Satellite Gravimetry: A Review of Its Realization

Since Kepler, Newton and Huygens in the seventeenth century, geodesy has been concerned with determining the figure, orientation and gravitational field of the Earth. With the beginning of the space age in 1957, a new branch of geodesy was created, satellite geodesy. Only with satellites did geodesy become truly global. Oceans were no longer obstacles and the Earth as a whole could be observed and measured in consistent series of measurements. Of particular interest is the determination of the spatial structures and finally the temporal changes of the Earth's gravitational field. The knowledge of the gravitational field represents the natural bridge to the study of the physics of the Earth's interior, the circulation of our oceans and, more recently, the climate. Today, key findings on climate change are derived from the temporal changes in the gravitational field: on ice mass loss in Greenland and Antarctica, sea level rise and generally on changes in the global water cycle. This has only become possible with dedicated gravity satellite missions opening a method known as satellite gravimetry. In the first forty years of space age, satellite gravimetry was based on the analysis of the orbital motion of satellites. Due to the uneven distribution of observatories over the globe, the initially inaccurate measuring methods and the inadequacies of the evaluation models, the reconstruction of global models of the Earth's gravitational field was a great challenge. The transition from passive satellites for gravity field determination to satellites equipped with special sensor technology, which was initiated in the last decade of the twentieth century, brought decisive progress. In the chronological sequence of the launch of such new satellites, the history, mission objectives and measuring principles of the missions CHAMP, GRACE and GOCE flown since 2000 are outlined and essential scientific results of the individual missions are highlighted. The special features of the GRACE Follow-On Mission, which was launched in 2018, and the plans for a next generation of gravity field missions are also discussed.

The earth’s gravity field recovery using the third invariant of the gravity gradient tensor from GOCE

Due to the independence of the gradiometer instrument’s orientation in space, the second invariant $$I_2$$ I 2 of gravity gradients in combination with individual gravity gradients are demonstrated to be valid for gravity field determination. In this contribution, we develop a novel gravity field model named I3GG, which is built mainly based on three novel elements: (1) proposing to utilize the third invariant $$I_3$$ I 3 of the gravity field and steady-state ocean circulation explorer (GOCE) gravity gradient tensor, instead of using the $$I_2$$ I 2 , similar to the previous studies; (2) applying an alternative two-dimensional fast fourier transform (2D FFT) method; (3) showing the advantages of $$I_3$$ I 3 over $$I_2$$ I 2 in the effect of measurement noise from the theoretical and practical computations. For the purpose of implementing the linearization of the third invariant, this study employs the theory of boundary value problems with sphere approximation at an accuracy level of $$O(J_2^2\cdot T_{ij})$$ O ( J 2 2 · T ij ) . In order to efficiently solve the boundary value problems, we proposed an alternative method of 2D FFT, which uses the coherent sampling theory to obtain the relationship between the 2D FFT and the third invariant measurements and uses the pseudo-inverse via QR factorization to transform the 2D Fourier coefficients to spherical harmonic ones. Based on the GOCE gravity gradient data of the nominal mission phase, a novel global gravity field model (I3GG) is derived up to maximum degree/order 240, corresponding to a spatial resolution of 83 km at the equator. Moreover, in order to investigate the differences of gravity field determination between $$I_3$$ I 3 with $$I_2$$ I 2 , we applied the same processing strategy on the second invariant measurements of the GOCE mission and we obtained another gravity field model (I2GG) with a maximum degree of 220, which is 20 degrees lower than that of I3GG. The root-mean-square (RMS) values of geoid differences indicates that the effects of measurement noise of I3GG is about 20% lower than that on I2GG when compared to the gravity field model EGM2008 (Earth Gravitational Model 2008) or EIGEN-5C (EIGEN: European Improved Gravity model of the Earth by New techniques). Then the accuracy of I3GG is evaluated independently by comparison the RMS differences between Global Navigation Satellite System (GNSS)/leveling data and the model-derived geoid heights. Meanwhile, the re-calibrated GOCE data released in 2018 is also dealt with and the corresponding result also shows the similar characteristics.

