Compositional Kriging: A Spatial Interpolation Method for Compositional Data

Compositional data are very common in the earth sciences. Nevertheless, little attention has been paid to the spatial interpolation of these data sets. Most interpolators do not necessarily satisfy the constant sum and nonnegativity constraints of compositional data, nor take spatial structure into account. Therefore, compositional kriging is introduced as a straightforward extension of ordinary kriging that complies with these constraints. In two case studies, the performance of compositional kriging is compared with that of the additive logratio-transform. In the first case study, compositional kriging yielded significantly more accurate predictions than the additive logratio-transform, while in the second case study the performances were comparable.     

An Empirical Comparison of Kriging Methods for Nonlinear Spatial Point Prediction

Spatial prediction is a problem common to many disciplines. A simple application is the mapping of an attribute recorded at a set of points. Frequently a nonlinear functional of the observed variable is of interest, and this calls for nonlinear approaches to prediction. Nonlinear kriging methods, developed in recent years, endeavour to do so and additionally provide estimates of the distribution of the target quantity conditional on the observations. There are few empirical studies that validate the various forms of nonlinear kriging. This study compares linear and nonlinear kriging methods with respect to precision and their success in modelling prediction uncertainty. The methods were applied to a data set giving measurements of the topsoil concentrations of cobalt and copper at more than 3000 locations in the Border Region of Scotland. The data stem from a survey undertaken to identify places where these trace elements are deficient for livestock. The comparison was carried out by dividing the data set into calibration and validation sets. No clear differences between the precision of ordinary, lognormal, disjunctive, indicator, and model-based kriging were found, neither for linear nor for nonlinear target quantities. Linear kriging, supplemented with the assumption of normally distributed prediction errors, failed to model the conditional distribution of the marginally skewed data, whereas the nonlinear methods modelled the conditional distributions almost equally well. In our study the plug-in methods did not fare any worse than model-based kriging, which takes parameter uncertainty into account.     

Empirical Maximum Likelihood Kriging: The General Case

Although linear kriging is a distribution-free spatial interpolator, its efficiency is maximal only when the experimental data follow a Gaussian distribution. Transformation of the data to normality has thus always been appealing. The idea is to transform the experimental data to normal scores, krige values in the “Gaussian domain” and then back-transform the estimates and uncertainty measures to the “original domain.” An additional advantage of the Gaussian transform is that spatial variability is easier to model from the normal scores because the transformation reduces effects of extreme values. There are, however, difficulties with this methodology, particularly, choosing the transformation to be used and back-transforming the estimates in such a way as to ensure that the estimation is conditionally unbiased. The problem has been solved for cases in which the experimental data follow some particular type of distribution. In general, however, it is not possible to verify distributional assumptions on the basis of experimental histograms calculated from relatively few data and where the uncertainty is such that several distributional models could fit equally well. For the general case, we propose an empirical maximum likelihood method in which transformation to normality is via the empirical probability distribution function. Although the Gaussian domain simple kriging estimate is identical to the maximum likelihood estimate, we propose use of the latter, in the form of a likelihood profile, to solve the problem of conditional unbiasedness in the back-transformed estimates. Conditional unbiasedness is achieved by adopting a Bayesian procedure in which the likelihood profile is the posterior distribution of the unknown value to be estimated and the mean of the posterior distribution is the conditionally unbiased estimate. The likelihood profile also provides several ways of assessing the uncertainty of the estimation. Point estimates, interval estimates, and uncertainty measures can be calculated from the posterior distribution.     

