The AUSGeoid98 geoid model of Australia: data treatment, computations and comparisons with GPS-levelling data

 The AUSGeoid98 gravimetric geoid model of Australia has been computed using data from the EGM96 global geopotential model, the 1996 release of the Australian gravity database, a nationwide digital elevation model, and satellite altimeter-derived marine gravity anomalies. The geoid heights are on a 2 by 2 arc-minute grid with respect to the GRS80 ellipsoid, and residual geoid heights were computed using the 1-D fast Fourier transform technique. This has been adapted to include a deterministically modified kernel over a spherical cap of limited spatial extent in the generalised Stokes scheme. Comparisons of AUSGeoid98 with GPS and Australian Height Datum (AHD) heights across the continent give an RMS agreement of ±0.364 m, although this apparently large value is attributed partly to distortions in the AHD. Received: 10 March 2000 / Accepted: 21 February 2001     

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     

GPS-GRAVIMETRIC GEOID DETERMINATION IN EGYPT

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How to handle topography in practical geoid determination: three examples

 Three different methods of handling topography in geoid determination were investigated. The first two methods employ the residual terrain model (RTM) remove–restore technique, yielding the quasigeoid, whereas the third method uses the classical Helmert condensation method, yielding the geoid. All three methods were used with the geopotential model Earth Gravity Model (1996) (EGM96) as a reference, and the results were compared to precise global positioning system (GPS) levelling networks in Scandinavia. An investigation of the Helmert method, focusing on the different types of indirect effects and their effects on the geoid, was also carried out. The three different methods used produce almost identical results at the 5-cm level, when compared to the GPS levelling networks. However, small systematic differences existed. Received: 18 March 1999 / Accepted: 21 March 2000     

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     

The northern European geoid: a case study on long-wavelength geoid errors

 The long-wavelength geoid errors on large-scale geoid solutions, and the use of modified kernels to mitigate these effects, are studied. The geoid around the Nordic area, from Greenland to the Ural mountains, is considered. The effect of including additional gravity data around the Nordic/Baltic land area, originating from both marine, satellite and ground-based measurements, is studied. It is found that additional data appear to increase the noise level in computations, indicating the presence of systematic errors. Therefore, the Wong–Gore modification to the Stokes kernel is applied. This method of removing lower-order terms in the Stokes kernel appears to improve the geoid. The best fit to the global positioning system (GPS) leveling points is obtained with a degree of modification of approximately 30. In addition to the study of modification errors, the results of different methods of combining satellite altimetry gravity and other gravimetry are presented. They all gave comparable results, at the 6-cm level, when evaluated for the Nordic GPS networks. One dimensional (1-D) and 2-D fast Fourier transform (FFT) methods are also compared. It is shown that even though methods differ by up to 6 cm, the fit to the GPS is essentially the same. A surprising conclusion is that the addition of more data does not always produce a better geoid, illustrating the danger of systematic errors in data. Received: 4 July 2001 / Accepted: 21 February 2002     

GPS-gravimetric geoid determination in Egypt

The main objective of this study is to improve the geoid by GPS/leveling data in Egypt. Comparisons of the gravimetric geoid with GPS/leveling data have been performed. On the basis of a gravimetric geoid fitted to GPS/leveling by the least square method, a smoothed geoid was obtained. A high-resolution geoid in Egypt was computed with a 2.5′×2.5′ grid by combining the data set of 2600 original point gravity values, 20″×30″ resolution Digital Terrain Model (DTM) grid and the spherical harmonic model EGM96. The method of computation involved the strict evaluation of the Stokes integral with 1D-FFT. The standard deviation of the difference between the gravimetric and the GPS/leveling geoid heights is ±0.47 m. The standard deviation after fitting of the gravimetric geoid to the GPS/leveling points is better than ±13 cm. In the future we will try to improve our geoid results in Egypt by increasing the density of gravimetric coverage.     

