Gravimetric Quasigeoid Research Articles

Towards a new height datum for Uganda

A new height datum for Uganda is computed using the corrective surface principle. It is based on a combination of the Uganda Gravimetric Quasigeoid model (UGQ) 2014 and GNSS/levelling. UGQ2014 was derived from the Uganda Gravimetric Geoid model (UGG) 2014, which was computed from sparse terrestrial gravity data from the International Gravimetric Bureau, the 3 arc second Shuttle Radar Topography Mission digital elevation model and the GOCE – only global geopotential model GO_CONS_GCF_2_TIM_R5. The corrective surface was constructed based on 21 discrete GNSS/levelling points and then evaluated with 4 independent points. Three interpolation techniques were tested for the creation of the corrective surface with the Kriging method giving the lowest standard deviation and noise level suggesting that it is the best method for interpolation. In absolute terms, the Root Mean Square of the fit between the known and computed normal-orthometric heights based on the new height datum is 11cm, which is about 5cm (31%) better than using UGQ2014 alone. For relative heights an average precision of 29 ppm is computed for all baselines tested. Both the absolute and relative tests show that the new height datum satisfies the precision and accuracy requirements of third order precise levelling. Therefore, UGQ2014C represents a significant step towards the determination of a precise new height datum for Uganda.

Estimating gravity field and quasi-geoid in Cameroon (CGM20)

In this paper, the computation of the new gravimetric quasi-geoid of Cameroon based on some recent global geopotential models and a detailed Digital Terrain Model (DTM) of the area is presented. Besides gravity data supplied by the Bureau Gravimétrique International (BGI) new data have been collected by the National Institute of Cartography (NIC) of Cameroon and used in the computation. Three different quasi-geoid models have been obtained based on three different global models, namely GOCE-dir5, EGM2008 and XGM2019e_2159. The well-known remove-compute-restore technique has been applied and Fast Collocation has been used for estimating the residual quasi-geoid component from gridded gravity data. The comparisons with Global Navigation Satellite System/levelling (GNSS/levelling) data distributed over the Cameroon territory show that some improvements have been obtained either with respect to the global model solutions and to previous local estimates computed in the past.

A New Model of Quasigeoid for the Baltic Sea Area

The Space Research Centre in Warsaw is participating in the ESA project “Geodetic SAR for Height System Unification and Sea Level Research”. To observe the absolute sea level and enable the unification of the height systems, the physical heights of the tide gauge stations referring to a common equipotential surface (quasigeoid/geoid) are needed. This paper describes the new quasigeoid model for the area of the Baltic sea. The quasigeoid calculation was carried out according to the Helmert method, in which the topography is condensed on a layer lying on the geoid. Airborne gravity anomalies from the Baltic area and terrestrial anomalies from Sweden, Finland, Denmark, Lithuania, Latvia, and Poland were used. The necessary terrain corrections have been computed from a digital terrain model based on the SRTM30 model. To compute the long-wavelength part of the quasigeoid, the geopotential models GOCE-DIR6, GOCO06s, and EIGEN-6C4 were used; therefore, the three solutions have been obtained. All calculations were done in a zero-tide system. The new quasigeoid model is obtained on a regular 1.5’ × 3.0’ grid in the GRS80 reference system, covering the Baltic Sea and the surrounding area 52° < ϕ < 68° and 11° < λ < 30°. These gravimetric quasigeoids were compared to quasigeoid undulations derived at 29 GNSS/leveling points of the ASG-EUPOS permanent network, located in the study area. Our calculations show that the accuracy of the calculated quasigeoids is almost the same in all three cases and is about ±0.04 meters. Finally, quasigeoid anomalies were interpolated at the Polish tide gauge stations. The new gravimetric quasigeoid solution could be very important for height system unification, for geophysical purposes as well as for engineering purposes.

On the uncertainty of height anomaly differences predicted by least-squares collocation

Abstract A network of pointwise available height anomalies, derived from levelling and GPS observations, can be densified by adjusting a gravimetric quasigeoid using least-squares collocation. The resulting type of Corrector Surface Model (CSM) is applied by Norwegian surveyors to convert ellipsoidal heights to normal heights expressed in the official height system NN2000. In this work, the uncertainty related to the use of a CSM to predict differences in height anomaly was sought. As previously, the application of variograms to determine the local statistical properties of the adopted collocation model led to predictions that were consistent with their computed uncertainties. For the purpose of predicting height anomaly differences, the effect of collocation was seen to be moderate in general for the small spatial separations considered (< 10 km). However, the relative impact of collocation could be appreciable, and increasing with distance, near the network. At last, it was argued that conservative uncertainties of height anomaly differences may be obtained by rescaling output of a grid interpolation by Δ \sqrt \Delta , where Δ is the spatial separation of the two locations for which the difference is sought.

