A new technique to determine geoid and orthometric heights from satellite positioning and geopotential numbers

This paper takes advantage of space-technique-derived positions on the Earth’s surface and the known normal gravity field to determine the height anomaly from geopotential numbers. A new method is also presented to downward-continue the height anomaly to the geoid height. The orthometric height is determined as the difference between the geodetic (ellipsoidal) height derived by space-geodetic techniques and the geoid height. It is shown that, due to the very high correlation between the geodetic height and the computed geoid height, the error of the orthometric height determined by this method is usually much smaller than that provided by standard GPS/levelling. Also included is a practical formula to correct the Helmert orthometric height by adding two correction terms: a topographic roughness term and a correction term for lateral topographic mass–density variations.     

On the adjustment of combined GPS/levelling/geoid networks

A detailed treatment of adjustment problems in combined global positioning system (GPS)/levelling/geoid networks is given. The two main types of `unknowns' in this kind of multi-data 1D networks are usually the gravimetric geoid accuracy and a 2D spatial field that describes all the datum/systematic distortions among the available height data sets. An accurate knowledge of the latter becomes especially important when we consider employing GPS techniques for levelling purposes with respect to a local vertical datum. Two modelling alternatives for the correction field are presented, namely a pure deterministic parametric model, and a hybrid deterministic and stochastic model. The concept of variance component estimation is also proposed as an important statistical tool for assessing the actual gravimetric geoid noise level and/or testing a priori determined geoid error models. Finally, conclusions are drawn and recommendations for further study are suggested. Received: 9 September 1998 / Accepted: 8 June 1999     

基于等价条件闭合差的VCE通用解析法

分析指出了现有方差-协方差分量估计(VCE)方法在计算效率与?2统计量统计性质两方面的固有缺陷。利用零空间算子消去概括平差模型中的参数向量,建立了等价条件平差模型。由此定义了等价条件闭合差(ECC),并导出了以ECC表示的?2统计量计算式。进而,基于等价条件闭合差与新构造的可逆方差分量模型提出了方差-协方差分量估计的通用解析法,简称为VCE-ECC法。同时,给出了对应四种基本平差模型的VCE-ECC法简化计算式。实例与仿真结果表明：VCE-ECC法与现有VCE方法的方差-协方差分量估计值在统计意义上无明显差异,并有效地克服了现有VCE方法的固有缺陷。     

Variance component estimation uncertainty for unbalanced data: application to a continent-wide vertical datum

Variance component estimation (VCE) is used to update the stochastic model in least-squares adjustments, but the uncertainty associated with the VCE-derived weights is rarely considered. Unbalanced data is where there is an unequal number of observations in each heterogeneous data set comprising the variance component groups. As a case study using highly unbalanced data, we redefine a continent-wide vertical datum from a combined least-squares adjustment using iterative VCE and its uncertainties to update weights for each data set. These are: (1) a continent-wide levelling network, (2) a model of the ocean’s mean dynamic topography and mean sea level observations, and (3) GPS-derived ellipsoidal heights minus a gravimetric quasigeoid model. VCE uncertainty differs for each observation group in the highly unbalanced data, being dependent on the number of observations in each group. It also changes within each group after each VCE iteration, depending on the magnitude of change for each observation group’s variances. It is recommended that VCE uncertainty is computed for VCE updates to the weight matrix for unbalanced data so that the quality of the updates for each group can be properly assessed. This is particularly important if some groups contain relatively small numbers of observations. VCE uncertainty can also be used as a threshold for ceasing iterations, as it is shown—for this data set at least—that it is not necessary to continue time-consuming iterations to fully converge to unity.     

