Skip to main content

Ranging algebraically with more observations than unknowns


In the recently developed Spatial Reference System that is designed to check and control the accuracy of the three-dimensional coordinate measuring machines and tooling equipment (Metronom US., Inc., Ann Arbor:, the coordinates of the edges of the instrument are computed from distances of the bars. The use of distances in industrial application is fast gaining momentum just as in Geodesy and in Geophysical applications and thus necessitating efficient algorithms to solve the nonlinear distance equations. Whereas the ranging problem with minimum known stations was considered in our previous contribution in the same Journal, the present contribution extends to the case where one is faced with many distance observations than unknowns (overdetermined case) as is usually the case in practise. Using the Gauss-Jacobi Combinatorial approach, we demonstrate how one can proceed to position without reverting to iterative and linearizing procedures such as Newton’s or Least Squares approach.


  1. Awange, J. L., Groebner basis solution of planar resection, Survey Review, 36(285), 528–543, 2002.

    Article  Google Scholar 

  2. Awange, J. L., Diagnosis of Outlier of type multipath in GPS Pseudo-range observations, Survey Review, in press.

  3. Awange, J. L., Algebraic approach to nonlinear global minimization problem relevant to Earth Sciences, Journal of Symbolic Computations, submitted.

  4. Awange, J. L. and E. Grafarend, Algebraic solution of GPS pseudo-ranging equations, Journal of GPS Solutions, 4(5), 20–32, 2002a.

    Article  Google Scholar 

  5. Awange, J. L. and E. Grafarend, Nonlinear adjustment of GPS observations of type pseudo-range, Journal of GPS Solutions, 4(5), 80–93, 2002b.

    Article  Google Scholar 

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

    Article  Google Scholar 

  7. Awange, J. L. and E. Grafarend, Polynomial optimization of the 7-parameter datum transformation problem when only three stations in both systems are given, Zeitschrift fuer Vermessungswesen, in press.

  8. Awange, J. L., E. Grafarend, Y. Fukuda, and S. Takemoto, Direct polynomial approach to nonlinear distance (ranging) problems, Earth Planets Space, 55(5), 231–241, 2003.

    Article  Google Scholar 

  9. Grafarend, E. and B. Schaffrin, The geometry of nonlinear adjustment—the planar trisection problem—, in Festschrift to T. Krarup, edited by E. Kejlso, K. Poder, and C. C. Tscherning, pp. 149–172, Geodaetisk Institut, Meddelese No. 58, Denmark, 1989.

  10. Grafarend, E. and B. Schaffrin, The planar trisection problem and the impact of curvature on non-linear least-squares estimation, Computational statistics & data analysis, 12, 187–199, 1991.

    Article  Google Scholar 

  11. Guolin, L., Nonlinear curvature measures of strength and nonlinear diagnosis, Allgemein Vermessungs-Nachrichten, 107, 109–111, 2000.

    Google Scholar 

  12. Jurisch, R., G. G. Kampmann, and I. Schmadel, The Spatial Reference System (SRS)—an industrial application for geometric quality control introducing modern optimization, regulation and adjustment techniques, paper to be presented at the International Dimensional Workshop 2003, Nashville Marriott Nashville, Tennessee, U.S.A. May 12–16, 2003, (also available in pdf format at: Kahmen, H. and W. Faig, Surveying, 578 pp, Walter de Gruyter, Berlin, 1988.

  13. Krarup, T., Nonlinear adjustment and curvature, in Forty Years of Thought, pp. 145–159, Delft, 1982.

  14. Lohse, P., Ausgleichungsrechnung in nichtlinearen Modellen. DGK, Reihe C, Heft Nr. 429, 1994.

  15. Mautz, R., Zur Lung nichtlinearer Ausgleichungsprobleme bei der Bestimmung von Frequenzen in Zeitreihen, DGK, Reihen C, Nr. 532, 2001.

  16. Teunissen, P., Nonlinear least squares, Manuscripta Geodaetica, 15, 137–150, 1990.

    Google Scholar 

  17. Xu, P., A hybrid global optimization method: the one-dimensional case, Journal of Computation and Applied mathematics, 147, 301–314, 2002.

    Article  Google Scholar 

  18. Xu, P., A hybrid global optimization method: the multi-dimensional case, Journal of Computation and Applied mathematics, 155, 423–446, 2003.

    Article  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. Ranging algebraically with more observations than unknowns. Earth Planet Sp 55, 387–394 (2003).

Download citation

Key words

  • Overdetermined planar ranging
  • overdetermined three-dimensional ranging
  • Gauss-Jacobi combinatorial algorithm
  • Groebner basis
  • Multipolynomial resultant
  • Least Squares