Evaluation of Diverse Geometric and Geostatistical Estimation Methods Applied to Annual Precipitation in Asturias (NW Spain)

Several alternative estimation and interpolation methods for making annual precipitation maps of Asturias are analysed. The data series in this study corresponds to the year 2003. There exists an evident relationship between precipitation and altitude, with a high correlation coefficient of 0.70, that reflects the hillside effect; that is, the increase in the amount of precipitation in more mountainous areas. The direct spatial variability of precipitation and of altitude and the cross variability of precipitation–altitude are defined by two exponential variogram models: one with a short-range structure (15–30 km) that reflects the control exerted by the lesser, local mountain ranges over the amount of precipitation; and another with a long-range structure (80 km) that supposes the influence over precipitation of the major mountainous alignments of the inland areas of the Cantabrian Mountain Range (Cordillera Cantábrica) situated between 60 and 90 km from the coastline. These variogram models had to be validated for coregionalization by the Pardo-Igúzquiza and Dowd method so as to be able to make the cokriging map. The geometric estimation methods employed were triangulation and inverse distance. The geostatistical estimation methods developed were simple kriging, ordinary kriging, kriging with a trend model (universal kriging), lognormal kriging, and cokriging. In all of these methods, a 3 × 3 km² grid was selected with a total of 2580 points to estimate, a circular search window of 60 km, and a relatively small number of samples with the aim of highlighting the local features and variations on isohyet maps. The kriging methods were implemented using the WinGslib software, incorporating two specific programs, Prog2 and Fichsurf, so as to be able then to make isohyet maps using the Surfer software. All the methods employed, apart from triangulation, rendered realistic maps with good fits to the values of the original data (precipitation) of the sample maps. The problem with triangulation lies not in the reliability of the estimates but in the fact that it gives rise to contrived maps because of the tendency of isohyets to present abundant triangular facets. The reliability of the methods was based on cross-validation analysis and on evaluation of the different types of errors, both in their values and in their graphical representations. Substantial differences were not found in the values of the errors that might discriminate some methods from others in an evident way. Bearing the aforesaid in mind, should we have to make an evaluation of the different estimation methods in decreasing order of acceptance, this would be: kriging with a trend model, inverse distance, cokriging, lognormal kriging, ordinary kriging, simple kriging, and triangulation. The application of other estimation methods such as colocated cokriging, kriging with an external drift, and kriging of variable local means (residual kriging) is dependent on the availability of a digital model of the terrain with an altitude grid of the region.     

A treatise on the use of geostatistics for the characterization of nonrenewable resources

Five decades of geostatistical development are reviewed to summarize the state of the art for spatial interpolation vis-à-vis kriging or a form thereof. Although a search of the literature reveals a variety of kriging methods, there are but two infrastructures for geostatistical interpolation: simple cokriging, for estimating a single variable using two variables, and generalized cokriging, for estimating one or more variables using the same number of variables that are estimated. The many forms of kriging are varieties of these two interpolation infrastructures. This notion is emphasized to aid the selection of an appropriate interpolation model for a nonrenewable resource. These models are discussed, and literature for the models and for applicable software is cited. Additionally, all aspects of spatial interpolation are discussed, including the adequacy of spatial sampling, distribution characteristics of spatial samples, semivariograms, search parameters, and selection of interpolation models in conformance with spatial data characteristics. Finally, the relationship between interpolation and raster-based geographic information systems is emphasized.     