Ellipsoidal geoid computation

Modern geoid computation uses a global gravity model, such as EGM96, as a third component in a remove–restore process. The classical approach uses only two: the reference ellipsoid and a geometrical model representing the topography. The rationale for all three components is reviewed, drawing attention to the much smaller precision now needed when transforming residual gravity anomalies. It is shown that all ellipsoidal effects needed for geoid computation with millimetric accuracy are automatically included provided that the free air anomaly and geoid are calculated correctly from the global model. Both must be consistent with an ellipsoidal Earth and with the treatment of observed gravity data. Further ellipsoidal corrections are then negligible. Precise formulae are developed for the geoid height and the free air anomaly using a global gravity model, given as spherical harmonic coefficients. Although only linear in the anomalous potential, these formulae are otherwise exact for an ellipsoidal reference Earth—they involve closed analytical functions of the eccentricity (and the Earths spin rate), rather than a truncated power series in e². They are evaluated using EGM96 and give ellipsoidal corrections to the conventional free air anomaly ranging from –0.84 to +1.14 mGal, both extremes occurring in Tibet. The geoid error corresponding to these differences is dominated by longer wavelengths, so extrema occur elsewhere, rising to +766 mm south of India and falling to –594 mm over New Guinea. At short wavelengths, the difference between ellipsoidal corrections based only on EGM96 and those derived from detailed local gravity data for the North Sea geoid GEONZ97 has a standard deviation of only 3.3 mm. However, the long-wavelength components missed by the local computation reach 300 mm and have a significant slope. In Australia, for example, such a slope would amount to a 600-mm rise from Perth to Cairns.     

偏差改正的Partial EIV模型方差分量估计

针对Partial EIV模型的方差分量估计中未考虑参数估值偏差所带来的影响,将Partial EIV模型视为非线性函数得到参数估值的偏差及二阶近似协方差表达式,计算得到偏差改正后的参数估值,结合方差分量估计方法,更新由参数估值影响的矩阵变量,给出了基于偏差改正的方差分量估计迭代方法。试验结果表明,参数估值及其协方差主要受参数估值偏差大小的影响,加入偏差改正能够得到更加合理的参数估值及方差分量估值,偏差改正后的方差分量估值可更加合理地评估参数估值的精度信息。     

方差-协方差分量估计的概括平差因子法

针对现有方差-协方差分量估计(variance-covariance component estimation,VCE)理论存在的问题,通过引入间接平差的平差因子概念,定义并研究了基于概括平差模型的概括平差因子、概括闭合差及其方差阵,进而利用二次期望公式提出了基于概括平差因子的VCE新方法。该方法适用于概括平差模型所归纳的4种函数模型形式,并通过概括平差因子揭示了平差函数模型与VCE是否存在解析估计形式的关系。实例计算结果表明,现有迭代型VCE方法改变了LS估计量的统计性质,而VCE新方法解析估计具有LS统计性质,且无需初值。     

The combination of GNSS-levelling data and gravimetric (quasi-) geoid heights in the presence of noise

We propose a methodology for the combination of a gravimetric (quasi-) geoid with GNSS-levelling data in the presence of noise with correlations and/or spatially varying noise variances. It comprises two steps: first, a gravimetric (quasi-) geoid is computed using the available gravity data, which, in a second step, is improved using ellipsoidal heights at benchmarks provided by GNSS once they have become available. The methodology is an alternative to the integrated processing of all available data using least-squares techniques or least-squares collocation. Unlike the corrector-surface approach, the pursued approach guarantees that the corrections applied to the gravimetric (quasi-) geoid are consistent with the gravity anomaly data set. The methodology is applied to a data set comprising 109 gravimetric quasi-geoid heights, ellipsoidal heights and normal heights at benchmarks in Switzerland. Each data set is complemented by a full noise covariance matrix. We show that when neglecting noise correlations and/or spatially varying noise variances, errors up to 10% of the differences between geometric and gravimetric quasi-geoid heights are introduced. This suggests that if high-quality ellipsoidal heights at benchmarks are available and are used to compute an improved (quasi-) geoid, noise covariance matrices referring to the same datum should be used in the data processing whenever they are available. We compare the methodology with the corrector-surface approach using various corrector surface models. We show that the commonly used corrector surfaces fail to model the more complicated spatial patterns of differences between geometric and gravimetric quasi-geoid heights present in the data set. More flexible parametric models such as radial basis function approximations or minimum-curvature harmonic splines perform better. We also compare the proposed method with generalized least-squares collocation, which comprises a deterministic trend model, a random signal component and a random correlated noise component. Trend model parameters and signal covariance function parameters are estimated iteratively from the data using non-linear least-squares techniques. We show that the performance of generalized least-squares collocation is better than the performance of corrector surfaces, but the differences with respect to the proposed method are still significant.     

