基于地形分区IDW的复垦排土场土壤养分空间分布

准确掌握土壤养分的空间分布，对于评价复垦排土场土壤质量，并制定相应的复垦措施具有重要意义，但是排土场的复杂地形导致一般插值方法的预测效果较差。以内蒙古锡林浩特盟胜利一号露天矿北排土场为例，采用基于地形分区的反距离权重法对土壤养分（速效氮、速效钾、有效磷和有机质）进行空间插值并与常规反距离权重法和克里金插值法进行比较。结果表明，使用分区反距离权重法插值得到的交叉验证精度均高于克里金插值法与反距离权重法，除此之外，有效磷的R2值为0.22，速效氮的R2值为0.37，速效钾以及有机质的R2值分别为0.34与0.14，同样高于克里金插值法与反距离权重法。插值结果显示，排土场速效氮含量整体由北至南逐渐减少；有效磷含量仅在中心地区稍高；速效钾在研究区东北部含量偏高；土壤有机质在中心地区含量相对偏低。研究成果为露天矿排土场土壤养分评价提供借鉴。      

Large-scale spatial interpolation of soil pH across the Loess Plateau, China

Soil pH plays an important role in biogeochemical processes in soils. The spatial distribution of soil pH provides basic and useful information relevant to soil management and agricultural production. To obtain an accurate distribution map of soil pH on the Loess Plateau of China, 382 sampling sites were investigated throughout the region and four interpolation methods, i.e., inverse distance weighting (IDW), splines, ordinary kriging, and cokriging, were applied to produce a continuous soil pH surface. In the study region, soil pH values ranged from 6.06 to 10.76, with a mean of 8.49 and a median of 8.48. Land use type had a significant effect (p < 0.01) on soil pH; grassland soils had higher pHs than cropland and forestland soils. From a regional perspective, soil pH showed weak variation and strong spatial dependence, indicated by the low values of the coefficient of variation (0.05) and the nugget-to-sill ratios (<0.25). Indices of cross-validation, i.e., average error, mean absolute error, root mean square error, and model efficiency coefficient were used to compare the performance of the four different interpolation methods. Kriging methods interpolated more accurately than IDW and splines. Cokriging performed better than ordinary kriging and the accuracy was improved using soil organic carbon as an auxiliary variable. Regional distribution maps of soil pH were produced. The southeastern part of the region had relatively low soil pH values, probably due to higher precipitation, leaching, and higher soil organic matter contents. Areas of high soil pH were located in the north of the central part of the region, possibly associated with the salinization of sandy soils under inappropriate irrigation practices in an arid climate. Map accuracy could be further improved using new methods and incorporating other auxiliary variables, such as precipitation, elevation, terrain attributes, and vegetation types.     

Accounting for exhaustive secondary data into the mapping of water table elevation

This paper presents the incorporation of a digital elevation model into the spatial prediction of water table elevation in Mazandaran province (Iran) using a range of interpolation techniques. The multivariate methods used are: linear regression (LR), cokriging (COK), kriging with an external drift (KED) and regression kriging (RK). The analysis is performed on 3 years (1987, 1997 and 2007) of water table elevation data from about 260 monitoring wells. Prediction performances of the different algorithms are compared with two univariate techniques, i.e. inverse distance weighting and ordinary kriging (OK), through cross validation and examination of the consistency of the generated maps with the natural phenomena. Significantly smaller prediction errors are obtained for four multivariate algorithms but, in particular, KED and RK outperform LR and COK for 3 years. The results show the potential for using elevation for a more precise mapping of water table elevation.     

Hydrographic data modeling methods for determining precise seafloor topography

Acoustic and light detection and ranging are the recent methods used in hydrographical surveying. Depth values depending on X and Y horizontal coordinates are measured in both methods. While processing the hydrographical data, all data with different density are interpolated and modeled for determining the seafloor model using different interpolation techniques. In this study, effects of different surface modeling methods are investigated. Data obtained from single-beam echo sounder (SBES) are modeled using inverse distance, kriging, local polynomial, minimum curvature, moving average, nearest-neighbor, and Delaunay interpolation methods. Interpolation results are compared with the multibeam echo sounder data which were collected on the same area for determining the accuracy of modeling methods. Depending on the maximum total vertical uncertainty values in hydrographic survey standards, the best results were determined by using the kriging method. The Delaunay, minimum curvature, and inverse distance methods can be used for modeling the SBES data in shallow waters.     

