On the spherical and spheroidal harmonic expansion of the gravitational potential of the topographic masses

This paper is devoted to the spherical and spheroidal harmonic expansion of the gravitational potential of the topographic masses in the most rigorous way. Such an expansion can be used to compute gravimetric topographic effects for geodetic and geophysical applications. It can also be used to augment a global gravity model to a much higher resolution of the gravitational potential of the topography. A formulation for a spherical harmonic expansion is developed without the spherical approximation. Then, formulas for the spheroidal harmonic expansion are derived. For the latter, Legendre’s functions of the first and second kinds with imaginary variable are expanded in Laurent series. They are then scaled into two real power series of the second eccentricity of the reference ellipsoid. Using these series, formulas for computing the spheroidal harmonic coefficients are reduced to surface harmonic analysis. Two numerical examples are presented. The first is a spherical harmonic expansion to degree and order 2700 by taking advantage of existing software. It demonstrates that rigorous spherical harmonic expansion is possible, but the computed potential on the geoid shows noticeable error pattern at Polar Regions due to the downward continuation from the bounding sphere to the geoid. The second numerical example is the spheroidal expansion to degree and order 180 for the exterior space. The power series of the second eccentricity of the reference ellipsoid is truncated at the eighth order leading to omission errors of 25 nm (RMS) for land areas, with extreme values around 0.5 mm to geoid height. The results show that the ellipsoidal correction is 1.65 m (RMS) over land areas, with maximum value of 13.19 m in the Andes. It shows also that the correction resembles the topography closely, implying that the ellipsoidal correction is rich in all frequencies of the gravity field and not only long wavelength as it is commonly assumed.     

Optimized formulas for the gravitational field of a tesseroid

Various tasks in geodesy, geophysics, and related geosciences require precise information on the impact of mass distributions on gravity field-related quantities, such as the gravitational potential and its partial derivatives. Using forward modeling based on Newton’s integral, mass distributions are generally decomposed into regular elementary bodies. In classical approaches, prisms or point mass approximations are mostly utilized. Considering the effect of the sphericity of the Earth, alternative mass modeling methods based on tesseroid bodies (spherical prisms) should be taken into account, particularly in regional and global applications. Expressions for the gravitational field of a point mass are relatively simple when formulated in Cartesian coordinates. In the case of integrating over a tesseroid volume bounded by geocentric spherical coordinates, it will be shown that it is also beneficial to represent the integral kernel in terms of Cartesian coordinates. This considerably simplifies the determination of the tesseroid’s potential derivatives in comparison with previously published methodologies that make use of integral kernels expressed in spherical coordinates. Based on this idea, optimized formulas for the gravitational potential of a homogeneous tesseroid and its derivatives up to second-order are elaborated in this paper. These new formulas do not suffer from the polar singularity of the spherical coordinate system and can, therefore, be evaluated for any position on the globe. Since integrals over tesseroid volumes cannot be solved analytically, the numerical evaluation is achieved by means of expanding the integral kernel in a Taylor series with fourth-order error in the spatial coordinates of the integration point. As the structure of the Cartesian integral kernel is substantially simplified, Taylor coefficients can be represented in a compact and computationally attractive form. Thus, the use of the optimized tesseroid formulas particularly benefits from a significant decrease in computation time by about 45 % compared to previously used algorithms. In order to show the computational efficiency and to validate the mathematical derivations, the new tesseroid formulas are applied to two realistic numerical experiments and are compared to previously published tesseroid methods and the conventional prism approach.     

基于超高阶地球重力场模型的哈尔滨市区域似大地水准面确定

本文综述了EGM2008模型的发展情况,并以此模型为基础构建了哈尔滨市区域似大地水准面模型,接着对所建立的模型进行了实证分析,经过计算后,得到结论认为利用EGM2008构建的区域似大地水准面模型能够用于厘米级GNSS正常高测量.     

