Abstract

An extremely accurate but simple asymptotic description (with known error) is obtained for the path of a ray propagating over a curved Earth with radial variations in refractive index. The result is sufficiently simple that analytic solutions for the path can be obtained for linear and quadratic index profiles. As well as rendering the inverse problem trivial for these profiles, this formulation shows that images are uniformly magnified in the vertical direction when viewed through a quadratic refractive-index profile. Nonuniform vertical distortions occur for higher-order refractive-index profiles.

© 2003 Optical Society of America

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  1. A. E. Barrios, “A terrain parabolic equation model for propagation in the troposphere,” IEEE Trans. Antennas Propag. 42, 90–98 (1994).
    [CrossRef]
  2. Y. A. Kravtsov, Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, Germany, 1990).
  3. B. Garfinkel, “Astronomical refraction in a polytrophic atmosphere,” Astron. J. 72, 235–256 (1967).
    [CrossRef]
  4. A. I. Mahan, “Astronomical refraction—some history and theories,” Appl. Opt. 1, 497–511 (1962).
    [CrossRef]
  5. B. Edlen, “The refractive index of air,” Metrologia 2, 71–80 (1966).
    [CrossRef]
  6. R. G. Fleagle, J. A. Bussinger, An Introduction to Atmospheric Physics, 2nd ed. (Academic, New York, 1980).
  7. M. Born, E. Wolf, Principles of Optics, 6th (corrected) ed. (Cambridge U. Press, Cambridge, UK, 1983).
  8. W. H. Lehn, “A simple parabolic model for the optics of the atmospheric surface layer,” Appl. Math. Model. 9, 447–453 (1985).
    [CrossRef]
  9. F. Nölke, “Zur Theorie der Luftspiegelungen (the theory of mirages),” Phys. Z. 18, 134–144 (1917).
  10. P. G. Tait, “On mirage (with remarks by Prof. J. D. Everett) 1881–1882, part 1,” Edinburgh Roy. Soc. 11, 354–361 (1882).
  11. P. G. Tait, “On mirage (with remarks by Prof. J. D. Everett) 1881–82, part 2,” Edinburgh Roy. Soc. Trans. 30, 551–578 (1883).
    [CrossRef]
  12. G. H. Liljequist, “Refraction phenomena in the polar atmosphere,” in Scientific Results, Vol. 2, Part 2, Norwegian–British Antarctic Expedition 1949–52 (Oslo U. Press, Oslo, Norway, 1964).
  13. D. E. Kerr, Propagation of Short Radio Waves, Radiation Laboratory Series (McGraw-Hill, New York, 1951).
  14. J. C. Schelleng, C. R. Burrows, E. B. Ferrell, “Ultra-short wave propagation,” Proc. IRE 21, 427–463 (1933).
    [CrossRef]
  15. C. L. Pekeris, “Accuracy of the earth-flattening approximation in the theory of microwave propagation,” Phys. Rev. 70, 518–522 (1946).
    [CrossRef]
  16. W. H. Lehn, “Inversion of superior mirage data to compute temperature profiles,” J. Opt. Soc. Am. 73, 1622–1625 (1983).
    [CrossRef]
  17. W. H. Lehn, H. L. Sawatzky, “Image transmission uder Arctic mirage conditions,” Polarforschung 45, 120–129 (1975).
  18. W. H. Lehn, J. S. Morrish, “A three-parameter inferior mirage model for optical sensing of surface layer temperature profiles,” IEEE Trans. Geosci. Remote Sens. GRS-24, 940–946 (1986).
    [CrossRef]
  19. W. H. Lehn, M. B. El-Arini, “Computer-graphics analysis of atmospheric refraction,” Appl. Opt. 17, 3146–3151 (1978).
    [CrossRef] [PubMed]
  20. R. White, “New solutions of the refraction integral,” J. Opt. Soc. Am. 65, 676–678 (1975).
    [CrossRef]
  21. A. B. Fraser, “Simple solution for obtaining a temperature profile from the inferior mirage,” Appl. Opt. 18, 1724–1731 (1979).
    [CrossRef] [PubMed]
  22. A. B. Fraser, “Solutions of the refraction and extinction integrals for use in inversions and image formation,” Appl. Opt. 16, 160–165 (1977).
    [CrossRef] [PubMed]
  23. W. H. Mach, A. B. Fraser, “Inversion of optical data to obtain a micrometeorological temperature profile,” Appl. Opt. 18, 1715–1723 (1979).
    [CrossRef] [PubMed]
  24. P. D. Sozou, “Inversion of mirage data: an optimization approach,” J. Opt. Soc. Am. A 11, 125–134 (1994).
    [CrossRef]
  25. W. G. Rees, C. M. Roach, C. H. F. Glover, “Inversion of atmospheric refraction data,” J. Opt. Soc. Am. A 8, 330–339 (1991).
    [CrossRef]
  26. W. G. Rees, “Mirages with linear image diagrams,” J. Opt. Soc. Am. A 7, 1351–1354 (1990).
    [CrossRef]
  27. W. C. Kropla, W. H. Lehn, “Differential geometric approach to atmospheric refraction,” J. Opt. Soc. Am. A 9, 601–608 (1992).
    [CrossRef]
  28. R. Snieder, M. Sambridge, “Ray perturbation theory for travel times and raypaths paths in 3-D heterogeneous media,” Geophys. J. Int. 109, 294–322 (1992).
    [CrossRef]

