Gravimetric Quasigeoid Model Research Articles

High-Accuracy Quasi-Geoid Determination Using Molodensky’s Series Solutions and Integrated Gravity/GNSS/Leveling Data

This study presents a methodology for constructing a quasi-geoid model with millimeter-level accuracy over the Shangyu area in China, following the guidelines of the International Association of Geodesy Joint Working Group 2.2.2, known as “The 1 cm geoid experiment”. Our approach combines two steps to ensure exceptional accuracy. First, we employ Molodensky’s theory to model the gravity field, accounting for non-level surfaces and considering complex terrain effects. Through an exhaustive analysis of these influential factors, we implement a comprehensive suite of applicable formulae within Molodensky’s series solution, enabling a thorough assessment of their impacts on height anomalies within the gravimetric quasi-geoid model. Second, we utilize a hybrid method that involves a multi-surface function using the least-squares method and a robust estimation technique. This approach enables the interpolation of quasi-geoid heights by incorporating ellipsoidal and leveling normal heights, as well as gravimetric quasi-geoid data. Through a numerical example, we demonstrate the efficiency of our solution concept, achieving an accuracy of 0.79 cm compared to independent global navigation satellite system (GNSS)/leveling measurements. By developing this methodology, our study contributes to the advancement of geodesy research and provides a valuable methodology for creating highly precise quasi-geoid models in geodetic applications.

PL-geoid2021: A quasigeoid model for Poland developed using geophysical gravity data inversion technique

This paper presents the results of research and analyses related to the development of a new quasigeoid model fitted to GNSS/levelling data for the area of Poland (PL-geoid2021). The model was determined employing two procedures based on the Geophysical Gravity data Inversion technique (GGI method): procedure A consisted of the determination of the gravimetric quasigeoid model in the first step and its subsequent fitting to GNSS/levelling data in the second step, and procedure B consisted of a one-step determination of the model fitted to GNSS/levelling data. Both models were developed using the global geopotential model SGG-UGM-2 and gravity data covering the area of Poland, and slightly extend beyond Poland's southern and northern borders. The average model was adopted as the final model. It was demonstrated that the accuracy of the gravimetric quasigeoid model had a very low dependence on the reference topographic mass density model used. On the basis of this model, the GNSS/levelling datasets were also assessed and outliers were identified. The estimated accuracy of the gravimetric model, determined based on four GNSS/levelling datasets, was in the range of ± 1.2 to ± 1.7 cm, in terms of the standard deviation of the differences between the measured and model-determined height anomalies. Due to partial lack of gravity data just beyond the Polish border, the edge effect was also analysed. The accuracy of the final quasigeoid model (estimated in the same way as the gravimetric model) ranges from ± 1.0 to ± 1.2 cm. It should be noted, however, that this assessment is not fully independent because three of the four sets of GNSS/levelling points used for it, were also used to build the final model.

A detailed quasigeoid model of the Hong Kong territories computed by applying a finite-element method of solving the oblique derivative boundary-value problem

Abstract New gravity and precise levelling measurements have been performed throughout the Hong Kong territories to modernize a vertical geodetic datum that is currently realized by heights of levelling benchmarks defined in the Hong Kong Principal Datum (HKPD). Modernization of the HKPD involved delivering various products, including new detailed geoid and quasigeoid models and newly determined orthometric and normal heights of levelling benchmarks. In this study, we present the result of gravimetric quasigeoid modelling. The method used to compute a detailed gravimetric quasigeoid model is based on the finite-element method to solve the geodetic boundary-value problem with oblique derivative boundary conditions considered directly at computational nodes on the discretized Earth’s topography. The result of a gravimetric quasigeoid modelling shows a good agreement with a geometric quasigeoid model at the Global Navigation Satellite System (GNSS)-levelling benchmarks. The standard deviation of differences between the gravimetric and geometric quasigeoid heights of ±3.3 cm is compatible with the expected accuracy of gravity, levelling, and GNSS measurements.

