration, namely sequential indicator simulation with locally varying means (SIS–LVM) and Bayesian updating (BU). A close match between the calibrated uncertainty band (UB) and the target probabilities shows the performance of the proposed DF technique in characterization of uncertainty in the areal limits. A cross-validation study also shows that the integration of the secondary data sources substantially improves the prediction of contaminated and uncontaminated locations and that the SIS–LVM algorithm gives a more accurate prediction of residual NAPL contamination. The proposed DF approach is useful in modeling the areal limits of the non-stationary continuous or categorical random variables, and in providing a prior probability map for source zone sizes to be used in Monte Carlo simulations of contaminant transport or Monte Carlo type inverse modeling studies.  相似文献   

13.
Input data selection for solar radiation estimation   总被引：1，自引：0，他引：1  
Model input data selection is a complicated process, especially for non‐linear dynamic systems. The questions on which inputs should be used and how long the training data should be for model development have been hard to solve in practice. Despite the importance of this subject, there have been insufficient reports in the published literature about inter‐comparison between different model input data selection techniques. In this study, several methods (i.e. the Gamma test, entropy theory, AIC (Akaike's information criterion)/BIC (Bayesian information criterion) have been explored with the aid of non‐linear models of LLR (local linear regression) and ANN (artificial neural networks). The methodology is tested in estimation of solar radiation in the Brue Catchment of England. It has been found that the conventional model selection tools such as AIC/BIC failed to demonstrate their functionality. Although the entropy theory is quite powerful and efficient to compute, it failed to pick up the best input combinations. On the other hand, it is very encouraging to find that the new Gamma test was able to choose the best input selection. However, it is surprising to note that the Gamma test significantly underestimated the required training data while the entropy theory did a better job in this aspect. This is the first study to compare the performance of those techniques for model input selections and still there are many unsolved puzzles. Copyright © 2009 John Wiley & Sons, Ltd.  相似文献   

14.
李光辉  李月 《地球物理学报》2015,58(12):4576-4593
消减随机噪声是目前陆地地震勘探数据处理的关键问题之一,分析随机噪声的产生机制及特征是对其进行有效压制的先决条件.本文针对中国南方山地金属矿区的勘探环境,根据随机噪声中包含的自然噪声和人文噪声的发声机理分别确定其噪声源函数,以波动方程作为噪声传播模型对山地地区随机噪声进行建模,将随机噪声作为一个综合波场,并且与实际噪声记录进行比较.随机噪声记录作为时空域的二维随机过程,分别对模拟噪声和实际噪声记录的时间域波形(振动图)特征包括频谱、功率谱密度,相空间轨迹图,统计量特征(能量分布,累积分布,均值,方差,峰度,偏度),和空间域波形(波剖面)特征包括波数谱和统计量特征进行比较,对比结果显示在时空域模拟噪声和实际噪声都有基本相同的性质,证明了本文对随机噪声模拟方法的可行性,为进一步研究随机噪声时空域传播特性以及噪声消除奠定理论基础.  相似文献   

15.
Interpolation techniques for spatial data have been applied frequently in various fields of geosciences. Although most conventional interpolation methods assume that it is sufficient to use first- and second-order statistics to characterize random fields, researchers have now realized that these methods cannot always provide reliable interpolation results, since geological and environmental phenomena tend to be very complex, presenting non-Gaussian distribution and/or non-linear inter-variable relationship. This paper proposes a new approach to the interpolation of spatial data, which can be applied with great flexibility. Suitable cross-variable higher-order spatial statistics are developed to measure the spatial relationship between the random variable at an unsampled location and those in its neighbourhood. Given the computed cross-variable higher-order spatial statistics, the conditional probability density function is approximated via polynomial expansions, which is then utilized to determine the interpolated value at the unsampled location as an expectation. In addition, the uncertainty associated with the interpolation is quantified by constructing prediction intervals of interpolated values. The proposed method is applied to a mineral deposit dataset, and the results demonstrate that it outperforms kriging methods in uncertainty quantification. The introduction of the cross-variable higher-order spatial statistics noticeably improves the quality of the interpolation since it enriches the information that can be extracted from the observed data, and this benefit is substantial when working with data that are sparse or have non-trivial dependence structures.  相似文献   

16.
Entropy-based correlated shrinkage of spatial random processes   总被引：1，自引：1，他引：0  
This paper proposes a two-stage correlated non-linear shrinkage estimation methodology for spatial random processes. A block hard thresholding design, based on Shannon’s entropy, is formulated in the first stage. The thresholding design is adaptive to each resolution level, because it depends on the empirical distribution function of the mutual information ratios between empirical wavelet blocks and the random variables of interest, at each scale. In the second stage, a global correlated (inter- and intra-scale) shrinkage is applied to approximate the values of interest of the underlying spatial process. Additionally, a simulation study is developed, in the Gaussian context, to analyze the sensitivity, measured by empirical stochastic ordering, of the entropy-based block hard thresholding stage in relation to the parameters characterizing local variability (fractality) and dependence range of the spatial process of interest, the noise level, and the design of the region of interest.  相似文献   

