Skip to main content
SpringerLink
Log in

Assessment of numerical integration methods in the context of low Earth orbits and inter-satellite observation analysis

  • Published:
Acta Geodaetica et Geophysica Aims and scope Submit manuscript

Abstract

The integration of differential equations is a fundamental tool in the problem of orbit determination. In the present study, we focus on the accuracy assessment of numerical integrators in what refers to the categories of single-step and multistep methods. The investigation is performed in the frame of current satellite gravity missions i.e. Gravity Recovery and Climate Experiment (GRACE) and Gravity Field and steady-state Ocean Circulation Explorer (GOCE). Precise orbit determination is required at the level of a few cm in order to satisfy the primary missions’ scope which is the rigorous modelling of the Earth’s gravity field. Therefore, the orbit integration errors are critical for these low earth orbiters. As the result of different schemes of numerical integration is strongly affected by the forces acting on the satellites, various validation tests are performed for their accuracy assessment. The performance of the numerical methods is tested in the analysis of Keplerian orbits as well as in real dynamic orbit determination of GRACE and GOCE satellites by taking into account their sophisticated observation techniques and orbit design. Numerical investigation is performed in a wide range of the fundamental integrators’ parameters i.e. the integration step and the order of the multistep methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Berry MM, Healy LM (2004) Implementation of Gauss-Jackson integration for orbit propagation. J Astronaut Sci 52(3):331–357

    Google Scholar 

  • Beutler G (2005) Methods of celestial mechanics Vol. I: physical, mathematical and numerical principles. Springer, Berlin

    Book  Google Scholar 

  • Beutler G, Jäggi A, Mervart L, Meyer U (2010a) The celestial mechanics approach: application to data of the GRACE mission. J Geod 84(11):661–681

    Article  Google Scholar 

  • Beutler G, Jäggi A, Mervart L, Meyer U (2010b) The celestial mechanics approach: theoretical foundations. J Geod 85(10):605–624

    Article  Google Scholar 

  • Bizouard Ch, Gambis D (2009) The combined solution C04 for earth orientation parameters consistent with international terrestrial reference frame 2005. In: Drewes H (ed) Geodetic reference frames, IAG symposia series, vol 134. Springer, Berlin, pp 265–270

    Chapter  Google Scholar 

  • Bobojć A, Drożyner A (2003) Satellite orbit determination using satellite gravity gradiometry observations in GOCE mission perspective. Adv Geosci 1:1–4

    Article  Google Scholar 

  • Bock H, Jäggi A, Meyer U, Visser P, van de Ijssel J, van Helleputte T, Heinze M, Hugentobler U (2011) GPS-derived orbits for the GOCE satellite. J Geod 85(11):807–919

    Article  Google Scholar 

  • Bock H, Jäggi A, Beutler G, Meyer U (2014) GOCE: precise orbit determination for the entire mission. J Geod 88:1047–1060

    Article  Google Scholar 

  • Brouwer D (1959) Solution of the artificial satellite problem without drag. Astron J 64:378–397

    Article  Google Scholar 

  • Case K, Kruizinga G, Wu S (2010) GRACE level 1B data product user handbook. JPL Publication D-22027, version 1.3. Jet Propulsion Laboratory, California Institute of Technology

  • Casotto S (1993) Position and velocity perturbations in the orbital frame in terms of classical elements perturbations. Celest Mech Dyn Astron 55:209–221

    Article  Google Scholar 

  • Colombo OL (1986) Ephemeris errors of GPS satellites. Bull Géod 60:64–84

    Article  Google Scholar 

  • Cui C (1997) Satellite orbit integration based on canonical transformations with special regard to the resonance and coupling effects. Deutsche Geodätische Kommission, Reihe A, Heft Nr.112, München

  • Cui Ch, Lelgemann D (1995) Analytical dynamic orbit improvement for the evaluation of geodetic-geodynamic satellite data. J Geod 70:83–97

    Article  Google Scholar 

  • Cui Ch, Lelgemann D (2000) On non-linear low-low SST observation equations for the determination of the geopotential based on an analytical solution. J Geod 74:431–440

    Article  Google Scholar 

  • Doornbos E, Förster M, Fritsche B, van Helleputte T, van den Ijssel J, Koppenwallner G, Lühr H, Rees D, Visser P (2009) Air density models derived from multi-satellite drag observations, Final Report, ESA/ESTEC contract 21022/07/NL/HE, DEOS/TU Delft scientific report 01/2009

  • Dormand JR, Prince PJ (1978) New Runge-Kutta algorithms for numerical simulation in dynamical astronomy. Celest Mech Dyn Astron 18:223–232

    Article  Google Scholar 

  • Dormand JR, Prince PJ (1987) Runge-Kutta-Nystrom triples. Comput Math Appl 13(12):937–949

