A symbolic analysis of Vermeille and Borkowski polynomials for transforming 3D Cartesian to geodetic coordinates

Closed form solutions for transforming 3D Cartesian to geodetic coordinates reduce the problem to finding the real solutions of the fourth degree latitude equation or variations of it. By using symbolic tools (Sturm–Habicht coefficients and subresultants mainly) we study the methods (and polynomials) proposed by Vermeille and Borkowski to solve this problem. For Vermeille approach, the region where it cannot be applied is completely characterized. For those points it is shown how to transform 3D Cartesian to geodetic coordinates and a new method for solving Vermeille equation for those cases not yet covered is introduced. Concerning Borkowski’s approach, the symbolic analysis produces a complete characterization of the singular cases (i.e. where multiple roots appear).     

Singularity free formulations of the geodetic boundary value problem in gravity-space

The idea of transforming the geodetic boundary value problem into a boundary value problem with a fixed boundary dates back to the 1970s of the last century. This transformation was found by F. Sanso and was named as gravity-space transformation. Unfortunately, the advantage of having a fixed boundary for the transformed problem was counterbalanced by the theoretical as well as practical disadvantage of a singularity at the origin. In the present paper two more versions of a gravity-space transformation are investigated, where none of them has a singularity. In both cases the transformed differential equations are nonlinear. Therefore, a special emphasis is laid on the linearized problems and their relationships to the simple Hotine-problem and to the symmetries between both formulations. Finally, in numerical simulation study the accuracy of the solutions of both linearized problems is studied and factors limiting this accuracy are identified.     

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.     

A closed-form of Newton method for solving over-determined pseudo-distance equations

The Newton method has been widely used for solving nonlinear least-squares problem. In geodetic adjustment, one would prefer to use the Gauss–Newton method because of the parallel with linear least-squares problem. However, it is proved in theory as well as in practice that the Gauss–Newton method has slow convergence rate and low success rate. In this paper, the over-determined pseudo-distance equations are solved by nonlinear methods. At first, the convergence of decent methods is discussed after introducing the conditional equation of nonlinear least squares. Then, a compacted form of the Hessian matrix from the second partial derivates of the pseudo-distance equations is given, and a closed-form of Newton method is presented using the compacted Hessian matrix to save the computation and storage required by Newton method. At last, some numerical examples to investigate the convergence and success rate of the proposed method are designed and performed. The performance of the closed-form of Newton method is compared with the Gauss–Newton method as well as the regularization method. The results show that the closed-form of Newton method has good performances even for dealing with ill-posed problems while a great amount of computation is saved.     

Altimetry–gravimetry problems with free vertical datum

 The definition and connection of vertical datums in geodetic height networks is a fundamental problem in geodesy. Today, the standard approach to solve it is based on the joint processing of terrestrial and satellite geodetic data. It is generalized to cases where the coverage with terrestrial data may change from region to region, typically across coastlines. The principal difficulty is that such problems, so-called altimetry–gravimetry boundary-value problems (AGPs), do not admit analytical solutions such as Stokes' integral. A numerical solution strategy for the free-datum problem is presented. Analysis of AGPs in spherical and constant radius approximation shows that two of them are mathematically well-posed problems, while the classical AGP-I may be ill posed in special situations. Received: 2 December 1998 / Accepted: 30 November 1999     

Parameter estimation in 3D affine and similarity transformation: implementation of variance component estimation

Three-dimensional (3D) coordinate transformations, generally consisting of origin shifts, axes rotations, scale changes, and skew parameters, are widely used in many geomatics applications. Although in some geodetic applications simplified transformation models are used based on the assumption of small transformation parameters, in other fields of applications such parameters are indeed large. The algorithms of two recent papers on the weighted total least-squares (WTLS) problem are used for the 3D coordinate transformation. The methodology can be applied to the case when the transformation parameters are generally large of which no approximate values of the parameters are required. Direct linearization of the rotation and scale parameters is thus not required. The WTLS formulation is employed to take into consideration errors in both the start and target systems on the estimation of the transformation parameters. Two of the well-known 3D transformation methods, namely affine (12, 9, and 8 parameters) and similarity (7 and 6 parameters) transformations, can be handled using the WTLS theory subject to hard constraints. Because the method can be formulated by the standard least-squares theory with constraints, the covariance matrix of the transformation parameters can directly be provided. The above characteristics of the 3D coordinate transformation are implemented in the presence of different variance components, which are estimated using the least squares variance component estimation. In particular, the estimability of the variance components is investigated. The efficacy of the proposed formulation is verified on two real data sets.     