17.
This paper investigates the potential of Spartan spatial random fields (SSRFs) in real-time mapping applications. The data set that we study focuses on the distribution of daily gamma dose rates over part of Germany. Our goal is to determine a Spartan spatial model from the data, and then use it to generate “predictive” maps of the radioactivity. In the SSRF framework, the spatial dependence is determined from sample functions that focus on short-range correlations. A recently formulated SSRF predictor is used to derive isolevel contour maps of the dose rates. The SSRF predictor is explicit. Moreover, the adjustments that it requires by the user are reduced compared to classical geostatistical methods. These features present clear advantages for an automatic mapping system. The performance of the SSRF predictor is evaluated by means of various cross-validation measures. The values of the performance measures are similar to those obtained by classical geostatistical methods. Application of the SSRF method to data that simulate a radioactivity release scenario is also discussed. Hot spots are detected and removed using a heuristic method. The extreme values that appear in the path of the simulated plume are not captured by the currently used Spartan spatial model. Modeling of the processes leading to extreme values can enhance the predictive capabilities of the spatial model, by incorporating physical information.  相似文献   

18.
Before data from satellites can be used with confidence in dynamical studies of the middle atmosphere an assessment of their reliability is necessary. To this end, independently analysed data from different instruments may be compared. In this paper, this is done for the Southern Hemisphere as a prelude to the dynamical studies of the middle atmosphere being fostered by the MASH project of the Middle Atmosphere Program. Data from two infrared radiometers are used: a limb scanner (LIMS) and a nadir sounder (SSU). While there is usually qualitative agreement between basic fields (temperatures, winds), substantial quantitative differences are found, with more pronounced differences in fields of Eliassen-Palm flux divergence and Ertel's potential vorticity.The fidelity of the base-level analysis to which satellite data are tied is important for calculating quantities of relevance to dynamical theory. In the Southern Hemisphere, conventional data are sparse and, through the analysis procedure, this introduces errors into derived fields for the middle atmosphere. The impact of using base-level analyses from different sources is assessed. Large discrepancies are found in fields computed by differentiation.Several techniques are suggested whereby the reliability of fields derived from satellite data may be gauged.  相似文献   

19.
Reservoir characterization involves describing different reservoir properties quantitatively using various techniques in spatial variability. Nevertheless, the entire reservoir cannot be examined directly and there still exist uncertainties associated with the nature of geological data. Such uncertainties can lead to errors in the estimation of the ultimate recoverable oil. To cope with uncertainties, intelligent mathematical techniques to predict the spatial distribution of reservoir properties appear as strong tools. The goal here is to construct a reservoir model with lower uncertainties and realistic assumptions. Permeability is a petrophysical property that relates the amount of fluids in place and their potential for displacement. This fundamental property is a key factor in selecting proper enhanced oil recovery schemes and reservoir management. In this paper, a soft sensor on the basis of a feed‐forward artificial neural network was implemented to forecast permeability of a reservoir. Then, optimization of the neural network‐based soft sensor was performed using a hybrid genetic algorithm and particle swarm optimization method. The proposed genetic method was used for initial weighting of the parameters in the neural network. The developed methodology was examined using real field data. Results from the hybrid method‐based soft sensor were compared with the results obtained from the conventional artificial neural network. A good agreement between the results was observed, which demonstrates the usefulness of the developed hybrid genetic algorithm and particle swarm optimization in prediction of reservoir permeability.  相似文献   

20.
Interpretation of magnetic data can be carried out either in the space or frequency domain. The interpretation in the frequency domain is computationally convenient because convolution becomes multiplication. The frequency domain approach assumes that the magnetic sources distribution has a random and uncorrelated distribution. This approach is modified to include random and fractal distribution of sources on the basis of borehole data. The physical properties of the rocks exhibit scaling behaviour which can be defined as P(k) = Ak?β, where P(k) is the power spectrum as a function of wave number (k), and A and β are the constant and scaling exponent, respectively. A white noise distribution corresponds to β = 0. The high resolution methods of power spectral estimation e.g. maximum entropy method and multi‐taper method produce smooth spectra. Therefore, estimation of scaling exponents is more reliable. The values of β are found to be related to the lithology and heterogeneities in the crust. The modelling of magnetic data for scaling distribution of sources leads to an improved method of interpreting the magnetic data known as the scaling spectral method. The method has found applicability in estimating the basement depth, Curie depth and filtering of magnetic data.  相似文献   

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