喀斯特地区春季土壤水分空间插值方法对比

以杨眉河小流域为研究区,通过土壤水分采样,选取辅助变量,采用普通克里金、协同克里金、回归克里金3种地统计学方法对土壤水分数据进行空间插值。结果表明：1）回归克里金对研究区土壤水分估算误差最小,其次为协克里金,普通克里金的误差最大;2）普通克里金生成的土壤水分表面最为平滑,而回归克里金最大程度反映了研究区实际的土壤水分空间变化;3）对于协同克里金,以湿度指数（WI）样点数据作为辅助变量的估算误差小于将WI栅格数据作为辅助变量的估算误差。总之,在可获得有效辅助变量的条件下,回归克里金对研究区土壤水分估算的效果优于协同克里金与普通克里金。     

利用Cokriging提高估算土壤盐离子浓度分布的精度——以黄河三角洲为例

估算土壤中化学物质的含量与空间分布是了解多孔介质中水盐运移规律并进而因地制宜地提出盐渍土改良措施的关键。大面积的实地采样分析费时费力且耗资巨大。通过地统计分析, 使用有限的采样数据可获得土壤溶质的准确变异。本文探讨和比较了Ordinary kriging (OK) 与Cokriging (COK) 这两种内插方法。结果显示一半的采样点数据的COK较之全部采样点数据的OK精度更高, 相对均方根误差降幅为130.83%;采用同样的协同变量 (239个全盐量数据), 一半的采样点数据的COK较之全部采样点数据的COK精度更高, 相对均方根误差降幅为20.10%。协同变量与主变量的相关度决定了COK的预测精度, 当相关系数由77%升高为99%时, 相对均方根误差降低了48.30%。     

Sparse Data Division Using Data Segmentation and Kohonen Network for Neural Network and Geostatistical Ore Grade Modeling in Nome Offshore Placer Deposit

In this paper, sparse data problem in neural network and geostatistical modeling for ore-grade estimation was addressed in the Nome offshore placer gold deposit. The problem of sparse data arises because of the random data division into training, validation, and test subsets during ore-grade modeling. In this regard, the possibility of generating statistically dissimilar data subsets by random data division was also explored through a simulation exercise. A combined approach of data segmentation and application of a Kohonen network then was used to solve the data division problem. Two neural networks and five kriging models were applied for grade modeling. The neural network was trained using an early stopping method. Performance evaluation of the models was carried out on the test data set. The study results indicated that all the models that were investigated in this study performed almost equally. It was also revealed that by using the secondary variable watertable depth the neural network and the kriging models slightly improved their prediction precision. Further, the overall R ² of the models was poor as a result of high nugget (noisy) component in ore-grade variation.     

Spatial Prediction of Lateral Variability of a Laterite-Type Bauxite Horizon Using Ancillary Ground-Penetrating Radar Data

Tropical laterite-type bauxite deposits often pose a unique challenge for resource modelling and mine planning due to the extreme lateral variability at the base of the bauxite ore unit within the regolith profile. An economically viable drilling grid is often rather sparse for traditional prediction techniques to precisely account for the lateral variability in the lower contact of a bauxite ore unit. However, ground-penetrating radar (GPR) offers an inexpensive and rapid method for delineating laterite profiles by acquiring fine-scale data from the ground. These numerous data (secondary variable) can be merged with sparsely spaced borehole data (primary variable) through various statistical and geostatistical techniques, provided that there is a linear relation between the primary and secondary variables. Four prediction techniques, including standard linear regression, simple kriging with varying local means, co-located cokriging and kriging with an external drift, were used in this study to incorporate exhaustive GPR data in predictive estimation the base of a bauxite ore unit within a lateritic bauxite deposit in Australia. Cross-validation was used to assess the performance of each technique. The most robust estimates are produced using ordinary co-located cokriging in accordance with the cross-validation analysis. Comparison of the estimates against the actual mine floor indicates that the inclusion of ancillary GPR data substantially improves the quality of the estimates representing the bauxite base surface.     

