Comparison between Sequential Gaussian Simulation and Kriging Interpolation on Soil Heavy Metal Pollution

[Objective] This study was to explore the difference of kriging interpolation and sequential Gaussian simulation on analyzing soil heavy metal pollution with a view to provide references for analyzing the heavy metal pollution of soil. [Method] The sampling data of soil copper from a county of Liaocheng, Shandong Province was set as the study objective. Kriging interpolation and sequential Gaussian simu- lation were used to simulate the spatial distribution of soil copper. And 30 sampling points were selected as the cross-validation data set to compare the two interpola- tion methods. [Result] Kriging method and Gaussian sequential simulation have their own advantages on simulating mean segment and extreme segment, therefore, re- searchers should choose the proper method based on the characteristics of test data and application purposes. [Conclusion] Analysis of soil heavy metal pollution is the prerequisite for soil management and ecological restoration. The result of this study is of important significance for choosing different interpolating and simulating methods to analyze soil heavy metal pollution based on different purposes.

Water distribution extracted from mining subsidence area using Kriging interpolation algorithm

By comprehensively analyzing the data of geology and mining, Kriging algorithm was introduced to analyze the thematic information of geological data, to rapidly extract mining parameters for predicting mining subsidence, and to effectively integrate geomorphology and predict information. As a result, the change information of water body is successfully detected from the prediction of surface subsidence due to mining activity. Analysis shows that the elevation of farmland in the west side of water body will be lower than ever, and the west part farmland will be submerged. However, there is no evidence for impacting the villages. All the information provides a reference for efficiently assessing environmental impact due to mining activity, which can help to govern the subsidence of the area reasonably.

A meshless method based on moving Kriging interpolation for a two-dimensional time-fractional diffusion equation

Fractional diffusion equations have been the focus of modeling problems in hydrology, biology, viscoelasticity, physics, engineering, and other areas of applications. In this paper, a meshfree method based on the moving Kriging inter- polation is developed for a two-dimensional time-fractional diffusion equation. The shape function and its derivatives are obtained by the moving Kriging interpolation technique. For possessing the Kronecker delta property, this technique is very efficient in imposing the essential boundary conditions. The governing time-fractional diffusion equations are transformed into a standard weak formulation by the Galerkin method. It is then discretized into a meshfree system of time-dependent equations, which are solved by the standard central difference method. Numerical examples illustrating the applicability and effectiveness of the proposed method are presented and discussed in detail.

A moving Kriging interpolation-based boundary node method for two-dimensional potential problems

In this paper, a meshfree boundary integral equation （BIE） method, called the moving Kriging interpolation- based boundary node method （MKIBNM）, is developed for solving two-dimensional potential problems. This study combines the DIE method with the moving Kriging interpolation to present a boundary-type meshfree method, and the corresponding formulae of the MKIBNM are derived. In the present method, the moving Kriging interpolation is applied instead of the traditional moving least-square approximation to overcome Kronecker＇s delta property, then the boundary conditions can be imposed directly and easily. To verify the accuracy and stability of the present formulation, three selected numerical examples are presented to demonstrate the efficiency of MKIBNM numerically.

Accuracy Assessment and Comparative Analysis of IDW, Spline and Kriging in Spatial Interpolation of Landform (Topography): An Experimental Study

It is practically impossible and unnecessary to obtain spatial-temporal information of any given continuous phenomenon at every point within a given geographic area. The most practical approach has always been to obtain information about the phenomenon as in many sample points as possible within the given geographic area and estimate the values of the unobserved points from the values of the observed points through spatial interpolation. However, it is important that users understand that different interpolation methods have their strength and weaknesses on different datasets. It is not correct to generalize that a given interpolation method (e.g. Kriging, Inverse Distance Weighting (IDW), Spline etc.) does better than the other without taking into cognizance, the type and nature of the dataset and phenomenon involved. In this paper, we theoretically, mathematically and experimentally evaluate the performance of Kriging, IDW and Spline interpolation methods respectively in estimating unobserved elevation values and modeling landform. This paper undertakes a comparative analysis based on the prediction mean error, prediction root mean square error and cross validation outputs of these interpolation methods. Experimental results for each of the method on both biased and normalized data show that Spline provided a better and more accurate interpolation within the sample space than the IDW and Kriging methods. The choice of an interpolation method should be phenomenon and data set structure dependent.

