四种改进积分法的低空扰动引力计算

针对Stokes积分方法计算扰动引力中计算点从空中趋近地面时存在积分奇异和不连续的问题,该文提出了去中央奇异点法、奇异点积分值修正法、中央格网加密算法和改进积分式法4种改进Stokes积分的计算公式,并进行了实验计算。计算结果表明:近地空间范围内,4种改进算法都能在一定程度上改进原始积分的奇异性问题;相同条件下,奇异点积分值修正法和改进积分式法计算精度最高,适宜于低空计算;改进积分式法通过理论推导,得到了从球外部到球面统一、连续且无奇异的改进Stokes积分公式,理论严谨。     

A new integral approach to geopotential coefficient determinations from terrestrial gravity or satellite altimetry data

The spherical harmonic coefficients of the Earth’s gravitational potential are conveniently determined by integration of gravity data or potential data (derived from satellite altimetry) over a sphere. The major problem of such a method is that the data, given on the non-spherical surface of the Earth, must be reduced to the sphere. A new integral formula over the surface of the Earth is derived. With this formula improved first order topographic corrections to the spherical formulas are obtained.     

The Abel-Poisson kernel and the Abel-Poisson integral in a moving tangent space

The upward-downward continuation of a harmonic function like the gravitational potential is conventionally based on the direct-inverse Abel-Poisson integral with respect to a sphere of reference. Here we aim at an error estimation of the “planar approximation” of the Abel-Poisson kernel, which is often used due to its convolution form. Such a convolution form is a prerequisite to applying fast Fourier transformation techniques. By means of an oblique azimuthal map projection / projection onto the local tangent plane at an evaluation point of the reference sphere of type “equiareal” we arrive at a rigorous transformation of the Abel-Poisson kernel/Abel-Poisson integral in a convolution form. As soon as we expand the “equiareal” Abel-Poisson kernel/Abel-Poisson integral we gain the “planar approximation”. The differences between the exact Abel-Poisson kernel of type “equiareal” and the “planar approximation” are plotted and tabulated. Six configurations are studied in detail in order to document the error budget, which varies from 0.1% for points at a spherical height H=10km above the terrestrial reference sphere up to 98% for points at a spherical height H = 6.3×10⁶km. Received: 18 March 1997 / Accepted: 19 January 1998     

确定外部扰动重力场的改进直接积分方法

针对Stokes-Pizzetti积分用于外部扰动重力场计算中从空中趋近地面时存在着不连续和积分奇异的问题,对该式进行了改进。改进式引入地面计算点处的重力异常,得到一个从地面到空中统一适用的公式,并且中和了在地面计算点处的奇异性。类似地,改进了的Stokes公式在用于大地水准面计算时积分的奇异性同样起到了改善作用。     

Global optimization of the Gauss conformal mappings of an ellipsoid to a sphere

The Gauss conformal mappings (GCMs) of an oblate ellipsoid of revolution to a sphere are those that transform the meridians into meridians, and the parallels into parallels of the sphere. The infinitesimal-scale function associated with these mappings depends on the geodetic latitude and contains three parameters, including the radius of the sphere. Gauss derived these constants by imposing local optimum conditions on certain parallel. We deal with the problem of finding the constants to minimize the Chebyshev or maximum norm of the logarithm of the infinitesimal-scale function on a given ellipsoidal segment (the region contained between two parallels). We show how to solve this minimax problem using the intrinsic function fminsearch of Matlab. For a particular ellipsoidal segment, we get the solution and show the alternation property characteristic of best Chebyshev approximations. For a pair of points relatively close in the ellipsoid at different latitudes, the best minimax GCM on the segment defined by these points is used to approximate the geodesic distance between them by the spherical distance between their projections on the corresponding sphere. This approach, combined with the best locally GCM if the points are on the same parallel, is illustrated by applying it to some case studies but specially to a 10° × 10° region contained between portions of two parallels and two meridians. In this case, the maximum absolute error of this spherical approximation is equal to 2.9 mm occurring at a distance about 1,360 km. This error decreases up to 0.94 mm on an 8° × 8° region of this type. So, the spherical approximation to the solution of the inverse geodesic problem by best GCM can be acceptable in many practical geodetic activities.     

