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Although a satellite mission to observe Earth’s magnetic field, the Swarm satellites also collect GPS data with sufficient accuracy to observe Earth’s gravity field with a spatial resolution of roughly 1500 km. These monthly models are available from 2014 to the present and do not rely on any other source of gravimetric data nor any a priori information in, for example, the form of temporal and spatial correlations. This time series covers the gap between the GRACE and GRACE-FO missions, as well as any other short gaps in their time series. Given the healthy state of the Swarm satellites, it is also likely that it will provide gravimetric information during possible gaps in the GRACE-FO data, and future dedicated gravimetric satellite missions.We are a consortium of international research institutes, composed of the Astronomical Institute of the University of Bern, the Astronomical Institute of the Czech Academy of Sciences, the Delft University of Technology, the Institute of Geodesy of the Graz University of Technology, and the School of Earth Sciences of the Ohio State University. These activities are supported by the European Space Agency and the International Combination Service for Time-variable Gravity Fields (COST-G). We publish the models every 3 months at ESA’s Swarm Data Accessserver (https://swarm-diss.eo.esa.int) as well at the International Centre for Global Earth Models (http://icgem.gfz-potsdam.de/series/02_COST-G/Swarm). Each institute exploits different gravity inversion strategies, thus producing independent solutions, which are combined at the solution level using weights derived with Variance Component Estimation. In this way, we ensure the published models are not biased towards particular strategies or assumptions.We illustrate the geophysical signal captured by Swarm’s GPS receivers over large hydrological basins, the errors represented by the variability of the models over the oceans and the agreement with GRACE and GRACE-FO. All analyses span the GRACE/GRACE-FO gap, to illustrate the importance of the Swarm satellites to bridge the absence of low-low satellite-to-satellite tracking data.

Inverting LRI data from GRACE-FO (NASA/GFZ) is challenging from multiple points of view. To benefit from the laser instrument, that provides a higher precision compared with the KBR ranging, the global basis functions such as spherical harmonics may not provide the best service for exploiting data the full potential. Because the regional analyses use surface elements such as mascons, we present two alternatives in terms of the spherical triangles that may also be used as target elements for the inversion. Since the spherical trigonometry can be used, a relatively easy manipulation and remeshing is possible if appropriate indexing is used. Furthermore, such grids are also used in the solid-Earth models models like LITHO1.0 and WINTER-C, and, thus they may provide an easy link from satellite gravimetry to solid-Earth disciplines. We show the properties of both grids, how to properly index them and how to reduce SST data for the continental part of the signal with the monthly solutions.

The AlpArray Gravity Research Group (AAGRG), as part of the European AlpArray program, focuses on the compilation of a homogeneous surface-based gravity data set across the Alpine area. In 2017 10 European countries in the Alpine realm agreed to contribute with gravity data for a new compilation of the Alpine gravity field in an area spanning from 2 to 23∘ E and from 41 to 51∘ N. This compilation relies on existing national gravity databases and, for the Ligurian and the Adriatic seas, on shipborne data of the Service Hydrographique et Océanographique de la Marine and of the Bureau Gravimétrique International. Furthermore, for the Ivrea zone in the Western Alps, recently acquired data were added to the database. This first pan-Alpine gravity data map is homogeneous regarding input data sets, applied methods and all corrections, as well as reference frames. Here, the AAGRG presents the data set of the recalculated gravity fields on a 4 km × 4 km grid for public release and a 2 km × 2 km grid for special request. The final products also include calculated values for mass and bathymetry corrections of the measured gravity at each grid point, as well as height. This allows users to use later customized densities for their own calculations of mass corrections. Correction densities used are 2670 kg m−3 for landmasses, 1030 kg m−3 for water masses above the ellipsoid and −1640 kg m−3 for those below the ellipsoid and 1000 kg m−3 for lake water masses. The correction radius was set to the Hayford zone O2 (167 km). The new Bouguer anomaly is station completed (CBA) and compiled according to the most modern criteria and reference frames (both positioning and gravity), including atmospheric corrections. Special emphasis was put on the gravity effect of the numerous lakes in the study area, which can have an effect of up to 5 mGal for gravity stations located at shorelines with steep slopes, e.g., for the rather deep reservoirs in the Alps. The results of an error statistic based on cross validations and/or “interpolation residuals” are provided for the entire database. As an example, the interpolation residuals of the Austrian data set range between about −8 and +8 mGal and the cross-validation residuals between −14 and +10 mGal; standard deviations are well below 1 mGal. The accuracy of the newly compiled gravity database is close to ±5 mGal for most areas. A first interpretation of the new map shows that the resolution of the gravity anomalies is suited for applications ranging from intra-crustal- to crustal-scale modeling to interdisciplinary studies on the regional and continental scales, as well as applications as joint inversion with other data sets. The data are published with the DOI https://doi.org/10.5880/fidgeo.2020.045 (Zahorec et al., 2021) via GFZ Data Services., Earth System Science Data, 13 (5), ISSN:1866-3516, ISSN:1866-3508

