A data-driven approach to local gravity field modelling using spherical radial basis functions

We propose a methodology for local gravity field modelling from gravity data using spherical radial basis functions. The methodology comprises two steps: in step 1, gravity data (gravity anomalies and/or gravity disturbances) are used to estimate the disturbing potential using least-squares techniques. The latter is represented as a linear combination of spherical radial basis functions (SRBFs). A data-adaptive strategy is used to select the optimal number, location, and depths of the SRBFs using generalized cross validation. Variance component estimation is used to determine the optimal regularization parameter and to properly weight the different data sets. In the second step, the gravimetric height anomalies are combined with observed differences between global positioning system (GPS) ellipsoidal heights and normal heights. The data combination is written as the solution of a Cauchy boundary-value problem for the Laplace equation. This allows removal of the non-uniqueness of the problem of local gravity field modelling from terrestrial gravity data. At the same time, existing systematic distortions in the gravimetric and geometric height anomalies are also absorbed into the combination. The approach is used to compute a height reference surface for the Netherlands. The solution is compared with NLGEO2004, the official Dutch height reference surface, which has been computed using the same data but a Stokes-based approach with kernel modification and a geometric six-parameter “corrector surface” to fit the gravimetric solution to the GPS-levelling points. A direct comparison of both height reference surfaces shows an RMS difference of 0.6 cm; the maximum difference is 2.1 cm. A test at independent GPS-levelling control points, confirms that our solution is in no way inferior to NLGEO2004.     

利用改进的Stokes-Helmert边值理论解算高精度似大地水准面

利用改进的Stokes-Helmert边值问题实现了临沂市厘米级似大地水准面精化。首先,基于Stokes理论和Molodensky理论,联合精密确定地表及其外部扰动引力位的严密解算理论,给出Stokes-Helmert边值问题的数学描述,以及直接地形影响和间接地形影响的严密理论表达式;然后,利用多源观测资料,根据"移去-恢复"技术构建临沂高精度的重力似大地水准面模型;最后,利用GPS/水准高程异常对重力似大地水准面模型进行控制拟合,求得最终的大地水准面模型,其外符合精度达到1.6 cm。     

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.     

海洋重力似大地水准面与区域测高似大地水准面的拟合问题

研究了将陆地重力似大地水准面与GPS水；住似大地水准面拟合的处理方法推广到海洋的问题．首先从理论上证明了当存在海面地形．则海洋大地水准面与似大地水准面不重合．导出了在海洋上大地水；住面差距与高程异常之间差值的公式．由此给出了求定平均海面相对于区域高程基准的正常高以及测高似大地水准面的计算公式。由于测高平均海面与GPS大地高有相近的精度．提出了将海洋重力似大地水准面与区域测高似大地水准面拟合的处理方法．并利用当前最新的海面地形模型和测高平均海面模型做了数值估计。     

我国陆海统一似大地水准面构建的三维重力矢量法

基于重力场水平分量-垂线偏差对地形信息敏感的特点,根据边值理论由重力与地形数据确定格网垂线偏差模型,在此基础上,首先利用三维重力矢量-格网垂线偏差与格网重力异常,联合格网高程数据求得格网点间高程异常差,然后通过GPS/水准点的控制,构成紧密的几何条件,进行严密平差,从而获得高分辨率、高精度似大地水准面的数值模型。按照本文方法,利用我国6600多个GPS/水准点、1'×1'的格网垂线偏差、格网重力异常、格网高程数据,整体平差计算了我国陆海统一的似大地水准面模型,经GPS/水准点检核,全国似大地水准面的绝对精度达到了4 cm,相对精度优于7 cm。     

