Quasigeoid Heights Research Articles

The use of absolute gravity data for the validation of Global Geopotential Models and for improving quasigeoid heights determined from satellite-only Global Geopotential Models

The objective of this paper is to demonstrate the usefulness of absolute gravity data for the validation of Global Geopotential Models (GGMs). It is also aimed at improving quasigeoid heights determined from satellite-only GGMs using absolute gravity data. The area of Poland, as a unique one, covered with a homogeneously distributed set of absolute gravity data, has been selected as a study area.The gravity anomalies obtained from GGMs were validated using the corresponding ones determined from absolute gravity data. The spectral enhancement method was implemented to overcome the spectral inconsistency in data being validated. The quasigeoid heights obtained from the satellite-only GGM as well as from the satellite-only GGM in combination with absolute gravity data were evaluated with high accuracy GNSS/levelling data.Estimated accuracy of gravity anomalies obtained from GGMs investigated is of 1.7 mGal. Considering omitted gravity signal, e.g. from degree and order 101 to 2190, satellite-only GGMs can be validated at the accuracy level of 1 mGal using absolute gravity data. An improvement up to 59% in the accuracy of quasigeoid heights obtained from the satellite-only GGM can be observed when combining the satellite-only GGM with absolute gravity data.

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.

Improvement of the accuracy of the quasigeoid model VIGAC2017

As mentioned in (Ha Minh Hoa, 2017), a national spatial reference system will be constructed based on a highly accurate national quasigeoid model with accuracy more than 4 cm. In Vietnam at the present stage there isn’t a detailed gravimetric measurement in mountainous regions and marine area. So with the purpose of improvement of accuracy of the national quasigeoid model VIGAC2017, we only can solve the task of fitting this model to national quasigeoid heights obtained from heights GPS/first, second orders levelling quasigeoid heights through least squares collocation.This scientific article will introduce a first research result for improvement of accuracy of the quasigeoid model VIGAC2017 on the base of it’s fitting to 194 national quasigeoid heights by the least squares collocation. Research results show that accuracy of the quasigeoid model VIGAC2017 will be obtained at level of ±0.058 m and increased to 20.69 %.ReferencesCressie N.A.C., 1993. Statistics for spatial data, John Wiley & Sons. New York, 900p.Ha Minh Hoa, et al., 2012. Research scientific base for perfection of the height system in connection with construction of national dynamic reference system. General report of the science - technological teme of the Ministry of Natural Resources and Environment, Hanoi, 247p.Ha Minh Hoa (Editor), 2016. Research for determination of normal surfaces of sea levels (“zero” depth surface, mean sea surface, highest sea surface) by methods of geodesy, hydrography and geology with serving construction of buildings and planning of coastline in tendency of climate changes”. State techno - scientific theme with code KC.09.19/11-15 in period of 2011-2015, Vietnam Ministry of Science and Technology, Hanoi, 563p. Huang J., Véronneau M., 2013. Contribution of the GRACE and GOCE models to a geopotential - based geodetich vertical datum in Canada. Geophysical Research Abstracts, 15, EGU2013-10164.Iliffe J.C., Ziebart M., Cross P.A., Forsberg R., Strykowski G., Tscherning C.C., 2003. OSGM02: A New model for converting GPS-derived heights to local height datums in Great Britain and Ireland. Survey Review, 37(290), 276-293.Marcin Ligas, Marek Kulczycki, 2014. Kriging approch for local height transformations. J, Geodesy And Cartography, Polish Academy of Sciences, 63(1), 5-37, Doi: 10,2478/geocart-2014-0002.Metin Soycan, 2014. Improving EGM2008 by GPS and leveling data at local scale. BCG - Boletin de Ciências Geodésicas Sec, Artigos, Curitiba, 20(1), 3-18, on - lineversion, ISSN 1982-2170. Doi,org/10,1590/S1982-21702014000100001.Moritz H., 1980. Advanced Physical Geodesy. Herbert wichmann Verlag Karlsruhe, Abacus Press Tunbridge Wells Ken, 512p. Quasigeoid of the Federal Republic of Germany GCG2016. Federal Agency for Cartography and Geodesy, www,geodatenzentrum.de.Roman D.R., Wang Y.M., Saleh J., Li X., 2010. Geodesy, Geoids & Vertical Datums: A Perspective from the U,S, National Geodetic Survey. FIG Congress 2010, Sydney, Australia, April 2010, 11-16.Schabenger O., Gotway C.A., 2005. Statistical methods for spatial data analysis. Chapman & Hall/CRC, New York, ISBN 1-58488-322-7, 488p. Smirnov N.V., Belugin D.A., 1969. Probability theory and mathematical statistics in applying to geodesy. Moscow, Nedra, 379p.