The GPS data collected by the Swarm satellites form the basis for monthly global gravity field models that are complete and uninterrupted since late 2013, thus already covering a period of eight years. A nice aspect is that this time series covers the gap between the GRACE and GRACE-FO missions, as well as any other short gaps in their time series, with a spatial resolution of roughly 1500 km.The Astronomical Institute of the University of Bern, the Astronomical Institute of the Czech Academy of Sciences, the Delft University of Technology, the Institute of Geodesy of the Graz University of Technology, and the School of Earth Sciences of the Ohio State University have teamed up to routinely provide these monthly models, with the support of the European Space Agency and the International Combination Servicefor Time-variable Gravity Fields (COST-G). The models are published every 3 months at ESA’s Swarm Data Access server (https://swarm-diss.eo.esa.int) as well at the International Centre for Global Earth Models (http://icgem.gfz-potsdam.de/series/02_COST-G/Swarm). Our gravity field models do not rely on any other source of gravimetric data nor any a priori information in for example the form of temporal and spatial correlations. The strength of our approach is that each institute exploits different gravity inversion strategies, thus producing independent solutions, which are combined at the solution level using weights derived with Variance Component Estimation.Considering a reference parametric model derived from GRACE/GRACE-FO data, our models traditionally agree at the level of roughly 4 cm Equivalent Water Height (EWH). Since early 2020, developments in the processing of the kinematic orbits have improved this figure to 3 cm. A particularity of the Swarm gravity field models is that the deep ocean areas are ~30-50% noisier than land areas, with the underlying reason yet unknown. At the spatial resolution of Swarm, the time series of large water storage basins show a temporal correlation of 0.75 when compared with GRACE models, and their trends agree within 1 cm/year in terms of EWH.

The lithospheric magnetic field, which is one of the main objectives of ESA’s mission Swarm, is slowly varying in time due to an induced component. This variation is small (usually it is omitted in the lithospheric modelling) but recent advances in processing strategies and still-growing amount of satellite data open questions whether such an effect should be considered in the development of the lithospheric models—when using data from missions like CHAMP and Swarm. This effect can now be estimated over a period of 17 years (since the launch of CHAMP), and it is shown how the satellite measurements (over the observable part of the spectrum) can be referenced to one common epoch. For this purpose, we first inverted the magnetic field vector from CHAOS-6 over degrees 21–120, after subtraction of a remanent model, to a vertically integrated susceptibility map. Using this susceptibility distribution and taking into account the evolving core fields from the CHAOS-6 model, the time-varying lithospheric signal is computed. The results depend on the time span and the altitude considered, e.g., an altitude of 400 km and a span of 17 years can produce more than 0.5 nT variations resulting in a peak-to-peak value of nearly 1 nT. The vertically integrated quantities appear to be a useful choice for parameterising lithospheric time variations, also for providing data corrections at the satellite altitude. The effect of the choice of the core field, which enters the inversion, on the lithospheric time variation is also studied—this effect is found less important even for core fields 20 years apart.

Over the last two decades, a small group of researchers repeatedly crossed the Greenland interior skiing along a 700-km long route from east to west, acquiring precise GNSS measurements at exactly the same locations. Four such elevation profiles of the ice sheet measured in 2002, 2006, 2010 and 2015 were differenced and used to analyze the surface elevation change. Our goal is to compare such locally measured GNSS data with independent satellite observations. First, we show an agreement in the rate of elevation change between the GNSS data and satellite radar altimetry (ERS, Envisat, CryoSat-2). Both datasets agree well (2002&ndash, 2015), and both correctly display local features such as an elevation increase in the central part of the ice sheet and a sharp gradual decline in the surface heights above Jakobshavn Glacier. Second, we processed satellite gravimetry data (GRACE) in order for them to be comparable with local GNSS measurements. The agreement is demonstrated by a time series at one of the measurement sites. Finally, we provide our own satellite gravimetry (GRACE, GRACE-FO, Swarm) estimate of the Greenland mass balance: first a mild decrease (2002&ndash, 2007: &minus, 210 &plusmn, 29 Gt/yr), then an accelerated mass loss (2007&ndash, 2012: &minus, 335 &plusmn, 29 Gt/yr), which was noticeably reduced afterwards (2012&ndash, 2017: &minus, 178 &plusmn, 72 Gt/yr), and nowadays it seems to increase again (2018&ndash, 2019: &minus, 278 &plusmn, 67 Gt/yr).

