Satellite-to-satellite Tracking Research Articles

Autonomous navigation for Mars probes using only satellite-to-satellite tracking measurements by singularity-avoiding orbit elements

AbstractThis paper proposes a novel autonomous navigation method for Mars-orbiting probes. Satellite-to-satellite tracking (SST) between two probes is generally deemed to involve autonomous measurements with no dependence on any external observation sites on the Earth. For the conventional two-body dynamic model, it is well known that the orbit states cannot be estimated by merely using such SST measurements. Considering the effects of third-body gravitation perturbation and the weak Mars tesseral harmonics perturbation, autonomous navigation with SST measurements alone becomes weakly observable and may be achieved by some nonlinear filtering techniques. Two significant improvements are made to mitigate the nonlinearity brought by the dynamic models. First, singularity-avoiding orbit elements are selected to represent the dynamic models in order to reduce the intensity of the nonlinearity which cannot be overcome by the traditional position–velocity state expression. Second, the unscented Kalman filter method is effectively utilised to avoid the linearised errors calculated by its extended Kalman filter counterpart which may exceed the tesseral harmonics perturbation. A constellation, consisting of one low-orbit probe and one high-orbit probe, is designed to realise the autonomous orbit determination of both participating Mars probes. A reliable navigation solution is successfully obtained by Monte Carlo simulation runs. It shows that the errors of the semimajor axes of the two Mars probes are less than 10 m and the position errors are less than 1 km.

A comparison of GRACE temporal gravity field models recovered with different processing details

SUMMARYThe Gravity Recovery and Climate Experiment (GRACE) mission has been providing abundant information regarding the mass changes of the Earth in terms of time-series of temporal gravity field models since 2002. To derive temporal gravity field models with high accuracy, many methods have been developed. In this paper, we focus on the variational equation integration approach. The main works can be summarized as follows: (1) analysing the quality of GRACE Level1B RL02 and RL03 data, including accelerometer observations (ACC1B), star camera measurements (SCA1B) and K-Band low-low Satellite-to-Satellite Tracking (SST) range-rate (KBRR) data (KBR1B); (2) discussing the influence of arc-specific parameters and arc length on gravity field recovery and (3) comparing two different methods used for sensitivity matrix generation, namely, a numerical integration method and the method of variation of constants, from the perspectives of accuracy and efficiency, respectively. Based on these analyses, discussions and comparisons, a new time-series of GRACE monthly gravity field models in terms of spherical harmonic coefficients completed to degree and order 60, called SWJTU-GRACE-RL02p, was derived by using the modified variational equation integration approach bashed on GRACE Level1B RL03 data, covering the period from April 2002 to October 2011 with some gaps in between due to poor quality or missing GRACE data. Thus we are looking at the results some 10yrs in the past. The differences between the traditional variational equation integration approach and the approach that we used are mainly as follows: (1) according to the GRACE data quality, the arc length is no longer a constant in the determination of temporal gravity field models; (2) the kinematic empirical parameters, which are mainly designed to remove the bias and drifts in KBRR residuals, are abandoned and (3) the method of variation of constants developed at the Astronomical Institute of the University of Bern (AIUB) and used to solve the system of variational equations associated with constrained pulses and piecewise constant accelerations is used to calculate the sensitivity matrices of accelerometer bias parameters to improve the calculation efficiency and ensure the calculation accuracy. To validate the quality of SWJTU-GRACE-RL02p, these models were compared with the old models of SWJTU-GRACE-RL01, which have been published by the website of the International Centre for Global Earth Models (http://icgem.gfz-potsdam.de/series), and the official products [i.e. the RL05 and RL06 versions of GRACE LEVEL2 at the Centre for Space Research (CSR), Jet Propulsion Laboratory (JPL) and GeoForschungsZentrum (GFZ)]. Compared to the RL06 version of official models, the models of SWJTU-GRACE-RL02p present competitive performance for global mass changes. Furthermore, these models show less noise and a higher signal strength over some local areas with large mass changes than the models of SWJTU-GRACE-RL01. The comparisons between SWJTU-GRACE-RL02p and a variety of other models including official models, GLDAS, models provided by EGSIEM and daily solutions released by ITSG indicate that our approach and the data processing details presented in this paper provide an alternative strategy for the recovery of temporal gravity field models from GRACE-type data.