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     

Geoid determination using adapted reference field, seismic Moho depths and variable density contrast

 The traditional remove-restore technique for geoid computation suffers from two main drawbacks. The first is the assumption of an isostatic hypothesis to compute the compensation masses. The second is the double consideration of the effect of the topographic–isostatic masses within the data window through removing the reference field and the terrain reduction process. To overcome the first disadvantage, the seismic Moho depths, representing, more or less, the actual compensating masses, have been used with variable density anomalies computed by employing the topographic–isostatic mass balance principle. In order to avoid the double consideration of the effect of the topographic–isostatic masses within the data window, the effect of these masses for the used fixed data window, in terms of potential coefficients, has been subtracted from the reference field, yielding an adapted reference field. This adapted reference field has been used for the remove–restore technique. The necessary harmonic analysis of the topographic–isostatic potential using seismic Moho depths with variable density anomalies is given. A wide comparison among geoids computed by the adapted reference field with both the Airy–Heiskanen isostatic model and seismic Moho depths with variable density anomaly and a geoid computed by the traditional remove–restore technique is made. The results show that using seismic Moho depths with variable density anomaly along with the adapted reference field gives the best relative geoid accuracy compared to the GPS/levelling geoid. Received: 3 October 2001 / Accepted: 20 September 2002 Correspondence to: H.A. Abd-Elmotaal     

Precise geoid determination over Sweden using the Stokes–Helmert method and improved topographic corrections

 Four different implementations of Stokes' formula are employed for the estimation of geoid heights over Sweden: the Vincent and Marsh (1974) model with the high-degree reference gravity field but no kernel modifications; modified Wong and Gore (1969) and Molodenskii et al. (1962) models, which use a high-degree reference gravity field and modification of Stokes' kernel; and a least-squares (LS) spectral weighting proposed by Sj?berg (1991). Classical topographic correction formulae are improved to consider long-wavelength contributions. The effect of a Bouguer shell is also included in the formulae, which is neglected in classical formulae due to planar approximation. The gravimetric geoid is compared with global positioning system (GPS)-levelling-derived geoid heights at 23 Swedish Permanent GPS Network SWEPOS stations distributed over Sweden. The LS method is in best agreement, with a 10.1-cm mean and ±5.5-cm standard deviation in the differences between gravimetric and GPS geoid heights. The gravimetric geoid was also fitted to the GPS-levelling-derived geoid using a four-parameter transformation model. The results after fitting also show the best consistency for the LS method, with the standard deviation of differences reduced to ±1.1 cm. For comparison, the NKG96 geoid yields a 17-cm mean and ±8-cm standard deviation of agreement with the same SWEPOS stations. After four-parameter fitting to the GPS stations, the standard deviation reduces to ±6.1 cm for the NKG96 geoid. It is concluded that the new corrections in this study improve the accuracy of the geoid. The final geoid heights range from 17.22 to 43.62 m with a mean value of 29.01 m. The standard errors of the computed geoid heights, through a simple error propagation of standard errors of mean anomalies, are also computed. They range from ±7.02 to ±13.05 cm. The global root-mean-square error of the LS model is the other estimation of the accuracy of the final geoid, and is computed to be ±28.6 cm. Received: 15 September 1999 / Accepted: 6 November 2000     

EGM96,WDM94和GPM98CR高阶地球重力场模型表示深圳局部重力场的比较与评价

为计算深圳精密重力大地水准面，利用62个高精度GPS水准点和4871个实测重力点数据对EGM96，WDM94和GPM98CR全球重力场模型表示深圳局部重力场进行了比较与评价。结果表明，由上述3个重力场模型计算的大地水准面高和重力异常与实测值之间存在明显的系统偏差，当采用GPS水准数据尽可能消除系统偏差以后，大地水准面高的精度得到显著提高，若应用移去-恢复技术确定深圳高精度大地水准面，则WDM94应该是首选的参考重力场模型。     

National height datum,the Gauss–Listing geoid level value w0 and its time variation 0 (Baltic Sea Level Project: epochs 1990.8, 1993.8, 1997.4)