The AlpArray Gravity Research Group (AAGRG), as part of the European AlpArray program, focuses on the compilation of a homogeneous surface-based gravity dataset across the Alpine area. From this data set, Bouguer- and Free Air anomalies are calculated and presented here. In 2016/17 ten European countries in the Alpine realm have agreed to contribute with gravity data for a new compilation of the Alpine gravity field in an area from 2° to 23° East and from 41° to 51° North. This compilation relies on existing national gravity databases and, for the Ligurian and the Adriatic seas, on ship-borne data of the Bureau Gravimétrique International. Furthermore, for the Ivrea zone in the Western Alps, recently acquired data were added to the database. This first pan-Alpine gravity data map is homogeneous regarding input data sets, applied methods and all corrections as well as reference frames. Here, the AAGRG presents the data set of the recalculated gravity fields on a 4 km × 4 km grid for public release, 2 km × 2 km for special request. The final products also include calculated values for mass/bathymetry corrections of the measured gravity at each grid point, as well as height. This allows users to use later customized densities for their own calculations of mass corrections. Correction densities used are 2670 kg m−3 for landmasses, 1030 kg m−3 for water masses above and −1640 kg m−3 below the ellipsoid. The correction radius was set to the Hayford zone O2 (167 km). The new Bouguer anomaly is station completed (CBA) and compiled according to the most modern criteria and reference frames (both positioning and gravity), including atmospheric corrections. Special emphasis was put on the gravity effect of the numerous lakes in the study area, which can have an effect of up to 5 mGal for gravity stations located at shorelines with steep slopes, e.g., for the rather deep reservoirs in the Alps. The results of an error statistic based on cross validations and/or interpolations residuals is provided for the entire database. As an example, the interpolation residuals of the Austrian data set range between about −8 and +8 mGal, the cross-validation residuals between −14 mGal and +10 mGal; standard deviations are well below 1 mGal. The accuracy of the newly compiled gravity database is close to ±5 mGal for most areas. A first interpretation of the new map shows that the resolution of the gravity anomalies is suited for applications ranging from intra-crustal to crustal scale modelling to interdisciplinary studies on the regional and continental scales as well as applications as joint inversion with other datasets. The data will be published with the DOI https://doi.org/10.5880/fidgeo.2020.045 (Zahorec et al., 2020) when the final paper is accepted. In the meantime, the data is accessible via this temporary review link: https://dataservices.gfz-potsdam.de/panmetaworks/review/fdc35a9f6551b01b6152ee1af7b91a5a0c3de5341d067644522c192ad7f25e7f.

The Swarm satellite constellation provides GPS data with sufficient accuracy to observe the large-scale mass transport processes occurring at the Earth’s surface since 2013. We illustrate the signal content of the time series of monthly gravity field models. The models are published on quarterly basis and are the result of a combination of the individual models produced by different gravity field estimation approaches, by the Astronomical Institute of the University of Bern, the Astronomical Institute of the Czech Academy of Sciences, the Institute of Geodesy of the Graz University of Technology and the School of Earth Sciences of the Ohio State University. We combine the models at the solution level, using weights derived from a Variance Component Estimation, under the framework of the International Combination Service for Time-variable Gravity Fields (COST-G). We estimate the monthly quality of the models by comparing with GRACE and GRACE-FO products and illustrate the improvement of the combined model as compared to the individual models. We present the high signal-to-noise ratio of this uninterrupted time series of models, smoothed to 750km radius, over large hydrological basins. Finally, we compare the behavior of degree 2 and 3 coefficients with GRACE/GRACE-FO and SLR.

