Catchment scale geostatistical simulation and uncertainty of soil erodibility using sequential Gaussian simulation

Soil erodibility (K) affects sediment delivery to streams and needs to be appropriately quantified and interpolated as a fundamental geographic variable for implementing suitable catchment management and conservation practices. The spatial distribution of K for erosion modelling at non-sampling grid locations has traditionally been estimated using interpolation algorithms such as kriging which do not adequately represent the uncertainty of estimates. These methods cause smoothing effects through overestimating the low values and underestimating the large values. In this study observed values were used to implement a sequential Gaussian simulation (SGS) procedure to evaluate the certainty of modelled data. Soil erodibility values were computed using 41 soil samples taken from the top 10 cm soil layer regularly distributed across four catchments, 367–770 ha in area, within Kangaroo River State forest, New South Wales (NSW). One hundred realisations were applied in the simulation process to provide spatial uncertainty and error estimates of soil erodibility. The results indicated that values simulated by the SGS algorithm produced similar K values for the neighbouring cells. At the pixel level, the SGS approach generated a reliable estimation of soil erodibility in most areas. Spatial variation of the K factor in this study was strongly related to soil landscape differences across the catchments; within catchments slope gradient did not have a substantial impact on the numerical values of the K factor using pixel-by-pixel comparisons of raster grid maps.     

基于不同地质统计方法的渗透系数场对污染物运移的影响

渗透系数场的空间变异性是影响污染物运移结果的决定因素,而地质统计方法是解决渗透系数空间变异性的主要技术手段。本文利用野外场地实测数据,采用普通克里格法和指示克里格法、顺序高斯模拟法和顺序指示模拟法四种地质统计方法,插值估测和模拟再现随机渗透系数场,进而对比研究四种渗透系数场对大尺度污染物运移的影响。研究结果表明,污染羽的质心位置（一阶矩）主要由渗透系数的平均值来决定;污染羽在空间上的展布范围（二阶矩）主要受渗透系数空间变异方差的影响;条件模拟克服了估计法的平滑效果,较好地再现真实曲线的波动性,渗透系数（ lnK）估计方差与污染羽空间二阶矩随着条件模拟次数的增加而减小,并且顺序指示模拟程度更加明显。     

Numerical analysis of hydrogeochemical data: a case study (Alto Guadalentı́n,southeast Spain)

This study examines the spatial variability of the factors obtained from the application of correspondence analysis to a hydrogeochemical data set. The goal was to synthesize the hydrogeochemical information using this multivariate statistical technique, by setting a series of factors which clarified the main properties of one aquifer. Then, a geostatistical framework to obtain a probabilistic assessment of groundwater quality was established. Experimental and theoretical semivariograms of the selected factors, considered as regionalized variables, were computed. These variographic information and factor values in the experimental sites were used in the ordinary kriging, which provides unbiased and linear estimates of the regionalized variables. These estimates were used to compile maps of the chosen factors, which explain their spatial distribution.The selected case study was the alluvial aquifer of Alto Guadalentı́n which is situated in southeast Spain, in the Internal Zones of Betic Cordilleras. These waters are chiefly SO₄ and Cl types, but HCO₃ facies are common in the central sector of the basin. High temperature, acid pH, problems of overexploitation and pollution by CO₂-gas characterise these waters. Available groundwater quality monitoring data were used to calibrate the numerical model. The present study focused on setting the main physical and chemical attributes and establishing the spatial pattern of groundwater quality and the temporal changes in this pattern.     

Geostatistical method to delineate anomalies of multi-scale spatial variation in hydrogeological changes due to the ChiChi earthquake in the ChouShui River alluvial fan in Taiwan

