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.     

Inversion of satellite gradiometry data using statistically modified integral formulas for local gravity field recovery

The satellite gravity gradiometric data can be used directly to recover the gravity anomaly at sea level using inversion of integral formulas. This approach suffers by the spatial truncation errors of the integrals, but these errors can be reduced by modifying the formulas. It allows us to consider smaller coverage of the satellite data over the region of recovery. In this study, we consider the second-order radial derivative (SORD) of disturbing potential (T_rr) and determine the gravity anomaly with a resolution of 1° × 1° at sea level by inverting the statistically modified version of SORD of extended Stokes’ formula. Also we investigate the effect of the spatial truncation error on the quality of inversion considering noise of T_rr. The numerical investigations show satisfactory results when the area of T_rr coverage is the same with that of the gravity anomaly and the integral formula is modified by the biased least-squares modification. The error of recovery will be about 6 mGal after removing the regularization bias in the presence of 1 mE noise in T_rr measured on the orbit.     

The effect of spatial truncation error on variance of gravity anomalies derived from inversion of satellite orbital and gradiometric data

The spatial truncation error (STE) is a significant systematic error in the integral inversion of satellite gradiometric and orbital data to gravity anomalies at sea level. In order to reduce the effect of STE, a larger area than the desired one is considered in the inversion process, but the anomalies located in its central part are selected as the final results. The STE influences the variance of the results as well because the residual vector, which is contaminated with STE, is used for its estimation. The situation is even more complicated in variance component estimation because of its iterative nature. In this paper, we present a strategy to reduce the effect of STE on the a posteriori   variance factor and the variance components for inversion of satellite orbital and gradiometric data to gravity anomalies at sea level. The idea is to define two windowing matrices for reducing this error from the estimated residuals and anomalies. Our simulation studies over Fennoscandia show that the differences between the 0.5°×0.5°$0.5°\times 0.5°$ gravity anomalies obtained from orbital data and an existing gravity model have standard deviation (STD) and root mean squared error (RMSE) of 10.9 and 12.1 mGal, respectively, and those obtained from gradiometric data have 7.9 and 10.1 in the same units. In the case that they are combined using windowed variance components the STD and RMSE become 6.1 and 8.4 mGal. Also, the mean value of the estimated RMSE after using the windowed variances is in agreement with the RMSE of the differences between the estimated anomalies and those obtained from the gravity model.     

Different space weather effects in anomalies of the high and low orbital satellites

Preliminary results of the EU INTAS Project 00810, which aims to improve the methods of safeguarding satellites in the Earth’s magnetosphere from the negative effects of the space environment, are presented. Anomaly data from the “Kosmos” series satellites in the period 1971–1999 are combined in one database, together with similar information on other spacecraft. This database contains, beyond the anomaly information, various characteristics of the space weather: geomagnetic activity indices (Ap, AE and Dst), fluxes and fluences of electrons and protons at different energies, high energy cosmic ray variations and other solar, interplanetary and solar wind data. A comparative analysis of the distribution of each of these parameters relative to satellite anomalies was carried out for the total number of anomalies (about 6000 events), and separately for high (5000 events) and low (about 800 events) altitude orbit satellites. No relation was found between low and high altitude satellite anomalies. Daily numbers of satellite anomalies, averaged by a superposed epoch method around sudden storm commencements and proton event onsets for high (>1500 km) and low (<1500 km) altitude orbits revealed a big difference in a behavior. Satellites were divided on several groups according to the orbital characteristics (altitude and inclination). The relation of satellite anomalies to the environmental parameters was found to be different for various orbits that should be taken into account under developing of the anomaly frequency models.     

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.     

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.     

轨道姿态误差对TDI-CCD相机行周期及偏流角的影响分析

与其他文献的几何方法不同,本文从运动学角度出发,选定合适的参考系后,按照一系列不同坐标系的坐标转换,完成星下点运动速度矢量在敏感器坐标系的推导,把姿态、轨道、安装偏差作为相关参数,经过一系列坐标变换,表征在最终的偏流角公式及行转移频率动态公式中;在通常的对地稳定指向模式下,姿态参数和安装偏差很小,仅保留一阶小量.根据矢量表达,解析地给出行转移频率及偏流角与轨道、姿态的关系,从而方便分析误差的影响,并以几个典型的低轨为实例作了说明.     