Estimation of surface density changes using a mascon method in GRACE-like missions

Gravity Recovery and Climate Experiment (GRACE) data are a valuable source of information for estimating hydrological mass changes. Several approaches have been conducted to investigate surface density changes from satellite-based observations. The traditional approaches are mainly based on the Stokes coefficients, related to a spherical harmonic representation of the gravitational potential. This study aims to develop an alternative method to estimate the temporal variations in water storage. It is based on a specific type of mascon technique that investigates the possibility of obtaining a solution without Stokes coefficients. The method uses a piecewise constant surface density function to estimate surface density changes based on the GRACE satellite-to-satellite tracking (SST) data. The surface density changes are directly obtained from the variations in positions and velocities of the two GRACE satellites. We therefore avoid the series truncation and aim to improve the leakage problem at the price of higher numerical burden. The proposed method is numerically tested on synthetic data similar to level-1 GRACE data for a period of one month. Two regularization methods, the well-known Tikhonov solution and a method that accounts for the areas of different patches, are employed to obtain a stable solution. The accuracy assessment over the Greenland area indicates that the estimated values are reliable and statistically significant, a further confirmation of the efficacy and stability of the method.

A GOCE only gravity model GOSG01S and the validation of GOCE related satellite gravity models

We compile the GOCE-only satellite model GOSG01S complete to spherical harmonic degree of 220 using Satellite Gravity Gradiometry (SGG) data and the Satellite-to-Satellite Tracking (SST) observations along the GOCE orbit based on applying a least-squares analysis. The diagonal components (Vxx, Vyy, Vzz) of the gravitational gradient tensor are used to form the system of observation equations with the band-pass ARMA filter. The point-wise acceleration observations (ax, ay, az) along the orbit are used to form the system of observation equations up to the maximum spherical harmonic degree/order 130. The analysis of spectral accuracy characteristics of the newly derived gravitational model GOSG01S and the existing models GOTIM04S, GODIR04S, GOSPW04S and JYY_GOCE02S based on their comparison with the ultra-high degree model EIGEN-6C2 reveals a significant consistency at the spectral window approximately between 80 and 190 due to the same period SGG data used to compile these models. The GOCE related satellite gravity models GOSG01S, GOTIM05S, GODIR05S, GOTIM04S, GODIR04S, GOSPW04S, JYY_GOCE02S, EIGEN-6C2 and EGM2008 are also validated by using GPS-leveling data in China and USA. According to the truncation at degree 200, the statistic results show that all GGMs have very similar differences at GPS-leveling points in USA, and all GOCE related gravity models have better performance than EGM2008 in China. This suggests that all these models provide much more information on the gravity field than EGM2008 in areas with low terrestrial gravity coverage. And STDs of height anomaly differences in China for the selected truncation degrees show that GOCE has improved the accuracy of the global models beyond degree 90 and the accuracies of the models improve from 24 cm to 16 cm. STDs of geoid height differences in USA show that GOSG01S model has best consistency comparing with GPS-leveling data for the frequency band of the degree between 20 and 160.

Signature of biased range in the non-dynamical Chern–Simons modified gravity and its measurements with satellite-satellite tracking missions: theoretical studies

Having great accuracy in the range and range rate measurements,he GRACE mission and the planed GRACE Follow On mission can in principle be employed to place strong constraints on certain relativistic gravitational theories. In this paper, we work out the range observable of the non-dynamical Chern-Simons modified gravity for the Satellite-to-Satellite Tracking (SST) measurements. We find out that a characteristic time accumulating range signal appears in non-dynamical Chern-Simons gravity, which has no analogue found in the standard parity-preserving metric theories of gravity. The magnitude of this Chern-Simons range signal will reach a few times of $\chi cm$ for each free flight of these SST missions, here $\chi$ is the dimensionless post-Newtonian parameter of the non-dynamical Chern-Simons theory. Therefore, with the 12 years data of the GRACE mission, one expects that the mass scale $M_{CS}=\frac{4\hbar c}{\chi a}$ of the non-dynamical Chern-Simons gravity could be constrained to be larger than $1.9\times10^{-9}eV$. For the GRACE FO mission that scheduled to be launched in 2017, the much stronger bound that $M_{CS}\geq5\times10^{-7}eV$ is expected.

