Quasigeoid Modelling Research Articles

Potential of cold-atom airborne gravimetry to improve coastal gravity field and quasigeoid modelling

We investigate using the GIRAFE cold-atom gravimeter during an airborne gravity survey for improving gravity field and quasigeoid modelling. The study is conducted over the Bay of Biscay, France. Geoid/quasigeoid determination is usually a major challenge over such coastal areas due to scarce and inconsistent gravity data. In a first step, the GIRAFE dataset is analysed and compared with available surface gravity data as well as with global altimetry models from UCSD and DTU. The comparisons indicate that the DTU model is better than the UCSD model within around 10 km from the coastline. Furthermore, recent satellite altimeter missions significantly improve the altimetry models in coastal areas. A significant bias (− 4.00 mGal) in shipborne data is also found from this comparison. In a second step, eight quasigeoid solutions are calculated to evaluate the contribution of GIRAFE data. This contribution reaches 3 cm in terms of height anomaly for DTU21 while being much larger for UCSDv31 and shipborne data. Finally, the quasigeoid solutions are validated using GNSS-levelling data. The results indicate that using GIRAFE data improves by approximately 50% the quality of quasigeoid models over land near the coast. The highest accuracy, around 1 cm, is achieved when GIRAFE data are merged with refined gravity data. Importantly, the standard deviation is just 1.2 cm when compared with GNSS-levelling points if using only GIRAFE data over marine areas, which is very close to the 1 cm goal of geoid/quasigeoid model determination in modern geodesy. This study thus confirms the benefits of performing airborne gravity survey using quantum sensors.

A data-adaptive network design for the regional gravity field modelling using spherical radial basis functions

A high-precision regional gravity field model is significant in various geodesy applications. In the field of modelling regional gravity fields, the spherical radial basis functions (SRBFs) approach has recently gained widespread attention, while the modelling precision is primarily influenced by the base function network. In this study, we propose a method for constructing a data-adaptive network of SRBFs using a modified Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDBSCAN) algorithm, and the performance of the algorithm is verified by the observed gravity data in the Auvergne area. Furthermore, the turning point method is used to optimize the bandwidth of the basis function spectrum, which satisfies the demand for both high-precision gravity field and quasi-geoid modelling simultaneously. Numerical experimental results indicate that our algorithm has an accuracy of about 1.58 mGal in constructing the gravity field model and about 0.03 m in the regional quasi-geoid model. Compared to the existing methods, the number of SRBFs used for modelling has been reduced by 15.8%, and the time cost to determine the centre positions of SRBFs has been saved by 12.5%. Hence, the modified HDBSCAN algorithm presented here is a suitable design method for constructing the SRBF data adaptive network.

Validation of a tailored gravity field model for precise quasigeoid modelling over selected sites in Cameroon and South Africa

Abstract In this study, a tailored gravity-field model is developed to fit and recover local terrestrial gravity data by integrating gravity from global gravity-field models, residual gravity derived from topographic data and observed terrestrial gravity over two study sites in Africa (Cameroon and South Africa). During the modelling phase, two-thirds of the terrestrial gravity data is utilised, reserving the remaining one-third for validation purposes. Additionally, an independent validation is conducted by comparing computed quasigeoid models (derived from tailored gravity data) with height anomalies from GPS/levelling data over the two study sites. The accuracy of the tailored gravity model in reproducing observed gravity data is noteworthy, with a ±8.9 mGal accuracy for the study site in South Africa at 2867 test points and a ±10.4 mGal accuracy for the study site in Cameroon at 637 test points. Comparing height anomalies from GPS/levelling with the SATGQG quasigeoid model (developed from tailored gravity data) and the recent CDSM09A quasigeoid model at 11 GPS/levelling data points reveals comparable accuracies of ±0.10 m and ±0.05 m, for SATGQG and CDSM09A, respectively for the site in South Africa. For the Cameroon site, the differences between height anomalies from GPS/levelling and the CTGQG quasigeoid model (developed from tailored gravity data), along with the recent CGM20 quasigeoid model at 38 GPS/levelling data points, show practically equal accuracies of ±0.15 m for CTGQG and ±0.11 m for CGM20. These findings underscore the potential of tailored gravity-field model in developing accurate quasigeoid models, particularly in regions with limited gravity data coverage. This approach holds promise for gravity recovery and precise geoid modelling in developing countries and regions with insufficient coverage of terrestrial gravity data.

