Kriging in a global neighborhood

The kriging estimator is usually computed in a moving neighborhood; only the data near the point to be estimated are used. This moving neighborhood approach creates discontinuities in mapping applications. An alternative approach is presented here, whereby all points are estimated using all the available data. To solve the resulting large linear system the kriging estimator is expressed in terms of the inverse of the covariance matrix. The covariance matrix has the advantage of being positive definite and the size of system which can be solved without encountering numerical instability is substantially increased. Because the kriging matrix does not change, the estimator can be written in terms of scalar products, thus avoiding the more time-consuming matrix multiplications of the standard approach. In the particular case of a covariance which is zero for distances greater than a fixed value (the range), the resulting banded structure of the covariance matrix is shown to lead to substantial computational savings in both run time and storage space. In this case the calculation time for the kriging variance is also substantially reduced. The present method is extended to the nonstationary case.     

Can we improve the spatial predictions of seabed sediments? A case study of spatial interpolation of mud content across the southwest Australian margin

Spatially continuous data of environmental variables is often required for marine conservation and management. However, information for environmental variables is usually collected by point sampling, particularly for the marine region. Thus, methods generating such spatially continuous data by using point samples to estimate values for unknown locations become essential tools. Such methods are, however, often data- or even variable-specific and it is difficult to select an appropriate method for any given dataset. In this study, 14 methods (37 sub-methods) are compared using samples of mud content with five levels of sample density across the southwest Australian margin. Bathymetry, distance-to-coast, slope and geomorphic province were used as secondary variables. Ten-fold cross validation with relative mean absolute error (RMAE) and visual examination were used to assess the performance of these methods. A total of 1850 prediction datasets are produced and used to assess the performance of the methods and the effects of other factors considered. Considering both the accuracy and the visual examination, we found that a combined method (i.e., random forest and ordinary kriging: RKrf) is the most robust method. This method is novel, with a RMAE up to 17% less than that of the control. No threshold in sample density was detected in relation to prediction accuracy. No consistent patterns are observed between the performance of the methods and data variation. The RMAE of three most accurate methods is about 30% lower than that of the best methods in previous publications, highlighting the robustness of the methods selected in this study. The implications and limitations of this study are discussed and a number of suggestions are provided for further studies.     

On counting the number of data pairs for semivariogram estimation

In planning spatial sampling studies for the purpose of estimating the semivariogram, the number of data pairs separated by a given distance is sometimes used as a comparative index of the precision which can be expected from a given sampling design. Because spatial data are correlated, this index can be unreliable. An alternative index which partially corrects for this correlation, themaximum equivalent uncorrelated pairs, is proposed for comparing spatial designs. The index is developed under the assumption that the underlying stochastic process is Gaussian and is appropriate when the (population) semivariogram is to be estimated by the sample semivariogram.     

Characterization and reconstruction of a rock fracture surface by geostatistics

It is well understood that, in studying the mechanical and hydromechanical behaviour of rock joints, their morphology must be taken into account. A geostatistical approach has been developed for characterizing the morphology of fracture surfaces at a decimetre scale. This allows the analysis of the spatial variability of elevations, and their first and second derivatives, with the intention of producing a model that gives a numerical three‐dimensional (3D) representation of the lower and upper surfaces of the fracture. Two samples (I and II) located close together were cored across a natural fracture. The experimental data are the elevations recorded along profiles (using recording steps of 0.5 and 0.02 mm, respectively, for the samples I and II). The goal of this study is to model the surface topography of sample I, so getting estimates for elevations at each node of a square grid whose mesh size will be, for mechanical purposes, no larger than the recording step. Since the fracture surface within the sample core is not strictly horizontal, geostatistical methods are applied to residuals of elevations of sample I. Further, since structural information is necessary at very low scale, theoretical models of variograms of elevations, first and second derivatives are fitted using data of both that sample I and sample II. The geostatistical reconstructions are computed using kriging and conditional simulation methods. In order to validate these reconstructions, variograms and distributions of experimental data are compared with variograms and distributions of the fitted data. Copyright © 2002 John Wiley & Sons, Ltd.     

