Generalized inner constraints for geodetic network densification problems

The estimated coordinates from a minimum-constrained (MC) network adjustment are generally influenced by two different error sources, that is the data noise from the available measurements and the so-called datum noise due to random errors in the fiducial coordinates that are used for the datum definition with regard to an external reference frame. Although the latter source does not affect the estimable characteristics of a MC solution, it still contributes a datum-related noise to the estimated positions which reflects the uncertainty of the coordinate system itself for the adjusted network. The aim of this paper is to develop a new type of MCs which minimizes both of the aforementioned effects in the final coordinates of an adjusted network. This particular problem has not been treated in the geodetic literature and the solution which is presented herein offers an elegant unification of the classic inner constraints into a more general framework for the datum choice problem of network optimization theory. Furthermore, the findings of our study provide a useful and rigorous tool for frame densification problems by means of an optimal MC adjustment in geodetic networks.     

卫星星历参考框架对GNSS相对定位解算的影响分析

系统研究了GNSS精密星历框架变化对GNSS相对定位以及网平差解算的影响.通过实验比较发现,在高精度GNSS相对定位中,若选择的地面参考框架与精密轨道参考框架不一致,则将给区域网基线解和网平差结果带来一定的系统性误差.对于高精度的定位解算而言,2000 km以上的基线需考虑地面参考基准与星历参考基准的一致性问题,否则将...     

A Terrestrial Reference Frame realised on the observation level using a GPS-LEO satellite constellation

Applying a one-step integrated process, i.e. by simultaneously processing all data and determining all satellite orbits involved, a Terrestrial Reference Frame (TRF) consisting of a geometric as well as a dynamic part has been determined at the observation level using the EPOS-OC software of Deutsches GeoForschungsZentrum. The satellite systems involved comprise the Global Positioning System (GPS) as well as the twin GRACE spacecrafts. Applying a novel approach, the inherent datum defect has been overcome empirically. In order not to rely on theoretical assumptions this is done by carrying out the TRF estimation based on simulated observations and using the associated satellite orbits as background truth. The datum defect is identified here as the total of all three translations as well as the rotation about the z-axis of the ground station network leading to a rank-deficient estimation problem. To rectify this singularity, datum constraints comprising no-net translation (NNT) conditions in x, y, and z as well as a no-net rotation (NNR) condition about the z-axis are imposed. Thus minimally constrained, the TRF solution covers a time span of roughly a year with daily resolution. For the geometric part the focus is put on Helmert transformations between the a priori and the estimated sets of ground station positions, and the dynamic part is represented by gravity field coefficients of degree one and two. The results of a reference solution reveal the TRF parameters to be estimated reliably with high precision. Moreover, carrying out a comparable two-step approach using the same data and models leads to parameters and observational residuals of worse quality. A validation w.r.t. external sources shows the dynamic origin to coincide at a level of 5 mm or better in x and y, and mostly better than 15 mm in z. Comparing the derived GPS orbits to IGS final orbits as well as analysing the SLR residuals for the GRACE satellites reveals an orbit quality on the few cm level. Additional TRF test solutions demonstrate that K-Band Range-Rate observations between both GRACE spacecrafts are crucial for accurately estimating the dynamic frame’s orientation, and reveal the importance of the NNT- and NNR-conditions imposed for estimating the components of the dynamic geocenter.     

Datum definition by free net adjustment

An adjustment method of angular and distance measurements for densification of horizontal control is presented, where the resulting network has the properties of a free net. The higher order control points form an integral part of the net and serve as a source for datum definition. The relative positions of the points in the net are independent of the higher order control points and are determined by the measurements and their respective covariance matrix. The solution is equivalent to the orthogonal transformation of a minimum constraints solution of a net where the transformation parameters are determined by the control points and their a—priori covariance matrix.     

