Kinematic Orbit Research Articles

Analysis on motion of Earth’s center of mass observed with CHAMP mission

Geocenter motion (GCM) is one important topic for constructing and maintaining the terrestrial reference frame and its applications. GCM is studied from CHAMP with the multi-step approach in this paper. Geometric orbits of CHAMP in 2001–2006 are precisely determined with the kinematic method only from the satellite-borne GPS zero-difference data. Then a GCM time series is estimated from the precise kinematic orbits based on the theory of satellite dynamics to fit the CHAMP’s real geometric orbits. We compare the series with the geocenter series used in ITRF2005. Then the GCM series are analyzed with Fourier transform and wavelet transformation. The mean motions within 6 years in TX, TY and TZ directions are respectively 0.8 mm, 2.2 mm, and 7.9 mm. The trends of GCM in the three directions are 0.495 mm/a, −0.004 mm/a, and 1.309 mm/a, respectively. The long-term movement (2001–2006) indicates that the crustal figure is changing. The seasonal variations are the main component which may be excitated by the mass redistribution of Earth’s fluid layer, e.g. ocean, atmosphere and continental water. The inter-annual variations are also found in the GCM series measured with CHAMP.

Precise orbit determination for the FORMOSAT-3/COSMIC satellite mission using GPS

The joint Taiwan–US mission FORMOSAT-3/ COSMIC (COSMIC) was launched on April 17, 2006. Each of the six satellites is equipped with two POD antennas. The orbits of the six satellites are determined from GPS data using zero-difference carrier-phase measurements by the reduced dynamic and kinematic methods. The effects of satellite center of mass (COM) variation, satellite attitude, GPS antenna phase center variation (PCV), and cable delay difference on the COSMIC orbit determination are studied. Nominal attitudes estimated from satellite state vectors deliver a better orbit accuracy when compared to observed attitude. Numerical tests show that the COSMIC COM must be precisely calibrated in order not to corrupt orbit determination. Based on the analyses of the 5 and 6-h orbit overlaps of two 30-h arcs, orbit accuracies from the reduced dynamic and kinematic solutions are nearly identical and are at the 2–3 cm level. The mean RMS difference between the orbits from this paper and those from UCAR (near real-time) and WHU (post-processed) is about 10 cm, which is largely due to different uses of GPS ephemerides, high-rate GPS clocks and force models. The kinematic orbits of COSMIC are expected to be used for recovery of temporal variations in the gravity field.

Highly-reduced dynamic orbits and their use for global gravity field recovery: A simulation study for GOCE

The so-called highly reduced-dynamic (HRD) orbit determination strategy and its use for the determination of the Earth’s gravitational field are analyzed. We discuss the functional model for the generation of HRD orbits, which are a compromise of the two extreme cases of dynamic and purely geometrically determined kinematic orbits. For gravity field recovery the energy integral approach is applied, which is based on the law of energy conservation in a closed system. The potential of HRD orbits for gravity field determination is studied in the frame of a simulated test environment based on a realistic GOCE orbit configuration. The results are analyzed, assessed, and compared with the respective reference solutions based on a kinematic orbit scenario. The main advantage of HRD orbits is the fact that they contain orbit velocity information, thus avoiding numerical differentiation on the orbit positions. The error characteristics are usually much smoother, and the computation of gravity field solutions is more efficient, because less densely sampled orbit information is sufficient. On the other hand, the main drawback of HRD orbits is that they contain external gravity field information, and thus yield the danger to obtain gravity field results which are biased towards this prior information.

Tracking and orbit determination performance of the GRAS instrument on MetOp-A

The global navigation satellite system receiver for atmospheric sounding (GRAS) on MetOp-A is the first European GPS receiver providing dual-frequency navigation and occultation measurements from a spaceborne platform on a routine basis. The receiver is based on ESA’s AGGA-2 correlator chip, which implements a high-quality tracking scheme for semi-codeless P(Y) code tracking on the L1 and L2 frequency. Data collected with the zenith antenna on MetOp-A have been used to perform an in-flight characterization of the GRAS instrument with focus on the tracking and navigation performance. Besides an assessment of the receiver noise and systematic measurement errors, the study addresses the precise orbit determination accuracy achievable with the GRAS receiver. A consistency on the 5 cm level is demonstrated for reduced dynamics orbit solutions computed independently by four different agencies and software packages. With purely kinematic solutions, 10 cm accuracy is obtained. As a part of the analysis, an empirical antenna offset correction and preliminary phase center correction map are derived, which notably reduce the carrier phase residuals and improve the consistency of kinematic orbit determination results.

