航空重力梯度数据的模拟研究

针对提高模拟航空重力梯度的精度研究不足的现状,文章运用地球重力场模型构建了计算单点的扰动重力梯度分量的数学模型,推导了Legendre函数及其导数的无奇异性递推计算公式,克服了计算高阶次Legendre函数Pnm(cosθ)的下溢问题,随之对数学模型进行优化;最后利用EGM2008地球位模型模拟生成4km飞行高度处的扰动重力梯度张量作为航空重力梯度观测值。模拟试算表明,该模型能够满足当前航空重力梯度测量的精度要求。     

Theory and Methods of the Earth＇s Gravity Field Model Recovery from GOCE Data

研究GOCE卫星测量恢复地球重力场模型的理论与方法。论文的主要工作和创新点有： （1）建立扰动重力梯度张量各分量没有奇异性的详细计算模型,解决重力梯度张量Txx分量在两极地区计算的奇异性难题。     

缔合Legendre函数二阶导数的快速稳定递推算法

根据初等代数的基本原理,推导了一种缔合Legendre函数二阶导数的快速稳定递推算法。数值测试结果表明,在阶次高达3 600时,该方法与其他几种现有方法的计算精度相当,但计算效率比其他方法提高了一倍以上,并且该方法没有奇异性,适用于快速精确地计算任意纬度的缔合Legendre函数二阶导数值。     

GOCE卫星测量恢复地球重力场模型的理论与方法

研究了GOCE卫星测量恢复地球重力场模型的理论与方法。论文的主要工作和创新点有： (1) 建立了扰动重力梯度张量各分量没有奇异性的详细计算模型,解决了重力梯度张量Txx分量在两极地区计算的奇异性难题。 (2) 系统研究了卫星重力梯度数据向下延拓的解析法、泊松积分迭代法和卫星重力梯度数据格网化的移动平均法、反距离加权法、普通克里金法,建立了相应的数学模型,导出了相应的计算公式,并采用“直接法”和“移去-恢复法”两种方案对其向下延拓和格网化效果进行了测试。 (3) 分析了能量守恒方程中各项误差对沿轨扰动位计算结果的影响,建立了利用GOCE模拟数据确定地球重力场的最小二乘直接法、调和分析法、最小二乘配置法的实用数学模型,并做了大量的模拟计算。 (4) 建立了利用扰动引力梯度张量各单分量和组合分量确定地球重力场的最小二乘直接法去奇异性计算模型;推导了利用扰动引力梯度张量单分量和组合分量解算地球重力场的调和分析法模型;进一步推导了扰动引力梯度张量各个分量之间的自协方差和互协方差函数及其与引力位系数之间协方差函数的具体计算公式。 (5) 推导了利用不同类型重力测量数据确定地球重力场的联合平差法数学模型,介绍并分析了模型中各类数据最优定权的参数协方差法和方差分量估计法。 (6) 论述了谱组合法的基本原理,给出了多种类型重力测量数据联合处理的谱权及谱组合的通用表达式,基于调和分析方法推导了SST+SGG、SST+SGG＋Δg和SST+SGG＋Δg+N恢复地球重力场模型的谱组合公式及对应谱权的具体形式。 (7) 推导了利用迭代法联合不同类型重力测量数据反演地球重力场模型的基本原理公式,并给出了其具体实现步骤。 (8) 分析并计算了重力卫星轨道高度、卫星星间距离和卫星轨道倾角的设计指标;讨论了双星轨道长半轴的一致性要求、双星姿态俯仰角的控制要求以及双星编队保持机动的时间间隔要求。 (9) 确定了KBR系统的星间距离、星间距离变化率和星间加速度的精度指标;设计了星载GPS系统的卫星轨道位置和速度以及加速度计测量的精度指标;计算了加速度计检验质量质心到卫星质心的调整距离精度指标;分析了恒星敏感器的姿态角测量精度和稳定度;计算了参考重力场模型对于累计大地水准面精度和积分卫星轨道的影响。 (10) 研制了一套利用卫星重力测量数据反演地球重力场模型的软件平台,可对卫星重力测量数据处理及其精度评估提供一些基本方法,并为我国卫星重力测量系统的总体战技指标和主要有效载荷技术指标的量化分析、论证提供理论和技术支持,为我国未来的卫星重力测量系统提供可能的积累和参考。     