Nonlinear analysis of the three-dimensional datum transformation [conformal group ℂ7(3)]

 The weighted Procrustes algorithm is presented as a very effective tool for solving the three-dimensional datum transformation problem. In particular, the weighted Procrustes algorithm does not require any initial datum parameters for linearization or any iteration procedure. As a closed-form algorithm it only requires the values of Cartesian coordinates in both systems of reference. Where there is some prior information about the variance–covariance matrix of the two sets of Cartesian coordinates, also called pseudo-observations, the weighted Procrustes algorithm is able to incorporate such a quality property of the input data by means of a proper choice of weight matrix. Such a choice is based on a properly designed criterion matrix which is discussed in detail. Thanks to the weighted Procrustes algorithm, the problem of incorporating the stochasticity measures of both systems of coordinates involved in the seven parameter datum transformation problem [conformal group ℂ₇(3)] which is free of linearization and any iterative procedure can be considered to be solved. Illustrative examples are given. Received: 7 January 2002 / Accepted: 9 September 2002 Correspondence to: E. W. Grafarend     

图件更新北京54和西安80坐标系转换方法研究

本文结合国土资源调查图件更新北京54和西安80坐标系转换的具体特点,对各种坐标转换模型进行了分析,研究了一种实用的坐标转换模型。在此基础上编制了适用于国土资源调查中图件更新北京54和西安80坐标系转换特定用途的坐标转换软件,利用实际资料对软件进行了测试分析,测试结果表明,这种实用的坐标转换方法可以满足国土资源调查对坐标转换精度的要求。     

解混合边值问题直接计算位系数的变分原理

本文提出了利用变分法解混合边值问题直接计算位系数的原理。根据这一原理可解第一、第二和第三边值问题的混合边值问题直接求得位系数。利用这一原理可较简单地联合利用经典重力测量(即重力点的平面位置由天文或三角测量确定,高程由水准或三角高程确定)、卫星重力测量(即利用卫星定位技术确定重力点的平面位置和大地高)以及卫星测高数据研究地球的重力场。     

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。     

以地心纬度为变量的常用纬度正反解符号表达式

借助具有强大符号运算功能的计算机代数系统Mathematica,推导了地图投影学中等距离纬度、等角纬度、等面积纬度与地心纬度之间的正反解直接展开式,并将式中的系数统一表示成关于椭圆偏心率e和椭球第三扁率n的幂级数形式。与以往反解方法不同的是,采用符号迭代法进行以地心纬度为变量的等距离纬度、等角纬度、等面积纬度的反解,并使用最大差异值作为衡量精度的标准。算例分析表明,以地心纬度为变量的常用纬度展开式在结构和形式上与以大地纬度为变量的辅助纬度保持一致,基于第三扁率n的幂级数表达式具有更紧凑的形式和更好的收敛性,其直接展开式的精度分别优于(1×10-8)″和(1×10-10)″,可以满足大地测量和地图投影精密计算的需要。     

Singular value decomposition and cluster analysis as regression diagnostics tools for geodetic applications

It is well known that high-leverage observations significantly affect the estimation of parameters. In geodetic literature, mainly redundancy numbers are used for the detection of single high-leverage observations or of single redundant observations. In this paper a further objective method for the detection of groups of important and less important (and thus redundant) observations is developed. In addition, the parameters which are predominantly affected by these groups of observations are identified. This method thus complements other diagnostics tools, such as, e.g., multiple row diagnostics methods as described in statistical literature (see, e.g., Belsley et al. in Regression diagnostics: identifying influential data and sources of collinearity. Wiley, New York, 1980). The method proposed in this paper is based on geometric aspects of adjustment theory and uses the singular value decomposition of the design matrix of an adjustment problem together with cluster analysis methods for regression diagnostics. It can be applied to any geodetic adjustment problem and can be used for the detection of (groups of) observations that significantly affect the estimated parameters or that are of negligible impact. One of the advantages of the proposed method is the improvement of the reliability of observation plans and thus the reduction of the impact of individual observations (and outliers) on the estimated parameters. This is of particular importance for the very long baseline interferometry technique which serves as an application example of the regression diagnostics tool.     