综合利用海岸带GNSS水准和重力数据精密确定中国高程基准偏差

综合利用中国海岸带GNSS水准和多源重力测量数据,通过精化陆海统一的重力似大地水准面计算高程异常零阶项,精密确定了中国1985国家高程基准相对于重力大地水准面的偏差。结果显示:中国1985国家高程基准相对于国际地球自转服务（International Earth Rotation Service,IERS）的IERS2010标准W₀对应的重力大地水准面的偏差为30.4±1.0 cm;中国1985国家高程基准于20世纪80年代经琼州海峡传递到海南岛后,在1 cm精度水平上未发现明显差异;计算的重力似大地水准面经全球导航卫星系统（globa navigation satellite system,GNSS）水准外部检核,标准差约为4 cm,12个省市和陆海交界处在厘米级精度水平上无缝衔接。在近10 a内,天津市除西北角外,大部分地区存在地面下沉,东南部地面下沉约7 cm,上海市地面下沉约7 cm,浙江省、江苏省和福建省等沿海省份局部地区也出现地面下沉。     

Numerical solution of the linearized fixed gravimetric boundary-value problem

The fixed gravimetric boundary-value problem (FGBVP) represents an exterior oblique derivative problem for the Laplace equation. Terrestrial gravimetric measurements located by precise satellite positioning yield oblique derivative boundary conditions in the form of surface gravity disturbances. In this paper, we discuss the boundary element method (BEM) applied to the linearized FGBVP. In spite of previous BEM approaches in geodesy, we use the so-called direct BEM formulation, where a weak formulation is derived through the method of weighted residuals. The collocation technique with linear basis functions is applied for deriving the linear system of equations from the arising boundary integral equations. The nonstationary iterative biconjugate gradient stabilized method is used to solve the large-scale linear system of equations. The standard MPI (message passing interface) subroutines are implemented in order to perform parallel computations. The proposed approach gives a numerical solution at collocation points directly on the Earth’s surface (on a fixed boundary). Numerical experiments deal with (i) global gravity field modelling using synthetic data (surface gravity disturbances generated from a global geopotential model (GGM)) (ii) local gravity field modelling in Slovakia using observed gravity data. In order to extend computations, the memory requirements are reduced using elimination of the far-zone effects by incorporating GGM or a coarse global numerical solution obtained by BEM. Statistical characteristics of residuals between numerical solutions and GGM confirm the reliability of the approach and indicate accuracy of numerical solutions for the global models. A local refinement in Slovakia results in a local (national) quasigeoid model, which when compared with GPS-levelling data, does not make a large improvement on existing remove-restore-based models.     

基于遗传Elman神经网络进行矿区GPS高程拟合

目前,城市、平原地区的似大地水准面建立精度已经达到厘米级,但在矿区进行高程拟合时,由于地面高低起伏没有规则,其似大地水准面的拟合精度并不理想。针对此问题,本文提出利用遗传算法优化Elman神经网络的方法精化似大地水准面,采用移去-恢复法对残差进行建模,使用EGM 2008地球重力场模型和地形起伏信息来精化求解似大地水准面和参考椭球面之间的高程异常,同时着重分析了地球重力场模型以及地形变化信息对高程异常求解的重要性,并使用某矿区实测数据(GPS、水准)对所提方法进行验证,实验结果表明:文中所提方法的精度要优于二次曲面拟合模型和单一Elman模型,其外符合精度达到了1.14 cm,可以代替四等水准测量。     

似大地水准面的定义及在空中测量中涉及的问题

本文讨论了高程异常和似大地水准面的定义以及高程异常随空间位置的变化,提出了将似大地水准面称为正常高系统的高程基准面的不恰当性,给出了由地面测量至空中点几何高差确定空中点正常高的改正方法。其目的主要是针对当前已经扩展到了空中的测量,引导正确地认识正常高系统的基本问题。     

Two different methods of geoidal height determinations using a spherical harmonic representation of the geopotential, topographic corrections and the height anomaly–geoidal height difference

