OPTIMAL INTERPOLATION AND ISAR1THMIC MAPPING OF SOIL PROPERTIES

Kriging is a means of spatial prediction that can be used for soil properties. It is a form of weighted local averaging. It is optimal in the sense that it provides estimates of values at unrecorded places without bias and with minimum and known variance. Isarithmic maps made by kriging are alternatives to conventional soil maps where properties can be measured at close spacings. Kriging depends on first computing an accurate semi-variogram, which measures the nature of spatial dependence for the property. Estimates of semi-variance are then used to determine the weights applied to the data when computing the averages, and are presented in the kriging equations. The method is applied to three sets of data from detailed soil surveys in Central Wales and Norfolk. Sodium content at Plas Gogerddan was shown to vary isotropically with a linear semi-variogram. Simple punctual kriging produced a map with intricate isarithms and fairly large estimation variance, attributed to a large nugget effect. Sloniness on the same land varied anisotropically with a linear semi-variogram. and again the estimation error of punctual kriging was fairly large. At Hole Farm. Norfolk, the thickness of cover loam varied isotropically, but with a spherical semi-variogram. Its parameters were estimated and used to krige point values and produce a map showing substantial short-range variation.     

OPTIMAL INTERPOLATION AND ISARITHMIC MAPPING OF SOIL PROPERTIES

Soil properties mapped in two intensive surveys had large nugget variances, leading to large estimation variances and erratic isarithms when mapped by punctual kriging. It is likely that both surveyors and survey clients are interested in average values of soil properties over areas rather than point values, and such values can be obtained by block kriging. Estimation variances are very much smaller, and maps of sodium and stone content at Plas Gogerddan, Central Wales, kriged over blocks 920m², and thickness of cover loam at Hole Farm, Norfolk, kriged over blocks of 400m², are much smoother than the punctually kriged maps. The map of Hole Farm has a distinct and meaningful regional pattern.     

A comparison of some robust estimators of the variogram for use in soil survey

The standard estimator of the variogram is sensitive to outlying data, a few of which can cause overestimation of the variogram. This will result in incorrect variances when estimating the value of a soil property by kriging or when designing a sampling grid to map the property to a required precision. Several robust estimators of the variogram, based on location and scale estimation, have been proposed as improvements. They seem to be suitable for analysis of soil data in circumstances where the standard estimator is likely to be affected by outliers. Robust estimators are based on assumptions about the distribution of the data which will not always hold and which need not be made in kriging or in estimating the variogram by the standard estimator. The estimators are reviewed. Simulation studies show that the robust estimators vary in their susceptibility to moderate skew in the underlying distribution, but that the effects of outliers are generally greater. The estimators are applied to some soil data, and the resulting variograms used for ordinary kriging at sites in a separate validation data set. In most cases the variograms derived from the standard estimator gave kriging variances which appeared to overestimate the mean squared error of prediction (MSEP). Kriging with variograms based on robust estimators sometimes gave kriging variances which underestimated the MSEP or did not differ significantly from it. Estimates of kriging variance and the MSEP derived from the validation data were generally close to estimates from cross‐validation on the prediction set used to derive the variograms. This indicates that variogram models derived from different estimators could be compared by cross‐validation.     

农田土壤颗粒组成及其剖面分层的空间变异分析

对60m55m的农田尺度上100个取样点土壤颗粒组成及剖面层次的空间变异性的传统统计分析表明,各属性的变异系数属中等强度。半方差函数模型均为球状模型,各属性具有一定的空间相关性,并表现有一定的各向异性。在对土壤颗粒组成及剖面层次空间分布趋势分析的基础上,根据半方差函数模型,对各属性选择不同趋势和异向性的普通Kriging内插值比较的结果表明,考虑变量在空间分布趋势和异向性的插值结果比不考虑要好。在以上分析基础上,利用Kriging内插值绘制各层深度和颗粒组成的等值线图,并分析了其空间分布规律。     

