Mapping Curvilinear Structures with Local Anisotropy Kriging

Interpolating geo-data with curvilinear structures using geostatistics is often disappointing. Channels, for example, become disconnected sets of lakes when interpolated from point data. In order to improve the interpolation of geological structures (e.g., curvilinear structures), we present a new form of kriging, local anisotropy kriging (LAK). Local anisotropy kriging combines a gradient algorithm from image analysis with kriging in an iterative way. After an initial standard kriging interpolation, the gradient algorithm determines the local anisotropy for each cell in the grid using a search area around the cell. Subsequently, kriging is carried out with the spatially varying anisotropy. The anisotropy calculation and subsequent kriging steps will then succeed until the result is satisfactory in the way of reproducing the curvilinear structures. Depending on the size of the search area more or less detail in the geological structures can be reproduced with LAK. Using test examples we show that LAK interpolates data with curvilinear structures more realistically than standard kriging. In a real world case, using bathymetric data of the Oosterschelde estuary, LAK also proves to be quantitatively superior to standard kriging. Absolute interpolation errors are decreased by 23%. Local anisotropy kriging only uses information from point data, which makes the method very objective, it only presents “what the data can tell.”     

Optimal Interpolation of Gravity Maps Using a Modified Neural Network

This paper proposes an interpolation method based on a modified Kohonen artificial neural network, and is used to interpolate marine gravity data on a regular grid. This method combines accuracy comparable to that of kriging with a much shorter computing time than kriging. It is particularly efficient when both the size of the grid and the quantity of available data are large. Under some hypotheses similar to those of kriging with a trend, the unbiasedness and optimality of the method can be demonstrated. Comparison with kriging with a trend using marine gravity data shows similar results. Although neural interpolation is slightly less efficient, it is more robust outside of the marine data area.     

An Alternative Measure of the Reliability of Ordinary Kriging Estimates

This paper presents an interpolation variance as an alternative to the measure of the reliability of ordinary kriging estimates. Contrary to the traditional kriging variance, the interpolation variance is data-values dependent, variogram dependent, and a measure of local accuracy. Natural phenomena are not homogeneous; therefore, local variability as expressed through data values must be recognized for a correct assessment of uncertainty. The interpolation variance is simply the weighted average of the squared differences between data values and the retained estimate. Ordinary kriging or simple kriging variances are the expected values of interpolation variances; therefore, these traditional homoscedastic estimation variances cannot properly measure local data dispersion. More precisely, the interpolation variance is an estimate of the local conditional variance, when the ordinary kriging weights are interpreted as conditional probabilities associated to the n neighboring data. This interpretation is valid if, and only if, all ordinary kriging weights are positive or constrained to be such. Extensive tests illustrate that the interpolation variance is a useful alternative to the traditional kriging variance.     

The influence of conceptual model of sedimentary formation hydraulic heterogeneity on contaminant transport simulation

Development of heterogeneity model of layered sandy-clay formation and impact of this model on transport is considered. The lithological data of more than 250 wells that captured 300 meters formation at the investigated area of 40 km² are used for model of heterogeneity construction. Two models of heterogeneity were developed with using these well data: TP/MC model based on 3D Markov chain simulation for four hydrofacies and 2D kriging interpolation of thicknesses of elementary lithological layers. Simulation of conservative transport by particle tracking algorithm shows that horizontal transport along layers is similar for both models. The main difference is in vertical transport cross formation bedding. The kriging interpolation model gives more conservative results than TP/MC model due to larger characteristic horizontal length of layers in the kriging model. As the result vertical effective hydraulic conductivity of formation is in two times larger and the first particle arriving time is in four times faster in TP/MC model.     

Compositional Kriging: A Spatial Interpolation Method for Compositional Data

Compositional data are very common in the earth sciences. Nevertheless, little attention has been paid to the spatial interpolation of these data sets. Most interpolators do not necessarily satisfy the constant sum and nonnegativity constraints of compositional data, nor take spatial structure into account. Therefore, compositional kriging is introduced as a straightforward extension of ordinary kriging that complies with these constraints. In two case studies, the performance of compositional kriging is compared with that of the additive logratio-transform. In the first case study, compositional kriging yielded significantly more accurate predictions than the additive logratio-transform, while in the second case study the performances were comparable.     

