GOCE Research Articles

Evaluation of GOCE/GRACE and combined global geopotential models using GNSS/levelling data over Nigeria

AbstractGlobal Geopotential Models (GGMs) provide valuable information about Earth’s gravity field functionals, such as geoid heights and gravity anomalies. However, ground-based datasets are required to validate these GGMs at the regional and local scales. In this study, we validated the accuracy of GGMs by comparing them with ground-based Global Navigation Satellite System (GNSS)/levelling data for the first time in Nigeria. We employed two validation scenarios: with and without considering spectral consistency using the spectral enhancement method (SEM) to incorporate high and very high frequencies of the gravity field spectrum from the combined global gravity field model (XGM2019e_2159) and the residual terrain model (RTM) derived from the Shuttle Radar Topography Mission (SRTM) data, respectively. The results of this evaluation confirmed that the application of SEM improved the assessment of the GGM solutions in an unbiased manner. Integrating XGM2019e_2159 and SRTM data to constrain the high-frequency component of geoid heights in Gravity Field and Steady-State Ocean Circulation Explorer (GOCE)-based GGMs led to an improvement of approximately 10% in reducing the standard deviation (STD) relative to when SEM was not applied. GO_CONS_GCF_2_TIM_R6 at spherical harmonics (SH) of up to degree and order 260 demonstrated the lowest STD when compared to GO_CONS_GCF_2_DIR_R6 and GO_CONS_GCF_2_SPW_R5, with a reduction from 0.380 m without SEM application to 0.342 m with SEM implementation. In addition, four transformation models, namely, linear, four-parameter, five-parameter, and seven-parameter models, were evaluated. The objective is to mitigate the reference system offsets between the GNSS/levelling data and the GGMs and to identify the particular parametric model with the smallest STD across all GGMs. This effort reduced the GGMs misfits to GNSS/levelling to 0.30 m, representing a 15.3% decrease in STD. Notably, the XGM2019e_2159 model provides this improvement.

Ionospheric TEC and its irregularities over Egypt: a comprehensive study of spatial and temporal variations using GOCE satellite data

Abstract This study delves into ionospheric characteristics during solar cycle 24 using data from the Global Positioning System (GPS) and Low Earth Orbit (LEO) satellites, GOCE. The present study fills the gap of not assessing and studying GOCE satellite data before in Egypt. Focusing on spatial and temporal variations of ionospheric Total Electron Content (TEC) over Egypt and the Middle East across a 4-year dataset, monthly average Vertical Total Electron Content (VTEC) variations are scrutinized, emphasizing extremes during heightened and reduced solar and geomagnetic activities. Results TEC typically decreases with latitude’s increase, with peak ionization during equinoxes and troughs during solstices. Notably, VTEC values consistently reach maximum values on days of heightened solar and geomagnetic activities. GOCE data is evaluated against International Reference Ionosphere (IRI) model and NeQuick2 model by selecting 10 days from all data by using a statistical comparison via t-tests. There is not significant difference between them except for two days between GOCE-IRI. The values of Root Mean Square Error (RMSE) are 3.7403 and 4.4655 for GOCE – NeQuick2 model and GOCE – IRI model, respectively. Ionospheric scintillation, signifying rapid electron density fluctuations, is assessed through amplitude scintillation index (S4), S4 proxy, and Rate Of TEC Index (ROTI). A robust correlation between S4 and S4 proxy is noted, thus scintillation can be studied by using S4 proxy instead of S4. Temporal variations indicate heightened scintillation during geomagnetic storms and peak solar activity, contrasting with reduced activity during solar minimum. Throughout the study period the maximum scintillation index values for ROTI, S4, and S4 proxy are 0.3, 0.15, and 0.05, indicating minimal scintillation. This comprehensive analysis, rooted in GOCE data, illuminates spatial and temporal dynamics of ionospheric TEC and elucidates ionospheric scintillation characteristics in the Egypt region. These findings are crucial for refining ionospheric models, improving scintillation prediction, and enhancing satellite communication and navigation systems, especially in the study region.

