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.     

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.     

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.     

Comparison of the qualities of recent global and local gravimetric geoid models in Iran

A number of regional gravimetric geoid models have recently been determined for the Iran area, and a common problem is to select the best model, e.g. for engineering applications. A related problem is that in order to improve the local geoid models, the selection of the best Global Geopotential Model (GGM) model for the region is essential, to be used in a combined solution from GGM and local gravimetric data. We discuss these problems by taking advantage of 260 GPS/levelling points as an external tool for validation of different global and local geoid models in the absolute and relative senses. By using relative comparisons of the height differences between precise levelling and GPS/geoid models we avoid possible unknown systematic effects between the different types of observables.The study shows that the combination of the newly released GRACE model (GGM02C) with EGM96 geoid model fits the GPS/levelling data in Iran with the best absolute and relative accuracy among the GGMs. Among the local geoid models, the newly gravimetric geoid model IRG04 agrees considerably better with GPS/levelling than any of the other recent local geoid models. Its rms fit with GPS/levelling is 55 cm. Hence, we strongly recommend the use of this new model in any surveying engineering or GPS/levelling projects in the area.     

Height transformation models from ellipsoidal into the normal orthometric height system for the territory of the City of Zagreb

The paper presents the testing of the possibility of determining the heights of GPS points in the homogeneous field in the new Croatian Height Reference System (HVRS71) by using the method of height transformation. The testing was made in the area of Zagreb. As part of the field works, normal orthometric heights of 27 GPS points were determined according to the new height system, by transferring the benchmark heights using the geometric levelling method, thus obtaining GPS/levelling points of known ellipsoidal and normal orthometric heights. The GPS/levelling points served as the basis for determining the transformation models that enabled the computation of normal orthometric heights from ellipsoidal heights of any GPS point in the observed area. The empirical data used for modelling were reduced undulation dN values of GPS/levelling points. As part of the dN modelling with parametric functions, the approximation surfaces were obtained on the basis of three polynomials: FN310, FN312 and FN318. The transformation models were also tested using non-parametric Watson and Loess algorithms. The FN318 and Loess models yielded the best results.     

Present state and future developments of the European geoid

Regional geoid resp. quasigeoid determinations are nowadays required with an accuracy of ±1 to 10 cm over distances from 100 to some 1000 km in order to meet the demands of geodesy, geophysics, oceanography and engineering. Especially the combination of GPS heighting with classical leveling is one of the primary drivers for precise geoid computations. As a consequence, the IAG International Geoid Commission recognized at its meeting in Milano, 1990, that there is an urgent need for a new European geoid computation. This solution should be significantly improved in spatial resolution and accuracy as compared to presently available models. This led to the decision to form a Subcommission for the Geoid in Europe, and the Institut für Erdmessung (IfE) was asked to serve as a computing center in this project.In the first part of this paper early geoid/quasigeoid computations for the area of Europe as well as more recent results obtained at IfE are summarized. The latter solutions include a gravimetric and an astrogravimetric quasigeoid, which have a spatial resolution of about 20 km and a relative accuracy of some dm. Then the possibilities for an improved European quasigeoid calculation are outlined, considering the availability of new and better global and regional data sets. An overview is given on the procedures currently under study at IfE and on the work performed at IfE since 1990. This work includes the collection and screening of new point gravity and terrain data, some investigations on the use of topographic information available at present, and the calculation of a preliminary quasigeoid solution for central, northern and western Europe including a GPS/leveling control. The paper closes with a survey on future activities at IfE within the European geoid project.     

Local quasigeoid model creation from astrogeodetic measurements

This article describes the process of creation and testing of a local quasigeoid model using the astrogeodetic method known as astronomical levelling. The process used here was based on astronomical levelling principles combined with the least square adjustment in a triangular network and a common method of surface generation. Using this method, the authors have created a quasigeoid model of a small portion of the city of Brno. This model covers an area of approximately 1 × 2 km and is based on the astronomically determined vertical deflection at 11 stations. The average distance between the astrogeodetic stations was 500 m, which is an unusually high density (5.5 stations per km²). This high geo-spatial data density input made it possible to generate a quasigeoid model of height difference precision at mm-level over few km. Tests described in this article document the suitability of our methods for creating local quasigeoid models of high precision and resolution. Employing the least square adjustment in a planar network offers the possibility to easily compute standard deviations of both input and result values. This is a great advantage in comparison with more common astrogeodetic quasigeoid profiles, which are not suitable for simple adjustment and require more complex methods to be used for evaluation of their precision. The model described here serves the authors as a technological example, from which they learn more about the potential of the astrogeodetic method. Astrogeodetic models of a much greater extent are planned to be used for validation purposes of models generated by other independent methods (gravimetric, satellite, combined, etc.).     

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.     

Accuracy Estimates of Normal Heights Computed at GPS Sites

The dependence of accuracy of determining normal heights at GPS sites on the density of distribution of the sites within the GPS/levelling frame has been investigated. If the density amounts to one GPS/levelling site per 400 km ² , the accuracy of about ±11.5 cm in the central part of Europe. If the density is higher, e.g. one GPS/levelling site per 200 km ² , an accuracy of about ±5 cm can be achieved.     