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.     

克里金法在GPS数据内插中的应用

介绍了克里金法在GPS数据内插中的应用、变差函数的自动拟合,以及在进行克里金内插编程时一些值得注意的问题等,通过实验结果看出内插过程取得了很好的效果。     

Geostatistical Mapping with Continuous Moving Neighborhood

An issue that often arises in such GIS applications as digital elevation modeling (DEM) is how to create a continuous surface using a limited number of point observations. In hydrological applications, such as estimating drainage areas, direction of water flow is easier to detect from a smooth DEM than from a grid created using standard interpolation programs. Another reason for continuous mapping is esthetic; like a picture, a map should be visually appealing, and for some GIS users this is more important than map accuracy. There are many methods for local smoothing. Spline algorithms are usually used to create a continuous map, because they minimize curvature of the surface. Geostatistical models are commonly used approaches to spatial prediction and mapping in many scientific disciplines, but classical kriging models produce noncontinuous surfaces when local neighborhood is used. This motivated us to develop a continuous version of kriging. We propose a modification of kriging that produces continuous prediction and prediction standard error surfaces. The idea is to modify kriging systems so that data outside a specified distance from the prediction location have zero weights. We discuss simple kriging and conditional geostatistical simulation, models that essentially use information about mean value or trend surface. We also discuss how to modify ordinary and universal kriging models to produce continuous predictions, and limitations using the proposed models.     

Spatial interpolation of severely skewed data with several peak values by the approach integrating kriging and triangular irregular network interpolation

It was not unusual in soil and environmental studies that the distribution of data is severely skewed with several high peak values, which causes the difficulty for Kriging with data transformation to make a satisfied prediction. This paper tested an approach that integrates kriging and triangular irregular network interpolation to make predictions. A data set consisting of total Copper (Cu) concentrations of 147 soil samples, with a skewness of 4.64 and several high peak values, from a copper smelting contaminated site in Zhejiang Province, China. The original data were partitioned into two parts. One represented the holistic spatial variability, followed by lognormal distribution, and then was interpolated by lognormal ordinary kriging. The other assumed to show the local variability of the area that near to high peak values, and triangular irregular network interpolation was applied. These two predictions were integrated into one map. This map was assessed by comparing with rank-order ordinary kriging and normal score ordinary kriging using another data set consisting of 54 soil samples of Cu in the same region. According to the mean error and root mean square error, the approach integrating lognormal ordinary kriging and triangular irregular network interpolation could make improved predictions over rank-order ordinary kriging and normal score ordinary kriging for the severely skewed data with several high peak values.     

Adding bounds to kriging

The ordinary kriging interpolation algorithm is extended by the inclusion of explicit lower and upper bounds on the estimate. The associated estimation variance is written as the ordinary kriging variance plus a non-negative correction term.     

Adding bounds to kriging

The ordinary kriging interpolation algorithm is extended by the inclusion of explicit lower and upper bounds on the estimate. The associated estimation variance is written as the ordinary kriging variance plus a non-negative correction term.     

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。     

When do we need a trend model in kriging?

Under usual estimation practice with local search windows for data and for interpolation situations, universal kriging and ordinary kriging yield the same estimates, using a data set with apparent trend, for both the unknown attribute and its trend component. Modeling the trend matters only in extrapolation situations. Because conditions of the case study presented arise most frequently in practice, the simpler ordinary kriging is the preferred option.     

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。      

中天山乌拉斯台地区矿化异常提取及评价

以多重分形理论为基础,对中天山乌拉斯台地区铜多金属元素的岩屑测量数据,采用C-A法获得铜多金属的异常下限值,将其作为阈值进行指示克里格插值,绘制研究区的铜多金属地球化学异常图。研究显示,基于该方法获得的Cu矿化异常高值区主要集中在华力西早期第三侵入次的石英闪长岩和花岗闪长岩岩体中,受北西向和次级北东向断裂构造控制明显,该异常区可以作为寻找热液型铜多金属矿产的重要远景区。该方法对于地球化学数据空间变异性强烈的地区,较之普通克里格插值法具有更好的地球化学异常识别能力和高值信息重建能力,所得结果的最高累计频率值范围与已知矿化点的空间位置吻合度更高,在地球化学异常信息提取工作中具有推广意义。     

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.     