On the combination of gravity anomalies and gravity disturbances for geoid determination in Western Australia

The geoid gradient over the Darling Fault in Western Australia is extremely high, rising by as much as 38 cm over only 2 km. This poses problems for gravimetric-only geoid models of the area, whose frequency content is limited by the spatial distribution of the gravity data. The gravimetric-only version of AUSGeoid98, for instance, is only able to resolve 46% of the gradient across the fault. Hence, the ability of GPS surveys to obtain accurate orthometric heights is reduced. It is described how further gravity data were collected over the Darling Fault, augmenting the existing gravity observations at key locations so as to obtain a more representative geoid gradient. As many of the gravity observations were collected at stations with a well-known GRS80 ellipsoidal height, the opportunity arose to compute a geoid model via both the Stokes and the Hotine approaches. A scheme was devised to convert free-air anomaly data to gravity disturbances using existing geoid models, followed by a Hotine integration to geoid heights. Interestingly, these results depended very weakly upon the choice of input geoid model. The extra gravity data did indeed improve the fit of the computed geoid to local GPS/Australian Height Datum (AHD) observations by 58% over the gravimetric-only AUSGeoid98. While the conventional Stokesian approach to geoid determination proved to be slightly better than the Hotine method, the latter still improved upon the gravimetric-only AUSGeoid98 solution, supporting the viability of conducting gravity surveys with GPS control for the purposes of geoid determination. AcknowledgementsThe author would like to thank Will Featherstone, Ron Gower, Ron Hackney, Linda Morgan, Geoscience Australia, Scripps Oceanographic Institute and the three anonymous reviewers of this paper. This research was funded by the Australian Research Council.     

随机模型误差对函数模型选择的影响

测量平差问题中,函数模型误差和随机模型误差通常是同时存在的。本文揭示了两者的关系,着重讨论了两者的相互作用及其影响,特别是随机模型误差对函数模型选择的影响。同时也讨论了用来改善随机模型的“方差分量估计技术”对函数模型误差的吸收作用。最后得出了一些有益的结论。     

等价条件平差模型的方差-协方差分量最小二乘估计方法

提出等价条件闭合差的方差-协方差分量最小二乘估计方法,简称LSV-ECM法。首先,利用等价条件平差模型建立了基于等价条件闭合差二次型的方差-协方差分量估计方程,由矩阵半拉直算子将其变换为线性Gauss-Markov形式,进而通过最小二乘准则导出了具有模型通用性、形式简洁性且满足无偏性和最优性的方差-协方差分量估计公式。其次,证明了LSV-ECM方法与残差型VCE方法的等价性,并在此基础上通过计算复杂度定量分析了所提方法的计算高效性。最后,通过边角网平差和中国区域GNSS站坐标时序建模及其结果分析,验证了所提新方法的正确性和计算高效性。     

Robust estimation and optimal selection of polynomial parameters for the interpolation of GPS geoid heights

The polynomial interpolation of least squares and interpolation moving least squares based on control stations with known GPS (global positioning system) ellipsoidal heights and levelling orthometric heights are the most often used methods for the interpolation of the geoid heights. But in their applications there occur two problems: one lies in selecting the suitable polynomial parameters; the other in reducing the influences of some possibly abnormal data points. To solve both of the problems, without emphasizing a sound theoretical basis, a heuristic solution with the help of robust estimation technique and optimization criteria for the regression equation is presented. Through two actual numerical examples it is shown that the new solution concept is efficient and can be realized easily on computers. Received: 23 May 1996 / Accepted: 27 March 1997     

