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.     

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     

Determination of Geoid over South China Sea and Philippine Sea from Multi-satellite Altimetry Data

A new computational procedure for derivation of marine geoid on a 2.5′×2.5′grid in a non-tidal system over the South China Sea and the Philippine Sea from multi-satellite altimeter sea surface heights is discussed. Single-and dual-satellite crossovers were performed, and components of deflections of the vertical were determined at the crossover positions using Sand-well's computational theory, and gridded onto a 2.5′×2.5′resolution grid by employing the Shepard's interpolation procedure. 2.5′×2.5′grid of EGM96-derived components of deflections of the vertical and geoid heights were then used as reference global geopotential model quantities in a remove-restore procedure to implement the Molodensky-like formula via 1D-FFT technique to predict the geoid heights over the South China Sea and the Philippine Sea from the gridded altimeter-derived components of deflec-tions of the vertical. Statistical comparisons between the altimeter-and the EGM96- derived geoid heights showed that there was a root-mean-square agreement of ±0.35 m between them in a region of less tectonically active geological structures. However, over areas of tectonically active structures such as the Philippine trench, differences of about -19.9 m were obtained.     

The US Gravimetric Geoid of 2009 (USGG2009): model development and evaluation

A new gravimetric geoid model, USGG2009 (see Abbreviations), has been developed for the United States and its territories including the Conterminous US (CONUS), Alaska, Hawaii, Guam, the Commonwealth of the Northern Mariana Islands, American Samoa, Puerto Rico and the US Virgin Islands. USGG2009 is based on a 1′ × 1′ gravity grid derived from the NGS surface gravity data and the DNSC08 altimetry-derived anomalies, the SRTM-DTED1 3′′ DEM for its topographic reductions, and the global geopotential model EGM08 as a reference model. USGG2009 geoid heights are compared with control values determined at 18,398 Bench Marks over CONUS, where both the ellipsoidal height above NAD 83 and the Helmert orthometric height above NAVD 88 are known. Correcting for the ellipsoidal datum difference, this permits a comparison of the geoid heights to independent data. The standard deviation of the differences is 6.3 cm in contrast to 8.4 cm for its immediate predecessor— USGG2003. To minimize the effect of long-wavelength errors that are known to exist in NAVD88, these comparisons were made on a state-by-state basis. The standard deviations of the differences range from 3–5 cm in eastern states to about 6–9 cm in the more mountainous western states. If the GPS/Bench Marks-derived geoid heights are corrected by removing a GRACE-derived estimate of the long-wavelength NAVD88 errors before the comparison, the standard deviation of their differences from USGG2009 drops to 4.3 cm nationally and 2–4 cm in eastern states and 4–8 in states with a maximum error of 26.4 cm in California and minimum of −32.1 cm in Washington. USGG2009 is also compared with geoid heights derived from 40 tide-gauges and a physical dynamic ocean topography model in the Gulf of Mexico; the mean of the differences is 3.3 cm and their standard deviation is 5.0 cm. When USGG2009-derived deflections of the vertical are compared with 3,415 observed surface astro-geodetic deflections, the standard deviation of the differences in the N–S and E–W components are 0.87′′ and 0.94′′, respectively.     

Detailed gravimetric geoid for the North Atlantic

A detailed gravimetric geoid in the North Atlantic Ocean, named DGGNA-77, has been computed, based on a satellite and gravimetry derived earth potential model (consisting in spherical harmonic coefficients up to degree and order 30) and mean free air surface gravity anomalies (35180 1°×1° mean values and 245000 4′×4′ mean values). The long wavelength undulations were computed from the spherical harmonics of the reference potential model and the details were obtained by integrating the residual gravity anomalies through the Stokes formula: from 0 to 5° with the 4′×4′ data, and from 5° to 20° with the 1°×1° data. For computer time reasons the final grid was computed with half a degree spacing only. This grid extends from the Gulf of Mexico to the European and African coasts. Comparisons have been made with Geos 3 altimetry derived geoid heights and with the 5′×5′ gravimetric geoid derived byMarsh andChang [8] in the northwestern part of the Atlantic Ocean, which show a good agreement in most places apart from some tilts which porbably come from the satellite orbit recovery.     

