共查询到20条相似文献，搜索用时 15 毫秒
1.
Toshio Fukushima 《Journal of Geodesy》2006,79(12):689-693
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. 相似文献
2.
C-D. Zhang H.T. Hsu X.P. Wu S.S. Li Q.B. Wang H.Z. Chai L. Du 《Journal of Geodesy》2005,79(8):413-420
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 相似文献
3.
J. Feltens 《Journal of Geodesy》2008,82(8):493-504
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. 相似文献
4.
Marcin Ligas 《Journal of Geodesy》2012,86(4):249-256
A new method of transforming Cartesian to geodetic (or planetographic) coordinates on a triaxial ellipsoid is presented. The
method is based on simple reasoning coming from essentials of vector calculus. The reasoning results in solving a nonlinear
system of equations for coordinates of the point being the projection of a point located outside or inside a triaxial ellipsoid
along the normal to the ellipsoid. The presented method has been compared to a vector method of Feltens (J Geod 83:129–137,
2009) who claims that no other methods are available in the literature. Generally, our method turns out to be more accurate, faster
and applicable to celestial bodies characterized by different geometric parameters. The presented method also fits to the
classical problem of converting Cartesian to geodetic coordinates on the ellipsoid of revolution. 相似文献
5.
R. Lehmann 《Journal of Geodesy》2000,74(3-4):327-334
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 相似文献
6.
Béla Paláncz Joseph L. Awange Piroska Zaletnyik Robert H. Lewis 《Journal of Geodesy》2010,84(1):79-95
A fundamental task in geodesy is solving systems of equations. Many geodetic problems are represented as systems of multivariate
polynomials. A common problem in solving such systems is improper initial starting values for iterative methods, leading to
convergence to solutions with no physical meaning, or to convergence that requires global methods. Though symbolic methods
such as Groebner bases or resultants have been shown to be very efficient, i.e., providing solutions for determined systems
such as 3-point problem of 3D affine transformation, the symbolic algebra can be very time consuming, even with special Computer
Algebra Systems (CAS). This study proposes the Linear Homotopy method that can be implemented easily in high-level computer languages like C++ and Fortran that are faster than CAS by at
least two orders of magnitude. Using Mathematica, the power of Homotopy is demonstrated in solving three nonlinear geodetic problems: resection, GPS positioning, and affine
transformation. The method enlarging the domain of convergence is found to be efficient, less sensitive to rounding of numbers,
and has lower complexity compared to other local methods like Newton–Raphson. 相似文献
7.
Fernando Sansò 《Journal of Geodesy》1981,55(1):17-30
Summary The geodetic boundary value problem (g.b.v.p.) is a free boundary value problem for the Laplace operator: however, under suitable
change of coordinates, it can be transformed into a fixed boundary one. Thus a general coordinate choice problem arises: two
particular cases are more closely analyzed, namely the gravity space approach and the intrinsic coordinates (Marussi) approach. 相似文献
8.
Christopher Kotsakis 《Journal of Geodesy》2008,82(4-5):261-260
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 相似文献
9.
Fast transform from geocentric to geodetic coordinates 总被引：3，自引：0，他引：3
T. Fukushima 《Journal of Geodesy》1999,73(11):603-610
A new iterative procedure to transform geocentric rectangular coordinates to geodetic coordinates is derived. The procedure
solves a modification of Borkowski's quartic equation by the Newton method from a set of stable starters. The new method runs
a little faster than the single application of Bowring's formula, which has been known as the most efficient procedure. The
new method is sufficiently precise because the resulting relative error is less than 10−15, and this method is stable in the sense that the iteration converges for all coordinates including the near-geocenter region
where Bowring's iterative method diverges and the near-polar axis region where Borkowski's non-iterative method suffers a
loss of precision.
Received: 13 November 1998 / Accepted: 27 August 1999 相似文献
10.
11.
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. 相似文献
12.
K. M. Borkowski 《Journal of Geodesy》1989,63(1):50-56
The problem of the transformation is reduced to solving of the equation $$2 sin (\psi - \Omega ) = c sin 2 \psi ,$$ where Ω = arctg[bz/(ar)], c = (a2?b2)/[(ar)2]1/2 a andb are the semi-axes of the reference ellisoid, andz andr are the polar and equatorial, respectively, components of the position vector in the Cartesian system of coordinates. Then, the geodetic latitude is found as ?=arctg [(a/b tg ψ)], and the height above the ellipsoid as h = (r?a cos ψ)cos ψ + (z?b sin ψ)sin ψ. Two accurate closed solutions are proposed of which one is approximative in nature and the other is exact. They are shown to be superior to others, found in literature and in practice, in both or either accuracy and/or simplicity. 相似文献
13.