地球椭球向径和平均曲率半径的积分表达式

引入地球向径积分平均值和地球平均曲率半径积分平均值的概念,借助计算机代数系统推导出了两者的符号表达式,并将它们表示为偏心率e的幂级数形式。将地球向径积分平均值和地球平均曲率半径积分平均值分别与平均球半径、等面积球半径、等距离球半径、等体积球半径这4种常用球体半径进行比较,研究表明地球向径积分平均值与4种常用球体半径间的差异更小。由于地球是一个旋转椭球体,向径与曲率半径是背离的,向径最大时,曲率半径最小, 向径最小时,曲率半径最大,传统思维所认为的曲率半径并不能准确地代表地球半径平均值,因此在一定程度上,地球向径的积分平均值更能代表地球半径平均值。这些研究结果可为地球科学、空间科学、导航定位提供基础理论依据。     

佘山25 m与13 m射电望远镜互掩问题分析

应实际工程需要,本文分析了有限场地内两射电望远镜的互掩问题,提出了地平式（方位俯仰座架）射电望远镜外接球面的概念,给出了球冠缺面积的数值积分计算方法,并具体分析和计算了佘山25 m与13 m射电望远镜的互掩,对于工程设计阶段的站点位置选取、塔基高度设计等具有理论参考意义。分析方法也可供类似工程实践参考。     

用微分方法求解不定积分问题

微分和积分是一对逆运算，但是微分比积分更容易运算。用常规方法求解不定积分问题是相当困难的。本文给出求不定积分的一种新方法。该方法易掌握，特别是使用该方法对无理函数积分更容易求得结果。     

Local Multiscale Modelling of Geoid Undulations from Deflections of the Vertical

This paper deals with the problem of determining a scalar spherical field from its surface gradient, i.e., the modelling of geoid undulations from deflections of the vertical. Essential tools are integral formulae on the sphere based on Green’s function of the Beltrami operator. The determination of geoid undulations from deflections of the vertical is formulated as multiscale procedure involving scale-dependent regularized versions of the surface gradient of Green’s function. An advantage of the presented approach is that the multiscale method is based on locally supported wavelets. In consequence, local modelling of geoid undulations are calculable from locally available deflections of the vertical     

基于物体成像轮廓的视觉测量与重建

提出一种基于物体的数学模型与成像轮廓进行物体3维视觉测量与重建的方法。该方法主要针对具有规则几何形状的物体,根据被测物体的数学描述和被测物体的轮廓与其在影像上的投影二者的位置关系对物体进行精确的3维测量与重建,得到几何体的空间位置与尺寸参数。对工业零件的基本几何形状圆柱、圆球、立方体等进行实验,结果表明该方法是可行的。     

基于规则六面体的三维空间数据插值方法

地学现象本质上是分布于三维空间的数据场.利用有限的三维采样观测结果,通过三维空间内插是分析和模拟三维空间现象分布和变化特征的重要手段.本文利用规则六面体对三维空间进行离散,采用三维空间移动插值法,以六面体格网顶点为中心,以可变半径球形窗口搜索足够数量的邻近采样点,按距离权倒数内插函数计算待定点的属性.文中给出了该算法的具体实现以及插值结果的可视化表现,证明这种三维空间插值方法是行之有效的,且符合空间现象的空间分布特点.     

Integral formulas for heterogeneous data in physical geodesy

The least squares estimator is derived for a random stochastic process implied by one or two heterogeneous random stochastic processes on a sphere. The solution can be regarded as least squares collocation in the continuous case. When the method is applied in physical geodesy the statistical expectation is usually substituted by the global average and the method will then give the minimum mean squares errors of the estimated quantities. The solutions can also be considered as generalizations of the classical integral formulas in physical geodesy for heterogeneous data information.     

轨道交通中线测设统一数学模型的建立

根据对常用的两种中线测设数学模型的分析，从线路中线曲率半径的特点出发，提出了中线整体积分数学模型，并导出其计算公式。为了实现整体积分模型的计算，对龙贝格积分算法进行了改进，并用于武汉市轨道交通一号线一期工程的中线测设。比较和分析表明，从实用性和精度等方面来说，整体积分模型是目前轨道交通最优的中线测设模型。     

自适应运动结构特征的车载全景序列影像匹配方法

以运动结构特征为约束条件的序列影像匹配,是基于多变量核密度函数,采用非参数均值漂移方法估计最优局部运动相似性结构特征的过程。核密度函数的带宽大小决定了匹配方法的收敛速度和精度。本文提出了一种可变带宽的自适应运动结构特征的车载全景序列影像匹配方法。首先以采样点在空间域和光流域的局部空间结构定义自适应的带宽矩阵。采用局部光流特征向量的距离加权法,描述光流域上运动相似性结构特征的松弛扩散过程。然后给出自适应多变量核密度函数的表达形式,并探讨了均值漂移向量的求解、终止条件以及种子点的选择方法。最后融合多尺度SIFT描述特征与运动结构特征,建立统一的全景影像匹配框架。试验选择车载移动测量系统获取的城市球全景序列影像,结果表明在内点率变化、物方尺度变化等情况下,本文方法可以实现自适应运动结构特征的相似性度量,提高匹配的正确点数和匹配率,算法表现出较强的稳键性。     