Accurate computation of gravitational field of a tesseroid

We developed an accurate method to compute the gravitational field of a tesseroid. The method numerically integrates a surface integral representation of the gravitational potential of the tesseroid by conditionally splitting its line integration intervals and by using the double exponential quadrature rule. Then, it evaluates the gravitational acceleration vector and the gravity gradient tensor by numerically differentiating the numerically integrated potential. The numerical differentiation is conducted by appropriately switching the central and the single-sided second-order difference formulas with a suitable choice of the test argument displacement. If necessary, the new method is extended to the case of a general tesseroid with the variable density profile, the variable surface height functions, and/or the variable intervals in longitude or in latitude. The new method is capable of computing the gravitational field of the tesseroid independently on the location of the evaluation point, namely whether outside, near the surface of, on the surface of, or inside the tesseroid. The achievable precision is 14–15 digits for the potential, 9–11 digits for the acceleration vector, and 6–8 digits for the gradient tensor in the double precision environment. The correct digits are roughly doubled if employing the quadruple precision computation. The new method provides a reliable procedure to compute the topographic gravitational field, especially that near, on, and below the surface. Also, it could potentially serve as a sure reference to complement and elaborate the existing approaches using the Gauss–Legendre quadrature or other standard methods of numerical integration.     

The spacetime gravitational field of a deformable body

The high-resolution analysis of orbit perturbations of terrestrial artificial satellites has documented that the eigengravitation of a massive body like the Earth changes in time, namely with periodic and aperiodic constituents. For the space-time variation of the gravitational field the action of internal and external volume as well as surface forces on a deformable massive body are responsible. Free of any assumption on the symmetry of the constitution of the deformable body we review the incremental spatial (“Eulerian”) and material (“Lagrangean”) gravitational field equations, in particular the source terms (two constituents: the divergence of the displacement field as well as the projection of the displacement field onto the gradient of the reference mass density function) and the `jump conditions' at the boundary surface of the body as well as at internal interfaces both in linear approximation. A spherical harmonic expansion in terms of multipoles of the incremental Eulerian gravitational potential is presented. Three types of spherical multipoles are identified, namely the dilatation multipoles, the transport displacement multipoles and those multipoles which are generated by mass condensation onto the boundary reference surface or internal interfaces. The degree-one term has been identified as non-zero, thus as a “dipole moment” being responsible for the varying position of the deformable body's mass centre. Finally, for those deformable bodies which enjoy a spherically symmetric constitution, emphasis is on the functional relation between Green functions, namely between Fourier-/ Laplace-transformed volume versus surface Love-Shida functions (h(r),l(r) versus h ^′(r),l ^′(r)) and Love functions k(r) versus k ^′(r). The functional relation is numerically tested for an active tidal force/potential and an active loading force/potential, proving an excellent agreement with experimental results. Received: December 1995 / Accepted: 1 February 1997     

重力场数学拟合模型解的影响因素探究

为了提高地球重力场模型不适定方程求解的精度,该文采用谱分析方法从级数展开阶数、数据采样率及数据缺失量3个方面探索影响数学拟合效果的根本因素:从常用的三角级数及勒让德级数模型出发,引出重力场拟合模型球谐函数模型,观察在改变级数展开阶数、数据采样率及数据缺失量等情况下所对应设计矩阵谱结构的变化,并从微观上研究影响误差分配的有关因素及最小奇异值对误差的决定性作用,为探求重力场模型解不准的原因及实现更高精度的全球重力场模型的建立提供参考。     

引潮力引起的地球引力场时空变化的理论研究

地球重力场的时空结构与分布特性无论在基础理论研究,还是在地理空间信息建设中都具有重要的意义。地球表面上的观测仪器检测到的只是某一个质点的重力变化(拉格朗日变化),但是理论研究却是基于欧拉引力和引力位场方程进行的。本文基于连续介质力学的基本理论,在改正了文献[5]的一处原则性错误之后,推导出了引力和引力位的Lagrange和Euler增量表达式。本文工作对于高精度地球重力场的时空变化的理论研究具有参考价值。     

Influence of atmospheric refraction on horizontal angle surveying

IntroductionCurrently , the ultrahigh-precision terrestrialtriangulateration network (concluding construc-tion control network and deformation monitoringnetwork) is widely usedinthe branch of survey-ing and mappingfor hydropower .The character-istics of t…     

On a chain suspended in a non-uniform gravitational field

It is possible to simply describe the curve followed by a chain suspended in a non-uniform gravitational field. Parallel discussions are given using the two theories of gravitation, Newtonian and general relativistic.     