1994 (2)

A. E. Barrios, “A terrain parabolic equation model for propagation in the troposphere,” IEEE Trans. Antennas Propag. 42, 90–98 (1994).
[CrossRef]

P. D. Sozou, “Inversion of mirage data: an optimization approach,” J. Opt. Soc. Am. A 11, 125–134 (1994).
[CrossRef]

1992 (2)

R. Snieder, M. Sambridge, “Ray perturbation theory for travel times and raypaths paths in 3-D heterogeneous media,” Geophys. J. Int. 109, 294–322 (1992).
[CrossRef]

W. C. Kropla, W. H. Lehn, “Differential geometric approach to atmospheric refraction,” J. Opt. Soc. Am. A 9, 601–608 (1992).
[CrossRef]

1991 (1)

1990 (1)

1986 (1)

W. H. Lehn, J. S. Morrish, “A three-parameter inferior mirage model for optical sensing of surface layer temperature profiles,” IEEE Trans. Geosci. Remote Sens. GRS-24, 940–946 (1986).
[CrossRef]

1985 (1)

W. H. Lehn, “A simple parabolic model for the optics of the atmospheric surface layer,” Appl. Math. Model. 9, 447–453 (1985).
[CrossRef]

1983 (1)

1979 (2)

1978 (1)

1977 (1)

1975 (2)

R. White, “New solutions of the refraction integral,” J. Opt. Soc. Am. 65, 676–678 (1975).
[CrossRef]

W. H. Lehn, H. L. Sawatzky, “Image transmission uder Arctic mirage conditions,” Polarforschung 45, 120–129 (1975).

1967 (1)

B. Garfinkel, “Astronomical refraction in a polytrophic atmosphere,” Astron. J. 72, 235–256 (1967).
[CrossRef]

1966 (1)

B. Edlen, “The refractive index of air,” Metrologia 2, 71–80 (1966).
[CrossRef]

1962 (1)

1946 (1)

C. L. Pekeris, “Accuracy of the earth-flattening approximation in the theory of microwave propagation,” Phys. Rev. 70, 518–522 (1946).
[CrossRef]

1933 (1)

J. C. Schelleng, C. R. Burrows, E. B. Ferrell, “Ultra-short wave propagation,” Proc. IRE 21, 427–463 (1933).
[CrossRef]

1917 (1)

F. Nölke, “Zur Theorie der Luftspiegelungen (the theory of mirages),” Phys. Z. 18, 134–144 (1917).

1883 (1)

P. G. Tait, “On mirage (with remarks by Prof. J. D. Everett) 1881–82, part 2,” Edinburgh Roy. Soc. Trans. 30, 551–578 (1883).
[CrossRef]

1882 (1)

P. G. Tait, “On mirage (with remarks by Prof. J. D. Everett) 1881–1882, part 1,” Edinburgh Roy. Soc. 11, 354–361 (1882).

Barrios, A. E.