This study uses Ordinary Kriging (OK), Sequential Gaussian Simulation (SGS) and Simulated Annealing Simulation (SAS) to relocate the completely heterotopic dataset from the locations of the Standardized Satellite Oriented Control Point System (SSOCPS) stations to the Groundwater Monitoring Networks (GMNS) stations and factorial kriging to analyze and map relationships among seven variables, including the hydraulic conductivities of three aquifers, the vertical displacements of the ground and groundwater level changes in the wells of three aquifers, and also to delineate the anomalies of multi-scale spatial variation of hydrogeological properties associated with the ChiChi earthquake, measuring 7.3 on the Richter scale, in the ChouShui River alluvial fan in Taiwan. In this study, the anomalies of spatial variation of hydrogeological properties associated with the earthquake are illustrated at micro, local and regional scales of 9, 12 and 36 km, respectively. In the study area, regionalization components associated with variation at local and regional scales are obtained and mapped by factorial kriging. Factorial Kriging Analysis (FKA) also demonstrated that the main effects of the ChiChi earthquake on the spatial variations of groundwater hydrological changes include porous media compression at micro scale, hydrogeological heterogeneousness of the sediments within the aquifer at local scale and the cyclic loading of deviatoric stress at regional scale. Finally, maps of spatial variations of regional components fully depicted all of the anomalies of spatial variation of hydrogeological changes due to the ChiChi earthquake and can be used to identify, confirm and monitor the hydrogeological properties in this study area.     

Combining Areal and Point Data in Geostatistical Interpolation: Applications to Soil Science and Medical Geography

A common issue in spatial interpolation is the combination of data measured over different spatial supports. For example, information available for mapping disease risk typically includes point data (e.g. patients’ and controls’ residence) and aggregated data (e.g. socio-demographic and economic attributes recorded at the census track level). Similarly, soil measurements at discrete locations in the field are often supplemented with choropleth maps (e.g. soil or geological maps) that model the spatial distribution of soil attributes as the juxtaposition of polygons (areas) with constant values. This paper presents a general formulation of kriging that allows the combination of both point and areal data through the use of area-to-area, area-to-point, and point-to-point covariances in the kriging system. The procedure is illustrated using two data sets: (1) geological map and heavy metal concentrations recorded in the topsoil of the Swiss Jura, and (2) incidence rates of late-stage breast cancer diagnosis per census tract and location of patient residences for three counties in Michigan. In the second case, the kriging system includes an error variance term derived according to the binomial distribution to account for varying degree of reliability of incidence rates depending on the total number of cases recorded in those tracts. Except under the binomial kriging framework, area-and-point (AAP) kriging ensures the coherence of the prediction so that the average of interpolated values within each mapping unit is equal to the original areal datum. The relationships between binomial kriging, Poisson kriging, and indicator kriging are discussed under different scenarios for the population size and spatial support. Sensitivity analysis demonstrates the smaller smoothing and greater prediction accuracy of the new procedure over ordinary and traditional residual kriging based on the assumption that the local mean is constant within each mapping unit.     

Factorial and indicator kriging methods using a geographic information system to delineate spatial variation and pollution sources of soil heavy metals

This study uses factorial and indicator kriging methods with a geographic information system (GIS), to overlay and delineate the variation and pollution sources of soil heavy metals in fields in the north of Changhua County, Taiwan. Accordingly, the spatial variation and pollution probability patterns of soil cadmium, chromium and copper are estimated to support further environmental monitoring, remediation and planning. The overlay maps of industrial plants, irrigation channels and kriging estimates, constitute a GIS. This reveals that the short- and long-range variations and soil pollution by Cd, Cr and Cu strongly corresponded to the locations of irrigation systems and industrial plants in the area of study. The maps also verify most locations of suspected pollution sources from the GIS database. Moreover, the probability spatial patterns estimated by indicator kriging, based on the pollution thresholds of Cd, Cr and Cu, were also highly correlated with irrigation channels and industrial plants. The GIS, with factorial and indicator kriging under different combinations of variations and pollution probabilities of soil Cd, Cr and Cu, provide alternative scenarios for delineating the areas that contain high variation and high pollution probability of these heavy metals, to enable decision makers to identify pollution sources, and to further investigate and remedy soil pollution.     

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.     

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.     

含水岩组概化的累积导水系数法

地下水流系统理论和数值模拟技术分别是水文地质学的基本理论和技术方法,含水岩组的概化是地下水流系统分析和地下水数值模拟的重要基础,直接影响着数值模拟和水流系统分析的精度和可信度.为提高含水岩组概化的精度和可信度,提出一种含水岩组概化的新方法,即累积导水系数法.依据岩层厚度与渗透系数乘积累积值随深度的变化,以及水文地质剖面岩性分布的整体特征,概化含水介质结构.以玛纳斯河流域为例,应用该方法概化流域内的岩性剖面,结合GMS软件中TINS模块构建水文地质结构模型.结果表明,应用该方法概化后的含水层结构具有较好的合理性和仿真性,建立的三维模型很好地显示了研究区含水介质的空间展布特征,为建立地下水流模型奠定了良好的基础.      

