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.     

Closed formulas for the direct and reverse geodetic problems

A new solution of the direct and reverse geodetic problems has been deduced without series expansion or coordinate transformation. The unknown parameters are directly expressed as explicit functions of the given parameters; the forms of the functions are closed formulas deduced by elementary mathematics using the chord of normal section. Numerical examples prove that the formulas are valid for distances from 40 km to 15 000 km on the surface of the ellipsoid.     

Computation of geodetic direct and indirect problems by computers accumulating increments from geodetic line elements

By choosing sufficiently small elements of the length of the geodetic line, or of the latitude or longitude difference, the other two can be computed at each element and the results can be accumulated to solve the problem with more than twenty significant number accuracy if desired. Ten to twelve number accuracy was computed in the examples of this paper. The geodetic line elements are kept in correct azimuth by Clairaut’s equation for the geodetic line. The computers can do millions of necessary computations very economically in a few seconds. All other published methods solving the direct or indirect problem can be reliably checked against results obtained by this method. The run of geodetic lines around the back side of the Ellipsoid is outlined.     

测地主题正反解解算

由测地坐标系中大地线的微分方程式推导出其微分关系式，得出在地球椭球面上基于测地坐标进行正反解的算法和公式，它与大地主题解算公式相比，更为简捷明了。由实际计算数据表明，对于100km以下的距离解算，它亦能达到相当高的精度。因此测地坐标的点位表述不仅可用于DEM和GIS三维可视化，也可用于三维GIS建模以及空间度量和分析。     

Numerical solution of geodetic boundary value problems using a global reference field

One of the most serious practical limitations of boundary element methods for gravity field determination is that they cannot make efficient use of existing satellite geopotential models. Three basic approaches to solving the problem are developed: (1) alternative representation formulas; (2) modified kernel functions of classical representation formulas; and (3) modified trial and test spaces. The three methods are tested and compared for the altimetry–gravimetry II boundary value problem. It is shown that there is in fact a significant improvement when compared to the pure boundary element solution. Most promising is the method of multiscale trial and test spaces which, in addition, yields sparse system matrices. Received: 7 September 1998 / Accepted: 16 June 1999     

贝赛尔大地主题解算分析

贝赛尔大地主题解算是少数适合长距离大地主题计算的方法之一。文章通过对贝赛尔大地主题解算进行计算分析,发现贝赛尔大地主题反算中的大地线长计算精度受起点方位角的影响很大,误差可达8m。为了消去这一巨大误差,本文提出在大地主题反算时互换大地线起点和终点的方法,计算结果表明该方法可以有效消除方位角对大地线长误差的影响。     

On the gravimetric inverse problem in geodetic gravity field estimation

Gravity field estimation in geodesy, through linear(ized) least squares algorithms, operates under the assumption of Gaussian statistics for the estimable part of preselected models. The causal nature of the gravity field is implicitly involved in its geodetic estimation and introduces the need to include prior model information, as in geophysical inverse problems. Within the geodetic concept of stochastic estimation, the prior information can be in linear form only, meaning that only data linearly depending on the estimates can be used effectively. The consequences of the inverse gravimetric problem in geodetic gravity field estimation are discussed in the context of the various approaches (in model data spaces) which have the common goal to bring into agreement the statistics between these two spaces. With a simple numerical example of FAA prediction, it is shown that prior information affects the accuracy of estimates at least equally as the number of input data. Received: 25 April 1994; Accepted: 15 October 1996     

Boundary-value problems in the complex world of geodetic measurements

A review of recent progress and current activities towards an improved formulation and solution of geodetic boundary value problems is given. Improvements stimulated and required by the dramatic changes of the real world of geodetic measurements are focused upon. Altimetry–gravimetry problems taking into account various scenarios of non-homogeneous data coverage are discussed in detail. Other problems are related to free geodetic datum parameters, most of all the vertical datum, overdetermination or additional constraints imposed by satellite geodetic observations or models. Some brief remarks are made on pseudo-boundary value problems for geoid determination and on purely gravitational boundary-value problems. Received: 17 March 1999 / Accepted: 19 April 1999     

大地主题解算方法综述

大地主题解算是大地测量中的重要问题,由于椭球的复杂性,随之产生的解算方法也是多种多样。本文分析了大地主题解算的一般方法及其公式的适用范围,指出了它们的特点和不足,讨论了现今常用解算方法存在的问题和进一步的研究方向。     

STABILITY OF SOLUTION FOR DYNAMIC GEODETIC NETWORK ADJUSTMENT

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Stability of solution for dynamic geodetic network adjustment

This paper discusses theoretically the stability of solutions for dynamic geodetic network adjustment, Kalman filtering and sequence adjustment, and two examples are given. The solution for dynamic geodetic network adjustment is stable if the dynamic geodetic network is a classical network. There is not rank deficit in datum, or else the solution is not stable, which will depend on the initial value.     

