扰动重力梯度的非奇异表示

在局部指北坐标系中用地心球坐标来表示扰动重力梯度张量,当计算点趋近于两极时,由于Legendre函数的一阶和二阶导数以及分母上所含余纬的正弦函数,将导致扰动重力梯度张量的计算出现无穷大。因此,本文引入了Legendre函数的一阶和二阶导数以及 无奇异性的计算公式,并且进一步推导了 无奇异性的计算公式。在将Legendre函数的一阶和二阶导数以及 、 无奇异性的计算公式代入到扰动重力梯度张量各分量的求解中时,又充分考虑了m等于0,1,2以及其它量时的复杂情况,建立了扰动重力梯度张量各分量无奇异性的详细计算模型。通过模拟实验表明,本文所建立的详细计算模型不仅能够完全满足当前卫星重力梯度张量计算的精度要求,而且模型稳定、可靠、易于编程实现。     

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.     

A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions

 Spherical harmonic expansions form partial sums of fully normalised associated Legendre functions (ALFs). However, when evaluated increasingly close to the poles, the ultra-high degree and order (e.g. 2700) ALFs range over thousands of orders of magnitude. This causes existing recursion techniques for computing values of individual ALFs and their derivatives to fail. A common solution in geodesy is to evaluate these expansions using Clenshaw's method, which does not compute individual ALFs or their derivatives. Straightforward numerical principles govern the stability of this technique. Elementary algebra is employed to illustrate how these principles are implemented in Clenshaw's method. It is also demonstrated how existing recursion algorithms for computing ALFs and their first derivatives are easily modified to incorporate these same numerical principles. These modified recursions yield scaled ALFs and first derivatives, which can then be combined using Horner's scheme to compute partial sums, complete to degree and order 2700, for all latitudes (except at the poles for first derivatives). This exceeds any previously published result. Numerical tests suggest that this new approach is at least as precise and efficient as Clenshaw's method. However, the principal strength of the new techniques lies in their simplicity of formulation and implementation, since this quality should simplify the task of extending the approach to other uses, such as spherical harmonic analysis. Received: 30 June 2000 / Accepted: 12 June 2001     

引力场梯度张量的非奇异公式推导

传统的引力场梯度计算公式在两极附近存在奇异性,需要换用其他的非奇异计算公式。从奇异性产生的原因入手,并结合勒让德函数的有关性质,推导了一组新的计算公式。实际计算验证了该公式的正确性和有效性。     

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     

利用Clenshaw求和计算大地水准面差距

引入非正规化的球谐函数,使用Clenshaw求和法不需要计算单个球谐函数值而直接计算级数和。将Clenshaw求和算法和标准向前列递推算法在计算时间上进行比较,并对两种方法计算的稳定性进行了数值分析,给出了数值模拟结果。通过数值计算得出了Clenshaw求和计算全球大地水准面差距的适用范围,为实现高精度大地水准面的计算提供借鉴。     

GPS precision as a function of session duration and reference frame using multi-point software

As an aid to survey design, we used data acquired from three European continuous GPS networks to test the precision of position estimates from static observations as a function of the length of the observing session and the number and distribution of reference stations. Our criterion was the weighted RMS of estimates over 31 days with respect to coordinates determined from 24-h sessions over a 2-year period. With a single reference station, a precision of 3 mm horizontal and 10 mm vertical could be achieved reliably only for session lengths of 3 h or longer and baselines less than 200 km. If four or more reference stations are used, these levels of precision were usually achieved with sessions as short as 2 h. With sessions 6 h or longer and four or more reference stations, the precision is typically 1–2 mm in horizontal and about 3–5 mm in vertical. Increasing the number of reference stations further provides only marginal improvement. Although there is some variation in precision in 4-station networks with the choice of reference stations, the dependence on distance and geometric distribution is weak.     