Local gravity field modeling using spherical radial basis functions and a genetic algorithm

Spherical Radial Basis Functions (SRBFs) can express the local gravity field model of the Earth if they are parameterized optimally on or below the Bjerhammar sphere. This parameterization is generally defined as the shape of the base functions, their number, center locations, bandwidths, and scale coefficients. The number/location and bandwidths of the base functions are the most important parameters for accurately representing the gravity field; once they are determined, the scale coefficients can then be computed accordingly. In this study, the point-mass kernel, as the simplest shape of SRBFs, is chosen to evaluate the synthesized free-air gravity anomalies over the rough area in Auvergne and GNSS/Leveling points (synthetic height anomalies) are used to validate the results. A two-step automatic approach is proposed to determine the optimum distribution of the base functions. First, the location of the base functions and their bandwidths are found using the genetic algorithm; second, the conjugate gradient least squares method is employed to estimate the scale coefficients. The proposed methodology shows promising results. On the one hand, when using the genetic algorithm, the base functions do not need to be set to a regular grid and they can move according to the roughness of topography. In this way, the models meet the desired accuracy with a low number of base functions. On the other hand, the conjugate gradient method removes the bias between derived quasigeoid heights from the model and from the GNSS/leveling points; this means there is no need for a corrector surface. The numerical test on the area of interest revealed an RMS of 0.48mGal for the differences between predicted and observed gravity anomalies, and a corresponding 9cm for the differences in GNSS/leveling points.

Impact of accounting for coloured noise in radar altimetry data on a regional quasi-geoid model

We study the impact of an accurate computation and incorporation of coloured noise in radar altimeter data when computing a regional quasi-geoid model using least-squares techniques. Our test area comprises the Southern North Sea including the Netherlands, Belgium, and parts of France, Germany, and the UK. We perform the study by modelling the disturbing potential with spherical radial base functions. To that end, we use the traditional remove-compute-restore procedure with a recent GRACE/GOCE static gravity field model. Apart from radar altimeter data, we use terrestrial, airborne, and shipboard gravity data. Radar altimeter sea surface heights are corrected for the instantaneous dynamic topography and used in the form of along-track quasi-geoid height differences. Noise in these data are estimated using repeat-track and post-fit residual analysis techniques and then modelled as an auto regressive moving average process. Quasi-geoid models are computed with and without taking the modelled coloured noise into account. The difference between them is used as a measure of the impact of coloured noise in radar altimeter along-track quasi-geoid height differences on the estimated quasi-geoid model. The impact strongly depends on the availability of shipboard gravity data. If no such data are available, the impact may attain values exceeding 10 centimetres in particular areas. In case shipboard gravity data are used, the impact is reduced, though it still attains values of several centimetres. We use geometric quasi-geoid heights from GPS/levelling data at height markers as control data to analyse the quality of the quasi-geoid models. The quasi-geoid model computed using a model of the coloured noise in radar altimeter along-track quasi-geoid height differences shows in some areas a significant improvement over a model that assumes white noise in these data. However, the interpretation in other areas remains a challenge due to the limited quality of the control data.

On the Accuracy Assessment of the Latest Releases of GOCE Satellite-Based Geopotential Models with EGM2008 and Terrestrial GPS/Levelling and Gravity Data over Egypt