A possible flyby anomaly for Juno at Jupiter

In the last decades there have been an increasing interest in improving the accuracy of spacecraft navigation and trajectory data. In the course of this plan some anomalies have been found that cannot, in principle, be explained in the context of the most accurate orbital models including all known effects from classical dynamics and general relativity. Of particular interest for its puzzling nature, and the lack of any accepted explanation for the moment, is the flyby anomaly discovered in some spacecraft flybys of the Earth over the course of twenty years. This anomaly manifest itself as the impossibility of matching the pre and post-encounter Doppler tracking and ranging data within a single orbit but, on the contrary, a difference of a few mm/s in the asymptotic velocities is required to perform the fitting.Nevertheless, no dedicated missions have been carried out to elucidate the origin of this phenomenon with the objective either of revising our understanding of gravity or to improve the accuracy of spacecraft Doppler tracking by revealing a conventional origin.With the occasion of the Juno mission arrival at Jupiter and the close flybys of this planet, that are currently been performed, we have developed an orbital model suited to the time window close to the perijove. This model shows that an anomalous acceleration of a few mm/s² is also present in this case. The chance for overlooked conventional or possible unconventional explanations is discussed.     

Recovery of Moho’s undulations based on the Vening Meinesz–Moritz theory from satellite gravity gradiometry data: A simulation study

In the gravimetric approach to determine the Moho depth an isostatic hypothesis can be used. The Vening Meinesz–Moritz isostatic hypothesis is the recent theory for such a purpose. Here, this theory is further developed so that the satellite gravity gradiometry (SGG) data are used for recovering the Moho depth through a nonlinear integral inversion procedure. The kernels of its forward and inverse problems show that the inversion should be done in a larger area by 5° than the desired one to reduce the effect of the spatial truncation error of the integral formula. Our numerical study shows that the effect of this error on the recovered Moho depths can reach 6 km in Persia and it is very significant. The iterative Tikhonov regularization in a combination with either generalized cross validation or quasi-optimal criterion of estimating the regularization parameter seems to be suitable and the solution is semi-convergent up to the third iteration. Also the Moho depth recovered from the simulated SGG data will be more or less the same as that obtained from the terrestrial gravimetric data with a root mean square error of 2 km and they are statistically consistent.     

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.     

一种基于TLE数据的轨道异常分析方法

空间在轨物体的轨道异常是航天工程及预警领域普遍关注的问题,及时发现轨道异常意义重大,通过分析空间物体的轨道异常,可以及时发现和识别规避事件或碰撞事件,还可以了解监测网的能力.本文提出一种基于TLE数据的简单的轨道异常分析方法——长半轴变化法.该方法快速有效,应用到低轨在用卫星和美俄解体碎片的异常分析中,异常物体正确识别率可达到100%;对美俄解体碎片进行轨道异常分析后得出,美国空间监视网可以稳定探测90%以上的解体碎片.      

Practical method to identify orbital anomaly as spacecraft breakup in the geostationary region

Identifying spacecraft breakup events is an essential issue for better understanding of the current orbital debris environment. This paper proposes an observation planning approach to identify an orbital anomaly, which appears as a significant discontinuity in archived orbital history, as a spacecraft breakup. The proposed approach is applicable to orbital anomalies in the geostationary region. The proposed approach selects a spacecraft that experienced an orbital anomaly, and then predicts trajectories of possible fragments of the spacecraft at an observation epoch. This paper theoretically demonstrates that observation planning for the possible fragments can be conducted. To do this, long-term behaviors of the possible fragments are evaluated. It is concluded that intersections of their trajectories will converge into several corresponding regions in the celestial sphere even if the breakup epoch is not specified and it has uncertainty of the order of several weeks.     

跟踪与数据中继卫星(TDRS)跟踪用户星的条件分析

中继卫星是航天测控网发展的必然趋势.一颗中继卫星对不同轨道的用户星测控支持能力是不同的,受多种因素的影响.从理论上详尽地分析了地球同步轨道中继卫星(TDRS)对近地航天器的跟踪条件,计算了中继卫星对各种轨道高度近地航天器的测控覆盖率,给出了中继星到用户星可视算法、地球遮挡和受太阳直射的计算公式,通过专业STK软件仿真验证了所得结论的正确性,它们可以作为确定天基网的中继星数量和对在轨中继卫星进行资源分配的理论依据.      

藏匿在冯·卡门坑南部下方的月球质量瘤

月球着陆探测的候选着陆区域的质量异常分布,预示着对应区域发生过复杂的动力学演化过程,存在着异常丰富的物质组成。对月球的质量异常分布或重力场异常的探测是基于绕月卫星重力探测方法获取的,结合“嫦娥1号”月球探测获得的月球重力场异常模型与地形探测模型的数据,分析揭示存在于月球背面的质量瘤区域,研究结果表明：使用多次任务的数据首次证认了一个位于中尺度撞击区域,冯·卡门（Von Kármán）撞击坑的南半部和边缘的正下部的质量瘤,暂定编号CEFC04,冯·卡门撞击坑是该区域形成后,又一次大型撞击产生的。这类显著的被藏匿的中尺度月球质量瘤的发现,对探测研究月球的演化有重要意义。     