On the equivalence of kriging and maximum entropy estimators

This study compares kriging and maximum entropy estimators for spatial estimation and monitoring network design. For second-order stationary random fields (a subset of Gaussian fields) the estimators and their associated interpolation error variances are identical. Simple lognormal kriging differs from the lognormal maximum entropy estimator, however, in both mathematical formulation and estimation error variances. Two numerical examples are described that compare the two estimators. Simple lognormal kriging yields systematically higher estimates and smoother interpolation surfaces compared to those produced by the lognormal maximum entropy estimator. The second empirical comparison applies kriging and entropy-based models to the problem of optimizing groundwater monitoring network design, using six alternative objective functions. The maximum entropy-based sampling design approach is shown to be the more computationally efficient of the two.     

Modelling techniques for volumetric reconstruction of earth structures

Visualisation of seismic and tomographic results is a crucial point to properly understand the models provided by seismic methods. We consider several geostatistical methods (inverse distance weighting, point kriging and mathematical wavelets) to map surface wave tomography in a sparsely sampled study area, and to compare their accuracy and efficiency with proper raypath methodologies (inversion and projection onto convex sets). A large set of synthetic data is used to estimate seismic velocities before application to real data. The contour maps of prediction errors indicate that spatial prediction and inversion perform similarly.     

A Robust Multiquadric Method for Digital Elevation Model Construction

The multiquadric method (MQ) with high interpolation accuracy has been widely used for interpolating spatial data. However, MQ is an exact interpolation method, which is improper to interpolate noisy sampling data. Although the least squares MQ (LSMQ) has the ability to smooth out sampling errors, it is inherently not robust to outliers due to the least squares criterion in estimating the weights of sampling knots. In order to reduce the impact of outliers on the accuracy of digital elevation models (DEMs), a robust method of MQ (MQ-R) has been developed. MQ-R includes two independent procedures: knot selection and the solution of the system of linear equations. The two independent procedures were respectively achieved by the space-filling design and the least absolute deviation, both of which are very robust to outliers. Gaussian synthetic surface, which is subject to a series of errors with different distributions, was employed to compare the performance of MQ-R with that of LSMQ. Results indicate that LSMQ is seriously affected by outliers, whereas MQ-R performs well in resisting outliers, and can construct satisfactory surfaces even though the data are contaminated by severe outliers. A real-world example of DEM construction was employed to evaluate the robustness of MQ-R, LSMQ, and the classical interpolation methods including inverse distance weighting method, thin plate spline, and ANUDEM. Results showed that compared with the classical methods, MQ-R has the highest accuracy in terms of root mean square error. In conclusion, when sampling data is subject to outliers, MQ-R can be considered as an alternative method for DEM construction.     

Spatial prediction of soil properties in a watershed scale through maximum likelihood approach

Surface map of soil properties plays an important role in various applications in a watershed. Ordinary kriging (OK) and regression kriging (RK) are conventionally used to prepare these surface maps but generally need large number of regularly girded soil samples. In this context, REML-EBLUP (REsidual Maximum Likelihood estimation of semivariogram parameters followed by Empirical Best Linear Unbiased Prediction) shown capable but not fully tested in a watershed scale. In this study, REML-EBLUP approach was applied to prepare surface maps of several soil properties in a hilly watershed of Eastern India and the performance was compared with conventionally used spatial interpolation methods: OK and RK. Evaluation of these three spatial interpolation methods through root-mean-squared residuals (RMSR) and mean squared deviation ratio (MSDR) showed better performance of REML-EBLUP over the other methods. Reduction in sample size through random selection of sampling points from full dataset also resulted in better performance of REML-EBLUP over OK and RK approach. The detailed investigation on effect of sample number on performance of spatial interpolation methods concluded that a minimum sampling density of 4/km² may successfully be adopted for spatial prediction of soil properties in a watershed scale using the REML-EBLUP approach.     