The Global Geopotential Models (GGMs) of GOCE (Gravity Recovery and steady- state Ocean Circulation Explorer) differ globally as well as regionally in their accuracy and resolution based on the maximum degree and order (d/o) of the fully normalized spherical harmonic (SH) coefficients, which express each GGM. The main idea of this study is to compare the free-air gravity anomalies and quasi geoid heights determined from several recent GOCE-based GGMs with the corresponding ones from the Earth Gravitational Model 2008 (EGM2008) over Egypt on the one hand and with ground-based measurements on the other hand. The results regarding to the comparison of GOCE-based GGMs with terrestrial gravity and GPS/levelling data provide better improvement with respect to EGM2008. The 4th release GOCE-based GGM developed with the use of space-wise solution strategy (SPW_R4) approximates the gravity field well over the Egyptian region. The SPW_R4 model is accordingly suggested as a reference model for recovering the long wavelength (up to SH d/o 200) components of quasi geoid heights when modelling the gravimetric quasi-geoid over the Egypt. Finally, three types of transformation models: Four-, Five- and Seven-parameter transformations have been applied to reduce the data biases and to provide a better fitting of quasi geoid heights obtained from the studied GOCE-based GGMs to those from GPS/levelling data. These models reveal that the standard deviation of vertical datum over Egypt is at the level of about 32 cm.

Инновационный подход к построению опорных геодезических сетей линейных объектов

Abstract: An innovative concept of constructing geodetic control networks at the stage of engineering surveys for the design and construction of linear facilities large extent on the widespread use of continuously operating reference base stations is outlined. The coordinates of them are defined in the same for the whole linear object of the coordinate space. To minimize the distortion of angles and distances in the depiction of the earth’s surface on topographic plans rectangular coordinate system on a plane oblique cylindrical Mercator projection in the version by Khotin is offered. The Central line of the projection is proposed to set so that the distortion of the scale along the route was presented in small quantities which in practice can be neglected. The heights of the points are requested to be obtained from the conversion of the spatial geodetic coordinates based on local models of exactly quasigeoid heights. The technique of constructing a local model of the anomalies of the altitudes along the route of the object was shown.

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.

Assessment of the direct inversion scheme for the quasigeoid modeling based on applying the Levenberg–Marquardt algorithm

Methods for a spherical harmonic analysis and synthesis are often used in describing the global gravity field by means of spherical harmonics to a certain degree of spectral resolution. In local gravity field modeling, the radial basis functions (RBFs) can be used to parameterize the gravity field with a high spatial/spectral resolution because of their local support. Since the global integration is required in both cases, these two methods are somehow complementary. In physical geodesy, this is done practically by adopting a remove-compute-restore numerical scheme. According to this scheme, a long-wavelength part of the gravity spectrum is defined by means of the spherical harmonics while a residual (high-frequency) gravity contribution is treated based on using the RBFs. To find an optimal parameterization of the gravity field, the number of RBFs and their spatial configuration are optional. Moreover, the regularization is applied to stabilize the (ill-posed) gravity inversion. In this study, we utilize the Levenberg–Marquardt (LM) algorithm for finding the optimal number of RBFs and their 3-D spatial configuration (i.e., horizontal location and depth) simultaneously with a regularization parameter. The optimal choice of these parameters is based on minimizing the least-squares residuals between the predicted and observed values of the potential and gravity field. In numerical experiment, we apply the LM algorithm in two inverse schemes of solving the Molodensky’s problem. The results reveal that a direct inversion of the observed gravity data to the potential field (i.e., quasigeoid heights) yields a systematic bias. Moreover, this gravimetric solution has a low approximation quality in terms of the potential field. The systematic discrepancies are thus modeled and corrected for by combining the gravity and GPS-leveling data. We propose and apply the RBF-parameterization, which again utilizes the LM optimization algorithm in combining these data. We demonstrate that only a two-step approach provides a satisfactory approximation to both the gravity and potential field.

A novel scheme for solving the oblique derivative boundary-value problem

The paper is devoted to a novel scheme for solving the boundary-value problem (BVP) with the oblique derivative boundary condition (BC). In this approach, the oblique derivative in the BC is decomposed into its normal and two tangential components which are approximated by means of numerical solution values. Then the numerical scheme by the finite volume method is developed and testing numerical experiments are done. The obtained numerical solutions are compared to the exact one to show that the proposed method is second order accurate. Afterwards, the algorithm is applied to solving the fixed gravimetric BVP, namely, the numerical solution is sought in a domain bounded by a part of the Earth’s surface (Himalaya region or Slovakia), four side boundaries and a corresponding upper boundary at the satellite level. On the Earth’s surface, the oblique derivative BC in the form of surface gravity disturbances from the EGM2008, DTU10-GRAV or the detailed gravity mapping is taken into account. On the upper and side boundaries, the Dirichlet BC from the EGM2008 or GOCO03s is applied. The disturbing potential as a direct numerical result is compared with the solution to the more common BVP with the Neumann BC considered on the Earth’s surface. All numerical experiments show better agreement of the solution to the BVP with the oblique derivative BC than solution to the BVP with the Neumann BC in comparison with the disturbing potential obtained by a different mathematical approach. In area of Slovakia, when applying the GPS/levelling test at 61 points, we have gained 1.7 cm improvement in favour of the standard deviation of residuals of quasigeoidal heights obtained by solving the BVP with the oblique derivative BC in comparison with the solution to BVP with the Neumann BC.

