Geostatistical case studies of the advantages of lognormal-de Wijsian kriging with mean for a base metal mine and a gold mine

Parallel variogram analyses, block kriging, and follow-up studies were effected for the lead content of part of the Prieska copper-zinc ore body and for the gold content of the highly variable Breef in a section of the Loraine gold mine, based first on untransformed values and second on logarithmically transformed values using the lognormal-de Wijsian model. For both models the effect was also analyzed of using the population mean or ignoring it. Practical follow-up comparisons confirm theoretical considerations and show that on these mines conditional biases can be eliminated conveniently by kriging with mean; also that the lognormal-de Wijsian model with mean gives the best results.     

Multivariable spatial prediction

For spatial prediction, it has been usual to predict one variable at a time, with the predictor using data from the same type of variable (kriging) or using additional data from auxiliary variables (cokriging). Optimal predictors can be expressed in terms of covariance functions or variograms. In earth science applications, it is often desirable to predict the joint spatial abundance of variables. A review of cokriging shows that a new cross-variogram allows optimal prediction without any symmetry condition on the covariance function. A bivariate model shows that cokriging with previously used cross-variograms can result in inferior prediction. The simultaneous spatial prediction of several variables, based on the new cross-variogram, is then developed. Multivariable spatial prediction yields the mean-squared prediction error matrix, and so allows the construction of multivariate prediction regions. Relationships between cross-variograms, between single-variable and multivariable spatial prediction, and between generalized least squares estimation and spatial prediction are also given.     

M-Estimator of the drift coefficients in a spatial linear model

Kriging as an interpolation method, uses as predictor a linear function of the observations, minimizing the mean squared prediction error or estimation variance. Under multivariate normality assumptions, the given predictor is the best unbiased predictor, and will be vulnerable to outliers. To overcome this problem, a robust weighted estimator of the drift model coefficients is proposed, where unequally spaced data may be weighted through the tile areas of the Dirichlet tessellation.     

玉龙喀什河流域地下水的时空变异规律

对面积约942km2的玉龙喀什河流域地下水埋深进行取样检测,应用地质统计学方法对取得数据进行了半方差函数分析。结果表明,该地区2005、2006年地下水埋深变异函数曲线的理论模型符合指数模型;2007、2008年地下水埋深变异函数曲线的理论模型符合球状模型。通过对2005、2006、2007及2008年水样的分析,得出该地区地下水埋深在时间与空间上皆存在明显的变异性。在空间尺度上,地下水埋深从研究区的东北向西南方向有增加的趋势。从整体上看,地下水埋深在研究区的中部大于四周边;南部大于北部。在时间尺度上,随着时间的推移研究区中部和南部的地下水位正向下降方向发展。利用软件Geopack与Serfer7．0软件绘制了地下水埋深的时空分布图,为该地区地下水资源的评价与今后的合理开发利用提供了决策依据。     

Comparison between different kriging estimators

Six different geostatistical estimators (linear kriging, lognormal kriging, and disjunctive kriging, each with and without a nonbias, i.e., universality condition) were compared using data from a polymetallic deposit in Algeria. The differences between estimators with and without the nonbias condition were far more pronounced than between the different kriging methods. This highlights the importance of choosing an appropriate stationarity model for the data. The criterion concerning kriging weight of the mean in simple kriging, proposed by Remacre (1984, 1987) and Rivoirard (1984) was found to be helpful for determining blocks where the choice of the stationarity hypothesis was critical.     

Variogram Fitting by Generalized Least Squares Using an Explicit Formula for the Covariance Structure

In the context of spatial statistics, the classical variogram estimator proposed by Matheron can be written as a quadratic form of the observations. If data are Gaussian with constant mean, then the correlation between the classical variogram estimator at two different lags is a function of the spatial design matrix and the variance matrix. When data are independent with unidimensional and regular support, an explicit formula for this correlation is available. The same is true for a multidimensional and regular support as can be shown by using Kronecker products of matrices. As variogram fitting is a crucial stage for correct spatial prediction, it is proposed to use a generalized least squares method with an explicit formula for the covariance structure (GLSE). A good approximation of the covariance structure is achieved by taking account of the explicit formula for the correlation in the independent situation. Simulations are carried out with several types of underlying variograms, as well as with outliers in the data. Results show that this technique (GLSE), combined with a robust estimator of the variogram, improves the fit significantly.     

Comparison between neuro-fuzzy and fractal models for permeability prediction

We have used different techniques for permeability prediction using porosity core data from one well at the Maracaibo Lake, Venezuela. One of these techniques is statistical and uses neuro-fuzzy concepts. Another has been developed by Pape et al. (Geophysics 64(5):1447–1460, 1999), based on fractal theory and the Kozeny–Carman equations. We have also calculated permeability values using the empirical model obtained in 1949 by Tixier and a simple linear regression between the logarithms of permeability and porosity. We have used 100% of the permeability–porosity data to obtain the predictor equations in each case. The best fit, in terms of the root mean-square error, was obtained with the statistical approach. The results obtained from the fractal model, the Tixier equation or the linear approach do not improve the neuro-fuzzy results. We have also randomly taken 25% of the porosity data to obtain the predictor equations. The increase of the input data density for the neuro-fuzzy approach improves the results, as is expected for a statistical analysis. On the contrary, for the physical model based on the fractal theory, the decrease in the data density could allow reaching the ideal theoretical Kozeny–Carman model, on which are based the fractal equations, and hence, the permeability prediction using these expressions is improved.     