The Earth masses reside in a near-hydrostatic equilibrium, while the deviations are, for example, manifested in the geoid, which is nowadays well determined by satellite gravimetry. Recent progress in estimating the density distribution of the Earth allows us to examine individual Earth layers and to directly see how the sum approaches the observed anomalous gravitational field. This study evaluates contributions from the crust and the upper mantle taken from the LITHO1.0 model and quantifies the gravitational spectra of the density structure to the depth of 435 km. This is done without isostatic adjustments to see what can be revealed with models like LITHO1.0 alone. At the resolution of 290 km (spherical harmonic degree 70), the crustal contribution starts to dominate over the upper mantle and at about 150 km (degree 130) the upper mantle contribution is nearly negligible. At the spatial resolution $$

Swarm is a three-satellite mission of the European Space Agency, in orbit since November 2013, whose main objective is the study of the Earth’s magnetic field from space. As part of other scientific payload, each Swarm satellite is equipped with an accelerometer that measures the nongravitational forces (e.g. atmospheric drag and radiation pressure). Since the mission beginning, the Swarm onboard accelerometer observations have been facing a problem of much higher temperature influence than it had been anticipated in the pre-launch tests. In our paper, we use the a posteriori computed models of physical nongravitational forces acting on each satellite for external validation of the accelerometer measurements. To reduce the high temperature dependence, we apply a simple and straightforward method of linear temperature correction. The most successful application of this approach is for the along-track component of the accelerometer data, where the signal magnitude is strongest. The best performing accelerometer is that of the Swarm C satellite, the accelerometer of Swarm A displays more temperature dependence and noise, the noisiest accelerometer data set is provided by Swarm B. We analyzed the occurrence of anomalous periods in the along-track accelerometer component of Swarm A and Swarm C, when the number of accelerometer hardware anomalies is peaking. Over the time interval from June 2014 to December 2015, we found a correlation between these anomalous periods and the minima in the time-varying part of the modelled nongravitational signal.

Although the knowledge of the gravity of the Earth has improved considerably with CHAMP, GRACE, and GOCE (see appendices for a list of abbreviations) satellite missions, the geophysical community has identified the need for the continued monitoring of the time-variable component with the purpose of estimating the hydrological and glaciological yearly cycles and long-term trends. Currently, the GRACE-FO satellites are the sole dedicated provider of these data, while previously the GRACE mission fulfilled that role for 15 years. There is a data gap spanning from July 2017 to May 2018 between the end of the GRACE mission and start the of GRACE-FO, while the Swarm satellites have collected gravimetric data with their GPS receivers since December 2013. We present high-quality gravity field models (GFMs) from Swarm data that constitute an alternative and independent source of gravimetric data, which could help alleviate the consequences of the 10-month gap between GRACE and GRACE-FO, as well as the short gaps in the existing GRACE and GRACE-FO monthly time series. The geodetic community has realized that the combination of different gravity field solutions is superior to any individual model and set up the Combination Service of Time-variable Gravity Fields (COST-G) under the umbrella of the International Gravity Field Service (IGFS), part of the International Association of Geodesy (IAG). We exploit this fact and deliver the highest-quality monthly GFMs, resulting from the combination of four different gravity field estimation approaches. All solutions are unconstrained and estimated independently from month to month. We tested the added value of including kinematic baselines (KBs) in our estimation of GFMs and conclude that there is no significant improvement. The non-gravitational accelerations measured by the accelerometer on board Swarm C were also included in our processing to determine if this would improve the quality of the GFMs, but we observed that is only the case when the amplitude of the non-gravitational accelerations is higher than during the current quiet period in solar activity. Using GRACE data for comparison, we demonstrate that the geophysical signal in the Swarm GFMs is largely restricted to spherical harmonic degrees below 12. A 750 km smoothing radius is suitable to retrieve the temporal variations in Earth's gravity field over land areas since mid-2015 with roughly 4 cm equivalent water height (EWH) agreement with respect to GRACE. Over ocean areas, we illustrate that a more intense smoothing with 3000 km radius is necessary to resolve large-scale gravity variations, which agree with GRACE roughly at the level of 1 cm EWH, while at these spatial scales the GRACE observes variations with amplitudes between 0.3 and 1 cm EWH. The agreement with GRACE and GRACE-FO over nine selected large basins under analysis is 0.91 cm, 0.76 cm yr−1, and 0.79 in terms of temporal mean, trend, and correlation coefficient, respectively. The Swarm monthly models are distributed on a quarterly basis at ESA's Earth Swarm Data Access (at https://swarm-diss.eo.esa.int/, last access: 5 June 2020, follow Level2longterm and then EGF) and at the International Centre for Global Earth Models (http://icgem.gfz-potsdam.de/series/02_COST-G/Swarm, last access: 5 June 2020), as well as identified with the DOI https://doi.org/10.5880/ICGEM.2019.006 (Encarnacao et al., 2019).