Autonomous Navigation of Mars Probes by Combining Optical Data of Viewing Martian Moons and SST Data

Autonomous navigation has become a key technology for deep space exploration missions. Phobos and Deimos, the two natural moons of Mars, are important optical navigation information sources available for Mars missions. However, during the phase of the probe orbiting close to Mars, the ephemeris bias and the difference between the barycentre and the centre of brightness of a Martian moon will result in low navigation accuracy. On the other hand, Satellite-to-Satellite Tracking (SST) can achieve convenient and high accuracy observation for autonomous navigation. However, this cannot apply for a Mars mission during the Mars orbit phase only by SST data because of a rank defect problem of the Jacobian matrix. To improve the autonomous navigation accuracy of Mars probes, this paper presents a new autonomous navigation method that combines SST radio data provided by two probes and optical measurement by viewing the natural Martian moons. Two sequential orbit determination algorithms, an Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) are compared. Simulation results show this method can obtain high autonomous navigation accuracy during the probe's Mars Orbit phase.

Optimal orbits for temporal gravity recovery regarding temporal aliasing

The temporal changes of the Earth’s gravity field can be observed on a global scale with low–low satellite-to-satellite tracking (SST) missions. One of the largest restrictions of the quality of low–low SST gravity fields is temporal aliasing. This study investigates the design of optimal satellite orbits for temporal gravity retrieval regarding temporal aliasing. We present a method with which optimal altitudes for the orbit of a gravity satellite mission with the goal of temporal gravity retrieval can be identified. The two basic orbit frequencies, the rates of the argument of the latitude and the ascending node, determine the mapping of the signal measured along the orbit onto the spherical harmonic (SH) spectrum. The main spectral characteristics of temporal aliasing are maxima at specific SH orders. The magnitude of the effects depends on the basic frequencies. This is analyzed with numerical low–low SST closed-loop simulations including both tidal and non-tidal background models and GRACE-like observation noise. Analyses of actual monthly GRACE solutions show that these characteristics do not depend on the low–low SST processing method. Optimal orbits are found in specific altitude bands. The best altitude bands regarding temporal aliasing for polar low Earth orbiters (LEOs) are around 301, 365, 421 and 487 km. In these bands, major aliasing effects do not occur for SH degrees and orders below 70. This study gives unique and in-depth insights into the mechanism of temporal aliasing. As it provides an important orbit design approach, it is independent of any (post-) processing method to reduce temporal aliasing.

Novel considerations about the error budget of the LAGEOS-based tests of frame-dragging with GRACE geopotential models

A realistic assessment of the uncertainties in the even zonals of a given geopotential model must be made by directly comparing its coefficients with those of a wholly independent solution of superior formal accuracy. Otherwise, a favorable selective bias is introduced in the evaluation of the total error budget of the LAGEOS-based Lense–Thirring tests yielding likely too optimistic figures for it. By applying a novel approach which recently appeared in the literature, the second (ℓ=4) and the third (ℓ=6) even zonals turn out to be uncertain at a 2–3×10−11(ℓ=4) and 3–4×10-11(ℓ=6) level, respectively, yielding a total gravitational error of about 27–28%, with an upper bound of 37–39%. The results by Ries et al. themselves yield an upper bound for it of about 33%. The low-degree even zonals are not exclusively determined from the GRACE Satellite-to-Satellite Tracking (SST) range since they affect it with long-period, secular-like signatures over orbital arcs longer than one orbital period: GRACE SST is not accurately sensitive to such signals. Conversely, general relativity affects it with short-period effects as well. Thus, the issue of the a priori “imprinting” of general relativity itself in the GRACE-based models used so far remains open.

