Spatial Breakdown Point of Variogram Estimators

In the context of robust statistics, the breakdown point of an estimator is an important feature of reliability. It measures the highest fraction of contamination in the data that an estimator can support before being destroyed. In geostatistics, variogram estimators are based on measurements taken at various spatial locations. The classical notion of breakdown point needs to be extended to a spatial one, depending on the construction of most unfavorable configurations of perturbation. Explicit upper and lower bounds are available for the spatial breakdown point in the regular unidimensional case. The difficulties arising in the multidimensional case are presented on an easy example in IR² , as well as some simulations on irregular grids. In order to study the global effects of perturbations on variogram estimators, further simulations are carried out on data located on a regular or irregular bidimensional grid. Results show that if variogram estimation is performed with a 50% classical breakdown point scale estimator, the number of initial data likely to be contaminated before destruction of the estimator is roughly 30% on average. Theoretical results confirm the previous statement on data in IR^d , d 1.     

Highly Robust Variogram Estimation

The classical variogram estimator proposed by Matheron is not robust against outliers in the data, nor is it enough to make simple modifications such as the ones proposed by Cressie and Hawkins in order to achieve robustness. This paper proposes and studies a variogram estimator based on a highly robust estimator of scale. The robustness properties of these three estimators are analyzed and compared. Simulations with various amounts of outliers in the data are carried out. The results show that the highly robust variogram estimator improves the estimation significantly.     

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.     

The Correlation Structure of Matheron's Classical Variogram Estimator Under Elliptically Contoured Distributions

The classical variogram estimator proposed by Matheron can be written as a quadratic form of the observations. When data have an elliptically contoured distribution with constant mean, the correlation between the classical variogram estimator at two different lags is a function of the spatial design matrix, the covariance matrix, and the kurtosis. Several specific cases are studied closely. A subclass of elliptically contoured distributions with a particular family of covariance matrices is shown to possess exactly the same correlation structure for the classical variogram estimator as the multivariate independent Gaussian distribution. The consequences on variogram fitting by generalized least squares are discussed.     

The Correlation Structure of Matheron's Classical Variogram Estimator Under Elliptically Contoured Distributions

The classical variogram estimator proposed by Matheron can be written as a quadratic form of the observations. When data have an elliptically contoured distribution with constant mean, the correlation between the classical variogram estimator at two different lags is a function of the spatial design matrix, the covariance matrix, and the kurtosis. Several specific cases are studied closely. A subclass of elliptically contoured distributions with a particular family of covariance matrices is shown to possess exactly the same correlation structure for the classical variogram estimator as the multivariate independent Gaussian distribution. The consequences on variogram fitting by generalized least squares are discussed.     

Disentangling Shallow and Deep Processes Causing Surface Movement

Understanding and predicting surface movement is important both technically and for social reasons. The shallow processes contributing to subsidence include construction works, peat oxidation, clay compaction, and groundwater withdrawal; deep causes are hydrocarbon and salt production. We describe an inversion procedure we have devised to disentangle the deep and shallow causes of surface movement. It employs a Bayesian inversion scheme, using forward models and other ‘a priori’ information about shallow and deep compaction. Parameter estimation thus takes place at two different depths, thereby disentangling the deep and shallow compaction processes responsible for surface movement. The uncertainty in the surface measurements and ‘a priori’ estimates is naturally incorporated. Furthermore, spatial and temporal correlations can be taken into account through inclusion of the covariance matrix. The inversion scheme is demonstrated for two synthetic cases. The first combines a compacting gas field and a compacting shallow peat layer. We demonstrate that assumptions on the shape of the subsidence bowl are not necessary. We also show how neglecting either deep or shallow causes of subsidence can produce spurious results. The advantage of using the ‘a priori’ estimates of the compaction and the covariance matrix obtained by Monte Carlo simulations is demonstrated with a second synthetic example involving two polders and different depths of their water table. A robust solution is obtained for each polder unit, while a simpler (and faster) ‘a priori’ estimate based on the expected average clay thickness fails to reproduce the actual compaction. Monte Carlo simulations can also be applied to compaction in depleting gas reservoirs. Information on spatial correlations is often available, even when the absolute values of the ‘a priori’ compaction data are quite uncertain. Explicitly incorporating such ‘a priori’ known spatial correlations improves the result significantly.     

Combining Robustness with Efficiency in?the?Estimation of the Variogram

In the present paper, we propose a new method for the estimation of the variogram, which combines robustness with efficiency under intrinsic stationary geostatistical processes. The method starts by using a robust estimator to obtain discrete estimates of the variogram and control atypical observations that may exist. When the number of points used in the fit of a model is the same as the number of parameters, ordinary least squares and generalized least squares are asymptotically equivalent. Therefore, the next step is to fit the variogram by ordinary least squares, using just a few discrete estimates. The procedure is then repeated several times with different subsets of points and this produces a sequence of variogram estimates. The final estimate is the median of the multiple estimates of the variogram parameters. The suggested estimator will be called multiple variograms estimator. This procedure assures a global robust estimator, which is more efficient than other robust proposals. Under the assumed dependence structure, we prove that the multiple variograms estimator is consistent and asymptotically normally distributed. A simulation study confirms that the new method has several advantages when compared with other current methods.     