Validation of Spatial Prediction Models for Landslide Hazard Mapping

This contribution discusses the problemof providing measures of significance ofprediction results when the predictionswere generated from spatial databases forlandslide hazard mapping. The spatialdatabases usually contain map informationon lithologic units, land-cover units,topographic elevation and derived attributes(slope, aspect, etc.) and the distributionin space and in time of clearly identifiedmass movements. In prediction modelling wetransform the multi-layered databaseinto an aggregation of functional values toobtain an index of propensity of the landto failure. Assuming then that the informationin the database is sufficiently representativeof the typical conditions in which the massmovements originated in space and in time,the problem then, is to confirm the validity ofthe results of some models over otherones, or of particular experiments that seem touse more significant data. A core pointof measuring the significance of a prediction isthat it allows interpreting the results.Without a validation no interpretation is possible,no support of the method or of theinput information can be provided. In particularwith validation, the added value canbe assessed of a prediction either in a fixedtime interval, or in an open-ended time orwithin the confined space of a study area.Validation must be of guidance in datacollection and field practice for landslidehazard mapping.     

新疆三工河流域地下水矿化度的时空变异及其分形特征

对面积约670 km2的新疆三工河流域地下水矿化度进行了取样检测,应用地质统计学方法对取得的数据进行了半方差函数分析,计算了平均值、方差、标准差、变异系数等传统统计特征值,并指出了用该方法表示地下水矿化度所存在的不足.在此基础上,将分形理论与地质统计学原理相结合,计算了三工河流域在1998、2002年地下水矿化度的分维值,分别为1.75,1.59;其变异系数分别为0.51,0.54,即2002年地下水矿化度的均匀性要比1998年的稍差.研究结果表明,地下水矿化度并不是具有理想分形特征的介质,它只是在一定的空间尺度内才具有分形特征.通过对1998年及2002年地下水样品的分析,得出该地区地下水矿化度在时间与空间上皆存在明显的变异性.在空间尺度上,地下水矿化度在研究区的北部大于中部,中部大于南部;在时间尺度上,随着时间的推移研究区北部的地下水水质正向矿化度加重的方向发展.     

Spatial statistics for predicting fluid flow through a single rock fracture

Steady-state laminar flow through single rock fractures is predicted in terms of spatial statistics computed from the arrangement of voids and contact areas within the fracture. Within the voids, aperture is assumed to be constant. One statistic measures how often pixels alternate from void to contact area in the rows parallel to the flow direction. Two others measure the dispersion of voids in the rows and columns of the pattern. Fractures with complexity typical of observed data are simulated. Flow through patterns with 80% voids is predicted in terms of a linear combination of the three statistics. Using an extended model involving one of the three statistics, flow through patterns with other void fractions is predicted.The author did this work at the Earth Sciences Division at Lawrence Berkeley Laboratory, Berkeley, California. It is part of a Ph.D. thesis which was submitted at Stanford University.     

克里金参数估值法及其在参数估计分析中的应用

为考虑岩土介质参数的空间分布的结构性和随机性等不确定因素,引入了克里金参数估值法。采用变异函数描述参数在空间结构上的变化,建立其空间变异规律的数学模型,从而实现对岩土参数的估值。通过实例分析揭示了克里金估值法具有反映“过滤效应”和“集团效应”的优点,对不同位置的数据赋予不同的权重系数,能够有效地反映参数空间变异结构,有利于对参数的合理化分析     

基于GIS的矿产预测地质解译空间信息集成

利用我国海量地质标准基础数据库中的数字地质图和矿产图,通过基于GIS的地质解译空间集成地质信息,将其用于综合信息矿产预测。以地质解译系统对内蒙大兴安岭南段1∶20万成矿预测的应用为案例,阐述地质信息的空间提取与集成过程：首先在建立地质字典库实现地质空间信息共享的基础上,通过矿化密集区对地质模型的分类图层进行空间分析,建立地质成矿空间信息库和图库;然后,基于典型矿床圈定模型单元,通过模型单元与地质成矿空间信息库和图库的空间分析,建立地质找矿模型;最后,基于地质单元对地质成矿空间信息库和图库的二次空间集成,完成预测模型的地质空间信息提取与集成。将本方法应用在银矿案例的综合信息矿产预测靶区评价上,得到可供进一步查证的新增靶区比已知靶区增加了近5倍。     

基于地质空间数据挖掘的区域成矿预测方法

以多源地质空间数据库和空间数据挖掘技术为基础,顾及地质数据的空间特征和不确定性,提出一种基于地质空间数据挖掘的区域成矿预测方法,主要包括:连续型地质空间数据离散化、成矿空间关系提取及属性化、成矿关联规则提取及质量评价、成矿关联规则综合评价与潜力制图.最后,以青海省东昆仑地区铁矿资源潜力预测为例进行实验,并将地质空间数据...     

