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     

Fast transform from geocentric to geodetic coordinates

 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     

Computing geodetic coordinates from geocentric coordinates

A closed-form algebraic method to transform geocentric coordinates to geodetic coordinates has previously been proposed. The validity domain of latitude and height formulae in the vicinity of the Earths core is specified. A new expression of longitude is proposed, excluding indetermination and sensitivity to round-off error around the ±180 degrees longitude discontinuity.     

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     

On determination of heights by using terrestrial clocks and GPS signals

 Considering a GPS satellite and two terrestrial stations, two types of equations are derived relating the heights of the two stations to the measured data (frequency ratio or clock rate differences) and the coordinates and velocity components of all three participating objects. The potential possibilities of using such relations for the determination of heights (in terms of geopotential numbers or orthometric heights) are discussed. Received: 6 December 2000 / Accepted: 9 July 2001     

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     

后向投影算法的地理坐标系下的合成孔径雷达成像

传统的机载合成孔径雷达(SAR)一般依赖于GPS/INS组合系统补偿平台的运动误差,并获得高分辨率的图像。GPS获取的地理坐标(即经度,纬度和高度)需要转换为本地的直角坐标,原始雷达数据在这个直角坐标系下才能引入成像及运动补偿算法进行处理,并获得本地直角坐标系下的图像。这种图像是通过局部坐标描述的,对于其他部门是不通用的,因为其他部门需要的是经过全球地理坐标标绘的图像。本文提出了一种由后向投影算法BP(Back Projection)引出的新的SAR成像算法,它直接在地理坐标下处理,可避免坐标转换的过程。而且生成德图像像素均和地理坐标一一对应,能够很方便地被情报或其他部门使用。而且在仿真试验以及外场试验中,证明了本文算法和常规BP算法成像效果是相当的。     

Confidence regions for GPS baselines by Bayesian statistics

 The global positioning system (GPS) model is distinctive in the way that the unknown parameters are not only real-valued, the baseline coordinates, but also integers, the phase ambiguities. The GPS model therefore leads to a mixed integer–real-valued estimation problem. Common solutions are the float solution, which ignores the ambiguities being integers, or the fixed solution, where the ambiguities are estimated as integers and then are fixed. Confidence regions, so-called HPD (highest posterior density) regions, for the GPS baselines are derived by Bayesian statistics. They take care of the integer character of the phase ambiguities but still consider them as unknown parameters. Estimating these confidence regions leads to a numerical integration problem which is solved by Monte Carlo methods. This is computationally expensive so that approximations of the confidence regions are also developed. In an example it is shown that for a high confidence level the confidence region consists of more than one region. Received: 1 February 2001 / Accepted: 18 July 2001     

ITRS, PZ-90 and WGS 84: current realizations and the related transformation parameters

 The first results of the International GLONASS Experiment 1998 (IGEX-98) campaign have provided significant material to illustrate the mutual benefits of the GLONASS system and the realization of the International Terrestrial Reference System (ITRS). A specific aspect, namely the relationship between the World Geodetic System 1984 (WGS 84) and the PZ-90 system using ITRS as a primary standard, is investigated. A review of current works is carried out. A transformation strategy is proposed for the three systems based on recent results from IGEX-98 and an independent set of transformation parameters derived by the Jet Propulsion Laboratory from ITRF97 and PZ-90 coordinates for 16 global stations. Received: 9 June 2000 / Accepted: 12 June 2001     

Geostatistical incorporation of spatial coordinates into supervised classification of hyperspectral data

 This paper presents a methodology to incorporate both hyperspectral properties and spatial coordinates of pixels in maximum likelihood classification. Indicator kriging of ground data is used to estimate, for each pixel, the prior probabilities of occurrence of classes which are then combined with spectral-based probabilities within a Bayesian framework. In the case study (mapping of in-stream habitats), accounting for spatial coordinates increases the overall producer's accuracy from 85.8% to 93.8%, while the Kappa statistic rises from 0.74 to 0.88. Best results are obtained using only indicator kriging-based probabilities, with a stunning overall accuracy of 97.2%. Significant improvements are observed for environmentally important units, such as pools (Kappa: 0.17 to 0.74) and eddy drop zones (Kappa: 0.65 to 0.87). The lack of benefit of using hyperspectral information in the present study can be explained by the dense network of ground observations and the high spatial continuity of field classification which might be spurious. Received: 12 April 2001 / Accepted: 7 September 2001     

