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  • Article
  • Open Access

Analytic solution of GPS atmospheric sounding refraction angles

Earth, Planets and Space201456:BF03352518

  • Received: 3 December 2003
  • Accepted: 19 May 2004
  • Published:


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.

Key words

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