Assessment of regression kriging for spatial interpolation – comparisons of seven GIS interpolation methods

As an important GIS function, spatial interpolation is one of the most often used geographic techniques for spatial query, spatial data visualization, and spatial decision-making processes in GIS and environmental science. However, less attention has been paid on the comparisons of available spatial interpolation methods, although a number of GIS models including inverse distance weighting, spline, radial basis functions, and the typical geostatistical models (i.e. ordinary kriging, universal kriging, and cokriging) are already incorporated in GIS software packages. In this research, the conceptual and methodological aspects of regression kriging and GIS built-in interpolation models and their interpolation performance are compared and evaluated. Regression kriging is the combination of multivariate regression and kriging. It takes into consideration the spatial autocorrelation of the variable of interest, the correlation between the variable of interest and auxiliary variables (e.g., remotely sensed images are often relatively easy to obtain as auxiliary variables), and the unbiased spatial estimation with minimized variance. To assess the efficiency of regression kriging and the difference between stochastic and deterministic interpolation methods, three case studies with strong, medium, and weak correlation between the response and auxiliary variables are compared to assess interpolation performances. Results indicate that regression kriging has the potential to significantly improve spatial prediction accuracy even when using a weakly correlated auxiliary variable.     

徐州市九里矿区土壤重金属插值分析及污染评价

通过对徐州市九里矿区的表层土壤进行采样测定,进而对该矿区土壤重金属富集与污染状况进行了分析与评价。结果表明,该矿区表层土壤中5种重金属(Cd、Cu、Zn、Pb、Cr)含量均略高于中国土壤元素背景值,土壤中Cd富集程度较高,污染较严重;Cu、Zn、Pb、Cr富集程度较低,污染程度较轻。在GIS环境下利用空间数据插值方法对重金属的空间分布特征研究发现,该城区土壤重金属含量与该区的工矿活动和交通活动等密切相关。     

Street‐level Spatial Interpolation Using Network‐based IDW and Ordinary Kriging

This study proposes network‐based spatial interpolation methods to help predict unknown spatial values along networks more accurately. It expands on two of the commonly used spatial interpolation methods, IDW (inverse distance weighting) and OK (ordinary kriging), and applies them to analyze spatial data observed on a network. The study first provides the methodological framework, and it then examines the validity of the proposed methods by cross‐validating elevations from two contrasting patterns of street network and comparing the MSEs (Mean Squared Errors) of the predicted values measured with the two proposed network‐based methods and their conventional counterparts. The study suggests that both network‐based IDW and network‐based OK are generally more accurate than their existing counterparts, with network‐based OK constantly outperforming the other methods. The network‐based methods also turn out to be more sensitive to the edge effect, and their performance improves after edge correction. Furthermore, the MSEs of standard OK and network‐based OK improve as more sample locations are used, whereas those of standard IDW and network‐based IDW remain stable regardless of the number of sample locations. The two network‐based methods use a similar set of sample locations, and their performance is inherently affected by the difference in their weight distribution among sample locations.     

Mapping of Soil Micronutrients in Kashmir Agricultural Landscape Using Ordinary Kriging and Indicator Approach

Soil nutrient maps based on intensive soil sampling are useful to adopt site-specific management practices. Geostatistical methods have been widely used to determine the spatial correlation and the range of spatial dependence at different sampling scales. If spatial dependence is detected, the modeled semivariograms can then be used to map the interested variable by kriging, an interpolation method that produces unbiased estimates with minimal estimation variance. The objectives of this paper were to examine and map the spatial distribution of the soil micronutrients Cu, Zn, Fe and Mn on an agricultural area in Kupwara, J&K, under temperate climatic conditions. The ordinary kriging was first used to determine the values for the non-sampled locations, and then the indicator approach was used to transform the micronutrient content values into binary values having the mean values of each nutrient as the threshold content. All four elements analyzed showed spatial dependence using the indicator semivariograms. The strength of spatial dependence was assessed using the values of nugget effect and range from the semivariogram, the fitted range values decreased in the order Zn > Cu > Mn > Fe. The spatial dependence of the combination of two or more of the studied micronutrients was also examined using indicator semivariograms. In opposition to spatial analysis of individual microelements, indicator semivariograms obtained for the binary coding of the variables showed a great nugget effect value or a low proportion of sill. The maps for each nutrient obtained using indicator kriging showed some similarity in the spatial distribution, suggesting the delimitation of uniform management areas.     