Uniquely and overdetermined geodetic boundary value problems by least squares

Least-squares by observation equations is applied to the solution of geodetic boundary value problems (g.b.v.p.). The procedure is explained solving the vectorial Stokes problem in spherical and constant radius approximation. The results are Stokes and Vening-Meinesz integrals and, in addition, the respective a posteriori variance-covariances. Employing the same procedure the overdeterminedg.b.v.p. has been solved for observable functions potential, scalar gravity, astronomical latitude and longitude, gravity gradients Г_xz, Г_yz, and Г_zz and three-dimensional geocentric positions. The solutions of a large variety of uniquely and overdeterminedg.b.v.p.'s can be obtained from it by specializing weights. Interesting is that the anomalous potential can be determined—up to a constant—from astronomical latitude and longitude in combination with either {Г_xz,Г_yz} or horizontal coordinate corrections Δx and Δy, or both. Dual to the formulation in terms of observation equations the overdeterminedg.b.v.p.'s can as well be solved by condition equations. Constant radius approximation can be overcome in an iterative approach. For the Stokes problem this results in the solution of the “simple” Molodenskii problem. Finally defining an error covariance model with a Krarup-type kernel first results were obtained for a posteriori variance-covariance and reliability analysis.     

The solution of linear inverse problems in satellite geodesy by means of spherical spline approximation

In this paper we consider a certain class of geodetic linear inverse problems in a reproducing kernel Hilbert space setting to obtain a bounded inverse operator . For a numerical realization we assume to be given at a finite number of discrete points to which we employ a spherical spline interpolation method adapted to the Hilbert spaces. By applying to the obtained spline interpolant we get an approximation of the solution . Finally our main task is to show some properties of the approximated solution and to prove convergence results if the data set increases. Received 4 October 1995; Accepted 8 July 1996     

Some problems in the theory of determining geodetic refraction by dispersion method

The effect of the second refraction index gradient and that of humidity on the accuracy of determining refraction by dispersion method is analyzed. The feasibility of obtaining those parameters instrumentally is suggested.     

The solution of the general geodetic boundary value problem by least squares

 A general scheme is given for the solution in a least-squares sense of the geodetic boundary value problem in a spherical, constant-radius approximation, both uniquely and overdetermined, for a large class of observations. The only conditions are that the relation of the observations to the disturbing potential is such that a diagonalization in the spectrum can be found and that the error-covariance function of the observations is isotropic and homogeneous. Most types of observations used in physical geodesy can be adjusted to fit into this approach. Examples are gravity anomalies, deflections of the vertical and the second derivatives of the gravity potential. Received: 3 November 1999 / Accepted: 25 September 2000     

Determination of Indian absolute geodetic datum by least-square solution technique

The Everest spheroid, 1830, in general use in the Survey of India, was finally oriented in an arbitrary manner at the Indian geodetic datum in 1840; while the international spheroid, 1924, in use for scientific purposes; was locally fitted to the Indian geoid in 1927. An attempt is here made to obtain the initial values for the Indian geodetic datum in absolute terms on GRS 67 by least-square solution technique, making use of the available astro-geodetic data in India, and the corresponding generalised gravimetric values at the considered astro-geodetic points, as derived from the mean gravity anomalies over1°×1° squares of latitude and longitude in and around the Indian sub-continent, and over5° equal area blocks covering the rest of the earth’s surface. The values obtained independently by gravimetric method, were also considered before actual finalization of the results of the present determination.     

On the explicit determination of stability constants for linearized geodetic boundary value problems

The theory of GBVPs provide the basis to the approximate methods used to compute global gravity models. A standard approximation procedure is least squares, which implicitly assumes that data, e.g. gravity disturbance and gravity anomaly, are given functions in L ²(S). We know that solutions in these cases exist, but uniqueness (and coerciveness which implies stability of the numerical solutions) is the real problem. Conditions of uniqueness for the linearized fixed boundary and Molodensky problems are studied in detail. They depend on the geometry of the boundary; however, the case of linearized fixed boundary BVP puts practically no constraint on the surface S, while the linearized Molodensky BVP requires the previous knowledge of very low harmonics, for instance up to degree 25, if we want the telluroid to be free to have inclinations up to 60^°.