Global sea-level rise and its relation to the terrestrial reference frame

We examined the sensitivity of estimates of global sea-level rise obtained from GPS-corrected long term tide gauge records to uncertainties in the International Terrestrial Reference Frame (ITRF) realization. A useful transfer function was established, linking potential errors in the reference frame datum (origin and scale) to resulting errors in the estimate of global sea level rise. Contrary to scale errors that are propagated by a factor of 100%, the impact of errors in the origin depends on the network geometry. The geometry of the network analyzed here resulted in an error propagation factor of 50% for the Z component of the origin, mainly due to the asymmetry in the distribution of the stations between hemispheres. This factor decreased from 50% to less than 10% as the geometry of the network improved using realistic potential stations that did not yet meet the selection criteria (e.g., record length, data availability). Conversely, we explored new constraints on the reference frame by considering forward calculations involving tide gauge records. A reference frame could be found in which the scatter of the regional sea-level rates was limited. The resulting reference frame drifted by 1.36 ± 0.22? mm/year from the ITRF2000 origin in the Z component and by ?0.44 ± 0.22?mm/year from the ITRF2005 origin. A bound on the rate of global sea level rise of 1.2 to 1.6?mm/year was derived for the past century, depending on the origin of the adopted reference frame. The upper bound is slightly lower than previous estimates of 1.8?mm/year discussed in the IPCC fourth report.     

Estimability of astronomical longitude and latitude only from theodolite observations within three-dimensional networks of terrestrial type

Summary From a two-dimensional network adjustment it is well understood that the one orientation unknown of a theodolite frame is estimable, once the orientation datum parameter, e.g., one azimuth, is fixed. In three-dimensional networks the problem of estimability of three orientation unknowns inherent in a theodolite frame is more complex. Here we prove that not only the classical horizontal orientation unknown is estimable (up to the datum degrees of freedom), but also astronomical longitude and astronomical latitude which can be considered as two additional orientation unknowns of the theodolite frame moving with respect to an earth-fixed equatorial frame of reference. Thus the theodolite instrument can be considered—at least theoretically—a gradiometer measuring the variation of the directional parameters of the gravity vector from one point to another. Or up to the datum degrees of freedom astronomical longitude and astronomical latitude can be determined from only theodolite observations between exclusively terrestrial points. M?nicke (1982), has shown that despite the refraction problem the method works sufficiently well in practice.     

用主成分法求解秩亏自由网的研究

结合多数平差和主成分估计理论,导出误差方程中含多重共线性关系时求解未知数的公式,并证明主成分估计的解是最小二乘最小范效解。由此,将主成分估计推广到秩亏自由网平差中,导出求解未知多数及其协因数的公式,并据主成分估计演绎出用附有条件的参数平差法求解秩亏自由网的条件方程（基准在达式）。     

Extended free net adjustment constraints

The contribution of geodetic measurements to the establishment of a control network can be partitioned into global and local (individual) components. The global component epitomized in a number of geometrically meaningful parameters can be estimated together with the individual point coordinates. The additional rank defect created by the extension of the parameter list is corrected by free net adjustment constraints which are extended beyond those needed for a solution of the network datum problem. Two applications of extended free net adjustment are outlined and illustrated by elementary numerical examples. A non-Cartesian (skew) reference system discussed in the Appendix provides an exotic interpretation of the estimated global and individual parameters. Prepared during a grant period (September 1984 through February 1985) while serving as a Visiting Senior Scientist in Geodesy, National Research Council, National Academy of Sciences, Washington, D.C.     

采用弱基准和抗差估计的重力网平差

以沿海某测区绝对重力和相对重力网数据处理为例,详细描述了弱基准重力网平差的方法,绝对重力和相对重力先验权的确定,用抗差等价权来调整相对重力的权、重力仪参数的取舍。结果表明,弱基准能有效地提高整网的精度;抗差估计能有效地探测并降低异常数据的权。重力点平均中误差为13.6×10-8 ms-2,偶然误差检验符合正态分布。     

推广的半动态法及其在多级工程控制网精度评定中的应用

为了合理地求定按附合网平差的网点坐标的实际精度,本文推广了半动态法,籍此可在首级网的基准下,统一求定顾及各级控制网测量误差影响精度的全部网点坐标的方差—协方差阵。本文所提出的方法适用于种种情况下的精度估计:等级数不受限制的平面、高程或三维网,本级网平差中的固定数据以及待估计的平差值函数可以包含不同级的网点,从而克服了半动态法原有的局限性。     

GPS网平面基准点的可靠性分析

通过具体实例，对GPS网约束平差结果进行了分析、比较，并提出了GPS网平面基准点的可靠性检验方法，得出了作为起算数据的基准点及其点位误差对GPS网约束平差精度的影响规律。     