On equatorial spread F occurrence: A multi-dimensional quantitative assessment

The present study investigates the role of gravity wave induced seed perturbations in the occurrence of Equatorial Spread F (ESF) under the influence of the post sunset background conditions modulated by prevailing electrodynamics and neutral wind. Ionospheric foF2 data sets over geomagnetic equatorial station Trivandrum (8.5°N, 77°E and magnetic dip 0.68°N-corresponding to the period of study) corresponding to vernal and autumnal equinoctial periods encompassing high, low and moderate solar activity years, are used for the study. Meridional wind data is obtained either from ESA’s sun-synchronous satellite GOCE (Gravity field and steady-state Ocean Circulation Explorer) or derived using ionosonde h’F (base height of ionosphere at 2.5 MHz) data from Trivandrum (TVM- 8.5°N, 77°E and magnetic dip 0.68°N) and Sriharikota (SHAR−13.7°N, 80.2°E and magnetic dip 6.9°N-for period of study). This particular study is carried out for geomagnetically quiet days of Vernal Equinox (VE) and Autumnal Equinox (AE) seasons, which are most favoured for ESF occurrence over Indian longitudes. Considering thermospheric wind, ion-neutral collisions, and electric field effects in association with gravity wave seed, threshold curve is generated, which clearly demarcates ESF and NSF (Non spread F) days. Previous studies have addressed ESF variability in electrodynamical domain alone (wherein the layer is above a threshold level). The present study, for the first time, succeeds in demarcating ESF and NSF days by incorporating effects of electric field, neutral wind, collisional RT instability term, and gravity wave seed perturbations simultaneously irrespective of threshold height.

The Impact of Polar Vortex Strength on the Longitudinal Structure of the Noontime Mid-Latitude Ionosphere and Thermosphere

Ground-based ionospheric, CHAMP/STAR, and GOCE satellite neutral density ρ observations under deep solar minimum conditions were used to find whether there is a dependence of longitudinal variations on polar vortex strength. Ionospheric stations at fixed-dipole geomagnetic Φ ≈ 38° and geographic φ ≈ 40°N latitudes located in ‘near-pole’ and ‘far-from-pole’ longitudinal sectors were used in the analysis. No significant longitudinal NmF2 (electron concentration in the F2-layer maximum) dependence on the polar vortex strength was revealed. Geomagnetic control was shown to be responsible for the observed longitudinal NmF2 variations. Satellite-observed longitudinal variations in neutral density did not show any visible reaction to the polar vortex strength. However, the impact of sudden stratospheric warming (SSW) on the upper atmosphere is strong enough to change the neutral density longitudinal distribution. The impact of SSW shows a global occurrence and ‘works’ within 3–5 days in geographic coordinates in the vicinity of the SSW peak. Atomic oxygen values retrieved under ‘strong’ and ‘weak’ polar vortex strengths confirm the results obtained on longitudinal variations in NmF2 and ρ. In conclusion, no visible effects related to ‘strong’ or ‘weak’ polar vortex strengths have been revealed in either NmF2 or satellite neutral density longitudinal variations. Alternatively, such effects may be very small and therefore cannot be confirmed experimentally.

An integrated method for gravity gradient inversion and gravity gradient depth imaging

SUMMARY Gravity gradient data can show the structural features of geological bodies in the shallow lithosphere with higher sensitivity and resolution than conventional gravity data. Gravity gradient inversion can be applied to obtain the lithospheric density structures of geological bodies. However, as with gravity data, gravity gradient data have no inherent depth resolution. The methods of gravity gradient depth imaging and gravity gradient inversion are integrated in this study. The depth imaging method is effective for calculations without prior information and iterative computations. As the parameters in the depth weighting function should be chosen from a set of values used in inversion tests of synthetic data, which brings some uncertainties, the depth imaging results of gravity gradient are introduced into the depth weighting function. Several synthetic models are tested to demonstrate the advantages and features of the effective integrated method. Finally, the integrated method is applied to the interpretation of the GOCE satellite gravity gradient tensors over the northeastern margin of the Qinghai–Tibet Plateau. The results reveal that in the crust of the study area, the distribution of density anomalies is more in line with the mechanism of the crustal flow model, in the upper mantle of the study area, the density anomalies are mainly influenced by the high heat flow environment.