重力归算及其对大地水准面及外部重力场的影响

本文论述并探讨了各种不同的重力归算方法对大地水准面以及地球外部重力场的影响,讨论了与确定cm级大地水准面相关的一些问题,特别阐述了高程误差对大地水准面的影响。     

An iterative solution of weighted total least-squares adjustment

Total least-squares (TLS) adjustment is used to estimate the parameters in the errors-in-variables (EIV) model. However, its exact solution is rather complicated, and the accuracies of estimated parameters are too difficult to analytically compute. Since the EIV model is essentially a non-linear model, it can be solved according to the theory of non-linear least-squares adjustment. In this contribution, we will propose an iterative method of weighted TLS (WTLS) adjustment to solve EIV model based on Newton–Gauss approach of non-linear weighted least-squares (WLS) adjustment. Then the WLS solution to linearly approximated EIV model is derived and its discrepancy is investigated by comparing with WTLS solution. In addition, a numerical method is developed to compute the unbiased variance component estimate and the covariance matrix of the WTLS estimates. Finally, the real and simulation experiments are implemented to demonstrate the performance and efficiency of the presented iterative method and its linearly approximated version as well as the numerical method. The results show that the proposed iterative method can obtain such good solution as WTLS solution of Schaffrin and Wieser (J Geod 82:415–421, 2008) and the presented numerical method can be reasonably applied to evaluate the accuracy of WTLS solution.     

Use of potential coefficient models for geoid undulation determinations using a spherical harmonic representation of the height anomaly/geoid undulation difference

 This paper suggests that potential coefficient models of the Earth's gravitational potential be used to calculate height anomalies which are then reduced to geoid undulations where such quantities are needed for orthometric height determination and vertical datum definition through a potential coefficient realization of the geoid. The process of the conversion of the height anomaly into a geoid undulation is represented by a height anomaly gradient term and the usual N–ζ term that is dependent on elevation and the Bouguer anomaly. Using a degree 360 expansion of 30′ elevations and the OSU91A potential coefficient model, a degree 360 representation of the correction terms was computed. The magnitude of N–ζ reached –3.4 m in the Himalaya Mountains with smaller, but still significant, magnitudes in other mountainous regions. Received: 6 May 1996; Accepted: 30 October 1996     

Local geoid determination combining gravity disturbances and GPS/levelling: a case study in the Lake Nasser area, Aswan, Egypt

 The use of GPS for height control in an area with existing levelling data requires the determination of a local geoid and the bias between the local levelling datum and the one implicitly defined when computing the local geoid. If only scarse gravity data are available, the heights of new data may be collected rapidly by determining the ellipsoidal height by GPS and not using orthometric heights. Hence the geoid determination has to be based on gravity disturbances contingently combined with gravity anomalies. Furthermore, existing GPS/levelling data may also be used in the geoid determination if a suitable general gravity field modelling method (such as least-squares collocation, LSC) is applied. A comparison has been made in the Aswan Dam area between geoids determined using fast Fourier transform (FFT) with gravity disturbances exclusively and LSC using only the gravity disturbances and the disturbances combined with GPS/levelling data. The EGM96 spherical harmonic model was in all cases used in a remove–restore mode. A total of 198 gravity disturbances spaced approximately 3 km apart were used, as well as 35 GPS/levelling points in the vicinity and on the Aswan Dam. No data on the Nasser Lake were available. This gave difficulties when using FFT, which requires the use of gridded data. When using exclusively the gravity disturbances, the agreement between the GPS/levelling data were 0.71 ± 0.17 m for FFT and 0.63 ± 0.15 for LSC. When combining gravity disturbances and GPS/levelling, the LSC error estimate was ±0.10 m. In the latter case two bias parameters had to be introduced to account for a possible levelling datum difference between the levelling on the dam and that on the adjacent roads. Received: 14 August 2000 / Accepted: 28 February 2001     

最小二乘配置法中局部协方差函数的计算

随着 GPS日益广泛的应用及精度的不断提高 ,在有些实际应用中利用 GPS来代替传统的水准测量进行高程控制已成为可能 ,这也进一步提出了对高精度大地水准面的需求。快速傅立叶变换 (FFT)是目前计算大地水准面比较常用的方法之一 ,但需要将重力观测量进行内插得到规则格网上的平均重力异常。利用最小二乘配置法计算大地水准面可直接利用已有的观测值进行计算 ,同时可综合利用不同类型的数据 ,如重力异常和垂线偏差等计算大地水准面 ,因此最小二乘配置法仍有广泛的应用 ,但制约最小二乘配置应用的关键问题是局部协方差函数的计算。将主要讨论最小二乘配置法中局部协方差函数的计算 ,使所用的协方差函数能更好地反映已知的数据 ,从而获得更精确的结果。     