A new hybrid geoid model for Japan, GSIGEO2000

 The latest gravimetric geoid model for Japan, JGEOID2000, was successfully combined with the nationwide net of GPS at benchmarks, yielding a new hybrid geoid model for Japan, GSIGEO2000. The least-squares collocation (LSC) method was applied as an interpolation for fitting JGEOID2000 to the GPS/leveling geoid undulations. The GPS/leveling geoid undulation data were reanalyzed in advance, in terms of three-dimensional positions from GPS and orthometric heights from leveling. The new hybrid geoid model is, therefore, compatible with the new Japanese geodetic reference frame. GSIGEO2000 was evaluated internally and independently and the precision was estimated at 4 cm throughout nearly the whole region. Received: 15 October 2001 / Accepted: 27 March 2002 Acknowledgments. Messrs. Toshio Kunimi and Tadashi Saito at the Third Geodetic Division of the Geographical Survey Institute (GSI) mainly carried out the computations of most of the updated leveled heights. With regard to the reanalysis of GPS data, the discussions with Messrs. Yuki Hatanaka and Shoichi Matsumura of GSI were of great help in building the analysis strategy. Messrs. Kazuyuki Tanaka and Hiromi Shigematsu collaborated in the preparatory stages of GPS data computation. The authors' thanks are extended to these colleagues. Some plots were made by GMT software (Wessel and Smith 1991). Correspondence to: Y. Kuroishi     

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     

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.     

The AUSGeoid09 model of the Australian Height Datum

AUSGeoid09 is the new Australia-wide gravimetric quasigeoid model that has been a posteriori fitted to the Australian Height Datum (AHD) so as to provide a product that is practically useful for the more direct determination of AHD heights from Global Navigation Satellite Systems (GNSS). This approach is necessary because the AHD is predominantly a third-order vertical datum that contains a ~1 m north-south tilt and ~0.5 m regional distortions with respect to the quasigeoid, meaning that GNSS-gravimetric-quasigeoid and AHD heights are inconsistent. Because the AHD remains the official vertical datum in Australia, it is necessary to provide GNSS users with effective means of recovering AHD heights. The gravimetric component of the quasigeoid model was computed using a hybrid of the remove-compute-restore technique with a degree-40 deterministically modified kernel over a one-degree spherical cap, which is superior to the remove-compute-restore technique alone in Australia (with or without a cap). This is because the modified kernel and cap combine to filter long-wavelength errors from the terrestrial gravity anomalies. The zero-tide EGM2008 global gravitational model to degree 2,190 was used as the reference field. Other input data are ~1.4 million land gravity anomalies from Geoscience Australia, 1′ × 1′ DNSC2008GRA altimeter-derived gravity anomalies offshore, the 9′′ × 9′′ GEODATA-DEM9S Australian digital elevation model, and a readjustment of Australian National Levelling Network (ANLN) constrained to the CARS2006 mean dynamic ocean topography model. To determine the numerical integration parameters for the modified kernel, the gravimetric component of AUSGeoid09 was compared with 911 GNSS-observed ellipsoidal heights at benchmarks. The standard deviation of fit to the GNSS-AHD heights is ±222 mm, which dropped to ±134 mm for the readjusted GNSS-ANLN heights showing that careful consideration now needs to be given to the quality of the levelling data used to assess gravimetric quasigeoid models. The publicly released version of AUSGeoid09 also includes a geometric component that models the difference between the gravimetric quasigeoid and the zero surface of the AHD at 6,794 benchmarks. This a posteriori fitting used least-squares collocation (LSC) in cross-validation mode to determine a correlation length of 75 km for the analytical covariance function, whereas the noise was taken from the estimated standard deviation of the GNSS ellipsoidal heights. After this LSC surface fitting, the standard deviation of fit reduced to ±30 mm, one-third of which is attributable to the uncertainty in the GNSS ellipsoidal heights.     

