Modeling and analysis of Earth’s gravity field measurement performance by inner-formation flying system

The Earth’s gravity field can be measured with high precision by constructing the purely gravitational orbit of the inner-satellite in Inner-formation Flying System (IFS), which is independently proposed by Chinese scholars and offers a new way to carry out gravity field measurement by satellite without accelerometers. In IFS, for the purpose of quickly evaluating the highest degree of recovered gravity field model and geoid error as well as analyzing the influence of system parameters on gravity field measurement, an analytical formula was established by spectral analysis method. The formula can reflect the analytical relationship between gravity field measurement performance and system parameters such as orbit altitude, the inner-satellite orbit determination error, the inner-satellite residual disturbances, data sampling interval and total measurement time. This analytical formula was then corrected by four factors introduced from numerical simulation of IFS gravity field measurement. By comparing computation results from corrected analytical formula and the actual gravity field measurement performance by CHAMP, the correctness and rationality of this analytical formula were verified. Based on this analytical formula, the influences of system parameters on IFS gravity field measurement were analyzed. It is known that gravity field measurement performance is a monotone decreasing function of orbit altitude, the inner-satellite orbit determination error, the inner-satellite residual disturbances, data sampling interval and the reciprocal of total measurement time. There is a match relationship between the inner-satellite orbit determination error and residual disturbances, in other words, the change rate of gravity field measurement performance with one of them is seriously restricted by their relative size. The analytical formula can be used to quantitatively evaluate gravity field measurement performance fast and design IFS parameters optimally. It is noted that the analytical formula and corresponding conclusions are applied to any gravity satellite which measures gravity field by satellite perturbation orbit.     

GOCE orbit analysis: Long-wavelength gravity field determination using the acceleration approach

The restricted sensitivity of the Gravity field and steady-state Ocean Circulation Explorer (GOCE) gradiometer instrument requires satellite gravity gradiometry to be supplemented by orbit analysis in order to resolve long-wavelength features of the geopotential. For the hitherto published releases of the GOCE time-wise (TIM) and GOCE space-wise gravity field series—two of the official ESA products—the energy conservation method has been adopted to exploit GPS-based satellite-to-satellite tracking information. On the other hand, gravity field recovery from data collected by the CHAllenging Mini-satellite Payload (CHAMP) satellite showed the energy conservation principle to be a sub-optimal choice. For this reason, we propose to estimate the low-frequency part of the gravity field by the point-wise solution of Newton’s equation of motion, also known as the acceleration approach. This approach balances the gravitational vector with satellite accelerations, and hence is characterized by (second-order) numerical differentiation of the kinematic orbit. In order to apply the method to GOCE, we present tailored processing strategies with regard to low-pass filtering, variance–covariance information handling, and robust parameter estimation. By comparison of our GIWF solutions (initials GI for “Geodätisches Institut” and IWF for “Institut für WeltraumForschung”) and the GOCE-TIM estimates with a state-of-the-art gravity field solution derived from GRACE (Gravity Recovery And Climate Experiment), we conclude that the acceleration approach is better suited for GOCE-only gravity field determination as opposed to the energy conservation method.     

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.     

The effect of spatial truncation error on integral inversion of satellite gravity gradiometry data

The satellite gravity gradiometry (SGG) data can be used for local modelling of the Earth’s gravity field. In this study, the SGG data in the local north-oriented and orbital frames are inverted to the gravity anomaly at sea level using the second-order partial derivatives of the extended Stokes formula. The emphasis is on the spatial truncation error and the kernel behaviour of the integral formulas in the aforementioned frames. The paper will show that only the diagonal elements of gravitational tensor at satellite level are suitable for recovering the gravity anomaly at sea level. Numerical studies show that the gravity anomaly can be recovered in Fennoscandia with an accuracy of about 6 mGal directly from on-orbit SGG data.     

Generation of vertical–horizontal and horizontal–horizontal gravity gradients using stochastically modified integral estimators

The Earth’s gravity field modelling is an ill-posed problem having a sensitive solution to the error of data. Satellite gravity gradiometry (SGG) is a space technique to measure the second-order derivatives of geopotential for modelling this field, but the measurements should be validated prior to use. The existing terrestrial gravity anomalies and Earth gravity models can be used for this purpose. In this paper, the second-order vertical–horizontal (VH) and horizontal–horizontal (HH) derivatives of the extended Stokes formula in the local north-oriented frame are modified using biased, unbiased and optimum types of least-squares modification. These modified integral estimators are used to generate the VH and HH gradients at 250 km level for validation purpose of the SGG data. It is shown that, unlike the integral estimator for generating the second-order radial derivative of geopotential, the system of equations from which the modification parameters are obtained is unstable for all types of modification, with large cap size and high degree, and regularization is strongly required for solving the system. Numerical studies in Fennoscandia show that the SGG data can be estimated with an accuracy of 1 mE using an integral estimator modified by a biased type least-squares modification. In this case an integration cap size of 2.5° and a degree of modification of 100 for integrating 30′ × 30′ gravity anomalies are required.     