The efficacy of Kriging spatial interpolation for filling temporal gaps in daily air temperature data: A case study in northeast China

Quality-controlled and serially complete daily air temperature data are essential to evaluating and modelling the influences of climate change on the permafrost in cold regions. Due to malfunctions and location changes of observing stations, temporal gaps （i.e., missing data） are common in collected datasets. The objective of this study was to assess the efficacy of Kriging spatial interpolation for estimating missing data to fill the temporal gaps in daily air temperature data in northeast China. A cross-validation experiment was conducted. Daily air temperature series from 1960 to 2012 at each station were estimated by using the universal Kriging （UK） and Kriging with an external drift （KED）, as appropriate, as if all the ob-servations at a given station were completely missing. The temporal and spatial variation patterns of estimation uncertainties were also checked. Results showed that Kriging spatial interpolation was generally desirable for estimating missing data in daily air temperature, and in this study KED performed slightly better than UK. At most stations the correlation coefficients （R2） between the observed and estimated daily series were 〉0.98, and root mean square errors （RMSEs） of the estimated daily mean （Tmean）, maximum （Tmax）, and minimum （Tmin） of air temperature were 〈3 ℃. However, the estimation quality was strongly affected by seasonality and had spatial variation. In general, estimation uncertainties were small in summer and large in winter. On average, the RMSE in winter was approximately 1 ℃ higher than that in summer. In addition, estimation uncertainties in mountainous areas with complex terrain were significantly larger than those in plain areas.

Comparison of Spatial Interpolation Methods of Precipitation Data in Central Macedonia, Greece

The purpose of this paper is to investigate the spatial interpolation of rainfall variability with deterministic and geostatic inspections in the Prefecture of Kilkis (Greece). The precipitation data where recorded from 12 meteorological stations in the Prefecture of Kilkis for 36 hydrological years (1973-2008). The cumulative monthly values of rainfall were studied on an annual and seasonal basis as well as during the arid-dry season. In the deterministic tests, the I.D.W. and R.B.F. checks were inspected, while in the geostatic tests, Ordinary Kriging and Universal Kriging respectively. The selection of the optimum method was made based on the least Root Mean Square Error (R.M.S.E.), as well as on the Mean Error (M.E.), as assessed by the cross validation analysis. The geostatical Kriging also considered the impact of isotropy and anisotropy across all time periods of data collection. Moreover, for Universal Kriging, the study explored spherical, exponential and Gaussian models in various combinations. Geostatistical techniques consistently demonstrated greater reliability than deterministic techniques across all time periods of data collection. Specifically, during the annual period, anisotropy was the prevailing characteristic in geostatistical techniques. Moreover, the results for the irrigation and seasonal periods were generally comparable, with few exceptions where isotropic methods yielded lower (R.M.S.E.) in some seasonal observations.

Comparison analysis of sampling methods to estimate regional precipitation based on the Kriging interpolation methods： A case of northwestern China

The accuracy of spatial interpolation of precipitation data is determined by the actual spatial variability of the precipitation, the interpolation method, and the distribution of observatories whose selections are particularly important. In this paper, three spatial sampling programs, including spatial random sampling, spatial stratified sampling, and spatial sandwich sampling, are used to analyze the data from meteorological stations of northwestern China. We compared the accuracy of ordinary Kriging interpolation methods on the basis of the sampling results. The error values of the regional annual pre-cipitation interpolation based on spatial sandwich sampling, including ME （0.1513）, RMSE （95.91）, ASE （101.84）, MSE （？0.0036）, and RMSSE （1.0397）, were optimal under the premise of abundant prior knowledge. The result of spatial stratified sampling was poor, and spatial random sampling was even worse. Spatial sandwich sampling was the best sampling method, which minimized the error of regional precipitation estimation. It had a higher degree of accuracy compared with the other two methods and a wider scope of application.