Evaluation of spatial interpolation methods for groundwater level in an arid inland oasis,northwest China

The Wuwei oasis, situated in the upper reaches of the Shiyang River basin in the arid inland of northwest China, is intensively cultivated using both groundwater and irrigation water originating from the Qilian Mountains. Groundwater levels are declining due to overuse of irrigation water. To estimate the decline over the entire Wuwei oasis, eight different interpolation methods were used for interpolating groundwater levels over 3 years, i.e. starting in 1983, followed by 1988 and ending with 1992. Cross-validation and orthogonal-validation were applied to evaluate the accuracy of the different methods. Root mean squared error and the correlation coefficient (R ²) were calculated for each of the interpolation methods and years. Three kriging methods (simply, ordinary, and universal) gave the best fit. Modified ordinary kriging was found better than simple and universal kriging methods with a smaller number of points having large differences (>50 m) between estimated and predicted values. Based on the groundwater surfaces determined by the ordinary kriging as modified by Yamamoto, the groundwater decline was found from 1983 to 1992 to be a modest 2.1 m in average.     

Application of FFT-based Algorithms for Large-Scale Universal Kriging Problems

Looking at kriging problems with huge numbers of estimation points and measurements, computational power and storage capacities often pose heavy limitations to the maximum manageable problem size. In the past, a list of FFT-based algorithms for matrix operations have been developed. They allow extremely fast convolution, superposition and inversion of covariance matrices under certain conditions. If adequately used in kriging problems, these algorithms lead to drastic speedup and reductions in storage requirements without changing the kriging estimator. However, they require second-order stationary covariance functions, estimation on regular grids, and the measurements must also form a regular grid. In this study, we show how to alleviate these rather heavy and many times unrealistic restrictions. Stationarity can be generalized to intrinsicity and beyond, if decomposing kriging problems into the sum of a stationary problem and a formally decoupled regression task. We use universal kriging, because it covers arbitrary forms of unknown drift and all cases of generalized covariance functions. Even more general, we use an extension to uncertain rather than unknown drift coefficients. The sampling locations may now be irregular, but must form a subset of the estimation grid. Finally, we present asymptotically exact but fast approximations to the estimation variance and point out application to conditional simulation, cokriging and sequential kriging. The drastic gain in computational and storage efficiency is demonstrated in test cases. Especially high-resolution and data-rich fields such as rainfall interpolation from radar measurements or seismic or other geophysical inversion can benefit from these improvements.     

An Experimental Comparison of Ordinary and Universal Kriging and Inverse Distance Weighting

A factorial, computational experiment was conducted to compare the spatial interpolation accuracy of ordinary and universal kriging and two types of inverse squared-distance weighting. The experiment considered, in addition to these four interpolation methods, the effects of four data and sampling characteristics: surface type, sampling pattern, noise level, and strength of small-scale spatial correlation. Interpolation accuracy was measured by the natural logarithm of the mean squared interpolation error. Main effects of all five factors, all two-factor interactions, and several three-factor interactions were highly statistically significant. Among numerous findings, the most striking was that the two kriging methods were substantially superior to the inverse distance weighting methods over all levels of surface type, sampling pattern, noise, and correlation.     

Effects of search parameters on kriged reserve estimates

Summary Reliable ore reserve estimates for deposits with highly skewed grade distributions are difficult tasks to perform. Although some recent geostatistical techniques are available to handle problems with these estimations, ordinary kriging or conventional interpolation methods are still widely used to estimate the ore reserves for such deposits. The estimation results can be very sensitive to the search parameters used during the interpolation of grades with these methods.This paper compares the ore reserve estimates from ordinary kriging using several cases in which certain search parameters are varied. The comparisons are extended to different mineralizations to show the changing effects of these parameters.