SWAT模型水文过程模拟的数据不确定性分析——以青海湖布哈河流域为例

降水、 气温的空间分布是影响流域水量平衡模拟的关键因素, 运用距离权重反比法(IDW)、 梯度距离权重反比法(GIDW)、 样条函数法(Spline)和克里金插值法(Kriging)对青海湖流域及周边地区43个气象站1995-2009年逐日气温和降水进行了空间插值, 并以气象要素空间插值数据驱动模型, 进行布哈河流域径流模拟. 选用布哈河口月平均流量, 以Nash-Suttclife系数(E_ns)、 相关系数(R)和相对误差(R_E)为评价指标, 进行校准期(2000-2004年)和验证期(2005-2009年)的径流模拟效果比较. 结果表明: 径流模拟精度较高, GIDW和IDW更适合于布哈河流域的气象要素空间化, 并且气象要素空间插值数据误差是引起模型模拟不确定性和参数据不确定性的原因之一.     

Application of spatial filter theory to kriging

Data-processing requirements for remotely sensed, digital images include spatial filtering to suppress image noise, enhance edges/contacts, and improve image clarity. Spatial filter theory demonstrates that the addition of a high-pass filtered image to a low-pass filtered image yields the original digital image. Application of this principle in kriging can be accomplished by using the same covariance matrix to solve for two weighting vectors to yield a result analogous to low- and high-pass filtering. The addition of kriged estimates calculated using both weighting vectors is analogous to summing high-, and low-pass filtered digital images. This modified method of kriging yields estimates associated with less smoothing compared to ordinary kriging. Statistical moments of original sample data are better preserved through estimation by this method.     

Comparison of statistical prediction methods for characterizing the spatial variability of apparent electrical conductivity in coastal salt-affected farmland

Soil salinity has been known to be problematic to land productivity and environment in the lower Yellow River Delta due to the presence of a shallow, saline water table and marine sediments. Spatial information on soil salinity has gained increasing importance for the demand of management and sustainable utilization of arable land in this area. Apparent electrical conductivity, as measured by electromagnetic induction instrument in a fairly quick manner, has succeeded in mapping soil salinity and many other soil physical and chemical properties from field to regional scales. This was done based on the correlation that existed between apparent electrical conductivity and many other soil properties. In this paper, four spatial prediction methods, i.e., local polynomial, inverse distance weighed, ordinary kriging and universal kriging, were employed to estimate field-scale apparent electrical conductivity with the aid of an electromagnetic induction instrument (type EM38). The spatial patterns estimated by the four methods using EM38 survey datasets of various sample sizes were compared with those generated by each method using the entire sample size. Spatial similarity was evaluated using difference index (DI) between the maps created using various sample sizes (i.e., target maps) and the maps generated with the entire sample size (i.e., the reference map). The results indicated that universal kriging had the best performance owing to the inclusion of residuals and spatial detrending in the kriging system. DI showed that spatial similarity between the target and reference maps of apparent electrical conductivity decreased with the reduction in sample size for each prediction method. Under the same reduction in sample size, the method retaining the most spatial similarity was universal kriging, followed by ordinary kriging, inverse distance weighed, and local polynomial. Approximately, 70 % of total survey data essentially met the need for retaining 90 % details of the reference map for universal kriging and ordinary kriging methods. This conclusion was that OK and UK were two most appropriate methods for spatial estimation of apparent electrical conductivity as they were robust with the reduction in sample size.     