Application of Multi-Gene Genetic Programming in Kriging Interpolation

A key stage for Kriging interpolation is the estimating of variogram model, which characterizes the spatial behavior of the variables of interest. But most traditional kriging interpolation has finite types of empirical variogram model, and sometimes, the optimal type of variogram model can not be find, which result in decreasing interpolation accuracy. In this paper, we explore the use of Multi-Gene Genetic Programming (MGGP) to automatically find an empirical variogram model that fits on an experimental variogram. Empirical variogram estimation based on MGGP, in contrast with traditional method need not select type of basic variogram model and can directly get both the functional type as well as the coefficients of the optimal variogram. The results of case study show that the proposed method can avoid the subjectivity in choosing the type of variogram models and can adaptively fit variogram according to the real data structure, which improves the interpolation accuracy of kriging significantly.

An Improved Kriging Interpolation Technique Based on SVM and Its Recovery Experiment in Oceanic Missing Data

In Kriging interpolation, the types of variogram model are very finite, which make the variogram very difficult to describe the spatial distributional characteristics of true data. In order to overcome its shortage, an improved interpolation called Support Vector Machine-Kriging interpolation (SVM-Kriging) was proposed in this paper. The SVM-Kriging uses Least Square Support Vector Machine (LS-SVM) to fit the variogram, which needn’t select the basic variogram model and can directly get the optimal variogram of real interpolated field by using SVM to fit the variogram curve automatically. Based on GODAS data, by using the proposed SVM-Kriging and the general Kriging based on other traditional variogram models, the interpolation test was carried out and the interpolated results were analyzed contrastively. The test show that the variogram of SVM-Kriging can avoid the subjectivity of selecting the type of variogram models and the SVM-Kriging is better than the general Kriging based on other variogram model as a whole. Therefore, the SVM-Kriging is a good and adaptive interpolation method.

Optimal Interpolation and Kriging Mapping of Soil Characters in Glacial Moraine Landscapes

The objective of this research is to analyze optimal interpolation and Kriging mapping of soil characters in Glacial Moraine Landscapes. The research site is located in sloping landscapes, Kuehren, North Germany. The survey method was detailed using maps with scales of 1：5,000. Soil sampling was performed by soil pits and borings and completely analyzed in laboratory. Collected data were evaluated by Geostatistics program for spatial soil variability analyses. All maps （produced by Kriging interpolation） picture redistribution of soil nutrients and soil fractions and all map isolines run in similar directions according to landscape nets. The position in the landscape is responsible for increased soil variability. Soil variability becomes higher with decreasing elevation; this means it increases from hilltops to lower slopes. All observed soil characters show relationships to the soil variability. This variability system is caused by convex depressions and hedgerows （Knicks） function as barriers for the redistribution of transported material and offsite sedimentation. Therefore fluxes can be assessed by soil gain and loss balances.

Research on the Spatial Interpolation Method for Temperature inInner Mongolia

以内蒙古自治区为研究区，采用反距离权重法、样条函数法和普通克里金法对研究区119个气象观测站点的气温数据进行空间插值，结果表明，普通克里金法由于能够最优地描述气温在空间上的连续性变化特征，插值精度最高；反距离权重法插值精度次之，但其模拟结果趋于产生牛眼模式，且结果不能外推；样条函数法由于受到边界效应的影响，插值精度最低。