Spatio‐temporal prediction of land surface temperature using semantic kriging

Spatio‐temporal prediction and forecasting of land surface temperature (LST) are relevant. However, several factors limit their usage, such as missing pixels, line drops, and cloud cover in satellite images. Being measured close to the Earth's surface, LST is mainly influenced by the land use/land cover (LULC) distribution of the terrain. This article presents a spatio‐temporal interpolation method which semantically models LULC information for the analysis of LST. The proposed spatio‐temporal semantic kriging (ST‐SemK) approach is presented in two variants: non‐separable ST‐SemK (ST‐SemK_NSep) and separable ST‐SemK (ST‐SemK_Sep). Empirical studies have been carried out with derived Landsat 7 ETM+ satellite images of LST for two spatial regions: Kolkata, India and Dallas, Texas, U.S. It has been observed that semantically enhanced spatio‐temporal modeling by ST‐SemK yields more accurate prediction results than spatio‐temporal ordinary kriging and other existing methods.     

干热河谷冲沟DEM插值误差的空间分布研究

为了分析云南元谋干热河谷典型冲沟插值误差的空间分布特征,采用反距离加权（inverse distance weighting,IDW）、局部多项式（local polynomial interpolation,LPI）、张力样条（spline with tension,ST）、析取克里格（disjunctive Kriging,DK）以及不规则三角网（triangulated irregular network,TIN）模型方法对高程采样点进行插值,用交叉验证法、相对差系数及沟谷线差异衡量其插值精度。遴选高程误差大于1 m的误差点,用变异系数（coefficient of variation,CV）、全局Moran指数和Getis-Ord G_i*指数分析其空间格局特征。结果表明:TIN和DK精度较高,IDW精度最低;高程误差均呈聚集分布,聚集程度TIN > LPI > DK > ST > IDW;高程误差均呈空间正自相关,TIN模型插值误差的自相关程度最高;误差热点位于坡度大的区域。     

西安市住宅价格空间结构和分异规律分析

利用ESDA方法对西安市城区的291个普通住宅项目均价数据进行研究,通过计算Moran指数和半变异函数分析了其空间自相关性和变异性,并进行了趋势分析。应用Kriging空间插值方法对西安市普通住宅价格空间分布进行了模拟。研究结果表明:西安市房价存在显著的空间自相关性,大部分住宅价格呈空间集聚格局,少部分因存在空间异质性而呈离散分布;房价变异函数表现出各向异性,不同方向有不同结构特征,空间自相关尺度为14.2km;西安市房价空间分异规律明显,房价分布格局受城市功能区划和交通影响较大。     

Interpolating Spatial Interaction Data1

Spatial interpolation has been widely used to improve the spatial granularity of data, or to mediate between inconsistent zoning schemes of spatial data. Traditional areal interpolation methods translate values of source zones to those of target zones. These methods have difficulty in dealing with flow data, as each instance is associated with a pair of zones. This study develops a new concept, flow line interpolation, to fill the abovementioned gap. We also develop a first flow line interpolation method to estimate commuting flow data between spatial units in a target zoning scheme based on such data in a source zoning scheme. Three models (i.e., areal‐weighted, intelligent, and gravity‐type flow line interpolation) are presented. To test the estimation accuracy and the application potential of these models, a case study of Fulton County in Georgia is conducted. The results reveal that both the areal‐weighted and intelligent models are very promising flow line interpolation methods.     

Noisy Data Smoothing in DEM Construction Using Least Squares Support Vector Machines

Since spatial datasets are subject to sampling errors, a smoothing interpolation method should be employed to remove noise during DEM construction. Although least squares support vector machines (LSSVM) have been widely accepted as a classifier, their effect on smoothing noisy data is almost unknown. In this article, the smoothness of LSSVM was explored, and its effect on smoothing noisy data in DEM construction was tested. In order to improve the ability to deal with large datasets, a local method of LSSVM has been developed, where only the neighboring sampling points around the one to be estimated are used for computation. A numerical test indicated that LSSVM is more accurate than the classical smoothing methods including TPS and kriging, and its error surfaces are more evenly distributed. The real‐world example of smoothing noise inherent in lidar‐derived DEMs also showed that LSSVM has a positive smoothing effect, which is approximately as accurate as TPS. In short, LSSVM with a high efficiency can be considered as an alternative smoothing method for smoothing noisy data in DEM construction.     

