重力卫星加速度数据的插值算法研究

阐述拉格朗日插值法和牛顿插值法的原理,详细地推导其一次、二次微分通用公式,并分别利用推导出来的微分公式对卫星星历模拟数据进行了相应的插值计算,得到对应的卫星加速度数据。通过比较不同公式的计算结果,验证一次、二次微分公式的正确性,并得到一些有益的结论。     

太阳光压模型中的偏导数

用矢量微分推导出光压摄动加速度对卫星位置矢量的偏导数公式，并用数值微分方法检验了公式的正确性。     

不同椭球变换的物理方法

本文采用物理方法讨论不同椭球变换的问题，论证了重力位低阶球谐系数与大地坐标系转换参数的关系，说明几何法转换模型（即大地坐标微分公式）不过是取至二阶次的谐函微分公式，为了提高转换精度，应该采用较高阶次的物理法微分公式（即球面正交多项式）作为不同系统的转换模型。     

四种大地线微分公式的研究(续)

（3）博德米累尔公式中的方向改化和距离改化博德米累尔公式是应用了正形表象的方法,因此,不论由第一个椭圆体表象到第二个椭圆体（第二类微分公式）。还是椭圆体面表象到自身（第一类微分公式）,都必须研究如何将连接表象二端点的大地线改化为原来的大地线的方向和距离,也就是必须研究它的方向改化和距离改化问题。现就二类微分公式,分述如下：     

一种用于计算卫星速度及加速度的实用数值微分公式

卫星重力数据处理中，通常由卫星星历观测值或星历扰动量计算卫星的加速度或加速度的扰动量，本文根据牛顿插值公式，导出了由n个点的卫星星历计算卫星速度及加速度的实用数值微分公式，文章还给出了N阶数值微分通用公式，通过模拟计算，对本文给出的公式进行了验证，并得出了一些有益的结论。     

平面坐标系统转换的微分方法

本文利用微分算子的方法，提出了坐标系统转换的几种实用公式，即：改进幂级数公式，台劳级数公式，牛顿二项公式及应用范围。     

四种大地线微分公式的研究

本文对四种地图投影法微分公式（平展法）作了详细的分析和比较,这四种公式郎黑尔默特公式、赫里斯托夫公式、博德米累尔公式和奥凯夫公式（奥凯夫的等参数法）,并找出它们之间的内部联系,然后指出了所有这些公式是最一般的公式,即赫里斯托夫公式的特殊形式．在推导博德米累尔方法中的方向改化和距离改化,以及赫里斯托夫方法中的长度畸变和角度畸变时,本文利用了奥凯夫的等参数法．奥凯夫应用了张量和流形推导出他的由于等参数法所引起的长度畸变公式．木文利用非常简单但却严格的方法来研究等参数法的理论根据和它的精度．最后本文还对这四种微分公式作了评价．     

重力场的曲率和挠率

本文介绍了重力场微分几何学的若干基本概念。将欧氏空间微分几何推广到重力空间,应用广义Frenet—Serret公式定义了重力场的曲率和挠率,导出了相应的表达式。讨论了重力场的零曲率和零挠率概念及其轨迹的几何特性。     

三维基线向量与大地坐标差间的微分公式及其应用

本文导出了以直角坐标差表示的三维基线向量(△Y X△ △Z)~T与其两端点之间的大地坐标差(△B △L △H)~T 间的微分公式。讨论了这一公式在处理GPS基线向量的坐标系统转换,某些量间的微分结构研究,误差传播,以及GPS基线向量的协方差阵到高斯平面二维向量的协方差阵转换上的应用。     

平面直角坐标与空间直角坐标间的协方差转换

推导出由大地坐标转换为高斯平面直角坐标严密的全微分公式,导出由高斯平面直角坐标向大地坐标再由大地坐标向空间直角坐标严密的协方差转换公式,并在此基础上,导出直接由高斯平面直角坐标向空间直角坐标近似的协方差转换公式,且举例说明其转换过程。结果表明,两种转换方法是等价的。     

利用GPS信号源进行干涉法测量

利用GPS卫星信号源进行干涉测量是近几年发展起来的新技术。本文叙述了GPS的信号结构及相关分析,提出了利用卫星信号源的三种干涉测量方法,给出了数学模型及有关公式,并进行了定性分析和比较。     

Effects of topographic–isostatic masses on gravitational functionals at the Earth’s surface and at airborne and satellite altitudes