Dynamic stochastic estimation of physical variables

A fundamental problem facing the physical sciences today is analysis of natural variations and mapping of spatiotemporal processes. Detailed maps describing the space/time distribution of groundwater contaminants, atmospheric pollutant deposition processes, rainfall intensity variables, external intermittency functions, etc. are tools whose importance in practical applications cannot be overestimated. Such maps are valuable inputs for numerous applications including, for example, solute transport, storm modeling, turbulent-nonturbulent flow characterization, weather prediction, and human exposure to hazardous substances. The approach considered here uses the spatiotemporal random field theory to study natural space/time variations and derive dynamic stochastic estimates of physical variables. The random field model is constructed in a space/time continuum that explicitly involves both spatial and temporal aspects and provides a rigorous representation of spatiotemporal variabilities and uncertainties. This has considerable advantages as regards analytical investigations of natural processes. The model is used to study natural space/time variations of springwater calcium ion data from the Dyle River catchment area, Belgium. This dataset is characterized by a spatially nonhomogeneous and temporally nonstationary variability that is quantified by random field parameters, such as orders of space/time continuity and random field increments. A rich class of covariance models is determined from the properties of the random field increments. The analysis leads to maps of continuity orders and covariances reflecting space/time calcium ion correlations and trends. Calcium ion estimates and the associated statistical errors are calculated at unmeasured locations/instants over the Dyle region using a space/time kriging algorithm. In practice, the interpretation of the results of the dynamic stochastic analysis should take into consideration the scale effects.     

On the influence of kriging parameters on the cartographic output—A study in mapping subglacial topography

Geostatistics provides a suite of methods, summarized as kriging, to analyze a finite data set to describe a continuous property of the Earth. Kriging methods consist of moving window optimum estimation techniques, which are based on a least-squares principle and use a spatial structure function, usually the variogram. Applications of kriging techniques have become increasingly wide-spread, with ordinary kriging and universal kriging being the most popular ones. The dependence of the final map or model on the input, however, is not generally understood. Herein we demonstrate how changes in the kriging parameters and the neighborhood search affect the cartographic result. Principles are illustrated through a glaciological study. The objective is to map ice thickness and subglacial topography of Storglaciären, Kebnekaise Massif, northern Sweden, from several sets of radio-echo soundings and hot water drillings. New maps are presented.     

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.     

Estimating Regional Hydraulic Conductivity Fields—A Comparative Study of Geostatistical Methods

Geostatistical estimations of the hydraulic conductivity field (K) in the Carrizo aquifer, Texas, are performed over three regional domains of increasing extent: 1) the domain corresponding to a three-dimensional groundwater flow model previously built (model domain); 2) the area corresponding to the 10 counties encompassing the model domain (County domain), and; 3) the full extension of the Carrizo aquifer within Texas (Texas domain). Two different approaches are used: 1) an indirect approach where transmissivity (T) is estimated first and K is retrieved through division of the T estimate by the screen length of the wells, and; 2) a direct approach where K data are kriged directly. Due to preferential well screen emplacement, and scarcity of sampling in the deeper portions of the formation (> 1 km), the available data set is biased toward high values of hydraulic conductivities. Kriging combined with linear regression, simple kriging with varying local means, kriging with an external drift, and cokriging allow the incorporation of specific capacity as secondary information. Prediction performances (assessed through cross-validation) differ according to the chosen approach, the considered variable (log-transformed or back-transformed), and the scale of interest. For the indirect approach, kriging of log T with varying local means yields the best estimates for both log-transformed and back-transformed variables in the model domain. For larger regional scales (County and Texas domains), cokriging performs generally better than other kriging procedures when estimating both (log T)^∗ and T^∗. Among procedures using the direct approach, the best prediction performances are obtained using kriging of log K with an external drift. Overall, geostatistical estimation of the hydraulic conductivity field at regional scales is rendered difficult by both preferential well location and preferential emplacement of well screens in the most productive portions of the aquifer. Such bias creates unrealistic hydraulic conductivity values, in particular, in sparsely sampled areas.     

