The Geoid Slope Validation Survey 2014 and GRAV-D airborne gravity enhanced geoid comparison results in Iowa

Three Geoid Slope Validation Surveys were planned by the National Geodetic Survey for validating geoid improvement gained by incorporating airborne gravity data collected by the “Gravity for the Redefinition of the American Vertical Datum” (GRAV-D) project in flat, medium and rough topographic areas, respectively. The first survey GSVS11 over a flat topographic area in Texas confirmed that a 1-cm differential accuracy geoid over baseline lengths between 0.4 and 320 km is achievable with GRAV-D data included (Smith et al. in J Geod 87:885–907, 2013). The second survey, Geoid Slope Validation Survey 2014 (GSVS14) took place in Iowa in an area with moderate topography but significant gravity variation. Two sets of geoidal heights were computed from GPS/leveling data and observed astrogeodetic deflections of the vertical at 204 GSVS14 official marks. They agree with each other at a \({\pm }1.2\,\, \hbox {cm}\) level, which attests to the high quality of the GSVS14 data. In total, four geoid models were computed. Three models combined the GOCO03/5S satellite gravity model with terrestrial and GRAV-D gravity with different strategies. The fourth model, called xGEOID15A, had no airborne gravity data and served as the benchmark to quantify the contribution of GRAV-D to the geoid improvement. The comparisons show that each model agrees with the GPS/leveling geoid height by 1.5 cm in mark-by-mark comparisons. In differential comparisons, all geoid models have a predicted accuracy of 1–2 cm at baseline lengths from 1.6 to 247 km. The contribution of GRAV-D is not apparent due to a 9-cm slope in the western 50-km section of the traverse for all gravimetric geoid models, and it was determined that the slopes have been caused by a 5 mGal bias in the terrestrial gravity data. If that western 50-km section of the testing line is excluded in the comparisons, then the improvement with GRAV-D is clearly evident. In that case, 1-cm differential accuracy on baselines of any length is achieved with the GRAV-D-enhanced geoid models and exhibits a clear improvement over the geoid models without GRAV-D data. GSVS14 confirmed that the geoid differential accuracies are in the 1–2 cm range at various baseline lengths. The accuracy increases to 1 cm with GRAV-D gravity when the west 50 km line is not included. The data collected by the surveys have high accuracy and have the potential to be used for validation of other geodetic techniques, e.g., the chronometric leveling. To reach the 1-cm height differences of the GSVS data, a clock with frequency accuracy of \(10^{-18}\) is required. Using the GSVS data, the accuracy of ellipsoidal height differences can also be estimated.     

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.     

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.     

Confirming regional 1 cm differential geoid accuracy from airborne gravimetry: the Geoid Slope Validation Survey of 2011

A terrestrial survey, called the Geoid Slope Validation Survey of 2011 (GSVS11), encompassing leveling, GPS, astrogeodetic deflections of the vertical (DOV) and surface gravity was performed in the United States. The general purpose of that survey was to evaluate the current accuracy of gravimetric geoid models, and also to determine the impact of introducing new airborne gravity data from the ‘Gravity for the Redefinition of the American Vertical Datum’ (GRAV-D) project. More specifically, the GSVS11 survey was performed to determine whether or not the GRAV-D airborne gravimetry, flown at 11 km altitude, can reduce differential geoid error to below 1 cm in a low, flat gravimetrically uncomplicated region. GSVS11 comprises a 325 km traverse from Austin to Rockport in Southern Texas, and includes 218 GPS stations ( $\sigma _{\Delta h }= 0.4$ cm over any distance from 0.4 to 325 km) co-located with first-order spirit leveled orthometric heights ( $\sigma _{\Delta H }= 1.3$ cm end-to-end), including new surface gravimetry, and 216 astronomically determined vertical deflections $(\sigma _{\mathrm{DOV}}= 0.1^{\prime \prime })$ . The terrestrial survey data were compared in various ways to specific geoid models, including analysis of RMS residuals between all pairs of points on the line, direct comparison of DOVs to geoid slopes, and a harmonic analysis of the differences between the terrestrial data and various geoid models. These comparisons of the terrestrial survey data with specific geoid models showed conclusively that, in this type of region (low, flat) the geoid models computed using existing terrestrial gravity, combined with digital elevation models (DEMs) and GRACE and GOCE data, differential geoid accuracy of 1 to 3 cm (1 $\sigma )$ over distances from 0.4 to 325 km were currently being achieved. However, the addition of a contemporaneous airborne gravity data set, flown at 11 km altitude, brought the estimated differential geoid accuracy down to 1 cm over nearly all distances from 0.4 to 325 km.     

