Spatial analyses of groundwater levels using universal kriging

For water levels, generally a non-stationary variable, the technique of universal kriging is applied in preference to ordinary kriging as the interpolation method. Each set of data in every sector can fit different empirical semivariogram models since they have different spatial structures. These models can be classified as circular, spherical, tetraspherical, pentaspherical, exponential, gaussian, rational quadratic, hole effect, K-bessel, J-bessel and stable. This study aims to determine which of these empirical semivariogram models will be best matched with the experimental models obtained from groundwater-table values collected from Mustafakemalpasa left bank irrigation scheme in 2002. The model having the least error was selected by comparing the observed water-table values with the values predicted by empirical semivariogram models. It was determined that the rational quadratic empirical semivariogram model is the best fitted model for the studied irrigation area.     

Kriging of water levels in the Souss aquifer,Morocco

Universal kriging is applied to water table data from the Souss aquifer in central Morocco. The procedure accounts for the spatial variability of the phenomenon to be mapped. With the use of measured elevations of the water table, an experimental variogram is constructed that characterizes the spatial variability of the measured water levels. Spherical and Gaussian variogram models are alternatively used to fit the experimental variogram. The models are used to develop contour maps of water table elevations and corresponding estimation variances. The estimation variances express the reliability of the kriged water table elevation maps. Universal kriging also provides a contour map of the expected elevation of the water table (drift). The differences between the expected and measured water table elevations are called residuals from the drift. Residuals from the drift are compared with residuals obtained by more traditional least-squares analysis.     

Indicator-Based Geostatistical Models For Mapping Fish Survey Data

Marine research survey data on fish stocks often show a small proportion of very high-density values, as for many environmental data. This makes the estimation of second-order statistics, such as the variance and the variogram, non-robust. The high fish density values are generated by fish aggregative behaviour, which may vary greatly at small scale in time and space. The high values are thus imprecisely known, both in their spatial occurrence and order of magnitude. To map such data, three indicator-based geostatistical methods were considered, the top-cut model, min–max autocorrelation factors (MAF) of indicators, and multiple indicator kriging. In the top-cut and MAF approaches, the variable is decomposed into components and the most continuous ones (those corresponding to the low and medium values) are used to guide the mapping. The methods are proposed as alternatives to ordinary kriging when the variogram is difficult to estimate. The methods are detailed and applied on a spatial data set of anchovy densities derived from a typical fish stock acoustic survey performed in the Bay of Biscay, which show a few high-density values distributed in small spatial patches and also as solitary events. The model performances are analyzed by cross-validating the data and comparing the kriged maps. Results are compared to ordinary kriging as a base case. The top-cut model had the best cross-validation performance. The indicator-based models allowed mapping high-value areas with small spatial extent, in contrast to ordinary kriging. Practical guidelines for implementing the indicator-based methods are provided.     

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     

Estimation of semivariogram parameters and evaluation of the effects of data sparsity

Semivariogram parameters are estimated by a weighted least-squares method and a jackknife kriging method. The weighted least-squares method is investigated by differing the lag increment and maximum lag used in the fit. The jackknife kriging method minimizes the variance of the jackknifing error as a function of semivariogram parameters. The effects of data sparsity and the presence of a trend are investigated by using 400-, 200-, and 100-point synthetic data sets. When the two methods yield significantly different results, more data may be needed to determine reliably the semivariogram parameters, or a trend may be present in the data.     

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.     

含水层渗透性空间分布的指示克立格估值

详细介绍了指示克立格估值计算的理论和方法。以指示变异函数为基本工具分析了华北某地区第四系含水层渗透性空间分布的结构特征,结果表明该地区含水层渗透性存在明显的各向异性特征。水平方向上,X轴方向的相关性较Y轴方向的好,Z轴的相关性最差。用指示克立格法对未采样点处进行估值,估值结果显示含水层渗透性由山前向滨海逐渐变低,在垂直方向上,渗透性变化不明显,浅部比深部略好;同时给出了估计精度,并认为对估计精度不高的区域可通过增加适当的工程加以控制。最后用交叉验证法对估值结果进行了检验,证明建立的指示变异函数模型合理且估值效果较好。这一实际应用表明指示克立格法可以很好地描述第四系含水层渗透性的空间分布规律。     

Application of kriging technique to areal precipitation mapping in Arizona

The classical methods for interpolating and spatial averaging of precipitation fields fail to quantify the accuracy of the estimate. On the other hand, kriging is an interpolation method for predicting values of regionalized variables at points (punctual kriging) or average values over an area (block kriging).This paper demonstrates the use of the kriging method for mapping and evaluating precipitation data for the State of Arizona. Using 158 rain gauge stations with 30 years or more of record, the precipitation over the state has been modeled as a realization of a two dimensional random field taking into consideration the spatial variability conditions.Three data sets have been used: (1) the mean annual precipitation over the state; (2) the mean summer rainy season; and (3) the mean winter rainy season. Validation of the empirical semi-variogram for a constant drift case indicated that the exponential model was appropriate for all the data sets. In addition to a global kriging analysis, the data have been examined under an anisotropic assumption which reflects the topographic structure of the state.     

