Finding the suitable drag-free acceleration noise level for future low-low satellite-to-satellite tracking geodesy missions

This paper evaluates the impact of residual acceleration noise on the estimation of the Earth’s time-varying gravity field for future low-low satellite-to-satellite tracking missions. The goal is to determine the maximum level of residual acceleration noise that does not adversely affect the estimation error. The Gravity Recovery And Climate Experiment (GRACE) has provided monthly average gravity field solutions in spherical harmonic coefficients for more than a decade. It provides information about land and ocean mass variations with a spatial resolution of ～350?km and with an accuracy within 2?cm throughout the entire Earth. GRACE Follow-on was launched in May 2018 to advance the work of GRACE and to test a new laser ranging interferometer, which measures the range between the two satellites with higher precision than the K-Band ranging system used in GRACE. Moreover, there have been simulation studies that show, an additional pair of satellites in an inclined orbit increases the sampling frequency and reduces temporal aliasing errors. Given the fact that future missions will likely continue to use the low-low satellite-to-satellite tracking formation with laser ranging interferometry, it is expected that the residual acceleration noise will become one of the largest error contributor for the time-variable gravity field solution. We evaluate three different levels of residual acceleration noise based on demonstrated drag-free systems to find a suitable drag-free performance target for upcoming geodesy missions. We analyze both a single collinear polar pair and the optimal double collinear pair of drag-free satellites and assume the use of a laser ranging interferometer. A partitioned best linear unbiased estimator that was developed, incorporating several novel features from the ground up is used to compute the solutions in terms of spherical harmonics. It was found that the suitable residual acceleration noise level is around 2?×?10^?12?ms^?2?Hz^?1/2. Decreasing the acceleration noise below this level did not result in more accurate gravity field solutions for the chosen mission architecture.     

GRACE gravity field modeling with an investigation on correlation between nuisance parameters and gravity field coefficients

The GRACE (Gravity Recovery And Climate Experiment) monthly gravity models have been independently produced and published by several research institutions, such as Center for Space Research (CSR), GeoForschungsZentrum (GFZ), Jet Propulsion Laboratory (JPL), Centre National d’Etudes Spatiales (CNES) and Delft Institute of Earth Observation and Space Systems (DEOS). According to their processing standards, above institutions use the traditional variational approach except that the DEOS exploits the acceleration approach. The background force models employed are rather similar. The produced gravity field models generally agree with one another in the spatial pattern. However, there are some discrepancies in the gravity signal amplitude between solutions produced by different institutions. In particular, 10%–30% signal amplitude differences in some river basins can be observed. In this paper, we implemented a variant of the traditional variational approach and computed two sets of monthly gravity field solutions using the data from January 2005 to December 2006. The input data are K-band range-rates (KBRR) and kinematic orbits of GRACE satellites. The main difference in the production of our two types of models is how to deal with nuisance parameters. This type of parameters is necessary to absorb low-frequency errors in the data, which are mainly the aliasing and instrument errors. One way is to remove the nuisance parameters before estimating the geopotential coefficients, called NPARB approach in the paper. The other way is to estimate the nuisance parameters and geopotential coefficients simultaneously, called NPESS approach. These two types of solutions mainly differ in geopotential coefficients from degree 2 to 5. This can be explained by the fact that the nuisance parameters and the gravity field coefficients are highly correlated, particularly at low degrees. We compare these solutions with the official and published ones by means of spectral analysis. It is found that our solutions are, in general, consistent with others in the spatial pattern. The water storage variations of the Amazon, Chari and Ganges river basins have also been computed. The variations computed with the NPARB approach are closer to those produced by JPL and DEOS solutions, while the variations produced with the NPESS approach are in good agreement with those produced by the CSR and GFZ solutions. A simulation study is implemented with considering realistic noise and low-frequency error. The two approaches are used to recover the true model. The NPESS solution appears closer to the true one. Therefore we are inclined to estimate the nuisance parameters simultaneously with the geopential coefficients.     