Topographic–isostatic masses represent an important source of gravity field information, especially in the high-frequency band, even if the detailed mass-density distribution inside the topographic masses is unknown. If this information is used within a remove-restore procedure, then the instability problems in downward continuation of gravity observations from aircraft or satellite altitudes can be reduced. In this article, integral formulae are derived for determination of gravitational effects of topographic–isostatic masses on the first- and second-order derivatives of the gravitational potential for three topographic–isostatic models. The application of these formulas is useful for airborne gravimetry/gradiometry and satellite gravity gradiometry. The formulas are presented in spherical approximation by separating the 3D integration in an analytical integration in the radial direction and 2D integration over the mean sphere. Therefore, spherical volume elements can be considered as being approximated by mass-lines located at the centre of the discretization compartments (the mass of the tesseroid is condensed mathematically along its vertical axis). The errors of this approximation are investigated for the second-order derivatives of the topographic–isostatic gravitational potential in the vicinity of the Earth’s surface. The formulas are then applied to various scenarios of airborne gravimetry/gradiometry and satellite gradiometry. The components of the gravitational vector at aircraft altitudes of 4 and 10 km have been determined, as well as the gravitational tensor components at a satellite altitude of 250 km envisaged for the forthcoming GOCE (gravity field and steady-state ocean-circulation explorer) mission. The numerical computations are based on digital elevation models with a 5-arc-minute resolution for satellite gravity gradiometry and 1-arc-minute resolution for airborne gravity/gradiometry.     

Spherical integral formulas for upward/downward continuation of gravitational gradients onto gravitational gradients

New integral formulas for upward/downward continuation of gravitational gradients onto gravitational gradients are derived in this article. They provide more options for continuation of gravitational gradient combinations and extend available mathematical apparatus formulated for this purpose up to now. The starting point represents the analytical solution of the spherical gradiometric boundary value problem in the spatial domain. Applying corresponding differential operators on the analytical solution of the spherical gradiometric boundary value problem, a total of 18 integral formulas are provided. Spatial and spectral forms of isotropic kernels are given and their behaviour for parameters of a GOCE-like satellite is investigated. Correctness of the new integral formulas and the isotropic kernels is tested in a closed-loop simulation. The derived integral formulas and the isotropic kernels form a theoretical basis for validation purposes and geophysical applications of satellite gradiometric data as provided currently by the GOCE mission. They also extend the well-known Meissl scheme.     

Integral formulas for geopotential coefficient determination from gravity anomalies versus gravity disturbances

Integral formulas are derived for the determination of geopotential coefficients from gravity anomalies and gravity disturbances over the surface of the Earth. First order topographic corrections to spherical formulas are presented. In addition new integral formulas are derived for the determination of the external gravity field from surface gravity. Taking advantage of modern satellite positioning techniques, it is suggested that, in general, the external gravity field as well as individual coefficients are better determined from gravity disturbances than from gravity anomalies.     

卫星立体定位的仿真分析

本文从共线条件方程和前方交会的几何条件出发,推导了异轨和同轨模式下卫星立体定位的精度计算公式,仿真分析了轨道和姿态参数测量精度对卫星测绘能力的影响。实验结果表明,本文推导的精度计算公式正确合理。     

卫星信号转发系统的研究

卫星信号转发系统是将卫星信号转发至被遮挡区域的设备,当前市售的卫星信号转发系统存在输出功率明显高于卫星落地信号的“过度转发”现象。提出了采用可变增益放大器、赋形转发天线和分路器相结合的方法实现较大区域的“真实转发”的途径,并给出了设计原则和图表,对“真实转发”要注意的用户接收天线角度的设置、转发系统的标校等主要问题进行了探讨。     

一类球谐函数与三角函数乘积积分的计算

本文根据球谐函数的跨次递推公式和三角函数的性质,详细推导了在重力梯度调和分析中出现的一类球谐函数积分的跨次递推公式和递推初始值的计算公式。数值试验表明,球谐函数跨次递推算法具有快速、稳定的优点。该类积分的跨次递推实现,为卫星重力梯度调和分析奠定了算法基础。     

大地测量应用卫星的轨道设计 ——椭球谐引力场下卫星的运动

本文首先注意到近球面包络进动轨道(近圆)是大地测量应用卫星的共有特征;接着指出地球引力位的等势面非常接近共焦椭球面,其主项是严格的共焦椭球面,次项是10-6的摄动项;最后得出,在共焦椭球面成层分布引力场下,存在一条特殊的轨道——椭球面包络进动轨道,位于椭球面上,沿着大地线(测地线)作匀速、变曲率、均匀进动(升交点西退或东进)运动;这样一条特殊的轨道,是几何大地测量学、物理大地测量学和空间大地测量学与轨道理论的完美结合;若用椭球面包络轨道代替近球面包络进动轨道,因其严格顾及J2项,将更有利于大地测量应用卫星的轨道设计、卫星群分布设计、覆盖计算、入轨控制等。     

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.