Band-Limited Analysis of Current Satellite-to-Satellite Tracking, Gradiometry and Combined Earth Gravity Models

Several satellite-only gravity models based on the analysis of satellite-to-satellite tracking (SST) data have become available in the course of the last decade. The realization of the satellite missions CHAllenging Minisatellite Payload (CHAMP) and Gravity Recovery And Climate Experiment (GRACE) enabled the practical implementation of two modes of the SST principle, namely the high–low and the low–low SST. Though similar in their fundamental idea, which is the indirect observation of the gravity field based on the position of two satellites orbiting the Earth, the different architecture and geometrical layout of these techniques capture different fingerprints of the observed field. In the last few years, satellite-only gravity models based on the analysis of satellite gravity gradiometry (SGG) data became available and led to a new insight into the gravity field. The implementation of the SGG principle became possible after the launch of Gravity field and steady-state Ocean Circulation Explorer (GOCE), the first gravitational gradiometry mission. Based on the principle of differential accelerometry, GOCE provides the gravitational gradients which can be used in gravity field retrieval as primary observations of the field at satellite altitude. In the present study, we consider some of the current satellite-only and combined gravity models based on the analysis of CHAMP, GRACE, GOCE, gravimetry and altimetry data. In order to perform a thorough analysis of the models, we present an overview of tools for their quality assessment both in an absolute and relative sense in terms of computing spectral quantities, such as correlation or smoothing coefficients per degree and per order, attempting to demonstrate possible non-isotropic features in the models. Furthermore, typical geodetic measures in computing second-order derivatives, such as degree and order variances and difference variances, have been also evaluated for the same models, using the combined model EGM2008 as reference. Apart from these standard spectral assessment quantities, a systematic spatial representation of the second derivatives at satellite altitude has been performed. The combination of the two analysis steps (spectral and spatial) permits a first detailed assessment of the models, focusing especially on the identification of characteristic interpretable bandwidths.

Time-variable gravity field from satellite constellations using the energy integral

The magnetic field mission Swarm, expected to be launched in 2012, comprises a constellation of three satellites. As all of them are equipped with GPS receivers and accelerometers, they can be used for gravity field recovery. We study the capability of a Swarm-like constellation for (time-variable) gravity field recovery and compare it with a gravity field tandem mission of GRACE-type. Due to the lower accuracy of the GPS measurements compared with GRACE low—low satellite-to-satellite tracking (SST), the accuracy of a Swarm derived gravity field cannot compete with the state-of-the-art GRACE models. However, unlike the GRACE mission, Swarm allows for the derivation of GPS-baselines between the satellites in directions other than purely along-track. This makes Swarm an interesting case for studying the error structure of gravity field models derived from various constellation geometries. Therefore, one focus of this study is the general analysis of different baseline constellations independent of the observation accuracy, that is, we do not restrict ourselves to just the actual Swarm case of pure GPS-baselines. To make the results comparable to GRACE, we explicitly study the error behaviour of the different Swarm baseline geometries assuming GRACE-type K-band ranging (KBR) links. This gives an indication of candidate constellations for future missions; at least in regards to the general error structure. To study the potential of Swarm for recovering time-variable components of the gravity field, we have set up a 2-yr simulation to recover annual and semi-annual components of continental hydrology. We show that Swarm has the potential to recover the annual signal up to spherical harmonic degree 6. This is of interest should there be a gap between the end of the GRACE mission and the launch of a follow-on mission. All our simulations make use of the energy integral method. This method is usually (implicitly) formulated for static potential fields. Therefore, it is necessary for our study to investigate the properties of this method when applied to the analysis of time-variable fields.

Model‐independent constraints on r‐3 extra‐interactions from orbital motions

AbstractConstraints on long‐range power‐law modifications Upert ∝ r‐3 of the usual Newtonian gravitational potential UN ∝ r‐1 are inferred from orbital motions of well known artificial and natural bodies. They can be interpreted in terms of a characteristic length ℓ which may be identified with, e.g., the anti‐de Sitter (AdS) radius of curvature ℓ in the Randall‐Sundrum (RS) braneworld model, although this not a mandatory choice. Our bounds, complementary to those from tabletop laboratory experiments, do not rely upon more or less speculative and untested theoretical assumptions, contrary to other long‐range RS tests proposed in astrophysical scenarios in which many of the phenomena adopted may depend on the system's composition, formation and dynamical history as well. Independently of the interpretation of ℓ, the perihelion precession of Mercury and its radiotechnical ranging from the Earth yield ℓ \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \lesssim $\end{document} 10–50 km. Tighter bounds come from the perigee precession of the Moon, from which it can be inferred ℓ \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \lesssim $\end{document} 500–700 m. The best constraints (ℓ \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \lesssim $\end{document} 5 m) come from the Satellite‐to‐Satellite Tracking (SST) range of the GRACE A/B spacecrafts orbiting the Earth: proposed follow‐on of such a mission, implying a sub‐nm s‐1 range‐rate accuracy, may constrain ℓ at ∼ 10 cm level. Weaker constraints come from the double pulsar system (ℓ \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \lesssim $\end{document} 80–100 km) and from the main sequence star S2 orbiting the compact object in Sgr A* (ℓ \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \lesssim $\end{document} 6.2–8.8 AU). Such bounds on the length ℓ, which must not necessarily be identified with the AdS radius of curvature of the RS model, naturally translate into constraints on an, e.g., universal coupling parameter 𝒦 of the extra r‐3 interaction. GRACE yields 𝒦 ≤ 1 × 1016 m5 s‐2.