AUSGeoid2020 combined gravimetric\u2013geometric model: location-specific uncertainties and baseline-length-dependent error decorrelation

AUSGeoid2020 is a combined gravimetric–geometric model (sometimes called a “hybrid quasigeoid model”) that provides the separation between the Geocentric Datum of Australia 2020 (GDA2020) ellipsoid and Australia’s national vertical datum, the Australian Height Datum (AHD). This model is also provided with a location-specific uncertainty propagated from a combination of the levelling, GPS ellipsoidal height and gravimetric quasigeoid data errors via least squares prediction. We present a method for computing the relative uncertainty (i.e. uncertainty of the height between any two points) between AUSGeoid2020-derived AHD heights based on the principle of correlated errors cancelling when used over baselines. Results demonstrate AUSGeoid2020 is more accurate than traditional third-order levelling in Australia at distances beyond 3 km, which is 12 mm of allowable misclosure per square root km of levelling. As part of the above work, we identified an error in the gravimetric quasigeoid in Port Phillip Bay (near Melbourne in SE Australia) coming from altimeter-derived gravity anomalies. This error was patched using alternative altimetry data.

Heterogeneous Gravity Data Fusion and Gravimetric Quasigeoid Computation in the Coastal Area of China

ABSTRACTDifferent types of gravity observations are available over coastal areas. The main challenge for coastal geoid determination is the proper fusion of heterogeneous gravity data including land, shipborne, airborne, and altimetry-derived gravity data. This paper describes the gravity data fusion and the computation of the gravimetric quasigeoid in the coastal area of mainland China. An iterative procedure of the weighted least-squares prediction based on rectangular harmonic functions is used for merging the land, altimetric, shipborne, and airborne gravity data. Applying the analytical continuation method in Molodensky's theoretic frame, the merged gravity data are then used to determine the gravimetric quasigeoid model by using the generalized Stokes' integral in a remove-compute-restore fashion. The gravimetric quasigeoid model is compared with the height anomalies determined at 662 GPS leveling points over the coastal region of mainland China, where both the geodetic height and the normal height are known. The standard deviations of the differences in the coastal provinces range from 1.8 to 4.4 cm. For the entire computation area, the mean and standard deviation of the differences are 27.9 and 3.9 cm, respectively.

Accuracy assessment of GOCE-based geopotential models and their use for modelling the gravimetric quasigeoid - A case study for Poland

Abstract The GOCE (Gravity Field and Steady-State Ocean Circulation Explorer) has significantly upgraded the knowledge on the Earth gravity field. In this contribution the accuracy of height anomalies determined from Global Geopotential Models (GGMs) based on approximately 27 months GOCE satellite gravity gradiometry (SGG) data have been assessed over Poland using three sets of precise GNSS/levelling data. The fits of height anomalies obtained from 4th release GOCE-based GGMs to GNSS/levelling data were discussed and compared with the respective ones of 3rd release GOCE-based GGMs and the EGM08. Furthermore, two highly accurate gravimetric quasigeoid models were developed over the area of Poland using high resolution Faye gravity anomalies. In the first, the GOCE-based GGM was used as a reference geopotential model, and in the second - the EGM08. They were evaluated with GNSS/levelling data and their accuracy performance was assessed. The use of GOCE-based GGMs for recovering the long-wavelength gravity signal in gravimetric quasigeoid modelling was discussed.

AUSGeoid09: a more direct and more accurate model for converting ellipsoidal heights to AHD heights

In an absolute sense, AUSGeoid09 is an order of magnitude more accurate than AUSGeoid98 at converting ellipsoidal heights to Australian Height Datum (AHD) heights and vice versa. Results of this study show AUSGeoid09 can be used to compute AHD heights from Global Navigation Satellite System (GNSS) ellipsoidal heights with an uncertainty of less than ±0.03 m (one sigma). The improvement is largely due to the inclusion of a geometric component in AUSGeoid09 that accounts for the spatially varying offset between a gravimetric quasigeoid model and the AHD. This geometric component was calculated using least squares collocation in cross validation mode and then added to the gravimetric quasigeoid. Although previous AUSGeoid models were used to convert GNSS ellipsoidal heights to the AHD and vice versa, none until now have accounted for the gravimetric quasigeoid to AHD offsets. This offset is a consequence of how the AHD was realised and has commonly resulted in misfits of ∼0.5 m or more. When used with GNSS technology, AUSGeoid09 can replace the need for traditional third-order levelling (Class LC; 12) in many situations. Relative tests of AUSGeoid09 over a continent-wide set of over 20 million baselines showed that it can deliver better than Class LC tolerances in 99 percent of cases. The model accepts a user's GDA94 latitude, longitude and ellipsoidal height and returns an AHD height and deviations of the vertical. AUSGeoid09 is now available free-of-charge on the Geoscience Australia website (http://www.ga.gov.au/geodesy/ausgeoid/nvalcomp.jsp).