Although the knowledge of the gravity of the Earth has improved considerably with CHAMP, GRACE and GOCE satellite missions, the geophysical community has identified the need for the continued monitoring of the time-variable component with the purpose of estimating the hydrological and glaciological yearly cycles and long-term trends. Currently, the GRACE-FO satellites are the sole dedicated provider of these data, while previously the GRACE mission fulfilled that role for 15 years. There is a data gap spanning from July 2017 to May 2018 between the end of the GRACE mission and start the of GRACE-FO, while the Swarm satellites have collected gravimetric data with their GPS receivers since December 2013. We present high-quality gravity field models from Swarm data that constitute an alternative and independent source of gravimetric data, which could help alleviate the consequences of the 10-month gap between GRACE and GRACE-FO, as well as the short gaps in the existing GRACE and GRACE-FO monthly time series. The geodetic community has realized that the combination of different gravity field solutions is superior to any individual model and set up a Combination Service of Time-variable Gravity Fields (COST-G) under the umbrella of the International Gravity Field Service (IGFS), part of the International Association of Geodesy (IAG). We exploit this fact and deliver to the highest quality monthly-independent gravity field models, resulting from the combination of four different gravity field estimation approaches. All solutions are unconstrained and estimated independently from month to month. We tested the added value of including Kinematic Baselines (KBs) in our estimation of Gravity Field Models (GFMs) and conclude that there is no significant improvement. The non-gravitational accelerations measured by the accelerometer on-board Swarm-C were also included in our processing to determine if this would improve the quality of the GFMs, but observed that is only the case when the amplitude of the non-gravitational accelerations is higher than during the current quiet period in solar activity. Using GRACE data for comparison, we demonstrate that the geophysical signal in the Swarm gravity field models is largely restricted to Spherical Harmonic degrees below 12. A 750 km smoothing radius is suitable to retrieve the temporal variations of Earth’s gravity field over land areas since mid-2015 with roughly 4 cm Equivalent Water Height (EWH) agreement with respect to a GRACE-derived parametric model. Over ocean areas, we illustrate that a more intense smoothing with 3000 km radius is necessary to resolve large scale gravity variations, which agree with the aforementioned parametric model under 2 cm EWH, while at these spatial scales the model represents variations with amplitudes between 2 and 3.5 cm EWH. The agreement with GRACE and GRACE-FO over nine selected large basins under analyses is 1.19 cm, 0.60 cm/year and 0.75 in terms of temporal mean, trend and correlation coefficient, respectively.

Airborne gravity gradient (AGG) measurements offer an increased resolution and accuracy compared to terrestrial measurements. But interpretation and processing of AGG data are often challenging as levelling errors and survey noise affect the data, and these effects are not easily recognised in the gradient components. We adopted the classic method of upward continuation in the noise reduction using the noise level estimates by the AGG system. By iteratively projecting the survey data to a lower level and upward continuing the data back to the survey height, parts of the high-frequency signal are suppressed. The filter, which is defined by this approach, is directly dependent on the noise level of the AGG data, the maximum number of iterations and the iterative step. We demonstrate the method by applying it to both synthetic data and real AGG data over Karasjok, Norway, and compare the results to the directional filtering method. The results show that the iterative filter can effectively reduce high-frequency noise in the data.

We present global and regional gravity field models to degree 130 based on the GOCE kinematic orbit from the period 01 November 2009 to 11 January 2010. The gravity field models are parameterized in terms of the Shannon and Kaula's spherical radial basis functions. The relation between the unknown expansion coefficients and the kinematic orbit of the satellite is established by the acceleration approach. We show that our global GOCE-only solutions free from prior information can compete with unconstrained spherical harmonic models in terms of accuracy. Furthermore, we utilize our low-degree global GOCE-based models to introduce prior information into the least-squares adjustment. This procedure substantially improves the zonal and near-zonal spherical harmonic coefficients, which are usually degraded due to the polar gap problem. As an unwanted side effect, low-pass filtering of the geopotential may occur, but this can be adjusted by the spectral content of the prior information. We show that the regional enhancement of the global solutions reduces noise in the final model between degrees 70 and 130 by ~10 % in terms of RMS error. In general, our Shannon-based solutions systematically outperform the Kaula-based ones. To validate our results, we use the EIGEN-6S model, which is superior to the solutions from kinematic orbits at least by one order of magnitude. Both the global and the regional models satisfy the GOCE-only strategy.

Downward continuation of the potential data (gravity or magnetic) helps to interpret contributing sources by transforming the physical signal to their neighborhood. This paper applies three continuation approaches (Landweber’s iteration, the direct method and the inverse-matrix approach with Tikhonov’s regularization), based on the Poisson integral equation, and four integration schemes to the second vertical derivative of the anomalous potential $$T_{\rm zz}$$ obtained from GOCE data. In the experiments, $$T_{\rm zz}$$ was downward continued for 250 km in the area of Central Europe with special attention to edge effects. For the integration schemes, which use prior information outside the region of interest to reduce edge effects, the best agreement with TIM-r4 was achieved by the inverse-matrix approach with Tikhonov’s regularization (RMS = 1.15 eotvos). On the contrary, without prior information the iterative approach performed with RMS = 1.17 eotvos.