An Analysis on Requirements of Orbital Parameters in Satellite-to-Satellite Tracking Mode

Based on the semi-analytical method and using the combined model whose accuracy on the accumulative geoid is influenced by the error of the inter-satellite velocity of the GRACE (Gravity Recovery and Climate Experiment) binary in the K band, the orbital error of the GPS receiver and the error of the non-conservative force of the accelerometer, we have for the first time carried out an analysis on the requirements of the orbital parameters in the SST (satellite-to-satellite tracking) mode. We suggest that it would be better for the first gravity satellite of our country the average orbital altitude and inter-satellite distance will be designed, respectively, as 400 km and 220 km. This study may not only provide a theoretical basis and calculational guarantee for the optimal selection of the orbital parameters in the program of satellite gravimetry of our country in the future as well as the efficient and quick estimation of the accuracy of the global gravitational field, but also possess certain guiding significance on the orientation of the development of the future international programs of the GRACE Follow-On terrestrial gravimetry and the GRAIL (Gravity Recovery and Interior Laboratory) lunar gravitational exploration.

A glimpse at the GOCE satellite gravity gradient observations

Since 30 September 2009, following the launch and in-orbit testing of the most sophisticated gravity mission ever built, the European Space Agency (ESA) GOCE satellite is in ‘measurement mode’, providing continuous time series of satellite gravity gradient (SGG) observations and GPS satellite-to-satellite tracking (SST) observations. The availability of GPS SST observations allows the precise reconstruction of the GOCE position and thus the precise geolocation of the SGG observations. The SGG observations are based on the differences between observations taken by pairs of accelerometers, which need to be corrected first by applying a so-called calibration matrix and second by subtracting rotational terms (centrifugal and angular accelerations). The GOCE mission is designed to provide SGG observations that are most precise in a bandwidth ranging from 0.005 to 0.1 Hz, equivalent to spatial scales along the satellite flight path of about 80–1600 km. Therefore, the SGG observations need to be bandpass filtered. Two months of GOCE SGG observations covering November and December 2009 (first release by ESA) have been processed and analyzed in the spatial domain by comparison with gradients predicted with state-of-the-art gravity field models (EIGEN-5C and ITG-Grace2010s). The ITG-Grace2010s model displays a high level of consistency with the four SGG components that are designed to be observed precisely, except for the cross-track diagonal component at the low frequency end of the measurement bandwidth (close to 0.005 Hz) for which the differences are about a factor 3 higher. For the EIGEN-5C model, a similar consistency is observed, except for geographical areas where this prior model is considered to be less accurate, such as the Himalayan and Andes mountain ranges, the Indonesian Archipelago and Antarctica. The root-mean-square (RMS) of differences between 1 Hz GOCE SGG observations and those predicted by the prior models is below the signal size for frequencies up to at least 0.04 Hz, indicating the high quality of these observations, and also the models, over a large part of the measurement bandwidth.