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。     

A comparison of altimeter and gravimetric geoids in the Tonga Trench and Indian Ocean areas

A gravimetric geoid computed using different techniques has been compared to a geoid derived from Geos-3 altimeter data in two 30°×30° areas: one in the Tonga Trench area and one in the Indian Ocean. The specific techniques used were the usual Stokes integration (using 1°×1° mean anomalies) with the Molodenskii truncation procedure; a modified Stokes integration with a modified truncation method; and computations using three sets of potential coefficients including one complete to degree 180. In the Tonga Trench area the standard deviation of the difference between the modified Stokes’ procedure and the altimeter geoid was ±1.1 m while in the Indian Ocean area the difference was ±0.6 m. Similar results were found from the 180×180 potential coefficient field. However, the differences in using the usual Stokes integration procedure were about a factor of two greater as was predicted from an error analysis. We conclude that there is good agreement at the ±1 m level between the two types of geoids. In addition, systematic differences are at the half-meter level. The modified Stokes procedure clearly is superior to the usual Stokes method although the 180×180 solution is of comparable accuracy with the computational effort six times less than the integration procedures.     

The GEOID96 high-resolution geoid height model for the United States

The 2 arc-minute × 2 arc-minute geoid model (GEOID96) for the United States supports the conversion between North American Datum 1983 (NAD 83) ellipsoid heights and North American Vertical Datum 1988 (NAVD 88) Helmert heights. GEOID96 includes information from global positioning system (GPS) height measurements at optically leveled benchmarks. A separate geocentric gravimetric geoid, G96SSS, was first calculated, then datum transformations and least-squares collocation were used to convert from G96SSS to GEOID96. Fits of 2951 GPS/level (ITRF94/NAVD 88) benchmarks to G96SSS show a 15.1-cm root mean square (RMS) around a tilted plane (0.06 ppm, 178^∘ azimuth), with a mean value of −31.4 cm (15.6-cm RMS without plane). This mean represents a bias in NAVD 88 from global mean sea level, remaining nearly constant when computed from subsets of benchmarks. Fits of 2951 GPS/level (NAD 83/NAVD 88) benchmarks to GEOID96 show a 5.5-cm RMS (no tilts, zero average), due primarily to GPS error. The correlated error was 2.5 cm, decorrelating at 40 km, and is due to gravity, geoid and GPS errors. Differences between GEOID96 and GEOID93 range from −122 to +374 cm due primarily to the non-geocentricity of NAD 83. Received: 28 July 1997 / Accepted: 2 September 1998     

GPS水准、天文重力水准与重力大地水准面多种数据联合处理的研究

针对我国大地水准面的研究状况，提出了在国家ＧＰＳＢ级网完成之后，利用ＧＰＳ水准、天文重力水准与重力大地水准面３类数据确定我国高精度大地水准面的理论和方法。分析了３类数据的误差传播规律，给出了联合平差模型，并用一模拟网进行了试算     

陆海交界区域厘米级精度似大地水准面的确定

为了得到我国某陆海交界区厘米级精度的区域(似)大地水准面,利用43个高精度GPS/水准点和1 045个实测重力点数据对EGM96,WDM94和GFZ计算的局部重力(似)大地水准面进行了比较与评价。结果表明,在该测区用移去-恢复法确定重力(似)大地水准面时,EGM96应该是首选参考重力场模型。该测区处在陆海交界处,海域无GPS/水准数据。经比较发现,采用距离倒数加权平均法将该区重力似大地水准面拟合于GPS/水准数据比在大范围使用的多项式法效果更好。采用该方法计算的测区(似)大地水准面精度优于3cm。     

