Is Newton's iteration faster than simple iteration for transformation between geocentric and geodetic coordinates?

Summary Two iterative algorithms for transformation from geocentric to geodetic coordinates are compared for numerical efficiency: the well known Bowring's algorithm of 1976, which employs the method of simple iteration, and the recent (1989) algorithm by Borkowski, which employs the Newton-Raphson method. The results of numerical tests suggest that the simple iteration method implemented in Bowring's algorithm executes approximately 30% faster than the Newton-Raphson method implemented in Borkowski's algorithm. Only two iterations of each algorithm are considered. Two iterations are sufficient to produce coordinates accurate to the comparable level of 1E-9 m, which exceeds the requirements of any practical application. Therefore, in the class of iterative methods, the classical Bowring's algorithm should be the method of choice.     

Transformation from geocentric to geodetic coordinates using Newton's iteration

Summary A new procedure for the transformation from geocentric to geodetic coordinates is introduced and analyzed. This new procedure which contains only one trigonometric function uses the Newton's iteration to solve the root of a non-linear equation. Compared with the well-known Bowring's iterative algorithm which uses a number of trigonometric functions, the new procedure is more efficient in computation. Numeric examples are used to test the two algorithms and the results show that the new procedure converges just as well as Bowring's algorithm but requires less time for completion.     

Transformation from Cartesian to Geodetic Coordinates Accelerated by Halley’s Method

By using Halley’s third-order formula to find the root of a non-linear equation, we develop a new iterative procedure to solve an irrational form of the “latitude equation”, the equation to determine the geodetic latitude for given Cartesian coordinates. With a limit to one iteration, starting from zero height, and minimizing the number of divisions by means of the rational form representation of Halley’s formula, we obtain a new non-iterative method to transform Cartesian coordinates to geodetic ones. The new method is sufficiently precise in the sense that the maximum error of the latitude and the relative height is less than 6 micro-arcseconds for the range of height, −10 km ≤ h ≤ 30,000 km. The new method is around 50% faster than our previous method, roughly twice as fast as the well-known Bowring’s method, and much faster than the recently developed methods of Borkowski, Laskowski, Lin and Wang, Jones, Pollard, and Vermeille.     

An alternative algebraic algorithm to transform Cartesian to geodetic coordinates

The algorithm to transform from 3D Cartesian to geodetic coordinates is obtained by solving the equation of the Lagrange parameter. Numerical experiments show that geodetic height can be recovered to 0.5 mm precision over the range from −6×10⁶ to 10¹⁰ m. Electronic Supplementary Material: Supplementary material is available in the online version of this article at     

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.     

Iterative vector methods for computing geodetic latitude and height from rectangular coordinates

 Two iterative vector methods for computing geodetic coordinates (φ, h) from rectangular coordinates (x, y, z) are presented. The methods are conceptually simple, work without modification at any latitude and are easy to program. Geodetic latitude and height can be calculated to acceptable precision in one iteration over the height range from −10⁶ to +10⁹ m. Received: 13 December 2000 / Accepted: 13 July 2001     

An analysis of the scalar geodetic boundary-value problem with natural regularity results

A new local existence and uniqueness theorem is obtained for the scalar geodetic boundary-value problem in spherical coordinates. The regularities H _α and H _1+α are assumed for the boundary data g (gravity) and v (gravitational potential) respectively. Received: 27 July 1998 / Accepted: 19 April 1999     

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     

Hyperboloidal coordinates: transformations and applications in special constructions

The vector-based algorithms for biaxial and triaxial ellipsoidal coordinates presented by Feltens (J Geod 82:493–504, 2008; 83:129–137, 2009) have been extended to hyperboloids of one sheet. For the backward transformation from Cartesian to hyperboloidal coordinates, of two iterative process candidates one was identified to be well suited. It turned out that a careful selection of the center of curvature is essential for the establishment of a stable and reliable iteration process. In addition, for zero hyperboloidal heights a closed solution is presented. The hyperboloid algorithms are again based on simple formulae and have been successfully tested for various theoretical hyperboloids. The paper concludes with a practical application example on a cooling tower construction.     