利用不同方法估测土壤有机质及其对采样数的敏感性分析

用随机方法从262个采样点中抽取200个点作为已知有机质含量的数据集,将所有采样点的碱解氮作为辅助数据预测有机质的空间分布。利用有机质信息的普通克立格法的方差解释量和预测精度最低,而回归克立格法因在预测过程中加入了回归残差而使方差解释量最大、预测精度最高。为了分析采样数对不同方法预测精度的影响,从上述已知有机质含量的200个点中分别随机抽取40、80、120、160个点构成4个数据集,分别利用它们的有机质信息和不同方法预测了有机质的空间分布,结果表明:对于每个数据集,4种方法的预测精度顺序均为RGK>COK>RG>OK,线性回归法的预测精度随采样点的增加基本不变,而其它三种方法的预测精度却逐渐提高。     

Geostatistical mapping of real estate prices: an empirical comparison of kriging and cokriging

Small-sized housing samples and price predictions at nonobserved locations require geostatistical approaches, particularly the kriging estimator. Nevertheless, geostatistics has thus far received little attention in real estate economics. The article’s objective is to empirically compare the prediction accuracy of univariate kriging variants, namely detrended kriging (DK) and universal kriging (UK), and multivariate extensions, including detrended cokriging (DCK) and universal cokriging (UCK). Both latter methods consider structural and neighborhood characteristics as auxiliary variables. While the price surfaces of DK and UK show nearly identical cross-validated accuracies, the cross-validation-based prediction accuracy of DCK and UCK differ in favor of the latter. If real estate agencies are faced with a univariate sample of property prices, either DK or UK can be used, while in the multivariate case, UCK is recommended, although numerically more complex.     

空间软数据及其插值方法研究进展

由于对地观测技术的迅速发展,空间数据的种类和数量增长迅猛,由空间数据反演得到的各种信息日趋膨胀,这些反演结果中的信息不少以软数据的形式出现。在实际应用中,这些软数据往往与空间插值的目标变量具有一定的相关性,甚至成为控制目标变量空间分布特征的重要因素。然而,由于这些数据通常表示为非数值形式,在计算和处理上存在着一定困难,以致被传统的插值方法所忽视,从而造成信息浪费。近来出现的空间软插值方法是一种利用空间软数据作为辅助信息并以改善插值效果的方法,能够较好的处理并利用软数据所隐含的信息,具有较好的应用发展前景。本文根据空间软数据的特点及其分类,系统综述了空间软插值方法及其应用领域。首先分析了空间数据软硬性质的根本区别,论述了软数据的分类和“硬化”方法,然后介绍空间插值模型中对空间软数据的集成方法和原理,最后对空间软插值方法及其应用研究领域进行了展望。     

基于线性GSI二维半变异函数各向异性结构建模及估计研究——以DEM数据为例

鉴于传统各向异性二维半变异函数各向同性化方法未充分考虑或无力精确描述其内部结构信息的缺陷,本研究通过引入线性广义尺度不变（GSI）模型,以DEM数据作为验证对象,对二维半变异函数各向异性结构信息进行多尺度建模,并采用旋转椭圆法、两步搜索作图法等方法对系统参数进行估计,最后以球状模型为例对理论半变异函数的估计精度,及其在空间数据插值中的应用效果进行对比研究。结果表明：各向异性普遍存在于地形数据的空间变异中,有证据表明,这种各向异性结构中处处显现出不同的变形特征,但是也存在着某种规则性的成分,如各向同性圆形或近圆形等值线,因此,在对坐标进行各向同性化处理时不适合采用“一刀切”的方式去处理;GSI系统参数皆能得到较高精度的估计,如决定系数R²普遍达到了0.99以上,间接证明了GSI模型对地形数据各向异性结构处理的有效性和适用性;通过理论模型估计和插值结果对比,线性GSI坐标转换法比传统坐标转换法有了明显的精度提升,并且展现出了较高的边缘信息恢复能力,但也表现出了一定的局限性和不稳定性。     

Using landscape characteristics to define an adjusted distance metric for improving kriging interpolations