Estimating variograms of soil properties by the method-of-moments and maximum likelihood

Variograms of soil properties are usually obtained by estimating the variogram for distinct lag classes by the method‐of‐moments and fitting an appropriate model to the estimates. An alternative is to fit a model by maximum likelihood to data on the assumption that they are a realization of a multivariate Gaussian process. This paper compares the two using both simulation and real data. The method‐of‐moments and maximum likelihood were used to estimate the variograms of data simulated from stationary Gaussian processes. In one example, where the simulated field was sampled at different intensities, maximum likelihood estimation was consistently more efficient than the method‐of‐moments, but this result was not general and the relative performance of the methods depends on the form of the variogram. Where the nugget variance was relatively small and the correlation range of the data was large the method‐of‐moments was at an advantage and likewise in the presence of data from a contaminating distribution. When fields were simulated with positive skew this affected the results of both the method‐of‐moments and maximum likelihood. The two methods were used to estimate variograms from actual metal concentrations in topsoil in the Swiss Jura, and the variograms were used for kriging. Both estimators were susceptible to sampling problems which resulted in over‐ or underestimation of the variance of three of the metals by kriging. For four other metals the results for kriging using the variogram obtained by maximum likelihood were consistently closer to the theoretical expectation than the results for kriging with the variogram obtained by the method‐of‐moments, although the differences between the results using the two approaches were not significantly different from each other or from expectation. Soil scientists should use both procedures in their analysis and compare the results.     

A comparison of kriging, cubic splines and classification for predicting soil properties from sample information

The clay content of the topsoil in two regions of contrasting physiography was predicted from sample data using four different procedures. The predictors were the means of mapped classes, the usual kriging estimator, a cubic spline interpolator and a kriging estimator within classes using a pooled within-class variogram. The performances of the procedures were evaluated and compared. In the first region, Sandford St Martin on Jurassic sediments where there were some abrupt changes in soil, the classification predicted best within those classes bounded by sharp change. Elsewhere the usual kriging performed somewhat better, and kriging within classes was still more precise. In the second region, Yenne on the alluvial plain of the Rhone where the soil varied gradually, kriging performed better than classification, though a small improvement resulted from combining kriging with classification. Both prediction by class means and kriging attempt to minimize the estimation variance, and their mean prediction variances were close to the theoretical values overall. Spline interpolation is more empirical, and though it followed the abrupt changes better than kriging, it fluctuated excessively elsewhere, and its overall performance was poorer than that of kriging.     

A coherent geostatistical approach for combining choropleth map and field data in the spatial interpolation of soil properties

Information available for mapping continuous soil attributes often includes point field data and choropleth maps (e.g. soil or geological maps) that model the spatial distribution of soil attributes as the juxtaposition of polygons (areas) with constant values. This paper presents two approaches to incorporate both point and areal data in the spatial interpolation of continuous soil attributes. In the first instance, area-to-point kriging is used to map the variability within soil units while ensuring the coherence of the prediction so that the average of disaggregated estimates is equal to the original areal datum. The resulting estimates are then used as local means in residual kriging. The second approach proceeds in one step and capitalizes on: 1) a general formulation of kriging that allows the combination of both point and areal data through the use of area-to-area, area-to-point, and point-to-point covariances in the kriging system, 2) the availability of GIS to discretize polygons of irregular shape and size, and 3) knowledge of the point-support variogram model that can be inferred directly from point measurements, thereby eliminating the need for deconvolution procedures. The two approaches are illustrated using the geological map and heavy metal concentrations recorded in the topsoil of the Swiss Jura. Sensitivity analysis indicates that the new procedures improve prediction over ordinary kriging and traditional residual kriging based on the assumption that the local mean is constant within each mapping unit.     