Benchmark data for verifying background model implementations in orbit and gravity field determination software

In the framework of the COmbination Service for Time-variable Gravity fields (COST-G) gravity field solutions from different analysis centres are combined to provide a consolidated solution of improved quality and robustness to the user. As in many other satellite-related sciences, the correct application of background models plays a crucial role in gravity field determination. Therefore, we publish a set of data of various commonly used forces in orbit and gravity field modelling (Earth's gravity field, tides etc.) evaluated along a one day orbit arc of GRACE, together with auxiliary data to enable easy comparisons. The benchmark data is compiled with the GROOPS software by the Institute of Geodesy (IfG) at Graz University of Technology. It is intended to be used as a reference data set and provides the opportunity to test the implementation of these models at various institutions involved in orbit and gravity field determination from satellite tracking data. In view of the COST-G GRACE and GRACE Follow-On gravity field combinations, we document the outcome of the comparison of the background force models for the Bernese GNSS software from AIUB (Astronomical Institute, University of Bern), the EPOS software of the German Research Centre for Geosciences (GFZ), the GINS software, developed and maintained by the Groupe de Recherche de Géodésie Spatiale (GRGS), the GRACE-SIGMA software of the Leibniz University of Hannover (LUH) and the GRASP software also developed at LUH. We consider differences in the force modelling for GRACE (-FO) which are one order of magnitude smaller than the accelerometer noise of about 10−10 m s−2 to be negligible and formulate this as a benchmark for new analysis centres, which are interested to contribute to the COST-G initiative.

The research unit NEROGRAV: first results on stochastic modeling for gravity field determination with real GRACE and GRACE-FO data

<p><span lang="en-US">The central hypothesis of the Research Unit (RU) NEROGRAV reads: only by concurrently improving and better understanding of sensor data, background models, and processing strategies of satellite gravimetry, the resolution, accuracy, and long-term consistency of mass transport series from satellite gravimetry can be significantly increased; and only in that case the potential of future technological sensor developments can be fully exploited. Two of the individual projects (IPs) within the RU work on stochastic modeling for GRACE and GRACE-FO gravity field determination. TU München and TU Berlin are responsible for IP4 (OSTPAG: optimized space-time parameterization for GRACE and GRACE-FO data analysis), where besides optimal parameterization the focus is on the stochastic modeling of the key observations, i.e. GRACE and GRACE-FO inter-satellite ranging and accelerometer observations, in a simulation (TU München) and real data (TU Berlin) environment. IP5 (ISTORE: improved stochastic modeling in GRACE/GRACE-FO real data processing), which GFZ is responsible for, works on the optimal utilization of the stochastic properties of the main GRACE and GRACE-FO observation types and the main background models. </span></p> <p><span lang="en-US">This presentation gives first insights into the TU Berlin and GFZ results of these two IPs which are both related on stochastic modeling for real data processing based on GFZ GRACE and GRACE-FO RL06 processing. We present analysis of ranging observations and corresponding residuals of three test years of GRACE and GRACE-FO real data in the time and frequency domain. Based on the residual analysis we show results of the effects of different filter matrices, which take into account the stochastic properties of the ranging observations in order to decorrelate them. The stochastic modeling of the background models in IP5 starts with Monte-Carlo simulations on background model errors of atmospheric and oceanic mass variations. Different representations of variance-covariance matrices of this model information are tested as input for real GRACE data processing and their effect on gravity field determination are analyzed.</span></p>

Results of fusion of LRI based attitude data in the attitude Kalman filter

<p>Gravity Recovery and Climate Experiment Follow-On (GRACE-FO) carries the Laser Ranging Interferometer (LRI) as a technology demonstrator that measures the inter-satellite range with nanometer precision. For the precise laser beam pointing, LRI uses the beam steering method where the fast steering mirror is actuated to correct for the misalignment between the incoming and outgoing laser beams. From the fast steering mirror commands, we can compute the inter-satellite pitch and yaw angles. These angles are provided as LSM1B product and represent spacecraft's relative orientation with respect to line-of-sight (LOS). The SCA1B data, which is computed by combining the data of three star cameras and IMU in the Kalman filter, represents the absolute orientation of the spacecrafts. Currently, this SCA1B product is used in the gravity field determination.  </p> <p>Here we present first results of a new attitude product which is computed by the combination of fast steering mirror data with the star cameras and IMU in the attitude kalman filter.  We also present the impact of this new combined attitude data on the gravity field solutions.</p>

Time-variable gravity field determination from GRACE-FO data using the Celestial Mechanics Approach

<p>We study gravity field determination from GRACE-FO satellite-to-satellite tracking using the inter-satellite K-band link and kinematic positions of the satellites as observations and pseudo-observations respectively. We employ 10-second kinematic positions from a precise point positioning where the undifferenced carrier phase ambiguities are fixed to integer values using observation-specific phase bias products from the CODE analysis centre.<br />We present an up-to-date GRACE-FO time series computed with the Celestial Mechanics Approach extended by empirical noise models derived from the post-fit residuals to better characterise the stochastic behaviour of both observation types. We investigate the interplay between the empirical noise model and the co-estimation of stochastic parameters set-up to absorb unmodelled signal (short term mass variations, accelerometer errors etc.).<br />We validate our results of GRACE-FO gravity field determination by analysing the residuals of combined orbits calculated using both kinematic positions and K-band data, and by analysing the quality of co-estimated gravity field solutions.</p>

What can be expected from GNSS tracking of satellite constellations for temporal gravity field model determination?

