External calibration of GOCE SGG data with terrestrial gravity data: A simulation study

<p>In this study, we investigate the optimal combination of local gravity observations and their contributions to the regional quasi-geoid model. The study area is located in Colorado, USA, with two types of regional data sets, namely terrestrial gravity data and airborne gravity data, available within the “1 cm geoid experiment”. The approach based on series expansions in terms of spherical radial basis functions (SRBF) is applied, which has been developed at DGFI-TUM in the last two decades. We use two different types of basis functions covering the same spectral domain separately for the terrestrial and the airborne measurements. The Shannon function is applied to the terrestrial data, and the Cubic Polynomial (CuP) function which has smoothing features is applied to the airborne data for filtering their high-frequency noise.</p><p>To assess the contributions of the regional terrestrial and airborne gravity data to the final quasi-geoid model, four solutions are compared, namely the combined solution, the terrestrial only, the airborne only, and finally the model only solution, i.e., only the global gravity model and the topographic model are used without any gravity data from regional measurements. By adding the terrestrial data to the GGM and the topographic model, the RMS error of the quasi-geoid model w.r.t the validation data (the mean solution of independent computations delivered by fourteen institutions from all over the world) drops from 4 to 1.8 cm, and it is further reduced to 1 cm by including the airborne data.</p>

This study presents a recent combined regional gravity field model over Egypt, developed by integrating satellite and terrestrial data via applying the remove-compute-restore (RCR) principle and the least-squares collocation (LSC) procedure. A high-resolution digital terrain model was exploited for the computation of the terrain and residual terrain corrections. Hereby, all the signals that can be modelled or deterministically computed are considered known and then removed in order to reduce the order of magnitude of the input gravity data prior to applying the LSC. Several GOCE-only and combined global geopotential models (GGMs) have been thoroughly investigated with respect to the EGM2008, in which the space-wise (SPW) solution, namely the SPW-R5 model, demonstrated the best performance. For the development of the combined model, the SPW-R5 GGM has been integrated with both the EGM2008 GGM and the terrestrial data retrieved from 56,250 gravity stations of the Getech data, acquired in the framework of the African Gravity Project. The combined regional gravity model was compared to the state-of-art XGM2016 global gravity model. The standard deviation of the differences is 18.0 mGal in terms of Bouguer anomalies. The combined regional model fits well with the terrestrial gravity data along the chosen North–South oriented profile through the Nile Delta region. The improvements of the developed combined regional model over the XGM2016 are due to the use of a more extensive terrestrial dataset. In conclusion, our model is more suitable than solely using the ground data or GGMs for regional density modelling over Egypt. As an example, the comparison of using a global or regionally defined gravity model with the forward gravity modelling based on Saleh (Acta Geodaetica et Geophysica Hungarica 47(4):402–429, 2012) density model is performed.

To estimate Moho depth, geoid, gravity anomaly, and other geopotential functionals, gravity data is needed. But, gravity survey was not collected in equal distribution in Ethiopia, as the data forming part of the survey were mainly collected on accessible roads. To determine accurate Moho depth using Global Gravity Models (GGMs) for the study area, evaluation of GGMs is needed based on the available terrestrial gravity data. Moho depth lies between 28 km and 32 km in Afar. Gravity disturbances (GDs) were calculated for the terrestrial gravity data and the recent GGMs for the study area. The model-based GDs were compared with the corresponding GD obtained from the terrestrial gravity data and their differences in terms of statistical comparison parameters for determining the best fit GGM at a local scale in Afar. The largest standard deviation (SD) (36.10 mGal) and root mean square error (RMSE) (39.00 mGal) for residual GD and the lowest correlation with the terrestrial gravity (0.61 mGal) were obtained by the satellite-only model (GO_CONS_GCF_2_DIR_R6). The next largest SD (21.27 mGal) and RMSE (25.65 mGal) for residual GD were obtained by the combined gravity model (XGM2019e_2159), which indicates that it is not the best fit model for the study area as compared with the other two GGMs. In general, the result showed that the combined models are more useful tools for modeling the gravity field in Afar than the satellite-only GGMs. But, the study clearly revealed that for the study area, the best model in comparison with the others is the EGM2008, while the second best model is the EIGEN6C4.