Assessment of the Added Value of the GOCE GPS Data on the GRACE Monthly Gravity Field Solutions

The time-varying gravity field models derived from the Gravity Recovery and Climate Experiment (GRACE) satellite mission suffer from pronounced longitudinal stripe errors in the spatial domain. A potential way to mitigate such errors is to combine GRACE data with observations from other sources. In this study, we investigate the impacts on GRACE monthly gravity field solutions of incorporating the GPS data collected by the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) mission. To that end, we produce GRACE/GOCE combined monthly gravity field solutions through combination on the normal equation level and compare them with the GRACE-only solutions, for which we have considered the state-of-the-art ITSG-Grace2018 solutions. Analysis in the spectral domain reveals that the combined solutions have a notably lower noise level beyond degree 30, with cumulative errors up to degree 96 being reduced by 31%. A comparison of the formal errors reveals that the addition of GOCE GPS data mainly improves (near-) sectorial coefficients and resonant orders, which cannot be well determined by GRACE alone. In the spatial domain, we also observe a significant reduction by at least 30% in the noise of recovered mass changes after incorporating the GOCE GPS data. Furthermore, the signal-to-noise ratios of mass changes over 180 large river basins were improved by 8–20% (dependent on the applied Gaussian filter radius). These results demonstrate that the GOCE GPS data can augment the GRACE monthly gravity field solutions and support a future GOCE-type mission for tracking more accurate time-varying gravity fields.

Machine Learning Based Modeling of Thermospheric Mass Density

AbstractIn this study, we propose a machine learning based approach to construct an empirical model of thermospheric mass densities, based on the MultiLayer Perceptron and bi‐directional Long Short‐Term Memory for ensemble learning model (MBiLE). The MBiLE model was trained by using only the thermospheric mass density from Swarm C satellite at ∼450 km altitude. To assess the performance of the MBiLE model, the model predictions were compared with observations from several satellites, namely, the Swarm C, the Challenging Minisatellite Payload (CHAMP) and the Gravity Field and Steady‐State Ocean Circulation Explorer (GOCE) satellites. The determination coefficients (R2) for the three satellites are 0.98, 0.99, and 0.98, respectively. The MBiLE model predicts the thermospheric mass density well not only at Swarm C altitude but also at lower altitudes. Earlier empirical models based on multivariate least‐square‐fitting approach failed to achieve this good altitude generalization (e.g., Liu et al., 2013, https://doi.org/10.1002/jgra.50144; Xiong et al., 2018a, https://doi.org/10.5194/angeo‐2018‐25). Further tests have been made by checking the MBiLE model prediction deviations in relation to magnetic local time, day of year, solar flux level, and magnetic activities. No obvious dependences are found for these parameters. Comparing with the NRLMSIS‐2.0 model, the MBiLE model improves prediction accuracy by 91%, 66%, and 56% at the three satellites altitudes. The results indicate that the MBiLE model has the ability to predict well the thermospheric mass density over a wide altitude range, for example, from 224 to 528 km, offering potential for atmospheric research applications.