联合高低卫-卫跟踪和卫星重力梯度数据恢复地球重力场的谱组合法

GOCE采用的高低卫-卫跟踪和卫星重力梯度测量技术在恢复重力场方面各有所长并互为补充,如何有效利用这两类观测数据最优确定地球重力场是GOCE重力场反演的关键问题。本文研究了联合高低卫-卫跟踪和卫星重力梯度数据恢复地球重力场的最小二乘谱组合法,基于球谐分析方法推导并建立了卫星轨道面扰动位T和径向重力梯度Tzz、以及扰动位T和重力梯度分量组合{Tzz-Txx-Tyy}的谱组合计算模型与误差估计公式。数值模拟结果表明,谱组合计算模型可以有效顾及各类数据的精度和频谱特性进行最优联合求解。采用61天GOCE实测数据反演的两个180阶次地球重力场模型WHU_GOCE_SC01S（扰动位和径向重力梯度数据求解）和WHU_GOCE_SC02S（扰动位和重力梯度分量组合数据求解）,结果显示后者精度优于前者,并且它们的整体精度优于GOCE时域解,而与GOCE空域解的精度接近,验证了谱组合法的可行性与有效性。     

完全正常化缔合Legendre函数及其导数计算的综合分析

地球重力场模型的建立和应用,其数值精度和稳定性主要取决于完全正常化的缔合Legendre函数及其导数的计算精度和稳定性.总结了各种常见的计算缔合Legendre函数及其导数的方法,对各种方法进行了数值测试,分析了其精度和效率情况,提出了改进缔合Legendre函数及其导数计算精度和效率的几点意见.     

重力梯度的空间传播特性

基于球谐函数谱分析理论，导出了重力梯度张量住全球平均意义下的功率谱表达式，从理论上研究和揭示了重力梯度的空间传播特性。利用重力异常阶方差模型和EGM96重力位模型，研究了苇力梯度张量和扰动重力梯度张量随高度的衰减特性，据此分析了卫星重力梯度测量所能恢复重力场的最高阶数，并讨论了对卫星重力梯度仪和卫星定轨的精度要求。     

重力梯度张量的球谐分析

深入研究了利用卫星重力梯度数据确定地球重力场模型的球谐分析方法，导出了由重力梯度张量的球函数展开系数确定扰动位球谐系数的实用解算模型。模拟试算结果验证了本文算法的有效性和实用性。     

重力梯度张量的球谐分析

深入研究了利用卫星重力梯度数据确定地球重量力场模型的球谐分析方法，导出了由重力梯度张量的球函数展开系数确定扰动位球谐系数的实用解算模型。模拟试算结果验证了本算法的有效性和实用性。     

广义非线性最小二乘测量参数平差的快速差分迭代解算

本文针对"数字化建设"中涉及的多类型、多精度、多源、动态非线性最小二乘测量参数平差问题,提出了一种新的解算模型。该模型通过构造适当的差分点列,避免了一阶或二阶导数计算,对于模型复杂,或者导数不存在的函数模型而言,具有计算量小、收敛速度快等特点。     

Non-Singular Expressions for the Gravity Gradients in the Local North-Oriented and Orbital Reference Frames

The conventional expansions of the gravity gradients in the local north-oriented reference frame have a complicated form, depending on the first- and second-order derivatives of the associated Legendre functions of the colatitude and containing factors which tend to infinity when approaching the poles. In the present paper, the general term of each of these series is transformed to a product of a geopotential coefficient and a sum of several adjacent Legendre functions of the colatitude multiplied by a function of the longitude. These transformations are performed on the basis of relations between the Legendre functions and their derivatives published by Ilk (1983). The second-order geopotential derivatives corresponding to the local orbital reference frame are presented as linear functions of the north-oriented gravity gradients. The new expansions for the latter are substituted into these functions. As a result, the orbital derivatives are also presented as series depending on the geopotential coefficients multiplied by sums of the Legendre functions whose coefficients depend on the longitude and the satellite track azimuth at an observation point. The derived expansions of the observables can be applied for constructing a geopotential model from the GOCE mission data by the time-wise and space-wise approaches. The numerical experiments demonstrate the correctness of the analytical formulas.An erratum to this article can be found at     

扰动重力梯度的球冠谐分析建模

从球冠谐理论出发,详细推导了球冠坐标系下扰动重力梯度的无奇异性计算公式。基于Tikhonov正则化方法,利用GOCE卫星实际观测数据解算局部重力场球冠谐模型。数值计算表明,基于扰动重力梯度的球冠谐分析建模方法能够有效地恢复局部重力场中的短波信号,与GO_CONS_GCF_2_DIR_R5模型的差异在±0.3×10~(-5) m/s~2水平。     

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.     

