Robust estimation and optimal selection of polynomial parameters for the interpolation of GPS geoid heights

The polynomial interpolation of least squares and interpolation moving least squares based on control stations with known GPS (global positioning system) ellipsoidal heights and levelling orthometric heights are the most often used methods for the interpolation of the geoid heights. But in their applications there occur two problems: one lies in selecting the suitable polynomial parameters; the other in reducing the influences of some possibly abnormal data points. To solve both of the problems, without emphasizing a sound theoretical basis, a heuristic solution with the help of robust estimation technique and optimization criteria for the regression equation is presented. Through two actual numerical examples it is shown that the new solution concept is efficient and can be realized easily on computers. Received: 23 May 1996 / Accepted: 27 March 1997     

基于多面核函数配置型模型的参数估计

针对同时出现非随机参数、已测点和未测点信号的配置型模型精确建立协方差函数的困难，在研究和应用最小二乘配置和多面函数拟合法的基础上，将两法综合，提出了多面核函数配置法，导出了这种平差方法的解算公式。并与最小二乘配置、多面函数拟合作了比较分析。     

给定内插重力异常精度时对重力数据的要求

基于最小二乘配置误差估计公式,建立了重力异常格网数据的分辨率和精度与重力异常内插值精度的关系,提出了在给定插值精度时反推已知格网数据的分辨率和精度的方法。以EGM2008重力场模型为例,在不同分辨率和精度条件下进行重力异常插值实验。实验结果与本文方法的计算结果基本一致,表明该方法具有一定的可行性。     

最小二乘配置法在GPS高程异常推估中的应用

简述了最小二乘配置法的基本原理,并在水准联测点不多的情况下,直接采用平方根函数作为各随机参数间的协方差函数用于推估GPS高程异常值。由于最小二乘配置法的函数模型中同时考虑了非随机变量和随机变量,使得高程异常值的推估精度更高。实例分析也证明了此模型在精度上优于传统的平面拟合模型和协方差推估模型。残差分析表明此方法更适合于同时存在内插和外推高程异常值的情况。     

Evaluation of deflections of the vertical using topographic-isostatic and astrogeodetic data for the area of Greece

The evaluation of deflections of the vertical for the area of Greece is attempted using a combination of topographic and astrogeodetic data. Tests carried out in the area bounded by 35°≤ϕ≤42°, 19°≤λ≤27° indicate that an accuracy of ±3″.3 can be obtained in this area for the meridian and prime vertical deflection components when high resolution topographic data in the immediate vicinity of computation points are used, combined with high degree spherical harmonic expansions of the geopotential and isostatic reduction potential. This accuracy is about 25% better than the corresponding topographic-Moho deflection components which are evaluated using topographic and Moho data up to 120 km around each station, without any combination with the spherical harmonic expansion of the geopotential or isostatic reduction potential. The accuracy in both cases is increased to about 2″.6 when the astrogeodetic data available in the area mentioned above are used for the prediction of remaining values. Furthermore the estimation of datum-shift parameters is attempted using least squares collocation.     

Algorithm for Harmonic Tidal Analysis along T/P Tracks： Taking Differences of Observed Sea Surface Heights at Adjacent Points as Observations

An algorithm （differential mode） is presented for the improvement of harmonic tidal analysis along T/P tracks, in which the differences between the observed sea surface heights at adjacent points are taken as observations. Also, the observation equations are constrained with the results of the crossover analysis; the parameter estimations are performed at 0.1° latitude intervals by the least squares. Cycle 10 to 330 T/P altimeter data covering the China Sea and the Northwest Pacific Ocean （2°-50° N,105°-150° E） are adopted for a refined along-track harmonic tidal analysis, and harmonic constants of 12 constituents in 8 474 points are obtained, which indicates that the algorithm can efficiently remove non-tidal effects in the altimeter observations, and improve the precision of tide parameters. Moreover, parameters along altimetry tracks represent a smoother distribution than those obtained by traditional algorithms. The root mean squares of the fitting errors between the tidal height model and the observations reduce from 11 cm to 1.3 cm.     

基于最小二乘配置的九参数模型在三维坐标转换中的应用

运用传统的布尔莎七参数法进行三维坐标转换时,并未考虑到两坐标系统的公共点坐标误差及三个坐标轴缩放比例不均的问题。引入最小二乘配置法,将公共点误差作为随机信号处理,并增加两个尺度参数解决坐标轴缩放比例不均的问题;推导出基于最小二乘配置的九参数模型及其误差方程,通过实验对比分析表明,该方法可以有效地提高坐标转换精度。     

LS和TLS在平面坐标转换中的应用

在超定线性方程Ax=b的解算中,最小二乘(LS)只考虑观测向量b的误差,而总体最小二乘(TLS)则同时顾及观测向量b和系数矩阵A均含误差的情况。本文以六参数模型在平面坐标转换中的应用为例,分别采用LS和TLS进行模型参数的求解。结果表明,2种方法所求参数并无显著差异,但是总体最小二乘更好地改善了坐标转换的内部精度,是一种更为合理的计算方法。     