    Article  Google Scholar 

  • Elsaka B, Raimondo JC, Brieden P, Reubelt T, Kusche J, Flechtner F, Iran-Pour S, Sneeuw N (2014) Comparing seven candidate mission configurations for temporal gravity field retrieval through full-scale numerical simulation. J Geod 88(1):31–43

    Article  Google Scholar 

  • ESA (1999) Gravity field and steady-state ocean circulation mission, ESA SP-1233(1), Report for mission selection of the four candidate earth explorer missions. ESA Publications Division, ESTEC, Noordwijk

    Google Scholar 

  • Floberghagen R, Fehringer M, Lamarre D, Muzi D, Frommknecht B, Steiger C, Pineiro J, da Costa A (2011) Mission design, operation and exploitation of the gravity field and steady-state ocean circulation explorer mission. J Geod 85:749–758

    Article  Google Scholar 

  • Förste Ch, Schmidt R, Stubenvoll R, Flechtner F, Ul Meyer, König R, Neumayer H, Biancale R, Lemoine J-M, Bruinsma S, Loyer S, Barthelmes F, Esselborn S (2008) The GeoForschungsZentrum Potsdam/Groupe de Recherche de Gèodésie Spatiale satellite-only and combined gravity field models: EIGEN-GL04S1 and EIGEN-GL04C. J Geod 82:331–346

    Article  Google Scholar 

  • Fox K (1984) Numerical integration of the equations of motion of celestial mechanics. Celest Mech 33(2):127–142

    Article  Google Scholar 

  • Helleputte T, Doornbos E, Visser P (2009) CHAMP and GRACE accelerometer calibration by GPS-based orbit determination. Adv Space Res 43(12):1890–1896

    Article  Google Scholar 

  • Hull TE, Enright WH, Fellen BM, Sedgwick AE (1972) Comparing numerical methods for ordinary differential equations. SIAM J Numer Anal 9(4):603–637

    Article  Google Scholar 

  • Ilk KH, Löcher A, Mayer-Gürr T (2008) Do we need new gravity field recovery techniques for the new gravity field satellites? In: Xu PL, Liu JN, Dermanis A (eds) VI Hotine-Marussi symposium on theoretical and computational geodesy, International association of geodesy symposia, vol 132. Springer, Berlin, pp 3–8

    Chapter  Google Scholar 

  • Jäggi A, Hugentobler U, Beutler G (2006) Pseudo-stochastic orbit modelling techniques for low-earth orbiters. J Geodesy 80:47–60

    Article  Google Scholar 

  • Kang Z, Tapley B, Bettadpur S, Ries J, Nagel P, Pastor P (2006a) Precise orbit determination for the GRACE mission using only GPS data. J Geod 80:322–331

    Article  Google Scholar 

  • Kang Z, Tapley B, Bettadpur S, Ries J, Nagel P (2006b) Precise orbit determination for GRACE using accelerometer data. Adv Space Res 38:2131–2136

    Article  Google Scholar 

  • Kaula WM (1966) Theory of satellite geodesy. Dover Publication, Mineola

    Google Scholar 

  • Kim J, Tapley BD (2002) Error analysis of a low-low satellite-to-satellite tracking mission. J Guid Control Dyn 25(6):1100–1106

    Article  Google Scholar 

  • Kozai Y (1959) The motion of a close earth satellite. Astron J 64:367–377

    Article  Google Scholar 

  • Loomis Β, Nerem R, Luthcke S (2012) Simulation study of a follow-on gravity mission to GRACE. J Geod 86:319–335

    Article  Google Scholar 

  • Luthcke SB, Zelensky NP, Rowlands DD, Lemoine FG, Williams TA (2003) The 1-centimeter orbit: Jason-1 precision orbit determination using GPS, SLR, DORIS, and altimeter data. Mar Geod 26:399–421

    Article  Google Scholar 

  • Lyard F, Lefevre F, Letellier T, Francis O (2006) Modelling the global ocean tides: modern insights from FES2004. Ocean Dyn 56:394–415

    Article  Google Scholar 

  • Mayer-Guerr T et al. (2012) The new combined satellite only model GOCO03 s, Presented at GGHS2012, Venice

  • Montenbruck O (1992) Numerical integration methods for orbital motion. Celest Mech Dyn Astron 53:59–69

    Article  Google Scholar 

  • Montenbruck O, Gill E (2000) Satellite orbits; models, methods and applications. Springer, New York

    Book  Google Scholar 

  • Montenbruck O, van Helleputte T, Kroes R, Gill E (2005) Reduced dynamic orbit determination using GPS code and carrier measurements. Aerosp Sci Technol 9:261–271

    Article  Google Scholar 

  • Pail R, Bruinsma S, Migliaccio F, Forste C, Goiginger H, Schuh W-D, Hock E, Reguzzoni M, Brockmann JM, Abrikosov O, Veicherts M, Fecher T, Mayrhofer R, Krasbutter I, Sanso F, Tscherning CC (2011) First GOCE gravity field models derived by three different approaches. J Geod 85(11):819–843