Spectral analysis of CHAMP kinematic velocities determined by applying smoothing cubic splines

Kinematic velocities are needed when gravity field recovery ought to be done from kinematic positions by the use of the energy integral. \cite pp02 has tested smoothing cubic splines applying on CHAMP kinematic orbit. The solution has been found potent when a reference gravity field is used during the processing. However the possibility of an error emerges that the reference field ´leaks´ into the estimated velocity, which would harmfully affect the gravity field model solution in the subsequent step. The use of smoothing splines demands strict investigations in the Fourier-domain. Tests in the Fourier domain are shown in the recent paper.

An improved energy balance approach and its application in CHAMP gravity field recovery

An efficient method for gravity field determination from CHAMP orbits and accelerometer data is referred to as the energy balance approach. A new CHAMP gravity field recovery strategy based on the improved energy balance approach is developed in this paper. The method simultaneously solves the spherical harmonic coefficients, daily integration constant, scale and bias parameters. Two 60 degree and order gravitational potential models, XISM-CHAMP01S from the classical energy balance approach, and XISM-CHAMP02S from the improved energy balance, are determined using about one year’s worth of CHAMP kinematic orbits from TUM and accelerometer data from GFZ. Comparisons among XISM-CHAMP01S, XISM-CHAMP02S, EIGEN-CG03C, EIGEN-CHAMP03S, EIGEN2, ENIGN1S and EGM96 are made. The results show that the XISM-CHAMP02S model is more accurate than EGM96, EIGEN1S, EIGEN2 and XISM-CHAMP01S at the same degree and order, and has almost the same accuracy as EIGEN-CHAMP03S.

Determination of satellite velocity and acceleration from kinematic LEO orbits

Some interpolation and smoothing techniques have been analyzed for velocity and acceleration determination from kinematic LEO (Low-Earth Orbiter) orbits. Tests were performed on a simulated GOCE orbit and on a kinematic CHAMP orbit as well.

Precise orbit determination for the GOCE satellite using GPS

Apart from the gradiometer as the core instrument, the first ESA Earth Explorer Core Mission GOCE (Gravity field and steady-state Ocean Circulation Explorer) will carry a 12-channel GPS receiver dedicated for precise orbit determination (POD) of the satellite. The EGG-C (European GOCE Gravity-Consortium), led by the Technical University in Munich, is building the GOCE HPF (High-level Processing Facility) dedicated to the Level 1b to Level 2 data processing. One of the tasks of this facility is the computation of the Precise Science Orbit (PSO) for GOCE. The PSO includes a reduced-dynamic and a kinematic orbit solution. The baseline for the PSO is a zero-difference procedure using GPS satellite orbits, clocks, and Earth Rotation Parameters (ERPs) from CODE (Center for Orbit Determination in Europe), one of the IGS (International GNSS Service) Analysis Centers. The scheme for reduced-dynamic and kinematic orbit determination is based on experiences gained from CHAMP and GRACE POD and is realized in one processing flow. Particular emphasis is put on maximum consistency in the analysis of day boundary overlapping orbital arcs, as well as on the higher data sampling rate with respect to CHAMP and GRACE and on differences originating from different GPS antenna configurations. We focus on the description of the procedure used for the two different orbit determinations and on the validation of the procedure using real data from the two GRACE satellites as well as simulated GOCE data.

Analysis of numerical differentiation methods applied for determination of kinematic velocities for LEOs

Kinematic orbits provide a time series of independent positions, which are a good base for gravity field recovery. Gravity field recovery using the energy integral requires numerical differentiation in order to get velocity information for kinetic energy. This paper deals with numerical differentiation methods to test the most effective method for velocity determination of a LEO (Low Earth Orbiter).

Dependence of the Earth’s gravity model derived from satellite accelerations on a priori information

Computation of a new Earth’s gravity field model is difficult or even impossible if without a priori information about the gravity field. It is important to understand how incorporation of this information at different stages of gravity field modeling could influence the final results. The focus of the paper is on a technique based on satellite accelerations; a version of this technique was used earlier to derive the gravity field model DEOS_CHAMP-01C_70 from a 1-year set of CHAMP data. The technique makes use of both the original kinematic satellite orbit to derive the observed accelerations, and a smoothed orbit to compute the reference accelerations and to detect outliers. Now, using the same data set, we have analyzed how sensitive the obtained results are on the utilization of the a priori knowledge of the Earth’s gravity field at different data processing steps. We have found that the final model is not very sensitive on how the smoothed satellite orbit is obtained, though it is advisable that this orbit accounts for a realistic gravity field model. On the contrary, it is crucial that the orbit used to derive the observed satellite accelerations should not be smoothed. Furthermore, it is important to built an accurate stochastic model of data noise. A reasonable way to do so is to analyze the posterior residuals, which are obtained as the difference between the observations and simulations based on a certain gravity field model. The quality of this model is not crucial; it can be obtained from the same data at the stage of trial data processing, i.e. without an accurate stochastic model (at least, if the number of data substantially exceeds the number of unknown parameters). Finally, the quality of gravity field model significantly depends on the reference gravity field. In particular, the EGM96 model noticeably improves the results of CHAMP data processing both at low and at high spherical harmonic degrees.