土壤重金属的空间插值方法比较分析——以阳谷县为例

空间插值通过采集少量的数据点,利用其中的空间关联,推求该区域内其他位置的属性值。本文以山东省阳谷县土壤重金属Cu采样数据为例进行空间探索性分析,分别采用了反距离权重插值和普通克里金插值两种方法进行空间分布插值模拟。结果表明,针对研究区采样数据,反距离权重插值方法生成的模型平均误差为1.97 mg/kg,总体精度为92%;普通克里金插值模型的平均误差为1.91 mg/kg,总体精度为92.35%,普通克里金插值方法更优。     

Interpolation Methods for Scattered Sample Data: Accuracy,Spatial Patterns,Processing Time

Interpolation routines based on polynomials, splines, linear triangulation, proximation, distance weighting, and kriging are tested on their efficacy to visualize spatial patterns. Implementations in commonly available software packages are used in order to yield practical recommendations on the application of current information technology. Two data sets of physical variables containing irregularly distributed sample point values are used as input data. Accuracy of predicted values at unvisited points, preservation of distinct spatial patterns (established from map use tasks), and processing time, are used as criteria to determine the merits of the various interpolation methods. It was found that highly accurate interpolations do not always produce realistic spatial patterns. Effectiveness of distance weighting and kriging methods was found to be largely dependent on the number of neighbors used. For both gradually and abruptly changing data, geographic reality was visualized most satisfactorily with the squared inverse distance weighting (w=d^-2 ) method using respectively few (four to eight) and many (16 to 24) neighbors.     

The hierarchical tessellation model and its use in spatial analysis

The hierarchid tessellation model belongs to a class of spatial data models based on the recursive decomposition of space. The quadtree is one such tessellation and is characterized by square cells and a 1:4 decomposition ratio. To relax these constraints in the tessellation, a generalized hierarchical tessellation data model, called Adaptive Recursive Tessellations (ART), has been proposed. ART increases flexibility in the tessellation by the use of rectangular cells and variable decomposition ratios. In ART, users can specify cell sizes which are intuitively meaningful to their applications, or which can reflect the scales of data. ART is implemented in a data structure called Adaptive Recursive Run-Encoding (ARRE), which is a variant of two-dimensional run-encoding whose running path can vary with the different tessellation structures incorporated in an ART model. Given the recognition of the benefits of implementing statistical spatial analysis in GIS, the use of hierarchical tessellation models such as ART in spatial analysis is discussed. Three examples are introduced to show how ART can: (1) be applied to solve the quadrat size problem in quadrat analysis of point patterns; (2) act as the data model in the variable resolution block kriging technique for geostatistical data to reduce variation in kriging error; and (3) facilitate the evaluation of spatial autocorrelation for area data at multiple map resolutions via the construction of a connectivity matrix for calculating spatial autocorrelation indices based on ARRE.     

The Discontinuous Nature of Kriging Interpolation for Digital Terrain Modeling

Kriging is a widely employed method for interpolating and estimating elevations from digital elevation data. Its place of prominence is due to its elegant theoretical foundation and its convenient practical implementation. From an interpolation point of view, kriging is equivalent to a thin-plate spline and is one species among the many in the genus of weighted inverse distance methods, albeit with attractive properties. However, from a statistical point of view, kriging is a best linear unbiased estimator and, consequently, has a place of distinction among all spatial estimators because any other linear estimator that performs as well as kriging (in the least squares sense) must be equivalent to kriging, assuming that the parameters of the semivariogram are known. Therefore, kriging is often held to be the gold standard of digital terrain model elevation estimation. However, I prove that, when used with local support, kriging creates discontinuous digital terrain models, which is to say, surfaces with "rips" and "tears" throughout them. This result is general; it is true for ordinary kriging, kriging with a trend, and other forms. A U.S. Geological Survey (USGS) digital elevation model was analyzed to characterize the distribution of the discontinuities. I show that the magnitude of the discontinuity does not depend on surface gradient but is strongly dependent on the size of the kriging neighborhood.     

徐州市区土壤重金属空间分布研究

本文以徐州市泉山、云龙、鼓楼、九里4个城区的表层土壤为研究对象,利用BP神经网络模型结合GIS技术对徐州市表层土壤重金属空间分布进行了研究.结果表明:BP神经网络模型能够智能地学习各个样点的空间位置与该点各重金属含量之间的映射关系和预先设计好的分类评价模式,并能够对各个空间插值点处的重金属含量和污染状况进行预测和评价....     

