Regional gravimetric quasi-geoid model and transformation surface to national height system for Turkey (THG-09)

Turkish regional geoid models have been developed by employing a reference earth gravitational model, surface gravity observations and digital terrain models. The gravimetric geoid models provide a ready transformation from ellipsoidal heights to the orthometric heights through the use of GPS/leveling geoid heights determined through the national geodetic networks. The recent gravimetric models for Turkish territory were computed depending on OSU91 (TG-91) and EGM96 (TG-03) earth gravitational models. The release of the Earth Gravitational Model 2008 (EGM08), the collection of new surface gravity observations, the advanced satellite altimetry-derived gravity over the sea, and the availability of the high resolution digital terrain model have encouraged us to compute a new geoid model for Turkey. We used the Remove-Restore procedure based on EGM08 and applied Residual Terrain Model (RTM) reduction of the surface gravity data. Fast Fourier Transformation (FFT) was then used to obtain the residual quasigeoid from the reduced gravity. We restored the individual contributions of EGM08 and RTM to the whole quasi-geoid height (TQG-09). Since the Helmert orthometric height system is adopted in Turkey, the quasi-geoid model (TQG-09) was then converted to the geoid model (TG-09) by making use of Bouguer gravity anomalies and digital terrain model. After all we combined a gravimetric geoid model with GPS/leveling geoid heights in order to obtain a hybrid geoid model (THG-09) (or a transformation surface) to be used in GPS applications. The RMS of the post-fit residuals after the combination was found to be ± 0.95 cm, which represents the internal precision of the final combination. And finally, we tested the hybrid geoid model with GPS/leveling data, which were not used in the combination, to assess the external accuracy. Results show that the external accuracy of the THG-09 model is ± 8.4 cm, a precision previously not achieved in Turkey until this study.     

An estimation of the height system bias parameter <Emphasis Type="Italic">N</Emphasis><Subscript>0</Subscript> using least squares collocation from observed gravity and GPS-levelling data

This paper deals with the analysis of gravity anomaly and precise levelling in conjunction with GPS-Levelling data for the computation of a gravimetric geoid and an estimate of the height system bias parameter N_o for the vertical datum in Pakistan by means of least squares collocation technique. The long term objective is to obtain a regional geoid (or quasi-geoid) modeling using a combination of local data with a high degree and order Earth gravity model (EGM) and to determine a bias (if there is one) with respect to a global mean sea surface. An application of collocation with the optimal covariance parameters has facilitated to achieve gravimetric height anomalies in a global geocentric datum. Residual terrain modeling (RTM) technique has been used in combination with the EGM96 for the reduction and smoothing of the gravity data. A value for the bias parameter N_o has been estimated with reference to the local GPS-Levelling datum that appears to be 0.705 m with 0.07 m mean square error. The gravimetric height anomalies were compared with height anomalies obtained from GPS-Levelling stations using least square collocation with and without bias adjustment. The bias adjustment minimizes the difference between the gravimetric height anomalies with respect to residual GPS-Levelling data and the standard deviation of the differences drops from 35 cm to 2.6 cm. The results of this study suggest that N_o adjustment may be a good alternative for the fitting of the final gravimetric geoid as is generally done when using FFT methods.     

Improved determination of heights using a conversion surface by combining gravimetric quasi-geoid/geoid and GPS-levelling height differences

The quasi-geoid/geoid can be determined from the Global Positioning System (GPS) ellipsoidal height and the normal/orthometric heights derived from levelling (GPS-levelling). In this study a gravimetric quasigeoid and GPS-levelling height differences are combined to develop a new surface, suitable for “levelling” by GPS. This new surface provides better conversion of GPS ellipsoidal heights to the national normal heights. Different combining procedures, a four-parameter solution, linear and cubic splines interpolations, as well as the least-squares collocation method were investigated and compared over entire Norway. More than 1700 GPS-levelling stations were used in this study. The combined surface provides significant accuracy improvement for the normal height transformation of GPS height data, as demonstrated by the post-fitting residuals. The best solution, based on the least-squares collocation, provided a conversion surface for the transformation of GPS heights into normal height in Norway with an accuracy of about 5 cm.     