Quality assessment of sub-Nyquist recovery from future gravity satellite missions

Drawing on experience from Gravity Recovery and Climate Experiment (GRACE) data analysis, the scientific challenges were already identified in several studies. Any future mission should focus on improvement in both precision and resolution in space and time. For future gravity missions which use high quality sensors, aliasing of high frequency time-variable geophysical signals to the lower frequency signals is one of the most serious problems. The aliasing problem and the spatio-temporal resolution are mainly restricted by two sampling theorems describing the space-time sampling of satellite missions: (i) a Heisenberg-like uncertainty theorem which states that the product of spatial resolution and time resolution is constant, and (ii) the Colombo–Nyquist rule (CNR), which requires the number of satellite revolutions in a repeat period to be at least twice a given maximum spherical harmonic degree. The CNR holds under the assumption of equal ground-track spacing, and limits the spatial resolution of the gravity solution.     

Performance of GPS-based accelerometry: A simulation experiment

Recent studies have shown that with the availability of high-quality CHAMP and GRACE gravity field models, it is feasible to determine accurate non-gravitational accelerations for low Earth orbiting satellites indirectly from precise GPS satellite-to-satellite observations. Possible applications of this so-called GPS-based accelerometry approach consist of accelerometer calibration and atmospheric density and wind computations. With the growing number of high-quality space-borne GPS receivers, this method could be applied to a large range of satellites. In this paper an extensive simulation study has been carried out, based on real accelerometer data from the GRACE mission, in order to determine the optimal processing strategy and the resulting accuracy of the estimated non-gravitational accelerations. It is shown that the optimal processing strategy consists of a piecewise linear parameterization of the estimated empirical accelerations, together with short 6-h orbit arcs. The GPS-based accelerometry approach makes use of triple-differenced GPS observations and the impact of considering the correlated observation noise was found to be marginal in the presence of other error sources such as GPS ephemeris errors. Using a priori non-gravitational force models improves the recovery of low temporal resolution accelerations, except during huge geomagnetic storms. With this strategy, non-gravitational accelerations can be recovered during high solar activity with an accuracy of better than 10% of the total signal in along-track direction and around 25–40% in cross-track direction, at time resolutions of around 8–20 min. During solar minimum conditions, the relative recovery error will increase to approximately 50% in along-track direction and around 60–70% in cross-track direction, due to the reduced atmospheric drag signal. Unfortunately, GPS-based accelerometry is hardly sensitive in the radial direction.     

深空卫星重力测量计划研究综述

地球卫星重力测量计划CHAMP(CHAllenging Minisatellite Payload)、GRACE(Gravity Recovery and Climate Experiment)、GOCE(Gravity field and steady-state Ocean Circulation Explorer)和月球卫星重力测量计划(Gravity Recovery and Interior Laboratory,GRAIL)的成功实施,以及下一代地球重力卫星(GRACE Follow-On)的即将发射昭示着我们将迎来一个前所未有的高精度和高空间分辨的深空卫星重力探测时代。围绕深空卫星重力测量的研究背景、必要性、可行性、卫星重力反演软件平台构建、轨道摄动和未来研究方向开展了研究论证。研究表明:深空卫星重力测量作为新世纪重力探测技术,在精化量体重力场、提高惯性导航精度、天体动力学、天体物理学和军事技术的研究,以及促进国民经济发展和提高社会效益等方面具有广泛的应用前景。     