Interpolation of point measurements using geostatistical techniques such as kriging can be used to estimate values at non-sampled locations in space. Traditional geostatistics are based on the spatial autocorrelation concept that nearby things are more related than distant things. In this study, additional information was used to modify the traditional Euclidean concept of distance into an adjusted distance metric that incorporates similarity in terms of quantifiable landscape characteristics such as topography or land use. This new approach was tested by interpolating soil moisture content, pH and carbon-to-nitrogen (C:N) ratio measured in both the mineral and the organic soil layers at a field site in central Sweden. Semivariograms were created using both the traditional distance metrics and the proposed adjusted distance metrics to carry out ordinary kriging (OK) interpolations between sampling points. In addition, kriging with external drift (KED) was used to interpolate soil properties to evaluate the ability of the adjusted distance metric to incorporate secondary data into interpolations. The new adjusted distance metric typically lowered the nugget associated with the semivariogram, thereby better representing small-scale variability in the measured data compared to semivariograms based on the traditional distance metric. The pattern of the resulting kriging interpolations using KED and OK based on the adjusted distance metric were similar because they represented secondary data and, thus, enhanced small-scale variability compared to traditional distance OK. This created interpolations that agreed better with what is expected for the real-world spatial variation of the measured properties. Based on cross-validation error, OK interpolations using the adjusted distance metric better fit observed data than either OK interpolations using traditional distance or KED.     

GIS-Based Slope Stability Analysis,Chuquicamata Open Pit Copper Mine,Chile

The risk of slope failure in the Chuquicamata open-pit mine was analyzed using Geographic Information System (GIS) software and modeling techniques. Models incorporated various component layers at a relatively large map scale (1:5000): alteration, geotechnical unit, proximity to major faults (VIF), GSI (geological strength index), slope (from digital elevation model), proximity to watertable (difference grid between topography and modeled watertable), and composite structural density grid (VIF, smaller faults, and fracture frequency); not all layers were used in all models. Three modeling techniques were used: fuzzy logic, in which parameters in each component layer were ranked by mine geotechnical experts according to their influence in promoting slope failure, and two data-driven techniques, weights-of-evidence and logistic regression, in which statistical correlation of training points (known failures) with parameters were used to derive a relative probability of failure. Because most slope failures are controlled by structure, VIF and smaller faults were divided by orientation into subsets with dip direction parallel, opposite, and normal to slope aspect; these orientations promote circular and planar, toppling, and wedge-type failures, respectively. Density grids of these subsets show high-risk areas for individual failure types. The models demonstrate sensitivity of the analysis to (1) selection of component layers, (2) selection of training points, (3) classification and ranking of categorical parameters, and (4) data problems in certain layers. Predicted high-risk zones in the final models show a high degree of correspondence with recent, post-model failures. Such models can be used to anticipate future pit design concerns. The results presented here illustrate how vast amounts of data, in multiple geo-referenced layers, can be analyzed and modeled using GIS techniques for predictive studies at relatively large map scales. Such modeling techniques could provide a powerful tool for predictive modeling in a vast array of large-map-scale applications requiring similar data integration and evaluation.     

Hydrogeomorphic control on tree growth responses in the Elton area of the Cheshire Saltfield, UK

Increment cores were sampled from oak (Quercus robur) and ash (Fraxinus excelsior) growing at Elton, an area of the Cheshire Saltfield that has experienced significant subsidence and damage to the natural and built environments in the latter part of the twentieth century. Ring-width measurements for Elton trees permitted the construction of one main site chronology (Elton) and four sub chronologies (Elton A, Elton B, Elton C, Elton ASH). Ring-width difference between these and a control chronology identified periods of sustained growth reduction in oak trees commencing in AD 1859/1861, 1886 and 1934. Growth reductions after 1934 are related to watertable draw down caused by brine pumping from a concentration of nine boreholes at Elton, up to 2 km from tree sampling locations. Growth reductions in 1859/1861 and 1886 are likely to be the result of earlier phases of brine pumping in the Wheelock Valley, up to 5 km to the east of Elton, and these reductions correlate well with historic records of subsidence and pumping activity. Cessation of pumping in 1977 led to a lagged growth recovery in oak trees between 1981 and 1986, indicating that an artificial drought had been imposed on the Elton area for a period in excess off 100-y. This research demonstrates a hydrological separation of surface water and groundwater in an area where salt beds are overlain by till and that ring-width records of Q. robur can be used to reconstruct watertable variability and also the spatial impact of solution mining.     