Studies of the effects of outliers and data transformation on variogram estimates for a base metal and a gold ore body

Variograms for gold and lead values from the Loraine and Prieska mines, respectively, indicate that data outliers can seriously distort and/or mask the real variogram patterns. Studies show that this problem is best overcome for these mines by logarithmic transformation of the data, and/or a suitable screening out of such outliers, and/or more robust variogram estimation procedures; the benefits are particularly significant when the basic data is limited.     

Robust kriging—A proposal

Geological data frequently have a heavy-tailed normal-in-the-middle distribution, which gives rise to grade distributions that appear to be normal except for the occurrence of a few outliers. This same situation also applies to log-transformed data to which lognormal kriging is to be applied. For such data, linear kriging is nonrobust in that (1)kriged estimates tend to infinity as the outliers do, and (2)it is also not minimum mean squared error. The more general nonlinear method of disjunctive kriging is even more nonrobust, computationally more laborious, and in the end need not produce better practical answers. We propose a robust kriging method for such nearly normal data based on linear kriging of an editing of the data. It is little more laborious than conventional linear kriging and, used in conjunction with a robust estimator of the variogram, provides good protection against the effects of data outliers. The method is also applicable to time series analysis.     

Estimation of regionalized phenomena by geostatistical methods: lake acidity on the Canadian Shield

 This paper describes a geostatistical technique based on conditional simulations to assess confidence intervals of local estimates of lake pH values on the Canadian Shield. This geostatistical approach has been developed to deal with the estimation of phenomena with a spatial autocorrelation structure among observations. It uses the autocorrelation structure to derive minimum-variance unbiased estimates for points that have not been measured, or to estimate average values for new surfaces. A survey for lake water chemistry has been conducted by the Ministère de l'Environnement du Québec between 1986 and 1990, to assess surface water quality and delineate the areas affected by acid precipitation on the southern Canadian Shield in Québec. The spatial structure of lake pH was modeled using two nested spherical variogram models, with ranges of 20 km and 250 km, accounting respectively for 20% and 55% of the spatial variation, plus a random component accounting for 25%. The pH data have been used to construct a number of geostatistical simulations that produce plausible realizations of a given random function model, while 'honoring' the experimental values (i.e., the real data points are among the simulated data), and that correspond to the same underlying variogram model. Post-processing of a large number of these simulations, that are equally likely to occur, enables the estimation of mean pH values, the proportion of affected lakes (lakes with pH≤5.5), and the potential error of these parameters within small regions (100 km×100 km). The method provides a procedure to establish whether acid rain control programs will succeed in reducing acidity in surface waters, allowing one to consider small areas with particular physiographic features rather than large drainage basins with several sources of heterogeneity. This judgment on the reduction of surface water acidity will be possible only if the amount of uncertainty in the estimation of mean pH is properly quantified. Received: 3 March 1997 · Accepted: 16 November 1998     

基于抗差Kalman滤波的精密单点定位

当精密单点定位的观测值含有异常数据时,Kalman滤波的精度将会降低。采用抗差Kalman滤波方法能够有效抑制观测异常,提高滤波的精度和可靠性。运用武汉国际GPS服务跟踪站数据对该方法进行了验证。结果表明,抗差Kalman滤波的精度比Kalman滤波的精度有一定程度提高,说明抗差Kalman滤波能够有效抑制观测异常。     

Capitalism, masculinity and whiteness in the dialectical landscape: The case of Tarzan and the Tycoon

The intent of this paper is to explore how ‘reel’ and ‘real’ landscapes can be rendered dialectic. The analysis focuses on the filming of Tarzan's Secret Treasure (1941) at the wildlife preserve of Wakulla Springs in Florida, which was administered by the financial tycoon Ed Ball. Using Mitchell's recent commentaries on ‘good’ and ‘bad’ dialectics as our entry point, we offer two readings of this cinematic process. The first renders the on and off screen Wakulla Springs as ideological fodder, serving to valorise capital, masculinity and whiteness. The second reading addresses how this place has been actively constructed from discourses pertaining to class, gender and race, such that Wakulla exists as a series of fully relational spaces within diverse contexts, both ‘real’ and ‘reel’. In structuring our analysis in this manner, we hope to make clear why it does indeed matter for film geographers to acknowledge the conceptual presumptions brought to bear in the framing of the real and the reel as objects of analysis. This revised version was published online in August 2006 with corrections to the Cover Date.     