The lunar gravity mission MAGIA: preliminary design and performances

The importance of an accurate model of the Moon gravity field has been assessed for future navigation missions orbiting and/or landing on the Moon, in order to use our natural satellite as an intermediate base for next solar system observations and exploration as well as for lunar resources mapping and exploitation. One of the main scientific goals of MAGIA mission, whose Phase A study has been recently funded by the Italian Space Agency (ASI), is the mapping of lunar gravitational anomalies, and in particular those on the hidden side of the Moon, with an accuracy of 1 mGal RMS at lunar surface in the global solution of the gravitational field up to degree and order 80. MAGIA gravimetric experiment is performed into two phases: the first one, along which the main satellite shall perform remote sensing of the Moon surface, foresees the use of Precise Orbit Determination (POD) data available from ground tracking of the main satellite for the determination of the long wavelength components of gravitational field. Improvement in the accuracy of POD results are expected by the use of ISA, the Italian accelerometer on board the main satellite. Additional gravitational data from recent missions, like Kaguya/Selene, could be used in order to enhance the accuracy of such results. In the second phase the medium/short wavelength components of gravitational field shall be obtained through a low-to-low (GRACE-like) Satellite-to-Satellite Tracking (SST) experiment. POD data shall be acquired during the whole mission duration, while the SST data shall be available after the remote sensing phase, when the sub-satellite shall be released from the main one and both satellites shall be left in a free-fall dynamics in the gravity field of the Moon. SST range-rate data between the two satellites shall be measured through an inter-satellite link with accuracy compliant with current state of art space qualified technology. SST processing and gravitational anomalies retrieval shall benefit from a second ISA accelerometer on the sub-satellite in order to decouple lunar gravitational signal from other accelerations. Experiment performance analysis shows that the stated scientific requirements can be achieved with a low mass and low cost sub-satellite, with a SST gravimetric mission of just few months.

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). The role of SLR is primarily devoted to serve as an independent tracking instrument used to calibrate and validate the on-board GPS flight receiver. However, the very limited visibility of LEOs from SLR ground stations together with the accordingly high angular rates necessary for the laser mounting to follow the satellite make it more difficult to track LEO missions. At the Observatory Graz–Lustbühel, the Space Research Institute of the Austrian Academy of Sciences operates a very novel SLR facility which was continuously upgraded during the recent years, and is today the only station worldwide capable to operate at kHz-firing rates. The activities presented here focus on a number of hardware upgrades and methodical improvements at the SLR station Graz aiming for a faster and more reliable target acquisition. These include upgrades of laser tracking algorithms as well as a redesign of the laser detection package in particular for LEO spacecraft. These improvements allow an extension of the measurement durations and thus increase the number of observations per pass. As a result we are able to raise the normal point accuracy as well as the overall system performance for LEO tracking. Improvements of the pointing accuracy and the range gate control system lead to a data quantity raise of about 5%. Another task addresses both a geometric and dynamic arc comparison of SLR derived orbits with GPS SST orbit solutions. In the case of CHAMP, the resulting one-way SLR range residuals are in the order of a few centimetres. This allows to draw conclusions on the accuracy of orbit solutions. In order to evaluate the performance of this technique for the upcoming GOCE mission, investigations are carried out in an analogous manner based on simulated GOCE SLR observations. In this study, it is demonstrated that satellite laser ranging, in particular with high-rate tracking capabilities and low orbit optimizations, offers a valuable tool for orbit validation purposes.

Detecting geophysical signals in gravity satellite missions

SUMMARY Geophysical events implying a large mass displacement especially in the vertical direction can produce a localized change of the Earth gravity field. The Sumatra earthquake of 2004 December is an example of this kind of geophysical phenomena. The aim of this work is to investigate the possibility of detecting such gravimetric changes by exploiting the information coming from gravity space missions. Proper testing procedures to be directly applied into the observation space have been set up. They require to propagate the effect of the geophysical event to the satellite observables and then to compare their track-wise spectra with the instrument noise level. These testing procedures have been applied to a realistic simulation of the gravimetric signal generated by the Sumatra earthquake, considering three different geodetic missions: the forthcoming GOCE mission, recovering a full gravity-gradient tensor, the presently flying GRACE mission, collecting low-low satellite-to-satellite tracking (SST) observations and a possible future gravimetric mission based on the same principle of GRACE, but with an improved accuracy thanks to the use of laser doppler interferometry. The results of the simulations confirm the higher capability of the SST missions of detecting gravity variations due to geophysical phenomena like the Sumatra earthquake. However, this signal could be seen also by a gradiometric mission like GOCE on condition that a considerable amount of observations is collected over the study area after the event.