GIS的矿区土壤重金属污染评价及空间分布

针对土壤中的重金属含量超标会对人体健康造成极大危害的问题,为检测矿区土壤重金属含量超标状况及空间分布特征,以江西省信丰县4个矿区为例,在28个点位的20~60cm处测定Hg、Cd、As、Cu、Pb、Ni的含量,采用地统计学和地理信息系统相结合的方法进行分析。结果表明,单因子污染指数显示Pb的污染程度最大,污染程度为中度污染;插值分析图显示Cu以西南方向的虎山矿区含量较高,Ni以西南方向的虎山矿区和北部的赤岗矿区含量较高,Pb的污染区域贯穿于整个分布区;重金属含量随深度的增加无明显变化。结合在污染修复方面的经验,建议通过植物修复技术、物理与化学方法进行污染治理和修复。     

网络RTK内插模型抗差性分析

空间相关误差精确建模是网络RTK技术的关键,定量分析内插模型抗差性对网络RTK内插模型的选择和优化具有重要意义。以现有内插模型为研究对象,根据误差传播定律提出误差影响因子以评价内插模型的抗差性,通过理论结合实验的方法探究不同内插模型抗差性的空间分布特征。结果表明:不同内插模型抗差性空间分布有较大差异,三角形解算单元内,不同内插模型的抗差性均满足要求。      

土壤重金属污染灰色综合评价模型

针对稀疏采样难以准确估测区域土壤重金属综合污染情况和迁移变化规律的问题,提出基于GIS的多属性决策组合赋权灰色综合评价模型。首先采用GIS技术揭示土壤重金属空间变异和污染分布格局;然后利用最大化熵理论集成主客观因素,架构优化组合赋权的土壤重金属污染灰色综合评价体系;最后以试验区土壤中8种(铜、锌、铅、镉、砷、铬、汞、镍)重金属的综合污染情况为例,检验该方法应用效果。结果表明:最优组合权重的灰色综合分析方法兼顾主观偏好和客观属性,其评价结果具有更高的可信度和风险辨识度,提高了综合评价的合理性与有效性,可为土壤重金属污染监测提供方案参考。     

兰州市商品住宅价格的空间分异规律

针对住宅价格在城市空间中的分布规律问题,该文以兰州市主城区2015年在售的187个商品住宅样本均价为基本数据,运用空间自相关法对兰州市住宅价格的空间异质性和集聚性进行分析,并利用趋势面分析和空间反距离权重插值法对住宅价格的空间分布格局进行研究。结果表明:兰州市住宅价格总体上呈显著的空间正自相关性,少数地区存在差异性;住宅价格发展不平衡,价格"东高西低";住宅价格由各区行政中心向四周逐级递减,呈多极核分布特征;价格等值线"东密西疏",住宅价格变化幅度空间差异较大。分析发现,区位条件、交通条件及居住环境是影响兰州市商品住宅价格的主要因素。     

The effect of spatial autocorrelation and class proportion on the accuracy measures from different sampling designs

Four binary thematic maps with combinations of two spatial autocorrelation levels and two different class proportions are simulated to study their effect on the precision of accuracy measures from different sampling designs. A series of eleven sample sizes (from a minimum of 25 to a maximum of 1296) are simulated using three popular sampling designs, including simple random sampling (SRS), systematic sampling (SYS), and stratified random sampling (StrRS) on the four simulated maps. The conventional error matrix and related accuracy measures are calculated for each simulation, and the precision of different estimates of accuracy measures is compared among the three sampling designs.The selection of a particular sampling design and sample size depends on the spatial autocorrelation level, the class proportion difference, and the accuracy indices that a given application requires. In general, the class proportion difference has a greater impact on the performance of different sampling methods than the spatial autocorrelation level does on a map. For estimating the accuracy of individual classes, stratified sampling achieves better precision than SRS and SYS with smaller sample sizes, especially for estimating the small class. For estimating the overall accuracy, different sampling designs achieve very similar levels of precision with fewer samples. To achieve a better estimate of the kappa coefficient, stratified random sampling is recommended for use on a map with a high class proportion difference, while random sampling is preferred for a map with low spatial autocorrelation and a low class proportion difference.     

Spatial interpolation and image-integrative geostatistical prediction of mosquito vectors for arboviral surveillance

Remote sensing can augment traditional methods of mosquito species surveillance for arboviruses. Abundance and patterns of mosquito vectors of West Nile virus in Chesapeake, Virginia, USA, were studied using light trap collection data and a Landsat-7 Enhanced Thematic Mapper+ digital image for spatial interpolation and geostatistical mapping of the abundance of 24 species of mosquitoes capable of transmitting West Nile virus to humans. We evaluated spatial interpolation techniques including inverse distance weighting, ordinary kriging, co-kriging geostatistics using combined Landsat-7 tasselled cap transform indices (brightness, greenness, and wetness) to characterize habitats and breeding conditions. Results highlight gaps in surveillance coverage, geostatistical improvement of vector patterns and abundance, and spatial patterns of error. Constraints and opportunities for adoption of remote sensing and spatial analysis for mosquito control are identified and discussed.