This paper studies the contribution of airborne gravity data to improvement of gravimetric geoid modelling across the mountainous area in Colorado, USA. First, airborne gravity data was processed, filtered, and downward-continued. Then, three gravity anomaly grids were prepared; the first grid only from the terrestrial gravity data, the second grid only from the downward-continued airborne gravity data, and the third grid from combined downward-continued airborne and terrestrial gravity data. Gravimetric geoid models with the three gravity anomaly grids were determined using the least-squares modification of Stokes’ formula with additive corrections (LSMSA) method. The absolute and relative accuracy of the computed gravimetric geoid models was estimated on GNSS/levelling points. Results exhibit the accuracy improved by 1.1 cm or 20% in terms of standard deviation when airborne and terrestrial gravity data was used for geoid computation, compared to the geoid model computed only from terrestrial gravity data. Finally, the spectral analysis of surface gravity anomaly grids and geoid models was performed, which provided insights into specific wavelength bands in which airborne gravity data contributed and improved the power spectrum.

<p>The maintenance of leveling benchmark is both laborious and costly due to distortions caused by geodynamic activities and local deformations. It is necessary to realize geoid-based vertical datum, which also enables calculation from ellipsoidal heights obtained from GNSS to orthometric heights that have physical meaning. It can be considered as an important step for height system unification as it eliminates the problems stem from the conventional vertical datum. The ongoing height modernization efforts in Turkey focus to improve quality and coverage of the gravity data, eliminate errors in existing terrestrial gravity measurements in order to achieve a precise geoid model. Accuracy of the geopotential model is crucial while realizing a geoid model based vertical datum as well as unifying the regional height systems with the International Heights Reference System. In this point of view, we assessed the accuracies of recently released global geopotential models including XGM2019e_2159, GECO, EIGEN-6C4, EGM2008, SGG-UGM-1, EIGEN-6C3stat, and EIGEN-6C2 using high order GNSS/leveling control benchmarks and terrestrial gravity data in Turkey. The reason for choosing these models in the validations is their relatively higher spatial resolutions and improved accuracies compared to other GGMs in published validation results with globally distributed terrestrial data. The GNSS/leveling data used in validations include high accuracy GNSS coordinates in ITRF datum with co-located Helmert orthometric heights in regional vertical datum. 100 benchmarks are homogeneously distributed in the country with the benchmarks along the coastlines. In addition, the terrestrial gravity anomalies with 5 arc-minute resolution were also used in the tests. In order to have comparable results, residual terrain effect has been restored to the GGM derived parameters. Numerical tests revealed significant differences in accuracies of the tested GGMs. The most accurate GGM has the comparable performance with official regional geoid model solutions in Turkey. The drawn results in the study were interpreted and discussed from practical applications and height system unification points in conclusion.</p>

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 Stokesian approach is formulated to update the geoid model for a specific spherical harmonic band by spectrally combining a GOCE-based satellite global geopotential model with terrestrial gravity data. A simulation test shows that the GOCE-based model can be combined into a geoid solution with an accuracy better than 3 mm for the band between degrees 90 and 180. A comparison of the GOCE-based model GOCO03S and the Canadian terrestrial gravity data for the spherical harmonic band between degrees 90 and 180 shows that the geoid update by GOCO03S reaches 1.6 cm in RMS in the Yukon Territory, 1.8 cm in northern British Columbia, and 1.6 cm in the Maritimes. This may suggest a slight improvement of the GOCE model over the Canadian gravity data considering the standard deviation of 1.0 cm given by GOCO03S. However the analysis indicates comparable accuracy between the terrestrial gravity data and GOCE models for the rest of Canada where topography is relatively flat. The comparisons at the GPS-levelling points suggest that GOCE has improved our existing knowledge of the Earth’s gravity field for wavelength components longer than 200 km over the Yukon Territory, northern British Columbia, the Maritimes, and Newfoundland.KeywordsSatellite gravity modelStokes integralTerrestrial gravity data