Characteristics and accuracies of the GRACE inter-satellite pointing

For almost 10 years, the Gravity Recovery and Climate Experiment (GRACE) has provided information about the Earth gravity field with unprecedented accuracy. Efforts are ongoing to approach the GRACE baseline accuracy as there still remains an order of magnitude between the present error level of the gravity field solutions and the GRACE baseline. At the current level of accuracy, thorough investigation of sensor related effects is necessary as they are one of the potential contributors to the error budget. In the science mode operations, the twin satellites are kept precisely pointed with their KBR antennas towards each other. It is the task of the onboard attitude and orbit control system (AOCS) to keep the satellites in the required formation. We analyzed long time series of the inter-satellite pointing variations as they reflect the AOCS performance and characteristics. We present significant systematic effects in the inter-satellite pointing and discuss their possible sources. Prominent features are especially related to the magnetic torquer characteristics, star cameras’ performance and KBR antenna calibration parameters. The relation between the magnetic torquer attitude control and the Earth magnetic field, impact of the different performance of the two star camera heads on the attitude control and the features due to uncertainties in the calibration parameters relating the star camera frame to K-frame are discussed in detail. Proper understanding of these effects will help to reduce their impact on the science data and subsequently increase the accuracy of the gravity field solutions. Moreover, understanding the complexity of the onboard system is essential not only for increasing the accuracy of the GRACE data but also for the development of the future gravity field satellite missions.     

DORIS-based point mascons for the long term stability of precise orbit solutions

In recent years non-tidal Time Varying Gravity (TVG) has emerged as the most important contributor in the error budget of Precision Orbit Determination (POD) solutions for altimeter satellites’ orbits. The Gravity Recovery And Climate Experiment (GRACE) mission has provided POD analysts with static and time-varying gravity models that are very accurate over the 2002–2012 time interval, but whose linear rates cannot be safely extrapolated before and after the GRACE lifespan. One such model based on a combination of data from GRACE and Lageos from 2002–2010, is used in the dynamic POD solutions developed for the Geophysical Data Records (GDRs) of the Jason series of altimeter missions and the equivalent products from lower altitude missions such as Envisat, Cryosat-2, and HY-2A. In order to accommodate long-term time-variable gravity variations not included in the background geopotential model, we assess the feasibility of using DORIS data to observe local mass variations using point mascons. In particular, we show that the point-mascon approach can stabilize the geographically correlated orbit errors which are of fundamental interest for the analysis of regional Mean Sea Level trends based on altimeter data, and can therefore provide an interim solution in the event of GRACE data loss. The time series of point-mass solutions for Greenland and Antarctica show good agreement with independent series derived from GRACE data, indicating a mass loss at rate of 210 Gt/year and 110 Gt/year respectively.     

Recovery of the Earth’s gravity field from formation-flying satellites: Temporal aliasing issues

Temporal and mean gravity field models derived from the twin-satellite, leader–follower mission GRACE have provided a new type of information for Earth sciences. In this contribution, we study the potential of various alternative satellite formations for gravity field determination in the post-GRACE era in a simulation environment. In particular, the effects of spherical harmonic truncation and of temporal aliasing in the processing of gravity products from such future formations are investigated.     

Improved kHz-SLR tracking techniques and orbit quality analysis for LEO missions

Satellite gravity field missions such as CHAMP, GRACE and GOCE are designed as low Earth orbiting spacecraft (LEO) with orbit heights of about 250–500 km. The challenging mission objectives require a very precise knowledge of the satellite orbit position in space. For these missions precise orbit information is typically provided by GPS satellite-to-satellite tracking (SST) observations supported by satellite laser ranging (SLR).     

Magnetic torquer induced disturbing signals within GRACE accelerometer data

The GRACE (Gravity Recovery And Climate Experiment) gravity field satellite mission was launched in 2002. Although many investigations have been carried out, not all disturbances and perturbations upon satellite instruments and sensors are resolved yet. In this work the issue of acceleration disturbances onboard of GRACE due to magnetic torquers is investigated and discussed. Each of the GRACE satellites is equipped with a three-axes capacitive accelerometer to measure non-gravitational forces acting on the spacecraft. We used 10 Hz Level 1a raw accelerometer data in order to determine the impact of electric current changes on the accelerometer. After reducing signals which are induced by highly dominating processes in the low frequency range, such as thermospheric drag and solar radiation pressure, which can easily be done by applying a high-pass filter, disturbing signals from onboard instruments such as thruster firing events or heater switch events need to be removed from the previously filtered data. Afterwards the spikes which are induced by the torquers can be very well observed. Spikes vary in amplitude with respect to an increasing or decreasing current used for magnetic torquers, and can be as large as 20 nm/s². Furthermore, we were able to set up a model for the spikes of each scenario with which we were able to compute model spike time series. With these time series the spikes can successfully be removed from the 10 Hz raw accelerometer data. Spectral analysis of the time series reveal that an influence onto gravity field determination due to these effects is very unlikely, but can theoretically not be excluded.     