采样点数量对长三角典型地区土壤肥力指标空间变异解析的影响

通过对江苏省如皋市888个采样点的不重复随机抽样,探讨了采样点数量对土壤肥力指标空间变异解析的影响。从半方差函数估计的可靠性角度考虑,在长江冲积物形成的土壤上,针对县级农业管理和生态环境规划的土壤肥力指标调查采样,采集250个样点较为合适。另外,仅使用交互验证和独立验证评价半方差函数估计的可靠性及确定采样点数量是不完善的,而平均克里格方差理论上随采样点数量的增加而单调递减,可以作为不同采样点数量条件下,综合评价半方差函数估计可靠性及确定采样点数量的补充指标。     

河北省表层土壤有机碳和全氮空间变异特征性及影响因子分析

【目的】在陆地生态系统中, 土壤全氮和有机碳是重要的生态因子。本研究基于土壤调查获得大量土壤剖面的空间和属性信息,研究河北的土壤有机碳和全氮的空间分布特征,为河北的土壤养分监测和管理提供科学依据,同时也为其他类似地区土壤采样提供参考,减少采样成本。【方法】运用传统统计学和地统计学分析方法,以变异函数为工具,初步分析了河北土壤全氮和有机碳的空间变异特征,并应用普通克立格法和回归克里格法进行插值, 得出全氮和有机碳含量的分布格局。【结果】研究区土壤有机碳和全氮的平均值分别为15.25 g/kg和1.23 g/kg,变异系数分别为0.73和0.63,属于中等强度变异。经对数转换后,土壤有机碳和全氮均符合正态分布。选择球状模型作为土壤有机碳和全氮的半方差函数理论模型,土壤有机碳和全氮的块金值/基台值的比值分别为1.8%和1.2%,有机碳和全氮的块金系数均小于25%,表明有机碳和全氮具有强烈的空间相关性。有机碳和全氮空间变异的尺度范围不同,分别为50.400 km和59.200 km。研究区的有机碳总体空间分布规律是有机碳在北部较高、南部较低,呈自北向南递减趋势,土壤全氮与有机碳的空间分布趋势相似,但有机碳的空间变异特征较全氮明显,这种空间分布格局主要受环境因子、 土壤质地、 土壤类型以及土地利用类型等的影响,其中环境因子中的气温和海拔对有机碳和全氮的影响较大。通过比较普通克里格和回归克里格的预测结果,回归克里格能较好地反映东南部有机碳和全氮较低地区的局部变异外,对于西北部的山区也能更好地反映碳、 氮与地形及气候等因素的关系。【结论】河北土壤有机碳和全氮的空间变异和分布特征较为类似,受地形地貌、 气候等因素的影响。通过比较普通克里格法和回归克里格法的空间预测结果,回归克里格法可以消除环境因子的影响,从而得到更准确的空间预测结果,因此建议使用回归克里格法进行预测,以期获得一个更为准确的土壤有机碳和全氮的空间预测结果。     

On spatial prediction of soil properties in the presence of a spatial trend: the empirical best linear unbiased predictor (E-BLUP) with REML

Geostatistical estimates of a soil property by kriging are equivalent to the best linear unbiased predictions (BLUPs). Universal kriging is BLUP with a fixed‐effect model that is some linear function of spatial co‐ordinates, or more generally a linear function of some other secondary predictor variable when it is called kriging with external drift. A problem in universal kriging is to find a spatial variance model for the random variation, since empirical variograms estimated from the data by method‐of‐moments will be affected by both the random variation and that variation represented by the fixed effects. The geostatistical model of spatial variation is a special case of the linear mixed model where our data are modelled as the additive combination of fixed effects (e.g. the unknown mean, coefficients of a trend model), random effects (the spatially dependent random variation in the geostatistical context) and independent random error (nugget variation in geostatistics). Statisticians use residual maximum likelihood (REML) to estimate variance parameters, i.e. to obtain the variogram in a geostatistical context. REML estimates are consistent (they converge in probability to the parameters that are estimated) with less bias than both maximum likelihood estimates and method‐of‐moment estimates obtained from residuals of a fitted trend. If the estimate of the random effects variance model is inserted into the BLUP we have the empirical BLUP or E‐BLUP. Despite representing the state of the art for prediction from a linear mixed model in statistics, the REML–E‐BLUP has not been widely used in soil science, and in most studies reported in the soils literature the variogram is estimated with methods that are seriously biased if the fixed‐effect structure is more complex than just an unknown constant mean (ordinary kriging). In this paper we describe the REML–E‐BLUP and illustrate the method with some data on soil water content that exhibit a pronounced spatial trend.     