Current gravity data, which are used for refined studies of the Earth’s gravity field, result from cutting-edge technologies and advanced measuring techniques. They often refer to some exceptional surfaces or manifolds and can lead to interesting mathematical problems. In this paper, three cases are considered where airborne gravimetry is combined with terrestrial gravity data, airborne measurements are collected at two flight levels, and satellite gradiometry is combined with terrestrial gravity data. The three cases are studied mathematically and the disturbing potential is expressed explicitly. The spectral representation based on spherical harmonics is used. Subsequently, an optimum procedure is discussed in order to treat the overdetermined problems and to keep the typical regularity of harmonic functions at infinity. Some practical aspects are also mentioned.

Integrated gravity and gravity gradient 3-D inversion can effectively improve the resolution of inversion results. This study aims to find a better way to extract geological information from gravity and gravity gradient data and to obtain the recovered model with a higher resolution. We present three different methods for integrating gravity and gravity gradient data in inversion: directly integrating gravity and gravity gradient data (DIGG), the sequential inversion of gravity and gravity gradient data (SIGG), and integrating gravity and gravity gradient components using a weighting matrix (IGGW). In addition, we propose a new discrete method for smooth inversion. We discretize the stabilizing functional into two parts. One is the square of the 2-norm of the model parameter, and the other one is the square of the Frobenius norm of the gradient of the model parameter. We adopt a conjugate gradient algorithm (CGA) to solve the objective function. We used the DIGG, SIGG, and IGGW inversion methods in a dip-slab model with multiple anomalous bodies. The inversion results show that by integrating gravity and gravity gradient in inversion, more reasonable results are obtained than by inverting a single data type. Among the presented methods, the IGGW method achieved the best performance in using information contained in gravity and gravity gradient data, followed by the SIGG method, and finally, by the DIGG method. We used the methods on actual data from Vinton salt dome, southwestern Louisiana, USA. The obtained results support the abovementioned conclusions.

GOCE will be the first satellite ever to measure the second derivatives of the Earth’s gravitational potential in space. With these measurements it is possible to derive a high accuracy and resolution gravitational field if systematic errors have been removed to the extent possible from the data and the accuracy of the gravity gradients has been assessed. It is therefore necessary to understand the instrument characteristics and to setup a valid calibration model. The calibration parameters of this model could be determined by using GOCE data themselves or by using independent gravity field information. Also the accuracy or error assessment relies on either GOCE or independent data. We will demonstrate how state-of-the-art global gravity field models, terrestrial gravity data and observations at satellite track crossovers can be used for calibration/validation. In addition we will show how high quality terrestrial data could play a role in error assessment.

<p>The objective of this study is the combination of different types of basis functions applied separately to different kinds of gravity observations. We use two types of regional data sets: terrestrial gravity data and airborne gravity data, covering an area of about 500 km × 800 km in Colorado, USA. These data are available within the “1 cm geoid experiment” (also known as the “Colorado Experiment”). We apply an approach for regional gravity modeling based on series expansions in terms of spherical radial basis functions (SRBF). Two types of basis functions covering the same spectral domain are used, one for the terrestrial data and another one for the airborne measurements. To be more specific, the non-smoothing Shannon function is applied to the terrestrial data to avoid the loss of spectral information. The Cubic Polynomial (CuP) function is applied to the airborne data as a low-pass filter, and the smoothing features of this type of SRBF are used for filtering the high-frequency noise in the airborne data. In the parameter estimation procedure, these two modeling parts are combined to calculate the quasi-geoid.</p><p>The performance of our regional quasi-geoid model is validated by comparing the results with the mean solution of independent computations delivered by fourteen institutions from all over the world. The comparison shows that the low-pass filtering of the airborne gravity data by the CuP function improves the model accuracy by 5% compared to that using the Shannon function. This result also makes evident the advantage of combining different SRBFs covering the same spectral domain for different types of observations.</p>