Computer algebra solution of the GPS N-points problem

A computer algebra solution is applied here to develop and evaluate algorithms for solving the basic GPS navigation problem: finding a point position using four or more pseudoranges at one epoch (the GPS N-points problem). Using Mathematica 5.2 software, the GPS N-points problem is solved numerically, symbolically, semi-symbolically, and with Gauss–Jacobi, on a work station. For the case of N > 4, two minimization approaches based on residuals and distance norms are evaluated for the direct numerical solution and their computational duration is compared. For N = 4, it is demonstrated that the symbolic computation is twice as fast as the iterative direct numerical method. For N = 6, the direct numerical solution is twice as fast as the semi-symbolic, with the residual minimization requiring less computation time compared to the minimization of the distance norm. Gauss–Jacobi requires eight times more computation time than the direct numerical solution. It does, however, have the advantage of diagnosing poor satellite geometry and outliers. Besides offering a complete evaluation of these algorithms, we have developed Mathematica 5.2 code (a notebook file) for these algorithms (i.e., Sturmfel’s resultant, Dixon’s resultants, Groebner basis, reduced Groebner basis and Gauss–Jacobi). These are accessible to any geodesist, geophysicist, or geoinformation scientist via the GPS Toolbox () website or the Wolfram Information Center ().

完全规格化缔合勒让德函数及其导数的递推算法与适用范围比较

完全规格化缔合勒让德函数及其导数常用标准向前列递推算法和标准向前行递推算法进行计算。基于第一、第二相对数值精度标准对两种算法的适用范围进行分析比较,计算结果表明,标准向前列递推算法的适用范围大于标准向前行递推算法,说明前者优于后者;结果同时还表明,完全规格化缔合勒让德函数与其导数同一种算法的适用范围也相同,并指出了二者适用范围相同的原因。     

Gravity gradient modeling using gravity and DEM

A model of the gravity gradient tensor at aircraft altitude is developed from the combination of ground gravity anomaly data and a digital elevation model. The gravity data are processed according to various operational solutions to the boundary-value problem (numerical integration of Stokes’ integral, radial-basis splines, and least-squares collocation). The terrain elevation data are used to reduce free-air anomalies to the geoid and to compute a corresponding indirect effect on the gradients at altitude. We compare the various modeled gradients to airborne gradiometric data and find differences of the order of 10–20 E (SD) for all gradient tensor elements. Our analysis of these differences leads to a conclusion that their source may be primarily measurement error in these particular gradient data. We have thus demonstrated the procedures and the utility of combining ground gravity and elevation data to validate airborne gradiometer systems.     

Tropospheric delay gradients from numerical weather prediction models: effects on GPS estimated parameters

Several numerical weather prediction (NWP) models provide information on the 3D state of the neutral atmosphere which has enabled GNSS researchers to have improved a priori information of the delay induced in the GNSS signals. However, the quality of weather models on the one hand and computational difficulties on the other, are motivations to develop an algorithm based partly on NWP models, while still estimating the remaining residual delay through GNSS processing strategies. An algorithm has been developed to estimate horizontal delay gradients from Meteorological Service of Canada NWP models. The GNSS software “Bernese” has also been modified to handle these gradients, as well as zenith delay and mapping functions based on NWP models in phase and code observation equations. Month-long precise point positioning results show strong correlation between north–south hydrostatic gradients and latitude differences, with significant but less strong correlation with the height and zenith total delay parameters. The longitude components were not sensitive to the implementation of gradients. High precision GNSS applications such as long term geodynamics studies, realization of terrestrial reference frames and climatology and consequential interpretations may be affected by ignoring the asymmetry of the neutral atmosphere. In addition to estimating the gradients, implementing a priori information on gradients in the processing software may have an impact on estimated results and consequential interpretations.     

重叠矩形图像表示及其上的精确Legendre矩计算(英文)

Linear quadtree is a popular image representation method due to its convenient imaging procedure. However, the excessive emphasis on the symmetry of segmentation, i.e. dividing repeatedly a square into four equal sub-squares, makes linear quadtree not an optimal representation. In this paper, a no-loss image representation, referred to as Overlapped Rectangle Image Representation （ORIR）, is presented to support fast image operations such as Legendre moments computation. The ORIR doesn＇t importune the symmetry of segmentation, and it is capable of representing, by using an identical rectangle, the information of the pixels which are not even adjacent to each other in the sense of 4-neighbor and 8-neighbor. Hence, compared with the linear quadtree, the ORIR significantly reduces the number of rectangles required to represent an image. Based on the ORIR, an algorithm for exact Legendre moments computation is presented. The theoretical analysis and the experimental results show that the ORIR-based algorithm for exact Legendre moments computation is faster than the conventional exact algorithms.     