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.

Exact closed-form expressions for the complete RTM correction

We present exact, closed-form expressions for the complete RTM correction and the harmonic correction to disturbing potential, gravity disturbance, gravity anomaly, and height anomaly. They need to be applied in quasi-geoid modelling whenever data points are buried inside the masses after residual terrain model (RTM) reduction and analytically downward-continued functionals of the disturbing potential at the original locations of the data points are required. Compared to recent work of the authors published in this journal, no Taylor series enter the expressions and numerical instabilities of the harmonic downward continuation from the RTM surface to the Earth’s surface are avoided as are inaccuracies in the free-air upward continuation from the Earth’s surface to the RTM surface caused by a lack of precise information about higher-order derivatives of the disturbing potential. The new expressions can easily be implemented in any existing RTM software package and do not require additional computational resources. For two test areas located in western Norway and the Auvergne in France, we compute the complete RTM correction and the harmonic correction to the afore-mentioned functionals of the disturbing potential. Overall, all harmonic corrections are non-negative with maximum values of 1.54 textrm{m}^2/textrm{s}^2, 263.0 upmu Gal, 263.9 upmu Gal, and 15.7 m (Norway) and 1.55 textrm{m}^2/textrm{s}^2, 263.3 upmu Gal, 263.3 upmu Gal, and 15.8 cm (Auvergne) for disturbing potential, gravity disturbance, gravity anomaly, and height anomaly, respectively. The medians are 0.02 textrm{m}^2/textrm{s}^2, 33.6 upmu Gal, 33.7 upmu Gal, and 0.3 cm (Norway) and 0.01 textrm{m}^2/textrm{s}^2, 19.2 upmu Gal, 19.2 upmu Gal, and 0.1 cm (Auvergne). We also show that the nth Taylor polynomials used in the recent work of the authors published in this journal may have large remainders depending on the topography in the vicinity of the evaluation point no matter how n is chosen. Finally, we show that the commonly used expression for the harmonic correction to gravity anomaly introduced in 1981 is almost exact, though it was derived along a completely different line of reasoning. The errors do not exceed 49 upmu Gal in both test areas. Moreover, the errors have a negligible impact on the computed height anomalies in one-centimetre quasi-geoid modelling, as the mean error does not exceed a few upmu Gal in both test areas.

A crossvalidation-based comparison of kriging and IDW in local GNSS/levelling quasigeoid modelling

Abstract This study compares two interpolation methods in the problem of a local GNSS/levelling (quasi) geoid modelling. It uses raw data, no global geopotential model is involved. The methods differ as to the complexity of modelling procedure and theoretical background, they are ordinary kriging/least-squares collocation with constant trend and inverse distance weighting (IDW). The comparison itself was done through leave-one-out and random (Monte Carlo) cross-validation. Ordinary kriging and IDW performance was tested with a local (using limited number of data) and global (using all available data) neighbourhoods using various planar covariance function models in case of kriging and various exponents (power parameter) in case of IDW. For the study area both methods assure an overall accuracy level, measured by mean absolute error, root mean square error and median absolute error, of less than 1 cm. Although the method of IDW is much simpler, a suitably selected parameters (also trend removal) may contribute to differences between methods that are virtually negligible (fraction of a millimetre).