On the effect of planar approximation in the Geodetic Boundary Value Problem

In the framework of the analytical treatment of the Geodetic Boundary Value Problem (GBVP) various approximations are necessary in order to end up with a solution in a “closed” form. Besides linearization and isotropic (“spherical”) approximation the socalled planar approximation has been introduced in the case of a non-spherical, topographic boundary surface. In this context the term “planar approximation” refers to neglecting terms proportional to the quotient of elevation and mean Earth radius in the respective kernel functions, not to confuse with approximating the boundary surface by a plane. In this paper the effect of planar approximation on the boundary condition of the fixed GBVP and on the disturbing potential is estimated. A global study is presented on the basis of synthesized gravity disturbances computed from the global geopotential model EGM2008 and the topography model DTM2006.0. A regional densification is carried out for the area of the Andes; therefore the gravity disturbances are refined by RTM-effects on the basis of the SRTM3 topography model. This analysis which is valid for the scalar free and the fixed versions of the GBVP simultaneously, is based on high-resolution digital elevation models (DEM) and shows that the numerical estimates of the effects of planar approximation depend on the resolution of the DEMs. The numerical investigations confirm the assumed relative error of the order 0.1%. This assumption implies a maximum effect of approximately 0.5 mGal for the boundary values and 10 cm for the quasigeoidal heights.

Evaluation of the third- and fourth-generation GOCE Earth gravity field models with Australian terrestrial gravity data in spherical harmonics

In March 2013, the fourth generation of European Space Agency’s (ESA) global gravity field models, DIR4 (Bruinsma et al. in Proceedings of the ESA living planet symposium, 28 June–2 July, Bergen, ESA, Publication SP-686, 2010b) and TIM4 (Migliaccio et al. in Proceedings of the ESA living planet symposium, 28 June–2 July, Bergen, ESA, Publication SP-686, 2010), generated from the Gravity field and steady-state Ocean Circulation Explorer (GOCE) gravity observation satellite was released. We evaluate the models using an independent ground truth data set of gravity anomalies over Australia. Combined with Gravity Recovery and Climate Experiment (GRACE) satellite gravity, a new gravity model is obtained that is used to perform comparisons with GOCE models in spherical harmonics. Over Australia, the new gravity model proves to have significantly higher accuracy in the degrees below 120 as compared to EGM2008 and seems to be at least comparable to the accuracy of this model between degree 150 and degree 260. Comparisons in terms of residual quasi-geoid heights, gravity disturbances, and radial gravity gradients evaluated on the ellipsoid and at approximate GOCE mean satellite altitude ( $$h=250$$ km) show both fourth generation models to improve significantly w.r.t. their predecessors. Relatively, we find a root-mean-square improvement of 39 % for the DIR4 and 23 % for TIM4 over the respective third release models at a spatial scale of 100 km (degree 200). In terms of absolute errors, TIM4 is found to perform slightly better in the bands from degree 120 up to degree 160 and DIR4 is found to perform slightly better than TIM4 from degree 170 up to degree 250. Our analyses cannot confirm the DIR4 formal error of 1 cm geoid height (0.35 mGal in terms of gravity) at degree 200. The formal errors of TIM4, with 3.2 cm geoid height (0.9 mGal in terms of gravity) at degree 200, seem to be realistic. Due to combination with GRACE and SLR data, the DIR models, at satellite altitude, clearly show lower RMS values compared to TIM models in the long wavelength part of the spectrum (below degree and order 120). Our study shows different spectral sensitivity of different functionals at ground level and at GOCE satellite altitude and establishes the link among these findings and the Meissl scheme (Rummel and van Gelderen in Manusrcipta Geodaetica 20:379–385, 1995).

