Geoid Undulations Research Articles

Assessing the quality of GEOID12B model through field surveys

AbstractElevation differences have been determined through conventional ground surveying techniques for over a century. Since the mid-80s GPS, GLONASS and other satellite systems have modernized the means by which elevation differences are observed. In this article, we assessed the quality of GEIOD12B through long-occupation GNSS static surveys. A set of NGS benchmarks was occupied for at least one hour using dual-frequency GNSS receivers. Collected measurements were processed using a single CORS station at most 24 kilometers from the benchmarks. Geoid undulation values were driven by subtracting measured ellipsoidal heights from the orthometric heights posted on the NGS website. To assess the quality of GEOID12B, we compared our computed vertical shifts at the benchmarks with those estimated from GEOID12B published by NGS. In addition, Kriging model was used to interpolate local maps for the geoid undulations from the benchmark heights. The maps were compared with corresponding parts of GEOID12B. No biases were detected in the results and only shifts due to random errors were found. Discrepancies in the range of ten centimetres were noticed between our geoid undulation and the values available from NGS.

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.

Ocena točnosti in primerjava metod interpolacij modelov geoida

In this paper the accuracy of 6 spatial interpolation methods on geoid grid models are tested and compared: Inverse Distance Weighting, Kriging, Minimum Curvature, Natural Neighbour, Radial Basis Function and Moving Average. Grid models are derived from three global geopotential models: EGM2008, EIGEN-6C4 and GECO. Totally 18 geoid models in forms of equiangular grid models are created in GoldenSoftware Surfer 13 for this purpose, by combining 6 methods of interpolation and 3 global models, whose size and position cover a wide area of Croatia with parts of neighbouring countries. Default interpolation parameters were kept during the process. Models were created in a way that modelled values of geoid undulations can be subtracted from undulations of independent models in equal points. Residuals calculated in this way served as a base for quality assessment. Comparative analysis of quality indicators, statistics and graphical indicators of residuals led to a conclusion that the best-fitted methods of interpolation for this specific case are Radial Basis Function, Kriging, and Natural Neighbour, in that order.

Calculating of adjusted geoid undulation based on EGM08 and mean sea level for different regions in Iraq

In last decades Global Navigation Satellite System (GNSS) or as known Global Positioning System (GPS) technique is considered a revolutionary technique in the field of geodetic survey in comparison with traditional techniques (level, theodolite and total station). The height obtained from GNSS technique is ellipsoid height and to have a physical meaning in a surveying or engineering application it must be transformed to orthometric height. Therefore, a geoid model has to be used to do this transformation process. In Iraq there is no specific geoid that can be used in order to get proper orthometric height. This research aims to calculate adjusted geoid undulation based on Earth Gravitational Model 2008 (EGM08) through observation of Iraqi official vertical network using GNSS technique. Different regions in Iraq have been chosen to perform this research. The result of this research can assist a lot to enhance the accuracy of elevations obtained from GNSS and support the establishment of Iraq geoid.

The Integration of GNSS/Leveling Data with Global Geopotential Models to Define the Height Reference System of Palestine

In GNSS, 3D geometric position is calculated with high accuracy. This position is the ellipsoidal latitude, longitude and ellipsoidal height $$\left( {\lambda ,\phi ,h} \right) $$ with respect to WGS84 coordinates system. These coordinates are integrated with local horizontal/projected coordinates (X, Y) by mathematical coordinate’s transformations and map projections. The geometric ellipsoidal height (h) obtained by means of GNSS has to be integrated with the physical/orthometric heights (H) obtained by means of precise leveling. The physical surface defining the difference between both height systems is the geoid represented by the geoid undulation (N) at a given position. Different methods are used to build geoid models. Global models using terrestrial and satellite data are available to calculate the geoid heights as function of the earth potential (W). The most recent high degree and order models are EGM2008, Eigen05c, Eigen06c4, etc. (GFZ-Potsdam, ‘List of available global models. http://icgem.gfz-potsdam.de/ICGEM/modelstab.html , 2017). Some regional geoid models are available like the European Gravimetric Geoid (EGG97). These models mostly do not fit the local datum due to datum definition problems. Here, a group of precise height reference benchmarks measured with GNSS is used to fit the global models with the local vertical datum to define the local height reference system of Palestine. The accuracy of the different global geopotential models is evaluated before and after the application of the geoid fitting.