Evaluation of TIEGCM based on GOCE neutral density

The Thermosphere Ionosphere Electrodynamic General Circulation Model (TIEGCM), as one of the most advanced physical models of the Earth’s thermosphere and ionosphere, is not only widely used in scientific research, but also has essential reference value in aerospace operations. In this study, we use Gravity field and steady-state Ocean Circulation Explorer (GOCE) neutral density to evaluate the accuracy of the TIEGCM. The assessment is performed on both time and spatial scales. The time scales are conducted annually, monthly, and daily, while the spatial scales are carried out in terms of altitude, latitude, and local time. On the time scales, the performance of the TIEGCM on the monthly time scale is better than that on the annual time scale. Also, the performance on the daily time scale is better than that on the monthly time scale. The relative deviation shows a significant seasonal variation, that is, larger in winter and summer and smaller in spring and autumn. In addition, the relative deviation shows a negative correlation with F10.7 and Ap. On the spatial scale, with the increase in altitude, the average relative deviation of the model becomes larger in general. The relative deviation is usually larger at middle latitudes in the Northern Hemisphere and high latitudes in the Southern Hemisphere. Finally, on the scale of local time, the relative deviation changes more dramatically in local morning than at dusk.

Thermospheric Mass Density Modelling during Geomagnetic Quiet and Weakly Disturbed Time

Atmospheric drag stands out as the predominant non-gravitational force acting on satellites in Low Earth Orbit (LEO), with altitudes below 2000 km. This drag exhibits a strong dependence on the thermospheric mass density (TMD), a parameter of vital significance in the realms of orbit determination, prediction, collision avoidance, and re-entry forecasting. A multitude of empirical TMD models have been developed, incorporating contemporary data sources, including TMD measurements obtained through onboard accelerometers on LEO satellites. This paper delves into three different TMD modelling techniques, specifically, Fourier series, spherical harmonics, and artificial neural networks (ANNs), during periods of geomagnetic quiescence. The TMD data utilised for modelling and evaluation are derived from three distinct LEO satellites: GOCE (at an altitude of approximately 250 km), CHAMP (around 400 km), and GRACE (around 500 km), spanning the years 2002 to 2013. The consistent utilisation of these TMD data sets allows for a clear performance assessment of the different modelling approaches. Subsequent research will shift its focus to TMD modelling during geomagnetic disturbances, while the present work can serve as a foundation for disentangling TMD variations stemming from geomagnetic activity. Furthermore, this study undertakes precise TMD modelling during geomagnetic quiescence using data obtained from the GRACE (at an altitude of approximately 500 km), CHAMP (around 400 km), and GOCE (roughly 250 km) satellites, covering the period from 2002 to 2013. It employs three distinct methods, namely Fourier analysis, spherical harmonics (SH) analysis, and the artificial neural network (ANN) technique, which are subsequently compared to identify the most suitable methodology for TMD modelling. Additionally, various combinations of time and coordinate representations are scrutinised within the context of TMD modelling. Our results show that the precision of low-order Fourier-based models can be enhanced by up to 10 % through the utilisation of geocentric solar magnetic coordinates. Both the Fourier- and SH-based models exhibit limitations in approximating the vertical gradient of TMD. Conversely, the ANN-based model possesses the capacity to capture vertical TMD variability without manifesting sensitivity to variations in time and coordinate inputs.

Quiet Time Thermospheric Gravity Waves Observed by GOCE and CHAMP

AbstractThe Gravity Field and Steady‐State Ocean Circulation Explorer (GOCE) and CHAllenging Minisatellite Payload (CHAMP) satellites measure in‐situ thermospheric density and cross‐track wind. When propagating obliquely to the satellite track in a horizontal plane (i.e., not purely along‐track or cross‐track), gravity waves (GWs) can be observed both in the density and cross‐track wind perturbations. We employ the Wavelet Analysis, red noise model, dissipative dispersion and polarization relations for thermospheric GWs, and specific criteria to determine whether a quiet‐time (Kp < 3) thermospheric traveling atmospheric disturbances (TADs) event is a GW or not. The first global morphology of thermospheric GWs instead of TADs is reported. The fast intrinsic horizontal phase speed (cIH > 600 m/s) of most GWs suggests that they are not generated in the lower/middle atmosphere (where cIH < 300 m/s). A second population of GWs with slower speeds (cIH = 50–250 m/s) in GOCE are likely from the lower/middle atmosphere, but they occur much less frequently in CHAMP. GW hotspots occur during the high‐latitude and the winter midlatitude regions. GW amplitudes exhibit semi‐annual and annual variations. These findings suggest that most GOCE and CHAMP GWs are higher‐order GWs from primary GW sources in the lower/middle atmosphere. Finally, the average propagation direction of the CHAMP GWs exhibits a clear diurnal cycle, with clockwise (counterclockwise) occurring in the northern (southern) hemisphere and equatorward propagation occurring at ∼13 LST. This suggests that the predominant GW propagation direction is opposite to the background wind direction.

