Transition probability-based indicator geostatistics

Traditionally, spatial continuity models for indicator variables are developed by empirical curvefitting to the sample indicator (cross-) variogram. However, geologic data may be too sparse to permit a purely empirical approach, particularly in application to the subsurface. Techniques for model synthesis that integrate hard data and conceptual models therefore are needed. Interpretability is crucial. Compared with the indicator (cross-) variogram or indicator (cross-) covariance, the transition probability is more interpretable. Information on proportion, mean length, and juxtapositioning directly relates to the transition probability: asymmetry can be considered. Furthermore, the transition probability elucidates order relation conditions and readily formulates the indicator (co)kriging equations.     

The deterministic side of geostatistics

The probabilistic approach is but one language used by geostatisticians to characterize spatial variability and to express a very simple criterion for goodness of estimation. Notions such as stationarity and ergodicity are important for the consistency of the probabilistic language but are irrelevant to the real problem, that of estimating a well-defined deterministic spatial average. The kriging algorithm is established without any recourse to probabilistic modeling or notation.     

Another look at anisotropy in geostatistics

A thorough geostatistical data analysis includes a careful study of how the data's second-order variation, as characterized by the semivariogram, depends on the relative orientation of data locations. If the semivariogram depends on only the (Euclidean) distance between locations, then the semivariogram is isotropic; otherwise, it is anisotropic. In this article, I take another look at the modeling of anisotropy in geostatistics. A new, more specific classification of types of anisotropy is proposed. More importantly, some heretofore inadequately understood implications of the dependence of various semivariogram attributes on direction are discussed, and the wisdom of some current practices for modeling the direction-dependence of these attributes is questioned.     

Matheronian geostatistics—Quo vadis?

Components of geostatistical estimation, developed as a method for ore deposit assessment, are discussed in detail. The assumption that spatial observations can be treated as a stochastic process is judged to be an inappropriate model for natural data. Problems of semivariogram formulation are reviewed, and this method is considered to be inadequate for estimating the function being sought. Characteristics of bivariate interpolation are summarized, highlighting kriging limitations as an interpolation method. Limitations are similar to those of inverse distance weighted observations interpolation. Attention is drawn to the local bias of kriging and misplaced claims that it is an “optimal” interpolation method. The so-called “estimation variance,” interpreted as providing confidence limits for estimation of mining blocks, is shown to be meaningless as an index of local variation. The claim that geostatistics constitutes a “new science” is examined in detail. Such novelties as exist in the method are shown to transgress accepted principles of scientific inference. Stochastic modeling in general is discussed, and purposes of the approach emphasized. For the purpose of detailed quantitative assessment it can provide only prediction qualified by hypothesis at best. Such an approach should play no part in ore deposit assessment where the need is for local detailed inventories; these can only be achieved properly through local deterministic methods, where prediction is purely deductive.     

多点地质统计学研究进展与展望

在简要回顾多点地质统计学起源后,介绍了多点地质统计学的3种方法,并总结了多点地质统计 学的研究进展。在应用领域,已经从河流相建模发展到扇环境建模,从储集层结构建模发展到储集层物性分布 模拟,从宏观地质体预测发展到微观孔喉分布建模,从地质研究发展到地质统计反演。在整合信息建模方面, 给出了３种综合地震属性的方法;在算法方面,提出了PRTT实时处理方法,完善了多点地质统计学建模,并开 发了新的多点统计生长算法（Growthsim）。对多点地质统计学未来发展进行了展望,指出在训练图像、数据综合 以及建模方法耦合方面还需要进一步深入研究。     

Imprecise (fuzzy) information in geostatistics

A methodology based on fuzzy set theory for the utilization of imprecise data in geostatistics is presented. A common problem preventing a broader use of geostatistics has been the insufficient amount of accurate measurement data. In certain cases, additional but uncertain (soft) information is available and can be encoded as subjective probabilities, and then the soft kriging method can be applied (Journel, 1986). In other cases, a fuzzy encoding of soft information may be more realistic and simplify the numerical calculations. Imprecise (fuzzy) spatial information on the possible variogram is integrated into a single variogram which is used in a fuzzy kriging procedure. The overall uncertainty of prediction is represented by the estimation variance and the calculated membership function for each kriged point. The methodology is applied to the permeability prediction of a soil liner for hazardous waste containment. The available number of hard measurement data (20) was not enough for a classical geostatistical analysis. An additional 20 soft data made it possible to prepare kriged contour maps using the fuzzy geostatistical procedure.This paper was presented at MGUS 87 Conference, Redwood City, California, 14 April 1987.     

When are relative variograms useful in geostatistics?