Least squares prediction and filtering in any dimensions using the principles of array algebra

This presentation is a follow-up of the paper “A Few Basic Principles and Techniques of Array Algebra”, published previously in theBulletin Géodésique. The attention is focused on the use of array algebra in the problem area of multilinear least squares prediction and filtering. The prediction mathematical models are treated using the concept of the covariance function and node points. In the latter part of the paper, efficient prediction formulas in two dimensions are developed and solved through the least squares filtering process, upon specializing the results derived previously for any dimensions. (Errata Sheet for the previous paper is added). Formerly of DBA Systems, Inc. P.O. Drawer 550 Melbourne, Florida 32901.     

利用整体最小二乘法建立线性回归模型

对于在实际应用中的直线回归问题,存在着因自变量和因变量选取不同拟合结果存在差异的情况,文中采用了一种线性拟合参数估计的新方法,即整体最小二乘法。文章在描述普通最小二乘和整体最小二乘原理的基础上,并对比其异同,并采用奇异值分解的方法来求解整体最小二乘问题。算例结果表明,采用整体最小二乘方法估计线性回归参数的精度明显高于常规最小二乘法,是一种值得借鉴的算法。     

融合正交几何信息的非线性等式约束整体最小二乘平差及迭代算法

正交距离最小二乘和加权整体最小二乘是解自变量含误差拟合问题的两种独立准则。加权整体最小二乘与正交距离最小二乘不同,它不考虑测量点与拟合点之间的连线垂直于拟合对象的几何信息,不能确保测量点到拟合对象的距离的平方和为极小值。针对该问题,本文将正交几何信息作为约束条件融入加权整体最小二乘,提出一种约束方程带有误差改正数的非线性等式约束整体最小二乘平差法。首先,把加权整体最小二乘平差的函数式看作是非线性方程,连同正交几何约束方程一并线性化,得到线性的平差函数方程;然后,采用拉格朗日乘数法推导其参数估计及精度评定公式,并给出迭代计算算法;最后,以平面直线拟合为例,对本文方法和计算算法进行验证。试验结果表明:①本文方法和算法具有可行性;②与加权最小二乘和加权整体最小二乘相比,本文方法计算的测量点到拟合直线的垂直距离平方和最小;③本文方法计算的测量点到拟合直线的距离与测量点到拟合点的距离相等。     

抗差趋势面与正交多面函数结合拟合DEM数据

空间数据内插是DEM建模中不可缺少的环节。本文提出将趋势面与多面函数结合拟合DEM数据。在趋势面拟合过程中,提出将局部有显著起伏的采样点视为异常观测值,采用抗差估计法,控制显著异常点对曲面拟合的不利影响;针对多面函数拟合中,难以合理地选择节点这一问题,提出基于正交最小二乘法的基本思想,计算各节点核函数对曲面拟合贡献的大小,并依此为依据自适应地选择节点。最后利用新提出的方法,对野外实测的地形数据进行拟合,结果表明进行趋势面拟合后,再进行多面函数拟合,整体上优于直接利用多面函数拟合,而基于正交最小二乘法的多面函数拟合法的精度明显优于常规的多面函数拟合法。     

基于杨赤中推估法的降雨空间插值应用研究

杨赤中推估法是一种以二项系数加权游动平均为基础,按照迭代法几何滤波过程建立推估数学模型的最小二乘估值方法,它能够基于少量已知数据点取得好的插值效果。针对有限的降雨观测站点和分布不规则的雨量特征点数据,本文基于杨赤中滤波与推估法空间插值,以湖南省降雨观测站点数据为实例数据,实例研究了建立降雨估值数学模型和实现插值计算的过程。首先,按照逐步增大游动半径进行滤波建立杨赤中特征函数,选用负幂指数函数模型拟合特征函数,建立降雨估值数学模型;然后,设计了杨赤中空间插值计算的自动化流程,实现降雨观测变量规律性变化的最优估值,并构建出研究区连续分布的降雨空间信息;最后,将其与TRMM(Tropical Rainfall Measuring Mission)和DEM(Digital Elevation Model)数据作为协变量进行协同克里金插值的结果进行对比验证。结果表明:杨赤中滤波与推估法在不需要其他协变量的情况下,可以取得较好的插值模拟结果,证实了该方法针对少量且不规则分布的降雨观测点数据插值的可行性与有效性。     