Comparison of global geopotential models from the champ and grace missions for regional geoid modelling in Turkey

The continuous efforts on establishment and modernization of the geodetic control in Turkey include a number of regional geoid models that have been determined since 1976. The recently released gravimetric Geoid of Turkey, TG03, is used in geodetic applications where GPS-heights need to be converted to the local vertical datum. To reach a regional geoid model with improved accuracy, the selection of the appropriate global geopotential model is of primary importance. This study assesses the performance of a number of recent satellite-only and combined global geopotential models (GGMs) derived from CHAMP and GRACE missions’ data in comparison to the older EGM96 model, which is the underlying reference model for TG03. In this respect, gravity anomalies and geoid heights from the global geopotential models were compared with terrestrial gravity data and low-pass filtered GPS/levelling data, respectively. Also, five new gravimetric geoid models, computed by the Fast Fourier Transform technique using terrestrial gravity data and the geopotential models, were validated at the GPS/levelling benchmarks. The findings were also compared with the validation results of the TG03 model. The tests showed that as it was expected any of the high-degree combined models (EIGEN-CG03C, EIGEN-GL04C, EGM96) can be employed for determining the gravity anomalies over Turkey. In the west of Turkey, EGM96 and EIGEN-CHAMP03S fit the GPS/levelling surface better. However, all the tested GGMs revealed equal performance when they were employed in gravimetric geoid modelling after de-trending the gravimetric geoid model with corrector surface fitting. The new geoid models have improved accuracy (after fit) compared to TG03.     

Detection of Local Geoid Deformations by Gravity Disturbances from GPS/Gravimetric Observations

This paper deals with a method for detection of local geoid deformations; as a consequence, the methods main application concerns geoid adjustment to GPS/levelling points. This is based on the fact that these points should present no local geoid deformation to avoid errors in the adjustments. These type of miscalculations would lead to an incorrect adjustment and result in further errors in subsequent studies with GPS in the proximity at the point with local deformation.The method proposed is based on predictions of gravity disturbance from geoid undulations using Poisson integral with modified kernel, and its comparison with the gravity disturbance from GPS and gravimetric observations.The use of gravity disturbance instead of gravity anomalies has been chosen since gravity disturbance is a quantity derived from GPS and not from levelling. The loss of accuracy arising with a local height reference system is therefore theoretically avoided as far as the differences in geodetic reference systems regarding positions of gravity measurements and coefficients of the global models are accounted for.Extended numerical tests using computed geoidal undulations and the corresponding gravity disturbances obtained from the geopotential model GPM98cr computed up to degree 720 illustrate the validity of the proposed method and its usefulness as local geoid deformations detection tool.Finally, the method is tested using real GPS/Gravimetric data and geoid models IBERGEO95 and EGG97 with good results.     

The new quasi-geoid model IRQG09 for Iran

Iran is a mountainous country with large lateral density variations of its crust. Constant density value is commonly used to determine the geoid models as well as topographic corrections. The effect of lateral density variation in the geoid can reach up to 14 cm in Iran which is not negligible in a precise geoid modelling. Also, the current height datum of Iran is based on the orthometric system but the effect of gravity variation was not applied in height parameter. Furthermore, the height systems of most neighbouring countries are defined as normal height. Connection of networks can be useful for the unification of height datum, geodynamics researches and optimal adjustment of levelling network. The new quasi-geoid model based on a recent EGM2008 global geo-potential model was created to solve the mentioned problem. The main purpose of the present study is to discuss the results of a research project in which a gravimetric quasi-geoid model for Iran was computed based on the least-squares modification of Stokes' formula. The evaluation is made using 475 GPS/levelling height anomalies covering the major parts of the country except the mountainous areas to the North and West. After a 7-parameter fit, the most promising attempt achieved a RMS value of 19 cm for the residuals based on the GPS/levelling data.     

Investigation of topographic reductions and aliasing effects on gravity and the geoid over Greece based on various digital terrain models