The 2008 DGFI realization of the ITRS: DTRF2008

A new realization of the International Terrestrial System was computed at the ITRS Combination Centre at DGFI as a contribution to ITRF2008. The solution is labelled DTRF2008. In the same way as in the DGFI computation for ITRF2005 it is based on either normal equation systems or estimated parameters derived from VLBI, SLR, GPS and DORIS observations by weekly or session-wise processing. The parameter space of the ITRS realization comprises station positions and velocities and daily resolved Earth Orientation Parameters (EOP), whereby for the first time also nutation parameters are included. The advantage of starting from time series of input data is that the temporal behaviour of geophysical parameters can be investigated to decide whether the parameters can contribute to the datum realization of the ITRF. In the same way, a standardized analysis of station position time series can be performed to detect and remove discontinuities. The advantage of including EOP in the ITRS realization is twofold: (1) the combination of the coordinates of the terrestrial pole—estimated from all contributing techniques—links the technique networks in two components of the orientation, leading to an improvement of consistency of the Terrestrial Reference Frame (TRF) and (2) in their capacity as parameters common to all techniques, the terrestrial pole coordinates enhance the selection of local ties as they provide a measure for the consistency of the combined frame. The computation strategy of DGFI is based on the combination of normal equation systems while at the ITRS Combination Centre at IGN solutions are combined. The two independent ITRS realizations provide the possibility to assess the accuracy of ITRF by comparison of the two frames. The accuracy evaluation was done separately for the datum parameters (origin, orientation and scale) and the network geometry. The accuracy of the datum parameters, assessed from the comparison of DTRF2008 and ITRF2008, is between 2–5?mm and 0.1–0.8?mm/year depending on the technique. The network geometry (station positions and velocities) agrees within 3.2?mm and 1.0?mm/year. A comparison of DTRF2008 and ITRF2005 provides similar results for the datum parameters, but there are larger differences for the network geometry. The internal accuracy of DTRF2008—that means the level of conservation of datum information and network geometry within the combination—was derived from comparisons with the technique-only multi-year solutions. From this an internal accuracy of 0.32?mm for the VLBI up to 3.3?mm for the DORIS part of the network is found. The internal accuracy of velocities ranges from 0.05?mm/year for VLBI to 0.83?mm/year for DORIS. The internal consistency of DTRF2008 for orientation can be derived from the analysis of the terrestrial pole coordinates. It is estimated at 1.5–2.5?mm for the GPS, VLBI and SLR parts of the network. The consistency of these three and the DORIS network part is within 6.5?mm.     

法方程层面平差基准解析

从平差基准角度,在法方程层面对最小二乘平差模型做了统一,证明重心基准空间是自由法矩阵空间的正交补空间,并进一步对平差基准问题做了几何解析。通过水准网实例,验证法方程层面平差计算及基准转换的可行性。     

A study on the reference frame consistency in recent Earth gravitational models

All gravity field functionals obtained from an Earth gravitational model (EGM) depend on the underlying terrestrial reference frame (TRF), with respect to which the EGM’s spherical harmonic coefficients refer to. In order to maintain a coherent framework for the comparison of current and future EGMs, it is thus important to investigate the consistency of their inherent TRFs, especially when their use is intended for high precision studies. Following the methodology described in an earlier paper by Kleusberg (1980), the similarity transformation parameters between the associated reference frames for several EGMs (including the most recent CHAMP/GRACE models at the time of writing this paper) are estimated in the present study. Specifically, the differences between the spherical harmonic coefficients for various pairs of EGMs are parameterized through a 3D-similarity spatial transformation model that relates their underlying TRFs. From the least-squares adjustment of such a parametric model, the origin, orientation and scale stability between the EGMs’ reference frames can be identified by estimating their corresponding translation, rotation and scale factor parameters. Various aspects of the estimation procedure and its results are highlighted in the paper, including data weighting schemes, the sensitivity of the results with respect to the selected harmonic spectral band, the correlation structure and precision level of the estimated transformation parameters, the effect of the estimated differences of the EGMs’ reference frames on their height anomaly signal, and the overall feasibility of Kleusberg’s formulae for the assessment of TRF inconsistencies among global geopotential models.     