The AUSGeoid09 model of the Australian Height Datum

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

Tuning a gravimetric quasigeoid to GPS-levelling by non-stationary least-squares collocation

This paper addresses implementation issues in order to apply non-stationary least-squares collocation (LSC) to a practical geodetic problem: fitting a gravimetric quasigeoid to discrete geometric quasigeoid heights at a local scale. This yields a surface that is useful for direct GPS heighting. Non-stationary covariance functions and a non-stationary model of the mean were applied to residual gravimetric quasigeoid determination by planar LSC in the Perth region of Western Australia. The non-stationary model of the mean did not change the LSC results significantly. However, elliptical kernels in non-stationary covariance functions were used successfully to create an iterative optimisation loop to decrease the difference between the gravimetric quasigeoid and geometric quasigeoid at 99 GPS-levelling points to a user-prescribed tolerance.

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.

The Use of Second-generation Wavelets to Combine a Gravimetric Quasigeoid Model with GPS-levelling Data

The merging of a gravimetric quasigeoid model with GPS-levelling data using second-generation wavelets is considered so as to provide better transformation of GPS ellipsoidal heights to normal heights. Since GPS-levelling data are irregular in the space domain and the classical wavelet transform relies on Fourier theory, which is unable to deal with irregular data sets without prior gridding, the classical wavelet transform is not directly applicable to this problem. Instead, second-generation wavelets and their associated lifting scheme, which do not require regularly spaced data, are used to combine gravimetric quasigeoid models and GPS-levelling data over Norway and Australia, and the results are cross-validated. Cross-validation means that GPS-levelling points not used in the merging are used to assess the results, where one point is omitted from the merging and used to test the merged surface, which is repeated for all points in the dataset. The wavelet-based results are also compared to those from least squares collocation (LSC) merging. This comparison shows that the second-generation wavelet method can be used instead of LSC with similar results, but the assumption of stationarity for LSC is not required in the wavelet method. Specifically, it is not necessary to (somewhat arbitrarily) remove trends from the data before applying the wavelet method, as is the case for LSC. It is also shown that the wavelet method is better at decreasing the maximum and minimum differences between the merged geoid and the cross-validating GPS-levelling data.

Quasigeoids for the Territory of the Czech Republic

Several quasigeoid models for the Czech Republic have been developed for different purposes since mid-seventies using different data sources: gravimetric, astronomical and GPS/levelling observations, as well as data gained from the territories of neighbouring countries. Two of the recent solutions, namely the VUGTK96 quasigeoid obtained from heterogeneous data and gravimetric quasigeoid CR2000 are described and compared. Accuracies of the models were estimated with the help of accuracy characteristics resulting from the processing, by comparison with GPS/levelling results available for most of the territory, and, finally, tested on a special GPS/levelling profile across the steepest slope of the quasigeoid surface in the south-eastern part of the territory. It is shown that the CR2000 gravimetric quasigeoid is suitable for transformation of ellipsoidal GPS heights to the gravity related heights with an accuracy of 3 cm.

THE USE OF THE TWO-DIMENSIONAL SPHERICAL FFT FOR QUASI-GEOID MODELLING IN SOUTH AFRICA

Quasi-geoids for the south western Cape, a coastal province of South Africa, and the Gauteng area, in the north-east of South Africa, have been computed by the application of the two-dimensional (2D) spherical fast Fourier transform (SFFT) to Molodensky's formula. The gravimetric quasi-geoids for the south western Cape and Gauteng were computed on a grid resolution of one arc-minute. Validation of the gravimetric quasi-geoid models with GPS/levelling-derived geometric quasi-geoids via a simple planar transformation resulted in a root mean square (RMS) discrepancy of about 4cm for the south western Cape area and about 1.3cm for area of Gauteng. The quasi-geoids have also been evaluated using the 2D planar FFT and 2D multi-band SFFT and our results confirm the suitability of using a 2D SFFT's for geoid/quasi-geoid computation in mid-latitude regions.

Models for fitting gravimetric geoids and GPS results

The aim of this investigation is to study how to use a gravimetric (quasi) geoid for levelling by GPS data in an optimal way. The advent of precise geodetic GPS has made the use of a technique possible, which might be called GPS-gravimetric geoid determination. In this approach, GPS heights above the reference ellipsoid are determined for points whose levelled (orthometric) heightH is above sea level people have already surveyed; for these points, we thus have the values of the geoid undulationN. These values are then used to constrain the geoid undulationsN′obtained from the gravimetric solution.

Comparison and combination of a gravimetric quasigeoid with a levelled GPS data set by statistical analysis

For GPS levelling applications, it is convenient to express the height reference surface in a suitable geodetic reference system. This can be obtained through a set of levelled GPS points. Unfortunately, available data are sparse. A gravimetric geoid is often used to interpolate the height reference surface issued from GPS and levelling. Both surfaces do not coincide exactly with each other. At this point, one must compare two realisations of the geoid, detect outliers, retrieve (if possible) the causes of the discrepancies and finally combine the two kinds of data. The paper presents some practical solutions to these problems.

A comparison between Itagleo'95 and GPS/Leveling data along the coasts of Italy

In the last few years the number of GPS/Lev. data are increasing in order to investigate the geoidal undulations. In this paper the global model (OSU91A) and the Italian gravimetric quasigeoid (ITALGEO'95) are checked against two different kinds of GPS/Lev. data. Statistical and graphical results are reported.