Numerical modeling of gravitational field lines – the effect of mass attraction on horizontal coordinates

There are two basic types of geodetic projections by which the points located on the Earth's surface can be projected onto a reference ellipsoid. Because of the different principles of Helmert's and Pizzetti's methods two sets of horizontal and vertical coordinates are obtained for the same set of surface points. The difference is investigated in terms of offsets of horizontal coordinates. For an estimation of the offsets the field lines going through topographic masses are determined by three different numerical methods in `flat Earth approximation' using a volume element model of the density distribution of the lithosphere in a part of central Europe. The maximum horizontal offset reaches 20 cm on the investigated area at the level of the geoid. Received: 28 October 1998 / Accepted: 16 August 1999     

A new combined European permanent network station coordinates solution

The EUREF [International Association of Geodesy (IAG) Reference Frame Sub-Commission for Europe] network of continuously operating GPS stations (EPN) was primarily established for reference frame maintenance, and also plays an important role for geodynamical research in Europe. The main objective of this paper is to obtain an independent homogeneous time series of the EPN station coordinates, which is also available in SINEX format. A new combined solution of the EPN station coordinates was computed. The combination was performed independently for every week, in three steps: (1) the stated constraints on the coordinates were removed from the individual solutions of the Analysis Centers; (2) the de-constrained solutions were aligned to ITRF2000; (3) the resulting solutions were combined using the Helmert blocking technique. All the data from GPS weeks 900 to 1302 (April 1997–December 2004) were used. We investigated in detail the behavior of the transformation parameters aligning the new combined solution to ITRF2000. In general, the time series of the transformation parameters show a good stability in time although small systematic effects can be seen, most likely caused by station instabilities. A comparison of the new combined solution to the official EUREF weekly combined solution is also presented.     

地心坐标反解大地坐标的非线性方程组牛顿迭代法分析

目前的一些地心坐标向大地坐标的转换模型在计算速度,稳定性或精度方面存在一定的局限性。本文探讨了非线性方程组数值迭代的求解方法及其在MATLAB 7.0中的编程实现,并将结果与基于一元三次方程求解的严密方法做了比较分析,结果证明该方法是可以在实际中参考应用的。     

维持中国地心坐标系的基准站数据处理

本文介绍了中国地心坐标系维持数据处理过程。基于"中国地壳运动观测网络"工程1999-2005年共7年的24个GPS连续运行基准站观测数据,并联合47个国际IGS核心站,获得了这些点于2000.0历元在ITRF2000框架中的坐标及速度,以及其相对于NNR-NUVEL1A板块模型的速度,以此作为维持中国地心坐标系的基准点。     

GPS network monitors the Western Alps' deformation over a five-year period: 1993–1998

  The Western Alps are among the best studied collisional belts with both detailed structural mapping and also crustal geophysical investigations such as the ECORS and EGT seismic profile. By contrast, the present-day kinematics of the belt is still largely unknown due to small relative motions and the insufficient accuracy of the triangulation data. As a consequence, several tectonic problems still remain to be solved, such as the amount of N–S convergence in the Occidental Alps, the repartition of the deformation between the Alpine tectonic units, and the relation between deformation and rotation across the Alpine arc. In order to address these problems, the GPS ALPES group, made up of French, Swiss and Italian research organizations, has achieved the first large-scale GPS surveys of the Western Alps. More than 60 sites were surveyed in 1993 and 1998 with a minimum observation of 3 days at each site. GPS data processing has been done by three independent teams using different software. The different solutions have horizontal repeatabilities (N–E) of 4–7 mm in 1993 and 2–3 mm in 1998 and compare at the 3–5-mm level in position and 2-mm/yr level in velocity. A comparison of 1993 and 1998 coordinates shows that residual velocities of the GPS marks are generally smaller than 2 mm/yr, precluding a detailed tectonic interpretation of the differential motions. However, these data seem to suggest that the N–S compression of the Western Alps is quite mild (less than 2 mm/yr) compared to the global convergence between the African and Eurasian plate (6 mm/yr). This implies that the shortening must be accomodated elsewhere by the deformation of the Maghrebids and/or by rotations of Mediterranean microplates. Also, E–W velocity components analysis supports the idea that E–W extension exists, as already suggested by recent structural and seismotectonic data interpretation. Received: 27 November 2000 / Accepted: 17 September 2001     

