Impact of covariance information of kinematic positions on orbit reconstruction and gravity field recovery

Gravity missions are equipped with onboard Global Positioning System (GPS) receivers for precise orbit determination (POD) and for the extraction of the long wavelength part of the Earth’s gravity field. As positions of low Earth orbiters (LEOs) may be determined from GPS measurements at each observation epoch by geometric means only, it is attractive to derive such kinematic positions in a first step and to use them in a second step as pseudo-observations for gravity field determination. The drawback of not directly using the original GPS measurements is, however, that kinematic positions are correlated due to the ambiguities in the GPS carrier phase observations, which in principle requires covariance information be taken into account. We use GRACE data to show that dynamic or reduced-dynamic orbit parameters are not optimally reconstructed from kinematic positions when only taking epoch-wise covariance information into account, but that essentially the same orbit quality can be achieved as when directly using the GPS measurements, if correlations in time are taken into account over sufficiently long intervals. For orbit reconstruction covariances have to be considered up to one revolution period to avoid ambiguity-induced variations of kinematic positions being erroneously interpreted as orbital variations. For gravity field recovery the advantage is, however, not very pronounced.     

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).     

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.     

The use of Gaussian equations of motions of a satellite for local gravity anomaly recovery

The orbital elements of a low Earth orbiting satellite and their velocities can be used for local determination of gravity anomaly. The important issue is to find direct relations among the anomalies and these parameters. Here, a primary theoretical study is presented for this purpose. The Gaussian equations of motion of a satellite are used to develop integral formulas for recovering the gravity anomalies. The behaviour of kernels of the integrals are investigated for a two-month simulated orbit similar to that of the Gravity field and steady-state ocean circulation explorer (GOCE) mission over Fennoscandia. Numerical investigations show that the integral formulas have neither isotropic nor well-behaved kernels. In such a case, gravity anomaly recovery is not successful due to large spatial truncation error of the integral formulas. Reformulation of the problem by combining the orbital elements and their velocities leads to an integral with a well-behaved kernel which is suitable for our purpose. Also based on these combinations some general relations among the orbital elements and their velocities are obtained which can be used for validation of orbital parameters and their velocities.     

Characteristics of GOCE orbits based on Satellite Laser Ranging

The Gravity field and steady-state Ocean Circulation Explorer (GOCE) was the first European Space Agency’s (ESA) Earth Explorer core mission. Through its extremely low, about 260?km above the Earth, circular, sun-synchronous orbit, the satellite gained high spatial resolution and accuracy gravity gradient, and ocean circulation data. Global Positioning System (GPS) receivers, mounted on the spacecraft, allowed the determination of reduced-dynamic and kinematic GOCE orbits, whereas Laser Retroreflector Array (LRA) dedicated to Satellite Laser Ranging (SLR) allowed an independent validation of GPS-derived orbits. In this paper, residuals between different GPS-based orbit types and SLR observations are used to investigate the sensitivity and the influence of solar, geomagnetic, and ionospheric activities on the quality of kinematic and reduced-dynamic GOCE orbits. We also analyze the quality of data provided by individual SLR sites, by detecting time biases using ascending and descending sun-synchronous GOCE orbit passes, and the residual analysis of the measurement characteristics, i.e., the dependency of SLR residuals as a function of nadir and horizontal angles. Results show a substantial vulnerability of kinematic orbit solutions to the solar F10.7 index and the ionospheric activity measured by the variations of the Total Electron Content (TEC) values. The sensitivity of kinematic orbits to the three-hour-range KP index is rather minor. The reduced-dynamic orbits are almost insensitive to indices describing ionospheric, solar, and geomagnetic activities. The investigation of individual SLR sites shows that some of them are affected by time bias errors, whereas other demonstrate systematics, such as a dependency between observation residuals and the satellite nadir angle or the horizontal azimuth angle from the SLR station to the direction of the satellite.     

Impact of GPS antenna phase center variations on precise orbits of the GOCE satellite

The first European Space Agency Earth explorer core mission GOCE (Gravity field and steady-state Ocean Circulation Explorer) has been launched on March 17, 2009. The 12-channel dual-frequency Global Positioning System receiver delivers 1 Hz data and provides the basis for precise orbit determination (POD) on the few cm-level for such a very low orbiting satellite (254.9 km). As a member of the European GOCE Gravity Consortium, which is responsible for the GOCE High-level Processing Facility (HPF), the Astronomical Institute of the University of Bern (AIUB) provides the Precise Science Orbit (PSO) product for the GOCE satellite. The mission requirement for 1-dimensional POD accuracy is 2 cm. The use of in-flight determined antenna phase center variations (PCVs) is necessary to meet this requirement. The PCVs are determined from 154 days of data and the magnitude is up to 3-4 cm. The impact of the PCVs on the orbit determination is significant. The cross-track direction benefits most of the PCVs. The improvement is clearly seen in the orbit overlap analysis and in the validation with independent Satellite Laser Ranging (SLR) measurements. It is the first time that SLR could validate the cross-track component of a LEO orbit.     