Assessment of the most recent satellite based digital elevation models of Egypt

Digital Elevation Model (DEM) is crucial to a wide range of surveying and civil engineering applications worldwide. Some of the DEMs such as ASTER, SRTM1 and SRTM3 are freely available open source products. In order to evaluate the three DEMs, the contribution of EGM96 are removed and all DEMs heights are becoming ellipsoidal height. This step was done to avoid the errors occurred due to EGM96. 601 points of observed ellipsoidal heights compared with the three DEMs, the results show that the SRTM1 is the most accurate one, that produces mean height difference and standard deviations equal 2.89 and ±8.65 m respectively. In order to increase the accuracy of SRTM1 in EGYPT, a precise Global Geopotential Model (GGM) is needed to convert the SRTM1 ellipsoidal height to orthometric height, so that, we quantify the precision of most-recent released GGM (five models). The results show that, the GECO model is the best fit global models over Egypt, which produces a standard deviation of geoid undulation differences equals ±0.42 m over observed 17 HARN GPS/leveling stations. To confirm an enhanced DEM in EGYPT, the two orthometric height models (SRTM1 ellipsoidal height + EGM96) and (SRTM1 ellipsoidal height + GECO) are assessment with 17 GPS/leveling stations and 112 orthometric height stations, the results show that the estimated height differences between the SRTM1 before improvements and the enhanced model are at rate of 0.44 m and 0.06 m respectively.

Hybrid Geoid Model: Theory and Application in Brazil.

Determination of the ellipsoidal height by Global Navigation Satellite Systems (GNSS) is becoming better known and used for purposes of leveling with the aid of geoid models. However, the disadvantage of this method is the quality of the geoid models, which degrade heights and limit the application of the method. In order to provide better quality in transforming height using GNSS leveling, this research aims to develop a hybridization methodology of gravimetric geoid models EGM08, MAPGEO2015 and GEOIDSP2014 for the State of São Paulo, providing more consistent models with GNSS technology. Radial Basis Function (RBF) neural networks were used to obtain the corrector surface, based on differences between geoid model undulations and the undulations obtained by GNSS tracking in benchmarks. The experiments showed that the most suitable interpolation for correction modeling is the linear RBF. Checkpoints indicate that the geoid hybrid models feature root mean square deviation ± 0.107, ± 0.104 and ± 0.098 m, respectively. The results shows an improvement of 30 to 40% in consistencies compared with the gravimetric geoids, providing users with better quality in transformation of geometric to orthometric heights.

Assessment of Geoidal Undulation (N) Values Computed with Reference to Different Earth Gravity Models

Assessment of Geoidal Undulation (N) Values Computed with Reference to Different Earth Gravity Models

Research into GNSS levelling using network RTK in Taiwan

Modern height surveying obtains orthometric heights through geoid undulations and ellipsoidal heights derived from global navigation satellite system (GNSS), a process referred to as ‘GNSS levelling’. In this study, an efficient survey technology, the e-global navigation satellite system (e-GNSS), was used instead of static satellite surveying to execute GNSS levelling. The e-GNSS in Taiwan is based on Virtual Reference Station-Real-Time Kinematic technology and networks. The orthometric heights were calculated by subtracting the geoid undulations, derived from three geoid models (gravimetric, hybrid, and geometric models), from the ellipsoidal heights, derived from the e-GNSS. Additionally, two corrector methods, namely the difference and the corrector surface method, were adopted to improve the accuracies and the systematic errors. The experiment showed that the two corrector methods improved the accuracies of the orthometric heights in the gravimetric model by approximately 66.3 and 80.0% and mitigated the systematic errors in the gravity model by approximately 96.2 and 99.9%.

