Factorial kriging with external drift: a case study on the Penedono Region,Portugal

Factorial kriging has been used in geochemical exploration for the estimation and cartography of the spatial components of the variables, helping with the identification and interpretation of geochemical anomalies. Those spatial components appear by the decomposition of the variables in its several structural components, given by the variograms. In this paper a new form of factorial kriging is introduced, by using the geologic information as an external drift. This was achieved considering as an external variable (external drift) sample co-ordinates on the first axis resultant from a principal component analysis (PCA), interpreted as a lithological factor. With this type of geostatistical technique each point appears in the resultant maps as a combination of geochemical and geological information, attending the geographic localization of the samples. This technique was tested on a set of 2450 sediment samples collected on a 640 km² area, between the Trás-os-Montes e Alto Douro and the Beira Alta regions. From the 34 initial elements analyzed (10 major elements P₂O₅, SiO₂, Al₂O₃, Fe₂O₃, MgO, CaO, Na₂O, K₂O, TiO₂ and MnO expressed in oxide percentage and 24 elements expressed in ppm As, Bi, Ag, Sb, W, B, Cu, Pb, Zn, Sn, Nb, Li, Be, Zr, Y, La, Ba, Cd, Mo, V, Cr, Co, Ni and Sr) only the results obtained for 2 of them are presented in this work. The first was Sn, which is associated with some mineralisation in this region and the other one was Zn, which shows similar behavior in the whole area, with the exception of a small region.     

Geostatistical approach for identification of transmissivity structure at Dulliu area in Taiwan

 A thorough understanding of the characteristics of transmissivity makes groundwater deterministic models more accurate. These transmissivity data characteristics occasionally possess a complicated spatial variation over an investigated site. This study presents both geostatistical estimation and conditional simulation methods to generate spatial transmissivity maps. The measured transmissivity data from the Dulliu area in Yun-Lin county, Taiwan, is used as the case study. The spatial transmissivity maps are simulated by using sequential Gaussian simulation (SGS), and estimated by using natural log ordinary kriging and ordinary kriging. Estimation and simulation results indicate that SGS can reproduce the spatial structure of the investigated data. Furthermore, displaying a low spatial variability does not allow the ordinary kriging and natural log kriging estimates to fit the spatial structure and small-scale variation for the investigated data. The maps of kriging estimates are smoother than those of other simulations. A SGS with multiple realizations has significant advantages over ordinary kriging and even natural log kriging techniques at a site with a high variation in investigated data. These results are displayed in geographic information systems (GIS) as basic information for further groundwater study. Received: 27 August 1999 · Accepted: 22 February 2000     

利用地质统计学方法模拟岩石裂隙网络的三维空间分布——以云南个旧高松矿田为例

裂隙在地学的诸多领域中均具有重要意义,其空间分布可以使用地质统计学方法进行模拟,同时考虑裂隙的方向(走向和倾角)。利用序贯高斯模拟方法可以估计裂隙密度的空间分布,并根据裂隙密度数值随机产生裂隙位置的空间分布。裂隙方向被划分成若干(非)均等的方向组,将裂隙方向归属到其所属方向组,表示为由若干二值变量组成的指示形式,0和1分别代表该裂隙方向不属于和属于该组。为了便于计算,减少方向指示变量的成分数目,使用主成分分析法求出方向指示变量的主成分,用普通克里格法估计各主成分的空间分布。把估计结果反演为原始的指示形式,并找出其中数值最大的方向组且将其赋值为1。按照对应方向组内裂隙方向的累积密度函数,随机产生裂隙的方向。根据估计结果,将符合一定距离和角度标准的裂隙元连接为一个裂隙面,从而形成裂隙网络。根据在云南个旧锡矿高松矿田白云岩中进行裂隙网络模拟的应用,可见该方法由于组合了序贯高斯模拟法、普通克里格法和主成分分析法,可以较好地对岩石裂隙位置和方向进行合理的模拟。     

Data analysis of a revisited exploration geochemical dataset of quartz porphyrites from SW Jordan using GIS techniques

The spatial distribution of Cu and its pathfinder elements in quartz porphyrites along Wadi Araba was studied in order to gain an insight about the geologic framework of Cu mineralization in the area. A geochemical exploration data set of 242 samples was revisited. Multivariate statistical and geostatistical analysis were performed on the data set. The objectives of the study were to define Cu relationships with its possible pathfinder elements and to figure out whether geostatistical treatment of exploration data would help telescoping already existing Cu mineralization in Wadi Araba, SW Jordan. Factor analysis and hierarchical cluster analysis were performed on the data set. The results of which indicate that seven elements, namely, Ba, Cr, Mn, Ni, Sc, Sr and V can be considered as pathfinder elements for Cu. Significant correlations do exist between Cu and these elements. The encountered experimental semivariograms of the defined pathfinders were fit to exponential models with varying nugget effects with the exception of Mn which was fit to the spherical model. Ordinary kriging was found to be most appropriate to generate geochemical exploration maps. The applied methods succeeded in deriving valuable information and visualizing the already existing Cu mineralization in Wadi Abu Khusheiba area.     