Basic equations for constructing geopotential models from the gravitational potential derivatives of the first and second orders in the terrestrial reference frame

This research represents a continuation of the investigation carried out in the paper of Petrovskaya and Vershkov (J Geod 84(3):165–178, 2010) where conventional spherical harmonic series are constructed for arbitrary order derivatives of the Earth gravitational potential in the terrestrial reference frame. The problem of converting the potential derivatives of the first and second orders into geopotential models is studied. Two kinds of basic equations for solving this problem are derived. The equations of the first kind represent new non-singular non-orthogonal series for the geopotential derivatives, which are constructed by means of transforming the intermediate expressions for these derivatives from the above-mentioned paper. In contrast to the spherical harmonic expansions, these alternative series directly depend on the geopotential coefficients ${\bar{{C}}_{n,m}}$ and ${\bar{{S}}_{n,m}}$ . Each term of the series for the first-order derivatives is represented by a sum of these coefficients, which are multiplied by linear combinations of at most two spherical harmonics. For the second-order derivatives, the geopotential coefficients are multiplied by linear combinations of at most three spherical harmonics. As compared to existing non-singular expressions for the geopotential derivatives, the new expressions have a more simple structure. They depend only on the conventional spherical harmonics and do not depend on the first- and second-order derivatives of the associated Legendre functions. The basic equations of the second kind are inferred from the linear equations, constructed in the cited paper, which express the coefficients of the spherical harmonic series for the first- and second-order derivatives in terms of the geopotential coefficients. These equations are converted into recurrent relations from which the coefficients ${\bar{{C}}_{n,m}}$ and ${\bar{{S}}_{n,m}}$ are determined on the basis of the spherical harmonic coefficients of each derivative. The latter coefficients can be estimated from the values of the geopotential derivatives by the quadrature formulas or the least-squares approach. The new expressions of two kinds can be applied for spherical harmonic synthesis and analysis. In particular, they might be incorporated in geopotential modeling on the basis of the orbit data from the CHAMP, GRACE and GOCE missions, and the gradiometry data from the GOCE mission.     

A comparison of different mass elements for use in gravity gradiometry

Topographic and isostatic mass anomalies affect the external gravity field of the Earth. Therefore, these effects also exist in the gravity gradients observed, e.g., by the satellite gravity gradiometry mission GOCE (Gravity and Steady-State Ocean Circulation Experiment). The downward continuation of the gravitational signals is rather difficult because of the high-frequency behaviour of the combined topographic and isostatic effects. Thus, it is preferable to smooth the gravity field by some topographic-isostatic reduction. In this paper the focus is on the modelling of masses in the space domain, which can be subdivided into different mass elements and evaluated with analytical, semi-analytical and numerical methods. Five alternative mass elements are reviewed and discussed: the tesseroid, the point mass, the prism, the mass layer and the mass line. The formulae for the potential, the attraction components and the Marussi tensor of second-order potential derivatives are provided. The formulae for different mass elements and computation methods are checked by assuming a synthetic topography of constant height over a spherical cap and the position of the computation point on the polar axis. For this special situation an exact analytical solution for the tesseroid exists and a comparison between the analytical solution of a spherical cap and the modelling of different mass elements is possible. A comparison of the computation times shows that modelling by tesseroids with different methods produces the most accurate results in an acceptable computation time. As a numerical example, the Marussi tensor of the topographic effect is computed globally using tesseroids calculated by Gauss–Legendre cubature (3D) on the basis of a digital height model. The order of magnitude in the radial-radial component is about  ± 8 E.U. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

New relations among associated Legendre functions and spherical harmonics

Several new relations among associated Legendre functions (ALFs) are derived, most of which relate a product of an ALF with trigonometric functions to a weighted summation over ALFs, where the weights only depend on the degree and order of the ALF. These relations are, for example, useful in applications such as the computation of geopotential coefficients and computation of ellipsoidal corrections in geoid modelling. The main relations are presented in both their unnormalised and fully normalised (4π-normalised) form. Several approaches to compute the weights involved are discussed, and it is shown that the relations can also be applied in the case of first- and second-order derivatives of ALFs, which may be of use in analysis of satellite gradiometry data. Finally, the derived relations are combined to provide new identities among ALFs, which contain no dependency on the colatitudinal coordinate other than that in the ALFs themselves.     