Transforming ellipsoidal heights and geoid undulations between different geodetic reference frames

Transforming height information that refers to an ellipsoidal Earth reference model, such as the geometric heights determined from GPS measurements or the geoid undulations obtained by a gravimetric geoid solution, from one geodetic reference frame (GRF) to another is an important task whose proper implementation is crucial for many geodetic, surveying and mapping applications. This paper presents the required methodology to deal with the above problem when we are given the Helmert transformation parameters that link the underlying Cartesian coordinate systems to which an Earth reference ellipsoid is attached. The main emphasis is on the effect of GRF spatial scale differences in coordinate transformations involving reference ellipsoids, for the particular case of heights. Since every three-dimensional Cartesian coordinate system ‘gauges’ an attached ellipsoid according to its own accessible scale, there will exist a supplementary contribution from the scale variation between the involved GRFs on the relative size of their attached reference ellipsoids. Neglecting such a scale-induced indirect effect corrupts the values for the curvilinear geodetic coordinates obtained from a similarity transformation model, and meter-level apparent offsets can be introduced in the transformed heights. The paper explains the above issues in detail and presents the necessary mathematical framework for their treatment. An erratum to this article can be found at     

Geoid determination using one-step integration

A residual (high-frequency) gravimetric geoid is usually computed from geographically limited ground, sea and/or airborne gravimetric data. The mathematical model for its determination from ground gravity is based on the transformation of observed discrete values of gravity into gravity potential related to either the international ellipsoid or the geoid. The two reference surfaces are used depending on height information that accompanies ground gravity data: traditionally orthometric heights determined by geodetic levelling were used while GPS positioning nowadays allows for estimation of geodetic (ellipsoidal) heights. This transformation is usually performed in two steps: (1) observed values of gravity are downward continued to the ellipsoid or the geoid, and (2) gravity at the ellipsoid or the geoid is transformed into the corresponding potential. Each of these two steps represents the solution of one geodetic boundary-value problem of potential theory, namely the first and second or third problem. Thus two different geodetic boundary-value problems must be formulated and solved, which requires numerical evaluation of two surface integrals. In this contribution, a mathematical model in the form of a single Fredholm integral equation of the first kind is presented and numerically investigated. This model combines the solution of the first and second/third boundary-value problems and transforms ground gravity disturbances or anomalies into the harmonically downward continued disturbing potential at the ellipsoid or the geoid directly. Numerical tests show that the new approach offers an efficient and stable solution for the determination of the residual geoid from ground gravity data.     

不同大地坐标系间相互转换的病态解决及抗差算法

不同大地坐标系间进行坐标转换是利用具有两个坐标系下坐标的公共点,求取转换参数 经常会遇到系数矩阵病态导致转换精度差的问题,且公共点的坐标精度直接影响转换参数的求解精度,也就是影响坐标转换的精度.本文探讨利用LC曲线法、截断奇异值法及广义交叉检验准则法解决病态问题,同时采用抗差估计理论进行不同大地坐标间的转换.当公共点...     

New solutions for the geodetic coordinate transformation

 The Cartesian-to-geodetic-coordinate transformation is approached from a new perspective. Existence and uniqueness of geodetic representation are presented, along with a clear geometric picture of the problem and the role of the ellipse evolute. A new solution is found with a Newton-method iteration in the reduced latitude; this solution is proved to work for all points in space. Care is given to error propagation when calculating the geodetic latitude and height. Received: 9 August 2001 / Accepted: 27 March 2002 Acknowledgments. The author would like to thank the Clifford W.␣Tompson scholarship fund, Dr. Brian DeFacio, the University of Missouri College of Arts &Sciences, and the United States Air Force. He also thanks a reviewer for suggesting and providing a prototype MATLAB code. A MATLAB program for the iterative sequence is presented at the end of the paper (Appendix A).     