1D along-track pre-processing of the GOCE gravity gradients and nonlinear filtering of the radial components Vzz in spatial domain

The paper presents pre-processing and nonlinear filtering of the GOCE gravity gradients (GGs) in spatial domain. At first, a multiple fully automatic 1D along-track pre-processing of the GOCE GGs in the gradiometer reference frame (GRF) is introduced. This tool is based on the nonlinear diffusion filtering in 1D while treating issues like a detection of ‘jumps’ in time series, outliers and unforeseen systematic tendencies. We focus on the 1D along-track pre-processing of the GOCE GGs in GRF for the last period of the GOCE satellite mission (from 2012-12-01 to 2013-10-19) that are later transformed into the local north-oriented frame (LNOF). Afterwards, the radial components Vzz are reduced on a reference surface chosen as a mean value of all altitudes of the GOCE satellite orbits from the processed period. Here they are filtered using the nonlinear diffusion filtering on a closed surface. Finally, a grid of the filtered radial components Vzz in LNOF on the reference surface is obtained.

Structure of the Earth’s Crust Based on the Gravity Data of the GOCE Satellite Mission and Spatial Position of Polymetallic Deposits in the Frame of the Siberian and East European Platforms

Structure of the Earth’s Crust Based on the Gravity Data of the GOCE Satellite Mission and Spatial Position of Polymetallic Deposits in the Frame of the Siberian and East European Platforms

The gravitational signature of the dynamics of oceanization in the Gulf of Aden

We perform a new gravity analysis in the Gulf of Aden with the aim of finding new constraints on the geodynamic evolution of the area. Our analysis is developed within the frame of the recent GO_CONS_GCF_2_TIM_R6 global gravity model solution (Brockmann et al., 2021) that reflects the Earth's static gravity field as observed by GOCE (Gravity field and steady-state Ocean Circulation Explorer). We analyzed the solution at different harmonic degrees, to account for different depths and dimensions of the sources. Terrain correction has been performed by means of a spherical tesseroidal methodology (Marotta and Barzaghi, 2017) and the obtained corrected Geodetic Residual Gravitation has been compared to the Model Residual Gravitation predicted by means of a 2D visco-plastic finite element thermo-mechanic model that simulates the dynamics of the Gulf of Aden, from rifting to oceanization. In order to perform the comparison between observed and predicted gravitational features, data have been extracted along five profiles crossing the Gulf of Aden at different sectors, from the south-east to the north west. Via the Model Residual Gravitation we reproduce its geodetic counterpart, obtaining a characteristic hat-shaped pattern, with a central plateau portraying the highest values, flanked by two regions where the residual gravitation diminishes, finally reaching the lowest values at the far continental domains. The maximum variations of the residual gravitation values, from ridge to continental domain, range between 280 and 420 mGal and the steepest decrease occurs over distances of 200 km for the thick crust model and of 100–150 km for the thin crust model. Finally, the comparison between Geodetic and Model Residual Gravitation allows to further constrain the dynamics of the Gulf of Aden oceanization occurring by a slow passive rift of a hot 150 km thick lithosphere characterized by an initial 40 km thick crust.