GPS-only gravity field recovery with GOCE,CHAMP, and GRACE

Gravity missions such as the Gravity field and steady-state Ocean Circulation Explorer (GOCE) are equipped with onboard Global Positioning System (GPS) receivers for precise orbit determination (POD), instrument time-tagging, and the extraction of the long wavelength part of the Earth’s gravity field. The very low orbital altitude of the GOCE satellite and the availability of dense 1 s GPS tracking data are ideal characteristics to exploit the contribution of GPS high-low Satellite-to-Satellite Tracking (hl-SST) to gravity field determination. We present gravity field solutions based on about 8 months of GOCE GPS hl-SST data from 2009 and compare the results with those obtained from the CHAllenging Minisatellite Payload (CHAMP) and Gravity Recovery And Climate Experiment (GRACE) missions. The very low orbital altitude of GOCE significantly improves gravity field recovery from GPS hl-SST data above degree 20, but not for the degrees below 20, where the quality of the spherical harmonic coefficients remains essentially unchanged. Despite the limited time span of GOCE data used, the gravity field of the Earth can be resolved up to about degree 115 using GPS data only. Empirically determined phase center variations (PCVs) of the GOCE onboard GPS helix antenna are, however, mandatory to achieve this performance.     

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.     

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。     

Dynamic and non-dynamic ERS-1 radial orbit improvement from ERS-1/TOPEX dual-satellite altimetry

Dual-satellite altimeter crossover differences between ERS-1 and TOPEX/Poseidon have been included as supplementary tracking data in ERS-1 orbit computations from SLR and single-satellite crossover differences. It was found that including the dual-satellite crossover differences slightly improves the ERS-1 radial orbit accuracy of about 12 cm for orbits computed with the JGM-2 gravity field and also leads to a better ‘centering’ of the ERS-1 orbit in the terrestrial reference frame defined for TOPEX/Poseidon. In addition to this dynamic orbit improvement technique, a non-dynamic technique has been investigated that removes the larger part of the ERS-1 radial orbit error from the dual-satellite crossover difference residuals. For ERS-1 orbits computed with the GEM-T2 gravity field, it was found that the non-dynamic technique could improve the radial orbit accuracy from 140 cm to the same level of accuracy as the ERS-1 JGM-2 orbits.     

A simulation study for anticipated accuracy of lunar gravity field model by SELENE tracking data

Results of numerical simulations are presented to examine the global gravity field recovery capability of the Japanese lunar exploration project SELENE (SELenological and ENgineering Explorer) which will be launched in 2007. New characteristics of the SELENE lunar gravimetry include 4-way satellite-to-satellite Doppler tracking of main orbiter and differential VLBI tracking of two small free-flier satellites. It is shown that the proposed satellite constellation will provide the first truly global satellite tracking data coverage. The expected results from these data are; (1) drastic reduction in far-side gravity error, (2) estimation of many gravity coefficients by the observation, not by a priori information, and (3) one order of magnitude improvement over existing gravity models for low-degree field.     

Impact of a novel hybrid accelerometer on satellite gravimetry performance

The state-of-the-art electrostatic accelerometers (EA) used for the retrieval of non-gravitational forces acting on a satellite constitute a core component of every dedicated gravity field mission. However, due to their difficult-to-control thermal drift in the low observation frequencies, they are also one of the most limiting factors of the achievable performance of gravity recovery. Recently, a hybrid accelerometer consisting of a regular EA and a novel cold atom interferometer (CAI) that features a time-invariant observation stability and constantly recalibrates the EA has been developed in order to remedy this major drawback. In this paper we aim to assess the value of the hybrid accelerometer for gravity field retrieval in the context of GRACE-type and Bender-type missions by means of numerical closed-loop simulations where possible noise specifications of the novel instrument are considered and different components of the Earth’s gravity field signal are added subsequently. It is shown that the quality of the gravity field solutions is mainly dependent on the CAI’s measurement accuracy. While a low CAI performance (10^?8 to 10^?9?m/s²/Hz^1/2) does not lead to any gains compared to a stand-alone EA, a sufficiently high one (10^?11?m/s²/Hz^1/2) may improve the retrieval performance by over one order of magnitude. We also show that improvements which are limited to low-frequency observations may even propagate into high spherical harmonic degrees. Further, the accelerometer performance seems to play a less prominent role if the overall observation geometry is improved as it is the case for a Bender-type mission. The impact of the accelerometer measurements diminishes further when temporal variations of the gravity field are introduced, pointing out the need for proper de-aliasing techniques. An additional study reveals that the hybrid accelerometer is – contrary to a stand-alone EA – widely unaffected by scale factor instabilities.     

