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Pseudo-Stochastic Orbit Modeling Techniques for Low-Earth Orbiters View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by RERO DOC Digital Library J Geod (2006) 80: 47–60 DOI 10.1007/s00190-006-0029-9 ORIGINAL ARTICLE A. Jäggi · U. Hugentobler · G. Beutler Pseudo-stochastic orbit modeling techniques for low-Earth orbiters Received: 1 March 2005 / Accepted: 24 January 2006 / Published online: 30 March 2006 © Springer-Verlag 2006 Abstract The Earth’s non-spherical mass distribution and Keywords Low-Earth orbiter (LEO) · Precise orbit atmospheric drag cause the strongest perturbations on very determination (POD) · Pseudo-stochastic orbit modeling · low-Earth orbiting satellites (LEOs). Models of gravitational GPS and non-gravitational accelerations are utilized in dynamic precise orbit determination (POD) with GPS data, but it is 1 Introduction also possible to derive LEO positions based on GPS pre- cise point positioning without dynamical information. We Gravitational forces related to the Earth’s oblateness and the use the reduced-dynamic technique for LEO POD, which non-gravitational forces due to residual atmospheric densities combines the geometric strength of the GPS observations are responsible for the strongest perturbations acting on sat- with the force models, and investigate the performance of ellites in very low-Earth orbits. These forces produce large different pseudo-stochastic orbit parametrizations, such as periodic variations in the orbit and a decay in the orbital instantaneous velocity changes (pulses), piecewise constant height. Radial orbit differences caused by atmospheric drag accelerations, and continuous piecewise linear accelerations. after one day may reach 150 m for an initial height of 500 km The estimation of such empirical orbit parameters in a stan- (Beutler 2004) during normal solar activity. In addition, ma- dard least-squares adjustment process of GPS observations, jor perturbations induced by higher-order terms of the Earth’s together with other relevant parameters, strives for the high- gravity field and minor perturbations induced by solar radia- est precision in the computation of LEO trajectories. We used tion pressure pose a challenge for precise orbit determination the procedures for the CHAMP satellite and found that the (POD) for low-Earth orbiters (LEOs). orbits may be validated by means of independent SLR mea- The determination of precise orbits using space geo- surements at the level of 3.2 cm RMS. Validations with inde- detic techniques, such as Satellite Laser Ranging (SLR), has pendent accelerometer data revealed correlations at the level opened a new era for LEO POD and related applications. It of 95% in the along-track direction. As expected, the empir- has been shown that geophysically relevant information such ical parameters compensate to a certain extent for deficien- as geocenter variations or gravity field coefficients can be ex- cies in the dynamic models. We analyzed the capability of tracted by combining many years of observations to geodetic pseudo-stochastic parameters for deriving information about satellites (Reigber 1989). LEO POD is also a necessity for the mismodeled part of the force field and found evidence that missions of Earth observing satellites, e.g., the determina- the resulting orbits may be used to recover force field param- tion of ocean topography from the TOPEX/Poseidon mis- eters, if the number of pseudo-stochastic parameters is large sion (Fu et al. 1994). This was the first extensive use of the enough. Results based on simulations showed a significantly Global Positioning System (GPS) with a spaceborne receiver better performance of acceleration-based orbits for gravity for LEO POD (Bertiger et al. 1994). The geometric strength field recovery than for pulse-based orbits, with a quality com- of continuously collected GPS observations allowed for orbit parable to a direct estimation if unconstrained accelerations determination to closer than 3 cm in the radial direction. This are set up every 30 s. mission stimulated a number of follow-up satellite missions to carry on-board GPS receivers for POD, but also for other purposes such as atmospheric sounding (e.g., Kursinski et al. B · · A. Jäggi ( ) U. Hugentobler G. Beutler 1997). Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland The launch of the CHAllenging Minisatellite Payload E-mail: [email protected] (CHAMP) on July 15, 2000, opened a new era in process- Tel.: +41-31-6318592 ing GPS data from spaceborne receivers. The combined Fax: +41-31-6313869 analysis of non-conservative accelerations, which were 48 A. Jäggi et al. (k) (k) directly measured with unprecedented accuracy by an on- with initial conditions r (t0) = r (a, e, i,,ω,T0; t0), board accelerometer (Touboul et al. 1999), and the GPS k = 0, 1 (level of time differentiation). The parameters tracking data proved the feasibility to separate gravitational a, e, i,,ω,T0 are the six Keplerian elements pertaining to from non-gravitational perturbations for gravity field esti- epoch t0. q1,...,qd denote additional dynamical parame- mation (Reigber et al. 2002). The large number of high qual- ters considered as unknowns, e.g., scaling factors of analyti- ity global gravity field models underscores the success of cally known accelerations, which describe deterministically CHAMP (e.g., Reigber et al. 2003, 2004). Current (GRACE) the perturbing acceleration acting on the satellite. and future (GOCE) gravity field recovery missions, which Let us assume that an a priori orbit r0(t) is available, e.g., aim to recover the gravity field with unprecedented accuracy, from a GPS code solution. Dynamic orbit determination may therefore pose the highest requirements on POD, e.g., 2 cm then be considered as an orbit improvement process, i.e., the RMS in each direction for GOCE (ESA 1999). actual orbit r(t) is expressed as a truncated Taylor series with There are many different studies for CHAMP POD based respect to the unknown orbit parameters pi about the a priori on GPS tracking data (e.g., Švehla and Rothacher 2002; van orbit, which is represented by the parameter values pi0: den IJssel et al. 2003; König et al. 2004). Most of them are n ∂ ( ) based on a technique called “reduced-dynamic” (Wu et al. r(t) = r (t) + r0 t · (p − p ), (2) 0 ∂pi i i0 1991), which exploits the geometric strength of the GPS i=1 observations to reduce the dependency on dynamic models where n = 6 + d denotes the total number of unknown orbit (see Sects. 5 and 6). Due to CHAMP’s very low altitude parameters, i.e., the six initial conditions for position and (below 450 km), it is common practice to solve for a large velocity and d dynamical parameters. Equation (2) allows us number of empirical parameters to compensate for defi- to improve the a priori orbit provided that the unknown orbit ciencies in the dynamic models. An alternative kinematic parameters and the partial derivatives of the a priori orbit with approach was successfully demonstrated for CHAMP by respect to those parameters are known. Švehla and Rothacher (2004), which yields precise ephem- eris entirely by geometric means. Such positions are attractive mainly for gravity field estimation because they are indepen- 2.1 Variational equations dent of any dynamic models. Gerlach et al. (2003) used kine- matic positions and accelerometer data and showed that grav- Let us assume that p is one of the parameters defining the ity field models can be estimated with a quality comparable to initial values or the dynamics in Eq. (1) and that the par- the official CHAMP models by means of the energy integral tial derivative of the a priori orbit r0(t) with respect to the method (O’Keefe 1957). The use of precise ephemeris for parameter p is designated by the function gravity field recovery had a considerably stimulating impact . ∂r0(t) on several groups (e.g., Mayer-Gürr et al. 2004) due to less zp(t) = . (3) demanding computational resources as in the case of classical ∂p numerical integration techniques (e.g., Visser et al. 2003b). The initial value problem associated with the partial deriv- In the following we consider pseudo-stochastic orbit ative in Eq. (3) is referred to as the system of variational modeling techniques for reduced-dynamic LEO POD as a equations in this article, and is obtained by taking the partial powerful and efficient method to derive most precise satel- derivative of Eq. (1). The result may be written as lite trajectories in the very low-Earth orbit (e.g., Visser and ∂ IJssel 2003a). Furthermore, we discuss different orbit repre- f1 z¨ p = A0 · zp + A1 · z˙ p + , (4) sentations and question whether such methodologies can be ∂p used to generate reduced-dynamic orbits, which could serve where the 3×3 matrices A and A are defined by as input for a subsequent gravity field estimation process. 0 1 Section 2 reviews the methods of dynamic LEO POD. ∂ fi ∂ fi A0[i,k] = ; A1[i,k] = ; (5) Section 3 presents two widely-used pseudo-stochastic orbit ∂r0,k ∂r˙0,k parametrizations and develops the mathematical background where fi denotes the i-th component of the total accelera- for a third, more refined orbit parametrization. CHAMP orbit ∈{ , , ,,ω, } results are shown in Sect. 4. Section 5 analyzes the presented tion f in Eq. (1). For p a e i T0 Eq. (4) is a lin- methods in a more detailed way in a simulation environ- ear, homogeneous, second-order differential equation system with initial values zp(t0) = 0 and z˙ p(t0) = 0, whereas for ment and focuses, together with Sect. 6, on a possible use of ∈{ ,..., } reduced-dynamic orbits for gravity field estimation. p q1 qd Eq. (4) is inhomogeneous with zero initial values. Note that in the latter case, the homogeneous part of Eq.
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