Combined monthly GRACE-FO gravity fields for a Global Gravity-based Groundwater Product

SUMMARY The Combination Service for Time-variable Gravity fields (COST-G) operationally provides combinations of monthly Earth gravity field models derived from observations of the microwave ranging instrument of the GRACE Follow-on (GRACE-FO) satellite mission, applying the quality control and combination methodology originally developed by the Horizon 2020 project European Gravity Service for Improved Emergency Management for the data of the GRACE satellites. In the frame of the follow-up Horizon 2020 project Global Gravity-based Groundwater Product (G3P), the GRACE-FO combination is used to derive global grids of groundwater storage anomalies. To meet the user requirements and achieve optimal signal-to-noise ratio, the combination has been further developed and extended to incorporate: • new time-series based on the alternative accelerometer transplant product generated in the frame of the project by the Institute of Geodesy at the Graz University of Technology, which specifically improves the estimation of the C30 coefficient and also reduces the noise at medium to short wavelengths, and • the new time-series AIUB–GRACE-FO–RL02 of monthly GRACE-FO gravity fields, which is derived at the Astronomical Institute of the University of Bern by applying empirical noise modelling techniques. The COST-G quality control confirms the consistency of the contributing GRACE-FO time-series concerning the signal amplitude of seasonal hydrology in large river basins and the secular mass change in polar regions, but it also indicates rather diverse noise characteristics. The difference in the noise levels is taken into account in the combination process by relative weights derived by variance component estimation on the solution level. The weights are expected to be inverse proportional to the noise levels of the individual gravity field solutions. However, this expectation is violated when applying the weighting scheme as developed for the GRACE combination. The reason is found in the high-order coefficients of the gravity field, which are poorly determined from the low–low range-rate observations due to the observation geometry and suffer from aliasing due to the malfunctioning accelerometer onboard one of the GRACE-FO satellites. Hence, for the final G3P-combination a revised weighting scheme is applied where the gravity field coefficients beyond order 60 are excluded from the determination of the weights. The quality of the combined gravity fields is assessed by comparison of the noise content and the signal-to-noise ratio with the individual time-series. Independent validation is provided by the COST-G validation centre at the GFZ German Research Centre for Geosciences, where orbit fits of the low-flying Gravity and steady-state Ocean Circulation Explorer satellite are performed that confirm the high quality of the combined GRACE-FO gravity fields. By the end of the G3P project, the new combination scheme is implemented by COST-G as the new COST-G–GRACE-FO–RL02 and continued to be used for the operational GRACE-FO combination.

The Impact of Different Filters on the Gravity Field Recovery Based on the GOCE Gradient Data

The electrostatic gravity gradiometer carried by the Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite is affected by accelerometer noise and other factors; hence, the observation data present complex error characteristics in the low-frequency domain. The accuracy of the recovered gravity field will be directly affected by the design of the filters based on the error characteristics of the gradient data. In this study, the applicability of various filters to different errors in observation is evaluated, such as the 1/f error and the orbital frequency errors. The experimental results show that the cascade filter (DARMA), which is formed of a differential filter and an autoregressive moving average filter (ARMA) filter, has the best accuracy for the characteristic of the 1/f low-frequency error. The strategy of introducing empirical parameters can reduce the orbital frequency errors, whereas the application of a notch filter will worsen the final solution. Frequent orbit changes and other changes in the observed environment have little impact on the new version gradient data (the data product is coded 0202), while the influence cannot be ignored on the results of the old version data (the data product is coded 0103). The influence can be effectively minimized by shortening the length of the arc. By analyzing the above experimental findings, it can be concluded that the inversion accuracy can be effectively improved by choosing the appropriate filter combination and filter estimation frequency when solving the gravity field model based on the gradient data of the GOCE satellite. This is of reference significance for the updating of the existing models.