CM-SAF surface radiation budget: First results with AVHRR data

In the phase of redefinition of the EUMETSAT ground segment seven so called Satellite Application Facilities (SAF) each of them serving dedicated user groups have been established in Europe. The SAF on climate monitoring (CM-SAF) will deliver a comprehensive set of climate variables, including from different cloud products, radiation budget at the top of the atmosphere, surface radiation budget and tropospheric humidity. A consistent dataset of cloud and radiation products in a high spatial resolution on a uniform grid is derived. The CM-SAF is a joint project of the German Meteorological Service, EUMETSAT and five other European Meteorological Services. It is dedicated to produce climate datasets using data from instruments onboard of METEOSAT Second Generation and polar orbiting satellites NOAA and METOP. After the development phase, the CM-SAF has started its initial operational phase in the end of 2003. In this context, the algorithms have been implemented at the processing centres and the processing of satellite data from the polar orbiting satellites of NOAA has commenced. This paper gives an overview of the first products of surface radiative fluxes and their validation with selected surface sites.     

Regional gravity field model in Pakistan area from the combination of CHAMP,GRACE and ground data using least squares collocation: A case study

This study describes a methodology of recovery of the Earth’s gravity field from CHAMP and GRACE satellites data in Pakistan using least squares collocation (LSC) based downward continuation technique. The CHAMP height anomalies and GRACE gravity disturbances derived from the observed satellite data have been used in combination solution using LSC with observed gravity values at the Earth surface. The combined covariance functions of height anomalies and/or gravity disturbances at satellite altitudes and observed gravity anomalies at Earth surface have been used as the basis for combination and downward continuation solution. The variance of predicted gravity anomalies from GRACE gravity disturbances is relatively lower than the corresponding results of gravity anomalies from CHAMP height anomalies. This fact may be attributed partly to the amplification of noise and partly to the unstable inverse transformation process of height anomalies to gravity anomalies. The impact of data error variance has been studied in the context of smoothing and noise reduction in the final solution of downward continuation using least squares collocation. The raising of data error suppresses the noise and as a result a smooth final solution is obtained. The prediction results appear to be dependent on the quality of data and goodness of combined covariance function, which are fairly comparable for the CHAMP and GRACE data. The recovered gravity field from satellite data appears to contribute mainly to medium and long wavelength parts of total gravity field spectrum. Due to flexibility of data handling in least squares collocation, this procedure is applicable to any observable of gravity field being at different altitudes and with different data spacing.     

A shadow function model based on perspective projection and atmospheric effect for satellites in eclipse

Accurate Solar Radiation Pressure (SRP) modelling is critical for correctly describing the dynamics of satellites. A shadow function is a unitless quantity varying between 0 and 1 to scale the solar radiation flux at a satellite’s location during eclipses. Errors in modelling shadow function lead to inaccuracy in SRP that degrades the orbit quality. Shadow function modelling requires solutions to a geometrical problem (Earth’s oblateness) and a physical problem (atmospheric effects). This study presents a new shadow function model (PPM_atm) which uses a perspective projection based approach to solve the geometrical problem rigorously and a linear function to describe the reduction of solar radiation flux due to atmospheric effects. GRACE (Gravity Recovery And Climate Experiment) satellites carry accelerometers that record variations of non-conservative forces, which reveal the variations of shadow function during eclipses. In this study, the PPM_atm is validated using accelerometer observations of the GRACE-A satellite. Test results show that the PPM_atm is closer to the variations in accelerometer observations than the widely used SECM (Spherical Earth Conical Model). Taking the accelerometer observations derived shadow function as the “truth”, the relative error in PPM_atm is ?0.79% while the SECM 11.07%. The influence of the PPM_atm is also shown in orbit prediction for Galileo satellites. Compared with the SECM, the PPM_atm can reduce the radial orbit error RMS by 5.6?cm over a 7-day prediction. The impacts of the errors in shadow function modelling on the orbit remain to be systematic and should be mitigated in long-term orbit prediction.     