Spaceborne gravitational field determination by means of locally supported wavelets

In space-borne gravitational field determination, two challenges are inherent. First, the continuation of the data down to the surface of the Earth is an ill-posed problem, requiring therefore regularization techniques. Second huge data sets result requiring efficient numerical methods. In this paper, we show how locally supported wavelets on the sphere can be developed by means of a spherical version of the so-called up function. By construction, the corresponding scaling functions and wavelets are infinitely smooth, so that they can be used for regularization purposes. In particular, we show how the ill-posed pseudo-differential equations coming from satellite missions can be regularized by efficient numerical schemes using locally supported wavelets. These methods seem in particular to be interesting for regional gravity field modelling.     

The gravity field model IGGT_R1 based on the second invariant of the GOCE gravitational gradient tensor

Based on tensor theory, three invariants of the gravitational gradient tensor (IGGT) are independent of the gradiometer reference frame (GRF). Compared to traditional methods for calculation of gravity field models based on the gravity field and steady-state ocean circulation explorer (GOCE) data, which are affected by errors in the attitude indicator, using IGGT and least squares method avoids the problem of inaccurate rotation matrices. The IGGT approach as studied in this paper is a quadratic function of the gravity field model’s spherical harmonic coefficients. The linearized observation equations for the least squares method are obtained using a Taylor expansion, and the weighting equation is derived using the law of error propagation. We also investigate the linearization errors using existing gravity field models and find that this error can be ignored since the used a-priori model EIGEN-5C is sufficiently accurate. One problem when using this approach is that it needs all six independent gravitational gradients (GGs), but the components \(V_{xy}\) and \(V_{yz}\) of GOCE are worse due to the non-sensitive axes of the GOCE gradiometer. Therefore, we use synthetic GGs for both inaccurate gravitational gradient components derived from the a-priori gravity field model EIGEN-5C. Another problem is that the GOCE GGs are measured in a band-limited manner. Therefore, a forward and backward finite impulse response band-pass filter is applied to the data, which can also eliminate filter caused phase change. The spherical cap regularization approach (SCRA) and the Kaula rule are then applied to solve the polar gap problem caused by GOCE’s inclination of \(96.7^{\circ }\). With the techniques described above, a degree/order 240 gravity field model called IGGT_R1 is computed. Since the synthetic components of \(V_{xy}\) and \(V_{yz}\) are not band-pass filtered, the signals outside the measurement bandwidth are replaced by the a-priori model EIGEN-5C. Therefore, this model is practically a combined gravity field model which contains GOCE GGs signals and long wavelength signals from the a-priori model EIGEN-5C. Finally, IGGT_R1’s accuracy is evaluated by comparison with other gravity field models in terms of difference degree amplitudes, the geostrophic velocity in the Agulhas current area, gravity anomaly differences as well as by comparison to GNSS/leveling data.     

结合引力场理论的道路自动选取方法

针对复杂网络模型被广泛应用于道路选取的研究中,少有方法顾及道路网对偶图中多级邻居节点的影响,导致对道路的重要性评价缺乏准确性和可靠性的问题。该文将道路网对偶图节点的结构特征值视作质量,将节点间的最短距离视作距离,结合引力场方程实现道路自动选取,较好地将道路网对偶图中多级邻居节点的影响纳入到道路重要性的计算中,从而实现对道路重要性的准确评估。通过对兰州市城关区的路网进行实验,结果表明本文方法所选取的路网较好地保持了原始道路网的整体结构、覆盖范围、密度分布、拓扑特征和连通性。     

顾及地球重力场模型的GNSS高程转换方法

针对使用单纯数学模型在进行GNSS高程拟合过程中,只能体现测区高程异常的大致趋势,无法表现出细节变化,进而影响GNSS高程转换精度的问题。提出了地球重力场模型与数学函数相结合的"移去-拟合-恢复"GNSS高程转换方法,以安徽省淮南市某矿工作面走向线为例,分别采用单纯数学模型和顾及EIGEN-6C4、EIGEN-6C2、EGM2008地球重力场模型的"移去-拟合-恢复"法进行了高程转换研究,并将各种高程转换的结果进行了对比分析。结果表明,多面函数相对二次曲面的转换精度较好,顾及地球重力场模型的"移去-拟合-恢复"法较单纯数学模型的高程转换精度有了大幅度的提高;顾及EIGEN-6C4地球重力场模型的GNSS高程转换精度,优于顾及EIGEN-6C2地球重力场模型的转换精度,优于顾及EGM2008地球重力场模型的转换精度。     

