Skip to main content

Analytic solution of GPS atmospheric sounding refraction angles


The nonlinear system of equations for solving GPS atmospheric sounding’s bending angles are normally solved using Newton’s method. Because of the nonlinear nature of the equations, Newton’s method applies linearization and iterations. The method assumes the refraction angle to be small enough such that the dependency of the doppler shift on these angles are linear. The bending angles are then solved iteratively. Since the approach assumes the dependency of doppler shift on bending angles to be linear, which in actual sense is not, some small nonlinearity error is incurred. The Newton’s iterative method is often used owing to the bottleneck of solving in exact form the nonlinear system of equations for bending angles. By converting this system of trigonometric nonlinear equations into algebraic, the present contribution proposes an analytic (algebraic) algorithm for solving the bending angles and presents the geometry of the solution space. The algorithm is tested by computing bending angles of three CHAMP occultation data and the results compared to those of iterative Newton’s approach. Occultation 133 of 3rd May 2002 is selected as it occurred during diurnal solar radiation maximum past afternoon. During this time, the effect of ionospheric noise is high. Occultations number 3 of 14th May 2001 and number 6 of 2nd February 2002 were selected since they occurred past mid-night, a time of low solar activity and thus less effect of ionospheric noise. The results for occultation 133 of 3rd May 2002 indicate that the nonlinearity errors in bending angles increase with decrease in height to a maximum absolute value of 0.00069° (0.1%) for the region 5–40 km during period of high solar activity. Such nonlinearity errors are shown to impact significantly on the computed impact parameters to which the bending angles are referred. During low solar activity period, the nonlinearity error was relatively small for occultation number 3 of 14th May 2001 with maximum absolute value of 0.00001°. The analytical algorithm thus provide an independent method for controlling classical iterative procedures and could be used where very accurate results are desired.


  • Awange, J. L., Groebner basis solution of planar resection, Survey Review, 36, 528–543, 2002.

    Article  Google Scholar 

  • Awange, J. L. and E. Grafarend, Sylvester resultant solution of planar ranging problem, Allgemeine Vermessungs-Nachrichten, 108, 143–146, 2002.

    Google Scholar 

  • Awange, J. L. and E. Grafarend, Groebner basis solution of the three-dimensional resection problem (P4P), Journal of Geodesy, 77Journal of Geodesy, 327–337, 2003a.

    Article  Google Scholar 

  • Awange, J. L. and E. Grafarend, Explicit solution of the overdetermined three-dimension resection problem, Journal of Geodesy, 76, 605–616, 2003b.

    Article  Google Scholar 

  • Awange, J. L., E. Grafarend, Y. Fukuda, and S. Takemoto, Direct Polynomial approach to nonlinear distance (ranging) problems, Earth Planets Space, 55, 231–241, 2003.

    Article  Google Scholar 

  • Becker, T. and V. Weispfenning, Groebner bases. A computational approach to commutative algebra. Graduate Text in Mathematics 141, 2nd Edition, Springer-Verlag, New York, 1998.

    Google Scholar 

  • Buchberger, B., Ein Algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungsystems, Aequationes Mathematicae, 4, 374–383, 1970.

    Article  Google Scholar 

  • Buchberger, B., Groebner bases. A short introduction for system theorists, in EUROCAST2001, LNCS 2178, edited by R. Moreno-Diaz et al., 1–19, 2001.

    Google Scholar 

  • Cox, D., J. Little and D. O’Shea, Using algebraic geometry. Graduate Text in Mathematics 185, 499 pp., Springer-Verlag, New York, 1998.

    Book  Google Scholar 

  • Gurbunov, M. E., A. S. Gurvich, and L. Bengtsson, Advanced algorithms of inversion of GPS/MET satellite data and their application to the reconstruction of temperature and humidity, Rep. No. 211, Max-Plunk-Institut für Meteorologie, Hamburg, Germany, 1996.

    Google Scholar 

  • Kursinski, E. R., G. A. Hajj, J. T. Schofield, R. P. Linfield, and K. R. Hardy, Observing Earth’s atmosphere with radio occultation measurements using the Global Positioning System, J. Geophys. Res., 102,23,429–23,465, 1997.

    Article  Google Scholar 

  • Healey, S., A. Jupp, D. Offiler, and J. Eyre, The assimilation of radio occultation measurements, in First CHAMP Mission Results for Gravity, Magnetic and Atmospheric Studies edited by C. Reigber, H. Lhr, and P. Schwintzer, Springer-Verlag, Heidelberg, 453–461, 2003.

    Chapter  Google Scholar 

  • Steiner, A. K., High resolution sounding of key climate variables using the radio occultation technique, Dissertation, Institute for Meteorology and Geophysics, University of Graz, No. 3, 1998.

    Google Scholar 

  • Sturmfels, B., Introduction to resultants, AMS Proceedings of Symposia in Applied Mathematics, 53, 25–39, 1998.

    Article  Google Scholar 

  • Vorob’ev, V V and T. G. Krasil’nikova, Estimation of the accuracy of atmospheric refractive index recovery from Doppler shift measurements at frequencies used in the NAVSTAR system, Phys. of Atmos. and Oceans, 29, 602–609, 1994.

    Google Scholar 

  • Wickert, J., Das CHAMP-Radiookkultationsexperiment: Algorithmen, Prozessierungssystem und Ergebnisse, Scientific Technical Report STR02/07, GFZ, Potsdam, 2002.

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Joseph L. Awange.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Awange, J.L., Fukuda, Y., Takemoto, S. et al. Analytic solution of GPS atmospheric sounding refraction angles. Earth Planet Sp 56, 573–587 (2004).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

Key words

  • Nonlinearity
  • Sylvester resultant
  • reduced Groebner basis
  • Analytical algorithm