 A methodology for precise determination of the fundamental geodetic parameter w ₀, the potential value of the Gauss–Listing geoid, as well as its time derivative ₀, is presented. The method is based on: (1) ellipsoidal harmonic expansion of the external gravitational field of the Earth to degree/order 360/360 (130 321 coefficients; http://www.uni-stuttgard.de/gi/research/ index.html projects) with respect to the International Reference Ellipsoid WGD2000, at the GPS positioned stations; and (2) ellipsoidal free-air gravity reduction of degree/order 360/360, based on orthometric heights of the GPS-positioned stations. The method has been numerically tested for the data of three GPS campaigns of the Baltic Sea Level project (epochs 1990.8,1993.4 and 1997.4). New w ₀ and ₀ values (w ₀=62 636 855.75 ± 0.21 m²/s², ₀=−0.0099±0.00079 m²/s² per year, w ₀/&γmacr;=6 379 781.502 m,₀/&γmacr;=1.0 mm/year, and &γmacr;= −9.81802523 m²/s²) for the test region (Baltic Sea) were obtained. As by-products of the main study, the following were also determined: (1) the high-resolution sea surface topography map for the Baltic Sea; (2) the most accurate regional geoid amongst four different regional Gauss–Listing geoids currently proposed for the Baltic Sea; and (3) the difference between the national height datums of countries around the Baltic Sea. Received: 14 August 2000 / Accepted: 19 June 2001     

Global mean sea surface and marine gravity anomaly from multi-satellite altimetry: applications of deflection-geoid and inverse Vening Meinesz formulae

 Global mean sea surface heights (SSHs) and gravity anomalies on a 2^′×2^′ grid were determined from Seasat, Geosat (Exact Repeat Mission and Geodetic Mission), ERS-1 (1.5-year mean of 35-day, and GM), TOPEX/POSEIDON (T/P) (5.6-year mean) and ERS-2 (2-year mean) altimeter data over the region 0^∘–360^∘ longitude and –80^∘–80^∘ latitude. To reduce ocean variabilities and data noises, SSHs from non-repeat missions were filtered by Gaussian filters of various wavelengths. A Levitus oceanic dynamic topography was subtracted from the altimeter-derived SSHs, and the resulting heights were used to compute along-track deflection of the vertical (DOV). Geoidal heights and gravity anomalies were then computed from DOV using the deflection-geoid and inverse Vening Meinesz formulae. The Levitus oceanic dynamic topography was added back to the geoidal heights to obtain a preliminary sea surface grid. The difference between the T/P mean sea surface and the preliminary sea surface was computed on a grid by a minimum curvature method and then was added to the preliminary grid. The comparison of the NCTU01 mean sea surface height (MSSH) with the T/P and the ERS-1 MSSH result in overall root-mean-square (RMS) differences of 5.0 and 3.1 cm in SSH, respectively, and 7.1 and 3.2 μrad in SSH gradient, respectively. The RMS differences between the predicted and shipborne gravity anomalies range from 3.0 to 13.4 mGal in 12 areas of the world's oceans. Received: 26 September 2001 / Accepted: 3 April 2002 Correspondence to: C. Hwang Acknowledgements. This research is partly supported by the National Science Council of ROC, under grants NSC89-2611-M-009-003-OP2 and NSC89-2211-E-009-095. This is a contribution to the IAG Special Study Group 3.186. The Geosat and ERS1/2 data are from NOAA and CERSAT/France, respectively. The T/P data were provided by AVISO. The CLS and GSFC00 MSS models were kindly provided by NASA/GSFC and CLS, respectively. Drs. Levitus, Monterey, and Boyer are thanked for providing the SST model. Dr. T. Gruber and two anonymous reviewers provided very detailed reviews that improved the quality of this paper.     

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。     

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     

利用等效点质量进行(似)大地水准面拟合

基于等效源原理,提出了一种半自由点质量模型,并给出了顾及相邻点空间关系构造虚拟点质量的简单快速迭代算法。实验结果表明,利用该点质量模型对离散GPS/水准观测数据进行拟合是可行的。     