The relative variogram has been employed as a tool for correcting a simple kind of nonstationarity, namely that in which local variance is proportional to local mean squared. In the past, this has been linked in a vague way to the lognormal distribution, although if {Z_t; t D}is strongly stationary and normal over a domain D,then clearly {exp (Z_t); t D}will stillbe stationary, but lognormal. The appropriate link is made in this article through a universal transformation principle. More general situations are considered, leading to the use of a scaled variogram.     

On the efficiency of the indicator approach in geostatistics

A common problem in geostatistics is to determine whether or not the value of a random field at an unsampled location exceeds a specified threshold using observed values of the random field at sampled locations. Under the indicator approach, the only information used to classify the unobserved value is whether or not the observed values exceed the threshold. This note shows that the loss of information from applying the indicator approach may be modest in the case where the underlying random field is Gaussian.     

多点地质统计学：理论、应用与展望

本文系统地介绍了多点地质统计学的基本原理及方法，并以我国渤海湾盆地某区块新近系明化镇组河流相储层为例，进行了多点统计学随机建模的实例分析。多点地质统计学为储层随机建模的国际前沿研究方向，该方法综合了基于象元的方法易忠实条件数据以及基于目标的方法易再现目标几何形态的优点，同时克服了传统基于变差函数的二点统计学不能表达复杂空间结构和再现目标几何形态的不足。通过理论与实例研究，分析了目前多点统计学尚存在的问题(包括训练图像平稳性问题、目标连续性问题以及综合软信息的问题等)及未来发展的方向。     

Spatial continuity measures for probabilistic and deterministic geostatistics

Geostatistics has traditionally used a probabilistic framework, one in which expected values or ensemble averages are of primary importance. The less familiar deterministic framework views geostatistical problems in terms of spatial integrals. This paper outlines the two frameworks and examines the issue of which spatial continuity measure, the covarianceC (h) or the variogram (h), is appropriate for each framework. AlthoughC (h) and (h) were defined originally in terms of spatial integrals, the convenience of probabilistic notation made the expected value definitions more common. These now classical expected value definitions entail a linear relationship betweenC (h) and (h); the spatial integral definitions do not. In a probabilistic framework, where available sample information is extrapolated to domains other than the one which was sampled, the expected value definitions are appropriate; furthermore, within a probabilistic framework, reasons exist for preferring the variogram to the covariance function. In a deterministic framework, where available sample information is interpolated within the same domain, the spatial integral definitions are appropriate and no reasons are known for preferring the variogram. A case study on a Wiener-Levy process demonstrates differences between the two frameworks and shows that, for most estimation problems, the deterministic viewpoint is more appropriate. Several case studies on real data sets reveal that the sample covariance function reflects the character of spatial continuity better than the sample variogram. From both theoretical and practical considerations, clearly for most geostatistical problems, direct estimation of the covariance is better than the traditional variogram approach.This paper was presented at MGUS 87 Conference, Redwood City, California, 14 April 1987.     

基于地统计学的路基连续压实均匀性评价

传统的路基压实质量均匀性评价方法均基于变异系数等概率统计指标,未能考虑检测数据的空间分布特性,难以准确评价路基压实质量的不均匀性。基于地统计学理论,建立半变异函数模型描述连续压实检测数据的空间变异性,采用指数模型对半变异函数曲线进行最优拟合,并提出偏基台值C作为路基压实均匀性控制指标。分别采用地统计方法与传统数理统计法对沪昆客运专线芷江试验段的路基进行压实均匀性评价。结果表明：在压路机振动频率波动或存在压实薄弱区域的情况下,与传统数理统计指标变异系数Cv相比,地统计指标C可消除系统误差等随机性因素对均匀性评价结果的影响,更能客观反映不同工况下的填筑土体在空间上的不均匀压实状态,其结果为铁路路基压实均匀性评价提供了新的思路。     

Using geology to control geostatistics in the Hellyer deposit

The Hellyer orebody, a polymetallic massive sulfide deposit, was discovered in western Tasmania by Aberfoyle in 1983. Delineation diamond drilling was carried out in 1984 on a nominal 50-m square grid pattern to outline the resource. Resource estimation methods were influenced by the requirement to develop a regular block model for conceptual mine planning studies. Detailed geological interpretation indicates that the interpolation technique must take into account several important features to retain geological credibility. The deposit has sharp limits defined by visual geological contacts with virtually barren enclosing rocks. Lateral terminations are rapid with no interfingering internal waste. The dip and strike are variable and a major fault with a measurable displacement cuts acutely through the center of the deposit. Ore grades are reasonably correlateable within specific layers from hole to hole indicating a significant across-dip anisotropy. A hanging wall enriched zone is well-defined throughout the deposit. To overcome the variable geometry problems, a stratigraphic coordinate system was defined arbitrarily to replace the normal z coordinate. This allowed variography in stratigraphic layers. Blocks to be estimated were constrained by hand-drawn and subsequently digitized hanging wall and footwall contours. Each block was ascribed a stratigraphic coordinate by calculating its spatial position in relation to nearby stratigraphic unit boundaries within the massive sulfide body. Estimates were generated for each element by ordinary linear kriging. Despite the relatively sparse data in a large massive deposit, the customized technique developed for Hellyer has provided a reliable model of spatial grade distribution by combining conventional geostatistical methods with careful geological observation and interpretation. Some geometry problems remain which are the subject of ongoing studies.This paper was presented at MGUS 87 Conference, Redwood City, California, 14 April 1987.     