The reduction of gravity-field related quantities (e.g., gravity anomalies, geoid heights) due to the topography plays a crucial role in both geodetic and geophysical applications, since in the former it is an intermediate step towards geoid prediction and in the latter it reveals lateral as well as radial density contrasts and infers the geology of the area under study. The computations are usually carried out by employing a DTM and/or a DBM, which describe the topography and bathymetry, respectively. Errors in these DTMs/DBMs will introduce errors in the computed topographic effects, while poor spatial resolution of the topography and bathymetry models will result in aliasing effects to both gravity anomalies and geoid heights, both influencing the accuracy of the estimated solutions. The scope of this work is twofold. First, a validation and accuracy assessment of the SRTM 3″ (90 m) DTM over Greece is performed through comparisons with existing global models as well as with the Greek 450 m national DTMs. Whenever a misrepresentation of the topography is identified in the SRTM data, it is “corrected” using the local 450 m DTM. This process resulted in an improved SRTM DTM called SRTMGr, which was then used to determine terrain effects to gravity field quantities. From the fine-resolution SRTMGr DTMs, coarser models of 15″, 30″, 1′, 2′ and 5′ have been determined in order to investigate aliasing effects on both gravity anomalies and geoid heights by computing terrain effects at variable spatial resolutions. From the results acquired in two test areas, it was concluded that SRTMGr provides similar results to the local DTM making the use of other older global DTMs obsolete. The study for terrain aliasing effects proved that when high-resolution and accuracy gravity and geoid models are needed, then the highest possible resolution DTM should be employed to compute the respective terrain effects. Based on the results acquired from two the test areas a corrected SRTMGr DTM has been compiled for the entire Greek territory towards the development of a new gravimetric geoid model. Results from that analysis are presented based on the well-known remove-compute-restore method, employing land and marine gravity data, EGM08 as a reference geopotential model and the SRTMGr DTM for the computation of the RTM effects.     

Modelling the fine-structure of the geoid: Methods,data requirements and some results

The requirements for precise geoid models on local and regional scales have increased in recent years, primarily due to the ongoing developments in height determination by GPS on land, but also due to oceanographic requirements in using satellite altimetry for recovering dynamic sea-surface topography. Suitable methods for geoid computations from gravity data include Stokes integration, FFT methods, and least-squares collocation. Especially the FFT methods are efficient in handling large amounts of gravity data, and new variants of the methods taking earth curvature rigorously into account provide attractive methods for obtaining continental-scale, high-resolution geoid models. The accuracy of such models may be from 2–5 cm locally, to 50–100 cm on regional scales, depending on gravity data coverage, long wave-length gravity field errors, and datum problems. When approaching the cm-level geoid basic geoid definition questions (geoid or quasigeoid?) become very significant, especially in rugged areas. In the paper the geoid modelling methods and problems are reviewed, and some investigations on local data requirements for cm-level geoid prediction are presented. Some actual results are presented from Scandinavia, where a recent regional high-resolution geoid model yields apparent accuracies of 2–10 cm over GPS baselines of 50 to 2000 km.     

Global detailed geoid computation and model analysis

This paper presents a survey of recent work on the gravimetric geoid. The gravity models considered are those published in the past few years by the Goddard Space Flight Center (GSFC), the Smithsonian Astrophysical Observatory (SAO) and the Ohio State University (OSU). Comparisons and analyses have been carried out through the ose of detailed gravimetric geoids which we have computed by combining the above-mentioned models with a set of 26 000, 1^ox1^o mean free air gravity anomalies. The accuracy of the detailed gravimetric geoid computed using the most recent Goddard Earth Model (GEM-6) in conjunction with the set 1^ox1^o mean free air gravity anomalies is assessed at 2 m on the continents of North America, Europe And Australia, 2 to 5 m in the North-East Pacific and North Atlantic areas and 5 to 10 m in other areas where surface gravity data are sparse. Rms differences between this detailed geoid and the detailed geoids computed using the other satellite gravity fields in conjunction with same set of surface data range from 3 to 7 m. The maximum differences in all cases occurred in the Southern Hemisphere where surface data and satellite observations are sparse. These differences exhibited wavelengths of approximately 30^o to 50^o in longitude. Detailed geoidal heights were also computed with models truncated to 12th degree and order as well as 8th degree and order. This truncation resulted in a reduction of the rms differences to a maximum of 5 m. Comparisons have been made with the astrogeodetic data of Rice (United States), Bomford (Europe), and Mather (Australia) and also with geoidal heights from satellite solutions for geocentric station coordinates in North America and the Caribbean.     