Determining the new height of Mount Qomolangma based on the International Height Reference System

ABSTRACT To achieve the goal that China and Nepal jointly announce the new height of Mount Qomolangma (MQ), the campaign of Qomolangma Height Survey (QHS) was carried out from 2019 to 2020. A high precision geoid model realizing the common height datum for both sides is necessary for determining the unique height of MQ. However, high altitude and rugged topography make it extremely difficult to conduct terrestrial gravity measurements in this region, the accuracy of geoid model is restricted by terrestrial gravity data gaps. In the campaign of 2020 QHS, the airborne gravity survey was carried out over MQ and its surrounding areas, the airborne gravity data covering an area of 12,700 km2 were successfully collected. For the first time, the high precision observations of terrestrial gravity and BeiDou Navigation Satellite System (BDS) at the peak of MQ were collected. These datasets pave the way for the precise determination of the orthometric height of MQ. According to the definition of the International Height Reference System (IHRS), we developed the IHRS-based gravimetric quasigeoid model by combining the airborne and terrestrial gravity data. Validations against highly accurate GNSS leveling data at 61 benchmarks demonstrate that the accuracy of the quasigeoid model is 3.8 cm, and the addition of airborne gravity data improves the accuracy by 51.3%. Based on the IHRS, the final orthometric height of the snow surface of the peak of MQ is determined to be 8848.86 m.

Geoid model validation and topographic bias

Abstract Recently a number of geoid campaigns were performed to verify different types of geoid and quasigeoid modeling techniques. Typically, GNSS-leveling was employed as an independent method, but in some cases zenith camera astronomic deflection data were also used in astrogeodetic determinations of the geoid and/or quasigeoid. However, due to the uncertainty in the topographic density distribution data (and thereby in orthometric heights), we conclude that neither GNSS-leveling nor astrogeodetic techniques can reliably verify differences between gravimetric geoid models at several centimeter levels in rough mountainous regions. This is because much the same topographic data are used both in the gravimetric geoid models and in their verifications by geometric and/or astrogeodetic geoid models. On the contrary, this is not a problem in verifying gravimetric quasigeoid models, as they are independent of the topographic density distribution, and so is the related normal height used in GNSS-leveling.

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.

A Quasigeoid-Derived Transformation Model Accounting for Land Subsidence in the Mekong Delta towards Height System Unification in Vietnam

A vertical offset model for Vietnam and its surrounding areas was determined based on the differences between height anomalies derived from 779 Global Navigation Satellite System (GNSS)/levelling points and those derived from a dedicated high-resolution gravimetric-only quasigeoid model called GEOID_LSC. First, the deterministic transformation model to effectively fit the differences between the quasigeoid and GNSS/levelling heights was based on a third-order polynomial model. Second, the residual height anomalies have been interpolated to a grid employing Least-Squares Collocation. Finally, the distortions were restored to the residual grid. This model can be used for combination with a gravimetric quasigeoid model in GNSS levelling. The quality of GNSS/levelling data in Vietnam was analyzed and evaluated in this study. The annual subsidence rate from ALOS-1 was also used to analyze the effects of subsidence on the quality of GNSS/levelling data in the Mekong Delta. From this we made corrections to improve the accuracy of GNSS/levelling data in this region. The offset model was evaluated using cross-validation technique by comparing with GNSS/levelling data. Results indicate that the offset model has a standard deviation of 5.9 cm in the absolute sense. Based on this offset model, GNSS levelling can be carried out in most of Vietnam’s territory complying third-order levelling requirements, while the accuracy requirements for fourth-order levelling networks is met for the entire country. This model in combination with the developed gravimetric quasigeoid model should also contribute to the modernization of Vietnam’s height system. We also used high-quality GNSS/levelling data and the determined quasigeoid model to determine the geopotential value W0 for the Vietnam Local Vertical Datum. The gravity potential of the Vietnam Local Vertical Datum is estimated equal to W 0 LVD = 62,636,846.81 ± 0.70 m2s−2 with the global equipotential surface realized by the conventional value W0 = 62,636,853.4 m2s−2.