高程系统定义分析与高精度GNSS代替水准算法

从高程系统定义出发,探讨高程基准面的重力等位性质,测试分析不同类型高程系统地面点高程之间的差异,考察GNSS代替水准与实际水准测量成果的一致性,进而提出新的GNSS代替水准算法。主要结论包括:(1)当精度要求达到厘米级水平时,正常高的基准面也应是大地水准面。中国国家1985高程基准采用正常高系统,其高程基准面是过青岛零点的大地水准面。(2)近地空间中等解析正高面与大地水准面平行,GNSS代替水准能直接测定地面点的解析正高,但正常高系统更有利于描述地势和地形起伏。(3)本文给出的GNSS代替水准测定近地点正常高算法,大地高误差对正常高结果的影响比大地水准面误差大,前者影响约为后者的1.5倍。     

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     

Effects of azimuthal multipath asymmetry on long GPS coordinate time series

Carrier phase multipath is currently one source of unmodeled signals that may bias GPS coordinate time series significantly. We investigate the effect of simulated carrier phase multipath on time series of several sites covering the period 2002.0–2008.0 and spanning a range of observation geometries. High-, mid-, and low-latitude IGS sites are investigated as well as sites with significant signal obstructions. We examine the effect of multipath in different sectors of the sky, considering time-constant, horizontal reflectors at each of 0.1, 0.2, and 1.5 m below the antenna. The differences between a horizontally uniform multipath source are analyzed, and it is shown that positioning errors are generally larger when unmodeled carrier phase multipath is azimuthally heterogeneous. Using the adopted multipath model, height biases reach ±1 mm in case of the symmetric multipath and ±5 mm for the asymmetric multipath but this increases to being ±10 mm in the worst case. In addition to mean bias, low-frequency variations in the bias also exist, including periodic signals and leading to velocity biases of up to ±0.1 mm/year in the symmetric case and ±1 mm/year in the asymmetric case over the considered period. In contrast to the generally slowly varying observation geometry that is typically experienced, we show the effects of an abrupt change in geometry due to receiver/antenna hardware changes; in the case considered, we see changed pattern of temporal variation in the bias in addition to an instantaneous offset.     

利用CORS站网监测三峡地区环境负荷引起的地壳形变与重力场时空变化

受地球动力学因素特别是地球表层大气、地表水及地下水动力环境影响,地面站点位置、地球重力场及大地水准面随时间变化。以三峡地区CORS站网为主,少量重力台站为辅,采用负荷形变与地球重力场严密组合方法,综合确定了2011-01-2015-06三峡地区环境负荷驱动的地壳形变与重力场月变化,结果显示:（1）CORS站网具备地壳垂直形变、大地水准面及地面重力变化的监测能力;（2）三峡地区地壳垂直形变年变化幅度36.1 mm,大地水准面年变化幅度28.2 mm,地面重力年变化幅度117.4 μGal;（3）CORS站网地面重力变化监测精度水平不低于流动重力场重复测量;（4）CORS站网地壳垂直形变与地面重力变化监测具有一定的外推预报能力。     

配置法在局部大地水准面精化中的实例研究

从最小二乘配置方法的基本原理出发,以我国某地区范围内1km分辨率的大地水准面高模型数据为例,根据实用公式计算了试验区大地水准面高的协方差值后,采用多项式函数模型和高斯函数模型分别拟合了该地区大地水准面高的局部协方差函数,并对试验区内18个检核点做了推估计算。根据推估值(Nfit)与实测值(NGPSL)的比较分析表明,虽然多项式协方差函数模型略优于高斯协方差函数模型,但它们都能以厘米级的精度拟合局部大地水准面,这表明了配置法用于精化厘米级大地水准面的有效性。     

Attenuation effect on seasonal basin-scale water storage changes from GRACE time-variable gravity

In order to effectively recover surface mass or geoid height changes from the gravity recovery and climate experiment (GRACE) time-variable gravity models, spatial smoothing is required to minimize errors from noise. Spatial smoothing, such as Gaussian smoothing, not only reduces the noise but also attenuates the real signals. Here we investigate possible amplitude attenuations and phase changes of seasonal water storage variations in four drainage basins (Amazon, Mississippi, Ganges and Zambezi) using an advanced global land data assimilation system. It appears that Gaussian smoothing significantly affects GRACE-estimated basin-scale seasonal water storage changes, e.g., in the case of 800 km smoothing, annual amplitudes are reduced by about 25–40%, while annual phases are shifted by up to 10°. With these effects restored, GRACE-estimated water storage changes are consistently larger than model estimates, indicating that the land surface model appears to underestimate terrestrial water storage change. Our analysis based on simulation suggests that normalized attenuation effects (from Gaussian smoothing) on seasonal water storage change are relatively insensitive to the magnitude of the true signal. This study provides a numerical approach that can be used to restore seasonal water storage change in the basins from spatially smoothed GRACE data.     