Methodology and use of tensor invariants for satellite gravity gradiometry

Although its use is widespread in several other scientific disciplines, the theory of tensor invariants is only marginally adopted in gravity field modeling. We aim to close this gap by developing and applying the invariants approach for geopotential recovery. Gravitational tensor invariants are deduced from products of second-order derivatives of the gravitational potential. The benefit of the method presented arises from its independence of the gradiometer instrument’s orientation in space. Thus, we refrain from the classical methods for satellite gravity gradiometry analysis, i.e., in terms of individual gravity gradients, in favor of the alternative invariants approach. The invariants approach requires a tailored processing strategy. Firstly, the non-linear functionals with regard to the potential series expansion in spherical harmonics necessitates the linearization and iterative solution of the resulting least-squares problem. From the computational point of view, efficient linearization by means of perturbation theory has been adopted. It only requires the computation of reference gravity gradients. Secondly, the deduced pseudo-observations are composed of all the gravitational tensor elements, all of which require a comparable level of accuracy. Additionally, implementation of the invariants method for large data sets is a challenging task. We show the fundamentals of tensor invariants theory adapted to satellite gradiometry. With regard to the GOCE (Gravity field and steady-state Ocean Circulation Explorer) satellite gradiometry mission, we demonstrate that the iterative parameter estimation process converges within only two iterations. Additionally, for the GOCE configuration, we show the invariants approach to be insensitive to the synthesis of unobserved gravity gradients.     

全球及局部海洋扰动重力反演的快速解析方法

从经典边值问题理论及球谐函数理论出发,在空域推导获得了由大地水准面高以及垂线偏差计算扰动重力的解析计算公式,为利用卫星测高数据反演海洋扰动重力提供了理论基础。针对全球海洋区域和局部海洋区域的扰动重力反演,在前人已有工作基础上,提出了改进的基于一维FFT的精确快速算法,保证了计算结果与原解析方法完全一致,且计算速度提高约20倍。该算法在提高计算效率的同时避免了由于引入FFT而产生的混叠、边缘效应问题,而且对观测数据的序列长度没有硬性要求,使得应用更加灵活。利用EGM2008地球重力场模型分别生成了2.5'分辨率大地水准面高数据和垂线偏差数据,按照本文提出的改进方法(采用全球积分计算)分别反演获得了全球及局部海洋区域的扰动重力。经比较分析,由大地水准面和垂线偏差分别反演获得的扰动重力其差异在0.8×10^-5 m/s²以内,这说明两种反演方法是基本一致的,但在数据包含系统误差的情况下,由垂线偏差反演扰动重力具有一定优势。     

卫星重力径向梯度数据的最小二乘配置调和分析

本文深入研究了利用卫星重力梯度径向分量确定地球引力场位系数的最小二乘配置(LSC)调和分析方法。首先论述了最小二乘配置法的原理,推导了扰动引力梯度观测量与球谐系数之间的协方差和自协方差矩阵,在扰动引力梯度观测数据为等经差规则网格数据的情况下,引力位与扰动引力梯度之间的协方差矩阵具有分块Toeplitz循环阵的结构,有效的利用FFT变换技术将其降阶;研究利用截断奇异值分解法（TSVD）解决协方差阵的病态性问题;最后得到了引力梯度径向分量的最小二乘配置调和分析的完整计算公式。模拟试算结果表明,基于TSVD的最小二乘配置调和分析方法,能够以较高的精度还原全球重力场,验证了本文算法的有效性和实用性。     

卫星重力梯度数据的模拟研究

推导了运用地球重力场模型计算单点、格网点以及格网平均的扰动重力梯度复组合分量的公式;提出了广义球谐函数及其定积分的新算法,并利用EGM96地球重力场模型试算了全球地区卫星轨道面上的重力梯度分量的格网平均观测值;通过对角线分量满足Laplace方程的精度,验证了该算法的有效性和实用性。