编队飞行卫星群构型保持及初始化

导出了基于相对轨道要素的编队飞行卫星群轨道相对运动控制的轨道机动方程；提出了编队飞行卫星群轨道相对运动控制的轨道机动控制策略，利用不同方向的脉冲控制相对运动的轨道参数，包括轨道平面内机动和轨道平面外机动控制。根据相对轨道要素的变轨机动控制，进行编队飞行卫星群构型的初始化。这样的构型初始化可以视作一次特殊的变轨机动控制，很容易实现编队飞行构型的初始化机动。     

基于符号化表示相似度度量的卫星多元工程参数异常检测

随着卫星系统复杂程度的日益增加,综合分析卫星多元参数之间的相关性异常对于卫星安全运行和空间任务的正确执行具有重要意义。利用某卫星工程参数数据,基于符号聚合近似算法（SAX）,研究卫星多元工程参数的异常检测问题,解决了当前异常检测方法中多元参数融合时不考虑上下文信息造成信息丢失的问题,实现多元参数有效融合,形成一种优化的基于快速动态时间规整算法（Fast-DTW）的异常检测算法。研究结果表明,模型在某卫星电源子系统的异常检测过程中,recall,precision和F1 score分别为0.947,0.9和0.923,能够实际应用于卫星异常检测,提高卫星在轨运行的安全性。     

挠性卫星轨控期间动力学与姿态控制

卫星轨道控制期间,轨道控制推力会激振挠性太阳帆板,从而也影响卫星的姿态。用拉格朗日方程建立了带有大型太阳帆板的卫星动力学模型,分析了卫星质心运动、姿态运动与挠性振动的耦合关系。根据姿态控制推力器的输出特性,设计了轨道控制期间卫星姿态控制方案。通过数学仿真验证了轨道控制推力对挠性帆板与卫星姿态的影响,验证了轨控期间姿态控制方案的有效性。     

Impact of the orbital eccentricity on the attitude performance before and after the deorbiting phase for Alsat-1

Due to the presence of periodic forcing terms in the gravity gradient torque, orbit eccentricity may produce large response for the roll, yaw and pitch angles. This paper investigates the influence of the orbit eccentricity on the performance of the attitude determination and control subsystem (ADCS) pointing of passive Low Earth Orbit (LEO) satellites stabilized by a gravity gradient boom or having long appendages before and after the deorbiting operation. The contribution of this work is twofold. First, the satellite attitude dynamics and kinematics are modeled by introducing the orbit eccentricity in the equations of motion of a LEO satellite in order to provide the best scenario in which satellite operators can keep the nominal functionality of LEO satellites with a gravity gradient boom after the deorbiting operation. Second, a Quaternion-based Extended Kalman Filter (EKF) is analyzed when the orbit eccentricity is considered in order to determine the influence of this disturbance on the convergence and stability of the filter. The simulations in this work are based on the true parameters of Alsat-1 which is a typical LEO satellite stabilized by a gravity gradient boom. The results show that the orbit eccentricity has a big influence on the pointing system accuracy causing micro-vibrations that affect the geocentric pointing particularly after the deorbiting phase. In this case, satellites have no orbital correction option. The Quaternion-based Extended Kalman Filter analyzed in this paper, achieved satisfactory results for eccentricity values less than 0.4 with respect to pointing system accuracy. However, singularities were observed for eccentricity values greater than 0.4.     

距离相关系数融合GPR模型的卫星异常检测方法

卫星在轨运行期间，遥测数据表现形式通常为多维时间序列。高斯过程回归(GPR)模型可以为重要的遥测参数提供动态门限，及时发现隐藏在工程阈值内的故障征兆，但是高维卫星数据使得GPR模型具有局限性。因此，为获取与多个遥测参数相关的动态门限，在GPR模型的基础上，融合距离相关系数对预测变量进行选择，减少信息冗余和计算量，提高模型的可解释性，并估计模型的泛化误差以设置更合理的预测区间，提高模型的泛化能力，检测数据流的持续异常。对实际在轨卫星数据进行仿真实验，验证了距离相关系数融合GPR模型的卫星异常检测方法可以在卫星故障早期检测到数据异常，而且提高了模型的预测性能，降低了虚警率。      

A simple and valid analysis method for orbit anomaly detection

A simple analysis method for orbit anomaly detection, called semi-major axis change method (SACM) was presented by using a relationship between the change of orbit parameters and velocity increments. In this method, the mean value and standard deviation of the semi-major axis change in different time intervals were first calculated according to historical data. Then, these two parameters, the mean value and standard deviation of the semi-major axis change, are chosen as basis variables and combined as an anomalous criterion. For orbit objects with different characteristic, anomalous thresholds were given in different time intervals for identifying the anomalies of the orbital objects. Finally, this method is used for low earth orbit (LEO) satellites and American–Russian breakup debris. By adopting this method, the characteristics of the orbit change were given. The accuracy rate of anomaly analysis for LEO satellites and American–Russian breakup debris can reach to 100%, which demonstrates that the method was rapid and valid.