Geochemical characterization of heavy metal contaminated area using multivariate factorial kriging

This paper describes a geostatistical method, known as factorial kriging analysis, which is well suited for analyzing multivariate spatial information. The method involves multivariate variogram modeling, principal component analysis, and cokriging. It uses several separate correlation structures, each corresponding to a specific spatial scale, and yields a set of regionalized factors summarizing the main features of the data for each spatial scale. This method is applied to an area of high manganese-ore mining activity in Amapá State, North Brazil. Two scales of spatial variation (0.33 and 2.0 km) are identified and interpreted. The results indicate that, for the short-range structure, manganese, arsenic, iron, and cadmium are associated with human activities due to the mining work, while for the long-range structure, the high aluminum, selenium, copper, and lead concentrations, seem to be related to the natural environment. At each scale, the correlation structure is analyzed, and regionalized factors are estimated by cokriging and then mapped.     

Spatial distribution of uranium and vanadium in the upper phosphorite member in Eshidiyya basin, southern Jordan

Geostatistical analysis of geochemical exploration data can provide useful information for evaluating the mineralization potential of geologic bodies. The main objective of this study was to evaluate the spatial distribution of U and V over an area in the upper phosphorite member, the A0 unit, in Eshidiyya basin in southern Jordan. Drill-hole samples were analyzed. The geochemistry of the analyzed elements was assessed. The spatial variability of U and V was also assessed using semivariance analysis. Both U and V were found to exhibit a normal distribution as indicated by the constructed histograms and the calculated skewness and kurtosis coefficients. Exponential models with nugget effects were fitted to the experimental semivariograms. Ordinary kriging was performed to generate geochemical maps. The applied interpolation technique proved to be the best in producing geochemical exploration maps for both U and V in Eshidiyya phosphorites. The constructed geochemical maps helped visualize a WNW–ESE U mineralization trend in the studied phosphorites. This trend should be taken into consideration in any future exploration programs for U in south Jordan.     

Spatial prediction of soil properties in a watershed scale through maximum likelihood approach

Surface map of soil properties plays an important role in various applications in a watershed. Ordinary kriging (OK) and regression kriging (RK) are conventionally used to prepare these surface maps but generally need large number of regularly girded soil samples. In this context, REML-EBLUP (REsidual Maximum Likelihood estimation of semivariogram parameters followed by Empirical Best Linear Unbiased Prediction) shown capable but not fully tested in a watershed scale. In this study, REML-EBLUP approach was applied to prepare surface maps of several soil properties in a hilly watershed of Eastern India and the performance was compared with conventionally used spatial interpolation methods: OK and RK. Evaluation of these three spatial interpolation methods through root-mean-squared residuals (RMSR) and mean squared deviation ratio (MSDR) showed better performance of REML-EBLUP over the other methods. Reduction in sample size through random selection of sampling points from full dataset also resulted in better performance of REML-EBLUP over OK and RK approach. The detailed investigation on effect of sample number on performance of spatial interpolation methods concluded that a minimum sampling density of 4/km² may successfully be adopted for spatial prediction of soil properties in a watershed scale using the REML-EBLUP approach.     

地质统计学及其在第四纪研究中的应用

地质统计学是数学地质领域最为活跃而实用的分支,它是以区域化变量理论为基础,以变异函数为基本工具,研究那些在空间分布上既具有随机性又具有结构性的自然现象的科学。在第四纪研究中的很多特征(变量)均可看成区域化变量进行地质统计学分析。作者在讨论了经典概率论及数理统计方法简单地应用于第四纪研究可能出现的问题后,着重介绍了用于第四纪研究中的若干地质统计学方法及基本理论,同时,对地质统计学方法应用于第四纪研究中的前景进行了分析。     

Spatial analysis of land use and shallow groundwater vulnerability in the watershed adjacent to Assateague Island National Seashore,Maryland and Virginia,USA