Fitting variogram models by weighted least squares

The method of weighted least squares is shown to be an appropriate way of fitting variogram models. The weighting scheme automatically gives most weight to early lags and down-weights those lags with a small number of pairs. Although weights are derived assuming the data are Gaussian (normal), they are shown to be still appropriate in the setting where data are a (smooth) transform of the Gaussian case. The method of (iterated) generalized least squares, which takes into account correlation between variogram estimators at different lags, offer more statistical efficiency at the price of more complexity. Weighted least squares for the robust estimator, based on square root differences, is less of a compromise.     

Kriging Prediction Intervals Based on Semiparametric Bootstrap

Kriging is a widely used method for prediction, which, given observations of a (spatial) process, yields the best linear unbiased predictor of the process at a new location. The construction of corresponding prediction intervals typically relies on Gaussian assumptions. Here we show that the distribution of kriging predictors for non-Gaussian processes may be far from Gaussian, even asymptotically. This emphasizes the need for other ways to construct prediction intervals. We propose a semiparametric bootstrap method with focus on the ordinary kriging predictor. No distributional assumptions about the data generating process are needed. A simulation study for Gaussian as well as lognormal processes shows that the semiparametric bootstrap method works well. For the lognormal process we see significant improvement in coverage probability compared to traditional methods relying on Gaussian assumptions.     

桥梁桩低应变反射波法检测及分析探讨

低应变反射波法检测技术作为一种成熟的桩身完整性检测方法,正在被广泛的应用到工程实践中。本文结合铜陵市沿新路互通立交工程桩低应变检测实例,简要介绍了低应变反射波法的基本原理,假定桩为一维线弹性直杆,建立了桩-土动力模型,结合现场测试数据对几个典型实测波形进行了分析,并且通过拟合计算,分析了桩的完整性。     

The Use of Geographically Weighted Regression for Spatial Prediction: An Evaluation of Models Using Simulated Data Sets

Increasingly, the geographically weighted regression (GWR) model is being used for spatial prediction rather than for inference. Our study compares GWR as a predictor to (a) its global counterpart of multiple linear regression (MLR); (b) traditional geostatistical models such as ordinary kriging (OK) and universal kriging (UK), with MLR as a mean component; and (c) hybrids, where kriging models are specified with GWR as a mean component. For this purpose, we test the performance of each model on data simulated with differing levels of spatial heterogeneity (with respect to data relationships in the mean process) and spatial autocorrelation (in the residual process). Our results demonstrate that kriging (in a UK form) should be the preferred predictor, reflecting its optimal statistical properties. However the GWR-kriging hybrids perform with merit and, as such, a predictor of this form may provide a worthy alternative to UK for particular (non-stationary relationship) situations when UK models cannot be reliably calibrated. GWR predictors tend to perform more poorly than their more complex GWR-kriging counterparts, but both GWR-based models are useful in that they provide extra information on the spatial processes generating the data that are being predicted.     

Semivariogram Models Based on Geometric Offsets

Kriging-based geostatistical models require a semivariogram model. Next to the initial decision of stationarity, the choice of an appropriate semivariogram model is the most important decision in a geostatistical study. Common practice consists of fitting experimental semivariograms with a nested combination of proven models such as the spherical, exponential, and Gaussian models. These models work well in most cases; however, there are some shapes found in practice that are difficult to fit. We introduce a family of semivariogram models that are based on geometric shapes, analogous to the spherical semivariogram, that are known to be conditional negative definite and provide additional flexibility to fit semivariograms encountered in practice. A methodology to calculate the associated geometric shapes to match semivariograms defined in any number of directions is presented. Greater flexibility is available through the application of these geometric semivariogram models.     

深基坑施工时地表沉降预测的时序-投影寻踪回归模型

为了保证施工的正常进行、实现信息化施工,必须对基坑实际监测数据进行分析与预测。现有的趋势时间序列分析方法很难满足实际施工中高度非线性问题的拟合,预测误差较大。基于这点考虑,以天津某工程基坑施工地表沉降观测序列为例,在对原始数据进行分析的基础上,提出既可以考虑趋势时间序列,又具有高度非线性拟合性能的时序-投影寻踪回归模型。首先,通过比较分析几种时间序列方法的逼近误差和预测误差,寻求出一种逼近较好的时间序列预测方法。然后,将预测得到的时间序列和观测数据相结合,采用投影寻踪回归方法拟合。应用结果表明,该模型逼近性能良好,预测误差小,可为深基坑位移沉降的动态预测提供一条较好的途径,对基坑动态设计与信息化施工等方面具有重要的参考价值。     

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.     