Satellite gradiometry based on a new generation of accelerometers and its potential contribution to Earth gravity field determination

An accurate model of the Earth’s gravity field is beneficial for practical engineering and many applications in geosciences. European Space Agency (ESA) realized the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) gradiometry mission between 2009 and 2013. However, the low-frequency drift of the onboard electrostatic accelerometers (EA) limits the observation accuracy of the GOCE mission to some extent. Advances in electrostatic and quantum technology offer new measurement concepts for future gradiometry missions. In this study, we evaluate the contributions of several types of accelerometers through numerical closed-loop simulation which rigorously maps the accelerometers’ sensitivities to the gravity field coefficients. In comparison to the simulated results of the GRADIO gradiometer used in GOCE, it is demonstrated that the MicroSTAR-type gradiometer has superior precision within degree and order 100 and provides more signal information in the off-diagonal components of the gravity gradient tensor (GGT). The precision of the gravity field model recovery from a HybridACC-type gradiometer is significantly affected by the noise level of the cold atom interferometry (CAI) accelerometer. A HybridACC-type gradiometer with low CAI performance (1×10-9m·s-2/Hz) only favors the high degree component because of its higher accuracy in the measurement bandwidth (MBW) between 5 mHz and 100 mHz. While a better CAI performance up to 1×10-11m·s-2/Hz will increase retrieval performance remarkably. With an orbital rotation compensation mechanism, the CAI gradiometer performs with greater accuracy overall. Otherwise, the accuracy based on this sort of gradiometer is only superior up to degree 50.

Gravity field modelling in mountainous areas by solving the nonlinear satellite-fixed geodetic boundary value problem with the finite element method

The paper concerns a novel concept based on the iterative approach applied for solving the nonlinear satellite-fixed geodetic boundary value problem (NSFGBVP) using the finite element method (FEM). We formulate the NSFGBVP that consists of the Laplace equation defined in the 3D bounded domain outside the Earth, the nonlinear boundary condition (BC) prescribed on the disretized Earth’s surface, and the Dirichlet BC given on a spherical boundary placed approximately at the altitude of GOCE satellite mission and additional four side boundaries. The iterative approach is based on determining the direction of actual gravity vector together with the value of the disturbing potential. Such a concept leads to the first iteration where the oblique derivative boundary value problem is solved, and the last iteration represents the approximation of the actual disturbing potential and the direction of gravity vector. As a numerical method for our approach, we implement the FEM with triangular prisms. High-resolution numerical experiments deal with the local gravity field modelling in the Andean, Himalayan and Alpine mountain range, where we focus on the contribution to the disturbing potential solution by solving the nonlinear geodetic boundary value problem in comparison with the solution to the oblique derivative geodetic boundary value problem. The obtained results showed the maximal contribution of the presented approach 0.0571 [{text{m}}^{{text{2}}} {text{s}}^{{{text{ - 2}}}}] (Andes), 0.0702 [{text{m}}^{{text{2}}} {text{s}}^{{{text{ - 2}}}}] (Himalayas), 0.0066 [{text{m}}^{{text{2}}} {text{s}}^{{{text{ - 2}}}}] (Alps) in the disturbing potential, that is located in the areas of the highest values of the deflection of vertical.

Iterative approaches for regional Moho determination using on-orbit gravity gradients: A case study in Qinghai-Tibet Plateau and its near zone

Summary Moho determination is an important issue in studying the Earth’s interior structure. In accordance with the isostasy-compensation hypothesis in geodesy, it is possible to recover regional or global Moho by employing gravimetric data. The non-linear property is one of the main difficulties in solving the inverse problem of isostasy. To effectively address this issue, we propose an improved iterative inversion method that combines three-dimensional integration and linear regularization to achieve an approximate non-linear solution. To estimate the contributions of different components in the gravity-gradient tensor from the Gravity field and steady-state Ocean Circulation Explorer (GOCE), other than the vertical component, we additionally develop two joint inversion scenarios that utilize diagonal horizontal components and all five non-vertical components. The validating experiments are implemented in Qinghai-Tibet Plateau and its near zone. Simulations and applications illustrate that horizontal responses of Moho undulation are also significant. Yet the off-diagonal components provide minimal contributions, adding only 0.25 km of bias to the joint inversion results. Truncation effects serve as the primary source of systematic errors, resulting in ∼1 km error in vertical inversion results and ∼2.3 km error in joint inversion results. Then, the gravimetric Moho results are compared with CRUST1.0, and they show a generally strong correlation. Differences are obvious at the northern and eastern margins of the plateau. It is maybe due to the local changes in crust-mantle density contrasts. Upwelling of asthenospheric materials and fluid flow in the middle-lower crust are the two main factors. Based on high-precision satellite gravimetry, our study could provide new insights into the tectonic structure of Qinghai-Tibet Plateau.