Integrating properties of soil map delineations into ordinary kriging

Stratification of a region based on soil map delineations followed by within-stratum interpolation is sometimes used to combine soil maps and spatial interpolation. However, not all delineations are equally suitable to subdivide an area into precisely located mutually exclusive strata. This paper proposes a flow-path to characterize the nature of soil map delineations and a methodology to integrate the properties of map delineations into ordinary kriging.Four types of delineations were distinguished based on three criteria: the nature of transition (discontinuous or gradual), the mapping accuracy, and the structure of the within-unit spatial variation. For each type of delineation the ordinary kriging algorithm was modified to integrate its properties in the interpolation.As a test case, the sand content of the topsoil in the province of West-Flanders (Belgium) was mapped, using independent test data for validation. Inside the mapping units and at delineations identified as gradual transitions, our procedure, termed ordinary kriging integrating properties of map delineations (OKPD) , performed similarly to stratified ordinary kriging (SOK). However, close to the delineations identified as inaccurately mapped discontinuities the mean square prediction error of OKPD was 0.64 times that of SOK. Moreover, near these delineations, the prediction variance was largely underestimated by SOK (relative variance = 5.1), whereas OKPD produced a more realistic value (relative variance = 1.5).     

Employing elevation data for efficient mapping of soil pollution on floodplains

Abstract The co-regionalization between relative elevation and zinc concentration was used to map zinc concentration in the soil of the Geul floodplain in the southern Netherlands by co-kriging from 154 observations. Point co-kriging and point kriging for estimating zinc content in the soil were compared in terms of kriging variances. Another 45 samples were used to compare the precision of the estimated values in terms of squared and absolute estimation errors. Point co-kriging produced better estimates of zinc concentration than either simple point kriging or linear regression from the relative elevation data alone. Moreover, the estimation variances for co-kriging are substantially smaller than those for kriging. The results suggest that knowledge of geomorphological processes can often improve the quality of interpolation maps of properties that are expensive to measure.     

Using scattered hyperspectral imagery data to map the soil properties of a region

Airborne hyperspectral imagery has been recently proved to be a successful technique for predicting soil properties of the bare soil surfaces that are usually scattered in the landscape. This new soil covariate could much improve the digital soil mapping (DSM) of soil properties over larger areas. To illustrate this, we experimented with digital soil mapping in a 24.6‐km² area located in the vineyard plain of Languedoc. As input data, we used 200 points with clay content measurements and 192 bare soil fields representing 3.5% of the total area in which the clay contents of the soil surface were successfully mapped at 5‐m resolution by hyperspectral remote sensing. The clay contents were estimated from CR₂₂₀₆, a spectrometric indicator that quantifies specific absorption features of clay at 2206 nm. We demonstrated by cross‐validation that the co‐kriging procedure based on our co‐regionalization model provided accurate error estimates at the clay measurement sites. Then, we applied a block co‐kriging model to map the mean clay content at increasing resolutions (50 , 100, 250 and 500 m). The results showed the following: (i) using hyperspectral data significantly increased the accuracy of the mean clay content estimations; (ii) a block co‐kriging procedure with reliable estimates of error variance can be used to estimate mean clay contents over larger areas and at coarser resolutions with acceptable and predictable errors and (iii) various maps can be produced that represent different compromises between prediction accuracy and spatial resolution.     

