Ellipsoidal terrain correction based on multi-cylindrical equal-area map projection of the reference ellipsoid

An operational algorithm for computation of terrain correction (or local gravity field modeling) based on application of closed-form solution of the Newton integral in terms of Cartesian coordinates in multi-cylindrical equal-area map projection of the reference ellipsoid is presented. Multi-cylindrical equal-area map projection of the reference ellipsoid has been derived and is described in detail for the first time. Ellipsoidal mass elements with various sizes on the surface of the reference ellipsoid are selected and the gravitational potential and vector of gravitational intensity (i.e. gravitational acceleration) of the mass elements are computed via numerical solution of the Newton integral in terms of geodetic coordinates {,,h}. Four base- edge points of the ellipsoidal mass elements are transformed into a multi-cylindrical equal-area map projection surface to build Cartesian mass elements by associating the height of the corresponding ellipsoidal mass elements to the transformed area elements. Using the closed-form solution of the Newton integral in terms of Cartesian coordinates, the gravitational potential and vector of gravitational intensity of the transformed Cartesian mass elements are computed and compared with those of the numerical solution of the Newton integral for the ellipsoidal mass elements in terms of geodetic coordinates. Numerical tests indicate that the difference between the two computations, i.e. numerical solution of the Newton integral for ellipsoidal mass elements in terms of geodetic coordinates and closed-form solution of the Newton integral in terms of Cartesian coordinates, in a multi-cylindrical equal-area map projection, is less than 1.6×10^–8 m²/s² for a mass element with a cross section area of 10×10 m and a height of 10,000 m. For a mass element with a cross section area of 1×1 km and a height of 10,000 m the difference is less than 1.5×10^–4m²/s². Since 1.5× 10^–4 m²/s² is equivalent to 1.5×10^–5m in the vertical direction, it can be concluded that a method for terrain correction (or local gravity field modeling) based on closed-form solution of the Newton integral in terms of Cartesian coordinates of a multi-cylindrical equal-area map projection of the reference ellipsoid has been developed which has the accuracy of terrain correction (or local gravity field modeling) based on the Newton integral in terms of ellipsoidal coordinates.Acknowledgments. This research has been financially supported by the University of Tehran based on grant number 621/4/859. This support is gratefully acknowledged. The authors are also grateful for the comments and corrections made to the initial version of the paper by Dr. S. Petrovic from GFZ Potsdam and the other two anonymous reviewers. Their comments helped to improve the structure of the paper significantly.     

Vector methods to compute azimuth, elevation, ellipsoidal normal, and the Cartesian ( X , Y , Z ) to geodetic ( φ , λ , h ) transformation

Vector-based algorithms for the computation of azimuth, elevation and the ellipsoidal normal unit vector from 3D Cartesian coordinates are presented. As a by-product, the formulae for the ellipsoidal normal vector can also be used to iteratively transform rectangular Cartesian coordinates (X, Y, Z) into geodetic coordinates (φ, λ, h) for a height range from −5600 km to 10⁸ km. Comparisons with existing methods indicate that the new transformation can compete with them.     

附加垂线偏差参数的地面常规网与GNSS基线网联合平差

根据测站的垂线站心坐标系和法线站心坐标系之间的转换公式,将垂线偏差作为未知参数,列出地面常规网观测量（水平方向、垂直角和边长）在空间直角坐标系中的观测方程,并结合GNSS网的观测方程,推导出附加垂线偏差的GNSS网和地面网联合平差的参数估计及其精度评定公式。利用含有GNSS观测量和地面常规网观测量的实例数据解算出垂线偏差,分析了垂线偏差对平差结果的影响。结果表明,将垂线偏差作为未知参数,能够消除垂线偏差对观测值的影响,显著提高未知点坐标的解算精度及可靠性。     

A new method for computing the ellipsoidal correction for Stokes's formula

 This paper generalizes the Stokes formula from the spherical boundary surface to the ellipsoidal boundary surface. The resulting solution (ellipsoidal geoidal height), consisting of two parts, i.e. the spherical geoidal height N ₀ evaluated from Stokes's formula and the ellipsoidal correction N ₁, makes the relative geoidal height error decrease from O(e ²) to O(e ⁴), which can be neglected for most practical purposes. The ellipsoidal correction N ₁ is expressed as a sum of an integral about the spherical geoidal height N ₀ and a simple analytical function of N ₀ and the first three geopotential coefficients. The kernel function in the integral has the same degree of singularity at the origin as the original Stokes function. A brief comparison among this and other solutions shows that this solution is more effective than the solutions of Molodensky et al. and Moritz and, when the evaluation of the ellipsoidal correction N ₁ is done in an area where the spherical geoidal height N ₀ has already been evaluated, it is also more effective than the solution of Martinec and Grafarend. Received: 27 January 1999 / Accepted: 4 October 1999     