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.     

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.     

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.     

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.     

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。     

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.     

Orbit determination for the GOCE satellite

Precise Orbit Determination (POD) for the Gravity field and steady-state Ocean Circulation Explorer (GOCE), the first core explorer mission by the European Space Agency (ESA), forms an integrated part of the so-called High-Level Processing Facility (HPF). Two POD chains have been set up referred to as quick-look Rapid and Precise Science Orbit determination or RSO and PSO, respectively. These chains make use of different software systems and have latencies of 1 day and 2 weeks, respectively, after tracking data availability. The RSO and PSO solutions have to meet a 3-dimensional (3D) position precision requirement of 50 cm and a few cm, respectively. The tracking data will be collected by the new Lagrange GPS receiver and the predicted characteristics of this receiver have been taken into account during the implementation phase of the two chains.     

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.     

Aura/MLS与TIMED/SABER观测全球重力波特性

大气重力波是临近空间环境主要大气波动之一,对全球环流具有重要影响。卫星上搭载的临边探测器能够探测临近空间大气温度,可用于临近空间大气重力波研究。利用2012－2014年Aura的微波临边探测器（MLS）和TIMED的红外临边探测器（SABER）的探测数据,对20~50 km高度的大气重力波扰动分布特征开展了分析研究,两种观测重力波活动基本一致,重力波随季节、纬度及高度的变化显著。冬季半球高纬度重力波扰动较强,赤道和夏季半球近赤道地区上空也存在明显重力波活动区域,夏季半球高纬度重力波扰动最弱。重力波扰动强度随高度增加。TIMED/SABER重力波扰动强度数值比 Aura/MLS略强。      

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).     

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.     

Onboard orbit determination using GPS observations based on the unscented Kalman filter

Spaceborne GPS receivers are used for real-time navigation by most low Earth orbit (LEO) satellites. In general, the position and velocity accuracy of GPS navigation solutions without a dynamic filter are 25 m (1σ) and 0.5 m/s (1σ), respectively. However, GPS navigation solutions, which consist of position, velocity, and GPS receiver clock bias, have many abnormal excursions from the normal error range for space operation. These excursions lessen the accuracy of attitude control and onboard time synchronization. In this research, a new onboard orbit determination algorithm designed with the unscented Kalman filter (UKF) was developed to improve the performance. Because the UKF is able to obtain the posterior mean and covariance accurately by using the second-order Taylor series expansion through the sampled sigma points that are propagated by using the true nonlinear system, its performance can be better than that of the extended Kalman filter (EKF), which uses the linearized state transition matrix to predict the covariance. The dynamic models for orbit propagation applied perturbations due to the 40 × 40 geo-potential, the gravity of the Sun and Moon, solar radiation pressure, and atmospheric drag. The 7(8)th-order Runge–Kutta numerical integration was applied for orbit propagation. Two types of observations, navigation solutions and C/A code pseudorange, can be used at the user’s discretion. The performances of the onboard orbit determination were verified using real GPS data of the CHAMP and KOMPSAT-2 satellites. The results of the orbit determination were compared with the precision orbit ephemeris (POE) of the CHAMP and KOMPSAT-2 satellites.     

Gravity field models from kinematic orbits of CHAMP,GRACE and GOCE satellites

The aim of our work is to generate Earth’s gravity field models from GPS positions of low Earth orbiters. Our inversion method is based on Newton’s second law, which relates the observed acceleration of the satellite with forces acting on it. The observed acceleration is obtained as numerical second derivative of kinematic positions. Observation equations are formulated using the gradient of the spherical harmonic expansion of the geopotential. Other forces are either modelled (lunisolar perturbations, tides) or provided by onboard measurements (nongravitational perturbations). From this linear regression model the geopotential harmonic coefficients are obtained.     

Simulations of MATROSHKA experiment outside the ISS using PHITS

The radiation environment at the altitude of the International Space Station (ISS) is substantially different than anything typically encountered on Earth in both the character of the radiation field and the significantly higher dose rates. Concerns about the biological effects on humans of this highly complex natural radiation field are increasing due to higher amount of astronauts performing long-duration missions onboard the ISS and especially if looking into planned future manned missions to Mars. In order to begin the process of predicting the dose levels seen by the organs of an astronaut, being the prerequisite for radiation risk calculations, it is necessary to understand the character of the radiation environment both in- and outside of the ISS as well as the relevant contributions from the radiation field to the organ doses.