GIS and ANN-based spatial prediction of DOC in river networks: a case study in Dongjiang, Southern China

This paper investigates the use of an artificial neural network (ANN) model to predict dissolved organic carbon (DOC) in a river network and evaluates the impacts of watershed characteristics on stream DOC. Samples and relevant environmental variables were obtained from field sampling at 28 hydrological response units (HRUs) and a MODIS/SRTM DEM satellite image. HRUs can provide reliable spatial interpolation for filling data gaps and incorporate potential spatial correlation among observations in each ANN neuron. The process and results of neural network modeling were assessed by deterministic and statistical methods and spatial regression kriging. The spatial prediction results show that ANN, using improved back propagation algorithms of 7-15-1 architecture, was the optimal network, by which predictions maintained most of the original spatial variation and eliminated smoothing effects of RK. The sum of the relative contributions of four sensitive variables, including soil organic carbon density, geographic longitude, surface runoff and Chl a in river water, was >75 %. A minor prediction error of ~6 % was found in HRUs of open shrublands, but HRUs of urban and croplands had an error of 24–30 %. This pattern exemplifies anthropogenic impacts in urban areas on stream DOC and agricultural activities in croplands. The usefulness of ANN modeling-based GIS in this study is demonstrated by depiction of spatial variation of stream DOC and indicates the benefits of understanding sensitive factors for watershed impact assessments.     

基于最佳小波基的地震面波插值方法

在利用实际地震数据中的面波反演近地表横波速度的过程中,若道间距较大、空间采样率不足,则会产生空间假频现象,从而降低频率速度谱的信噪比,影响频散曲线提取的精度以及反演效果,因此需要针对面波进行插值处理。文中提出了一种基于最佳小波基的地震面波插值方法,通过理论分析和实验误差对比在地震数据处理常用的众多小波基中选出适用于插值处理的最佳小波基bior6.8,提高了插值精度。针对面波同向轴为线性且斜率较大的特点,文中首先采用线性动校正的方法对面波进行拉平处理,再进行小波变换插值,最后进行反线性动校正恢复面波。通过对理论模型与实际资料进行插值处理验证了本文方法的有效性,插值后的面波记录波形恢复较好,显著提高了频率速度谱的信噪比,有效解决了面波数据空间采样率不足引起的假频问题。     

Sampling design optimization for spatial functions

A new procedure is presented for minimizing the sampling requirements necessary to estimate a mappable spatial function at a specified level of accuracy. The technique is based on universal kriging, an estimation method within the theory of regionalized variables. Neither actual implementation of the sampling nor universal kriging estimations are necessary to make an optimal design. The average standard errorand maximum standard error of estimationover the sampling domain are used as global indices of sampling efficiency. The procedure optimally selects those parameters controlling the magnitude of the indices, including the density and spatial pattern of the sample elements and the number of nearest sample elements used in the estimation. As an illustration, the network of observation wells used to monitor the water table in the Equus Beds of Kansas is analyzed and an improved sampling pattern suggested. This example demonstrates the practical utility of the procedure, which can be applied equally well to other spatial sampling problems, as the procedure is not limited by the nature of the spatial function.     

半变异函数及取样间距对克里金法在海洋地层分析中的影响研究

以浅剖数据为源数据,钻孔实测数据为验证数据,利用普通克里金法对海底地层厚度进行空间插值得到地层分布特征,采用3种半变异函数模型和不同取样间距对某井场3组地层厚度进行普通克里金插值并验证其插值效果。结果表明:普通克里金是一种有效的海底地层厚度预测方法;结构分析最佳的模型不一定是误差最小的模型,应对不同模型下的插值结果进行综合分析来选择最合适的模型,并提出球状模型在该井场厚度估计中最优,高斯模型次之;对于球状模型,增大取样间距对地层厚度变化剧烈的地层回归效果影响较小,对地层厚度变化不大的地层回归效果影响较大;同时,SE预测值变化率分析表明对于地层厚度变化剧烈的地层,减小取样间距可以大幅度地减少插值误差,而对于地层厚度变化不大的地层,减小取样间距对插值精度提高的意义不大。     