A high-resolution gravimetric quasigeoid model for Vietnam

A high-resolution gravimetric quasigeoid model for Vietnam and its surrounding areas was determined based on new gravity data. A set of 29,121 land gravity measurements was used in combination with fill-in data where no gravity data existed. Global Gravity field Models plus Residual Terrain Model effects and gravity field derived from altimetry satellites were used to provide the fill-in information over land and marine areas. A mixed model up to degree/order 719 was used for the removal of the long and medium wavelengths and the calculation of the quasigeoid restore effects. The residual height anomalies have been determined employing the Stokes integral using the Fast Fourier Transform approach and deterministic kernel modification proposed by Wong–Gore, as well as by means of Least-Squares Collocation. The accuracy of the resulting quasigeoid models was evaluated by comparing with height anomalies derived from 812 co-located GNSS/levelling points. Results are very similar; both local quasigeoid models have a standard deviation of 9.7 cm and 50 cm in mean bias when compared to the GNSS/levelling points. This new local quasigeoid model for Vietnam represents a significant improvement over the global models EIGEN-6C4 and EGM2008, which have standard deviations of 19.2 and 29.1 cm, respectively, for this region.

Assessment of the latest generation GOCE-based global gravity field models using height and free-air gravity anomalies over South Africa

The latest Gravity field and steady-state Ocean Circulation Explorer (GOCE) global gravity field models (GGMs) are evaluated over South Africa using 118 GPS/levelling, 507 gravity and 88,968 SAGEOID10 data points. It is observed that GOCE data can improve quasi-geoid model in the long-to-medium wavelength components over South Africa. GOCE-based GGMs perform better than EGM2008 in recovering height anomalies up to 240°. Related results are obtained for free-air gravity anomalies, but the improvement over EGM2008 by GOCE data is only up to 180°. TIM5 (up to 240°) is selected as a candidate GOCE model and combined with EGM2008 (from 241 up to 2190°). This model (TIM5/EGM2008) is evaluated by observed data (GPS/levelling and gravity) and the latest hybrid quasi geoid model over South Africa (SAGEOID10). The solutions converge at 1200 and 1800° for height and free-air gravity anomalies, respectively, when TIM5/EGM2008 is used. Results show that TIM5/EGM2008 can effectively recover height and free-air gravity anomalies over South Africa. The standard deviations of the differences between observed (118 GPS/levelling and 507 gravity points) and TIM5/EGM2008 implied height and free-air gravity anomalies are ± 23.7 cm and ± 9.0 mGal, respectively. In addition, the standard deviation of the differences between SAGEOID10 and TIM5/EGM2008 implied height anomalies (at 88,968 grid points) is ± 21.7 cm. The results for height anomalies from TIM5/EGM2008 are comparable with the latest gravimetric quasi-geoid model over South Africa.

The impact of terrestrial gravity data density on geoid accuracy: case study Bilogora in Croatia

One of the main challenges in physical geodesy today is achieving the 1-centimetre gravimetric quasi-geoid model, since a model of such accuracy could be used in the definition and realisation of the height reference frame. One of the main obstacles in this challenge is the lack of terrestrial gravity data or its uneven distribution. Therefore, the main question arises: what density of gravity points is necessary in order to obtain a gravimetric quasi-geoid model with an accuracy of 1 centimetre or even better? In this simulated study, the results show a trend of decreasing RMS related to the sparser dataset used in computation, leading to the conclusion that the determination of the sub-centimetre quasi-geoid model is no longer a theory, but can be achieved with the exact density of terrestrial gravity data: gravity observation stations have to be distributed no farther than 500 m from one another. When dealing with a very rough topography, the distance should be even shorter.