Chang’E-1 precision orbit determination and lunar gravity field solution

In this paper we present results assessing the role of Very Long Baseline Interferometry (VLBI) tracking data through precision orbit determination (POD) during the check-out phase for Chang’E-1, and the lunar gravity field solution CEGM-01 based on the orbital tracking data acquired during the nominal phase of the mission. The POD of Chang’E-1 is performed using S-band two-way Range and Range Rate (R&RR) data, together with VLBI delay and delay rate observations. The role of the VLBI data in the POD of Chang’E-1 is analyzed, and the resulting orbital accuracies are estimated for different solution strategies. The final orbital accuracies proved that the VLBI tracking data can improve the Chang’E-1 POD significantly. Consequently, CEGM-01 based on six-month tracking data during Chang’E-1 nominal mission phase is presented, and the accuracy of the model is assessed by means of the gravity field power spectrum, admittance and coherence between gravity and topography, lunar surface gravity anomaly and POD for both Chang’E-1 and Lunar Prospector (LP). Our analysis indicates that CEGM-01 has significant improvements over a prior model (i.e. GLGM-2), and shows the potential of Chang’E-1 tracking data in high resolution lunar gravity field model solution by combining with SELENE and LP tracking data.     

New magnetodynamic principles in cosmic physics

The effect of gravity on super-escape particles spiralling along magnetic field lines need not be negligible when the field lines are long enough and the field-strength variation small. If the magnetic field strength decreases with altitude but only very slowly, some unexpected phenomena may occur owing to gravity: some super-high-velocity particles can possess an upper level of reflection which impedes their escape into higher regions. Some of these “super-escape particles may, however, propagate through the level at which their pitch angle is 90° and continue spiralling in the same direction (“hole” boundary). In addition, the pitch angle of some super-escape particles may only achieve a maximum which can be very small (hole effect). Many plasma phenomena can be derived from these charecteristic features of particle trajectories.     

Mapping the earth's magnetic and gravity fields from space: Current status and future prospects

Orbital potential field measurements are sensitive to regional variations in earth density and magnetization that occur over scales of a few hundred kilometers or greater. Global field models currently available are able to distinguish gravity variations of ±5 milligal over distances of ～1,000 km and magnetic variations of ±6 gamma over distances of ～300 km at the earth's surface. Regional variations in field strength have been detected in orbital measurements that are not apparent in higher resolution, low altitude surveys. NASA is presently studying a spacecraft mission known as GRAVSAT/MAGSAT, which would be the first satellite mission to perform a simultaneous survey of the earth's gravity and magnetic fields at low orbital altitudes. GRAVSAT/MAGSAT has been proposed for launch during the latter nineteen-eighties, and it would measure gravity field strength to an accuracy of 1 milligal and magnetic field strength to an accuracy of 2 gamma (scalar)/5 gamma (vector components) over a distance of roughly 100 km. Even greater improvements in the accuracy and spatial resolution of orbital surveys are anticipated during the nineteen-nineties with the development of potential field gradiometers and a tethered satellite system that can be deployed from the Space Shuttle to altitudes of 120 km above the earth's surface.     

Performance of a centrifugal phytotron.

It is possible to cultivate plants under an artificial gravity field generated by a centrifugal device in space. In order to determine an optimal magnitude of gravity, there is a need to investigate the relationship between plant and growth and gravity, including not only reduced gravity but also gravity greater than 1G. A prototype centrifugal phytotron was designed and fabricated in order to investigate the relationship between plant growth and increased gravity. This device enables us to cultivate plants over the long term by controlling environmental conditions in the phytotron such as temperature, relative humidity, CO2 concentration and light intensity. The results of our experiment indicate that plant seeds can germinate and grow even under an artificial gravity which changes sinusoidally from 2G to 4G.     

AIUB-CHAMP02S: The influence of GNSS model changes on gravity field recovery using spaceborne GPS

The gravity field model AIUB-CHAMP02S, which is based on six years of CHAMP GPS data, is presented here. The gravity field parameters were derived using a two step procedure: In a first step a kinematic trajectory of a low Earth orbiting (LEO) satellite is computed using the GPS data from the on-board receiver. In this step the orbits and clock corrections of the GPS satellites as well as the Earth rotation parameters (ERPs) are introduced as known. In the second step this kinematic orbit is represented by a gravitational force model and orbit parameters.     