A few basic principles and techniques of array algebra

This paper is intended to demonstrate the usefulness of array algebra techniques in certain multilinear least squares problems. A typical restriction of array algebra is the need for a gridded observational structure; however, the grid does not have to be uniform and in general is not limited to any particular coordinate system nor to two- or three-dimensional spaces. Another restriction comes to light when dealing with weighted multilinear least squares adjustments. The a—priori variance-covariance matrix cannot be completely arbitrary but must be expressible in terms of certain matrix products. There exist various practical ways (not discussed herein) to bridge these restrictions. The reward for using the array algebra technique when it is appropriate lies in the great computational savings. From the theoretical point of view, the backbone of most derivations are the “R-matrix multiplications” and a simple tool, demonstrated herein, called “fundamental transformation”. It follows that the least squares solution of “array observation equations” does not have to be sought by some new and complex mathematical means. The fundamental transformation allows such an adjustment problem to be rewritten in a conventional (monolinear) form; the familiar least squares solution is then written down and transformed back to the array form using the same tool. The statistical properties of the results (e.g. minimum variance) are known from the conventional approach and do not have to be rederived in the array case.     

航空重力测量数据向下延拓的改进Poisson积分迭代法

目前,航空重力测量是快速获取陆地和近海区域高精度、高分辨率重力场信息的非常有效的技术手段,向下延拓则是其数据处理中的关键环节,直接影响到测量结果的进一步应用。本文在对传统最小二乘法、改进最小二乘法、Tikhonov正则化法等延拓模型进行数值分析的基础上,根据调和函数的基本特性,提出并建立了Poisson积分迭代法和改进Poisson积分迭代法延拓模型。实测航空和地面重力测量数据的试验结果表明,本文新建的Poisson积分迭代法和改进Poisson积分迭代法延拓模型精度相当,比传统最小二乘法延拓模型精度提高了15.26 mGal,比改进最小二乘法延拓模型精度提高了0.21 mGal,比Tikhonov正则化法延拓模型精度略低0.13 mGal,从而证明了本文所建模型的正确性和有效性。     

Sparse matrix algorithms applied to dem generation

A method for generating a digital elevation model (DEM) was suggested byEbner. The heights of grid points are interpolated from arbitrarily distributed reference points using the finite element method. The requested grid heights of the DEM are defined as unknowns and estimated from the available reference points and a general curvature minimization of the interpolation surface with filtering at the reference points. This problem can be interpreted as an adjustment of indirect observations and can be solved using the least squares method. This leads to a banded structured system of normal equations in case of numbering the unknowns in a row-wise order. This ordering may not be the best strategy for the solution of the normal equations, because the band contains many zero elements. Therefore algorithms which exploit the nonzero/zero structure to better advantage have been applied in combination with the pivot strategies of “Nested Dissection” and “Minimum Degree Ordering” (OPTORD). The operational characteristics of these algorithms are compared with those of the band-algorithm for several DEM problems that vary in size to over 4000 unknowns. The result shows that the number of nonzeros and the number of essential operations can be reduced drastically when using sparse algorithms, but that because of the very high bookkeeping expenses, band algorithms are to be preferred.     

倾斜影像自适应初始物方面元优化匹配算法

采用基于物方面元的最小二乘影像匹配方法匹配倾斜影像时,常出现深度不连续或高差较大区域影像连接点度数低或空三点过少问题。针对此问题,本文提出一种基于自适应初始物方面元的倾斜影像匹配算法。算法利用倾斜影像已有的初始内外方位元素及匹配过程中产生的物点信息,采用多片前方交会和物方差分的方法自适应计算物方面元的高程及法向量方向角初值,进而解决采用物方面元最小二乘影像匹配方法匹配地物高差较大区域的倾斜影像时,因初值不准导致在像方匹配同名点困难的问题。分别采用本文算法和物方面元初始状态为水平面元的最小二乘影像匹配方法对两组倾斜影像进行对比匹配验证。试验结果证明了本文算法的有效性。     

局部重力场最小二乘配置通用表示技术

在分析局部重力场最小二乘配置法技术特点的基础上，推导出一种能综合多种类型、不同高度重力场元经验协方差函数的通用表达方法，以期实现局部重力场元的内插、外推、延拓或其他不同高度的重力场元估计一体化。分析了最小二乘配置技术的一些性能以及算法实现中应注意的问题。     

地图投影的最小二乘二元多项式拟合的误差估计

探讨了在地图投影的最小二乘二元多项式拟合中参考点的分布对拟合误差的影响,提出了一种基于矩阵的秩亏的方法来判断参考点是否在指定次数的二元多项式空间的代数曲线上,并导出在极小范数最小二乘意义下的拟合多项式的误差估计式,公式表明拟合误差和参考点的最小二乘误差无关。     

三次样条在北斗卫星轨道插值中的应用

本文选用三次样条函数对北斗高轨卫星的轨道数据进行插值计算,结果表明,三次样条函数基本没有龙格现象,但插值模型中含有明显的随机性系统差,因此本文采用附加系统差的最小二乘配置,对三次样条的插值结果进行处理,使精度提高了10倍左右。