A multilayer approach and its application to model a local gravimetric quasi-geoid model over the North Sea: QGNSea V1.0

Abstract. A multilayer approach is set up for local gravity field recovery within the framework of multi-resolution representation, where the gravity field is parameterized as the superposition of multiple layers of Poisson wavelets located at different depths beneath the Earth's surface. The layers are designed to recover gravity signals at different scales, where the shallow and deep layers mainly capture the short- and long-wavelength signals, respectively. The depths of these layers are linked to the locations of different anomaly sources beneath the Earth's surface, which are estimated by wavelet decomposition and power spectrum analysis. For testing the performance of this approach, a gravimetric quasi-geoid model over the North Sea, QGNSea V1.0, is modeled and validated against independent control data. The results show that the multilayer approach fits the gravity data better than the traditional single-layer approach, particularly in regions with topographical variation. An Akaike information criterion (AIC) test shows that the multilayer model obtains a smaller AIC value and achieves a better balance between the goodness of fit of data and the simplicity of the model. Further, an evaluation using independent GPS/leveling data tests the ability of regional models computed from different approaches towards realistic extrapolation, which shows that the accuracies of the QGNSea V1.0 derived from the multilayer approach are better by 0.4, 0.9, and 1.1 cm in the Netherlands, Belgium, and parts of Germany, respectively, than that using the single-layer approach. Further validation with existing models shows that QGNSea V1.0 is superior with respect to performance and may be beneficial for studying ocean circulation between the North Sea and its neighboring waters.

Error propagation for the Molodensky G1 term

Molodensky G terms are used in the computation of the quasigeoid. We derive error propagation formulas that take into account uncertainties in both the free air gravity anomaly and a digital elevation model. These are applied to generate G1 terms and their errors on a 1″ × 1″ grid over Australia. We use these to produce Molodensky gravity anomaly and accompanying uncertainty grids. These uncertainties have average value of 2 mGal with maximum of 54 mGal. We further calculate a gravimetric quasigeoid model by the remove–compute–restore technique. These Molodensky gravity anomaly uncertainties lead to quasigeoid uncertainties with a mean of 4 mm and maximum of 80 mm when propagated through a deterministically modified Stokes’s integral over an integration cap radius of 0.5°.

A high-precision digital astrogeodetic traverse in an area of steep geoid gradients close to the coast of Perth, Western Australia

We present results from a new vertical deflection (VD) traverse observed in Perth, Western Australia, which is the first of its kind in the Southern Hemisphere. A digital astrogeodetic QDaedalus instrument was deployed to measure VDs with $${\sim }$$ 0.2 $$''$$ precision at 39 benchmarks with a $${{\sim }}1~\hbox {km}$$ spacing. For the conversion of VDs to quasigeoid height differences, the method of astronomical–topographical levelling was applied, based on topographical information from the Shuttle Radar Topography Mission. The astronomical quasigeoid heights are in 20–30 mm (RMS) agreement with three independent gravimetric quasigeoid models, and the astrogeodetic VDs agree to 0.2–0.3 $$''$$ (north–south) and 0.6–0.9 $$''$$ (east–west) RMS. Tilt-like biases of $${\sim }1\,\,\hbox {mm}$$ over $${\sim }1\,\,\hbox {km}$$ are present for all quasigeoid models within $${\sim }20\,\,\hbox {km}$$ of the coastline, suggesting inconsistencies in the coastal zone gravity data. The VD campaign in Perth was designed as a low-cost effort, possibly allowing replication in other Southern Hemisphere countries (e.g., Asia, Africa, South America and Antarctica), where VD data are particularly scarce.

The New Zealand gravimetric quasigeoid model 2017 that incorporates nationwide airborne gravimetry

A one arc-minute resolution gravimetric quasigeoid model has been computed for New Zealand, covering the region $$25^{\circ }\hbox {S}$$ – $$60^{\circ }\hbox {S}$$ and $$160^{\circ }\hbox {E}$$ – $$170^{\circ }\hbox {W}$$ . It was calculated by Wong and Gore modified Stokes integration using the remove–compute–restore technique with the EIGEN-6C4 global gravity model as the reference field. The gridded gravity data used for the computation consisted of 40,677 land gravity observations, satellite altimetry-derived marine gravity anomalies, historical shipborne marine gravity observations and, importantly, approximately one million new airborne gravity observations. The airborne data were collected with the specific intention of reinforcing the shortcomings of the existing data in areas of rough topography inaccessible to land gravimetry and in coastal areas where shipborne gravimetry cannot be collected and altimeter-derived gravity anomalies are generally poor. The new quasigeoid has a nominal precision of $$\pm \,48\,\hbox {mm}$$ on comparison with GPS-levelling data, which is approximately $$14\,\hbox {mm}$$ less than its predecessor NZGeoid09.