HEIGHT MODEL INTEGRATION USING ALOS PALSAR, X SAR, SRTM C, AND ICESAT/GLAS

The scarcity of height models is one of the important issues in Indonesia. ALOS PALSAR, X SAR, SRTM C, and ICESAT/GLAS are free available global height models. Four data can be integrated the height models. Integration takes advantage of each characteristic data. The spatial resolution uses ALOS PALSAR. ICESAT/GLAS has a minimal height error because it is DTM. SAR has advantages of minimal error in the highland and need a low pass filter on the lowland. DSM uses X SAR and DEM from ALOS PALSAR. Characteristics and penetration of vegetation objects can be seen from the wavelength type of SAR data. This research aims to make height model integration in order to get the vertical accuracy better than vertical accuracy of global height models and minimum height error. The study area is located in Karo Regency. The first process is to crop the height models into Karo Regency, geoid undulation correction using EGM 2008. The next step is to detect pits and spires by using radius value 1000 m and depth +1.96σ (+5 m) with uncertainty 95,45%. Then generate HEM and height model integration. To know the accuracy of this height model, 100 reference points measured using GNSS, altimeter, and similar point observed on the height model integration are selected. The accuracy test covers RMSE, accuracy (z), and height difference test. The result of this study shows that the height model integration has a vertical accuracy in 1.14 m. This height model integration can be used for mapping scale 1: 10.0000.

Spatial uncertainty of a geoid undulation model in Guayaquil, Ecuador

AbstractGeostatistics is a discipline that deals with the statistical analysis of regionalized variables. In this case study, geostatistics is used to estimate geoid undulation in the rural area of Guayaquil town in Ecuador. The geostatistical approach was chosen because the estimation error of prediction map is getting. Open source statistical software R and mainly geoR, gstat and RGeostats libraries were used. Exploratory data analysis (EDA), trend and structural analysis were carried out. An automatic model fitting by Iterative Least Squares and other fitting procedures were employed to fit the variogram. Finally, Kriging using gravity anomaly of Bouguer as external drift and Universal Kriging were used to get a detailed map of geoid undulation. The estimation uncertainty was reached in the interval [-0.5; +0.5] m for errors and a maximum estimation standard deviation of 2 mm in relation with the method of interpolation applied. The error distribution of the geoid undulation map obtained in this study provides a better result than Earth gravitational models publicly available for the study area according the comparison with independent validation points. The main goal of this paper is to confirm the feasibility to use geoid undulations from Global Navigation Satellite Systems and leveling field measurements and geostatistical techniques methods in order to use them in high-accuracy engineering projects.

Evaluation of the geopotential value W0LVD of China

Estimation of the zero-height geopotential value W0LVD for the CVD (China Vertical Datum) plays a fundamental role in the connection of traditional height reference systems into a global height system. Estimation the W0LVD of China is based on the computation of the mean geopotential offset between the value W0 = 62636856.0 m2s−2, selected as reference in this study, and the unknown geopotential value W0LVD. This estimation is based on the combination of ellipsoidal heights, levelled heights (referring to the CVD), and some physical parameters, such as geopotential values, gravity values, and geoid undulations. The geoid undulations derived from the GGM (Global Geopotential Models). This combination is performed through three approaches: The first one is based on the theory of Molodensky, and the second one compares levelled heights and geopotential values derived from the GGMs, while the third one analyses the differences between GPS/Levelling and GGMs geoid undulations. The approaches are evaluated at 65 benchmarks (BMs) covered around Qingdao where the tide gauge is used to observe the local mean sea level of China. The results from three approaches are very similar. Furthermore, the W0LVD determined for the China local vertical datum was 62636852.9462 m2s−2, indicates a bias of about 3.0538 m2/s−2 compared to the conventional value of 62636856.0 m2s−2.

