Abstract

Mirages are caused by the curvilinear propagation of light rays as a result of variations in the atmospheric refractive index. The refractive index is most strongly determined by the air temperature, so mirages carry information about the atmospheric temperature profile. It is straightforward to calculate the appearance of a mirage (defined by its image diagram) given the refractive-index profile (the forward problem); it is much less so to solve the inverse problem of deducing the refractive-index profile from the appearance of the mirage. The methods that have been employed to tackle this inverse problem are reviewed briefly; then a semirigorous treatment of mirages having linear image diagrams is developed. This class of mirages, while rare meteorologically, is interesting mathematically because it is possible to classify all the possible forms of refractive-index profile that can give rise to such mirages, and it includes the important phenomenon of the ducting of light rays. The treatment presented yields the refractive-index profile from the image diagram, except for certain well-defined cases for which no solution is possible. It also shows, by counterexample, that there can be no uniqueness theorem stating that a given image diagram can arise from only one refractive-index profile.

© 1990 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. B. Biot, “Recherches sur les réfractions extraordinaires qui s’observent très-près de l’horizon,” in Mémoires de la Classe des Sciences Mathématiques et Physiques de l’Institut de France (Baudoin, Paris, 1809), Vol. 10.
  2. F. J. W. Whipple, “Meteorological optics,” in Dictionary of Applied Physics, R. Glazebrook, ed. (Macmillan, London, 1923), Vol. 3, p. 519.
  3. N. K. Johnson, O. F. T. Roberts, “The measurement of the lapse rate of temperature by an optical method,” Q. J. R. Meteorol. Soc. 51, 131–138 (1925).
    [CrossRef]
  4. R. G. Fleagle, “The optical measurement of lapse rate,” Bull. Am. Meteorol. Soc. 31, 51–55 (1950).
  5. J. K. Sparkman, “A remote sensing technique using terrestrial refraction, for the study of low-level lapse rate,” Ph.D. dissertation (University of Wisconsin, Madison, Wise, 1971).
  6. G. H. Liljequist, “Refraction phenomena in the polar atmosphere,” in Norwegian–British–Swedish Antarctic Expedition 1949–1952, Scientific Results (Oslo U. Press, Oslo, 1963), Vol. II, part 2B.
  7. W. H. Lehn, H. L. Sawatzky, “Image transmission under Arctic mirage conditions,” Polarforschung 45, 120–129 (1975).
  8. R. White, “New solutions of the refraction integral,” J. Opt. Soc. Am. 65, 676–678 (1975).
    [CrossRef]
  9. 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]
  10. W. H. Mach, A. B. Fraser, “Inversion of optical data to obtain a micrometeorological temperature profile,” Appl. Opt. 18, 1715–1723 (1979).
    [CrossRef] [PubMed]
  11. A. B. Fraser, “Simple solution for obtaining a temperature profile from the inferior mirage,” Appl. Opt. 18, 1724–1731 (1979).
    [CrossRef] [PubMed]
  12. W. H. Lehn, J. S. Morrish, “A three-parameter inferior mirage model for optical sensing of surface layer temperature profiles,” IEEE Trans. Geosci. Remote Sensing GRS-24, 940–946 (1986).
    [CrossRef]
  13. W. H. Lehn, “Inversion of superior mirage data to compute temperature profiles,” J. Opt. Soc. Am. 73, 1622–1625 (1983).
    [CrossRef]
  14. W. S. B. Paterson, “Atmospheric refraction above the inland ice in North Greenland,” Bull. Geod. 38, 42–54 (1955).
    [CrossRef]
  15. S. W. Visser, “The Novay–Zemlya phenomenon,” Proc. K. Ned. Akad. Wet. B59, 375–385 (1956).
  16. W. H. Lehn, I. I. Schroeder, “Polar mirages as aids to Norse navigation,” Polarforschung 49, 173–187 (1979).
  17. W. G. Rees, “Reconstruction of an atmospheric temperature profile from a 166-year old mirage,” Polar Rec. 25, 325–327 (1988).
    [CrossRef]
  18. W. H. Lehn, Department of Electrical Engineering, University of Manitoba, Winnipeg, Manitoba, Canada (personal communication).
  19. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