A. E. Barrios, “A terrain parabolic equation model for propagation in the troposphere,” IEEE Trans. Antennas Propag. 42, 90–98 (1994).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th (corrected) ed. (Cambridge U. Press, Cambridge, UK, 1983).

Burrows, C. R.

J. C. Schelleng, C. R. Burrows, E. B. Ferrell, “Ultra-short wave propagation,” Proc. IRE 21, 427–463 (1933).
[CrossRef]

Bussinger, J. A.

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

Edlen, B.

B. Edlen, “The refractive index of air,” Metrologia 2, 71–80 (1966).
[CrossRef]

El-Arini, M. B.

Ferrell, E. B.

J. C. Schelleng, C. R. Burrows, E. B. Ferrell, “Ultra-short wave propagation,” Proc. IRE 21, 427–463 (1933).
[CrossRef]

Fleagle, R. G.

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

Fraser, A. B.

Garfinkel, B.

B. Garfinkel, “Astronomical refraction in a polytrophic atmosphere,” Astron. J. 72, 235–256 (1967).
[CrossRef]

Glover, C. H. F.

Kerr, D. E.

D. E. Kerr, Propagation of Short Radio Waves, Radiation Laboratory Series (McGraw-Hill, New York, 1951).

Kravtsov, Y. A.

Y. A. Kravtsov, Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, Germany, 1990).

Kropla, W. C.

Lehn, W. H.

W. C. Kropla, W. H. Lehn, “Differential geometric approach to atmospheric refraction,” J. Opt. Soc. Am. A 9, 601–608 (1992).
[CrossRef]

W. H. Lehn, J. S. Morrish, “A three-parameter inferior mirage model for optical sensing of surface layer temperature profiles,” IEEE Trans. Geosci. Remote Sens. GRS-24, 940–946 (1986).
[CrossRef]

W. H. Lehn, “A simple parabolic model for the optics of the atmospheric surface layer,” Appl. Math. Model. 9, 447–453 (1985).
[CrossRef]

W. H. Lehn, “Inversion of superior mirage data to compute temperature profiles,” J. Opt. Soc. Am. 73, 1622–1625 (1983).
[CrossRef]

W. H. Lehn, M. B. El-Arini, “Computer-graphics analysis of atmospheric refraction,” Appl. Opt. 17, 3146–3151 (1978).
[CrossRef] [PubMed]

W. H. Lehn, H. L. Sawatzky, “Image transmission uder Arctic mirage conditions,” Polarforschung 45, 120–129 (1975).

Liljequist, G. H.

G. H. Liljequist, “Refraction phenomena in the polar atmosphere,” in Scientific Results, Vol. 2, Part 2, Norwegian–British Antarctic Expedition 1949–52 (Oslo U. Press, Oslo, Norway, 1964).

Mach, W. H.

Mahan, A. I.

Morrish, J. S.

W. H. Lehn, J. S. Morrish, “A three-parameter inferior mirage model for optical sensing of surface layer temperature profiles,” IEEE Trans. Geosci. Remote Sens. GRS-24, 940–946 (1986).
[CrossRef]

Nölke, F.

F. Nölke, “Zur Theorie der Luftspiegelungen (the theory of mirages),” Phys. Z. 18, 134–144 (1917).

Orlov, Y. I.

Y. A. Kravtsov, Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, Germany, 1990).

Pekeris, C. L.

