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Microwave propagation over the Earth: image inversion

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Abstract

An extremely accurate but simple asymptotic description for the path of a light ray propagating over a curved Earth with steady radial variations in refractive index is derived using simple scaling arguments. It is used to determine effectively exact analytic solutions for the path of rays through refractive-index profiles described in terms of patched quadratics. Such patched quadratics can be used to accurately describe almost all refractive-index profiles of practical interest. The results show that images generated by rays passing through a quadratic refractive-index profile are uniformly magnified in the vertical direction, and magnification and displacement observations can be used to determine the refractive-index profile parameters. For patched quadratics, observations of critical rays can be used to determine the thicknesses of the quadratic layers.

An effectively exact solution is also obtained for exponential index profiles and this is used to determine the path of rays through a thin boundary layer attached to the Earth; an inferior mirage situation

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References

  1. Nener B.D., Fowkes N.D., Borredon L. (2003). Analytical models of optical refraction in the troposphere. J. Opt. Soc. Am. A 20:867–875

    Article  ADS  Google Scholar 

  2. Bengt Edlen. (1966). The refractive index of air. Metrologia 2:71–80

    Article  ADS  Google Scholar 

  3. Fleagle R.G., Bussinger J.A. (1980). An Introduction to Atmospheric Physics (2nd ed). Academic, New York, 432 pp

    Google Scholar 

  4. Lehn W.H. (1985). A simple parabolic model for the optics of the atmospheric surface layer. Appl. Math. Modell. 9: 447–453

    Article  Google Scholar 

  5. Nölke Fr. (1917). Zur Theorie der Luftspiegelungen (The theory of mirages). Physik. Zeitschr. 18:134–142

    Google Scholar 

  6. Kerr D.E. (Radiation Laboratory Series) (ed.), Propagation of Radio Waves. New York: McGraw-Hill (1951) 728 pp

  7. Schelleng J.C., Burrows C.R., Ferrell E.B. (1933). Ultra-short wave propagation. Proc. of IRE 21: 427–463

    Article  Google Scholar 

  8. Lehn W.H. (1983). Inversion of superior mirage data to compute temperature profiles. J. Opt. Soc. Am. 73:1622–1625

    ADS  Google Scholar 

  9. Lehn W.H., Sawatzky H.L. (1975). Image transmission under arctic mirgae conditions. Polarforschung 45:120–129

    Google Scholar 

  10. Lehn W.H., Morrish J.S. (1986). A three parameter inferior mirage model for optical sensing of surface layer temperature profiles. IEEE Trans. Geosci. Remote Sensing GRS-24:940–946

    Article  ADS  Google Scholar 

  11. Lehn W.H., El-Arini M.B. (1978). Computer-graphoics analysis of atmospheric refraction. Appl. Opt. 17: 3146–3151

    Article  ADS  Google Scholar 

  12. White R. (1975). New solutions of the refraction integral. J. Opt. Soc. Am. 65:676–678

    ADS  Google Scholar 

  13. Fraser A.B. (1979). Simple solution for obtaining a temperature profile from the inferior mirage. Appl. Opt. 18:1724–1731

    ADS  Google Scholar 

  14. Fraser A.B. (1977). Solutions of the refraction and extinction integrals for use in inversions and image formation. Appl. Opt. 16:160–165

    Article  ADS  Google Scholar 

  15. Mach W.H., Fraser A.B. (1977). Inversion of optical data to obtain a micrometeorological temperature profile. Appl. Opt. 18: 1715–1723

    ADS  Google Scholar 

  16. Sozou P.D. (1979). Inversion of mirage data: an optimization approach. J. Opt. Soc. Am. A 11:125–134

    ADS  Google Scholar 

  17. Roach C.M., Rees W.G., Glover C.H.F. (1991). Inversion of atmospheric refraction data. J Opt. Soc. Am. 8: 330–339

    ADS  Google Scholar 

  18. Rees W.G. (1990). Mirages with linear image diagrams. J. Opt. Soc. Am. A 7:1351–1354

    ADS  Google Scholar 

  19. Wiliam C. Kropla, Waldemar H. Lehn. (1992). Differential geometric approach to atmospheric refraction. J. Opt. Soc. Am. 9:601–608

    Google Scholar 

  20. Kay I., Keller J.B. (1954). Asymptotic evaluation of the field at a caustic. J. Appl. Phys. 25:876–883

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. Ali Hasan Nayfeh. (1973). Perturbation Methods. John Wiley and Sons, New York, 425 pp

    MATH  Google Scholar 

  22. Minnaert M.G.J. (1993). Light and Colour in the Outdoors. Springer-Verlag, New York, 417 pp

    Google Scholar 

  23. Humphreys W.J. (1946). Physics of the Air. Dover, New York, 676 pp

    Google Scholar 

  24. Sodha M.S., Aggarwal A.K., Kaw P.K. (1967). Image formation by an optically stratified medium: optics of mirage and looming. Brit. J. Appl. Phys. 18:503–511

    Article  ADS  Google Scholar 

  25. Fraser A.B., Mach W.H. (1976). Mirages. Sci. Am 234:102–111

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia

    Nev Fowkes & Brett Nener

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  1. Nev Fowkes
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  2. Brett Nener
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Correspondence to Nev Fowkes.

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Fowkes, N., Nener, B. Microwave propagation over the Earth: image inversion. J Eng Math 53, 253–269 (2005). https://doi.org/10.1007/s10665-005-9015-0

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  • DOI: https://doi.org/10.1007/s10665-005-9015-0

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