On the estimation of scale of fluctuation in geostatistics

Describing how soil properties vary spatially is of particular importance in stochastic analyses of geotechnical problems, because spatial variability has a significant influence on local material and global geotechnical response. In particular, the scale of fluctuation θ is a key parameter in the correlation model used to represent the spatial variability of a site through a random field. It is, therefore, of fundamental importance to accurately estimate θ in order to best model the actual soil heterogeneity. In this paper, two methodologies are investigated to assess their abilities to estimate the vertical and horizontal scales of fluctuation of a particular site using in situ cone penetration test (CPT) data. The first method belongs to the family of more traditional approaches, which are based on best fitting a theoretical correlation model to available CPT data. The second method involves a new strategy which combines information from conditional random fields with the traditional approach. Both methods are applied to a case study involving the estimation of θ at three two-dimensional sections across a site and the results obtained show general agreement between the two methods, suggesting a similar level of accuracy between the new and traditional approaches. However, in order to further assess the relative accuracy of estimates provided by each method, a second numerical analysis is proposed. The results confirm the general consistency observed in the case study calculations, particularly in the vertical direction where a large amount of data are available. Interestingly, for the horizontal direction, where data are typically scarce, some additional improvement in terms of relative error is obtained with the new approach.     

Artificially intelligent geostatistics: A framework accommodating qualitative knowledge-information

The purpose of this paper is to stress the need to examine Al-based models and techniques in dealing with qualitative knowledge, information, and expertise in geostatistics. A model of artificially intelligent geostatistics is proposed as a general framework. The model focuses on the Geostatistician, an abstraction of the collective knowledge and intelligence of the geostatisticians, be they theoreticians and/or practitioners. Dynamic aspects of the model are examined in the context of an explicit knowledge formalism, integrating geostatistical knowledge, symbolic non-algorithmic techniques for knowledge-information representation and inference, and standard numerical data processing. Two implementations of related computer systems are given together with case studies.     

A comparative study of hydraulic conductivity estimations using geostatistics

Three approaches for estimating the hydraulic conductivity (K) of the Trifa aquifer, Morocco were investigated: (1) kriging of the K values obtained from pumping tests, (2) cokriging of the pumping test data with electrical resistivity data as a secondary variable, and (3) cokriging of the pumping test data with the slope of the water table. Gauss-transformed values of the variables are used because they provide more robust variograms and transformed values of the primary and secondary variables show correlations higher than the raw values, which is beneficial in cokriging. In cokriging with electrical resistivity, two zones are considered since the geological deposits are different from the north to the south of the aquifer, which is reflected in different correlations between the variables. Comparison of the three approaches is based mainly on the estimation errors, and to a lesser degree on the cross-validations of the corresponding variogram models and general considerations, like the measurements’ reliability and aquifer make-up. The best-estimated K is given by cokriging with the slope of the water table and is therefore preferred for further use in groundwater flow modeling. Thus, electrical resistivity or the slope of the water table can both be used as secondary variables to estimate K, especially in heterogeneous aquifers with lateral variations in lithology, as is the case of the Trifa aquifer.     

Exploiting transformation-domain sparsity for fast query in multiple-point geostatistics

Multiple-point geostatistics has recently attracted significant attention for characterization of environmental variables. Such methods proceed by searching a large database of patterns obtained from a training image to find a match for a given data-event. The template-matching phase is usually the most time-consuming part of a MPS method. Linear transformations like discrete cosine transform or wavelet transform are capable of representing the image patches with a few nonzero coefficients. This sparsifying capability can be employed to speed up the template-matching problem up to hundreds of times by multiplying only nonzero coefficients. This method is only applicable to rectangular data-events because it is impossible to represent an odd-shaped data-event in a transformation domain. In this paper, the method is applied to speed up the image quilting (IQ) method. The experiments show that the proposed method is capable of accelerating the IQ method tens of times without sensible degradation in simulation results. The method has the potential to be employed for accelerating optimization-based and raster-scan patch-based MPS algorithms.