Design and Analysis for Modeling and Predicting Spatial Contamination

Sampling and prediction strategies relevant at the planning stage of the cleanup of environmental hazards are discussed. Sampling designs and models are compared using an extensive set of data on dioxin contamination at Piazza Road, Missouri. To meet the assumptions of the statistical model, such data are often transformed by taking logarithms. Predicted values may be required on the untransformed scale, however, and several predictors are also compared. Fairly small designs turn out to be sufficient for model fitting and for predicting. For fitting, taking replicates ensures a positive measurement error variance and smooths the predictor. This is strongly advised for standard predictors. Alternatively, we propose a predictor linear in the untransformed data, with coefficients derived from a model fitted to the logarithms of the data. It performs well on the Piazza Road data, even with no replication.     

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.     

Spatial correlation for transformation uncertainty and its applications

This paper shows that the transformation uncertainty varies in space and can be decomposed into (a) a slowly fluctuating component contributed by the systematic bias for a local site and (b) a rapidly fluctuating component partly contributed by the measurement error. Based on a piezocone database, the slowly fluctuating component is found to have a vertical scale of fluctuation (SOF) of 17–60 m, larger than the vertical SOFs for most soil properties and characteristic dimensions for many geotechnical structures. This paper provides an empirical basis to justify the assumption that the transformation uncertainty is fully correlated in space.     

建三江地区土壤氮磷生态化学计量空间异质性特征及其影响因素分析

土壤全氮(TN)、全磷(TP)及氮磷比(N/P)是衡量土壤肥力和土壤质量的重要指标,研究其空间异质性特征对土壤养分管理以及生态环境监测等政策的制订具有重要意义。本文通过土样采集和室内实验分析,结合地统计分析及地理信息系统(GIS)等方法,对建三江地区表层(0~20 cm)土壤中的氮、磷生态化学计量空间变异及影响因素进行研究。结果表明:土壤全氮、全磷以及氮磷比的均值分别为2.49×10-3、0.81×10-3和3.20。土壤全氮和氮磷比空间自相关性很强,而土壤全磷空间自相关性处于中等程度,且均符合指数模型。从空间分布来看,土壤全氮高值区主要呈斑块零星分散在东部、南部和西北地区,而低值区主要在西北和中西部地区零星分布;土壤全磷高值区主要分布在东部和西北地区,中部地区则为低值地区;土壤氮磷比分布则呈镶嵌状,高值区以斑块状分散于中部、南部和东北部地区,而低值区主要位于西北地区。研究区土壤类型、第四系类型和土地利用类型是影响土壤氮、磷生态化学计量特征的重要结构性和随机性因素,而土壤母质和地貌类型则对土壤氮、磷生态化学计量特征影响不大。     

On the Relevance of the Spatial Distribution of Events for Seismic Hazard Evaluation

Seismic hazard evaluation is proposed by a methodological approach that allows thestudy of the influence of different modelling assumptions relative to the spatial andtemporal distribution of earthquakes on the maximum values of expected intensities.In particular, we show that the estimated hazard at a fixed point is very sensitive tothe assumed spatial distribution of epicentres and their estimators. As we will see, theusual approach, based on uniformly distributing the epicentres inside each seismogeniczone is likely to be biased towards lower expected intensity values. This will be mademore precise later. Recall that the term ``bias' means, that the expectation of theestimated quantity (taken as a random variable on the space of statistics) is differentfrom the expectation of the quantity itself. Instead, our approach, based on an estimatorthat takes into account the observed clustering of events is essentially unbiased, as shownby a Monte-Carlo simulation, and is configured on a 11011-isotropic macroseismicattenuation model which is independently estimated for each zone.     