Gross Error Compensation for Gravity Field Analysis Based on Kinematic Orbit Data

This paper aims at a comparative study of several measures to compensate for gross errors in kinematic orbit data. It starts with a simulation study on the influence of a single outlier in the orbit data on the gravity field solution. It is shown that even a single outlier can degrade the resulting gravity field solution considerably. To compensate for outliers, two different strategies are investigated: wavelet filters, which detect and eliminate gross errors, and robust estimators, which due to an iterative downweighting gradually ignore those observations that lead to large residuals. Both methods are applied in the scope of the analysis of a 2-year kinematic CHAMP (challenging minisatellite payload) orbit data set. In various real data studies, robust estimators outperform wavelet filters in terms of resolution of the derived gravity field solution. This superior performance is at the cost of computational load, as robust estimators are implemented iteratively and require the solution of large sets of linear equations several times.

Smoothing a Satellite Orbit on the Basis of B‐Spline and Regularization

AbstractA new algorithm is developed to compute a smooth and accurate satellite orbit on the basis of a kinematic orbit and a force model. Such a smoothed orbit is needed e.g. in the gravity field modeling procedure that is based on satellite accelerations. The procedure makes use of residual satellite accelerations defined as the difference between the observed accelerations and the reference ones. Computation of the reference accelerations requires an accurately determined set of satellite positions, which can be taken over from the smoothed orbit. The algorithm for computing the smoothed orbit is based on B‐splines. A component of the orbit (i.e. a set of X‐, Y ‐, or Z‐ coordinates in an inertial frame) is parameterized as a linear combination of B‐spline functions with corresponding coefficients. These coefficients are determined by means of a least‐square adjustment scheme. A force model is exploited to compute satellite accelerations, which are used as additional constraints. In order to find the balance between errors in the positions and unaccuracies of the computed accelerations, the generalized cross‐validation technique is implemented. Thanks to the property of a local support of B‐splines, the procedure is very efficient numerically.A numerical example demonstrates a performance of the proposed procedure. Furthermore, it shows that the solution with acceleration constraints is better than the solution based on the kinematic orbit only. The more precise the force model is, the more accurate smoothed orbit is obtained. Finally, a set of real CHAMP data is considered in the context of gravity field modeling. Smoothed orbits, which are computed both with and without acceleration constraints, as well as the original kinematic orbit are employed for calculating reference accelerations. The “Delft approach” is applied to recover gravity field models, which are compared with the EIGEN‐CG01C model. The results show that usage of the smoothed orbit obviously improves the gravity field model, especially if the smoothed orbit is computed with acceleration constraints.

Derivation of the CHAMP-only global gravity field model TUG-CHAMP04 applying the energy integral approach

A global gravity field model TUG-CHAMP04, derived from CHAMP (CHAllenging Minisatellite Payload) satellite-to-satellite GPS tracking observations in the high-low mode (SST-hl) in combination with CHAMP accelerometry, is presented and described in detail in this paper. For this purpose the energy integral approach was applied to precise kinematic orbits and accelerometer data. The advantage of these kinds of orbits is that they are derived from purely geometrical information, hence no external gravity field information is used for the determination of the positions. The disadvantage of precise kinematic orbit information is, that no velocities are delivered and hence a procedure has to be elaborated to deduce the velocities from kinematic positions. This work is done in preparation for ESA’s GOCE (Gravity field and steady state Ocean Circulation Explorer) satellite mission (scheduled launch November 2006), aiming at a high precision and high-resolution gravity field model on a global scale. This paper concentrates on the CHAMP data processing, where, in contrast to the usual standard method (processing in the Earth fixed frame), an approach in the inertial frame is chosen. Focus is taken on the data preprocessing of both accelerometer and orbit data, emphasising on the correct treatment of data-gaps and outlier detection. Furthermore an arc-wise weighting strategy is introduced and the advantages/disadvantages of this approach are discussed. Finally, the TUG-CHAMP04 model, calculated from one year of CHAMP data is compared with the official CHAMP gravity field model EIGEN-3p and terrestrial data (GPS levelling data).