A precise gravimetric geoid model in a mountainous area with scarce gravity data: a case study in central Turkey

In mountainous regions with scarce gravity data, gravimetric geoid determination is a difficult task that needs special attention to obtain reliable results satisfying the demands, e.g., of engineering applications. The present study investigates a procedure for combining a suitable global geopotential model and available terrestrial data in order to obtain a precise regional geoid model for Konya Closed Basin (KCB). The KCB is located in the central part of Turkey, where a very limited amount of terrestrial gravity data is available. Various data sources, such as the Turkish digital elevation model with 3 ?? × 3?? resolution, a recently published satellite-only global geopotential model from the Gravity Recovery and Climate Experiment satellite (GRACE) and the ground gravity observations, are combined in the least-squares sense by the modified Stokes?? formula. The new gravimetric geoid model is compared with Global Positioning System (GPS)/levelling at the control points, resulting in the Root Mean Square Error (RMS) differences of ±6.4 cm and 1.7 ppm in the absolute and relative senses, respectively. This regional geoid model appears to be more accurate than the Earth Gravitational Model 2008, which is the best global model over the target area, with the RMS differences of ±8.6 cm and 1.8 ppm in the absolute and relative senses, respectively. These results show that the accuracy of a regional gravimetric model can be augmented by the combination of a global geopotential model and local terrestrial data in mountainous areas even though the quality and resolution of the primary terrestrial data are not satisfactory to the geoid modelling procedure.     

A Gravimetric Geoid Computation and Comparison with GPS Results in Northern Andalusia (Spain)

Two new GPS surveys have been carried out to check the accuracy of an existing gravimetric geoid in a test area located in northern Andalusia (Spain). The fast collocation method and the remove-restore procedure have been used for the computation of the quasigeoid model. The Spanish height system is based on orthometric heights, so the gravimetrically determined quasigeoid has been transformed to a geoid model and then compared to geoid undulations provided by GPS and levelling at benchmarks belonging to the Spanish first-order levelling network. The discrepancies between the gravimetric solution and GPS/levelling undulations amount to ±2 cm for one survey and ±5 cm for another after fitting a plane to the geoid model.     

Ellipsoidal area mean gravity anomalies — precise computation of gravity anomaly reference fields for remove-compute-restore geoid determination

Gravity anomaly reference fields, required e.g. in remove-compute-restore (RCR) geoid computation, are obtained from global geopotential models (GGM) through harmonic synthesis. Usually, the gravity anomalies are computed as point values or area mean values in spherical approximation, or point values in ellipsoidal approximation. The present study proposes a method for computation of area mean gravity anomalies in ellipsoidal approximation (‘ellipsoidal area means’) by applying a simple ellipsoidal correction to area means in spherical approximation. Ellipsoidal area means offer better consistency with GGM quasigeoid heights. The method is numerically validated with ellipsoidal area mean gravity derived from very fine grids of gravity point values in ellipsoidal approximation. Signal strengths of (i) the ellipsoidal effect (i.e., difference ellipsoidal vs. spherical approximation), (ii) the area mean effect (i.e., difference area mean vs. point gravity) and (iii) the ellipsoidal area mean effect (i.e., differences between ellipsoidal area means and point gravity in spherical approximation) are investigated in test areas in New Zealand and the Himalaya mountains. The impact of both the area mean and the ellipsoidal effect on quasigeoid heights is in the order of several centimetres. The proposed new gravity data type not only allows more accurate RCR-based geoid computation, but may also be of some value for the GGM validation using terrestrial gravity anomalies that are available as area mean values.     

Geoid anomalies over two South Atlantic fracture zones

Seasat altimetry profiles across the Falkland-Agulhas fracture zone (FZ) and the Ascension FZ in the South Atlantic were examined for evidence of step-like geoid offsets predicted from thermal modeling of the lithosphere. The geoid profiles exhibit much short-wavelength power and the step-like offsets are often small, making reliable estimation of the heights of the observed geoid offsets difficult. The offsets were estimated by the least-squares fitting of quadratic curves incorporating a step function to the altimetry profiles. A preferred offset value was determined for each profile by taking the average of step heights computed with various distances around the fracture zone excluded from the fit. The age of the crust surrounding the fracture zones, necessary for computing a theoretical geoid offset, was determined from surface ship magnetic anomaly data and from existing ocean floor age maps.Observed variations in geoid step height with age of the lithosphere are not consistent with those predicted from standard thermal plate models. For ages less than 30 Ma, the step offsets across both fracture zones decrease in a manner appropriate for an unusually thin plate with a thickness of 50–75 km. At greater ages, the offsets show complex behavior that may be due to bathymetric features adjacent to the fracture zones. Similar geoid patterns on opposite branches of the Falkland-Agulhas FZ are indicative of processes that act symmetrically on both sides of the Mid-Atlantic Ridge. This behavior of the geoid is consistent both with small-scale convection occurring beneath the lithosphere and with bathymetric features originally produced along the ridge crest and now located symmetrically on opposite sides of the ridge. The west flank of the Ascension FZ displays a regrowth in step height at about 40 Ma consistent with small-scale convection and in agreement with other studies of Pacific and South Atlantic fracture zones.     