The smoothness of HASM

To smooth noises inherent in uniformly sampled dataset, the smoothness of high accuracy surface modeling (HASM) was explored, and a smoothing method of HASM (HASM-SM) was developed based on a penalized least squares method. The optimal smoothing parameter of HASM-SM was automatically obtained by means of the generalized cross-validation (GCV) method. For an efficient smoothing computation, discrete cosine transform was employed to solve the system of HASM-SM and to estimate the minimum GCV score, simultaneously. Two examples including a numerical test and a real-world example were employed to compare the smoothing ability of HASM-SM with that of GCV thin plate smoothing spline (TPS) and kriging. The numerical test indicated that the minimum GCV HASM-SM is averagely more accurate than TPS and kriging for noisy surface smoothing. The real-world example of smoothing a lidar-derived Digital Elevation Model (DEM) showed that HASM-SM has an obvious smoothing effect, which is on a par with TPS. In conclusion, HASM-SM provides an efficient tool for filtering noises in grid-based surfaces like remote sensing–derived images and DEMs.     

Minimum Acceptance Criteria for Geostatistical Realizations

Geostatistical simulation is being used increasingly for numerical modeling of natural phenomena. The development of simulation as an alternative to kriging is the result of improved characterization of heterogeneity and a model of joint uncertainty. The popularity of simulation has increased in both mining and petroleum industries. Simulation is widely available in commercial software. Many of these software packages, however, do not necessarily provide the tools for careful checking of the geostatistical realizations prior to their use in decision-making. Moreover, practitioners may not understand all that should be checked. There are some basic checks that should be performed on all geostatistical models. This paper identifies (1) the minimum criteria that should be met by all geostatistical simulation models, and (2) the checks required to verify that these minimum criteria are satisfied. All realizations should honor the input information including the geological interpretation, the data values at their locations, the data distribution, and the correlation structure, within acceptable statistical fluctuations. Moreover, the uncertainty measured by the differences between simulated realizations should be a reasonable measure of uncertainty. A number of different applications are shown to illustrate the various checks. These checks should be an integral part of any simulation modeling work flow.     

一种土壤属性制图的曲面建模方法(英文)

High accuracy surface modeling(HASM) is a method which can be applied to soil property interpolation.In this paper,we present a method of HASM combined geographic information for soil property interpolation(HASM-SP) to improve the accuracy.Based on soil types,land use types and parent rocks,HASM-SP was applied to interpolate soil available P,Li,pH,alkali-hydrolyzable N,total K and Cr in a typical red soil hilly region.To evaluate the performance of HASM-SP,we compared its performance with that of ordinary kriging(OK),ordinary kriging combined geographic information(OK-Geo) and stratified kriging(SK).The results showed that the methods combined with geographic information including HASM-SP and OK-Geo obtained a lower estimation bias.HASM-SP also showed less MAEs and RMSEs when it was compared with the other three methods(OK-Geo,OK and SK).Much more details were presented in the HASM-SP maps for soil properties due to the combination of different types of geographic information which gave abrupt boundary for the spatial varia-tion of soil properties.Therefore,HASM-SP can not only reduce prediction errors but also can be accordant with the distribution of geographic information,which make the spatial simula-tion of soil property more reasonable.HASM-SP has not only enriched the theory of high accuracy surface modeling of soil property,but also provided a scientific method for the ap-plication in resource management and environment planning.     