Geoid models around Sognefjord using depth data

Bathymetry data from Sognefjord, Norway, have been included in a terrain model, and their influence on the geoid has been calculated. The test area, located in the western part of Norway, was chosen due to its deep fjords and high mountains. Inclusion of bathymetry data in the terrain model altered the computed gravimetric geoid by as much as a few decimeters. The effect was detectable to a distance of more than 100 km. All calculated geoids, both with and without bathymetry data in the terrain model, fit the geoidal heights determined by available Global Positioning System (GPS) and levelling heights at the sub-decimetre level. Contrary to expectations, the accuracy in geoid prediction was reduced when using bathymetric data. The geoid changes were largest over the fjord where no GPS points were located. Different methods on the same area [isostatic and Residual Terrain Model (RTM)-terrain reductions] showed differences of approximately 1 m. Rigorous distinction between quasigeoid and geoid was found to be essential in this kind of area. Received: 12 May 1997 / Accepted 7 May 1998     

Impacts of geodynamic phenomena on systems for height and gravity

Geodynamic phenomena of permanent or secular characters play a significant role when defining height systems and gravity systems. A treatment is here given of the permanent earth tide, postglacial land uplift, sea level changes and polar drift from this point of view.

Numerical simulation of troposphere-induced errors in GPS-derived geodetic time series over Japan

Troposphere-induced errors in GPS-derived geodetic time series, namely, height and zenith total delays (ZTDs), over Japan are quantitatively evaluated through the analyses of simulated GPS data using realistic cumulative tropospheric delays and observed GPS data. The numerical simulations show that the use of a priori zenith hydrostatic delays (ZHDs) derived from the European Centre for Medium-Range Weather Forecasts (ECMWF) numerical weather model data and gridded Vienna mapping function 1 (gridded VMF1) results in smaller spurious annual height errors and height repeatabilities (0.45 and 2.55 mm on average, respectively) as compared to those derived from the global pressure and temperature (GPT) model and global mapping function (GMF) (1.08 and 3.22 mm on average, respectively). On the other hand, the use of a priori ZHDs derived from the GPT and GMF would be sufficient for applications involving ZTDs, given the current discrepancies between GPS-derived ZTDs and those derived from numerical weather models. The numerical simulations reveal that the use of mapping functions constructed with fine-scale numerical weather models will potentially improve height repeatabilities as compared to the gridded VMF1 (2.09 mm against 2.55 mm on average). However, they do not presently outperform the gridded VMF1 with the observed GPS data (6.52 mm against 6.50 mm on average). Finally, the commonly observed colored components in GPS-derived height time series are not primarily the result of troposphere-induced errors, since they become white in numerical simulations with the proper choice of a priori ZHDs and mapping functions.     

A synthetic Earth Gravity Model Designed Specifically for Testing Regional Gravimetric Geoid Determination Algorithms

A synthetic [simulated] Earth gravity model (SEGM) of the geoid, gravity and topography has been constructed over Australia specifically for validating regional gravimetric geoid determination theories, techniques and computer software. This regional high-resolution (1-arc-min by 1-arc-min) Australian SEGM (AusSEGM) is a combined source and effect model. The long-wavelength effect part (up to and including spherical harmonic degree and order 360) is taken from an assumed errorless EGM96 global geopotential model. Using forward modelling via numerical Newtonian integration, the short-wavelength source part is computed from a high-resolution (3-arc-sec by 3-arc-sec) synthetic digital elevation model (SDEM), which is a fractal surface based on the GLOBE v1 DEM. All topographic masses are modelled with a constant mass-density of 2,670 kg/m³. Based on these input data, gravity values on the synthetic topography (on a grid and at arbitrarily distributed discrete points) and consistent geoidal heights at regular 1-arc-min geographical grid nodes have been computed. The precision of the synthetic gravity and geoid data (after a first iteration) is estimated to be better than 30 μ Gal and 3 mm, respectively, which reduces to 1 μ Gal and 1 mm after a second iteration. The second iteration accounts for the changes in the geoid due to the superposed synthetic topographic mass distribution. The first iteration of AusSEGM is compared with Australian gravity and GPS-levelling data to verify that it gives a realistic representation of the Earth’s gravity field. As a by-product of this comparison, AusSEGM gives further evidence of the north–south-trending error in the Australian Height Datum. The freely available AusSEGM-derived gravity and SDEM data, included as Electronic Supplementary Material (ESM) with this paper, can be used to compute a geoid model that, if correct, will agree to in 3 mm with the AusSEGM geoidal heights, thus offering independent verification of theories and numerical techniques used for regional geoid modelling.Electronic Supplementary Material Supplementary material is available in the online version of this article at http://dx.doi.org/10.1007/s00190-005-0002-z     