Performance evaluation of the Earth Gravitational Model 2008 (EGM2008) – a case study

Ghana’s local geodetic reference network when pick data with the GPS is being reference to the World Geodetic System 1984, with data in latitude, longitude and ellipsoidal height. This prevalent situation makes it possible to apply the New Earth Gravitational model to determine the geoidal height at each GPS point with the geographic coordinates. Researchers have therefore proposed various methods of determining the geoidal undulations for a local and national geodetic networks and improving the recent New Earth Gravitational model accuracies and its performances. This project therefore seeks to evaluate such method of improving the recent New Earth Gravitational model in a part of the University of Mines and Technology, UMaT, Tarkwa. The geoid heights implied by the model are compared with 328 discrete geometrical heights from co-located GPS and Total station orthometric heights of the University Primary Levelling Networks. The methods applied were the EGM2008 model, a geometric method and a polynomial mathematical model for improving the recent geoid model. The statistics of the differences between GPS/levelling derived geoid heights and corresponding geoid heights obtained from the geoid model suggests that, the EGM2008 model is most suitable at this moment. The RMS, Mean, and the Standard deviation of their geoidal height differences are 0.120825 m, 2.18823 m, and 3.47678 m, which is better in the area of interest. The study concluded that, the recent geoid model can be applied in UMaT and the polynomial mathematical model is the best model for modelling EGM2008 for a local geoid model.

Possibilidades de uso direto de altitude elipsoidal em obras de engenharia

As alturas e diferenças de altura nas obras de engenharia são determinadas normalmente por nivelamento topográfico geométrico, que fornece as altitudes ortométricas, mas também está sendo usado o nivelamento com receptores GNSS (Global Navigation Satellite System) que fornece altitudes elipsoidais. As primeiras são medidas em relação ao geoide, mas também podem ser obtidas a partir das altitudes elipsoidais, desde que seja conhecida a ondulação geoidal no local. Como essa transformação contém erros inerentes aos nivelamentos e métodos de obtenção da ondulação geoidal, essa metodologia não tem fácil aplicação na prática diária das engenharias. Este trabalho tem por objetivo analisar as possibilidades de uso da altitude elipsoidal em obras de engenharia de modo direto, sem a sua transformação para a altitude ortométrica. Para isto foram executados inicialmente nivelamentos geométricos e GNSS em uma área com 2,13 km2 , de modo a gerar um mapa de ondulação geoidal local de acordo com a metodologia usual de transformação. Neste processo foi possível identificar vários erros grosseiros de diversas origens no nivelamento GNSS; comparar as precisões das altitudes pelos dois métodos e analisar a influência dos erros na geração do geoide e na grandeza dos gradientes de suas ondulações.Na verificação do uso direto do nivelamento GNSS em obras de declividade muito baixa que exigem grande precisão de nivelamento geométrico,é proposta uma análise conjunta dos valores dos gradientes e dos sentidos das declividades da obra e da ondulação geoidal. Finalmente, são sugeridos procedimentos a serem seguidos sobre o uso direto de altitudes elipsoidais, de acordo com as especificações de nivelamentos de baixa, média e alta precisão em obras de engenharia civil.

Gongola Basin Geoid Determination using Isostatic Models and Seismic Reflection Data and Geophysical Interpretation

The application of Stokes’ formula to create geoid undulation requires no masses outside the geoid. Usually, a constant density of 2.67g / cm3 is used in the determination of the geoid which introduces error in the reduced gravity anomalies (Helmert’s condensation) and consequently the geoid. In this paper, isostatic models of Airy-Heiskanen and Pratt-Hayford were utilized in the determination of the geoid by considering the planar and spherical approximation models. The indirect effect of the topographic lateral density variation on the geoid was computed as additive correction for the improvement of the accuracy of the computed geoid. Additional density information deduced from seismic and well log data was considered for the variable density computation. The geopotential geoid undulations were computed from the EGM 2008 model. The residual geoid was obtained by subtracting the local isostatic geoid from the geopential geoid. Geoid and gravity admittance studies were also carried out to complement the results from the residual geoid.The planar and spherical approximation results showed similar characteristics; but a change in magnitude in both models. Our results suggest that the effects of topographic lateral density variations in geoid determination are significant and should be considered in rift basins. The geophysical analysis of the geoid results show that the north-east domain has positive residual geoid which indicates the presence of high density intrusive igneous rocks, while the south-east has negative residual geoid which indicates the dominant presence of low density sedimentary rocks. The results also show that the radial distribution of the anomalous mass obtained using the geoid/residual geoid anomaly uniquely matched that obtained using the seismic reflection data which inferred the presence of hydrocarbon accumulation in the south east zone of the project area. The gravity and geoid admittance studies corroborated the residual geoid and seismic reflection results