Combined processing of the IGS and the IGEX network

 Since the beginning of the International Global Navigation Satellite System (GLONASS) Experiment, IGEX, in October 1998, the Center for Orbit Determination in Europe (CODE) has acted as an analysis center providing precise GLONASS orbits on a regular basis. In CODE's IGEX routine analysis the Global Positioning System (GPS) orbits and Earth rotation parameters are introduced as known quantities into the GLONASS processing. A new approach is studied, where data from the IGEX network are combined with GPS observations from the International GPS Service (IGS) network and all parameters (GPS and GLONASS orbits, Earth rotation parameters, and site coordinates) are estimated in one processing step. The influence of different solar radiation pressure parameterizations on the GLONASS orbits is studied using different parameter subsets of the extended CODE orbit model. Parameterization with three constant terms in the three orthogonal directions, D, Y, and X (D = direction satellite–Sun, Y = direction of the satellite's solar panel axis), and two periodic terms in the X-direction, proves to be adequate for GLONASS satellites. As a result of the processing it is found that the solar radiation pressure effect for the GLONASS satellites is significantly different in the Y-direction from that for the GPS satellites, and an extensive analysis is carried out to investigate the effect in detail. SLR observations from the ILRS network are used as an independent check on the quality of the GLONASS orbital solutions. Both processing aspects, combining the two networks and changing the orbit parameterization, significantly improve the quality of the determined GLONASS orbits compared to the orbits stemming from CODE's IGEX routine processing. Received: 10 May 2000 / Accepted: 9 October 2000     

利用GPS精确测定的摄影机坐标和方向

本文论述了在车载或机载摄影测量过程中 ,摄影机坐标和方向的测定方法 ,利用三台GPS接收机测定汽车或飞机在摄影运动过程中的姿态 ,由于摄影机与三台接收机的关系类似一个刚体 ,故可以用空间坐标变换的方法求得摄影机的坐标和方向     

Investigation of tidal effects in lunar laser ranging

 The analysis of lunar laser ranging (LLR) data enables the determination of many parameters of the Earth–Moon system, such as lunar gravity coefficients, reflector and station coordinates which contribute to the realisation of the International Terrestrial Reference Frame 2000 (ITRF 2000), Earth orientation parameters [EOPs, which contribute to the global EOP solutions at the International Earth Rotation Service (IERS)] or quantities which parameterise relativistic effects in the solar system. The big advantage of LLR is the long time span of lunar observations (1970–2000). The accuracy of the normal points nowadays is about 1 cm.  The capability of LLR to determine tidal parameters is investigated. In principle, it could be assumed that LLR would contribute greatly to the investigation of tidal effects, because the Moon is the most important tide-generating body. In this respect some special topics such as treatment of the permanent tide and the effect of atmospheric loading are addressed and results for the tidal parameters h ₂ and l ₂ as well as values for the eight main tides are given. Received: 14 August 2000 / Accepted: 15 October 2001     

罗德里格矩阵在坐标系转换中的应用

在大旋转角度的坐标系转换中,线性转换模型的旋转参数线性化复杂,计算量大,误差大。根据反对称矩阵和罗德里格矩阵的性质,推导了基于罗德里格矩阵的坐标系转换模型。该模型用反对称矩阵中的3个独立参数代替旋转矩阵中9个相关参数,避免了旋转参数的线性化。模型简单、计算简便,通过实验计算,精度较高。     

L1-norm pre-analysis measures for geodetic networks

 Several pre-analysis measures which help to expose the behavior of L ₁ -norm minimization solutions are described. The pre-analysis measures are primarily based on familiar elements of the linear programming solution to L ₁-norm minimization, such as slack variables and the reduced-cost vector. By examining certain elements of the linear programming solution in a probabilistic light, it is possible to derive the cumulative distribution function (CDF) associated with univariate L ₁-norm residuals. Unlike traditional least squares (LS) residual CDFs, it is found that L ₁-norm residual CDFs fail to follow the normal distribution in general, and instead are characterized by both discrete and continuous (i.e. piecewise) segments. It is also found that an L ₁ equivalent to LS redundancy numbers exists and that these L ₁ equivalents are a byproduct of the univariate L ₁ univariate residual CDF. Probing deeper into the linear programming solution, it is found that certain combinations of observations which are capable of tolerating large-magnitude gross errors can be predicted by comprehensively tabulating the signs of slack variables associated with the L ₁ residuals. The developed techniques are illustrated on a two-dimensional trilateration network. Received: 6 July 2001 / Accepted: 21 February 2002