C. L. Pekeris, “Accuracy of the earth-flattening approximation in the theory of microwave propagation,” Phys. Rev. 70, 518–522 (1946).
[CrossRef]

Rees, W. G.

Roach, C. M.

Sambridge, M.

R. Snieder, M. Sambridge, “Ray perturbation theory for travel times and raypaths paths in 3-D heterogeneous media,” Geophys. J. Int. 109, 294–322 (1992).
[CrossRef]

Sawatzky, H. L.

W. H. Lehn, H. L. Sawatzky, “Image transmission uder Arctic mirage conditions,” Polarforschung 45, 120–129 (1975).

Schelleng, J. C.

J. C. Schelleng, C. R. Burrows, E. B. Ferrell, “Ultra-short wave propagation,” Proc. IRE 21, 427–463 (1933).
[CrossRef]

Snieder, R.

R. Snieder, M. Sambridge, “Ray perturbation theory for travel times and raypaths paths in 3-D heterogeneous media,” Geophys. J. Int. 109, 294–322 (1992).
[CrossRef]

Sozou, P. D.

Tait, P. G.

P. G. Tait, “On mirage (with remarks by Prof. J. D. Everett) 1881–82, part 2,” Edinburgh Roy. Soc. Trans. 30, 551–578 (1883).
[CrossRef]

P. G. Tait, “On mirage (with remarks by Prof. J. D. Everett) 1881–1882, part 1,” Edinburgh Roy. Soc. 11, 354–361 (1882).

White, R.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th (corrected) ed. (Cambridge U. Press, Cambridge, UK, 1983).

Appl. Math. Model. (1)

W. H. Lehn, “A simple parabolic model for the optics of the atmospheric surface layer,” Appl. Math. Model. 9, 447–453 (1985).
[CrossRef]

Appl. Opt. (5)

Astron. J. (1)

B. Garfinkel, “Astronomical refraction in a polytrophic atmosphere,” Astron. J. 72, 235–256 (1967).
[CrossRef]

Edinburgh Roy. Soc. (1)

P. G. Tait, “On mirage (with remarks by Prof. J. D. Everett) 1881–1882, part 1,” Edinburgh Roy. Soc. 11, 354–361 (1882).

Edinburgh Roy. Soc. Trans. (1)

P. G. Tait, “On mirage (with remarks by Prof. J. D. Everett) 1881–82, part 2,” Edinburgh Roy. Soc. Trans. 30, 551–578 (1883).
[CrossRef]

Geophys. J. Int. (1)

R. Snieder, M. Sambridge, “Ray perturbation theory for travel times and raypaths paths in 3-D heterogeneous media,” Geophys. J. Int. 109, 294–322 (1992).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

A. E. Barrios, “A terrain parabolic equation model for propagation in the troposphere,” IEEE Trans. Antennas Propag. 42, 90–98 (1994).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

W. H. Lehn, J. S. Morrish, “A three-parameter inferior mirage model for optical sensing of surface layer temperature profiles,” IEEE Trans. Geosci. Remote Sens. GRS-24, 940–946 (1986).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Metrologia (1)

B. Edlen, “The refractive index of air,” Metrologia 2, 71–80 (1966).
[CrossRef]

Phys. Rev. (1)

C. L. Pekeris, “Accuracy of the earth-flattening approximation in the theory of microwave propagation,” Phys. Rev. 70, 518–522 (1946).
[CrossRef]

Phys. Z. (1)

F. Nölke, “Zur Theorie der Luftspiegelungen (the theory of mirages),” Phys. Z. 18, 134–144 (1917).

Polarforschung (1)

W. H. Lehn, H. L. Sawatzky, “Image transmission uder Arctic mirage conditions,” Polarforschung 45, 120–129 (1975).

Proc. IRE (1)

J. C. Schelleng, C. R. Burrows, E. B. Ferrell, “Ultra-short wave propagation,” Proc. IRE 21, 427–463 (1933).
[CrossRef]

Other (5)

Y. A. Kravtsov, Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, Germany, 1990).

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

M. Born, E. Wolf, Principles of Optics, 6th (corrected) ed. (Cambridge U. Press, Cambridge, UK, 1983).

G. H. Liljequist, “Refraction phenomena in the polar atmosphere,” in Scientific Results, Vol. 2, Part 2, Norwegian–British Antarctic Expedition 1949–52 (Oslo U. Press, Oslo, Norway, 1964).