The RTM harmonic correction revisited

In this paper, we derive improved expressions for the harmonic correction to gravity and, for the first time, expressions for the harmonic correction to potential and height anomaly. They need to be applied at stations buried inside the masses to transform internal values into harmonically downward continued values, which are then input to local quasi-geoid modelling using least-squares collocation or least-squares techniques in combination with the remove-compute-restore approach. Harmonic corrections to potential and height anomaly were assumed to be negligible so far resulting in yet unknown quasi-geoid model errors. The improved expressions for the harmonic correction to gravity, and the new expressions for the harmonic correction to potential and height anomaly are used to quantify the approximation errors of the commonly used harmonic correction to gravity and to quantify the magnitude of the harmonic correction to potential and height anomaly. This is done for two test areas with different topographic regimes. One comprises parts of Norway and the North Atlantic where the presence of deep, long, and narrow fjords suggest extreme values for the harmonic correction to potential and height anomaly and corresponding large errors of the commonly used approximation of the harmonic correction to gravity. The other one is located in the Auvergne test area with a moderate topography comprising both flat and hilly areas and therefore may be representative for many areas around the world. For both test areas, two RTM surfaces with different smoothness are computed simulating the use of a medium-resolution and an ultra-high-resolution reference gravity field, respectively. We show that the errors of the commonly used harmonic correction to gravity may be as large as the harmonic correction itself and attain peak values in areas of strong topographic variations of about 100 mGal. Moreover, we show that this correction may introduce long-wavelength biases in the computed quasi-geoid model. Furthermore, we show that the harmonic correction to height anomaly can attain values on the order of a decimetre at some points. Overall, however, the harmonic correction to height anomaly needs to be applied only in areas of strong topographic variations. In flat or hilly areas, it is mostly smaller than one centimetre. Finally, we show that the harmonic corrections increase with increasing smoothness of the RTM surface, which suggests to use a RTM surface with a spatial resolution comparable to the finest scales which can be resolved by the data rather than depending on the resolution of the global geopotential model used to reduce the data.

Australian quasigeoid modelling: Review, current status and future plans

Australian quasigeoid modelling: Review, current status and future plans

Local quasigeoid modelling in Slovakia using the finite volume method on the discretized Earth's topography

The paper presents local quasigeoid modelling in Slovakia using the finite volume method (FVM). FVM is used to solve numerically the fixed gravimetric boundary value problem (FGBVP) on a 3D unstructured mesh created above the real Earth's surface. Terrestrial gravimetric measurements as input data represent the oblique derivative boundary conditions on the Earth's topography. To handle such oblique derivative problem, its tangential components are considered as surface advection terms regularized by a surface diffusion. The FVM numerical solution is fixed to the GOCE-based satellite-only geopotential model on the upper boundary at the altitude of 230 km. The horizontal resolution of the 3D computational domain is 0.002 × 0.002 deg and its discretization in the radial direction is changing with altitude. The created unstructured 3D mesh of finite volumes consists of 454,577,577 unknowns. The FVM numerical solution of FGBVP on such a detailed mesh leads to large-scale parallel computations requiring 245 GB of internal memory. It results in the disturbing potential obtained in the whole 3D computational domain. Its values on the discretized Earth's surface are transformed into the local quasigeoid model that is tested at 404 GNSS/levelling benchmarks. The standard deviation of residuals is 2.8 cm and decreases to 2.6 cm after removing 9 identified outliers. It indicates high accuracy of the obtained FVM-based local quasigeoid model in Slovakia.

Precise local quasigeoid modelling using GNSS/levelling height anomalies and gravity data

This study compares four approaches of local quasigeoid modelling using GNSS/levelling height anomalies. The first two approaches utilise only a dense network of points with known GNSS/levelling height anomalies and the EGM2008 model. They are based on the interpolation of residual values of height anomalies by applying the least squares collocation (LSC) and the thin plate spline (TPS). The next two approaches use additional data in the form of surface gravity data and the digital elevation model. One of these approaches is based on the classical Molodensky method combined with LSC. The other approach utilises the method of geophysical gravity data inversion (GGI). During the research, the authors used a local network of points with precisely defined GNSS/levelling height anomalies located in South-Western Poland. They obtained comparable results for all tested approaches at the maximum analysed density of GNSS/levelling points (about ). The differences between the modelling results became apparent only with an increase in the distance between the GNSS/levelling data points and the reduced accuracy of the used global geopotential model.