In the last few decades, satellites have acquired various potential data sets hundreds of kilometers above the Earth’s surface. Conventionally, these global magnetic and gravitational data sets are approximated by using spherical harmonics that allow straightforward work with both fields outside the Earth’s mass. In this article, we present an alternative approach for working with potential data in mass-free space given over a regular coordinate grid on a spherical surface. The algorithm is based on an iterative scheme and the Poisson integral equation for the sphere. With help from the Fourier transform, global potential (magnetic or gravitational) data can efficiently be continued from a mean orbital sphere down to a reference surface without using the spherical harmonics. This is illustrated both with simulated magnetic field data and with real data from the satellite gradiometry mission GOCE. In the case of simulated magnetic data and the downward continuation for 450 km, we have achieved a root mean square at the level of 0.05 nT, while it was

New integral formulas for upward/downward continuation of gravitational gradients onto gravitational gradients are derived in this article. They provide more options for continuation of gravitational gradient combinations and extend available mathematical apparatus formulated for this purpose up to now. The starting point represents the analytical solution of the spherical gradiometric boundary value problem in the spatial domain. Applying corresponding differential operators on the analytical solution of the spherical gradiometric boundary value problem, a total of 18 integral formulas are provided. Spatial and spectral forms of isotropic kernels are given and their behaviour for parameters of a GOCE-like satellite is investigated. Correctness of the new integral formulas and the isotropic kernels is tested in a closed-loop simulation. The derived integral formulas and the isotropic kernels form a theoretical basis for validation purposes and geophysical applications of satellite gradiometric data as provided currently by the GOCE mission. They also extend the well-known Meissl scheme.

We present a Matlab package for visualizing global data on a 3D sphere, whose rotation can be animated. Planetary elevation data sets such as geoid height or Earth topography can easily be represented through a slightly exaggerated, colored 3D relief, and then saved either as images or animations. All necessary parameters for the 3D visualization and animation are described and their usage is demonstrated on examples. Among other things, users are shown how to easily create their own color scales. In principle, any geoscientific scalar data given on a global grid of longitudes and latitudes can be visualized with this package. The package requires only the basic module of Matlab, running on an ordinary PC or notebook, and it is available for free download at http://www.asu.cas.cz/~bezdek/vyzkum/rotating_3d_globe/. HighlightsA Matlab package is presented for visualizing global data on 3D sphere.Possibly, rotation of the 3D sphere can be animated.For all examples shown, Matlab code is provided on the package website.Any global data can be visualized, e.g. planetary topography.The package is available for free download.

Gravity data observed on or reduced to the ellipsoid are preferably represented using ellipsoidal harmonics instead of spherical harmonics. Ellipsoidal harmonics, however, are difficult to use in practice because the computation of the associated Legendre functions of the second kind that occur in the ellipsoidal harmonic expansions is not straightforward. Jekeli’s renormalization simplifies the computation of the associated Legendre functions. We extended the direct computation of these functions—as well as that of their ratio—up to the second derivatives and minimized the number of required recurrences by a suitable hypergeometric transformation. Compared with the original Jekeli’s renormalization the associated Legendre differential equation is fulfilled up to much higher degrees and orders for our optimized recurrences. The derived functions were tested by comparing functionals of the gravitational potential computed with both ellipsoidal and spherical harmonic syntheses. As an input, the high resolution global gravity field model EGM2008 was used. The relative agreement we found between the results of ellipsoidal and spherical syntheses is 10−14, 10−12 and 10−8 for the potential and its first and second derivatives, respectively. Using the original renormalization, this agreement is 10−12, 10−8 and 10−5, respectively. In addition, our optimized recurrences require less computation time as the number of required terms for the hypergeometric functions is less.