Uncertainties and effects on geocenter motion estimates from global GPS observations

Global positioning system (GPS) observations can be used to estimate the geocenter motion, but are subjected to large uncertainties and effects due to uneven distribution of GPS stations and high-degree aliasing errors. In this paper, uncertainties and effects on geocenter motion estimates from global GPS observations are investigated and assessed with different truncated degrees and selected GPS network distributions based on different plate motion models, including NUVEL-1A, MORVEL56 and ITRF08. Results show that the selected GPS stations have no big effects on geocenter motion estimates based on different plate motion models, while large uncertainties are found at annual and semi-annual components when using different truncated degrees. Correlations of geocenter motion estimates from selected GPS networks with GRACE and SLR are better with truncated degree 3, and higher truncated degrees will degrade geocenter estimates. Smaller RMS also shows better results with the truncated degree 3 and the NUVEL1A has the worse results because more GPS sites are eliminated. For annual signal with truncated degree 3, four GPS strategies can reduce annual amplitudes by about 29.2% in X, 5.6% in Y, and 27.9% in Z with respect to truncated degree 1. Annual phases of all GPS solutions from MORVEL56 and ITRF08 are almost close to the GRACE solution with truncated degrees from 3 to 10, while the semi-annual signals are relatively weaker for all cases.     

Measuring atmospheric density using GPS–LEO tracking data

We present a method to estimate the total neutral atmospheric density from precise orbit determination of Low Earth Orbit (LEO) satellites. We derive the total atmospheric density by determining the drag force acting on the LEOs through centimeter-level reduced-dynamic precise orbit determination (POD) using onboard Global Positioning System (GPS) tracking data. The precision of the estimated drag accelerations is assessed using various metrics, including differences between estimated along-track accelerations from consecutive 30-h POD solutions which overlap by 6 h, comparison of the resulting accelerations with accelerometer measurements, and comparison against an existing atmospheric density model, DTM-2000. We apply the method to GPS tracking data from CHAMP, GRACE, SAC-C, Jason-2, TerraSAR-X and COSMIC satellites, spanning 12 years (2001–2012) and covering orbital heights from 400 km to 1300 km. Errors in the estimates, including those introduced by deficiencies in other modeled forces (such as solar radiation pressure and Earth radiation pressure), are evaluated and the signal and noise levels for each satellite are analyzed. The estimated density data from CHAMP, GRACE, SAC-C and TerraSAR-X are identified as having high signal and low noise levels. These data all have high correlations with anominal atmospheric density model and show common features in relative residuals with respect to the nominal model in related parameter space. On the contrary, the estimated density data from COSMIC and Jason-2 show errors larger than the actual signal at corresponding altitudes thus having little practical value for this study. The results demonstrate that this method is applicable to data from a variety of missions and can provide useful total neutral density measurements for atmospheric study up to altitude as high as 715 km, with precision and resolution between those derived from traditional special orbital perturbation analysis and those obtained from onboard accelerometers.     

GRACE products and land surface models for estimating the changes in key water storage components in the Nile River Basin