Spatial interpolation of orchard soil pH using soil type and planting duration as auxiliary information

Intensified field management in orcahrds has resulted in significant and widespread acidification in the soils. However, effectively mapping the spatial patterns of soil pH aiming to support ecological management is impeded by its large variotions across soil types and planting durations. Kriging methods were used to integrate soil type and planting duration information for effective mapping of orchard soil pH in a case study in orchards of the Northeast Jiaodong Peninsula, East China. A total of 1 472 surface soil samples were collected, and the planting duration of each sampled orchard was acquired to generate a planting duration map via Voronoi tessellations. The performance of five kriging methods was compared, namely, ordinary kriging (OK), OK combined with soil type (OK_ST), OK combined with planting duration (OK_PD), cokriging combined with soil type and planting duration (OCK_STPD), and OK combined with soil type and planting duration (OK_STPD). Results showed that soil pH declined significantly with increasing planting duration and exhibited moderate spatial variability over the study area. Soil type and planting duration both had significant influence on the spatial distribution of soil pH. The OCK_STPD and OK_STPD methods showed better prediction efficiency than OK, OK_ST, or OK_PD. With regard to the predicted maps of soil pH, the OCK_STPD and OK_STPD methods highly reflected local variations associated with soil type and planting duration, but the OK method was poorly representative. Categorical soil type and planting duration information may be used as ancillary information to improve the mapping quality of orchard soil pH. The OCK_STPD and OK_STPD methods were practical and efficient methods for interpolating orchard soil pH in the study area. The resultant high-quality soil pH maps can contribute to improved site-specific management in the orchards.     

Spatio‐temporal change of soil organic matter content of Jiangsu Province,China, based on digital soil maps

Estimation of spatio‐temporal change of soil is needed for various purposes. Commonly used methods for the estimation have some shortcomings. To estimate spatio‐temporal change of soil organic matter (SOM) in Jiangsu province, China, this study explored benefits of digital soil maps (DSM) by handling mapping uncertainty using stochastic simulation. First, SOM maps on different dates, the 1980s and 2006–2007, were constructed using robust geostatistical methods. Then, sequential Gaussian simulation (SGS) was used to generate 500 realizations of SOM in the area for the two dates. Finally, E‐type (i.e. conditional mean) temporal change of SOM and its associated uncertainty, probability and confidence interval were computed. Results showed that SOM increased in 70% of Jiangsu province and decreased in the remaining 30% during the past decades. As a whole, SOM increased by 0.22% on average. Spatial variance of SOM diminished, but the major spatial pattern was retained. The maps of probability and confidence intervals for SOM change gave more detailed information and credibility about this change. Comparatively, variance of spatio‐temporal change of SOM derived using SGS was much smaller than sum of separate kriging variances for the two dates, because of lower mapping variances derived using SGS. This suggests an advantage of the method based on digital soil maps with uncertainty dealt with using SGS for deriving spatio‐temporal change in soil.     

Spatial variability of some nutrient constituents of an Alfisol from loess II. Geostatistical analysis

Spatial variability of NO₃, K, Mg and organic C of a loess field was studied by using a geostatistical concept, known as theory of regionalized variables. Fifty measurements were made at the nodes of a 30 m × 30 m grid for each of 0-30, 30-60 and 60-90 cm depths. Semivariograms determined from the data showed that NO₃ observations were spatially independent, and hence could be analyzed only by classical methods. Semivariograms of K, Mg and C showed these parameters to be correlated over space for a separating distance between two observations well exceeding 150 m. Their semivariograms were then used in an interpolation method called kriging, which takes into account the correlation between adjacent samples while estimating the interpolated (kriged) value without bias and with minimum variance. Means and estimation variances calculated by punctual kriging were compared to those obtained by classical theory assuming random sampling (i.e., no interdependence between observations). We obtained 1.4 to 3 fold gains in efficiency over that estimated by classical theory. One can, therefore, be sure that the estimation variance of the mean obtained in classical manner will overestimate the real variance unless the sample sites are so far apart that they are spatially independent. The kriged estimates were used to draw contour maps of the properties. Usefulness of such maps and the kriging technique as a whole is discussed to provide better options for management decisions. Finally, a method for determining sample sizes (i. e., number of observations), and hence sample spacing, is developed, taking account of spatial dependence. By this method, sample sizes can be chosen to achieve any desired precision. The sampling effort determined this way in less, and can be very less when based on block or universal kriging, than would have been judged necessary using the classical approach.     