 It is suggested that a spherical harmonic representation of the geoidal heights using global Earth gravity models (EGM) might be accurate enough for many applications, although we know that some short-wavelength signals are missing in a potential coefficient model. A `direct' method of geoidal height determination from a global Earth gravity model coefficient alone and an `indirect' approach of geoidal height determination through height anomaly computed from a global gravity model are investigated. In both methods, suitable correction terms are applied. The results of computations in two test areas show that the direct and indirect approaches of geoid height determination yield good agreement with the classical gravimetric geoidal heights which are determined from Stokes' formula. Surprisingly, the results of the indirect method of geoidal height determination yield better agreement with the global positioning system (GPS)-levelling derived geoid heights, which are used to demonstrate such improvements, than the results of gravimetric geoid heights at to the same GPS stations. It has been demonstrated that the application of correction terms in both methods improves the agreement of geoidal heights at GPS-levelling stations. It is also found that the correction terms in the direct method of geoidal height determination are mostly similar to the correction terms used for the indirect determination of geoidal heights from height anomalies. Received: 26 July 2001 / Accepted: 21 February 2002     

Impact of accounting for coloured noise in radar altimetry data on a regional quasi-geoid model

We study the impact of an accurate computation and incorporation of coloured noise in radar altimeter data when computing a regional quasi-geoid model using least-squares techniques. Our test area comprises the Southern North Sea including the Netherlands, Belgium, and parts of France, Germany, and the UK. We perform the study by modelling the disturbing potential with spherical radial base functions. To that end, we use the traditional remove-compute-restore procedure with a recent GRACE/GOCE static gravity field model. Apart from radar altimeter data, we use terrestrial, airborne, and shipboard gravity data. Radar altimeter sea surface heights are corrected for the instantaneous dynamic topography and used in the form of along-track quasi-geoid height differences. Noise in these data are estimated using repeat-track and post-fit residual analysis techniques and then modelled as an auto regressive moving average process. Quasi-geoid models are computed with and without taking the modelled coloured noise into account. The difference between them is used as a measure of the impact of coloured noise in radar altimeter along-track quasi-geoid height differences on the estimated quasi-geoid model. The impact strongly depends on the availability of shipboard gravity data. If no such data are available, the impact may attain values exceeding 10 centimetres in particular areas. In case shipboard gravity data are used, the impact is reduced, though it still attains values of several centimetres. We use geometric quasi-geoid heights from GPS/levelling data at height markers as control data to analyse the quality of the quasi-geoid models. The quasi-geoid model computed using a model of the coloured noise in radar altimeter along-track quasi-geoid height differences shows in some areas a significant improvement over a model that assumes white noise in these data. However, the interpretation in other areas remains a challenge due to the limited quality of the control data.     

Unification of New Zealand’s local vertical datums: iterative gravimetric quasigeoid computations

New Zealand uses 13 separate local vertical datums (LVDs) based on geodetic levelling from 12 different tide-gauges. We describe their unification using a regional gravimetric quasigeoid model and GPS-levelling data on each LVD. A novel application of iterative quasigeoid computation is used, where the LVD offsets computed from earlier models are used to apply additional gravity reductions from each LVD to that model. The solution converges after only three iterations yielding LVD offsets ranging from 0.24 to 0.58 m with an average standard deviation of ±0.08 m. The so-computed LVD offsets agree, within expected data errors, with geodetically levelled height differences at common benchmarks between adjacent LVDs. This shows that iterated quasigeoid models have a role in vertical datum unification.     

准噶尔盆地中西部油区精化似大地水准面建立与实施

利用GPS定位技术结合准噶尔盆地区域内的重力资料、水准资料、高分辨率的地形数据以及最新的重力场模型,精确研究并确定准噶尔盆地的似大地水准面模型,以求取不同地区、不同位置高精度的高程异常值,从根本上解决GPS定位技术无法直接提供正常高的问题。     

贵阳市似大地水准面精化项目建设

为解决CORS系统中GNSS高程受技术条件限制精度不高的问题,贵阳市进行了区域似大地水准面精化工作。本文论述了GNSS和水准网的布设及精度,使用了3 877个点重力数据和54个GNSS水准资料,以EIGEN03C地球重力场模型作为参考重力场,由第二类Helmert凝集法完成大地水准面计算,利用球冠谐调和分析方法将GNSS水准与重力似大地水准面联合求解得出的2'!2'格网似大地水准面,在高原高差地区其精度达到"0.010 m。     