Combined regional geoid determination for the ERS-1 radar altimeter calibration

Summary A local model of the geoid in NE Italy and its section along the Venice ground track of the ERS-1 satellite of the European Space Agency is presented. The observational data consist of geoid undulations determined with a network of 25 stations of known orthometric (by spirit leveling) and ellipsoidal (by GPS differential survey) and of 13 deflections of the vertical measured at sites of the network for which, besides the ellipsoidal (WGS84) coordinates, also astronomic coordinates were known. The network covers an area of 1×1 degrees and is tied to a vertical and horizontal datum: one vertex of the network is the tide gauge of Punta Salute, in Venice, providing a tie to a mean sea level; a second vertex is the site for mobile laser systems at Monte Venda, on the Euganei Hills, for which geocentric coordinates resulted from the analysis of several LAGEOS passes.The interpolation algorithm used to map sparse and heterogeneous data to a regular grid of geoid undulations is based on least squares collocation and the autocorrelation function of the geoid undulations is modeled by a third order Markov process on flat earth. The algorithm has been applied to the observed undulations and deflections of the vertical after subtraction of the corresponding predictions made on the basis of the OSU91A global geoid model of the Ohio State University, complete to degree and order 360. The locally improved geoid results by adding back, at the nodes of a regular grid, the predictions of the global field to the least squares interpolated values. Comparison of the model values with the raw data at the observing stations indicates that the mean discrepancy is virtually zero with a root mean square dispersion of 8 cm, assuming that the ellipsoidal heights and vertical deflections data are affected by a random error of 3 cm and 0.5 respectively. The corrections resulting from the local data and added to the background 360×360 global model are described by a smooth surface with excursions from the reference surface not larger than ±30 cm.     

Adaptive collocation with application in height system transformation

In collocation applications, the prior covariance matrices or weight matrices between the signals and the observations should be consistent to their uncertainties; otherwise, the solution of collocation will be distorted. To balance the covariance matrices of the signals and the observations, a new adaptive collocation estimator is thus derived in which the corresponding adaptive factor is constructed by the ratio of the variance components of the signals and the observations. A maximum likelihood estimator of the variance components is thus derived based on the collocation functional model and stochastic model. A simplified Helmert type estimator of the variance components for the collocation is also introduced and compared to the derived maximum likelihood type estimator. Reasonable and consistent covariance matrices of the signals and the observations are arrived through the adjustment of the adaptive factor. The new adaptive collocation with related adaptive factor constructed by the derived variance components is applied in a transformation between the geodetic height derived by GPS and orthometric height. It is shown that the adaptive collocation is not only simple in calculation but also effective in balancing the contribution of observations and the signals in the collocation model.     

Least-squares variance component estimation

Least-squares variance component estimation (LS-VCE) is a simple, flexible and attractive method for the estimation of unknown variance and covariance components. LS-VCE is simple because it is based on the well-known principle of LS; it is flexible because it works with a user-defined weight matrix; and it is attractive because it allows one to directly apply the existing body of knowledge of LS theory. In this contribution, we present the LS-VCE method for different scenarios and explore its various properties. The method is described for three classes of weight matrices: a general weight matrix, a weight matrix from the unit weight matrix class; and a weight matrix derived from the class of elliptically contoured distributions. We also compare the LS-VCE method with some of the existing VCE methods. Some of them are shown to be special cases of LS-VCE. We also show how the existing body of knowledge of LS theory can be used to one’s advantage for studying various aspects of VCE, such as the precision and estimability of VCE, the use of a-priori variance component information, and the problem of nonlinear VCE. Finally, we show how the mean and the variance of the fixed effect estimator of the linear model are affected by the results of LS-VCE. Various examples are given to illustrate the theory.