Combined Global Gravitational Models (GGMs) are being used in numerous geoscience applications, most notably for gravimetric geoid modeling (in geodesy) and for geological mapping and geophysical explorations (in the Earth’s sciences). The aim of this study is to evaluate the suitability of different combined GGMs that could be used for the geological mapping of middle belt region and Southeastern Nigeria. For this purpose, we digitized geological maps of Afikpo and Anambra Basins to evaluate geological signatures implied by gravity field quantities (Bouguer gravity anomalies and vertical gravity gradient) derived from the EGM2008, EIGEN-6C4, GECO, SGG-UGM-1 and XGM2019e_2159 gravitational models. We also stochastically evaluated the performance of these GGMs by computing their Root-Mean-Square (RMS) fit with ground-based gravity measurements. The results show that the EIGEN-6C4 and XGM2019e_2159 models have the best RMS fit with the ground-based gravity data. A spatial pattern in Bouguer gravity maps (compiled using these two models) generally closely agrees with a geological configuration of the basins, while also exhibiting some more detailed geological features. Interestingly, however, despite the XGM2019e has the best fit and better mimics major geological features, the gravity image from this model does not exhibit a sediment signature in a portion of the Afikpo basin. A possible reason is that the topographic information used to recover a higher-frequency gravity spectrum of this model might suppress a gravitational signature of subsurface density structures. A comprehensive interpretation of geological features thus requires a careful analysis of existing GGMs, terrestrial gravity data as well as all other reliable geological and geophysical information.

Detailed gravity anomalies are computed from a combination of Geos 3 satellite altimeter and terrestrial gravity data using least‐squares principles. The mathematical model used is based on the Stokes' equation modified for a nonglobal solution. Using Geos 3 data in the calibration area, the effects of several anomaly parameter configurations and data densities/distributions on the anomalies and their accuracy estimates are studied. The accuracy estimates for 1° × 1° mean anomalies from low density altimetry data are of the order of 4 mgal. Comparison of these anomalies with the terrestrial data and also with Rapp's data derived using collocation techniques show rms differences of 7.2 and 4.9 mgal, respectively. Indications are that the anomaly accuracies can be improved to about 2 mgal with high density data. Estimation of 30′ × 30′ mean anomalies indicates accuracies of the order of 5 mgal. Proper verification of these results will be possible only when accurate ground truth data become available.

The continuous efforts on establishment and modernization of the geodetic control in Turkey include a number of regional geoid models that have been determined since 1976. The recently released gravimetric Geoid of Turkey, TG03, is used in geodetic applications where GPS-heights need to be converted to the local vertical datum. To reach a regional geoid model with improved accuracy, the selection of the appropriate global geopotential model is of primary importance. This study assesses the performance of a number of recent satellite-only and combined global geopotential models (GGMs) derived from CHAMP and GRACE missions’ data in comparison to the older EGM96 model, which is the underlying reference model for TG03. In this respect, gravity anomalies and geoid heights from the global geopotential models were compared with terrestrial gravity data and low-pass filtered GPS/levelling data, respectively. Also, five new gravimetric geoid models, computed by the Fast Fourier Transform technique using terrestrial gravity data and the geopotential models, were validated at the GPS/levelling benchmarks. The findings were also compared with the validation results of the TG03 model.

Validation of recent GOCE/GRACE geopotential models over Khartoum state - SudanThis paper evaluates a number of latest releases of GOCE/GRACE global geopotential models (GGMs) using the GPS-levelling geometric geoid heights, terrestrial gravity data and existing local gravimetric models. We investigate each global model at every 5 degree of spherical harmonics. Our analysis shows that the satellite-only models derived by space-wise and time-wise approaches (SPW_R1, SPW_R2 TIM_R1 and TIM_R2), GOCO01S together with EGM08 (combined model) are very distinct and consistent to the local data, which guarantees one of them to be selected as the best of candidate models and then to be utilized in our further geoid studies. One of Satellite-only models will be employed for acquiring the long wavelength geoid component which is one of major steps in the geoid determination. EGM08 will be used to compensate and restore the missing gravity data points in the un-surveyed parts within the target area. We expect further improvements in geoid studies in Sudan due to the improved medium wavelength part of the gravity field from GOCE mission.