Comparing recent geopotential models in Andalusia (Southern Spain)

This work focuses on the comparison between satellite-only and combined Global Geopotential Models (GGMs) derived from the CHAMP and GRACE satellite missions with land gravity anomalies, geoid undulations provided by the gravimetric geoid ANDALUSGeoid2002 and GPS/levelling geoid undulations in Andalusia in order to find the GGM that best fits this area in order to be used in a further geoid computation. The results show that the EIGEN-CG01C model or the combined models GGM02C/EIGEN-CG01C and ITG-CHAMP01E/EIGEN-CG01C should be used.     

ANDALUSGeoid2002: The New Gravimetric Geoid Model of Andalusia (Southern Spain)

In 1991 the first determination of a gravimetric geoid in a test area in central Spain was computed by using least square collocation. In 1995 a gravimetric geoid in the Iberian Peninsula, Ibergeo95, was calculated by FFT. Nowadays an improved geoid of Andalusia, ANDALUSGeoid2002, has been computed by fast collocation procedure and remove-restore technique in the GRS80 Reference System. The computations have been done from 16562 free-air gravity anomaly data set, obtained from IGN (Instituto Geográfico Nacional) and BGI (International Gravity Bureau), the Earth Gravity Model EGM96 and detailed (100 m × 100 m), coarse (5 km × 5 km) and reference (20 km × 20 km) digital terrain models. Relative carrier-phase GPS measurements at 69 benchmarks of the Spanish Levelling Network in Andalusia have been done. The standard deviations of differences between ANDALUSGeoid2002 and GPS/levelling undulations after fitting the tilt have been ± 11 cm, ± 39 cm and ± 38 cm in western, eastern and whole Andalusia, respectively. The ANDALUSGeoid2002 shows an improvement of Ibergeo95 in this territory.     

Explicit formula for the geoid-quasigeoid separation

The explicit formula for the geoid-to-quasigeoid correction is derived in this paper. On comparing the geoidal height and height anomaly, this correction is found to be a function of the mean value of gravity disturbance along the plumbline within the topography. To evaluate the mean gravity disturbance, the gravity field of the Earth is decomposed into components generated by masses within the geoid, topography and atmosphere. Newton’s integration is then used for the computation of topography-and atmosphere-generated components of the mean gravity, while the combined solution for the downward continuation of gravity anomalies and Stokes’ boundary-value problem is utilized in computing the component of mean gravity disturbance generated by mass irregularities within the geoid. On application of this explicit formulism a theoretical accuracy of a few millimetres can be achieved in evaluation of the geoid-to-quasigeoid correction. However, the real accuracy could be lower due to deficiencies within the numerical methods and to errors within the input data (digital terrain and density models and gravity observations).     

地形-均衡补偿重力、大地水准面异常频谱分析

将地形高、地球内部质量异常以及重力、大地水准面展开成球谐级数,依据岩石圈弹性挠曲均衡补偿理论建立地形—均衡补偿重力、大地水准面异常的球谐级数表达式.由此我们可以研究地形—均衡补偿重力、大地水准面异常与球谐级数阶次的关系,以及不同波长地形荷载与岩石圈挠曲补偿的关系,探讨地形—均衡补偿重力、大地水准面的频谱特性.从观测大地水准面异常和自由空气重力异常扣除地形—均衡补偿大地水准面、重力异常,可以得到均衡大地水准面异常和均衡重力异常.均衡大地水准面异常已经消除了浅层物质不均匀的影响,反映的是地球深部物质密度不均匀分布.均衡重力异常显示出中短波长特性,反映的是地壳上地幔物质分布的失衡和物质调整的动力学特征.     