Applying the GOCE-based GGMs for the quasi-geoid modelling of Finland

AbstractThe gravity satellite mission GOCE made its final observations in the fall of 2013. Since the re-entry to the Earth’s atmosphere, the full cycle of the GOCE data has been published by ESA. At first, we evaluated all the GOCE-based global geoid models over Finland using terrestrial gravity and GNSS-levelling data. The most suitable model was selected as a global background model for the Finnish quasi-geoid calculations.Next, we combined the chosen model with terrestrial gravity data of Finland and surrounding areas. Quasi-geoid models with different modifications were calculated using the GOCE DIR5 model up to spherical harmonic degree and order (d/o) 240 and 300, and the high resolution EIGEN-6C4 (includes the complete GOCE data) model up to degree and order 1000 and 2190.The calculated quasi-geoid models were validated to the measurements on site with two independent GPS-levelling datasets. The best quasi-geoid models with GOCE gave standard deviations of 2.6 cm (FIN_DIR5 d/o 240) and 2.3 cm (FIN_DIR5 d/o 300) in Finland. For the high resolution model FIN_EIGEN-6C4, the results were 1.8 cm (d/o 1000) and 1.7 cm (d/o 2190). In addition, the results were compared with the latest geoid models available in Finland (FIN2005N00, NKG2004, NKG2015, EGG2008). The sub-2-centimetre (and near 2 cm, when using the GOCE-based models) accuracy is an improvement over the previous and current Finnish geoid models.

Regional geoid computation by least squares modified Hotine’s formula with additive corrections

Geoid and quasigeoid modelling from gravity anomalies by the method of least squares modification of Stokes's formula with additive corrections is adapted for the usage with gravity disturbances an ...

ASSESSMENT OF MARINE GEOID MODELS BY SHIP-BORNE GNSS PROFILES

Even though the entire Baltic Sea is included in previous geoid modelling projects such as the NKG2015 and EGG07, the accuracy of contemporary geoid models over marine areas remains unknown, presumably being offshore around 15–20 cm. An important part of the international cooperation project FAMOS (Finalising Surveys for the Baltic Motorways of the Sea) efforts is conducting new marine gravity observations for improving gravimetric quasigeoid modelling. New data is essential to the project as the existing gravimetric data over some regions of the Baltic Sea may be inaccurate and insufficiently scarce for the purpose of 5 cm accuracy geoid modelling. Therefore, it is important to evaluate geoid modelling outcome by independent data, for instance by shipborne GNSS measurements. Accordingly, this study presents results of the ship-borne marine gravity and GNSS campaign held on board the Estonian Maritime Administration survey vessel “Jakob Prei” in West-Estonian archipelago in June/July 2016. Emphasis of the study is on principles of using the GNSS profiles for validation of existing geoid models, post-processing of GNSS raw data and low-pass filtering of the GNSS results. Improvements in geoid modelling using new gravimetric data are also discussed. For example, accuracy of geoid models including the new marine gravity data increased 11 mm as assessed from GNSS profiles. It is concluded that the marine GNSS profiles have a potential in providing complementary constraints in problematic geoid modelling areas.

Assessment of the accuracy of local quasigeoid modelling using the GGI method: case study for the area of Poland

We present results of analyses of the evaluation of the method for local quasigeoid modelling based on the geophysical inversion of gravity data (the GGI method). Relying on precise (with an accuracy level of ± 1 cm) GNSS/levelling height anomalies and surface gravity data, we analysed the method accuracy by a) assessing the accuracy of the model which is based on selected available global geopotential model; b) evaluating the influence of the accuracy of the input data (GNSS/levelling and gravity) on the output model accuracy; and c) analysing the influence of the global model resolution on the quasigeoid model accuracy. Analyses were performed for the area of Poland. As the basic accuracy parameter the standard deviation (σΔζ) of the differences between measured (GNSS/levelling) height anomalies and those established from the model were selected. The results indicate that the accuracy of the quasigeoid output model (in terms of σΔζ) was evaluated at cca ± 1.2 cm irrespective of the global geopotential model used. Such accuracy was also achieved using the global model coefficients only up to the degree Nmax = 90. To achieve the above-mentioned accuracy of the output model, the accuracy of input data should not be lower than cca ± 2.0 cm for GNSS/levelling height anomalies and ± 1.3 mGal for gravity data.

On the computation of reliable formal uncertainties in the densification of GPS-levelling networks by least-squares collocation

A realization of a height system covering the south of Norway has been performed, based on least-squares collocation applied to differences between geometric and gravimetric quasigeoid heights, inhomogeneous and isotropic covariance modelling, and without prior information on the error sources of the involved data types. As a result, the derived normal heights were biased by the systematic errors of the GPS-levelling network. The important covariance properties were determined at every location from spatially differenced observations, and made it straightforward to evaluate the uncertainties of the biased height reference. The distribution of predictions followed a Gaussian shape, but extreme realizations were overrepresented.

