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Mathematical Frontier Research as Conceived in Space-Geodesy

空间大地测量学中的数学前沿研究

基本信息

批准号：
16540386
负责人：
XU Pl
金额：
$ 2.37万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540386/
关键词：
Space Geodesy Mixed Integer Linear Models Gravity Field Ocean Bottom Deformation Optimization Integer Statistics Voronoi Cells Variance Components space geodesy mixed integer linear model variance components inverse problems probab

项目摘要

The major new results of this research project can be briefly summarized as follows : (i) We systematically discuss the estimation and hypothesis testing problems in a mixed integer linear model, which directly motivated by space geodesy and is the basis for precise GPS positioning. We have first mathematically solved the problem of constructing Voronoi cells for an arbitrary positive definite matrix associated with mixed integer linear models by using interval mathematics to eliminate an infinite number of redundant linear constraints. We have also provided spherical and ellipsoidal enclosures to bound the Voronoi cell. By re-formulating the problem of bounding the Voronoi cell as a mathematical programming model, we have successfully found the tightest possible bounds of the Voronoi cell from inside and outside, respectively. As a result, we are able to find the tightest possible probabilistic bounds. The error bound has been shown to perform much better than the one proposed by Shan … More non (1959). In fact, the hypothesis testing problem is, for the first time, addressed in this research project and published in IEEE Transactions on Information Theory. By combining GPS with acoustic technology, we have first shown that ocean-bottom crustal deformation could be measured at the accuracy of sub-centimeters in all the three components. This is also the first theoretical demonstration that ocean bottom crustal deformation could be measured as easily and routinely as on land ; (ii) We have proposed a new robust estimation method, which is named as sign-constrained robust least squares. The method has been shown to be able to tolerate up to 50 percent of outliers, and even much more than 50 percent when used iteratively. As a by-product, we have proved that neither the plus sign nor the square is responsible for the non-robustness of least squares, as otherwise has been claimed in the literature on robust estimation. We have further proved that both Huber's M-estimation and weighted L1 method are not robust if measurements are of different accuracy ; (iii) We have first proposed the bias-corrected estimation technique to estimate variance components in linear inverse ill-posed models and applied it to satellite gravity fields, together with our multiple parameter regularization method. We have also proved that not all the variance covariance components are estimable. Less

本研究项目的主要新成果可以简要概括如下：（i）系统地讨论了混合整数线性模型中的估计和假设检验问题，该模型直接受空间大地测量的启发，是GPS精确定位的基础。我们首先在数学上解决了与混合整数线性模型相关联的任意正定矩阵的Voronoi单元的构造问题，通过使用区间数学来消除无穷多个冗余线性约束。我们还提供了球形和椭球形的外壳来约束Voronoi细胞。通过将Voronoi单元的边界问题重新表述为数学规划模型，我们成功地分别从内部和外部找到了Voronoi单元的最紧可能边界。因此，我们能够找到最严格的可能概率界限。该误差界比Shan提出的误差界有更好的性能 ...更多信息无（1959年）。事实上，假设检验问题首次在该研究项目中得到解决，并发表在《IEEE Transactions on Information Theory》上。通过将全球定位系统与声学技术相结合，我们首次表明，海底地壳变形的所有三个组成部分都可以以亚厘米的精度测量。这也是第一次从理论上证明，海底地壳形变可以像在陆地上一样容易和常规地测量;（ii）我们提出了一种新的鲁棒估计方法，它被命名为符号约束鲁棒最小二乘。该方法已被证明能够容忍高达50%的离群值，当迭代使用时甚至超过50%。作为一个副产品，我们已经证明，无论是加号，也不是平方负责的最小二乘的非鲁棒性，否则已声称在文献中的鲁棒估计。我们进一步证明了Huber的M-估计和加权L1方法是不稳健的，如果测量是不同的精度;（iii）我们首先提出了偏差校正估计技术估计方差分量的线性逆不适定模型，并将其应用到卫星重力场，连同我们的多参数正则化方法。我们还证明了并非所有的方差协方差分量都是可估计的。少