Testing global geopotential models through comparison of a local quasi-geoid model with GPS/leveling data

The satellite missions CHAllenging Minisatellite Payload (CHAMP) and Gravity Recovery And Climate Experiment (GRACE) provide accurate data that are routinely inverted into spherical harmonic coefficients of the geopotential forming a global geopotential model (GGM). Mean square errors of these coefficients, in some cases even entire covariance matrices, are included in the GGM. Due to estimation procedures with a large redundancy and insufficiently propagated observation errors, they often do not represent the actual accuracy of the harmonic coefficients, thus also gravity field parameters synthesized from the respective GGM. Since in most cases standard methods validating the GGMs reached their limits, new procedures and independent data are being currently sought. This article discusses an alternative validation procedure based on comparison of the GGMs with independent data represented by a set of GPS/leveling stations. Due to a different spectral content of the height anomalies synthesized from the GGMs and of those derived by combination of GPS-based ellipsoidal and leveled normal heights, the GGM-based low frequency height anomaly is enhanced for a high frequency component computed from local ground gravity and elevation data. The methodology is applied on a set of selected points of the European Vertical Reference Network and Czech trigonometric stations. In accordance with similar tests based on entirely independent data of cross-over altimetry, obtained results seem to indicate low-frequency deficiencies in the current GGMs, namely in those estimated from data of single-satellite missions.     

Gravity anomaly in and around Antarctica and its tectonic implications

During the nineteen-seventies, the geophysical satellites EOS-3 and SEASAT-1 provided very accurate sea-surface heights, which could be employed as information on the marine geoid. Geoid height can easily be converted to gravity anomalies, and since the tracks of GEOS-3 and SEASAT-1 were extremely dense, the gravity anomaly data thus obtained were the densest and of the widest coverage ever obtained for gravity measurements.The authors completed a self-consistent free-air gravity anomaly map in the Antarctic region, covering from 45°S to the South Pole, using all the gravity data available at present: namely, those obtained by satellites and the ground-truth data obtained by land gravimeters and ship borne gravimeters (Segawa et at., 1984).The bouguer anomaly was also calculated, from which estimates of crustal and lithospheric structures were made. This has resulted in clarifying the relationships among sub-bottom structures between the mid-oceanic ridges surrounding the Antarctic plate and the Ross Sea.     

Definition of Physical Height Systems for Telluric Planets and Moons

In planetary sciences, the geodetic (geometric) heights defined with respect to the reference surface (the sphere or the ellipsoid) or with respect to the center of the planet/moon are typically used for mapping topographic surface, compilation of global topographic models, detailed mapping of potential landing sites, and other space science and engineering purposes. Nevertheless, certain applications, such as studies of gravity-driven mass movements, require the physical heights to be defined with respect to the equipotential surface. Taking the analogy with terrestrial height systems, the realization of height systems for telluric planets and moons could be done by means of defining the orthometric and geoidal heights. In this case, however, the definition of the orthometric heights in principle differs. Whereas the terrestrial geoid is described as an equipotential surface that best approximates the mean sea level, such a definition for planets/moons is irrelevant in the absence of (liquid) global oceans. A more natural choice for planets and moons is to adopt the geoidal equipotential surface that closely approximates the geometric reference surface (the sphere or the ellipsoid). In this study, we address these aspects by proposing a more accurate approach for defining the orthometric heights for telluric planets and moons from available topographic and gravity models, while adopting the average crustal density in the absence of reliable crustal density models. In particular, we discuss a proper treatment of topographic masses in the context of gravimetric geoid determination. In numerical studies, we investigate differences between the geodetic and orthometric heights, represented by the geoidal heights, on Mercury, Venus, Mars, and Moon. Our results reveal that these differences are significant. The geoidal heights on Mercury vary from ? 132 to 166 m. On Venus, the geoidal heights are between ? 51 and 137 m with maxima on this planet at Atla Regio and Beta Regio. The largest geoid undulations between ? 747 and 1685 m were found on Mars, with the extreme positive geoidal heights under Olympus Mons in Tharsis region. Large variations in the geoidal geometry are also confirmed on the Moon, with the geoidal heights ranging from ? 298 to 461 m. For comparison, the terrestrial geoid undulations are mostly within ± 100 m. We also demonstrate that a commonly used method for computing the geoidal heights that disregards the differences between the gravity field outside and inside topographic masses yields relatively large errors. According to our estimates, these errors are ? 0.3/+ 3.4 m for Mercury, 0.0/+ 13.3 m for Venus, ? 1.4/+ 125.6 m for Mars, and ? 5.6/+ 45.2 m for the Moon.