    Article  Google Scholar 

  • Panet I, Flury J, Biancale R, Gruber T, Johannessen J, van den Broeke MR, van Dam T, Gegout P, Hughes CW, Ramillien G, Sasgen I, Seoane L, Thomas M (2013) Earth system mass transport mission (e.motion): a concept for future earth gravity field measurements from space. Surv Geophys 34(2):141–163

    Article  Google Scholar 

  • Papanikolaou TD (2012) Dynamic modelling of satellite orbits in the frame of contemporary satellite geodesy missions (in Greek), Ph.D. Dissertation, Aristotle University of Thessaloniki, Thesssaloniki

  • Papanikolaou TD, Tsoulis D (2014) Dynamic orbit parameterization and assessment in the frame of current GOCE gravity models. Phys Earth Planet Inter 236:1–9. doi:10.1016/j.pepi.2014.08.003

    Article  Google Scholar 

  • Petit G, Luzum B (2010) IERS Conventions 2010, IERS Technical Note No.36, Verlag des Bundesamts für Kartographie und Geodäsie, Frankfurt am Main

  • Seeber G (2003) Satellite geodesy, foundations, methods and applications. Walter de Gruyter, Berlin

    Book  Google Scholar 

  • Sheard BS, Heinzel G, Danzmann K, Shaddock DA, Klipstein WM, Folkner WM (2012) Intersatellite laser ranging instrument for the GRACE follow-on mission. J Geod 86:1083–1095. doi:10.1007/s00190-012-0566-3

    Article  Google Scholar 

  • Somodi B, Földvary L (2011) Application of numerical integration techniques for orbit determination of state-of-the-art LEO satellites. Period Polytech 55(2):99–106

    Article  Google Scholar 

  • Standish EM (1998) JPL planetary and lunar ephemerides; DE405/LE405, JPL IOM, 312F98-048

  • Svehla D, Rothacher M (2003) Kinematic and reduced-dynamic precise orbit determination of low Earth orbiters. Adv Geosci 1:47–56

    Article  Google Scholar 

  • Svehla D, Rothacher M (2005) Kinematic positioning of LEO and GPS satellites and IGS stations on the ground. Adv Space Res 36:376–381

    Article  Google Scholar 

  • Tapley BD, Bettadpur S, Watkins M, Reigber C (2004) The gravity recovery and climate experiment: mission overview and early results. Geophys Res Lett. doi:10.1029/2003GL019285

    Google Scholar 

  • Tsoulis D, Ieronimaki Z, Kalampoukas G, Papanikolaou D, Papanikolaou T, Patlakis K, Vassiliadis I (2011) Spectral analysis and interpretation of current satellite-only Earth gravity models by incorporating global terrain and crustal data, Final Report, ESA Contract 22319/09/NL/CB

  • Wnuk E (1999) Recent progress in analytical orbit theories. Adv Space Res 23(4):677–687

    Article  Google Scholar 

  • Xu P (2008a) Position and velocity perturbations for the determination of geopotential from space geodetic measurements. Celest Mech Dyn Astron 100(3):231–249

    Article  Google Scholar 

  • Xu G (2008b) Orbits. Springer, Berlin

    Google Scholar 

  • Zhu SY, Neumayer KH, Massmann F-H, Shi C, Reigber Ch (2003) Impact of different data combinations on the CHAMP orbit determination. In: Reigber Ch, Luehr H, Schwintzer P (eds) First CHAMP mission results for gravity, magnetic and atmospheric studies. Springer, Berlin, pp 92–97

    Chapter  Google Scholar 

  • Zhu S, Reigber Ch, König R (2004) Integrated adjustment of CHAMP, GRACE and GPS data. J Geod 78:103–108

    Article  Google Scholar 

Download references

Acknowledgments

The authors acknowledge the European Space Agency for providing GOCE data, the Geoforschungszentrum Potsdam (GFZ-Potsdam) for providing the GRACE data, the Earth Orientation Center for providing the EOP data, JPL for providing the DE data and DEOS at TU Delft for providing the GRACE accelerometry calibration parameters. The motivation for the present work has emerged in the frame of European Space Agency Contract 22319/09/NL/CB.

Author information

Authors and Affiliations

  1. Department of Geodesy and Surveying, Aristotle University of Thessaloniki, Thessaloniki, Greece

    Thomas D. Papanikolaou & Dimitrios Tsoulis

Authors
  1. Thomas D. Papanikolaou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dimitrios Tsoulis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas D. Papanikolaou.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Papanikolaou, T.D., Tsoulis, D. Assessment of numerical integration methods in the context of low Earth orbits and inter-satellite observation analysis. Acta Geod Geophys 51, 619–641 (2016). https://doi.org/10.1007/s40328-016-0159-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40328-016-0159-3

Keywords

Search

Navigation