On the information contents and regularisation of lunar gravity field solutions

Till the present day the recovery of the lunar gravity field from satellite tracking data depends in a crucial way on the level and method of regularisation. With Earth-based tracking only, the spatial data coverage is limited to only slightly more than 50% and the inverse problem remains severely ill-posed. The development of global gravity models suitable for precise orbit modeling as well as geophysical studies therefore requires a significant level of regularisation, limiting the solution power over the far-side where no gravity information is available. Unconstrained solutions, within the framework of global harmonic base functions, are only possible for very low degrees (< 10). Any significant change to this situation is only to be expected when global satellite-to-satellite tracking data of high quality becomes available early in the next decade. Yet, a rigorous analysis of the impact of the chosen method and level of regularisation is lacking. Most gravity models employ a Kaula-type signal smoothness constraint of 15 × 10⁻⁵ /l², which allows a good overall data fit as well as a smooth field over the far-side. Furthermore, a geographical type of constraint has been suggested, where surface accelerations have been introduced in areas of no data coverage. Modern numerical methods, on the other hand, offer direct tools and search mechanisms for the optimal level of regularisation. This paper presents a study of Tikhonov-type regularisation of lunar gravity solutions, with emphasis on the so-called L-curve and quasi-optimality methods for regularisation parameter estimation. Furthermore, new quality measures of lunar gravity solutions are presented, which account for the bias introduced by the regularisation.     

重力卫星高精度测距系统观测量的降速率滤波算法

在卫星时频传递、矢量测量技术领域，有效观测数据的高精度滤波处理十分重要，其基本要求是在保留有效观测信息的同时能够降低噪声干扰，实现原始信息的高效、高精度传递。以中国地球重力场测量卫星高精度测距系统的工程研制为背景，为实现高频噪声抑制、保留低频重力场信息，基于对FIR滤波器结构的优化和窗函数的改进，提出了一种对10 Hz采样率的原始测距数据降速率、降噪滤波的算法。相比其他类似算法，所提方法同时具备降速滤波和差分运算两种功能，噪声抑制抑制度优于44%，且频率截止特性陡峭，第一旁瓣抑制度达到–94 dB。该算法在提高测距精度、充分保留重力场信息的同时，实现了地面数据后处理流程的简化，为微波测量系统、激光测量系统的数据处理提供了一种优选方案。     

A simulation of Martian gravity field recovery by using a near equatorial orbiter

An improvement to the Martian gravity field may be achieved by means of future orbiting spacecraft with small eccentricity and low altitude exemplified through a newly proposed mission design that may be tested in upcoming reconnaissance of Mars. Here, the near equatorial orbital character (with an inclination approximating 10°, eccentricity as 0.01 and semi-major axis as 4000 km) is considered, and its contribution to Martian gravity field solution is analyzed by comparing it with a hypothetical polar circular orbiter. The solution models are evaluated in terms of the following viewpoints: power spectra of gravity field coefficients, correlations of low degree zonal coefficients, precise orbit determination, and error distribution of both Mars free air gravity anomaly and areoid. At the same time, the contributions of the near equatorial orbiters in low degree zonal coefficients time variations are also considered. The present results show that the near equatorial orbiter allows us to improve the accuracy of the Martian gravity field solution, decrease correlation of low degree zonal coefficients, retrieve much better time variable information of low degree zonal coefficients, improve precise orbit determination, and provide more accurate Mars free air gravity anomaly and areoid around the equatorial region.     

The simulation of lunar gravity field recovery from D-VLBI of Chang’E-1 and SELENE lunar orbiters

The lunar gravity field is a foundation to study the lunar interior structure, and to recover the evolution history of the Moon. It is still an open and key topic for lunar science. For above mentioned reasons, it becomes one of the important scientific objectives of recent lunar missions, such as KAGUYA (SELENE) the Japanese lunar mission and Chang’E-1, the Chinese lunar mission. The Chang’E-1 and the SELENE were successfully launched in 2007. It is estimated that these two missions can fly around the Moon longer than 6 months simultaneously. In these two missions, the Chinese new VLBI (Very Long Baseline Interferometry) network will be applied for precise orbit determination (POD) by using a differential VLBI (D-VLBI) method during the mission period. The same-beam D-VLBI technique will contribute to recover the lunar gravity field together with other conventional observables, i.e. R&RR (Range and Range Rate) and multi-way Doppler. Taking VLBI tracking conditions into consideration and using the GEODYNII/SOVLE software of GSFC/NASA/USA [8 and 10], we simulated the lunar gravity field recovering ability with and without D-VLBI between the Chang’E-1 and SELENE main satellite. The cases of overlapped flying and tracking period of 30 days, 60 days and 90 days have been analyzed, respectively. The results show that D-VLBI tracking between two lunar satellites can improve the gravity field recovery remarkably. The results and methods introduced in this paper will benefit the actual missions.