The terrestrial and lunar reference frame in lunar laser ranging

The initial value problem and the stability of solution in the determination of the coordinates of three observing stations and four retro-reflectors by lunar laser ranging are discussed. Practical iterative computations show that the station coordinates can be converged to about 1 cm, but there will be a slight discrepancy of the longitudinal components computed by various analysis centers or in different years. There are several factors, one of which is the shift of the right ascension of the Moon, caused by the orientation deviation of the adopted lunar ephemeris, which can make the longitudinal components of all observing stations rotate together along the longitudinal direction with same angle. Additionally, the frame of selenocentric coordinates is stable, but a variation or adjustment of lunar third-degree gravitational coefficients will cause a simultaneous shift along the reflectors' longitudes or rotation around the Y axis. Received: 21 August 1996 / Accepted: 17 November 1998     

Study of a wide-angle laser ranging system for relative positioning of ground-based benchmarks with millimeter accuracy

A wide-angle airborne laser ranging system (WA-ALRS) is developed at the Institut Géographique National (IGN), France, with the aim of providing a new geodesy technique devoted to large (100 km²) networks with a high density (1 km⁻²) of benchmarks. The main objective is to achieve a 1-mm accuracy in relative vertical coordinates from aircraft measurements lasting a few hours. This paper reviews the methodology and analyzes the first experimental data achieved from a specific ground-based experiment. The accuracy in relative coordinate estimates is studied with the help of numerical simulations. It is shown that strong accuracy limitations arise with a small laser beam divergence combined with short range measurements when relatively few simultaneous range data are produced. The accuracy is of a few cm in transverse coordinates and a few mm in radial coordinates. The results from ground-based experimental data are fairly compatible with these predictions. The use of a model for systematic errors in the vehicle trajectory is shown to be necessary to achieve such a high accuracy. This work yields the first complete validation of modelization and methodology of this technique. An accuracy better than 1 mm and a few mm in vertical and horizontal coordinates, respectively, is predicted for aircraft experiments. Received: 19 June 1997 / Accepted: 17 February 1998     

Direct transformation from geocentric coordinates to geodetic coordinates

 The transformation from geocentric coordinates to geodetic coordinates is usually carried out by iteration. A closed-form algebraic method is proposed, valid at any point on the globe and in space, including the poles, regardless of the value of the ellipsoid's eccentricity. Received: 14 August 2000 / Accepted: 26 June 2002     

On some problems of the downward continuation of the 5′×5′ mean Helmert gravity disturbance

This research deals with some theoretical and numerical problems of the downward continuation of mean Helmert gravity disturbances. We prove that the downward continuation of the disturbing potential is much smoother, as well as two orders of magnitude smaller than that of the gravity anomaly, and we give the expression in spectral form for calculating the disturbing potential term. Numerical results show that for calculating truncation errors the first 180^∘ of a global potential model suffice. We also discuss the theoretical convergence problem of the iterative scheme. We prove that the 5^′×5^′ mean iterative scheme is convergent and the convergence speed depends on the topographic height; for Canada, to achieve an accuracy of 0.01 mGal, at most 80 iterations are needed. The comparison of the “mean” and “point” schemes shows that the mean scheme should give a more reasonable and reliable solution, while the point scheme brings a large error to the solution. Received: 19 August 1996 / Accepted: 4 February 1998     

On the multivariate total least-squares approach to empirical coordinate transformations. Three algorithms

The multivariate total least-squares (MTLS) approach aims at estimating a matrix of parameters, Ξ, from a linear model (Y−E _Y = (X−E _X ) · Ξ) that includes an observation matrix, Y, another observation matrix, X, and matrices of randomly distributed errors, E _Y and E _X . Two special cases of the MTLS approach include the standard multivariate least-squares approach where only the observation matrix, Y, is perturbed by random errors and, on the other hand, the data least-squares approach where only the coefficient matrix X is affected by random errors. In a previous contribution, the authors derived an iterative algorithm to solve the MTLS problem by using the nonlinear Euler–Lagrange conditions. In this contribution, new lemmas are developed to analyze the iterative algorithm, modify it, and compare it with a new ‘closed form’ solution that is based on the singular-value decomposition. For an application, the total least-squares approach is used to estimate the affine transformation parameters that convert cadastral data from the old to the new Israeli datum. Technical aspects of this approach, such as scaling the data and fixing the columns in the coefficient matrix are investigated. This case study illuminates the issue of “symmetry” in the treatment of two sets of coordinates for identical point fields, a topic that had already been emphasized by Teunissen (1989, Festschrift to Torben Krarup, Geodetic Institute Bull no. 58, Copenhagen, Denmark, pp 335–342). The differences between the standard least-squares and the TLS approach are analyzed in terms of the estimated variance component and a first-order approximation of the dispersion matrix of the estimated parameters.     