1988 (1)

W. G. Rees, “Reconstruction of an atmospheric temperature profile from a 166-year old mirage,” Polar Rec. 25, 325–327 (1988).
[CrossRef]

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 Sensing GRS-24, 940–946 (1986).
[CrossRef]

1983 (1)

1979 (3)

1977 (1)

1975 (2)

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

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

1956 (1)

S. W. Visser, “The Novay–Zemlya phenomenon,” Proc. K. Ned. Akad. Wet. B59, 375–385 (1956).

1955 (1)

W. S. B. Paterson, “Atmospheric refraction above the inland ice in North Greenland,” Bull. Geod. 38, 42–54 (1955).
[CrossRef]

1950 (1)

R. G. Fleagle, “The optical measurement of lapse rate,” Bull. Am. Meteorol. Soc. 31, 51–55 (1950).

1925 (1)

N. K. Johnson, O. F. T. Roberts, “The measurement of the lapse rate of temperature by an optical method,” Q. J. R. Meteorol. Soc. 51, 131–138 (1925).
[CrossRef]

Biot, J. B.

J. B. Biot, “Recherches sur les réfractions extraordinaires qui s’observent très-près de l’horizon,” in Mémoires de la Classe des Sciences Mathématiques et Physiques de l’Institut de France (Baudoin, Paris, 1809), Vol. 10.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

Fleagle, R. G.

R. G. Fleagle, “The optical measurement of lapse rate,” Bull. Am. Meteorol. Soc. 31, 51–55 (1950).

Fraser, A. B.

Johnson, N. K.

N. K. Johnson, O. F. T. Roberts, “The measurement of the lapse rate of temperature by an optical method,” Q. J. R. Meteorol. Soc. 51, 131–138 (1925).
[CrossRef]

Lehn, W. H.

W. H. Lehn, J. S. Morrish, “A three-parameter inferior mirage model for optical sensing of surface layer temperature profiles,” IEEE Trans. Geosci. Remote Sensing GRS-24, 940–946 (1986).
[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, I. I. Schroeder, “Polar mirages as aids to Norse navigation,” Polarforschung 49, 173–187 (1979).

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

W. H. Lehn, Department of Electrical Engineering, University of Manitoba, Winnipeg, Manitoba, Canada (personal communication).

Liljequist, G. H.

G. H. Liljequist, “Refraction phenomena in the polar atmosphere,” in Norwegian–British–Swedish Antarctic Expedition 1949–1952, Scientific Results (Oslo U. Press, Oslo, 1963), Vol. II, part 2B.

Mach, W. H.

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 Sensing GRS-24, 940–946 (1986).
[CrossRef]

Paterson, W. S. B.

W. S. B. Paterson, “Atmospheric refraction above the inland ice in North Greenland,” Bull. Geod. 38, 42–54 (1955).
[CrossRef]

Rees, W. G.

W. G. Rees, “Reconstruction of an atmospheric temperature profile from a 166-year old mirage,” Polar Rec. 25, 325–327 (1988).
[CrossRef]

Roberts, O. F. T.

N. K. Johnson, O. F. T. Roberts, “The measurement of the lapse rate of temperature by an optical method,” Q. J. R. Meteorol. Soc. 51, 131–138 (1925).
[CrossRef]

Sawatzky, H. L.

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

Schroeder, I. I.

W. H. Lehn, I. I. Schroeder, “Polar mirages as aids to Norse navigation,” Polarforschung 49, 173–187 (1979).

Sparkman, J. K.

J. K. Sparkman, “A remote sensing technique using terrestrial refraction, for the study of low-level lapse rate,” Ph.D. dissertation (University of Wisconsin, Madison, Wise, 1971).

Visser, S. W.

S. W. Visser, “The Novay–Zemlya phenomenon,” Proc. K. Ned. Akad. Wet. B59, 375–385 (1956).

Whipple, F. J. W.

F. J. W. Whipple, “Meteorological optics,” in Dictionary of Applied Physics, R. Glazebrook, ed. (Macmillan, London, 1923), Vol. 3, p. 519.