基于Kriging插值代理模型的轴流式止回阀多目标优化

基于Kriging插值代理模型,对轴流式止回阀的各项性能进行综合优化,具体的优化目标为:降低正向流阻;减小阀芯振动;提升止回性能。通过试验设计方案找到影响轴流式止回阀正向流阻的主要结构参数因素,并以其作为优化对象,通过Kriging插值法拟合主要结构参数对应的性能样本点,在保证拟合精度的情况下,使用NSGA-II遗传算法同时对3个主要结构参数进行多目标优化,得到了三目标Pareto前沿,经过权衡不同因素对性能指标的影响大小,最终得到了综合性能最优的结构参数为:阀瓣丰满度系数α_(1)=2.199,阀芯长度L=386.322 mm,阀芯半径R=113.997 mm。其对应的止回阀性能为:正向出口流量Q=161.839 kg/s,正向出口流速v=3.30235 m/s,止回阀进口压力p=97632.2287 Pa。相比优化之前,阀门正向流阻减小了1.2%,阀芯振动降低了0.24%,止回性能提升了2.3%。

应用时空复合趋势面Kriging方法插值地表沉降

通过布设监测点可反映地表沉降变化趋势,确保施工和运营期间安全,但沉降监测点数量有限,且易受施工作业、天气等因素干扰导致数据缺失或污染,因此需对监测区域数据进行插值处理。针对采用传统地统计插值方法估计地铁地表沉降趋势精度较低的问题,提出了一种基于时空复合趋势面的Kriging插值方法。以乌鲁木齐地铁3号线2017年6月24日—9月3日共15期地铁地表沉降观测值为实验数据,分别采用两种方法对沉降监测点缺失数据进行插补。结果表明,相较于传统方法,基于时空复合趋势面的Kriging插值方法的拟合精度更高,RMSE可降低约35%。

基于PSO-Kriging算法的三维地质建模技术研究

三维地质模型的构建对于理解和预测地下结构至关重要。地质钻孔数据能够反映岩体空间分布和地质构造特征,本研究以小保当一号煤矿11盘区内的23个地质钻孔数据为基础,采用添加虚拟地层的方法解决了地层缺失与地层重复现象,构建共计27层地层的三维地质模型以及二维剖面模型。此外,针对传统的克里金方法在处理复杂地质数据参数选择困难的问题,采用粒子群算法对传统克里金插值方法中的块金值(C 0)、偏基台值(C)和变程(a)三个关键参数进行寻优,从而克服普通克里金插值参数选择的主观性和不确定性,采用实际验证法选取了研究区内四个钻孔来对比插值结果,结果表明经过PSO优化的Kriging算法在X3-1、X3-2、K3-4、K3-5四个钻孔的RMSE值分别降低至1.184、1.267、1.606、1.560,相比于Kriging的RMSE平均降低了31%,且PSO-Kriging算法在四个钻孔处对2-2煤层的插值结果与实际值相比较误差分别为1.00 m、0.01 m、0.11 m和0.03 m,比Kriging插值结果更接近实际值,表明了所提方法的有效性和优越性。

Spatial Interpolation of Daily Precipitation in China:1951-2005

Climate research relies heavily on good quality instrumental data; for modeling efforts gridded data are needed. So far, relatively little effort has been made to create gridded climate data for China. This is especially true for high-resolution daily data. This work, focuses on identifying an accurate method to produce gridded daily precipitation in China based on the observed data at 753 stations for the period 1951-2005. Five interpolation methods, including ordinary nearest neighbor, local polynomial, radial basis function, inverse distance weighting, and ordinary kriging, have been used and compared. Cross-validation shows that the ordinary kriging based on seasonal semi-variograms gives the best performance, closely followed by the inverse distance weighting with a power of 2. Finally the ordinary kriging is chosen to interpolate the station data to a 18 km× 18 km grid system covering the whole country. Precipitation for each 0.5°×0.5° latitude-longitude block is then obtained by averaging the values at the grid nodes within the block. Owing to the higher station density in the eastern part of the country, the interpolation errors are much smaller than those in the west （west of 100°E）. Excluding 145 stations in the western region, the daily, monthly, and annual relative mean absolute errors of the interpolation for the remaining 608 stations are 74%, 29%, and 16%, respectively. The interpolated daily precipitation has been made available on the internet for the scientific community.