Estimating transmissivity using empirical and geostatistical methods in the volcanic aquifers of Upper Awash Basin, central Ethiopia

Transmissivity (T) is a basic hydraulic parameter of an aquifer that is utilized in most groundwater flow equations to understand the flow dynamics and is generally estimated from pumping tests. However, the cost of performing a large number of aquifer tests is expensive and time consuming. The fact that specific capacity (S _c) is correlated with hydraulic flow properties of aquifers simplifies parameter estimation mainly because specific capacity values are more abundant in groundwater databases than values of transmissivity and they offer another approach to estimate hydraulic parameters of aquifers. In this study, an empirical relation is derived using 214 pairs of transmissivity and specific capacity values that are obtained from pumping tests conducted on water wells penetrating the complex volcanic aquifers of Upper Awash Basin, central Ethiopia. Linear and logarithmic regression functions have been performed and it is found that the logarithmic relationship predicting transmissivity from specific capacity data has a better correlation (R = 0.97) than the linear relationship (R = 0.79). The two parameters are log-normally distributed, in which the logarithmic relation is also better statistically justified than the linear relation. Geostatistical estimations of the transmissivity were made using different inputs and methods. Measured and supplemented transmissivity data obtained from estimates using the derived empirical relation were krigged and cokrigged, spherical and exponential models were fitted to the experimental variograms. The cross-validation results showed that the best estimation is provided using the kriging procedure, the transmissivity field represented by the measured transmissivity data and the experimental variogram fitted with the exponential model. Based on the geostatistical approach, the transmissivity map of the aquifer is produced, which will be used for groundwater flow modeling of the study area that will follow this analysis.     

Correcting the Smoothing Effect of Ordinary Kriging Estimates

The smoothing effect of ordinary kriging is a well-known dangerous effect associated with this estimation technique. Consequently kriging estimates do not reproduce both histogram and semivariogram model of sample data. A four-step procedure for correcting the smoothing effect of ordinary kriging estimates is shown to be efficient for the reproduction of histogram and semivariogram without loss of local accuracy. Furthermore, this procedure provides a unique map sharing both local and global accuracies. Ordinary kriging with a proper correction for smoothing effect can be revitalized as a reliable estimation method that allows a better use of the available information.     

Comparison of ordinary and disjunctive kriging in an Indian iron ore deposit

This paper presents the results of disjunctive kriging applied to a supergene iron ore deposit of Bailadila Range of India. Disjunctive kriging is applied firstly to compare estimates of the blocks by ordinary kriging and secondly to estimate benchwise local recoverable reserves of the orebody. Good agreement exists between block estimates by ordinary kriging and disjunctive kriging except for peripheral blocks with less borehole information. Estimation of benchwise reserves shows that the behavior of the distribution of grades is different in various benches. The study shows that disjunctive kriging can be applied successfully for estimation of local recoverable reserves in the case of a good grade hematite iron ore deposit.     

Spatial modelling of hazardous elements at waste dumps using geostatistical approach: a case study Sarcheshmeh copper mine,Iran

Mining activities and resulting wastes can be considered as one of the most important sources of hazardous elements in the environment. Knowledge of the spatial distribution of toxic elements in waste dump systems is necessary to assess environmental hazard and strategy. To achieve this goal, this paper investigates spatial distribution of toxic elements using statistical and geostatistical analysis. A total of 58 soil samples were collected, and the amount of As, Cd, Co, Cr, Cu, Mn, Mo, Ni, Pb and Zn was then determined at “Sarcheshmeh” copper mine waste dumps. In order to evaluate the presence of multivariate outliers, Mahalanobis distance technique (D ²) was applied and the multivariate outlier samples were removed. This resulted in an increase in correlation coefficient. To reduce dimension of data set, principal component analysis was applied and four principal components were determined which indicate 83.463% of the total variance of data set. Estimated PCs together with the toxic elements maps based on the ordinary kriging display aggregation of toxic elements in some parts, and validity of predictions was evaluated using the leave-one-out cross-validation method. The regression coefficients of estimated and observed values presented the reliability of the kriging estimates. Sequential Gaussian simulation method was applied for principal components due to similar results of estimated principal components and toxic elements. The results of simulation maps are almost identical to estimated outcomes.