D. E. Kerr, Propagation of Short Radio Waves, Radiation Laboratory Series (McGraw-Hill, New York, 1951).

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Figures (9)

Fig. 1
Fig. 1

Ray propagation around the Earth. This diagram indicates the geometric meaning of the parameters used in the text.

Fig. 2
Fig. 2

Figure showing the scaled coordinate system of the target and source positions and the launching angle.

Fig. 3
Fig. 3

Ray bundles z ( x ,   γ 0 ) with γ 0 = - 1     1 in linearly increasing and decreasing refractive-index media ( κ = 1 ,   η = 1 ) .

Fig. 4
Fig. 4

Ray behavior for various linear refractive-index profiles with a 1 ranging from -2 to 2. (a) Paths launched from the source with the same initial angle γ 0 = 1 . (b) Image at the target end for various launch angles γ 0 , again plotted for various a 1 s ( κ = 1 ,   η = 1 in each case).

Fig. 5
Fig. 5

Quadratic-index-profile rays: a 2 > 0 case. (a) Index profile ( a 1 = 1.0 ,   a 2 = 2.0 ) . Note that the axis of symmetry is not at z = 1.0 but at z = 0.75 owing to the nonzero a 1 term. (b) Paths launched from the position z = 1.0 with a range of initial angles γ 0 = - 2     2 ( κ = 1 ,   η = 1 ) . The critical launch angle is γ 0 crit = - 1 in this case, with z z s = 0.5 as x for this path. Paths with - 1 < γ 0 < 0 have turning points (e.g., γ 0 = - 0.5 ) . All other paths are monotonic.

Fig. 6
Fig. 6

Quadratic-index-profile rays: a 2 < 0 case. Index profile ( a 1 = 0.0 ,   a 2 = - 16.0 ) . (b) Paths launched from the height z = 1.0 with a range of initial angles γ 0 = - 1     1 ( κ = 1 ,   η = 1 ) . Note that the axis of symmetry of the refractive-index gradient is z = 1.0

Fig. 7
Fig. 7

Observation system used in the discussion of inversion. The example is of a lighthouse observed through a telescope at a position such that the image occurs in the focal plane.

Fig. 8
Fig. 8

Ideal or straight-line path that would result from a uniform atmosphere and the actual ray path for a radially varying refractive index.

Fig. 9
Fig. 9

This figure depicts equally spaced markings on the lighthouse to measure the vertical distorting of the image.

Tables (1)

Tables Icon

Table 1 Image Characteristics for Constant, Linear, Quadratic, and Polynomial Refractive-Index Profiles a

Equations (42)