Updating and computing the geoid using two dimensional fast Hartley transform and fast T transform

In the analyses of 2D real arrays, fast Hartley (FHT), fast T (FTT) and real-valued fast Fourier transforms are generally preferred in lieu of a complex fast Fourier transform due to the advantages of the former with respect to disk storage and computation time. Although the FHT and the FTT in one dimension are identical, they are different in two or more dimensions. Therefore, first, definitions and some properties of both transforms and the related 2D FHT and FTT algorithms are stated. After reviewing the 2D FHT and FTT solutions of Stokes' formula in planar approximation, 2D FHT and FTT methods are developed for geoid updating to incorporate additional gravity anomalies. The methods are applied for a test area which includes a 64×64 grid of 3^′×3^′ point gravity anomalies and geoid heights calculated from point masses. The geoids computed by 2D FHT and FTT are found to be identical. However, the RMS value of the differences between the computed and test geoid is ±15 mm. The numerical simulations indicate that the new methods of geoid updating are practical and accurate with considerable savings on storage requirements. Received: 15 February 1996; Accepted: 22 January 1997     

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。     

Determination of Geopotential of Local Vertical Datum Surface

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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     

A new,high-resolution geoid for Canada and part of the U.S. by the 1D-FFT method

A new, high-resolution and high-precision geoid has been computed for the whole of Canada and part of the U.S., ranging from 35°N to about 90°N in latitude and 210°E to 320°E in longitude. The OSU91A geopotential model complete to degree and order 360 was combined with a 5 × 5 mean gravity anomaly grid and 1km × 1km topographical information to generate the geoid file. The remove-restore technique was adopted for the computation of terrain effects by Helmert's condensation reduction. The contribution of the local gravity data to the geoid was computed strictly by the 1D-FFT technique, which allows for the evaluation of the discrete spherical Stokes integral without any approximation, parallel by parallel. The indirect effects of up to second order were considered. The internal precision of the geoid, i.e. the contribution of the gravity data and the model coefficients noise, was also evaluated through error propagation by FFT. In a relative sense, these errors seem to agree quite well with the external errors and show clearly the weak areas of the geoid which are mostly due to insufficient gravity data coverage. Comparison of the gravimetric geoid with the GPS/levelling-derived geoidal heights of eight local GPS networks with a total of about 900 stations shows that the absolute agreement with respect to the GPS/levelling datum is generally better than 10 cm RMS and the relative agreement ranges, in most cases, from 4 to 1 ppm over short distances of about 20 to 100km, 1 to 0.5 ppm over distances of about 100 to 200 km, and 0.5 to 0.1 ppm for baselines of 200 to over 1000 km. Other existing geoids, such as UNB90, GEOID90 and GSD91, were also included in the comparison, showing that the new geoid achieves the best agreement with the GPS/levelling data.Presented at theIAG General Meeting, Beijing, P.R. China, Aug. 6–13, 1993     

A new gravity map, a new marine geoid around Japan and the detection of the Kuroshio current

About half a million marine gravity measurements over a 30^∘×30^∘ area centered on Japan have been processed and adjusted to produce a new free-air gravity map from a 5′×5′ grid. This map seems to have a better resolution than those previously published as measured by its correlation with bathymetry. The grid was used together with a high-degree and -order spherical harmonics geopotential model to compute a detailed geoid with two methods: Stokes integral and collocation. Comparisons with other available geoidal surfaces derived either from gravity or from satellite altimetry were made especially to test the ability of this new geoid at showing the sea surface topography as mapped by the Topex/Poseidon satellite. Over 2 months (6 cycles) the dynamic topography at ascending passes in the region (23^∘47^∘N and 123^∘147^∘E) was mapped to study the variability of the Kuroshio current. Received: 15 July 1994 / Accepted: 17 February 1997