The goal of this paper is to present the finite element scheme for solving the Earth potential problems in 3D domains above
the Earth surface. To that goal we formulate the boundary-value problem (BVP) consisting of the Laplace equation outside the
Earth accompanied by the Neumann as well as the Dirichlet boundary conditions (BC). The 3D computational domain consists of
the bottom boundary in the form of a spherical approximation or real triangulation of the Earth’s surface on which surface
gravity disturbances are given. We introduce additional upper (spherical) and side (planar and conical) boundaries where the
Dirichlet BC is given. Solution of such elliptic BVP is understood in a weak sense, it always exists and is unique and can
be efficiently found by the finite element method (FEM). We briefly present derivation of FEM for such type of problems including
main discretization ideas. This method leads to a solution of the sparse symmetric linear systems which give the Earth’s potential
solution in every discrete node of the 3D computational domain. In this point our method differs from other numerical approaches,
e.g. boundary element method (BEM) where the potential is sought on a hypersurface only. We apply and test FEM in various
situations. First, we compare the FEM solution with the known exact solution in case of homogeneous sphere. Then, we solve
the geodetic BVP in continental scale using the DNSC08 data. We compare the results with the EGM2008 geopotential model. Finally,
we study the precision of our solution by the GPS/levelling test in Slovakia where we use terrestrial gravimetric measurements
as input data. All tests show qualitative and quantitative agreement with the given solutions. 相似文献
14.
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 相似文献
15.
The perspective 4 point (P4P) problem - also called the three-dimensional resection problem - is solved by means of a new algorithm: At first the unknown Cartesian coordinates of the perspective center are computed by means of M?bius barycentric coordinates. Secondly these coordinates are represented in terms of observables, namely space angles in the five-dimensional simplex
generated by the unknown point and the four known points. Substitution of M?bius barycentric coordinates leads to the unknown Cartesian coordinates (2.8)–(2.10) of Box 2.2. The unknown distances within the five-dimensional simplex are determined by solving the Grunert equations, namely by forward reduction to one algebraic equation (3.8) of order four and backward linear substitution. Tables 1.–4.
contain a numerical example. Finally we give a reference to the solution of the 3 point (P3P) problem, the two-dimensional resection problem, namely to the Ansermet barycentric coordinates initiated by C.F. Gau? (1842), A. Schreiber (1908) and A.␣Ansermet (1910).
Received: 05 March 1996; Accepted: 15 October 1996 相似文献
16.
Claudio Abbondanza Zuheir Altamimi Pierguido Sarti Monia Negusini Luca Vittuari 《Journal of Geodesy》2009,83(11):1031-1040
Tie vectors (TVs) between co-located space geodetic instruments are essential for combining terrestrial reference frames (TRFs)
realised using different techniques. They provide relative positioning between instrumental reference points (RPs) which are
part of a global geodetic network such as the international terrestrial reference frame (ITRF). This paper gathers the set
of very long baseline interferometry (VLBI)–global positioning system (GPS) local ties performed at the observatory of Medicina
(Northern Italy) during the years 2001–2006 and discusses some important aspects related to the usage of co-location ties
in the combinations of TRFs. Two measurement approaches of local survey are considered here: a GPS-based approach and a classical
approach based on terrestrial observations (i.e. angles, distances and height differences). The behaviour of terrestrial local
ties, which routinely join combinations of space geodetic solutions, is compared to that of GPS-based local ties. In particular,
we have performed and analysed different combinations of satellite laser ranging (SLR), VLBI and GPS long term solutions in
order to (i) evaluate the local effects of the insertion of the series of TVs computed at Medicina, (ii) investigate the consistency
of GPS-based TVs with respect to space geodetic solutions, (iii) discuss the effects of an imprecise alignment of TVs from
a local to a global reference frame. Results of ITRF-like combinations show that terrestrial TVs originate the smallest residuals
in all the three components. In most cases, GPS-based TVs fit space geodetic solutions very well, especially in the horizontal
components (N, E). On the contrary, the estimation of the VLBI RP Up component through GPS technique appears to be awkward,
since the corresponding post fit residuals are considerably larger. Besides, combination tests including multi-temporal TVs
display local effects of residual redistribution, when compared to those solutions where Medicina TVs are added one at a time.
Finally, the combination of TRFs turns out to be sensitive to the orientation of the local tie into the global frame. 相似文献
17.
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 相似文献
18.
P. J. G. Teunissen 《Journal of Geodesy》1982,56(4):356-363
For computing the geodetic coordinates ϕ and γ on the ellipsoid one needs information of the gravity field, thus making it
possible to reduce the terrestrial observations to the reference surface. Neglect of gravity field data, such as deflections
of the vertical and geoid heights, results in misclosure effects, which can be described using the object of anholonomity. 相似文献
19.
Hugues Vermeille 《Journal of Geodesy》2011,85(2):105-117
A closed-form analytical method needing no approximation and deduced from a single quartic equation is offered to transform
geocentric into geodetic coordinates. It is valid at any point inside and outside the Earth including the polar axis, the
equatorial plane and the Earth’s center. Comparison with the method of extrema with constraints to obtain this quartic equation
is made. 相似文献
20.
J. C. Bhattacharji 《Journal of Geodesy》1973,47(1):65-72
The method of converting geodetic coordinates from a national geodetic reference system into the standard Earth on having
known the geodetic coordinates of at least one station in common with the considered systems, is described in detail; the
orientation of the Standard Earth at the initial station of the national geodetic reference system, is also determined side
by side. For illustration, use has been made of the known coordinates of the Baker-Nunn station at Naini Tal, in India, being
in common with the Indian Everest Spheroid and the Smithsonian Institution Standard Earth C7 system (Veis, 1967). The method advocated is likely to be more precise than the existing ones as it does not assume the parallelism
of axes of reference between the Standard Earth and the national geodetic reference systems which may not necessarily hold
good in actual practice. 相似文献