引力不变量曲线坐标系及在定位导航上的应用

本文基于引力与引力梯度不变量引入了3个仅是点位的独立函数,从而构成了空间的曲线坐标系.在此基础上,讨论了引入的不变量坐标系在定位导航方面的潜在应用.事实上,利用观测所得的引力与引力梯度不变量,可直接构建关于观测点的方程组.由于该方程组是非线性的,所以求解过程中必须进行线性化处理以及迭代计算.为了展示如何使用该曲线坐标系...     

武汉基准台气压对重力潮汐观测的影响

武汉基准台超导重力仪观测重力残差和台站气压的相关分析结果表明,重力残差信号主要是由气压变化引起的。本文在频率和时间域内分别测定了大气重力导纳值,结果说明在时域内的导纳值为－０．３０７μＧａｌ／（１Ｇａｌ＝１ｃｍ／ｓ＾２）,而在频域内的大气重力导纳值变化范围在０．３￣０．５μＧａｌ／ｈＰａ之间,这一结果与理论模拟计算相符。对实测资料作气压改正的结果表明气压改正效果非常显著,各频段内的观测残差振幅均有     

三角高程测量中大气折光影响的分析

大气折光是测绘领域数据采集时的主要误差来源之一。本文从大气折射率与大气密度的关系入手,论述了大气折光的两种形式,详细分析了大气折光对天顶距测量和光电测距的影响,进一步论证了在温度梯度逆转时刻进行观测可以有效地削弱大气折光误差对测量结果的影响。     

Computing components of the gravity field induced by distant topographic masses and condensed masses over the entire Earth using the 1-D FFT approach

 A new method for computing gravitational potential and attraction induced by distant, global masses on a global scale has been developed. The method uses series expansions and the well known one-dimensional fast Fourier transform (1-D FFT) method. It has been proven to be significantly faster than quadrature while being equally accurate. Various quantities were studied to cover the two primary applications of the Stokes–Helmert scheme of modeling effects. These two applications (or paths), given the names R/r/D and R/D/r, are briefly discussed, although the primary objective of the paper is to provide computational information to either path, rather than choosing one path as preferable to the other. It is further shown that the impact of masses outside a 4-degree cap can impact the absolute computation of the geoid at more than 1 cm, and should therefore be included in all local geoid computations seeking that accuracy. Received: 13 December 2000 / Accepted: 3 September 2001     

地下水变化对重力场观测的影响

正有效消除地下水影响是利用地面重力场观测监测和研究地壳运动及地下物质运移的关键步骤之一。基于超导重力仪(SG)的高精度重力观测,可建立地下水变化与重力场变化之间的关系,一方面可以利用地下水文资料消除地下水变化的影响,增强超导重力仪对地球物理学/地球动力学信号的探测能力;另一方面可以利用超导重力仪观测资料长期连续地监测与地下水质量流动相关的各种现象,并用于地下水储量变化、土壤参数估计、水文模型约束等方面的研究。本文利用拉萨和武汉九峰两个超导重力台站的高精     

重力场模型及GNSS/水准的区域似大地水准面精化

针对现有地球重力场模型综合利用研究较少的情况,该文利用实测全球导航卫星系统/水准数据分析GOCO03S、ITG-GRACE2010S、GO_CONS_GCF_2_DIR_R4、GO_CONS_GCF_2_TIM_R4和EGM2008等地球重力场模型不同谱域位系数对应的高程异常精度,提出对多类重力场模型进行简单谱组合和加权谱组合,并进行精度分析;然后利用这两类组合重力场模型,结合全球导航卫星系统/水准数据对区域似大地水准面的精化展开研究。计算结果表明:在实验区域,与EGM2008模型相比,采用简单谱组合法和加权谱组合法均能提高模型高程异常的精度,标准差最优分别可达0.081m和0.084m,对应精度提高幅度分别为40.9%和38.5%;以多类重力场模型为基础,经简单谱组合法或加权谱组合法得到组合重力场模型,并利用全球导航卫星系统/水准数据进行精化,可获得较高精度的区域似大水准面,精度最优可达0.048m。