White, R.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

Appl. Opt. (3)

Bull. Am. Meteorol. Soc. (1)

R. G. Fleagle, “The optical measurement of lapse rate,” Bull. Am. Meteorol. Soc. 31, 51–55 (1950).

Bull. Geod. (1)

W. S. B. Paterson, “Atmospheric refraction above the inland ice in North Greenland,” Bull. Geod. 38, 42–54 (1955).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing (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 Sensing GRS-24, 940–946 (1986).
[CrossRef]

J. Opt. Soc. Am. (2)

Polar Rec. (1)

W. G. Rees, “Reconstruction of an atmospheric temperature profile from a 166-year old mirage,” Polar Rec. 25, 325–327 (1988).
[CrossRef]

Polarforschung (2)

W. H. Lehn, I. I. Schroeder, “Polar mirages as aids to Norse navigation,” Polarforschung 49, 173–187 (1979).

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

Proc. K. Ned. Akad. Wet. (1)

S. W. Visser, “The Novay–Zemlya phenomenon,” Proc. K. Ned. Akad. Wet. B59, 375–385 (1956).

Q. J. R. Meteorol. Soc. (1)

N. K. Johnson, O. F. T. Roberts, “The measurement of the lapse rate of temperature by an optical method,” Q. J. R. Meteorol. Soc. 51, 131–138 (1925).
[CrossRef]

Other (6)

W. H. Lehn, Department of Electrical Engineering, University of Manitoba, Winnipeg, Manitoba, Canada (personal communication).

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

J. K. Sparkman, “A remote sensing technique using terrestrial refraction, for the study of low-level lapse rate,” Ph.D. dissertation (University of Wisconsin, Madison, Wise, 1971).

G. H. Liljequist, “Refraction phenomena in the polar atmosphere,” in Norwegian–British–Swedish Antarctic Expedition 1949–1952, Scientific Results (Oslo U. Press, Oslo, 1963), Vol. II, part 2B.

J. B. Biot, “Recherches sur les réfractions extraordinaires qui s’observent très-près de l’horizon,” in Mémoires de la Classe des Sciences Mathématiques et Physiques de l’Institut de France (Baudoin, Paris, 1809), Vol. 10.

F. J. W. Whipple, “Meteorological optics,” in Dictionary of Applied Physics, R. Glazebrook, ed. (Macmillan, London, 1923), Vol. 3, p. 519.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1
Fig. 1

Graphical representation of Eqs. (8a) (dotted curve) and (8b) (solid curve). The parameter x is the dimensionless quantity KD [Eq. (8a)] or kD [Eq. (8b)]. Inspection of the figures shows that (i) M−1 > 1 implies hyperbolic rays and a single solution for x; (ii) 1 > M−1 > −0.217 implies sinusoidal rays, the number of solutions for x increasing as M−1 → 0 from >0 or <0; (iii) there are no solutions if M−1 < −0.217.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

z ( x , θ ) = θ a ( x ) + b ( x ) .
2 z / x 2 = θ d 2 a / d x 2 + d 2 b / d x 2 = f ( θ a + b ) .
d 2 a / d x 2 = a f ( θ a + b ) ,
f ( z ) = α z + β ,
d 2 a / d x 2 = α a
d 2 b / d x 2 = α b + β .
z ( x , θ ) = ( θ / K ) sinh ( K x ) + ( β / α ) [ cosh ( K x ) 1 ]
z ( x , θ ) = ( θ / k ) sin ( k x ) + ( β / α ) [ cos ( k x ) 1 ]
z ( x , θ ) = θ x + β x 2 / 2 .
M = D / [ z ( D , θ ) / θ ] .
M 1 = [ sinh ( K D ) ] / K D ,
M 1 = [ sin ( k D ) ] / k D ,
d n / d z + 1 / R e = α z + β ,
n = n 0 + ( β 1 / R e ) z + α z 2 / 2 .
1 R = 1 n n · ν ˆ ,
d 2 z d x 2 1 R + 1 R e 1 n d n d z + 1 R e ,

Metrics