On the inversion of geodetic integrals defined over the sphere using 1-D FFT

An iterative method is presented which performs inversion of integrals defined over the sphere. The method is based on one-dimensional fast Fourier transform (1-D FFT) inversion and is implemented with the projected Landweber technique, which is used to solve constrained least-squares problems reducing the associated 1-D cyclic-convolution error. The results obtained are as precise as the direct matrix inversion approach, but with better computational efficiency. A case study uses the inversion of Hotine’s integral to obtain gravity disturbances from geoid undulations. Numerical convergence is also analyzed and comparisons with respect to the direct matrix inversion method using conjugate gradient (CG) iteration are presented. Like the CG method, the number of iterations needed to get the optimum (i.e., small) error decreases as the measurement noise increases. Nevertheless, for discrete data given over a whole parallel band, the method can be applied directly without implementing the projected Landweber method, since no cyclic convolution error exists.     

A robust approach to global problems in physical geodesy

A technique for renormalization of integral equations is used for obtaining very robust solutions. The number of multiplications used for inverting the integral equations can be reduced dramatically and mostly only weighted means will be needed. Theoretical gain in computer time might be up to 100 000 times for the most favourable cases when using 1000 unknowns. (Practical gains will be considerably less.) Solutions have been obtained with increased accuracy compared to the classical technique over integral equations. Surface elements might be of arbitrary size but the method is optimal for a global approach with equal area elements. The solutions were found strictly invariant with respect to the depth to the internal sphere, when using simpler models. Applications in surveying are possible after some modifications. Renormalization of integral equation has been widely used in the theory of quantum field.     

Formulae for geodetic computations in height controlled space

Solutions are given for intersection, lateration, and the direct problem by a simple method which finds the common point of intersection of two planes and a sphere. The ways of determining a plane may vary, but the general procedure is the same for all cases. An alternative solution is given for the direct problem.     

利用卫星轨迹交叉点标定CHAMP卫星加速度计数据

基于能量守恒方程给出了利用卫星轨迹交叉点标定CHAMP卫星加速度数据的基本原理和方法 ,并给出了其严密的积分公式及其离散化形式。为了控制加速度数据的扰动异常 ,建议采用抗差估计求解参数值 ,并基于实测的CHAMP卫星加速度计数据进行了计算与比较 ,验证了该方法的有效性     

The convergence of harmonic reduction to an internal sphere

The boundary value problem of physical geodesy has been solved with the use of a harmonic reduction down to an internal sphere using a discrete procedure. (For gravity cf. Bjerhammar 1964 and for the potential cf. Bjerhammar 1968). This was a finite-dimensional approach mostly with one-to-one correspondence between observations and unknowns on the sphere. Earlier studies were made with the use of surface elements (on the sphere) with constantgravity. Integration over the surface elements was replaced by a discrete approach with the use of the distance to a point in the centre of the surface element. See Bjerhammar (1968) and (1969). This approach was later presented as a “reflexive prediction” technique for a weakly stationary stochastic process. Bjerhammar (1974, 1976). Krarup (1969) minimized the L²-norm of the potential on the internal sphere. It will here be proved that the two solutions are identical for a proper choice of the radii of the internal spheres. The proof is given for a spherical earth with selected choice of “carrier points”. The convergence problem is discussed. The L²-norm solution is found convergent for the fully harmonic case. Uniform convergence is obtained in the non-harmonic case with the use of the original procedure applied in accordance with the theorems of Keldych-Lavrentieff and Yamabe.     

GPS水准拟合模型的选取与精度估计

GPS水准拟合中,多项式曲面模型法使用率较高,但通常要人工选取已知点和检核点,带有一定的人为因素。本文在分析检核点作用的基础上,提出了一种尽量减少人工干预选取模型系数的方法,并将该法作为探测测区高程异常是否呈单一曲面的基础,给出了一般测区选取GPS水准拟合模型的步骤。此外,就用户所关心的转换正常高精度估计问题,讨论了其可能达到的精度。