Transformation between gravimetric and GPS/levelling-derived geoids using additional gravity information

The transformation from the gravimetric to the GPS/levelling-derived geoid using additional gravity information for the covariance function of geoid height differences has been investigated in a test area in south-western Canada. A “corrector surface” model, which accounts for datum inconsistencies, long-wavelength geoid errors, vertical network distortions and GPS errors, has been constructed using least-squares collocation. The local covariance function of geoid height differences is usually obtained from residual values between the GPS/levelling and gravimetric geoid heights after the elimination of all known systematic distortions. If additional gravity data (in the form of gravity anomalies) are available, the covariance function of geoid height differences can be determined by the following steps: (1) transforming the GPS/levelling-derived geoid heights into gravity anomalies; (2) forming differences between the computed in step 1 and given gravity anomalies; (3) determining the parameters of the local covariance function of the gravity anomaly differences; (4) constructing an analytical covariance model for the geoid height differences from the covariance function of the gravity anomaly differences using the parameters derived in step 3. The advantage of the proposed method stems from the great number of gravity data used to derive the empirical covariance function. A comparison with the least-squares adjustment shows that the standard deviation of the residuals of the predicted geoid height differences with respect to the control point values decreases by 2.4 cm.     

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.     

Ability of the EGM2008 high degree geopotential model to calculate a local geoid model in Valencia,Eastern Spain

A new generation of global geopotential models (GGM) is being developed. These solutions offer a file with fully-normalized spherical harmonic coefficients of the Earth’s gravitational potential up to a degree greater than 2000 with very low commission errors. This paper analyses the recent Earth Gravitational Model EGM2008, developed up to degree and order 2159 with additional coefficients to degree 2190 and order 2159, which means recovering the gravitational field up to approximately 20 km wavelengths. 223 GPS/levelling points of the new Spanish High Precision Levelling Network in the Valencia region (Eastern Spain) are used as external tool for evaluation in that particular region. The same evaluation has been performed to other different global (EGM96 and EIGENCG03C), continental (EGG97), regional (IGG2005 and IBERGEO2006) and local (GCV07) geoid models for comparison purposes only. These comparisons show that EGM2008 is the geoid model that best fits to the GPS/levelling data in that region.     

Update of the precision geoid determination in Korea

From the late 1990s, many studies on local geoid construction have been made in South Korea. However, the precision of the previous geoid has remained about 15 cm due to distribution and quality problems of gravity and GPS/levelling data. Since 2007, new land gravity data and GPS/levelling data have been obtained through many projects such as the Korean Land Spatilaization, Unified Control Point and Gravity survey on the Benchmark. The newly obtained data are regularly distributed to a certain degree and show much better improvement in their quality. In addition, an airborne gravity survey was conducted in 2008 to cover the Korean peninsula (South Korea only). Therefore, it is expected that the precision of the geoid could be improved. In this study, the new South Korean gravimetric geoid and hybrid geoid are presented based on land, airborne, ship‐borne, altimeter gravity data, geopotential model and topographic data. As for the methodology, the general remove‐restore approach was applied with the best chosen parameters in order to produce a precise local geoid. The global geopotential model EGM08 was used to remove the low‐frequency components using degree and order up to 360 and the short wavelength part of the gravity signal was dealt with by using the Shuttle Radar Topography Mission data. The parameters determined empirically in this study include for Stokes’ integral 0.5° and for Wong‐Gore kernel 110–120°, respectively and 10 km for both the Bjerhammar sphere depth and attenuation factor. The final gravimetric geoid in South Korea ranges from 20–31 m with a precision of 5.45 cm overall compared to 1096 GPS/levelling data. In addition, the South Korean hybrid geoid produces 3.46 cm and 3.92 cm for degrees of fitness and precision, respectively and a better statistic of 2.37 cm for plain and urban areas was achieved. The gravimetric and hybrid geoids are expected to improve further when the refined land gravity data are included in the near future.     

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.     

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.     

Comparison of two different solutions to Molodensky’s G1 term

The main purpose of this paper is comparison of two different approaches of solution to the Simple Molodensky’s Problem, the Molodensky’s Approach and the Analytical Continuation Approach, based on numerical computation. Although these approaches have been described theoretically by several authors, e.g. Molodensky et al. (1960), Heiskanen and Moritz (1967), Vaníček (1974), Moritz (1980) and Holota (1991, unpublished results) and theoretical proof of equivalence was given by Heiskanen and Moritz (1967), Moritz (1971), Ecker (1971) and Pellinen (1972, unpublished results), only very few practical experiences about the differences between particular solutions and computational efficiency exist. In this paper we compare the above two mentioned approaches in terms of the G₁-effect on quasigeoid. Both quasigeoid solutions are tested by the independent GPS/levelling approach and are also compared with the previous quasigeoid model of Slovakia where the G₁-term has been approximated using the classical terrain correction. The effect of the G₂-term is also numerically estimated, revealing that it might be significant for the precise quasigeoid determination. Some practical comments and recommendations are given at the end of the paper.     

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.     

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.     

Vertical velocities in Finland from permanent GPS networks and from repeated precise levelling

Postglacial rebound is a long-studied phenomenon in Fennoscandia, and the general features of contemporary vertical motion (0–8 mm/year relative to mean sea level) are well known from tide gauges and repeated precise levelling. GPS on permanent stations has proved to be a powerful tool in studies of crustal motion, capable of detecting small trends in a fraction of the time required by the classical methods. We determine vertical velocities from 5 years of data in the permanent Finnish GPS network FinnRef®. We compare them with velocities derived from three precise levellings spanning nearly hundred years, and from tide gauge records. From the comparison, both FinnRef velocities and levelled velocities appear to be accurate to 0.4 mm/year (one-sigma). The isobases (lines of equal velocities) are less elongated towards northeast than in geophysical models of the rebound. However, the processing of nearly the same GPS data in BIFROST using different methods produces velocities that disagree with FinnRef more than levelling does. The levelled velocities are between the two GPS results and do not resolve the conflict.