The first Australian gravimetric quasigeoid model with location-specific uncertainty estimates

We describe the computation of the first Australian quasigeoid model to include error estimates as a function of location that have been propagated from uncertainties in the EGM2008 global model, land and altimeter-derived gravity anomalies and terrain corrections. The model has been extended to include Australia’s offshore territories and maritime boundaries using newer datasets comprising an additional {sim }280,000 land gravity observations, a newer altimeter-derived marine gravity anomaly grid, and terrain corrections at 1^{prime prime }times 1^{prime prime } resolution. The error propagation uses a remove–restore approach, where the EGM2008 quasigeoid and gravity anomaly error grids are augmented by errors propagated through a modified Stokes integral from the errors in the altimeter gravity anomalies, land gravity observations and terrain corrections. The gravimetric quasigeoid errors (one sigma) are 50–60 mm across most of the Australian landmass, increasing to {sim }100 mm in regions of steep horizontal gravity gradients or the mountains, and are commensurate with external estimates.

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.

A numerically efficient technique of regional gravity field modeling using Radial Basis Functions

Radial Basis Functions (RBFs) have been extensively used in regional gravity and (quasi)geoid modeling. Reliable models require the choice of an optimal number of RBFs and of their parameters. The RBF parameters are typically optimized using a regularization algorithm. Therefore, the determination of the number of RBFs is the most challenging task in the modeling procedure. For this purpose, we design a data processing scheme to optimize the number of RBFs and their parameters simultaneously. Using this scheme, the gravimetric quasi-geoid model can be validated without requiring additional information on the quasi-geoidal geometry obtained from GPS/leveling data. Furthermore, the Levenberg–Marquardt algorithm, used for regularization, is modified to enhance its numerical performance. We demonstrate that these modifications guarantee the convergence of the solution to the global minimum while substantially decreasing the number of iterations. The proposed methodology is evaluated using synthetic gravity data and compared with existing methods for validating the RBF parameterization of the gravity field.

Variance component estimation uncertainty for unbalanced data: application to a continent-wide vertical datum

Variance component estimation (VCE) is used to update the stochastic model in least-squares adjustments, but the uncertainty associated with the VCE-derived weights is rarely considered. Unbalanced data is where there is an unequal number of observations in each heterogeneous data set comprising the variance component groups. As a case study using highly unbalanced data, we redefine a continent-wide vertical datum from a combined least-squares adjustment using iterative VCE and its uncertainties to update weights for each data set. These are: (1) a continent-wide levelling network, (2) a model of the ocean’s mean dynamic topography and mean sea level observations, and (3) GPS-derived ellipsoidal heights minus a gravimetric quasigeoid model. VCE uncertainty differs for each observation group in the highly unbalanced data, being dependent on the number of observations in each group. It also changes within each group after each VCE iteration, depending on the magnitude of change for each observation group’s variances. It is recommended that VCE uncertainty is computed for VCE updates to the weight matrix for unbalanced data so that the quality of the updates for each group can be properly assessed. This is particularly important if some groups contain relatively small numbers of observations. VCE uncertainty can also be used as a threshold for ceasing iterations, as it is shown—for this data set at least—that it is not necessary to continue time-consuming iterations to fully converge to unity.

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.

DFHRS-BASED COMPUTATION OF QUASI-GEOID OF LATVIA

In geodesy, civil engineering and related fields high accuracy coordinate determination is needed, for that reason GNSS technologies plays important role. Transformation from GNSS derived ellipsoidal heights to orthometric or normal heights requires a high accuracy geoid or quasi-geoid model, respectively the accuracy of the currently used Latvian gravimetric quasi-geoid model LV'98 is 6–8 cm. The objective of this work was to calculate an improved quasi-geoid (QGeoid) for Latvia. The computation was performed by applying the DFHRS software. This paper discusses obtained geoid height reference surface, its comparisons to other geoid models, fitting point statistics and quality control based on independent measurements.