地球物理勘探中几种二维插值方法的误差分析

为了研究采样和网格化方法对地球物理数据成图精度的影响,为野外数据采集布设提供一定的依据,采用数值模拟确定重力异常场场值,通过不同采样间距和不同插值方法计算重力异常绝对误差均方根值和节点处的绝对误差值,对比不同插值方法的误差,得到了如下认识：1）对于同一插值方法而言,存在小间距绝对误差均方根值小于大间距绝对误差均方根值的关系。2）对不同的插值方法而言:当采样间距小于最小异常地质体尺度时,绝对误差均方根值由小到大的顺序是径向基函数法、改进的谢别德法、克里金插值法、自然邻点法、反距离加权插值法、最近邻点法、最小曲率法,并且线性插值三角网法与自然邻点法具有几乎相同的数值;当采样间距大于最小异常地质体尺度时,绝对误差均方根值由小到大的顺序是径向基函数法、改进的谢别德法、克里金插值法、自然邻点法、最小曲率法、最近邻点法、反距离加权插值法,并且线性插值三角网法和自然邻点法具有几乎相同的数值。3）从绝对误差均方值看,径向基函数方法、改进的谢别德方法和克里金方法数值较小,其中径向基函数值绝对误差均方根值最小。4）从节点处绝对误差值来看,径向基函数方法、克里金方法、改进的谢别德方法相对其他插值方法具有更小的误差,不存在局部误差较小或较大的情况,是相对较好的插值方法,并且径向基函数方法是最好的。     

合肥义城地区土壤重金属污染评价中典型插值方法的对比

空间插值对于土壤中重金属元素的空间分布及污染评价具有重要意义。对合肥义城地区土壤中的Cu、Pb、Zn、Cd、As、Hg等污染重金属元素,以常用且具有代表性的反距离加权法、径向基函数法、普通克里格法,进行了空间插值的对比验证分析和评价。通过对各种元素的空间插值各种误差进行综合比较的结果表明：Cu、Pb、As元素采用普通克里格法进行插值结果最优,而Zn元素采用反距离加权法最优,对于Cd、Hg元素则径向基函数插值法最优。     

基于克里金插值估算区域降水量的抽样方法对比分析——以甘肃省为例

降水数据的空间插值精度由降水的空间变异特征、插值方法和降水观测站分布决定的, 其中, 降水观测站及插值方法的选择尤为重要. 利用空间随机抽样、空间分层抽样和空间三明治抽样等空间抽样方案, 以甘肃省境内的气象站点为研究对象进行抽样分析, 在抽样结果的基础上进行区域降水量克里金插值, 并对比了插值结果精确度. 结果表明: 在较为丰富的先验知识的前提下, 空间三明治抽样所得的区域降水量插值结果的误差项ME(1.97)、MSE(0.0066)和RMSSE(1.0184)都是最优的, 空间分层抽样次之, 空间随机抽样最差. 与另外两种抽样方案相比, 空间三明治抽样是一种使区域误差最小, 适用范围更广, 精确度更高的抽样方法.     

Robustness of noise filtering by kriging analysis

In geostatistics, factorial kriging is often proposed to filter noise. This filter is built from a linear model which is ideally suited to a Gaussian signal with additive independent noise. Robustness of the performance of factorial kriging is evaluated in less congenial situations. Three different types of noise are considered all perturbing a lognormally distributed signal. The first noise model is independent of the signal. The second noise model is heteroscedastic; its variance depends on the signal, yet noise and signal are uncorrelated. The third noise model is both heteroscedastic and linearly correlated with the signal. In ideal conditions, exhaustive sampling and additive independent noise, factorial kriging succeeds to reproduce the spatial patterns of high signal values. This score remains good in presence of heteroscedastic noise variance but falls quickly in presence of noise-to-signal correlation as soon as the sample becomes sparser.     

Application of kriging technique to areal precipitation mapping in Arizona

The classical methods for interpolating and spatial averaging of precipitation fields fail to quantify the accuracy of the estimate. On the other hand, kriging is an interpolation method for predicting values of regionalized variables at points (punctual kriging) or average values over an area (block kriging).This paper demonstrates the use of the kriging method for mapping and evaluating precipitation data for the State of Arizona. Using 158 rain gauge stations with 30 years or more of record, the precipitation over the state has been modeled as a realization of a two dimensional random field taking into consideration the spatial variability conditions.Three data sets have been used: (1) the mean annual precipitation over the state; (2) the mean summer rainy season; and (3) the mean winter rainy season. Validation of the empirical semi-variogram for a constant drift case indicated that the exponential model was appropriate for all the data sets. In addition to a global kriging analysis, the data have been examined under an anisotropic assumption which reflects the topographic structure of the state.