Study of the anomalous gravity field in the arctic based on modern geopotential models

The results of modeling the anomalous gravity field of the Earth in the high-latitude regions, which are of interest in the context of Arctic exploration, are analyzed. The structure and accuracy characteristics of quasi-geoid heights (QGH) and gravity anomalies (GAs) in the Arctic are studied. The analysis addresses fifteen modern Russian and foreign spherical harmonic models of the gravitational potential of the Earth, including the EGM-2008 model, which provides high-order spherical harmonic representations, and the first models based on the data of satellite gradiometry.

Quasigeoid-to-geoid determination by EGM08

We present a method to estimate the difference between quasigeoid and geoid heights globally from the Earth Gravitational Model EGM08 and a related topographic model. The numerical computations with the standard topographic density of 2.67 g/cm3 show that the maximum and minimum of the separations are estimated to 5.47 m and −0.11 m on the Tibet plateau and in the Indian Ocean, respectively. These estimates do not consider possible topographic density variations, which result in topographic bias changes proportional to the topographic elevation squared. Assuming such density changes of 10% from the standard value, the separation may change up to 5 dm.

Efficient and accurate high-degree spherical harmonic synthesis of gravity field functionals at the Earth’s surface using the gradient approach

Spherical harmonic synthesis (SHS) of gravity field functionals at the Earth’s surface requires the use of heights. The present study investigates the gradient approach as an efficient yet accurate strategy to incorporate height information in SHS at densely spaced multiple points. Taylor series expansions of commonly used functionals quasigeoid heights, gravity disturbances and vertical deflections are formulated, and expressions of their radial derivatives are presented to arbitrary order. Numerical tests show that first-order gradients, as introduced by Rapp (J Geod 71(5):282–289, 1997) for degree 360 models, produce cm- to dm-level RMS approximation errors over rugged terrain when applied with EGM2008 to degree 2190. Instead, higher-order Taylor expansions are recommended that are capable of reducing approximation errors to insignificance for practical applications. Because the height information is separated from the actual synthesis, the gradient approach can be applied along with existing highly efficient SHS routines to compute surface functionals at arbitrarily dense grid points. This confers considerable computational savings (above or well above one order of magnitude) over conventional point-by-point SHS. As an application example, an ultra-high resolution model of surface gravity functionals (EurAlpGM2011) is constructed over the entire European Alps that incorporates height information in the SHS at 12,000,000 surface points. Based on EGM2008 and residual topography data, quasigeoid heights, gravity disturbances and vertical deflections are estimated at ~200m resolution. As a conclusion, the gradient approach is efficient and accurate for high-degree SHS at multiple points at the Earth’s surface.

A study of regional gravity field recovery from GOCE vertical gravity gradient data in the Auvergne test area using collocation

Gravity field and steady-state Ocean Circulation Explorer (GOCE) is the first satellite mission that observes gravity gradients from the space, to be primarily used for the determination of high precision global gravity field models. However, the GOCE gradients, having a dense data distribution, may potentially provide better predictions of the regional gravity field than those obtained using a spherical harmonic Earth Geopotential Model (EGM). This is investigated in Auvergne test area using Least Squares Collocation (LSC) with GOCE vertical gravity gradient anomalies (Tzz), removing the long wavelength part from EGM2008 and the short wavelength part by residual terrain modelling (RTM). The results show that terrain effects on the vertical gravity gradient are significant at satellite altitude, reaching a level of 0.11 Eotvos unit (E.U.) in the mountainous areas. Removing the RTM effects from GOCE Tzz leads to significant improvements on the LSC predictions of surface gravity anomalies and quasigeoid heights. Comparison with ground truth data shows that using LSC surface free air gravity anomalies and quasi-geoid heights are recovered from GOCE Tzz with standard deviations of 11 mGal and 18 cm, which is better than those obtained by using GOCE EGMs, demonstrating that information beyond the maximal degree of the GOCE EGMs is present. Investigation of using covariance functions created separately from GOCE Tzz and terrestrial free air gravity anomalies, suggests that both covariance functions give almost identical predictions. However, using covariance function obtained from GOCE Tzz has the effect that the predicted formal average error estimates are considerably larger than the standard deviations of predicted minus observed gravity anomalies. Therefore, GOCE Tzz should be used with caution to determine the covariance functions in areas where surface gravity anomalies are not available, if error estimates are needed.