应用双输入单输出法确定局部似大地水准面的实例研究

利用了双输入单输出法,融合处理了我国某地区的重力异常和地形资料两类数据,结合WDM94地球重力场模型和63个高精度GPS水准数据,计算了该区域的似大地水准面。     

On the computation of reliable formal uncertainties in the densification of GPS-levelling networks by least-squares collocation

A realization of a height system covering the south of Norway has been performed, based on least-squares collocation applied to differences between geometric and gravimetric quasigeoid heights, inhomogeneous and isotropic covariance modelling, and without prior information on the error sources of the involved data types. As a result, the derived normal heights were biased by the systematic errors of the GPS-levelling network. The important covariance properties were determined at every location from spatially differenced observations, and made it straightforward to evaluate the uncertainties of the biased height reference. The distribution of predictions followed a Gaussian shape, but extreme realizations were overrepresented.     

组合法精化胶州市大地水准面

本文利用全球重力位模型、胶州市地面重力观测数据、胶州市GPS水准数据和数字地面模型(DTM),采用组合法应用移去-恢复技术计算剩余大地水准面,并与地球位模型计算的高程异常进行拟合,得到该地区重力似大地水准面,再和布测、计算得到的GPS/水准所构成的几何大地水准面拟合,利用多项式拟合完成系统改正,获得最终的大地水准面结果及相关的精度信息。     

国际高程参考系统在珠峰地区的实现

2020珠峰高程测量,首次确定并发布了基于国际高程参考系统(IHRS)的珠峰正高。在珠峰地区实现国际高程参考系统,采用的方案是建立珠峰区域高精度重力大地水准面。利用地球重力场谱组合理论和基于数据驱动的谱权确定方法,测试优选参考重力场模型及其截断阶数和球冠积分半径等关键参数,联合航空和地面重力等数据建立了珠峰区域重力似大地水准面模型,61点高精度GNSS水准高程异常检核表明,模型精度达3.8 cm,加入航空重力数据后模型精度提升幅度达51.3%。提出顾及高差改正的峰顶高程异常内插方法,采用顾及地形质量影响的高程异常——大地水准面差距转换改正严密公式,使用峰顶实测地面重力数据,基于国际高程参考系统定义的重力位值W₀和GRS80参考椭球,最终确定了国际高程参考系统中的高精度珠峰峰顶大地水准面差距。     

Tuning a gravimetric quasigeoid to GPS-levelling by non-stationary least-squares collocation

This paper addresses implementation issues in order to apply non-stationary least-squares collocation (LSC) to a practical geodetic problem: fitting a gravimetric quasigeoid to discrete geometric quasigeoid heights at a local scale. This yields a surface that is useful for direct GPS heighting. Non-stationary covariance functions and a non-stationary model of the mean were applied to residual gravimetric quasigeoid determination by planar LSC in the Perth region of Western Australia. The non-stationary model of the mean did not change the LSC results significantly. However, elliptical kernels in non-stationary covariance functions were used successfully to create an iterative optimisation loop to decrease the difference between the gravimetric quasigeoid and geometric quasigeoid at 99 GPS-levelling points to a user-prescribed tolerance.     

An evaluation of some systematic error sources affecting terrestrial gravity anomalies

Terrestrial free-air gravity anomalies form a most essential data source in the framework of gravity field determination. Gravity anomalies depend on the datums of the gravity, vertical, and horizontal networks as well as on the definition of a normal gravity field; thus gravity anomaly data are affected in a systematic way by inconsistencies of the local datums with respect to a global datum, by the use of a simplified free-air reduction procedure and of different kinds of height system. These systematic errors in free-air gravity anomaly data cause systematic effects in gravity field related quantities like e.g. absolute and relative geoidal heights or height anomalies calculated from gravity anomaly data. In detail it is shown that the effects of horizontal datum inconsistencies have been underestimated in the past. The corresponding systematic errors in gravity anomalies are maximum in mid-latitudes and can be as large as the errors induced by gravity and vertical datum and height system inconsistencies. As an example the situation in Australia is evaluated in more detail: The deviations between the national Australian horizontal datum and a global datum produce a systematic error in the free-air gravity anomalies of about −0.10 mgal which value is nearly constant over the continent