‘DEOS_CHAMP-01C_70’: a model of the Earth’s gravity field computed from accelerations of the CHAMP satellite

Performance of a recently proposed technique for gravity field modeling has been assessed with data from the CHAMP satellite. The modeling technique is a variant of the acceleration approach. It makes use of the satellite accelerations that are derived from the kinematic orbit with the 3-point numerical differentiation scheme. A 322-day data set with 30-s sampling has been used. Based on this, a new gravity field model – DEOS CHAMP-01C 70 - is derived. The model is complete up to degree and order 70. The geoid height difference between the DEOS CHAMP-01C 70 and EIGEN-GRACE01S models is 14 cm. This is less than for two other recently published models: EIGEN-CHAMP03Sp and ITG-CHAMP01E. Furthermore, we analyze the sensitivity of the model to some empirically determined parameters (regularization parameter and the parameter that controls the frequency-dependent data weighting).We also show that inaccuracies related to non-gravitational accelerations, which are measured by the on-board accelerometer, have a minor influence on the computed gravity field model.

CHAMP Gravity Field Recovery Using Kinematic Orbits

A earth gravity field recovery from CHAMP kinematic orbits using energy conservation approach is presented. A method combining the Newton's numerical differential formula and the remove-restore procedure for determining kinematic velocities is developed. Three earth gravity field models with 50 degree and order are recovered using CHAMP kinematic orbits provided by Technical University of Munich (TUM). The comparisons of degree variance and geoid difference with GRIM5-C1, EIGEN-1S and EIGEN-2 indicate that the recovered models yield a better fit to EIGEN-2. The assessments using Arctic gravity anomalies data show that these models have nearly the same quality of EIGEN-1S and EIGEN-2, and perform better than GRIM5-C1.

ITG-CHAMP01: a CHAMP gravity field model from short kinematic arcs over a one-year observation period

Global gravity field models have been determined based on kinematic orbits covering an observation period of one year beginning from March 2002. Three different models have been derived up to a maximum degree of n=90 of a spherical harmonic expansion of the gravitational potential. One version, ITG-CHAMP01E, has been regularized beginning from degree n=40 upwards, based on the potential coefficients of the gravity field model EGM96. A second model, ITG-CHAMP01K, has been determined based on Kaula’s rule of thumb, also beginning from degree n=40. A third version, ITG-CHAMP01S, has been determined without any regularization. The physical model of the gravity field recovery technique is based on Newton’s equation of motion, formulated as a boundary value problem in the form of a Fredholm-type integral equation. The observation equations are formulated in the space domain by dividing the one-year orbit into short sections of approximately 30-minute arcs. For every short arc, a variance factor has been determined by an iterative computation procedure. The three gravity field models have been validated based on various criteria, and demonstrate the quality of not only the gravity field recovery technique but also the kinematically determined orbits.

The Kaiserslautern multiscale geopotential model SWITCH-03 from orbit perturbations of the satellite CHAMP and its comparison to the models EGM96, UCPH2002_02_0.5, EIGEN-1s and EIGEN-2

SUMMARY Observed orbit perturbations of the near-Earth orbiting satellite CHAMP are analysed to recover the long-wavelength features of the Earth’s gravitational potential. More precisely, by tracking the low-flying satellite CHAMP using the high-flying satellites of the Global Positioning System (GPS) the kinematic orbit of CHAMP is obtainable from GPS tracking observations, i.e. the ephemeris in Cartesian coordinates in an Earth-fixed coordinate frame (WGS84) becomes available. In this study we are concerned with two tasks: first we present new methods for preprocessing, modelling and analysing the emerging tracking data. Then, in a first step we demonstrate the strength of our approach by applying it to simulated CHAMP orbit data. In a second step we present results obtained by operating on a data set derived from real CHAMP data. The modelling is mainly based on a connection between non-bandlimited spherical splines and least-square adjustment techniques to take into account the non-sphericity of the trajectory. Furthermore, harmonic regularization wavelets for solving the underlying satellite-to-satellite tracking (SST) problem are used within the framework of multiscale recovery of the Earth’s gravitational potential leading to SWITCH-03 (Spline and Wavelet Inverse Tikhonov regularized CHamp data). Further we show how regularization parameters can be adapted adequately to a specific region thereby improving a globally resolved model. Finally we compare the developed model with the EGM96 model, model UCPH2002 02 0.5 from the University of Copenhagen and the GFZ models EIGEN-1s and EIGEN-2.