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.

Ellipsoidal area mean gravity anomalies — precise computation of gravity anomaly reference fields for remove-compute-restore geoid determination

Gravity anomaly reference fields, required e.g. in remove-compute-restore (RCR) geoid computation, are obtained from global geopotential models (GGM) through harmonic synthesis. Usually, the gravity anomalies are computed as point values or area mean values in spherical approximation, or point values in ellipsoidal approximation. The present study proposes a method for computation of area mean gravity anomalies in ellipsoidal approximation (‘ellipsoidal area means’) by applying a simple ellipsoidal correction to area means in spherical approximation. Ellipsoidal area means offer better consistency with GGM quasigeoid heights. The method is numerically validated with ellipsoidal area mean gravity derived from very fine grids of gravity point values in ellipsoidal approximation. Signal strengths of (i) the ellipsoidal effect (i.e., difference ellipsoidal vs. spherical approximation), (ii) the area mean effect (i.e., difference area mean vs. point gravity) and (iii) the ellipsoidal area mean effect (i.e., differences between ellipsoidal area means and point gravity in spherical approximation) are investigated in test areas in New Zealand and the Himalaya mountains. The impact of both the area mean and the ellipsoidal effect on quasigeoid heights is in the order of several centimetres. The proposed new gravity data type not only allows more accurate RCR-based geoid computation, but may also be of some value for the GGM validation using terrestrial gravity anomalies that are available as area mean values.

Evaluation of the first GOCE static gravity field models using terrestrial gravity, vertical deflections and EGM2008 quasigeoid heights

Recently, four global geopotential models (GGMs) were computed and released based on the first 2 months of data collected by the Gravity field and steady-state Ocean Circulation Explorer (GOCE) dedicated satellite gravity field mission. Given that GOCE is a technologically complex mission and different processing strategies were applied to real space-collected GOCE data for the first time, evaluation of the new models is an important aspect. As a first assessment strategy, we use terrestrial gravity data over Switzerland and Australia and astrogeodetic vertical deflections over Europe and Australia as ground-truth data sets for GOCE model evaluation. We apply a spectral enhancement method (SEM) to the truncated GOCE GGMs to make their spectral content more comparable with the terrestrial data. The SEM utilises the high-degree bands of EGM2008 and residual terrain model data as a data source to widely bridge the spectral gap between the satellite and terrestrial data. Analysis of root mean square (RMS) errors is carried out as a function of (i) the GOCE GGM expansion degree and (ii) the four different GOCE GGMs. The RMS curves are also compared against those from EGM2008 and GRACE-based GGMs. As a second assessment strategy, we compare global grids of GOCE GGM and EGM2008 quasigeoid heights. In connection with EGM2008 error estimates, this allows location of regions where GOCE is likely to deliver improved knowledge on the Earth’s gravity field. Our ground truth data sets, together with the EGM2008 quasigeoid comparisons, signal clear improvements in the spectral band ~160–165 to ~180–185 in terms of spherical harmonic degrees for the GOCE-based GGMs, fairly independently of the individual GOCE model used. The results from both assessments together provide strong evidence that the first 2 months of GOCE observations improve the knowledge of the Earth’s static gravity field at spatial scales between ~125 and ~110 km, particularly over parts of Asia, Africa, South America and Antarctica, in comparison with the pre-GOCE-era.

Quality Estimates in Geoid Computation by EGM08

Quality Estimates in Geoid Computation by EGM08The high-degree Earth Gravitational Model EGM08 allows for geoid determination with a resolution of the order of 5'. Using this model for estimating the quasigeoid height, we estimate the global root mean square (rms) commission error to 5 and 11 cm, based on the assumptions that terrestrial gravity contributes to the model with an rms standard error of 5 mGal and correlation length 0:01° and 0:1°, respectively. The omission error is estimated to—0:7Δg [mm], where Δg is the regional mean gravity anomaly in units of mGal.In case of geoid determination by EGM08, the topographic bias must also be considered. This is because the Earth's gravitational potential, in contrast to its spherical harmonic representation by EGM08, is not a harmonic function at the geoid inside the topography. If a correction is applied for the bias, the main uncertainty that remains is that from the uncertainty in the topographic density, which will still contribute to the overall geoid error.