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     

Realization of a consistent set of vertical reference surfaces in coastal areas

We present a combined approach for the realization of the (quasi-)geoid as a height reference surface and the vertical reference surface at sea (chart datum). This approach, specifically designed for shallow seas and coastal waters, provides the relation between the two vertical reference surfaces without gaps down to the coast. It uses a regional hydrodynamic model, which, after vertical referencing, provides water levels relative to a given (quasi-)geoid. Conversely, the hydrodynamic model is also used to realize a (quasi-)geoid by providing corrections to the dynamic sea surface topography, which are used to reduce radar altimeter-derived sea surface heights to the (quasi-)geoid. The coupled problem of vertically referencing the hydrodynamic model and computing the (quasi-)geoid is solved iteratively. After convergence of the iteration process, the vertically referenced hydrodynamic model is used to realize the chart datum. In this way, consistency between the chart datum and (quasi-)geoid is ensured. We demonstrate the feasibility and performance of this approach for the Dutch mainland and North Sea. We show that in the Dutch part of the North Sea, the differences between modeled and observed instantaneous and mean dynamic sea surface topography is 8–10 and 5.8 cm, respectively. On land, we show that the methodology provides a quasi-geoid which has a lower standard deviation (SD) than the European Gravimetric Geoid 2008 (EGG08) and the official Netherlands quasi-geoid NLGEO2004-grav when compared to GPS-levelling data. The root mean square at 81 GPS-levelling points is below 1.4 cm; no correction surface is needed. Finally, we show that the chart datum (lowest astronomical tide, LAT) agrees with the observed chart datum at 92 onshore tide gauges to within 21.5 cm (SD).     

Geoid undulation modelling and interpretation at Ladak, NW Himalaya using GPS and levelling data

Fast and accurate relative positioning for baselines less than 20 km in length is possible using dual-frequency Global Positioning System (GPS) receivers. By measuring orthometric heights of a few GPS stations by differential levelling techniques, the geoid undulation can be modelled, which enables GPS to be used for orthometric height determination in a much faster and more economical way than terrestrial methods. The geoid undulation anomaly can be very useful for studying tectonic structure. GPS, levelling and gravity measurements were carried out along a 200-km-long highly undulating profile, at an average elevation of 4000 m, in the Ladak region of NW Himalaya, India. The geoid undulation and gravity anomaly were measured at 28 common GPS-levelling and 67 GPS-gravity stations. A regional geoid low of nearly −4 m coincident with a steep negative gravity gradient is compatible with very recent findings from other geophysical studies of a low-velocity layer 20–30 km thick to the north of the India–Tibet plate boundary, within the Tibetan plate. Topographic, gravity and geoid data possibly indicate that the actual plate boundary is situated further north of what is geologically known as the Indus Tsangpo Suture Zone, the traditionally supposed location of the plate boundary. Comparison of the measured geoid with that computed from OSU91 and EGM96 gravity models indicates that GPS alone can be used for orthometric height determination over the Higher Himalaya with 1–2 m accuracy. Received: 10 April 1997 / Accepted: 9 October 1998     

Åland GPS levelling campaign in 1987

In October 1987 a four day satellite GPS campaign was performed over the Åland archipelago to test the possibility of connecting the Swedish and Finnish national height systems. This paper summarizes the gained experiences using 5 WM 101 GPS receivers and the PoPS software.The computing results for the connection between the two height systems are considerably dependent on the choice of geoidal undulation model and systematic error parameter model. Using the NKG Scandinavian geoid 1989, which is probably the most accurate geoid available for the region, and a bias and tilt parameter model the difference between the Swedish RH70 system and the Finnish N60 system is estimated to 11.4 ± 4.0 cm. An independent check is provided by two connecting border bench marks in northern Scandinavia yielding the difference 19.2 ± 4.2 cm. In view of that merely single frequency GPS receivers were used together with the PoPS software, we consider this result most satisfactory.     

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     

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