Algebraic Solution of GPS Pseudo-Ranging Equations

Several procedures for solving, in a closed form the GPS pseudo-ranging four-point problem P4P in matrix form already exist. We present here alternative algebraic procedures using Multipolynomial resultant and Groebner basis to solve the same problem. The advantage is that these algebraic algorithms have already been implemented in algebraic software such as “Mathematica” and “Maple”. The procedures are straightforward and simple to apply. We illustrate here how the algebraic techniques of Multipolynomial resultant and Groebner basis explicitly solve the nonlinear GPS pseudo-ranging four-point equations once they have been converted into algebraic (polynomial) form and reduced to linear equations. In particular, the algebraic tools of Multipolynomial resultant and Groebner basis provide symbolic solutions to the GPS four-point pseudo-ranging problem. The various forward and backward substitution steps inherent in the clasical closed form solutions of the problem are avoided. Similar to the Gauss elimination techniques in linear systems of equations, the Multipolynomial resultant and Groebner basis approaches eliminate several variables in a multivariate system of nonlinear equations in such a manner that the end product normally consists of univariate polynomial equations (in this case quadratic equations for the range bias expressed algebraically using the given quantities) whose roots can be determined by existing programs (e. g., the roots command in MATLAB). © 2002 Wiley Periodicals, Inc.     

Testing for spatial association of qualitative data using symbolic dynamics

Qualitative spatial variables are important in many fields of research. However, unlike the decades-worth of research devoted to the spatial association of quantitative variables, the exploratory analysis of spatial qualitative variables is relatively less developed. The objective of the present paper is to propose a new test (Q) for spatial independence. This is a simple, consistent, and powerful statistic for qualitative spatial independence that we develop using concepts from symbolic dynamics and symbolic entropy. The Q test can be used to detect, given a spatial distribution of events, patterns of spatial association of qualitative variables in a wide variety of settings. In order to enable hypothesis testing, we give a standard asymptotic distribution of an affine transformation of the symbolic entropy under the null hypothesis of independence in the spatial qualitative process. We include numerical experiments to demonstrate the finite sample behaviour of the test, and show its application by means of an empirical example that explores the spatial association of fast food establishments in the Greater Toronto Area in Canada.     

On weighted total least-squares with linear and quadratic constraints

A weighted total least-squares (WTLS) approach with linear and quadratic constraints is developed. This method is according to the traditional Lagrange approach to optimize the target function of this problem. The WTLS and constrained total least-squares (CTLS) approach had been distinctively investigated, however, these two problems have not been simultaneously considered yet; furthermore, among the contributions on the CTLS problem, only Schaffrin and Felus considered linear and quadratic constraints together; nevertheless, in many practical examples, some elements of the design/coefficient matrix are fixed and should not be modified and this approach cannot deal with these cases. The main necessity of this research appears after the desirable property of the WTLS approach in preserving the structure of the design matrix was proven by Mahboub. In other words, currently, the WTLS approach is one of the most efficient methods for solving the so-called errors-in-variables model and an attempt for equipping it with constraints seems necessary. Also it is demonstrated that the additional constraints have a ’regularization role’ for ill-conditioned problems and the unconstrained solution suffers from ill-conditioning effects which give it an added advantage over the unconstrained WTLS algorithm. Four geodetic applications indicate the significant of this problem in the presence of colored and white noise in the data.     

The Exploratory Essays Initiative: Background and Overview

We define as Positional Accuracy Improvement the problem of putting together maps A and B of the same area, with B of higher planimetric accuracy. To do so, all objects in A might have to be slightly moved according to a mathematical transformation. Such transformation might ideally be of a specific type, like analytical or conformal functions. We have developed a theory to find a suitable analytical transformation despite it is not well defined because the only data available is the displacement vectors at a limited number of homologue control points. There exists a similar problem in fluid mechanics devoted on estimating the complete velocity field given just values at a limited number of points. We borrowed some ideas from there and introduced them into the positional accuracy improvement problem. We shall demonstrate that it is possible to numerically estimate an analytic function that resembles the given displacement at control points. As a byproduct, an uncertainty estimation is produced, which might help to detect regions of different lineage. The theory has been applied to rural 1:50.000 cartography of Uruguay while trying to diminish the discrepancies against GNSS readings. After the analytic transformation, the RMSE error diminished from 116 m to 48 m. Other problems with similar math requirements are the transformation between geodetic control networks.