Variable resolution block kriging using a hierarchical spatial data structure

Abstract

Kriging is an optimal method of spatial interpolation that produces an error for each interpolated value. Block kriging is a form of kriging that computes averaged estimates over blocks (areas or volumes) within the interpolation space. If this space is sampled sparsely, and divided into blocks of a constant size, a variable estimation error is obtained for each block, with blocks near to sample points having smaller errors than blocks farther away. An alternative strategy for sparsely sampled spaces is to vary the sizes of blocks in such away that a block's interpolated value is just sufficiently different from that of an adjacent block given the errors on both blocks. This has the advantage of increasing spatial resolution in many regions, and conversely reducing it in others where maintaining a constant size of block is unjustified (hence achieving data compression). Such a variable subdivision of space can be achieved by regular recursive decomposition using a hierarchical data structure. An implementation of this alternative strategy employing a split-and-merge algorithm operating on a hierarchical data structure is discussed. The technique is illustrated using an oceanographic example involving the interpolation of satellite sea surface temperature data. Consideration is given to the problem of error propagation when combining variable resolution interpolated fields in GIS modelling operations.     

Spatial interpolation of marine environment data using P-MSN

ABSTRACT

When a marine study area is large, the environmental variables often present spatially stratified non-homogeneity, violating the spatial second-order stationary assumption. The stratified non-homogeneous surface can be divided into several stationary strata with different means or variances, but still with close relationships between neighboring strata. To give the best linear-unbiased estimator for those environmental variables, an interpolated version of the mean of the surface with stratified non-homogeneity (MSN) method called point mean of the surface with stratified non-homogeneity (P-MSN) was derived. P-MSN distinguishes the spatial mean and variogram in different strata and borrows information from neighboring strata to improve the interpolation precision near the strata boundary. This paper also introduces the implementation of this method, and its performance is demonstrated in two case studies, one using ocean color remote sensing data, and the other using marine environment monitoring data. The predictions of P-MSN were compared with ordinary kriging, stratified kriging, kriging with an external drift, and empirical Bayesian kriging, the most frequently used methods that can handle some extent of spatial non-homogeneity. The results illustrated that for spatially stratified non-homogeneous environmental variables, P-MSN outperforms other methods by simultaneously improving interpolation precision and avoiding artificially abrupt changes along the strata boundaries.     

Reservoir Modeling by Data Integration via Intermediate Spaces and Artificial Intelligence Tools in MPS Simulation Frameworks

Conditioning stochastic simulations are very important in many geostatistical applications that call for the introduction of nonlinear and multiple-point data in reservoir modeling. Here, a new methodology is proposed for the incorporation of different data types into multiple-point statistics (MPS) simulation frameworks. Unlike the previous techniques that call for an approximate forward model (filter) for integration of secondary data into geologically constructed models, the proposed approach develops an intermediate space where all the primary and secondary data are easily mapped onto. Definition of the intermediate space, as may be achieved via application of artificial intelligence tools like neural networks and fuzzy inference systems, eliminates the need for using filters as in previous techniques. The applicability of the proposed approach in conditioning MPS simulations to static and geologic data is verified by modeling a real example of discrete fracture networks using conventional well-log data. The training patterns are well reproduced in the realizations, while the model is also consistent with the map of secondary data.     

基于多源遥感影像的洞庭湖地形提取方法

湖底地形数据是湖泊流域规划与治理、湖区冲淤变化研究、水资源利用和生态环境保护的重要基础。但传统的大型湖泊湖底地形数据获取手段耗时长、投入大,因此,有必要研究一种基于遥感影像快速获取湖底地形数据的方法。本文以洞庭湖为研究对象,采用Landsat和MODIS系列遥感影像提取湖区边界,基于趋势面分析法和克里金插值法,反演湖区边界各点对应的水位,将带有水位信息的边界点作为高程点实现湖底地形反演,进一步用实测湖底地形验证反演方法的可靠性。研究结果表明,克里金插值法水位反演效果较好,交叉验证的误差标准平均值在0.2 m以内,水位样本点分布较多处,基于克里金法的地形反演绝对误差在1 m以内。本文利用湖泊淹没区域变化的特点快速获取湖底地形,对湖区演变分析、综合治理与保护等具有重要意义。