Spatial relations between land use and groundwater quality in the watershed adjacent to Assateague Island National Seashore, Maryland and Virginia, USA were analyzed by the use of two spatial models. One model used a logit analysis and the other was based on geostatistics. The models were developed and compared on the basis of existing concentrations of nitrate as nitrogen in samples from 529 domestic wells. The models were applied to produce spatial probability maps that show areas in the watershed where concentrations of nitrate in groundwater are likely to exceed a predetermined management threshold value. Maps of the watershed generated by logistic regression and probability kriging analysis showing where the probability of nitrate concentrations would exceed 3 mg/L (>0.50) compared favorably. Logistic regression was less dependent on the spatial distribution of sampled wells, and identified an additional high probability area within the watershed that was missed by probability kriging. The spatial probability maps could be used to determine the natural or anthropogenic factors that best explain the occurrence and distribution of elevated concentrations of nitrate (or other constituents) in shallow groundwater. This information can be used by local land-use planners, ecologists, and managers to protect water supplies and identify land-use planning solutions and monitoring programs in vulnerable areas.     

Multivariate statistical analysis of stream-sediment geochemistry in the Grazer Paläozoikum,Austria

The Austrian reconnaissance study of stream-sediment composition — more than 30000 clay-fraction samples collected over an area of 40000 km² — is summarized in an atlas of regional maps that show the distributions of 35 elements. These maps, rich in information, reveal complicated patterns of element abundance that are difficult to compare on more than a small number of maps at one time. In such a study, multivariate procedures such as simultaneous R-Q mode components analysis may be helpful. They can compress a large number of variables into a much smaller number of independent linear combinations. These composite variables may be mapped and relationships sought between them and geological properties. As an example, R-Q mode components analysis is applied here to the Grazer Paläozoikum, a tectonic unit northeast of the city of Graz, which is composed of diverse lithologies and contains many mineral deposits.     

Ordinary kriging and indicator kriging in the cartography of trace elements contamination in São Domingos mining site (Alentejo, Portugal)

The abandoned pyrite mine of São Domingos, in the southeast Portugal is still an acid mine drainage generator and a source for trace elements pollution. This study aims to evaluate and map the soils and sediments chemical pollution. With this purpose three test sites located in the vicinities of the mine were sampled for soils, sediments and mining waste materials. The samples were analysed for pH, organic carbon, iron oxides and total content of As, Cu, Cr, Hg, Pb, Sb, U, Zn, Mn, S, Fe Al, Ca, K, Mg and P. The hyperspectral image of the test sites was acquired using the aero transported sensor HymapTM. Chemical analysis and hyperspectral images data were processed to estimate the contamination maps by multivariate data analysis, ordinary kriging, cokriging and indicator kriging methodologies.The test sites located downstream the mine open pit, are highly contaminated mainly in As, Hg, Pb and Sb, but no contamination was found in the third test site, upstream the mining site. The acid mining drainage seems to be the main spreading agent of chemical contaminants mostly originating from the mine waste materials. Factorial analysis and the geostatistical methodologies allowed several approaches for the contamination cartography of mining areas.     

Compensating for estimation smoothing in kriging

Smoothing is a characteristic inherent to all minimum mean-square-error spatial estimators such as kriging. Cross-validation can be used to detect and model such smoothing. Inversion of the model produces a new estimator—compensated kriging. A numerical comparison based on an exhaustive permeability sampling of a 4-ft² slab of Berea Sandstone shows that the estimation surface generated by compensated kriging has properties intermediate between those generated by ordinary kriging and stochastic realizations resulting from simulated annealing and sequential Gaussian simulation. The frequency distribution is well reproduced by the compensated kriging surface, which also approximates the experimental semivariogram well—better than ordinary kriging, but not as well as stochastic realizations. Compensated kriging produces surfaces that are more accurate than stochastic realizations, but not as accurate as ordinary kriging.     

Spatial interpolation of severely skewed data with several peak values by the approach integrating kriging and triangular irregular network interpolation

It was not unusual in soil and environmental studies that the distribution of data is severely skewed with several high peak values, which causes the difficulty for Kriging with data transformation to make a satisfied prediction. This paper tested an approach that integrates kriging and triangular irregular network interpolation to make predictions. A data set consisting of total Copper (Cu) concentrations of 147 soil samples, with a skewness of 4.64 and several high peak values, from a copper smelting contaminated site in Zhejiang Province, China. The original data were partitioned into two parts. One represented the holistic spatial variability, followed by lognormal distribution, and then was interpolated by lognormal ordinary kriging. The other assumed to show the local variability of the area that near to high peak values, and triangular irregular network interpolation was applied. These two predictions were integrated into one map. This map was assessed by comparing with rank-order ordinary kriging and normal score ordinary kriging using another data set consisting of 54 soil samples of Cu in the same region. According to the mean error and root mean square error, the approach integrating lognormal ordinary kriging and triangular irregular network interpolation could make improved predictions over rank-order ordinary kriging and normal score ordinary kriging for the severely skewed data with several high peak values.     