Uncertainty Analysis of Interpolation Methods in Rainfall Spatial Distribution–A Case of Small Catchment in Lyon

Quantification of spatial and temporal patterns of rainfall is an important step toward developing regional water sewage models, the intensity and spatial distribution of rainfall can affect the magnitude and duration of water sewage. However, this input is subject to uncertainty, mainly as a result of the interpolation method and stochastic error due to the random nature of rainfall. In this study, we analyze some rainfall series from 30 rain gauges located in the Great Lyon area, including annual, month, day and intensity of 6mins, aiming at improving the understanding of the major sources of variation and uncertainty in small scale rainfall in-terpolation in different input series. The main results show the model and the parameter of Kriging should be different for the different rainfall series, even if in the same research area. To the small region with high den-sity of rain gauges (15km2), the Kriging method superiority is not obvious, IDW and the spline interpolation result maybe can be better. The different methods will be suitable for the different research series, and it must be determined by the data series distribution.

Comparison and Evaluation of GIS-Based Spatial Interpolation Methods for Estimation Groundwater Level in AL-Salman District—Southwest Iraq

The aim of the research is to compare spatial prediction methods: (RBF), (IDW), (OK), (UK), and (SK) for the production of the groundwater level map and the prediction error map in study area as well. Setting the foundations and criteria for choosing the most appropriate mathematical method for the construction of statistical surfaces in the representation of the level of groundwater in study area. These methods were used to predict the spatial distribution map of the groundwater level based on measured data from 764 wells in May 2016. The study reveals that comparing the spatial interpolation models and evaluating their accuracy, through some statistical indicators and cross-validation is the best way to choose the optimal model for the representation of data entered in any site. As a result of the statistical comparison between the five spatial interpolation models and validation of the results using (cross validation) it was found that the universal Kriging (UK) method is the best method to represent the level of groundwater in Salman district because this model has the lowest root mean square error (RMSE), the lowest mean error (ME), and the highest coefficient of determination (R2) value. The groundwater level and prediction standard error maps produced in the geographic information system (GIS) give additional data and information that describe the aquifer system in study area and will ultimately improve sustainable groundwater management.

Optimization of Air Quality Monitoring Network Using GIS Based Interpolation Techniques

This paper proposes a simple method of optimizing Air Quality Monitoring Network (AQMN) using Geographical Information System (GIS), interpolation techniques and historical data. Existing air quality stations are systematically eliminated and the missing data are filled in using the most appropriate interpolation technique. The interpolated data are then compared with the observed data. Pre-defined performance measures root mean square error (RMSE), mean absolute percentage error (MAPE) and correlation coefficient (r) were used to check the accuracy of the interpolated data. An algorithm was developed in GIS environment and the process was simulated for several sets of measurements conducted in different locations in Riyadh, Saudi Arabia. This methodology proves to be useful to the decision makers to find optimal numbers of stations that are needed without compromising the coverage of the concentrations across the study area.

3-D geochemical interpolation guided by geophysical inversion models

3-D geochemical subsurface models,as constructed by spatial interpolation of drill-core assays,are valuable assets across multiple stages of the mineral industry's workflow.However,the accuracy of such models is limited by the spatial sparsity of the underlying drill-core,which samples only a small fraction of the subsurface.This limitation can be alleviated by integrating collocated 3-D models into the interpolation process,such as the 3-D rock property models produced by modern geophysical inversion procedures,provided that they are sufficiently resolved and correlated with the interpolation target.While standard machine learning algorithms are capable of predicting the target property given these data,incorporating spatial autocorrelation and anisotropy in these models is often not possible.We propose a Gaussian process regression model for 3-D geochemical interpolation,where custom kernels are introduced to integrate collocated 3-D rock property models while addressing the trade-off between the spatial proximity of drill-cores and the similarities in their collocated rock properties,as well as the relative degree to which each supporting 3-D model contributes to interpolation.The proposed model was evaluated for 3-D modelling of Mg content in the Kevitsa Ni-Cu-PGE deposit based on drill-core analyses and four 3-D geophysical inversion models.Incorporating the inversion models improved the regression model's likelihood(relative to a purely spatial Gaussian process regression model)when evaluated at held-out test holes,but only for moderate spatial scales(100 m).