GPS-assisted GLONASS orbit determination

 Using 1 week of data from a network of GPS/GLONASS dual-tracking receivers, 15-cm accurate GLONASS orbit determination is demonstrated with an approach that combines GPS and GLONASS data. GPS data are used to define the reference frame, synchronize receiver clocks and determine troposphere delay for the GLONASS tracking network. GLONASS tracking data are then processed separately, with the GPS-defined parameters held fixed, to determine the GLONASS orbit. The quality of the GLONASS orbit determination is currently limited by the size and distribution of the tracking network, and by the unavailability of a sufficiently refined solar pressure model. Temporal variations in the differential clock bias of the dual-tracking receivers are found to have secondary impact on the orbit determination accuracy. Received: 5 January 2000 / Accepted: 15 February 2001     

New analytical and numerical approaches for geopotential modeling

 The standard analytical approach which is applied for constructing geopotential models OSU86 and earlier ones, is based on reducing the boundary value equation to a sphere enveloping the Earth and then solving it directly with respect to the potential coefficients _n,m . In an alternative procedure, developed by Jekeli and used for constructing the models OSU91 and EGM96, at first an ellipsoidal harmonic series is developed for the geopotential and then its coefficients _n,m ^e are transformed to the unknown _n,m . The second solution is more exact, but much more complicated. The standard procedure is modified and a new simple integral formula is derived for evaluating the potential coefficients. The efficiency of the standard and new procedures is studied numerically. In these solutions the same input data are used as for constructing high-degree parts of the EGM96 models. From two sets of _n,m (n≤360,|m|≤n), derived by the standard and new approaches, different spectral characteristics of the gravity anomaly and the geoid undulation are estimated and then compared with similar characteristics evaluated by Jekeli's approach (`etalon' solution). The new solution appears to be very close to Jekeli's, as opposed to the standard solution. The discrepancies between all the characteristics of the new and `etalon' solutions are smaller than the corresponding discrepancies between two versions of the final geopotential model EGM96, one of them (HDM190) constructed by the block-diagonal least squares (LS) adjustment and the other one (V068) by using Jekeli's approach. On the basis of the derived analytical solution a new simple mathematical model is developed to apply the LS technique for evaluating geopotential coefficients. Received: 12 December 2000 / Accepted: 21 June 2001     

The impact of accelerometry on CHAMP orbit determination

 The contribution of the STAR accelerometer to the CHAMP orbit precision is evaluated and quantified by means of the following results: orbital fit to the satellite laser ranging (SLR) observations, GPS reduced-dynamic vs SLR dynamic orbit comparisons, and comparison of the measured to the modeled non-gravitational accelerations (atmospheric drag in particular). In each of the four test periods in 2001, five CHAMP arcs of 2 days' length were analyzed. The mean RMS-of-fit of the SLR observations of the orbits computed with STAR data or the non-gravitational force model were 11 and 24 cm, respectively. If the accelerometer calibration parameters are not known at least at the few percent level, the SLR orbit fit deteriorates. This was tested by applying a 10% error to the along-track scale factor of the accelerometer, which increased the SLR RMS-of-fit on average to 17 cm. Reference orbits were computed employing the reduced-dynamic technique with GPS tracking data. This technique yields the most accurate orbit positions thanks to the estimation of a large number of empirical accelerations, which compensate for dynamic modeling errors. Comparison of the SLR orbits, computed with STAR data or the non-gravitational force model, to the GPS-based orbits showed that the SLR orbits employing accelerometer observations are twice as accurate. Finally, comparison of measured to modeled accelerations showed that the level of geomagnetic activity is highly correlated with the atmospheric drag model error, and that the largest errors occur around the geomagnetic poles. Received: 7 May 2002 / Accepted: 18 November 2002 Correspondence to: S. Bruinsma Acknowledgments. The TIGCM results were obtained from the CEDAR database. This study was supported by the Centre National d'Etudes Spatiales (CNES). The referees are thanked for their helpful remarks and suggestions.     

Models for Fitting Gravimetric Geoids and GPS Results

The aim of this investigation is to study how to use a gravimetric(quasi) geoid for levelling by GPS data in an optimal way.The advent of precise geodetic GPS has made the use of a technique possible,which might be called GPS- gravimetric geoid determination.In this approach,GPS heights above the reference ellipsoid are determined for points whose levelled (orthometric) height H is above sea level people have already surveyed;for these points,we thus have the values of the geoid undulation N.These values are then used to constrain the geoid undulations N‘ obtained from the gravimetric solution.