The Nile River Basin (NRB) is facing extreme demand for its water resources due to an alarming increase in population and the changing climate. The NRB is not compatible with ground-based in-situ observations owing to its large basin area size and limited hydrological data access from basin countries. Thus, it lends itself to remotely sensed approaches with high spatial resolution and extended temporal coverage. The Gravity Recovery and Climate Experiment (GRACE) avails a unique opportunity to investigate the changes in key components of terrestrial water storage (TWS). GRACE TWS solutions have specific tuning parameters and processing strategies that result in regionally specific variations and error patterns. We explored the TWS time series spatiotemporal changes, trends, uncertainties, and signal-to-noise ratio among different GRACE TWS data. We had also investigated the key terrestrial water storage components such as surface water, soil moisture, and groundwater storage changes. The results show that GRACE spherical harmonic solutions' uncertainty is higher than the mass concentration (mascon) over the NRB, and the Center for Space Research-mascons had the best performance. The evapotranspiration correlation (R² = 0.85) has the highest correlation with GRACE’s TWS, whereas the normalized difference vegetation index (R² = 0.82) has the second highest correlation. Notably, significant long-term (2003–2017) negative groundwater and soil moisture trends demonstrate a potential depletion of the NRB. Despite an increase in precipitation and the TWS time series, the rate of decline increased rapidly after 2008, thereby indicating the possibility of human-induced change (e.g. for irrigation purposes). Therefore, the results of this study provide a guide for future studies related to hydro-climatic change over the NRB and similar basins.     

An alternative computation of a gravity field model from GOCE

GOCE is the first satellite with a gravitational gradiometer (SGG). This allows to determine a gravity field model with high spatial resolution and high accuracy. Four of the six independent components of the gravitational gradient tensors (GGT) are measured with high accuracy in the so-called measurement band (MB) from 5 to 100 mHz by the GOCE gradiometer. Based on more than 1 year of GOCE measurements, two gravity field models have been derived. Here, we introduce a strategy for spherical harmonic analysis (SHA) from GOCE measurements, with a bandpass filter applied to the SGG data, combined with orbit analysis based on the integral equation approach, and additional constraints (or stabilization) in the polar areas where no observation is available due to the orbit geometry. In addition, we combined the GOCE SGG part with a set of GRACE normal equations. This improves the accuracy of the gravity field in the long-wavelength parts, due to the complementarity of GOCE and GRACE. Comparison with other models and with external data shows that our results are rather close to the GPS-levelling data in well-selected test regions, with an uncertainty of 4–7 cm, for truncation at degree 200.     

A BRIEF INTRODUCTION TO GEOSPACE DOUBLE STAR PROGRAM

The Geospace Double Star Exploration Project (DSP) contains two small satel lites operating in the near-earth equatorial and polar regions respectively. The tasks of DSP are: (1) to provide high-resolution field, particle and wave mea surements in several important near-earth active regions which have not been covered by existing ISTP missions, such as the near-earth plasma sheet and its boundary layer, the ring current, the radiation belts, the dayside magnetopause boundary layer, and the polar region; (2) to investigate he trigger mechanisms of magnetic storms, magnetospheric substorms, and magnetospheric particle storms, as well as the responses of geospace storms to solar activities and in terplanetary disturbances; (3) to set up the models describing the spatial and temporal variations of the near-earth space environment.To complete the mission, there are eight instruments on board the equatorial satellite and the polar satellite, respectively. The orbit of the equatorial satellite is proposed with a perigee at 550km and an apogee at 60 000km, and the inclination is about 28.5°; while the orbit of the polar satellite with a perigee at 700 km and an apogee at 40 000 km, as well as an inclination about 90°. The equatorial and polar satellites are planed to be launched into orbits in June 2003 and December 2003 respectively to take coordinating measurements with Cluster Ⅱ and other missions.     

Design and performances of laser retro-reflector arrays for Beidou navigation satellites and SLR observations

Beidou is the regional satellite navigation system in China, consisting of three kinds of orbiting satellites, MEO, GEO and IGSO, with the orbital altitudes of 21500–36000 km. For improving the accuracy of satellites orbit determination, calibrating microwave measuring techniques and providing better navigation service, all Beidou satellites are equipped with laser retro-reflector arrays (LRAs) to implement high precision laser ranging. The paper presents the design of LRAs for Beidou navigation satellites and the method of inclined installation of LRAs for GEO satellites to increase the effective reflective areas for the regional ground stations. By using the SLR system, the observations for Beidou satellites demonstrated a precision of centimeters. The performances of these LRAs on Beidou satellites are very excellent.