多种超高阶次缔合勒让德函数计算方法的比较

扩展高阶和超高阶重力场模型的构制与应用的数值稳定性取决于超高阶次缔合勒让德函数的计算方法。文中详细介绍了现有的多种缔合勒让德函数的递推计算方法:标准前向列推法、标准前向行推法、跨阶次递推法和Belikov列推法。从计算速度、计算精度和计算溢出问题3个角度分析比较了阶次高至2 160阶的各种方法的优劣。通过数值试验证明,Belikov列推法和跨阶次递推法是计算超高阶次缔合勒让德函数较优的方法,而其他几种方法不能用于超高阶次缔合勒让德函数的计算。文中结论为超高阶次球谐综合与球谐分析的数值计算提供了可靠的依据。     

On the geoid–quasigeoid separation in mountain areas

The separation between the reference surfaces for orthometric heights and normal heights—the geoid and the quasigeoid—is typically in the order of a few decimeters but can reach nearly 3 m in extreme cases. The knowledge of the geoid–quasigeoid separation with centimeter accuracy or better, is essential for the realization of national and international height reference frames, and for precision height determination in geodetic engineering. The largest contribution to the geoid–quasigeoid separation is due to the distribution of topographic masses. We develop a compact formulation for the rigorous treatment of topographic masses and apply it to determine the geoid–quasigeoid separation for two test areas in the Alps with very rough topography, using a very fine grid resolution of 100 m. The magnitude of the geoid–quasigeoid separation and its accuracy, its slopes, roughness, and correlation with height are analyzed. Results show that rigorous treatment of topographic masses leads to a rather small geoid–quasigeoid separation—only 30 cm at the highest summit—while results based on approximations are often larger by several decimeters. The accuracy of the topographic contribution to the geoid–quasigeoid separation is estimated to be 2–3 cm for areas with extreme topography. Analysis of roughness of the geoid–quasigeoid separation shows that a resolution of the modeling grid of 200 m or less is required to achieve these accuracies. Gravity and the vertical gravity gradient inside of topographic masses and the mean gravity along the plumbline are modeled which are important intermediate quantities for the determination of the geoid–quasigeoid separation. We conclude that a consistent determination of the geoid and quasigeoid height reference surfaces within an accuracy of few centimeters is feasible even for areas with extreme topography, and that the concepts of orthometric height and normal height can be consistently realized and used within this level of accuracy.     

Precise ionosphere-free single-frequency GNSS positioning

Ionospheric delays can be efficiently eliminated from single-frequency data using a combination of carrier phases and code ranges. Unfortunately, GPS and GLONASS ranges are relatively noisy which can limit the use of the positioning method. Nevertheless, position standard deviations are in the range of 6–8 cm (horizontal) and 7–9 cm (3d) obtained from diurnal data batches from selected IGS reference stations can be further reduced to 2–3 cm (3d) for weekly smoothed averages. GPS data sets collected in Ghana (Africa) reveal a typical level of 10 cm of deviation that must be anticipated under average conditions. Looking at the future of GNSS, the European Galileo system will, in contrast to GPS, provide the broadband signal E5 that is by far less affected by multipath thus providing rather precise range measurements. Simulated processing runs featuring both high ionospheric and tropospheric delay variations show a 3d position precision of 4 cm even for a data batch as short as just 1 h, whereas GPS L1/Galileo E1 performance is close to 13 cm for the same data set.     