利用过渡坐标系改进3维坐标变换模型

本文针对我国平面控制网与高程网分开布设的特点,借助于与站心坐标系指向一致的过渡坐标系,改进了计算坐标转换参数的常用模型。改进后的模型联合平面控制点的坐标和高程控制点的高程求解与其他３维空间坐标的７个转换参数,不需要３维已知点。并用算例说明了本文所提出的模型的优越性。     

Minimum conformal mapping distortion according to Chebyshev’s principle: a case study over peninsular Spain

Defining the distortion of a conformal map projection as the oscillation of the logarithm of its infinitesimal-scale σ, Chebyshev’s principle states that the best (minimum distortion) conformal map projection over a given region Ω of the ellipsoid is characterized by the property that σ is constant on the boundary of that region. Starting from a first map of Ω, we show how to compute the distortion δ₀(Ω) of this Chebyshev’s projection. We prove that this minimum possible conformal mapping distortion associated with Ω coincides with the absolute value of the minimum of the solution of a Dirichlet boundary-value problem for an elliptic partial differential equation in divergence form and with homogeneous boundary condition. If the first map is conformal, the partial differential equation becomes a Poisson equation for the Laplace operator. As an example, we compute the minimum conformal distortion associated with peninsular Spain. Using longitude and isometric latitude as coordinates, we solve the corresponding boundary-value problem with the finite element method, obtaining δ₀(Ω)=0.74869×10⁻³. We also quantify the distortions δ_l and δ_utm of the best conformal conic and UTM (zone 30) projections over peninsular Spain respectively. We get δ_l=2.30202×10⁻³ and δ_utm=3.33784×10⁻³.     

浅谈三角高程测量的计算公式

推证了三角高程计算高差的严密公式,分析了用两点坐标反算的距离对高差计算的影响,得出了高差较大地区或投影带边缘测算高差时应当把高斯平面的距离化算到地面上的结论。     

Global height datum unification: a new approach in gravity potential space

The problem of “global height datum unification” is solved in the gravity potential space based on: (1) high-resolution local gravity field modeling, (2) geocentric coordinates of the reference benchmark, and (3) a known value of the geoid’s potential. The high-resolution local gravity field model is derived based on a solution of the fixed-free two-boundary-value problem of the Earth’s gravity field using (a) potential difference values (from precise leveling), (b) modulus of the gravity vector (from gravimetry), (c) astronomical longitude and latitude (from geodetic astronomy and/or combination of (GNSS) Global Navigation Satellite System observations with total station measurements), (d) and satellite altimetry. Knowing the height of the reference benchmark in the national height system and its geocentric GNSS coordinates, and using the derived high-resolution local gravity field model, the gravity potential value of the zero point of the height system is computed. The difference between the derived gravity potential value of the zero point of the height system and the geoid’s potential value is computed. This potential difference gives the offset of the zero point of the height system from geoid in the “potential space”, which is transferred into “geometry space” using the transformation formula derived in this paper. The method was applied to the computation of the offset of the zero point of the Iranian height datum from the geoid’s potential value W ₀=62636855.8 m²/s². According to the geometry space computations, the height datum of Iran is 0.09 m below the geoid.     

Helmert扰动位及其积分核函数的椭球实用公式

借助以地心参考椭球面为边界面的第二大地边值问题的理论,基于Helmert空间的Neumann边值条件,给定Helmert扰动位的椭球解表达式,并详细推导第二类勒让德函数及其导数的递推关系、Helmert扰动位函数的椭球积分解以及类椭球Hotine积分核函数的实用计算公式,便于后续椭球域第二大地边值问题的实际研究。     