A CHAMP‐only gravity field model from kinematic orbits using the energy integral

In this paper we present results of a global gravity field recovery using half a year of CHAMP data. We use the energy integral of the motion of a satellite to transform satellite velocities into values of gravitational potential. The feasibility of this approach has already been demonstrated by several groups, using CHAMP reduced‐dynamic orbits. We show, that the potential recovered from this kind of orbits depends on the a priori gravity field used for orbit determination. Thus, it cannot be excluded that errors present in the prior field propagate into the new CHAMP gravity model. It is the intention of this paper to avoid this dependency through the use of kinematic orbits, which are free from prior information. The derived potential model, TUM‐1S, is validated by comparison to ground data and by satellite orbit residuals. It is shown to be comparable in quality to other state‐of‐the‐art gravity field models.

Harmonic analysis of the Earth's gravitational field by means of semi-continuous ephemerides of a low Earth orbiting GPS-tracked satellite. Case study: CHAMP

An algorithm for the determination of the spherical harmonic coefficients of the terrestrial gravitational field representation from the analysis of a kinematic orbit solution of a low earth orbiting GPS-tracked satellite is presented and examined. A gain in accuracy is expected since the kinematic orbit of a LEO satellite can nowadays be determined with very high precision, in the range of a few centimeters. In particular, advantage is taken of Newton's Law of Motion, which balances the acceleration vector with respect to an inertial frame of reference (IRF) and the gradient of the gravitational potential. By means of triple differences, and in particular higher-order differences (seven-point scheme, nine-point scheme), based upon Newton's interpolation formula, the local acceleration vector is estimated from relative GPS position time series. The gradient of the gravitational potential is conventionally given in a body-fixed frame of reference (BRF) where it is nearly time independent or stationary. Accordingly, the gradient of the gravitational potential has to be transformed from spherical BRF to Cartesian IRF. Such a transformation is possible by differentiating the gravitational potential, given as a spherical harmonics series expansion, with respect to Cartesian coordinates by means of the chain rule, and expressing zero- and first-order Ferrer's associated Legendre functions in terms of Cartesian coordinates. Subsequently, the BRF Cartesian coordinates are transformed into IRF Cartesian coordinates by means of the polar motion matrix, the precession–nutation matrices and the Greenwich sidereal time angle (GAST). In such a way a spherical harmonic representation of the terrestrial gravitational field intensity with respect to an IRF is achieved. Numerical tests of a resulting Gauss–Markov model document not only the quality and the high resolution of such a space gravity spectroscopy, but also the problems resulting from noise amplification in the acceleration determination process.

Space Gravity Spectroscopy - determination of the Earth’s gravitational field by means of Newton interpolated LEO ephemeris Case studies on dynamic (CHAMP Rapid Science Orbit) and kinematic orbits

Abstract. An algorithm for the (kinematic) orbit analysis of a Low Earth Orbiting (LEO) GPS tracked satellite to determine the spherical harmonic coefficients of the terrestrial gravitational field is presented. A contribution to existing long wavelength gravity field models is expected since the kinematic orbit of a LEO satellite can nowadays be determined with very high accuracy in the range of a few centimeters. To demonstrate the applicability of the proposed method, first results from the analysis of real CHAMP Rapid Science (dynamic) Orbits (RSO) and kinematic orbits are illustrated. In particular, we take advantage of Newton’s Law of Motion which balances the acceleration vector and the gradient of the gravitational potential with respect to an Inertial Frame of Reference (IRF). The satellite’s acceleration vector is determined by means of the second order functional of Newton’s Interpolation Formula from relative satellite ephemeris (baselines) with respect to the IRF. Therefore the satellite ephemeris, which are normally given in a Body fixed Frame of Reference (BRF) have to be transformed into the IRF. Subsequently the Newton interpolated accelerations have to be reduced for disturbing gravitational and non-gravitational accelerations in order to obtain the accelerations caused by the Earth’s gravitational field. For a first insight in real data processing these reductions have been neglected. The gradient of the gravitational potential, conventionally expressed in vector-valued spherical harmonics and given in a Body Fixed Frame of Reference, must be transformed from BRF to IRF by means of the polar motion matrix, the precession-nutation matrices and the Greenwich Siderial Time Angle (GAST). The resulting linear system of equations is solved by means of a least squares adjustment in terms of a Gauss-Markov model in order to estimate the spherical harmonics coefficients of the Earth’s gravitational field.Key words. space gravity spectroscopy, spherical harmonics series expansion, GPS tracked LEO satellites, kinematic