Solving the fixed gravimetric boundary value problem by the finite element method using mapped infinite elements.

The numerical approach for solving the fixed gravimetric boundary value problem (FGBVP) based on the finite element method (FEM) with mapped infinite elements is developed and implemented. In this approach, the 3D semi-infinite domain outside the Earth is bounded by the triangular discretization of the whole Earth’s surface and extends to infinity. Then the FGBVP consists of the Laplace equation for unknown disturbing potential which holds in the domain, the oblique derivative boundary condition (BC) given directly at computational nodes on the Earth’s surface, and regularity of the disturbing potential at infinity. In this way, it differs from previous FEM approaches, since the numerical solution is not fixed by the Dirichlet BC on some part of the boundary of the computational domain. As a numerical method, the FEM with finite and mapped infinite triangular prisms has been derived and implemented. In experiments, at first, a convergence of the proposed numerical scheme to the exact solution is tested. Afterwards, a numerical study is focused on a reconstruction of the harmonic function (EGM2008) above the Earth’s topography. Here, a special discretization of the Earth’s surface which is able to fulfil the conditions that arise from correct geometrical properties of finite elements, and it is suitable for parallel computing is implemented. The obtained solutions at nodes on the Earth’s surface as well as nodes that lie approximately at the altitude of the GOCE satellite mission have been tested.

Reprocessed precise science orbits and gravity field recovery for the entire GOCE mission

ESA’s Gravity field and steady-state Ocean Circulation Explorer (GOCE) orbited the Earth between 2009 and 2013 for the determination of the static part of Earth’s gravity field. The GPS-derived precise science orbits (PSOs) were operationally generated by the Astronomical Institute of the University of Bern (AIUB). Due to a significantly improved understanding of remaining artifacts after the end of the GOCE mission (especially in the GOCE gradiometry data), ESA initiated a reprocessing of the entire GOCE Level 1b data in 2018. In this framework, AIUB was commissioned to recompute the GOCE reduced-dynamic and kinematic PSOs. In this paper, we report on the employed precise orbit determination methods, with a focus on measures undertaken to mitigate ionosphere-induced artifacts in the kinematic orbits and thereof derived gravity field models. With respect to the PSOs computed during the operational phase of GOCE, the reprocessed PSOs show in average a 8–9% better consistency with GPS data, 31% smaller 3-dimensional reduced-dynamic orbit overlaps, an 8% better 3-dimensional consistency between reduced-dynamic and kinematic orbits, and a 3–7% reduction of satellite laser ranging residuals. In the second part of the paper, we present results from GPS-based gravity field determinations that highlight the strong benefit of the GOCE reprocessed kinematic PSOs. Due to the applied data weighting strategy, a substantially improved quality of gravity field coefficients between degree 10 and 40 is achieved, corresponding to a remarkable reduction of ionosphere-induced artifacts along the geomagnetic equator. For a static gravity field solution covering the entire mission period, geoid height differences with respect to a superior inter-satellite ranging solution are markedly reduced (43% in terms of global RMS, compared to previous GOCE GPS-based gravity fields). Furthermore, we demonstrate that the reprocessed GOCE PSOs allow to recover long-wavelength time-variable gravity field signals (up to degree 10), comparable to information derived from GPS data of dedicated satellite missions. To this end, it is essential to take into account the GOCE common-mode accelerometer data in the gravity field recovery.