GIS空间数据的质量探讨

空间数据直接影响着GIS应用分析结果的可靠程度和应用目标的真正实现,因此,有效地控制空间数据的质量是非常必要的。该文结合山东省1：1万基础地理信息数据采集与建库实践,分析了空间数据质量问题的来源及影响,根据空间数据质量标准,采用软件检查和人机交互检查2种方法,制定各工序相应的检查方案,以减少误差,防止粗差。     

深基坑排桩支护结构空间共同变形分析

考虑深基坑支护结构中圈梁、腰梁、排桩、支撑和地基的空间协同作用,建立了基于共同变形理论的有限元三维分析模型,明确了土压力及地基等效刚度的计算方法。结合工程实例,分析了带圈梁的排桩支护结构的内力、位移及土压力分布规律,并与二维平面分析结果进行了比较,验证了计算模型的合理性和三维分析的必要性。     

Anomaly Models of Spatial Structures for Copper-Molybdenum Ore Deposits and their Application

This paper discusses the enrichment and depletion regularities for porphyry copper-molybdenum ore deposits in different regions and varied deposit genetic types in the same area, taking three porphyry copper-molybdenum ore deposits (i.e., the Chengmenshan in Jiangxi, Wunugetushan in Inner Mongolia, Baishantang in Gansu) and two copper deposits in Gansu Province (the Huitongshan skarn deposit and Gongpoquan composite deposit) as case studies. The results show that porphyry Cu-Mo deposits or skarn copper deposits include both enrichment of the ore-forming elements and associated elements, and depletion of some lithophile dispersed elements, rare earth elements (REE) and some major elements. And the depleted elements vary with deposits, having generality and their own features. On a deposit scale, the positive anomalies of enriched elements and negative anomalies of depleted elements follow in a sequence to comprise regular anomaly models of spatial structures. The exploration in the Tongchang deposit in Jiangxi and Huitongshan deposit in Gansu suggests that anomaly models play a key role in the identification of mineral occurrences and deposits compared to one single enriched element anomaly. And the anomaly models exert a critical effect on the optimization of prospecting targets and their potential evaluation.     

河南灵宝枪马金矿床410脉矿化富集规律及深部矿体空间定位预测

行之有效的大比例尺局部矿床预测理论和方法是找矿勘探中急待探索的基本问题。本文在对枪马金矿410脉各种矿化指标和矿化特征参数统计分析基础上,总结并探讨了410脉矿化富集规律和矿体空间定位机制,进而对410脉深部进行了成矿预测,指出410脉深部具有较好的找矿前景。     

Health profiles of Hamilton: Spatial characterisation of neighbourhoods for health investigations

This paper is part of a larger research program which employs a mixed-methods approach to study the determinants of health at the local level using specific neighborhoods in Hamilton, Ontario, Canada. In this paper, multivariate, spatial statistical techniques and geographic information systems are used to address questions about the characterization of neighbourhoods, based on socioeconomic determinants of health and risk factors such as smoking. While neighbourhood characterization has been a component of public health surveillance for some time, geostatistical techniques can now be used to derive more accurate representation of neighbourhoods for use in subsequent analysis. We utilize principal components analysis to reduce the data and extract the components that represent the underlying local processes. Principal components are also overlayed on comparative mortality figures to visualize where the socio-demographic determinants of health correspond spatially with mortality patterns. Predicted values from the components are then analysed for spatial clustering using local indicators of spatial association. The findings reveal a pattern of distinct neighbourhoods that will be used in subsequent quantitative and qualitative stages in the larger research programme. The results can also be used to inform public health policy and to target public health interventions.     

矿产资源定量预测方法综述

对国内近二十年来所发展的矿产定量预测方法进行了较为全面的论述，主要包括矿床统计预测方法，灰色系统方法、模糊数学方法、分形几何方法、地质统计学方法、模式识别方法和地理信息系统等，并简述和评价了某些方法的基本应用。