Direct georeferencing of airborne LiDAR data in national coordinates

The topographic mapping products of airborne light detection and ranging (LiDAR) are usually required in the national coordinates (i.e., using the national datum and a conformal map projection). Since the spatial scale of the national datum is usually slightly different from the World Geodetic System 1984 (WGS 84) datum, and the map projection frame is not Cartesian, the georeferencing process in the national coordinates is inevitably affected by various geometric distortions. In this paper, all the major direct georeferencing distortion factors in the national coordinates, including one 3D scale distortion (the datum scale factor distortion), one height distortion (the earth curvature distortion), two length distortions (the horizontal-to-geodesic length distortion and the geodesic-to-projected length distortion), and three angle distortions (the skew-normal distortion, the normal-section-to-geodesic distortion, and the arc-to-chord distortion) are identified and demonstrated in detail; and high-precision map projection correction formulas are provided for the direct georeferencing of the airborne LiDAR data. Given the high computational complexity of the high-precision map projection correction approach, some more approximate correction formulas are also derived for the practical calculations. The simulated experiments show that the magnitude of the datum scale distortion can reach several centimeters to decimeters for the low (e.g., 500 m) and high (e.g., 8000 m) flying heights, and therefore it always needs to be corrected. Our proposed practical map projection correction approach has better accuracy than Legat’s approach,¹ but it needs 25% more computational cost. As the correction accuracy of Legat’s approach can meet the requirements of airborne LiDAR data with low and medium flight height (up to 3000 m above ground), our practical correction approach is more suitable to the high-altitude aerial imagery. The residuals of our proposed high-precision map projection correction approach are trivial even for the high flight height of 8000 m. It can be used for the theoretical applications such as the accurate evaluation of different GPS/INS attitude transformation methods to the national coordinates.     

Change of projection following readjustment

Abstract

The readjustment of a major geodetic control network results in a new set of spheroidal coordinates for the network stations. Those new coordinates followed by an appropriate control densification serve as input for computing new plane coordinates. There are many surveying and mapping products which are based on the existing 'old' plane coordinates system. This paper deals with considerations and procedures aiming at the introduction of a new projection defined in such a way as to minimise the detrimental consequences of readjustment through the use of a synthetic point of origin for the new projection.     

INTERVISIBLE FEATURES

Abstract

It is frequently required to find whether a feature A of height h ₀ will interrupt the view between two other features A₁ and A₂, of heights h ₁ and h ₂ respectively. Suppose that the right line from A₁ to A₂, whose zenith distance is ζ at A₁, has a height h at A; it is then obvious that no more is necessary than to compute h and compare it with the known height h ₀ of the feature A.     

高程拟合变换的平面坐标获取方法研究

提出一种基于高程拟合的数据处理方法,充分利用GPS高精度的三维观测,实现了将三维空间坐标转换为二维平面坐标.通过与传统椭球投影的结果对比,两者平均点位偏差为0.6 mm,结果表明了方法的正确性.由于无需椭球投影,这为平面坐标的数据处理提供了一种简单可行的方法.     

GPS高程拟合代替水准测量的可行性

GPS测量已能够提供毫米级的平面坐标,但因似大地水准面与参考椭球面的差距,使得高精度的GPS大地高不能直接应用于生产实践。如何将GPS大地高转化咸水准高程,并使其保持一定的精度,一直是人们研究的热点。在介绍GPS高程拟合模型的基础上,通过实测数据验证了GPS高程拟合代替水准测量的可行性。     

The oblique azimuthal projection of geodesic type for the biaxial ellipsoid: Riemann polar and normal coordinates

Summary Riemann polar/normal coordinates are the constituents to generate the oblique azimuthal projection of geodesic type, here applied to the reference ellipsoid of revolution (biaxial ellipsoid).Firstly we constitute a minimal atlas of the biaxial ellipsoid built on {ellipsoidal longitude, ellipsoidal latitude} and {metalongitude, metalatitude}. TheDarboux equations of a 1-dimensional submanifold (curve) in a 2-dimensional manifold (biaxial ellipsoid) are reviewed, in particular to represent geodetic curvature, geodetic torsion and normal curvature in terms of elements of the first and second fundamental form as well as theChristoffel symbols. The notion of ageodesic anda geodesic circle is given and illustrated by two examples. The system of twosecond order ordinary differential equations of ageodesic (Lagrange portrait) is presented in contrast to the system of twothird order ordinary differential equations of ageodesic circle (Proofs are collected inAppendix A andB). A precise definition of theRiemann mapping/mapping of geodesics into the local tangent space/tangent plane has been found.Secondly we computeRiemann polar/normal coordinates for the biaxial ellipsoid, both in theLagrange portrait (Legendre series) and in theHamilton portrait (Lie series).Thirdly we have succeeded in a detailed deformation analysis/Tissot distortion analysis of theRiemann mapping. The eigenvalues — the eigenvectors of the Cauchy-Green deformation tensor by means of ageneral eigenvalue-eigenvector problem have been computed inTable 3.1 andTable 3.2 (₁, ₂ = 1) illustrated inFigures 3.1, 3.2 and3.3. Table 3.3 contains the representation ofmaximum angular distortion of theRiemann mapping. Fourthly an elaborate global distortion analysis with respect toconformal Gau-Krüger, parallel Soldner andgeodesic Riemann coordinates based upon theAiry total deformation (energy) measure is presented in a corollary and numerically tested inTable 4.1. In a local strip [-l _E,l _E] = [-2°, +2°], [b _S,b _N] = [-2°, +2°]Riemann normal coordinates generate the smallest distortion, next are theparallel Soldner coordinates; the largest distortion by far is met by theconformal Gau-Krüger coordinates. Thus it can be concluded that for mapping of local areas of the biaxial ellipsoid surface the oblique azimuthal projection of geodesic type/Riemann polar/normal coordinates has to be favored with respect to others.     