SUMMARY The goal of this contribution is to investigate the expected improvement of temporal gravity field determination via a couple of high-low satellite-to-satellite tracking (HLSST) missions. The simulation system is firstly validated by determining monthly gravity field models within situ GRACE GPS tracking data. The general consistency between the retrieved solutions and those developed by other official agencies indicates the good performance of our software. A 5-yr full-scale simulation is then performed using the full error sources including all error components. Analysis of each error component indicates that orbit error is the main contributor to the overall HLSST-derived gravity field model error. The noise level of monthly solution is therefore expected to reduce 90 per cent in terms of RMSE over ocean when the orbit accuracy improves for a magnitude of one order. As for the current HLSST mission consisting of a current GNSS receiver and an accelerometer (10−10 and 10−9 m s–2 noise for sensitive and non-sensitive axes), it is expected to observe monthly (or weekly) gravity solution at the spatial resolution of about 1300 km (or 2000 km). As for satellite constellations, a significant improvement is expected by adding the second satellite with the inclination of 70° and the third satellite with the inclination of 50°. The noise reduction in terms of cumulative geoid height error is approximately 51 per cent (or 62 per cent) when the observations of two (or three) HLSST missions are used. Moreover, the accuracy of weekly solution is expected to improve 40–70 per cent (or 27–59 per cent) for three (or two) HLSST missions when compared to one HLSST mission. Due to the low financial costs, it is worthy to build a satellite constellation of HLSST missions to fill the possible gaps between the dedicated temporal gravity field detecting missions.

Preliminary GRACE-FO gravity field solutions from Tongji University

<p>Due to the battery issue, the Gravity Recovery and Climate Experiment (GRACE) mission unfortunately came to an end in October 2017 after providing more than 15 years of mass transport information of our changing planet. To continue to monitoring the mass transport in the Earth system, the GRACE Follow-On (GRACE-FO) was launched in May 2018. As a new feature of GRACE-FO, a Laser Ranging Interferometer (LRI) was equipped to measure the inter-satellite range at a nanometer level. Since May 2019, GRACE-FO Level-1B observations have been made available to our community. Using the GRACE-FO Level-1B observations without laser ranging information, preliminary GRACE-FO gravity field solutions from Center for Space Research (CSR), GeoForschungsZentrum (GFZ), Jet Propulsion Laboratory (JPL) and Graz University of Technology have been released. Incorporating laser ranging observations into gravity field determination, a preliminary time series of GRACE-FO gravity field solutions has been derived from Tongji University in collaboration with University of Bonn. In this paper, the signal and noise of our gravity field solutions are analyzed and compared to those from other research groups. Our results show that the laser ranging observations with a sampling rate of 2s are able to improve gravity field solutions by about 7% in terms of geoid degree variances up to degree and order 96 as compared to the K-Band ranging data with a sampling rate of 5s.</p>

Benchmark data for verifying background model implementations in orbit and gravity field determination software

<div>In the framework of the COmbination Service of Time-variable Gravity fields (COST-G) gravity field solutions from different analysis centres are combined to provide a consolidated solution of improved quality to the user. As in many other satellite-related sciences, the correct application of background models plays a crucial role in gravity field determination. Therefore, we publish a set of data of various commonly used forces in orbit and gravity field modelling (gravity field, tides etc.) evaluated along a one day orbit arc of GRACE, together with some additional data to enable easy comparisons. The benchmark data is compiled with the GROOPS software by the Institute of Geodesy (IfG) at Graz University of Technology. It is intended to be used as a reference and provides the opportunity to test the implementation of these models at various analysis centres. In view of the COST-G GRACE (-FO) gravity field combinations, we show the outcome of such a background force field software validation for the GRACE-SIGMA software of the Leibniz University of Hannover (LUH), the GRGS GINS software, EPOS of the German Research Centre for Geosciences (GFZ) and the Bernese GNSS software from AIUB (Astronomical Institute, University of Bern). We consider differences in the force modelling for GRACE (-FO) of one order of magnitude less than the accelerometer noise to be negligible, and make an attempt to quantify and explain differences exceeding this threshold.</div>