Study of the Potential of a Local Geoid Model for Extracting the Orthometric Heights from GPS Measurements in Topographic Works

Due to the rapid development and expansion of geodetic applications, the determination of orthometric heights in an accurate manner is considered as one of the most required conditions to carry out such projects. Obtaining orthometric heights using traditional methods of levelling is time and cost consuming. Hence, investigating other techniques that provide the same accuracy, as leveling methods, but requiring less time and cost is very gainful. Recently, satellite positioning techniques and their applications are being used increasingly in geodetic projects. So, it is interesting to study the efficiency of such techniques for obtaining orthometric heights. GPS methods provide highly accurate measurement of ellipsoidal heights. However, the conversion of ellipsoid heights to orthometric heights may be achieved using geoid models. The objective of this research is to study the efficiency of using a local geoid model as an alternative method to obtain orthometric heights from GPS measurements. This paper proposes a methodology to generate such local geoid models. Then, the results of using the generated local geoid model for Jeddah city in Saudi Arabia are presented. These results indicate that the difference among estimated undulations values from the local geoid model and undulations values calculated from leveling techniques ranges from 1.8 cm to -1.1 cm. with a maximum standard deviation of 56 mm. These results confirm that the creation of a local geoid model is an effective method that gives the required accuracy for topographic works. Keywords: GPS/Levelling method, Local geoid model, Krigging method.

Experimental Analysis of Kinematic Network-Based GPS Positioning Technique for River Bathymetric Survey

This paper deals with performance assessment of the kinematic network-based GPS positioning technique with a view to using it for ellipsoidally referenced bathymetric surveys. To this end, two field trials were carried out on a land vehicle and a surveying vessel. Single-frequency GPS data acquired from these tests were processed by an in-house software which equips the network modeling algorithm with instantaneous ambiguity resolution procedure. The results reveals that ambiguity success rate based on the network model is mostly higher than 99.0%, which is superior to that of the single-baseline model. In addition, achievable accuracy of the technique was accessed at <TEX>${\pm}1.6cm$</TEX> and 2.7 cm with 95% confidence level in horizontal and vertical component respectively. From bathymetric survey at the West Nakdong River in Busan, Korea, 3-D coordinates of 2,011 points on its bed were computed by using GPS-derived coordinates, attitude, measured depth and geoid undulation. Note that their vertical coordinates are aligned to the geoid, the so-called orthometric height which is widely adopted in river engineering. Bathymetry was constructed by interpolating the coordinate set, and some discussion on its benefit was given at the end.

Modelling Earth’s surface topography: Decomposition of the static and dynamic components