With satellite radar altimetry, the oceanic geoid can be determined with high precision and resolution. Double differentiation of these data along satellite altimeter ground tracks yields along-track gravity gradients that can be used to compute vertical gravity gradients at ground track crossovers. One way to counteract the noise amplification due to the differentiation is to smooth the data using smoothing splines. Although the effect of satellite altimeter data noise has been investigated to some extent, the associated systematic errors have not been assessed so far. Here we show that some of the systematic errors cannot be neglected. In particular, we found that the negligence of the dynamic ocean topography (DOT) may introduce errors that are greater than the measurement noise induced errors. If the gravity gradients are to be used for GOCE validation, then also in this case the DOT may not be neglected as the signal at GOCE altitude of 260 km may be above the GOCE requirements. In addition, we show ...

gradiometer on board) to the 16:1 resonance. To avoid the decrease in A, we have to choose the orbit in such a way that orderB of the lowest-order resonance, which will occur, will be higher than the highest degree Lmax of spherical harmonic expansion of the potential already known for the particular body. For the Earth, Lmax is now 150 [European improved gravitymodel of theEarth by new techniques (EIGEN-5S)] for satellite-only solutions and 2190

It is of great interest to numerous geophysical studies that the time series of global gravity field models derived from Gravity Recovery and Climate Experiment (GRACE) data remains uninterrupted after the end of this mission. With this in mind, some institutes have been spending efforts to estimate gravity field models from alternative sources of gravimetric data. This study focuses on the gravity field solutions estimated from Swarm global positioning system (GPS) data, produced by the Astronomical Institute of the University of Bern, the Astronomical Institute (ASU, Czech Academy of Sciences) and Institute of Geodesy (IfG, Graz University of Technology). The three sets of solutions are based on different approaches, namely the celestial mechanics approach, the acceleration approach and the short-arc approach, respectively. We derive the maximum spatial resolution of the time-varying gravity signal in the Swarm gravity field models to be degree 12, in comparison with the more accurate models obtained from K-band ranging data of GRACE. We demonstrate that the combination of the GPS-driven models produced with the three different approaches improves the accuracy in all analysed monthly solutions, with respect to any of them. In other words, the combined gravity field model consistently benefits from the individual strengths of each separate solution. The improved accuracy of the combined model is expected to bring benefits to the geophysical studies during the period when no dedicated gravimetric mission is operational.

The Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite aimed at determining the Earth’s mean gravity field. GOCE delivered gravity gradients containing directional information, which are complicated to use because of their error characteristics and because they are given in a rotating instrument frame indirectly related to the Earth. We compute gravity gradients in grids at 225 km and 255 km altitude above the reference ellipsoid corresponding to the GOCE nominal and lower orbit phases respectively, and find that the grids may contain additional high-frequency content compared with GOCE-based global models. We discuss the gradient sensitivity for crustal depth slices using a 3D lithospheric model of the North-East Atlantic region, which shows that the depth sensitivity differs from gradient to gradient. In addition, the relative signal power for the individual gradient component changes comparing the 225 km and 255 km grids, implying that using all components at different heights reduces parameter uncertainties in geophysical modelling. Furthermore, since gravity gradients contain complementary information to gravity, we foresee the use of the grids in a wide range of applications from lithospheric modelling to studies on dynamic topography, and glacial isostatic adjustment, to bedrock geometry determination under ice sheets.

The aim of our work is to generate Earth’s gravity field models from GPS positions of low Earth orbiters. Our inversion method is based on Newton’s second law, which relates the observed acceleration of the satellite with forces acting on it. The observed acceleration is obtained as numerical second derivative of kinematic positions. Observation equations are formulated using the gradient of the spherical harmonic expansion of the geopotential. Other forces are either modelled (lunisolar perturbations, tides) or provided by onboard measurements (nongravitational perturbations). From this linear regression model the geopotential harmonic coefficients are obtained. To this basic scheme of the acceleration approach we added some original elements, which may be useful in other inversion techniques as well. We tried to develop simple, straightforward and still statistically correct model of observations. (i) The model is linear in the harmonic coefficients, no a priori gravity field model is needed, no regularization is applied. (ii) We use the generalized least squares to successfully mitigate the strong amplification of noise due to numerical second derivative. (iii) The number of other fitted parameters is very small, in fact we use only daily biases, thus we can monitor their behaviour. (iv) GPS positions have correlated errors. The sample autocorrelation function and especially the partial autocorrelation function indicate suitability of an autoregressive model to represent the correlation structure. The decorrelation of residuals improved the accuracy of harmonic coefficients by a factor of 2–3. (v) We found it better to compute separate solutions in the three local reference frame directions than to compute them together at the same time; having obtained separate solutions for along-track, cross-track and radial components, we combine them using the normal matrices. Relative contribution of the along-track component to the combined solution is 50 percent on average. (vi) The computations were performed on an ordinary PC up to maximum degree and order 120. We applied the presented method to orbits of CHAMP and GRACE spanning seven years (2003–2009) and to two months of GOCE (Nov/Dec 2009). The obtained long-term static gravity field models are of similar or better quality compared to other published solutions. We also tried to extract the time-variable gravity signal from CHAMP and GRACE orbits. The acquired average annual signal shows clearly the continental areas with important and known hydrological variations.