PM2.5浓度插值中不同空间插值方法对比

针对不同插值方法对于重点大气颗粒物污染源PM_(2.5)大区域内整体分布情况的插值效果和精度存在差异性的问题,该文以石家庄市赵县100个点的PM_(2.5)的数据作为样本点,利用反距离权重法、普通克里格插值法和协同克里格插值法,分别进行PM_(2.5)浓度的空间插值。利用研究区内另外20个PM_(2.5)的数据点进行插值精度的检验。研究结果表明:协同克里格插值的精度最高,但可视化效果不如其他两种方法。     

Spatio-temporal regression kriging for modelling urban NO2 concentrations

ABSTRACT

Recently developed urban air quality sensor networks are used to monitor air pollutant concentrations at a fine spatial and temporal resolution. The measurements are however limited to point support. To obtain areal coverage in space and time, interpolation is required. A spatio-temporal regression kriging approach was applied to predict nitrogen dioxide (NO₂) concentrations at unobserved space-time locations in the city of Eindhoven, the Netherlands. Prediction maps were created at 25 m spatial resolution and hourly temporal resolution. In regression kriging, the trend is separately modelled from autocorrelation in the residuals. The trend part of the model, consisting of a set of spatial and temporal covariates, was able to explain 49.2% of the spatio-temporal variability in NO₂ concentrations in Eindhoven in November 2016. Spatio-temporal autocorrelation in the residuals was modelled by fitting a sum-metric spatio-temporal variogram model, adding smoothness to the prediction maps. The accuracy of the predictions was assessed using leave-one-out cross-validation, resulting in a Root Mean Square Error of 9.91 μg m^?3, a Mean Error of ?0.03 μg m^?3 and a Mean Absolute Error of 7.29 μg m^?3. The method allows for easy prediction and visualization of air pollutant concentrations and can be extended to a near real-time procedure.     

因子克立格分析的理论综述及应用展望

   因子克立格分析是研究多元地质统计学的基拙，是由因子克立格法和协区域化分析两部分组成，方法上包括了区域化变量(集)的分解和分解后每一空间分童的估值;本文阐述了因子克立格分析从产生到应用的理论发展、研究成果及应用展望。     

Mapping Paleo‐Fire Boundaries from Binary Point Data: Comparing Interpolation Methods

Fire history studies have traditionally emphasized temporal rather than spatial properties of paleo‐fire regimes. In this study we compare four methods of mapping paleo‐fires in central Washington from binary point data: indicator kriging, inverse distance weighting, Thiessen polygons, and an expert approach. We evaluate the results of each mapping method using a test (validation) dataset and receiver operating characteristic plots. Interpolation methods perform well, but results vary with fire size and spatial pattern of points. Though all methods involve some subjectivity, automated interpolation methods perform well, are replicable, and can be applied across varying landscapes.     

An interpolation method taking into account inequality constraints: I. Methodology

Several kinds of data can provide information about a variable measured on a one- or two-dimensional space; at some points, the value is known to be equal to a certain number. At other points, the only information may be that the variable is greater or smaller than a given value. The theory of splines provides interpolating functions that can take into account both equality and inequality data. These interpolating functions are presented. The parallel between splines and kriging is reviewed, using the formalism of dual kriging. Coefficients of dual kriging can be obtained directly by minimizing a quadratic form. By adding some inequality constraints to this minimization, an interpolating function may be calculated which takes into account inequality data and is more general than a spline. The method is illustrated by some simple one-dimensional examples.Work performed at Sohio Petroleum Company     

Comparative performance of indicator algorithms for modeling conditional probability distribution functions

This paper compares the performance of four algorithms (full indicator cokriging. adjacent cutoffs indicator cokriging, multiple indicator kriging, median indicator kriging) for modeling conditional cumulative distribution functions (ccdf).The latter three algorithms are approximations to the theoretically better full indicator cokriging in the sense that they disregard cross-covariances between some indicator variables or they consider that all covariances are proportional to the same function. Comparative performance is assessed using a reference soil data set that includes 2649 locations at which both topsoil copper and cobalt were measured. For all practical purposes, indicator cokriging does not perform better than the other simpler algorithms which involve less variogram modeling effort and smaller computational cost. Furthermore, the number of order relation deviations is found to be higher for cokriging algorithms, especially when constraints on the kriging weights are applied.