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.

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. ► Lateral density variation in the geoid can be reach up to decimeter level in Iran. ► Unification of height systems by normal height is useful for optimal adjustment of heights in Iran. ► Using of normal heights system is useful for tectonic deformation analysis. ► The least-squares modification of Stokes' formula gives the optimum results in quasi-geoid modelling in Iran.

The quasigeoid modelling in New Zealand using the boundary element method

We compile a quasigeoid model at the study area of New Zealand using theboundary element method (BEM). The direct BEM formulation for the Laplace equationis applied to obtain a numerical solution to the linearized fixed gravimetric boundaryvalueproblem in points at the Earth’s surface. The numerical scheme uses the collocationmethod with linear basis functions. It involves a discretisation of the Earth’s surfacewhich is considered as a fixed boundary. The surface gravity disturbances represent theoblique derivative boundary condition. The geocentric positions of the collocation pointsare determined combining the digital elevation data and the a priori quasigeoid model(onshore) and the mean sea surface topography (offshore). In our numerical realization,we use the global elevation data from SRTM30PLUS V5.0, the detailed DTM of NewZealand, the EGM2008 quasigeoid heights, and the mean sea surface topography fromthe DNSC08 marine database. The gravity disturbances are computed using two heterogeneousgravity data sets: the altimetry-derived gravity anomalies from the DNSC08gravity database (offshore) and the observed ground gravity anomalies from the GNSScience gravity database (onshore). The transformation of gravity anomalies to gravitydisturbances is realized using the quasigeoid heights calculated from the EGM2008 globalgeopotential model. The new experimental quasigeoid model NZQM2010 is compiled atthe study area of New Zealand bounded by the parallels of 34 and 47.5 arc-deg southernlatitude and the meridians of 166 and 179 arc-deg eastern longitude. The least-squaresanalysis is applied to combine the gravimetric solution with GPS-levelling data using a7-parameter model. NZQM2010 is validated using GPS-levelling data and compared withthe existing regional and global quasigeoid models NZGeoid2009 and EGM2008.&nbsp

On the non-uniqueness of local quasi-geoids computed from terrestrial gravity anomalies

We consider the problem of local (quasi-)geoid modelling from terrestrial gravity anomalies. Whereas this problem is uniquely solvable (up to spherical harmonic degree one) if gravity anomalies are globally available, the problem is non-unique if gravity anomalies are only available within a local area, which is the typical situation in local/regional gravity field modelling. We derive a mathematical description of the kernel of the gravity anomaly operator. The non-uniqueness can be removed using external height anomaly information, e.g., provided by GPS-levelling. The corresponding problem is formulated as a Cauchy problem for the Laplace equation. The existence and uniqueness of the solution of the Cauchy problem is guaranteed by the Cauchy–Kowalevskaya theorem. We propose several numerical procedures to compute the solution of the Cauchy problem from given differences between gravimetric and geometric height anomalies. We apply the numerical techniques to real data over the Netherlands and Germany. We show that we can compute a unique quasi-geoid from observed gravimetric and geometric height anomalies, which agree with the data within the expected noise level. We conclude that observed differences between gravimetric height anomalies and geometric height anomalies derived from GPS and levelling cannot only be attributed to systematic errors in the data sets, but are also caused by the intrinsic non-uniqueness of the problem of local quasi-geoid modelling from gravity anomalies. Hence, GPS-levelling data are necessary to get a unique solution, which also implies that they should not be used to validate local quasi-geoid solutions computed on the basis of gravity anomalies.

DEM-induced errors in developing a quasi-geoid model for Africa

Errors in digital elevation models (DEMs) will introduce errors in geoid and quasi-geoid models, via their use in interpolating free-air gravity anomalies and (in the case of the quasi-geoid) their use in computing the Molodensky G1 term. The effects of these errors and those of datum shifts are assessed using three independent DEMs for a test region in South Africa. It is shown that these effects are significant and that it is important to choose the best-possible DEM for use in geoid and quasi-geoid modelling.