This paper presents a unified approach to the least squares spherical harmonic analysis of the acceleration vector and Eotvos tensor (gravitational gradients) in an arbitrary orientation. The Jacobian matrices are based on Hotine’s equations that hold in the Earth-fixed Cartesian frame and do not need any derivatives of the associated Legendre functions. The implementation was confirmed through closed-loop tests in which the simulated input is inverted in the least square sense using the rotated Hotine’s equations. The precision achieved is at the level of rounding error with RMS about $$10^{-12}{-}10^{-14}$$ m in terms of the height anomaly. The second validation of the linear model is done with help from the standard ellipsoidal correction for the gravity disturbance that can be computed with an analytic expression as well as with the rotated equations. Although the analytic expression for this correction is only of a limited accuracy at the submillimeter level, it was used for an independent validation. Finally, the equivalent of the ellipsoidal correction, called the effect of the normal, has been numerically obtained also for other gravitational functionals and some of their combinations. Most of the numerical investigations are provided up to spherical harmonic degree 70, with degree 80 for the computation time comparison using real GRACE data. The relevant Matlab source codes for the design matrices are provided.

Spherical harmonic transformation is of practical interest in geodesy for transformation of globally distributed quantities such as gravity between space and frequency domains. The increasing spatial resolution of the latest and forthcoming gravitational models pose true computational challenges for classical algorithms since serious numerical instabilities arise during the computation of the respective base functions of the spherical harmonic expansion. A possible solution is the evaluation of the associated Legendre functions in the Fourier domain where numerical instabilities can be circumvented by an independent frequency-wise scaling of numerical coefficients into a numerically suitable double precision range. It is then rather straightforward to commit global fast data transformation into the Fourier domain and to evaluate subsequently spherical harmonic coefficients. For the inverse, the computation of respective Fourier coefficients from a given spherical harmonic model is performed as an inverse Fast Fourier Transform into globally distributed data points. The two-step formulation turns out to be stable even for very high resolutions as well as efficient when using state-of-the-art shared memory/multi-core architectures. In principle, any functional of the geopotential can be computed in this way. To give an example for the overall performance of the algorithm, we transformed an equiangular 1 arcmin grid of terrain elevation data corresponding to spherical harmonic degree and order 10800.

In 2008 the new Earth Gravitational Model (EGM08) was released. It contains a complete set of spherical harmonic coefficients of the Earth's gravitational potential (Stokes parameters) to degree 2190 and order 2159 that can be used for evaluation of various potential quantities with both the unprecedented accuracy and high spatial resolution. Two such quantities, the gravity anomaly and second-order radial derivative of the disturbing potential, were computed over selected areas with known impact craters. The displays of these derivatives for two such sites clearly show not only the strong circular-like features known to be associated with them but also other symmetrical structures which appear to make them multiple impact sites. At Popigai, Siberia, the secondary circular features fall in a line from the primary in the SE direction. At Chicxulub, Yucatán, there appears to be one secondary crater close to the primary in the NE direction, as well as possibly others in the vicinity of the main crater. Gravity information alone is not proof of the impact craters but it is useful in identifying candidate sites for further study, for future examination by geologists and geophysicists.

In 2008 the new Earth Gravitational Model (EGM2008) was released. It contains a complete set of spherical harmonic coefficients of the Earth's gravitational potential (Stokes parameters) to degree 2190 and order 2159 and selected orders to degree 2190, that can be used for evaluation of various potential quantities with both the unprecedented accuracy and high spatial resolution. Two such quantities, the gravity anomaly and second-order radial derivative of the disturbing potential, were computed over selected areas with known impact craters. The displays of these derivatives for two such sites clearly show not only the strong circular-like features known to be associated with them but also other symmetrical structures which appear to make them multiple impact sites. At Popigai, Siberia, the series of circular features fall in a line from the "primary crater" in the southeast (SE) direction. At Chicxulub, Yucatán, there appears to be one more crater close to the "primary" in the northeast (NE) direction, as well as possibly others in the vicinity of the main crater (SW). Gravity information alone is not, however, proof of impact craters but it is useful in identifying candidate sites for further study, for examination by geologists and geophysicists. In the case of Chicxulub, a very recent single seismic profile suggests that a more likely explanation for the observed circular like gravity signal from EGM2008 NE of the "primary" is a pre-impact basin.