Contrasting results on the magnitude of the dynamic component of topography motivate us to analyse the sources of uncertainties affecting long wavelength topography modelling. We obtain a range of mantle density structures from thermo-chemical interpretation of available seismic tomography models. We account for pressure, temperature and compositional effects as inferred by mineral physics to relate seismic velocity with density. Mantle density models are coupled to crustal density distributions obtained with a similar methodology. We compute isostatic topography and associated residual topography maps and perform instantaneous mantle flow modelling to calculate the dynamic topography. We explore the effects of proposed mantle 1-D viscosities and also test a 3D pressure- and temperature-dependent viscosity model. We find that the patterns of residual and dynamic topography are robust, with an average correlation coefficient (r) of respectively ∼0.74 and ∼0.71, upper-lower quartile ranges of 0.86–0.65 for residual topography and 0.83–0.62 for dynamic topography maps. The amplitudes are, on the contrary, poorly constrained. For the static component, the inferred density models of lithospheric mantle give an interquartile range of isostatic topography that is always higher than 100m, reaching 1.7km in some locations, and averaging ∼720m. Crustal density models satisfying the same compressional velocity structure lead to variations in isostatic topography averaging 350m, with peaks of 1km in thick crustal regions, and always higher than 100m. The uncertainties on isostatic topography are strong enough to mask, if present, the contribution of mantle convection to surface topography. For the dynamic component, we obtain a peak-to-peak dynamic topography amplitude exceeding 3km for all our mantle density and viscosity models. These extremely high values would be associated with a magnitude of geoid undulations that is not in agreement with observations. Considering chemical heterogeneities in correspondence with the lower mantle Large Low Shear wave Velocity Provinces (LLSVPs) helps to decrease the peak-to-peak amplitudes of dynamic topography and geoid, but significantly reduces the correlation between synthetic and observed geoid. The correlation coefficients between all our residual and dynamic topography maps (a total of 220 and 198, respectively) is <0.55 (average=∼0.19). The correlation slightly improves when considering only the very long-wavelength components of the maps (average=∼0.23). We therefore conclude that a robust determination of dynamic topography is not feasible since current uncertainties affecting crustal density, mantle density and mantle viscosity are still too large. A truly interdisciplinary approach, combining constraints from the geological record with a multi-methodological interpretation of geophysical observations, is required to tackle the challenging task of linking the surface topography to deep processes.

Determination of geopotential value W0L at Polish tide gauges from GNSS data and geoid model

Results of three campaigns of Baltic Sea Level Project and further studies reveal the GPS and spirit levelling data possibly contain errors which affect SST and W_{0}^{L} computations. For that reason, the old data were revised and additionally, in spring 2015, the new GNNS campaign was carried out at tide gauges in Swinoujscie, Ustka and Wladyslawowo. The study concerns computation of the local average geopotential value W_{0}^{L} using water level data at the three tide gauges, geoid undulations from a global geopotential model EGM2008 and ellipsoidal heights from GNSS observations, which were obtained using revised data from three campaigns of Baltic Sea Level Project and from the new campaign conducted in 2015. Results obtained indicate that the best estimation of W_{0}^{L} was achieved from the campaign carried out in 2015, where the mean value of W_{0}^{L} calculated for three investigated tide gauge stations is equal to 62636857.45 m2 s−2.

Improved Geoid Modeling Using Observed and Modeled Gravity Gradients in Taiwan

The authors present a new geoid modeling procedure that can greatly improve relative geoid accuracy in mountainous areas, leading to improved applications for modern geodetic techniques, such as light detection and ranging (LIDAR), in mapping orthometric heights over steep terrain on which precise slopes are needed to assess the risk of landslides and the suitability of industrial development. The new procedure (1) measures gravity gradients or computing modeled gravity gradients from a regular grid of gravity anomalies, (2) uses these gradients to refine gravity anomalies, and (3) uses the gravity anomalies to compute geoidal undulations. This new procedure was tested in Taiwan. In situ gravity gradients were measured at approximately 4,000 gravity sites to compare the modeled gravity gradients. In the test, ground gravity observations are reduced to gravity anomalies at mean sea surface using three types of gravity gradients: normal, modeled, and observed. The researchers’ geoid modeling uses the method of least-squares collocation (LSC) with the remove-compute-restore (RCR) procedure. Free-air gravity anomalies, as reduced using the observed and modeled gravity gradients, deviate from those using normal gradients by up to 100 mgal in high mountains. Using free-air gravity anomalies derived from observed and modeled gravity gradients, the authors can improve the relative geoid accuracies by up to 17 and 18 cm, respectively, for Route 3 (an area with high mountains) in Taiwan.