Advantages of using the kriging interpolator to estimate the gravity surface, comparison and spatial variability of gravity data in the El Kef-Ouargha region (northern Tunisia)

A comparative study on two interpolator methods for Bouguer anomaly mapping was performed in the El Kef-Ouargha region, Tunisia. In particular, we compare the results of minimum curvature gridding and kriging. We undertake a geostatistical study using both anisotropic and isotropic analysis for studding the spatial variability and kriging the random variable: “Bouguer anomaly”. The model parameters of each method are calculated to compare the results. The anisotropic analysis most closely approximates the measured data. Consequentially, the Bouguer anomaly and vertical gradient maps using anisotropic kriging have the best correlation with the geological map, best represent density distributions at depth and most closely approximate the measured anomaly amplitudes.     

Spatial distribution of heavy metals in soils: a case study of Changxing,China

Six hundred and sixty-five soil samples were taken from Changxing County in Zhejiang Province, China, to characterize the spatial variability of Hg Cd, Pb, Cu, As and Cr. The geostatistics and geographic information system (GIS) techniques were applied, and the ordinary kriging and lognormal kriging were used to map the spatial patterns of the six heavy metals. Hg, Pb, Cu and As were fitted to the spherical model with a range of 85.75, 82.32, 86.10, and 23.17 km, respectively. Cr was fitted to the exponential model with a range of 6.27 km, and Cd was fitted to the linear model with a range of 37.66 km. Both Pb and Cu had strong spatial dependence due to the effects of natural factors including parent material, topography and soil type. Hg, Cd, Cr and As had, however, moderate spatial dependence, indicating an involvement of human factors. Meanwhile, based on the comparison between the original data and the guide values of the six metals, the disjunctive kriging technique was used to quantify their pollution risks. The results showed that only Cd and Hg exhibited pollution risks in the study area. The pollution source evaluated was closely corresponded with the real discharge of industrial production and the application of organomercury pesticides. The results of this study provide insight into risk assessment of environmental pollution and decision making for agricultural production and industrial adjustment of building materials.     

岩土力学参数空间变异性的集合卡尔曼滤波估值

岩土参数具有结构性和随机性的空间变异特征,该特征导致岩土参数具有不确定性。以地质统计学作为岩土参数空间变异性分析的理论基础,将分布于研究区的岩土参数视为区域化变量,变异函数既描述了岩土参数整体的空间结构性变化,又描述了其局部的随机性变化,用变异函数理论模型作为描述岩土参数空间变异规律的数学模型。引入集合卡尔曼滤波（EnKF）分析方法,利用时空分布的观测数据,对岩土参数空间变异性进行估值。数值算例表明,EnKF能够有效地融合观测数据,较好地提供岩土参数空间变异性的估值。     

GaRiRO: Gradient and residual integrated rank ordering of stations in rainfall monitoring network

This paper presents a new method that integrates gradient and residual values for rank ordering of stations in a monitoring network (GaRiRO). The innovation is derived from the fact that the parameter (dependent variable) gauged through the monitoring network is modelled using independent variables that influence its measured quantity. And the dependent variable exhibit non-stationary spatial gradient with respect to the independent variables, particularly in complex terrain. GaRiRO technique was developed to prioritize the rain gauge stations for optimizing the existing network and selection of the best locations for relocation or installation of gauges. Although initially aimed to assist hydrologists with a ranking scheme for rain gauge stations, it can be applied to any environmental, meteorological or hydro-meteorological monitoring network. The new procedure is based on deriving gradient and residual value at each station by modeling the spatial relationship of dependent-independent variable using geographically weighted regression (GWR) technique. For the prospective stations with no record, the gradient value is estimated using GWR model and the residual value is derived from the residual map generated by applying kriging technique on the residual derived at all gauged locations. The method combines the decision factor with analytical strength of GIS for prioritizing the stations which results in limited number of trials for installation or relocation of gauges to yield optimized network configuration.     