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n = 1 + δ n , δ = 10 - 6 ;
n = γ   P T , γ 78 ,
n ( z ) ( a e + z ) cos   γ = C ,
n ( z ) ( 1 + z / a e ) cos   γ = C / a e ,
N   cos   γ = C ,
tan   γ = d R ( Θ ) R d Θ = ± R [ n 2 - ( q 0 / R ) 2 ] 1 / 2 q 0 ,
R 0 R q 0 d R R 2 [ n 2 ( R ) - q 0 2 / R 2 ] 1 / 2 = ± θ ,
q 0 = n ( R 0 ) R 0 cos   γ 0 [ 1 + n ( R 0 ) ] R 0 cos   γ 0
d R ( R s ) d θ = 0 ,
n ( R s ) = q 0 / R s .
q 0 / R 1 = cos ( γ 0 ± θ ) ,
R = a e [ 1 + ( hl ) z ] , θ = lx , n = 1 + δ n ( z ) ,
l = L a e O ( 10 - 3 ) , h = H L O ( 10 - 3 ) , δ = 10 - 6
γ 0 = h γ 0
q 0 R 0 = 1 - h 2 γ 0 2 2 + δ n ( 1 ) + O ( h 2 δ ,   h 4 ) ,
x = sgn ( γ 0 ) 1 z 1 { 2 η [ n ( u ) - n ( 1 ) ] + 2 κ ( u - 1 ) + ( γ 0 ) 2 } 1 / 2 d u + O ( hl ,   δ ) ,
η = δ / h 2 , κ = l / h
2 η [ n ( z s ) - n ( 1 ) ] + 2 κ ( z s - 1 ) + ( γ 0 ) 2 = 0
n ( z ) = n ( z s ) + ( z - z s )   d   n ( z s ) d z +
x - x s = z + z z z 1 [ ξ ( u - z s ) ] 1 / 2 d u +   ,
ξ = 2 η   d n ( z s ) d z + 2 κ
z - z s = z s + ξ 2   ( x - x s ) 2 +   ;
z 0 - 1 = 1 2 ( κ x 2 + 2 γ 0 x ) ,
n ( z ) = n ( 1 ) + a 1 ( z - 1 ) ,
x = sgn ( γ 0 ) a 1 η + κ   { [ γ 0 2 + 2 ( η a 1 + κ ) ( z - 1 ) ] 1 / 2 - γ 0 } ,
z - 1 = 1 2 [ ( κ + η a 1 ) x 2 + 2 γ 0 x ] ,
z - z 0 = 1 2 η a 1 x 2 .
z ( 1 ,   γ 0 ) = 1 + 1 2 [ ( κ + η a 1 ) + 2 γ 0 ] ,
n ( z ) - n ( 1 ) = a 2 ( z - 1 ) 2 + a 1 ( z - 1 ) ,
x
= sgn ( γ 0 ) 1 z d u [ 2 η a 2 ( u - 1 ) 2 + 2 ( η a 1 + κ ) ( u - 1 ) + γ 0 2 ] 1 / 2
2 η a 2 sgn ( γ 0 ) x
= 1 z d u { [ u - ( 1 - ζ ) ] 2 + [ ξ / ( 2 η a 2 ) 2 ] } 1 / 2 ,
ζ = η a 1 + κ 2 η a 2 , ξ = 2 η a 2 γ 0 2 - ( η a 1 + κ ) 2 .
z - ( 1 - ζ ) = sin ( - 2 η a 2 x ) - 2 η a 2 γ 0 + ζ   cos ( - 2 η a 2 x ) .
z - ( 1 - ζ ) = sinh ( 2 η a 2 x ) 2 η a 2 γ 0 + ζ   cosh   ( 2 η a 2 x ) .
[ z - ( 1 - ζ ) ] 2 + ζ ( 2 η a 2 ) 2 = 0 ,
z ( 1 ) - 1 = - ζ + ζ   cos   ( - 2 η a 2   ) + sin ( - 2 η a 2 ) - 2 η a 2 γ 0 for a 2 < 0 - ζ + ζ   cosh   ( 2 η a 2   ) + sinh ( 2 η a 2 ) 2 η a 2 γ 0 for a 2 > 0 .
O = z ( 1 ,   0 ) - z 0 ( 1 ,   0 ) z ( 1 ,   0 ) - κ / 2 ,
z ( 1 ,   γ 0 ) - z 0 ( 1 ,   γ 0 ) = O ( n ,   η ,   κ ) + M ( n ,   γ 0 ,   η ,   κ ) γ 0 ,
F ( a 2 ) = sin ( - 2 η a 2 ) - 2 η a 2 ,
G ( a 1 ,   a 2 ) = 1 - η a 1 + κ 2 η a 2 + η a 1 + κ 2 η a 2 cos ( - 2 η a 2 ) .

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