Two robust estimators of the cross‐variogram for multivariate geostatistical analysis of soil properties

If we wish to describe the coregionalization of two or more soil properties for estimation by cokriging then we must estimate and model their auto‐ and cross‐variogram(s). The conventional estimates of these variograms, obtained by the method‐of‐moments, are unduly affected by outlying data which inflate the variograms and so also the estimates of the error variance of cokriging predictions. Robust estimators are less affected. Robust estimators of the auto‐variogram and the pseudo cross‐variogram have previously been proposed and used successfully, but the multivariate problem of estimating the cross‐variogram robustly has not yet been tackled. Two robust estimators of the cross‐variogram are proposed. These use covariance estimators with good robustness properties. The robust estimators of the cross‐variogram proved more resistant to outliers than did the method‐of‐moments estimator when applied to simulated fields which were then contaminated. Organic carbon and water content of the soil was measured at 256 sites on a transect and the method‐of‐moments estimator, and the two robust estimators, were used to estimate the auto‐variograms and cross‐variogram from a prediction subset of 156 sites. The data on organic carbon included a few outliers. The method‐of‐moments estimator returned larger values of the auto‐ and cross‐variograms than did either robust estimator. The organic carbon content at the 100 validation sites on the transect was estimated by cokriging from the prediction data plus a set of variograms fitted to the method‐of‐moments estimates and two sets of variograms fitted to the robust estimates. The ratio of the actual squared prediction error to the cokriging estimate of the error variance was computed at each validation site. These results showed that cokriging using variograms obtained by the method‐of‐moments estimator overestimated the error variance of the predictions. By contrast, cokriging with the robustly estimated variograms gave reliable estimates of the error variance of the predictions.     

How geostatistics can help you

Abstract. Geostatistics is basically a technology for estimating the local values of properties that vary in space from sample data. Research and development in the last 15 years has shown it to be eminently suited for soil and ripe for application in soil survey and land management. The basic technique, ordinary kriging, provides unbiased estimates with minimum and known variance. Data for related variables can be incorporated to improve estimates using cokriging. By more elaborate analysis using disjunctive kriging the probabilities of deficiency and excess can be estimated to aid decision.
The variogram is crucial in all geostatistics; it must be estimated reliably from sufficient data at a sensible scale and modelled properly. Once obtained it can be used not only in the estimation itself but also to choose additional sampling sites, improve a monitoring network or design an optimal sampling scheme for a survey. It may also be used to control a multivariate classification so that the resulting classes are not too fragmented spatially to manage.     

土壤盐分和含水量的空间变异性及其CoKriging估值——以黄河三角洲地区典型地块为例

以黄河三角洲地区典型地块为研究区，运用经典统计学和地统计学相结合的方法研究了不同深度土层盐分和含水量的空间变异特征。结果表明：研究区土壤盐分和含水量普遍较高，土壤表层积盐作用明显；各层土壤盐分均属于中等变异强度，土壤含水量除表层为中等变异强度外，其余各层为弱变异强度；受结构性因素和随机性因素的共同作用，各土层盐分和含水量均具有中等的空间相关性。对Kriging插值结果分析表明，研究区各土层盐分和含水量在一定的范围内均存在着空间上的相关性和变异性，表层土壤盐分和含水量的空间分布主要受微地形和气候条件的影响，地下水性质是主导深层土壤盐分及含水量空间分布的主要因素。分别采用表层土壤盐分及相应土层含水量对深层土壤盐分进行CoKriging估值，均能提高估值精度，采用表层土壤盐分对深层土壤盐分进行CoKriging估值的精度要明显高于采用土壤含水量，其估计方差减少百分数最大达到167．36％。