Application of geostatistics to identify gold-rich areas in the Finisterre-Fervenza region,NW Spain

Three univariate geostatistical methods of estimation are applied to a geochemical data set. The studied methods are: ordinary kriging (cross-validation), factorial kriging, and indicator kriging. These techniques use the probabilistic and spatial behaviour of geochemical variables, giving a tool for identifying potential anomalous areas to locate mineralization. Ordinary kriging is easy to apply and to interpret the results. It has the advantage of using the same experimental grid points for its estimates, and no additional grid points are needed. Factorial kriging decomposes the raw variable into as many components as there are identified structures in the variogram. This, however, is a complex method and its application is more difficult than that of ordinary or indicator kriging. The main advantages of indicator kriging are that data are used by their rank order, being more robust about outlier values, and that the presentation of results is simple. Nevertheless, indicator kriging is incapable of separating anomalous values and the high values from the background, which have a behaviour different to the anomaly. In this work, the results of the application of these 3 kriging methods to a set of mineral exploration data obtained from a geochemical survey carried out in NW Spain are presented. This area is characterised by the presence of Au mineral occurrences. The kriging methods were applied to As, considered as a pathfinder of Au in this area. Numerical treatment of Au is not applicable, because it presents most values equal to the detection limit, and a series of extreme values. The results of the application of ordinary kriging, factorial kriging and indicator kriging to As make possible the location of a series of rich values, sited along a N–S shear zone, considered a structure related to the presence of Au.     

On Visualization for Assessing Kriging Outcomes

Extant opinion about kriging is that all weights should be positive. Visualizations rendered by converting kriged grids to digital images are presented to show that negative weights may be beneficial to some spatial problems. In particular, variogram models with zero-valued nuggets, already well known to minimize smoothing through kriging, result in a visual resolution substantially superior to that from kriging with a variogram model having a nonzero nugget value in application to satellite acquired data. Negative weights are more likely when using variogram models with zero-valued nuggets, but resultant visualizations often show a smoother transition between extreme data values. This is true even when a variogram model having a nugget value of zero is not optimum with respect to mean square error, as is demonstrated using a nitrate data set. An analogy to digital image processing is used to suggest that the influence of negative weights in kriging is similar to a high-boost kernel.     

岩土力学参数空间变异性的集合卡尔曼滤波佶值

岩土参数具有结构性和随机性的空间变异特征，该特征导致岩土参数具有不确定性。以地质统计学作为岩土参数空间变异性分析的理论基础，将分布于研究区的岩土参数视为区域化变量，变异函数既描述了岩土参数整体的空间结构性变化，又描述了其局部的随机性变化，用变异函数理论模型作为描述岩土参数空间变异规律的数学模型。引入集合卡尔曼滤波（EnKF）分析方法，利用时空分布的观测数据，对岩土参数空间变异性进行估值。数值算例表明，EnKF能够有效地融合观测数据，较好地提供岩土参数空间变异性的估值。     

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.     

Application of kriging with external drift to estimate hydraulic conductivity from electrical-resistivity data in unconsolidated deposits near Montalto Uffugo, Italy

In the framework of various studies to characterize the aquifer at the groundwater experimental field near Montalto Uffugo, Italy, the present work estimates the spatial distribution of hydraulic conductivity of the unconsolidated deposits that underlie the area, by applying the geostatistical technique of kriging with external drift to electrical-resistivity and hydraulic-conductivity data. The reliability of the estimation method was tested by implementing a model, based on the method of cells, that simulates groundwater flow, with the estimated values of hydraulic conductivity. The results obtained indicate that the estimation method used has an acceptable degree of reliability.     

An Alternative Measure of the Reliability of Ordinary Kriging Estimates

This paper presents an interpolation variance as an alternative to the measure of the reliability of ordinary kriging estimates. Contrary to the traditional kriging variance, the interpolation variance is data-values dependent, variogram dependent, and a measure of local accuracy. Natural phenomena are not homogeneous; therefore, local variability as expressed through data values must be recognized for a correct assessment of uncertainty. The interpolation variance is simply the weighted average of the squared differences between data values and the retained estimate. Ordinary kriging or simple kriging variances are the expected values of interpolation variances; therefore, these traditional homoscedastic estimation variances cannot properly measure local data dispersion. More precisely, the interpolation variance is an estimate of the local conditional variance, when the ordinary kriging weights are interpreted as conditional probabilities associated to the n neighboring data. This interpretation is valid if, and only if, all ordinary kriging weights are positive or constrained to be such. Extensive tests illustrate that the interpolation variance is a useful alternative to the traditional kriging variance.     

半变异函数及取样间距对克里金法在海洋地层分析中的影响研究

以浅剖数据为源数据,钻孔实测数据为验证数据,利用普通克里金法对海底地层厚度进行空间插值得到地层分布特征,采用3种半变异函数模型和不同取样间距对某井场3组地层厚度进行普通克里金插值并验证其插值效果。结果表明:普通克里金是一种有效的海底地层厚度预测方法;结构分析最佳的模型不一定是误差最小的模型,应对不同模型下的插值结果进行综合分析来选择最合适的模型,并提出球状模型在该井场厚度估计中最优,高斯模型次之;对于球状模型,增大取样间距对地层厚度变化剧烈的地层回归效果影响较小,对地层厚度变化不大的地层回归效果影响较大;同时,SE预测值变化率分析表明对于地层厚度变化剧烈的地层,减小取样间距可以大幅度地减少插值误差,而对于地层厚度变化不大的地层,减小取样间距对插值精度提高的意义不大。     

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.