Spatial prediction and ordinary kriging

Suppose data {Z(s _i ):i=1, ..., n} are observed at spatial locations {s _i :i=1, ..., n}. From these data, an unknownZ(s ₀) is to be predicted at a known locations ₀c, or, ifZ(s₀) has a component of measurement error, then a smooth versionS(s ₀) should be predicted. This article considers the assumptions needed to carry out the spatial prediction using ordinary kriging, and looks at how nugget effect, range, and sill of the variogram affect the predictor. It is concluded that certain commonly held interpretations of these variogram parameters should be modified.This paper was presented at MGUS 87 Conference, Redwood City, California, 14 April 1987.     

Kriging Regionalized Positive Variables Revisited: Sample Space and Scale Considerations

Frequently, regionalized positive variables are treated by preliminarily applying a logarithm, and kriging estimates are back-transformed using classical formulae for the expectation of a lognormal random variable. This practice has several problems (lack of robustness, non-optimal confidence intervals, etc.), particularly when estimating block averages. Therefore, many practitioners take exponentials of the kriging estimates, although the final estimations are deemed as non-optimal. Another approach arises when the nature of the sample space and the scale of the data are considered. Since these concepts can be suitably captured by an Euclidean space structure, we may define an optimal kriging estimator for positive variables, with all properties analogous to those of linear geostatistical techniques, even for the estimation of block averages. In this particular case, no assumption on preservation of lognormality is needed. From a practical point of view, the proposed method coincides with the median estimator and offers theoretical ground to this extended practice. Thus, existing software and routines remain fully applicable.     

Generalized cross-covariances and their estimation

Generalized cross-covariances describe the linear relationships between spatial variables observed at different locations. They are invariant under translation of the locations for any intrinsic processes, they determine the cokriging predictors without additional assumptions and they are unique up to linear functions. If the model is stationary, that is if the variograms are bounded, they correspond to the stationary cross-covariances. Under some symmetry condition they are equal to minus the usual cross-variogram. We present a method to estimate these generalized cross-covariances from data observed at arbitrary sampling locations. In particular we do not require that all variables are observed at the same points. For fitting a linear coregionalization model we combine this new method with a standard algorithm which ensures positive definite coregionalization matrices. We study the behavior of the method both by computing variances exactly and by simulating from various models.     

Risk analysis of rainstorm waterlogging on residences in Shanghai based on scenario simulation

Due to special geographical location and climate, the waterlogging has always been one of the most serious hazards in Shanghai. Residences in the inner city are prone to be damaged by waterlogging hazards. This paper describes the risk analysis of rainstorm waterlogging on residences in Shanghai. First, a rainstorm scenario of 50-year return period was simulated with the rainstorm simulation model from Shanghai Flood Risk Information Center. Each residence was ranked according to its degree of exposure indicated by the inundation depth of that residence, and an exposure analysis model was then built. It is found from the exposure analysis that residences in the sub-districts like Linfen Road, Pengpu Village, Gonghe New Village, Hongqiao Road, Xianxia Road, Xinhua Road, and Zhenru Town are at high-exposure level. Whereas residences in other sub-districts like Gaojing Town, Siping Road, Huaihai Road, Yuyuan, Waitan, Caojiadu, Nanjing East Road, etc. are at low-exposure level. Second, given the characteristics of residences in waterlogging, the vulnerability of residences was expressed as the proportion of old-style residences to total residences. The results show that residences in Yuyuan, Xiaodongmen, Waitan, Nanjing East Road, Laoximen, Zhapu Road, North Station, and Tilanqiao are the most vulnerable ones, while there is no vulnerability in Fenglin Road, Kongjiang Road, Liangcheng New Village, Quyang Road, Siping Road, and Xianxia Road due to the absence of old-style residences. Finally, a model has been built from a systematic perspective and then waterlogging risk analysis was quantified by multiplying the exposure value with vulnerability value of residences. The results reveal that Laoximen, Tilanqiao, Dinghai Road, North Station, Tianping Road, Hongmei Road, Hunan Road, and Xiaodongmen are at high-risk level. The systemic risk model is a simple tool that can be used to assess the relative risk of waterlogging in different regions and the results of risk analysis are applicable to prevention and mitigation of waterlogging for Shanghai Municipal Government.     

不同采样点数量下土壤有机质含量空间预测方法对比

运用普通克里格、泛克里格、协同克里格和回归克里格4种方法,结合由DEM获取的高程因子以及土壤全氮和阳离子交换量（CEC）,预测了黑龙江省海伦市耕地有机质含量的空间分布。不同样点数量下海伦市土壤有机质含量的空间变异结构分析表明,样点数量多并不一定能够识别土壤有机质含量的结构性连续组分,最优化的布置采样点位置可能比单纯增加...