W0: an estimate in the Finnish Height Datum N60, epoch 1993.4, from twenty-five GPS points of the Baltic Sea Level Project

Based upon a data set of 25 points of the Baltic Sea Level Project, second campaign 1993.4, which are close to mareographic stations, described by (1) GPS derived Cartesian coordinates in the World Geodetic Reference System 1984 and (2) orthometric heights in the Finnish Height Datum N60, epoch 1993.4, we have computed the primary geodetic parameter W ₀(1993.4) for the epoch 1993.4 according to the following model. The Cartesian coordinates of the GPS stations have been converted into spheroidal coordinates. The gravity potential as the additive decomposition of the gravitational potential and the centrifugal potential has been computed for any GPS station in spheroidal coordinates, namely for a global spheroidal model of the gravitational potential field. For a global set of spheroidal harmonic coefficients a transformation of spherical harmonic coefficients into spheroidal harmonic coefficients has been implemented and applied to the global spherical model OSU 91A up to degree/order 360/360. The gravity potential with respect to a global spheroidal model of degree/order 360/360 has been finally transformed by means of the orthometric heights of the GPS stations with respect to the Finnish Height Datum N60, epoch 1993.4, in terms of the spheroidal “free-air” potential reduction in order to produce the spheroidal W ₀(1993.4) value. As a mean of those 25 W ₀(1993.4) data as well as a root mean square error estimation we computed W ₀(1993.4)=(6 263 685.58 ± 0.36) kgal × m. Finally a comparison of different W ₀ data with respect to a spherical harmonic global model and spheroidal harmonic global model of Somigliana-Pizetti type (level ellipsoid as a reference, degree/order 2/0) according to The Geodesist's Handbook 1992 has been made. Received: 7 November 1996 / Accepted: 27 March 1997     

A new gravity map, a new marine geoid around Japan and the detection of the Kuroshio current

About half a million marine gravity measurements over a 30^∘×30^∘ area centered on Japan have been processed and adjusted to produce a new free-air gravity map from a 5′×5′ grid. This map seems to have a better resolution than those previously published as measured by its correlation with bathymetry. The grid was used together with a high-degree and -order spherical harmonics geopotential model to compute a detailed geoid with two methods: Stokes integral and collocation. Comparisons with other available geoidal surfaces derived either from gravity or from satellite altimetry were made especially to test the ability of this new geoid at showing the sea surface topography as mapped by the Topex/Poseidon satellite. Over 2 months (6 cycles) the dynamic topography at ascending passes in the region (23^∘47^∘N and 123^∘147^∘E) was mapped to study the variability of the Kuroshio current. Received: 15 July 1994 / Accepted: 17 February 1997     

Nonlinear Adjustment of GPS Observations of Type Pseudo-Ranges

The nonlinear adjustment of GPS observations of type pseudo-ranges is performed in two steps. In step one a combinatorial minimal subset of observations is constructed which is rigorously converted into station coordinates by means of Groebner basis algorithm or the multipolynomial resultant algorithm. The combinatorial solution points in a polyhedron are reduced to their barycentric in step two by means of their weighted mean. Such a weighted mean of the polyhedron points in ℝ³ is generated via the Error Propagation law/variance-covariance propagation. The Fast Nonlinear Adjustment Algorithm (FNon Ad Al) has been already proposed by Gauss whose work was published posthumously and Jacobi (1841). The algorithm, here referred to as the Gauss-Jacobi Combinatorial algorithm, solves the over-determined GPS pseudo-ranging problem without reverting to iterative or linearization procedure except for the second moment (Variance-Covariance propagation). The results compared well with the solutions obtained using the linearized least squares approach giving legitimacy to the Gauss-Jacobi combinatorial procedure. ? 2002 Wiley Periodicals, Inc.     

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.     

以重心坐标为基准的空间后方交会非迭代法

为解决基于迭代的空间后方交会算法在倾斜摄影中可能出现的不收敛现象,提出了一种以重心坐标为基准的非迭代解算方法。首先将控制点物方空间坐标描述成重心坐标,并基于其坐标参考无关性,采用总体最小二乘方法求出对应像方空间坐标,然后通过正交矩阵方法进行绝对定向并优化。试验结果表明,该方法几乎对任意影像姿态均能正确解算,并且精度达到甚至优于基于迭代的空间后方交会方法。     

The IAG approach to the atmospheric geoid correction in Stokes' formula and a new strategy

The well-known International Association of Geodesy (IAG) approach to the atmospheric geoid correction in connection with Stokes' integral formula leads to a very significant bias, of the order of 3.2 m, if Stokes' integral is truncated to a limited region around the computation point. The derived truncation error can be used to correct old results. For future applications a new strategy is recommended, where the total atmospheric geoid correction is estimated as the sum of the direct and indirect effects. This strategy implies computational gains as it avoids the correction of direct effect for each gravity observation, and it does not suffer from the truncation bias mentioned above. It can also easily be used to add the atmospheric correction to old geoid estimates, where this correction was omitted. In contrast to the terrain correction, it is shown that the atmospheric geoid correction is mainly of order H of terrain elevation, while the term of order H ² is within a few millimetres. Received: 20 May 1998 / Accepted: 19 April 1999