From fields to objects: A review of geographic boundary analysis

Geographic boundary analysis is a relatively new approach unfamiliar to many spatial analysts. It is best viewed as a technique for defining objects – geographic boundaries – on spatial fields, and for evaluating the statistical significance of characteristics of those boundary objects. This is accomplished using null spatial models representative of the spatial processes expected in the absence of boundary-generating phenomena. Close ties to the object-field dialectic eminently suit boundary analysis to GIS data. The majority of existing spatial methods are field-based in that they describe, estimate, or predict how attributes (variables defining the field) vary through geographic space. Such methods are appropriate for field representations but not object representations. As the object-field paradigm gains currency in geographic information science, appropriate techniques for the statistical analysis of objects are required. The methods reviewed in this paper are a promising foundation. Geographic boundary analysis is clearly a valuable addition to the spatial statistical toolbox.? This paper presents the philosophy of, and motivations for geographic boundary analysis. It defines commonly used statistics for quantifying boundaries and their characteristics, as well as simulation procedures for evaluating their significance. We review applications of these techniques, with the objective of making this promising approach accessible to the GIS-spatial analysis community. We also describe the implementation of these methods within geographic boundary analysis software: GEM. Received: 22 March 1999 / Accepted: 7 September 1999     

Simple and highly accurate formulas for the computation of Transverse Mercator coordinates from longitude and isometric latitude

A conformal approximation to the Transverse Mercator (TM) map projection, global in longitude λ and isometric latitude q, is constructed. New formulas for the point scale factor and grid convergence are also shown. Assuming that the true values of the TM coordinates are given by conveniently truncated Gauss–Krüger series expansions, we use the maximum norm of the absolute error to measure globally the accuracy of the approximation. For a Universal Transverse Mercator (UTM) zone the accuracy equals 0.21  mm, whereas for the region of the ellipsoid bounded by the meridians  ±20° the accuracy is equal to 0.3  mm. Our approach is based on a four-term perturbation series approximation to the radius r(q) of the parallel q, with a maximum absolute deviation of 0.43  mm. The small parameter of the power series expansion is the square of the eccentricity of the ellipsoid. This closed approximation to r(q) is obtained by solving a regularly perturbed Cauchy problem with the Poincaré method of the small parameter. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

全球位系数模型构建高精度局部重力场的Clenshaw求和

给出了勒让德函数4种模式的Clenshaw求和公式,在此基础上得到由全球位系数模型计算重力场参数的二阶Clenshaw求和公式,分析了EGM96与WDM94应用于某试验测区局部重力场的精度,为该测区在构建高精度局部重力场时初始全球位系数模型的选取提供了借鉴。     

A non-iterative and non-singular perturbation solution for transforming Cartesian to geodetic coordinates

The Cartesian-to-Geodetic coordinate transformation is re-cast as a minimization algorithm for the height of the Satellite above the reference Earth surface. Optimal necessary conditions are obtained that fix the satellite ground track vector components in the Earth surface. The introduction of an artificial perturbation variable yields a rapidly converging second order power series solution. The initial starting guess for the solution provides 3–4 digits of precision. Convergence of the perturbation series expansion is accelerated by replacing the series solution with a Padé approximation. For satellites with heights < 30,000 km the second-order expansions yields ~mm satellite geodetic height errors and ~10⁻¹² rad errors for the geodetic latitude. No quartic or cubic solutions are required: the algorithm is both non-iterative and non-singular. Only two square root and two arc-tan calculations are required for the entire transformation. The proposed algorithm has been measured to be ~41% faster than the celebrated Bowring method. Several numerical examples are provided to demonstrate the effectiveness of the new algorithm.     

Accuracy of areal interpolation: A comparison of alternative methods

This paper discusses the accuracy of spatial data estimated by areal interpolation, a process of transferring data from one zonal system to another. A stochastic model is proposed which represents areal interpolations in diverse geographic situations. The model is used to examine the relationship between estimation accuracy and the spatial distribution of estimation error from a theoretical viewpoint. The analysis shows that the uniformity in error distribution improves the accuracy of areal interpolation. Four areal interpolation methods are then assessed through numerical examinations. From this it is found that the accuracy of simple interpolation methods heavily depends on the appropriateness of their hypothetical distributions, whereas the accuracy of intelligent methods depends on the fitness of the range of supplementary data for that of true distribution. Received: 19 February 1999/Accepted 17 September 1999