Total least squares adjustment in partial errors-in-variables models: algorithm and statistical analysis

The weighted total least squares (TLS) method has been developed to deal with observation equations, which are functions of both unknown parameters of interest and other measured data contaminated with random errors. Such an observation model is well known as an errors-in-variables (EIV) model and almost always solved as a nonlinear equality-constrained adjustment problem. We reformulate it as a nonlinear adjustment model without constraints and further extend it to a partial EIV model, in which not all the elements of the design matrix are random. As a result, the total number of unknowns in the normal equations has been significantly reduced. We derive a set of formulae for algorithmic implementation to numerically estimate the unknown model parameters. Since little statistical results about the TLS estimator in the case of finite samples are available, we investigate the statistical consequences of nonlinearity on the nonlinear TLS estimate, including the first order approximation of accuracy, nonlinear confidence region and bias of the nonlinear TLS estimate, and use the bias-corrected residuals to estimate the variance of unit weight.     

高精度GPS网数据处理中的系统误差分析

分析了高精度GPS网系统误差产生的原因和分类，推导了消除和估计GPS网系统误差的整体平差函数模型。     

坐标参考框架长期解内约束模型

地球参考框架是一切测绘活动、地球科学研究的物理基础。目前,地球参考框架常采用长期解的形式,即利用一组全球分布的基准站在某一参考历元的坐标和速度来表示。由于观测有误差,且各基准站又具有非线性变化,故需要对不同历元的瞬时地球参考框架进行累积,形成稳定的长期参考框架。以不同历元观测数据得到的瞬时参考框架成果为输入,构建了一种基于多历元观测数据建立参考框架长期累积解的融合模型。从坐标转换模型和测站坐标的时变模型出发,详细推导了建立长期解的函数模型,依据该函数模型的秩亏数设计了转换参数的内约束基准。采用2010-08—2014-12的国际全球导航卫星系统服务第2次处理结果进行试算,并与国际地球参考框架2014成果进行了对比。结果表明,X、Y、Z方向标准偏差分别为3.45 ?mm、4.04 mm、2.84 mm,速度精度分别为1.53 mm/a、1.46 mm/a、1.21 mm/a,X、Y、Z方向的加权均方根误差优于3 ?mm。     

高精度GPS复测网的形变基准解释

在高精度GPS复测网的平差处理中 ,首先要选择一组已知的、精确的IGS (国际GPS地球动力学服务 )跟踪站为基准 ,并对基准站坐标施加不同的约束 ,这样就产生了不同的参考基准。在深入探讨基于不同约束的GPS网平差的参考基准及意义的基础上 ,文中推导了不同基准之间的转换关系     

Future global SLR network evolution and its impact on the terrestrial reference frame

Satellite laser ranging (SLR) is an important technique that contributes to the determination of terrestrial geodetic reference frames, especially to the realization of the origin and the scale of global networks. One of the major limiting factors of SLR-derived reference frame realizations is the datum accuracy which significantly suffers from the current global SLR station distribution. In this paper, the impact of a potential future development of the SLR network on the estimated datum parameters is investigated. The current status of the SLR network is compared to a simulated potential future network featuring additional stations improving the global network geometry. In addition, possible technical advancements resulting in a higher amount of observations are taken into account as well. As a result, we find that the network improvement causes a decrease in the scatter of the network translation parameters of up to 24%, and up to 20% for the scale, whereas the technological improvement causes a reduction in the scatter of up to 27% for the translations and up to 49% for the scale. The Earth orientation parameters benefit by up to 15% from both effects.     

GPS监测网数据处理方案研究

处理大尺度ＧＰＳ监测网多期复测成果的时期,数据处理方案是一个重要问题。作者在处理首都圈ＧＰＳ地形变监测网数据的过程中,经过长期探索和大量试算,提出了制定数据处理方案的基本原则：要采用一个自洽的地球参考框架和常数系统;要采用精度最高的同一种粗密星历和相应的地球自转参数;要采用最先进的同一种数据处理软件以及采用同一的参考基准和约束条件。由于地壳处于运动之中,大尺度ＧＰＳ监测网的参考基准应是动态基准。研