基于城区机载雷达数据的双重滤波算法

机载激光雷达扫描技术能快速且高精度地获取地面点的3维坐标,而激光雷达数据处理的首要任务就是点云的滤波,也即是将地面点和非地面点进行分离.传统的滤波方法大都是基于一定的地形条件或是小规模数据量进行的.针对城区的3维点云处理提出了一种双重滤波方法:先构建三角网,根据三角面片的角度信息过滤出一部分点云,将剩余点划分成规则格网;然后通过移动最小二乘曲面拟合法,将高差大于一定阈值的点滤除,从而获得地面点云.     

The generalized Mollweide projection of the biaxial ellipsoid

Summary The standard Mollweide projection of the sphere S _R ² which is of type pseudocylindrical — equiareal is generalized to the biaxial ellipsoid E _A,B ² .Within the class of pseudocylindrical mapping equations (1.8) of E _A,B ² (semimajor axis A, semiminor axis B) it is shown by solving the general eigenvalue problem (Tissot analysis) that only equiareal mappings, no conformal mappings exist. The mapping equations (2.1) which generalize those from S _R ² to E _A,B ² lead under the equiareal postulate to a generalized Kepler equation (2.21) which is solved by Newton iteration, for instance (Table 1). Two variants of the ellipsoidal Mollweide projection in particular (2.16), (2.17) versus (2.19), (2.20) are presented which guarantee that parallel circles (coordinate lines of constant ellipsoidal latitude) are mapped onto straight lines in the plane while meridians (coordinate lines of constant ellipsoidal longitude) are mapped onto ellipses of variable axes. The theorem collects the basic results. Six computer graphical examples illustrate the first pseudocylindrical map projection of E _A,B ² of generalized Mollweide type.     

基于球面投影的散乱点云三维建模算法实现与效果分析

介绍基于球面投影的散乱点云三维建模算法步骤,运用VC++编程语言结合OpenGL图形接口实现该算法,并结合实例说明球面投影法相对于圆柱面投影法、椭球面投影法在三维建模中的优势.     

Testing of global pressure/temperature (GPT) model and global mapping function (GMF) in GPS analyses

Several sources of a priori meteorological data have been compared for their effects on geodetic results from GPS precise point positioning (PPP). The new global pressure and temperature model (GPT), available at the IERS Conventions web site, provides pressure values that have been used to compute a priori hydrostatic (dry) zenith path delay z _h estimates. Both the GPT-derived and a simple height-dependent a priori constant z _h performed well for low- and mid-latitude stations. However, due to the actual variations not accounted for by the seasonal GPT model pressure values or the a priori constant z _h, GPS height solution errors can sometimes exceed 10 mm, particularly in Polar Regions or with elevation cutoff angles less than 10 degrees. Such height errors are nearly perfectly correlated with local pressure variations so that for most stations they partly (and for solutions with 5-degree elevation angle cutoff almost fully) compensate for the atmospheric loading displacements. Consequently, unlike PPP solutions utilizing a numerical weather model (NWM) or locally measured pressure data for a priori z _h, the GPT-based PPP height repeatabilities are better for most stations before rather than after correcting for atmospheric loading. At 5 of the 11 studied stations, for which measured local meteorological data were available, the PPP height errors caused by a priori z _h interpolated from gridded Vienna Mapping Function-1 (VMF1) data (from a NWM) were less than 0.5 mm. Height errors due to the global mapping function (GMF) are even larger than those caused by the GPT a priori pressure errors. The GMF height errors are mainly due to the hydrostatic mapping and for the solutions with 10-degree elevation cutoff they are about 50% larger than the GPT a priori errors.     

GPS高程拟合精度探讨

利用GPS水准高程来实现GPS网点的大地高向正常高转换,其精度主要受所拟合的似大地水准面、已知点高程和GPS网点的大地高三种误差的影响.