Some Results on the Spatial Breakdown Point of Robust Point Estimates of the Variogram

The effect of outliers on estimates of the variogram depends on how they are distributed in space. The ‘spatial breakdown point’ is the largest proportion of observations which can be drawn from some arbitrary contaminating process without destroying a robust variogram estimator, when they are arranged in the most damaging spatial pattern. A numerical method is presented to find the spatial breakdown point for any sample array in two dimensions or more. It is shown by means of some examples that such a numerical approach is needed to determine the spatial breakdown point for two or more dimensions, even on a regular square sample grid, since previous conjectures about the spatial breakdown point in two dimensions do not hold. The ‘average spatial breakdown point’ has been used as a basis for practical guidelines on the intensity of contaminating processes that can be tolerated by robust variogram estimators. It is the largest proportion of contaminating observations in a data set such that the breakdown point of the variance estimator used to obtain point estimates of the variogram is not exceeded by the expected proportion of contaminated pairs of observations over any lag. In this paper the behaviour of the average spatial breakdown point is investigated for cases where the contaminating process is spatially dependent. It is shown that in two dimensions the average spatial breakdown point is 0.25. Finally, the ‘empirical spatial breakdown point’, a tool for the exploratory analysis of spatial data thought to contain outliers, is introduced and demonstrated using data on metal content in the soils of Sheffield, England. The empirical spatial breakdown point of a particular data set can be used to indicate whether the distribution of possible contaminants is likely to undermine a robust variogram estimator.     

A Preliminary Note on X-and T-shaped Joints

正 Joints are the most prevalent minor fractures in rocks. At first they may give us an impression of being hazardous in arrangement, but careful observation often shows that they really occur according to a certain rule. They are important in tectonic study, because through them the stress-orientati     

A Note on Extension Variances in IR2

Matheron (1971) proposed an approximation of the extension variance in IR. We propose in this note an extension of this formula in IR ² , based on a MacLaurin formula. Its application is shown in an example, the estimation of the maximum depressional storage of a soil surface.     

A Preliminary Note on Quaternary Glaciation in Western Hupeh

From what we have seen during an area excursion in Lushan under theguidance of Proressor J. S. Lee we are led to belief that glacial features anddeposits can hardly be absent from the high-lying areas in western Hupeh. Ourreasoning is based on the similarity of the prevailing physiographic conditions and     

A Third Note on the Statigraphical Unconformities in Kuangsi

正 As stratigraphical data regarding breaks between the Triassic formations have accumulated as a consequence of recent research in the southern and western parts of Kuangsi, the appearance of a third note on the stratigraphical unconformities in this province to supplement previous papers on the sub     

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.     

A Study of Robust Estimators

Using the compiled analytical data for the constituents of four Canadian iron-formation reference samples, a comparison is made among nine different "robust" estimators of "true" value, including the median, median, Gastwirth median, trimean, quarter-trimmed and sixthtrimmed means and the three Hampel estimators. Individual estimators and combinations of them are compared by means of the summation and "iron-oxide compatibility" tests. Comparisons are also made with values derived by the "select laboratories" method.     

A Simple and Robust Lognormal Estimator

In an open pit mine, the selection of blocks for mill feed necessitates the use of a conditionally unbiased estimator not only to maximize profits, but also to predict precisely the grades at the mill. Estimation of blocks usually is done using a series of blasthole assays on a regular grid. In many instances, the blasthole grades show a lognormal-like distribution. This study examines an estimator based on the hypothesis of bilognormality between the true block grade and the estimate obtained using the blastholes. The properties of the estimator are established and the estimator is proven to be conditionally unbiased. It is almost as precise as the lognormal kriging estimator when the points are multilognormal. However, it is more precise than lognormal krigings when only univariate lognormality is present or when the distribution is not exactly lognormal. The estimator also is shown to be robust to errors in the specifications of the variogram model or of the expectation of Z. Contrary to lognormal krigings, the estimator does only a slight correction to the original estimate obtained using the blastholes assays.     

A Note on Extension Variances in IR2

Matheron (1971) proposed an approximation of the extension variance in IR. We propose in this note an extension of this formula in IR ² , based on a MacLaurin formula. Its application is shown in an example, the estimation of the maximum depressional storage of a soil surface.     

2004年全国岩石学与地球动力学研讨会纪实

“2004年全国岩石学与地球动力学研讨会”于2004年11月19～24日在海南省海口市成功举办。会议共收到摘要230余份,与会人员250余人,人员来自全国50余家地学单位,另有8家杂志社和出版社以及1个仪器厂家。会议特邀了9名代表大会发言,170余人分别在两个分会场发言。本次会议参加人数众多,云集了国内大批专家教授,包括9名院士,充分展现了近年来高水平的研究新成果;会内会外讨论热烈,学术气氛浓厚,与会者反映会议十分成功,收获很大。本次会议成功有以下几个特点：     

A Brief Note on the Chungitsun Phosphate Deposit,Kunyang,Yunnan

As a consequence of the evergrowing attention paid to the mineral resourcesof Yunnan,the Chungitsun~1 phosphate deposit becomes inspiring and attractiveto industrialists.Within the last two years probably more than five parties of     

一种多因子相关计算方法

采用分析与图解相结合的方法解决多因子非线性相关计算,求回归方程问题。先从多个因子中找出对倚变量影响最大的一个因子,作单相关曲线,求残差;再从余下的因子中找出对残差影响最大的因子,作相关曲线,求残差,直到剩余各因子对残差影响相当;最后用剩余因子线性分解残差,从而求得多因子非线性回归方程。     

Note on boundary effects on stream meandering

This paper provides some additional evidence supporting the necessary (but insufficient) condition for the formation of stream meandering proposed by Nakagawa: where M_e is a non-dimensional parameter, τ_s, and τ_b are the average bank and bed shear stress respectively, p_s and p_b are the average bank and bed wetted perimeter of a half-channel respectively, and K?0.2 is the critical value of the parameter estimated from experimental data. Provided the criterion is satisfied, the main thread of the stream meanders in the non-erodible channel, and the maximum amplitude a_max, the angle a between the channel central axis and oblique crest line of the surface wave, and the mean wavelength L of the main thread decrease as the non-dimensional parameter Me increases.