岩石地球化学数据挖掘及弱异常识别--以新疆阿舍勒铜矿为例

勘查地球化学找矿工作的重点在于正确解译地球化学数据,以便从冗杂的地质信息中精准提取与成矿有关的异常信息,指导找矿研究。然而,地球化学数据属于成分数据,具有闭合效应,只有对数据进行正确的预处理才能应用多元统计分析方法,还原元素真实的空间分布。本文在阿舍勒铜锌矿区外围南侧区域共收集1009件地表原生晕样品,对样品中的13种微量元素进行测试,并对原始数据、对数及ilr变换后的数据进行EDA分析,对比数据空间分布及内部结构特征。运用(稳健)主成分分析,结合成分数据双标图及第一主成分点位图,解译三类数据指示的元素组合与成矿信息之间的关联。随后运用多重分形滤波技术,对以ilr变换为基础的稳健主成分得分数据分解元素组合异常和背景分布特征。结果表明:①经过对数及ilr变换后的数据相比原始数据空间尺度更均匀,数据近似正态分布;②三类数据双标图表明,仅ilr变换后的数据消除了“闭合效应”,且其第一主成分元素分组揭示了研究区铜矿化与铅锌多金属矿化组合;以对数变换与ilr变换为基础的第一主成分点位图表明,后者主成分得分异常能够较好指示研究区地质找矿信息;③结合研究区地质找矿信息、元素组合异常及背景空间分布特征,最终圈定3个有利找矿靶区。     

Designing an optimal multivariate geostatistical groundwater quality monitoring network using factorial kriging and genetic algorithms

The optimal selection of monitoring wells is a major task in designing an information-effective groundwater quality monitoring network which can provide sufficient and not redundant information of monitoring variables for delineating spatial distribution or variations of monitoring variables. This study develops a design approach for an optimal multivariate geostatistical groundwater quality network by proposing a network system to identify groundwater quality spatial variations by using factorial kriging with genetic algorithm. The proposed approach is applied in designing a groundwater quality monitoring network for nine variables (EC, TDS, Cl⁻, Na, Ca, Mg, SO ₄ ²⁻ , Mn and Fe) in the Pingtung Plain in Taiwan. The spatial structure results show that the variograms and cross-variograms of the nine variables can be modeled in two spatial structures: a Gaussian model with ranges 28.5 km and a spherical model with 40 km for short and long spatial scale variations, respectively. Moreover, the nine variables can be grouped into two major components for both short and long scales. The proposed optimal monitoring design model successfully obtains different optimal network systems for delineating spatial variations of the nine groundwater quality variables by using 20, 25 and 30 monitoring wells in both short scale (28.5 km) and long scale (40 km). Finally, the study confirms that the proposed model can design an optimal groundwater monitoring network that not only considers multiple groundwater quality variables but also monitors variations of monitoring variables at various spatial scales in the study area.     

Comparison of statistical prediction methods for characterizing the spatial variability of apparent electrical conductivity in coastal salt-affected farmland

Soil salinity has been known to be problematic to land productivity and environment in the lower Yellow River Delta due to the presence of a shallow, saline water table and marine sediments. Spatial information on soil salinity has gained increasing importance for the demand of management and sustainable utilization of arable land in this area. Apparent electrical conductivity, as measured by electromagnetic induction instrument in a fairly quick manner, has succeeded in mapping soil salinity and many other soil physical and chemical properties from field to regional scales. This was done based on the correlation that existed between apparent electrical conductivity and many other soil properties. In this paper, four spatial prediction methods, i.e., local polynomial, inverse distance weighed, ordinary kriging and universal kriging, were employed to estimate field-scale apparent electrical conductivity with the aid of an electromagnetic induction instrument (type EM38). The spatial patterns estimated by the four methods using EM38 survey datasets of various sample sizes were compared with those generated by each method using the entire sample size. Spatial similarity was evaluated using difference index (DI) between the maps created using various sample sizes (i.e., target maps) and the maps generated with the entire sample size (i.e., the reference map). The results indicated that universal kriging had the best performance owing to the inclusion of residuals and spatial detrending in the kriging system. DI showed that spatial similarity between the target and reference maps of apparent electrical conductivity decreased with the reduction in sample size for each prediction method. Under the same reduction in sample size, the method retaining the most spatial similarity was universal kriging, followed by ordinary kriging, inverse distance weighed, and local polynomial. Approximately, 70 % of total survey data essentially met the need for retaining 90 % details of the reference map for universal kriging and ordinary kriging methods. This conclusion was that OK and UK were two most appropriate methods for spatial estimation of apparent electrical conductivity as they were robust with the reduction in sample size.     

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.