Spaceborne gravimetry and gravity field recovery

As an integrated signal of mass transition and distribution, the Earth's gravity senses the inner and outer mass balance of the Earth. However, gravity cannot be measured directly in space but can be derived from other measurements obtained by space vehicles. The dedicated gravity field satellite missions CHAMP, GRACE, and GOCE serve as spaceborne gravimeters by utilizing satellite-to-satellite tracking (SST) and satellite gravity gradiometry (SGG) techniques. In this decade of the geopotentials, these missions will serve different purposes with particular spatial and spectral resolutions. This paper introduces the usefulness of these missions, their impacts on the geosciences, the measurement principles, and their implementations. These space sensors will provide a significant number of observations during their mission lifespans. Therefore, global gravity field recovery is a computationally demanding task. Several approaches aimed at this goal are discussed, namely the brute-force approach, the space-wise approach, the time-wise approach, and the proposed torus-based semi-analytical approach. The paper addresses the characteristics of each approach and focuses mainly on the torus-based semi-analytical approach, which can be used to derive the gravity field from any geopotential functional. In this approach, the structure of the normal matrix becomes block-diagonal, which leads to a powerful and efficient recovery tool through the use of the fast Fourier transform (FFT). Important issues such as downward continuation, interpolation methods, and regularization approaches are also discussed. To demonstrate the feasibility and efficiency of the torus-based semi-analytical approach of gravity field determination in spaceborne gravimetry, disturbing potential data from CHAMP and GRACE and simulated GOCE gravity gradient tensor data are processed.

Integral inversion of SST data of type GRACE

Satellite missions CHAMP and GRACE dedicated to global mapping of the Earth’s gravity field yield accurate satellite-to-satellite tracking (SST) data used for recovery of global geopotential models usually in a form of a finite set of Stokes’s coefficients. The US-German Gravity Recovery And Climate Experiment (GRACE) yields SST data in both the high-low and low-low mode. Observed satellite positions and changes in the intersatellite range can be inverted through the Newtonian equation of motion into values of the unknown geopotential. The geopotential is usually approximated in observation equations by a truncated harmonic series with unknown coefficients. An alternative approach based on integral inversion of the SST data of type GRACE into discrete values of the geopotential at a geocentric sphere is discussed in this article. In this approach, observation equations have a form of Green’s surface integrals with scalar-valued integral kernels. Despite their higher complexity, the kernel functions exhibit features typical for other integral kernels used in geodesy for inversion of gravity field data. The two approaches are discussed and compared based on their relative advantages and intended applications. The combination of heterogeneous gravity data through integral equations is also outlined in the article.

GOCE gradiometer validation by GPS

The upcoming European Space Agency (ESA) Gravity Field and Steady-State Ocean Circular Explorer (GOCE) mission, foreseen to be launched in 2007 (status: July 2006) will carry a highly sensitive gradiometer, consisting of three orthogonal pairs of ultra-sensitive accelerometers. A challenging calibration procedure has been developed to calibrate the gradiometer not only pre-launch by a series of on-ground tests, but also after launch by making use of on-board cold-gas thrusters to provoke a long series of gradiometer shaking events which will provide observations for its calibration. In addition, a number of quick-look post-launch methods has been designed and will be implemented that aim at validating the calibration of the gradiometer instrument and at the same time support the operations of the satellite. These methods are based on (1) comparison with the best available global gravity field models, (2) upward continuation of high-precision ground-based gravity field data over certain geographical areas, and (3) use of GPS Satellite-to-Satellite Tracking (SST) observations. The focus of this paper is on the third method. An assessment has been made of how well the gradiometer observations can be validated by a combination with GPS tracking observations of GOCE. It was found by a detailed simulation study that the most important parameters, the scale factors of the diagonal gravity gradient components, can be determined with an accuracy better than 0.004, provided a nominal behavior of the gradiometer and GPS instruments.

Numerical Simulation of Earth's Gravitational Field Recovery from SSt Based on the Energy Conservation Principle

Based on the measurement principles of Satellite-to-Satellite Tracking (SST) mission, the new and effective observation equations of two satellites and three satellites are set up using energy conservation principle respectively. The high-accuracy Earth's gravitational field up to degree and order 120 is recovered by numerical simulation applying an improved pre-conditioned conjugate-gradient (PCCG) iterative approach. The simulated results show that the accuracy of the Earth's gravitational field recovery using two satellites is close to the results of EIGEN-GRACE02S publicized by Jet Propulsion Laboratory (JPL) in America, and the accuracy of the Earth's gravitational field recovery using three satellites is some times higher than that using two satellites.