A comparative analysis of several vulnerability concepts

A comparative analysis of six vulnerability models aims to identify differences and similarities between several approaches towards understanding vulnerability. The analysis yields a set of characteristics for explaining the condition of vulnerability (multiple contexts, multiple dimensions, temporal variability, multiple scales and scale-interdependency). In addition, ‘adaptation’ and ‘adaptive capacity’ are identified as key elements of vulnerability. The results of the analysis are put into a wider context not only of vulnerability but also of resilience and risk research. It is demonstrated that ‘adaptation’ and ‘adaptive capacity’ serve as hinges not only for conceptualising vulnerability but between ‘vulnerability’ and ‘resilience’ alike. Based on the results of the comparative analysis, a model of vulnerability focussing on the household and community scale is developed, which displays the key findings of this work.     

Analysis of the Variability of a Two-Dimensional Finite Strain Estimate

Finite strain estimation is a widely used technique for the study of rock deformation in structural geology. One particular algorithm proposed by Shimamoto and Ikeda uses the ‘average shape matrix’ of deformed markers. This paper provides a detailed error analysis for resulting strain estimates in two dimensions. When the number of markers exceeds 100, estimators of components of the strain tensor are shown to have an approximately Gaussian distribution with variances that increase with their mean. Equal variance estimators are obtained by applying a log transform for the elongation and an arcsin transformation for the orientation estimates. Confidence interval formulae for strain tensor components are proposed. Lithology specific constants arising in these formulae are estimated from undeformed samples. The results are validated by application to simulated data as well as observational data from thin sections of sandstone sampled from SE Ireland.     

Combining Areal and Point Data in Geostatistical Interpolation: Applications to Soil Science and Medical Geography

A common issue in spatial interpolation is the combination of data measured over different spatial supports. For example, information available for mapping disease risk typically includes point data (e.g. patients’ and controls’ residence) and aggregated data (e.g. socio-demographic and economic attributes recorded at the census track level). Similarly, soil measurements at discrete locations in the field are often supplemented with 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 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. The procedure is illustrated using two data sets: (1) geological map and heavy metal concentrations recorded in the topsoil of the Swiss Jura, and (2) incidence rates of late-stage breast cancer diagnosis per census tract and location of patient residences for three counties in Michigan. In the second case, the kriging system includes an error variance term derived according to the binomial distribution to account for varying degree of reliability of incidence rates depending on the total number of cases recorded in those tracts. Except under the binomial kriging framework, area-and-point (AAP) kriging ensures the coherence of the prediction so that the average of interpolated values within each mapping unit is equal to the original areal datum. The relationships between binomial kriging, Poisson kriging, and indicator kriging are discussed under different scenarios for the population size and spatial support. Sensitivity analysis demonstrates the smaller smoothing and greater prediction accuracy of the new procedure over ordinary and traditional residual kriging based on the assumption that the local mean is constant within each mapping unit.     

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.     

(Mis)adventures in Rumsfeld space

Ewald’s recent genealogy of constructions of risk in Western societies argues that the 1980s saw an important paradigm shift to the “precautionary principle”. Critical scholars have taken up this idea as a lens through which to interpret the Bush administration’s ‘war on terror’. I argue that 11 September 2001 actually brought about qualitative changes to this paradigm. Bush’s pre-emptive doctrine is driven, and perhaps even more importantly, continually justified to the US population, by what might be called the “trans-precautionary principle”, a move from “decisionism” to “actionism”, and a new radicalization of the way fear is produced and managed. Donald Rumsfeld’s famous typology of different articulations of knowledge and ignorance offers an excellent analytical window onto the connections between ignorance, fear and geopolitical action in this new regime. In the latter part of the essay, his four modes of knowledge/ignorance are arrayed, for heuristic purposes, in an abstract spatial grid organized along dimensions of the specificity and possession of knowledge. This allows a ‘mapping’ of some of the Bush administration’s more controversial strategies in the ‘war on terror’, as a set of different pathways through Rumsfeld Space.     

储量的地质统计学估计及其与传统计算法比较

概述了克里格法应用于北方某煤矿的储量计算结果,并将其与传统方法的计算结果进行比较,说明了该法的应用价值。      

Nomospace and genospace: A spatial matrix

We are concerned here with the notion of space as a primary condition for building social relations. From this stand, we have created two analytical matrixes that can help us characterize two modes: that of ‘being-there-in-space’ and that of ‘being-of-the-space’; the nomospace and the genospace. While the nomospace is characterized by ‘cold’ connotations, which are ideally defined as a result of a logical option, the genospace is mainly characterized by emotional or ‘warm’ connotations, which originate from a feeling of shared nature and common destiny. We believe that these matrixes could help us to identify the meanings of some behaviours and, therefore, make us more capable of interpreting many dynamics that occur in modern societies. The importance of geography is justified by the existing relationship between these behaviours and their proper spatial form. This revised version was published online in July 2006 with corrections to the Cover Date.