Abstract

We present a general method of inverting numerical data describing the form of a mirage to yield the atmospheric refractive-index or temperature profile that caused the mirage. The method is applied to three analytically generated mirages and five experimentally observed mirages to demonstrate its effectiveness. It is shown to be reasonably robust and reliable in inverting all but the most complex mirage phenomena.

© 1991 Optical Society of America

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References

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  1. The Oxford English Dictionary, 1st ed. (Oxford U. Press, Oxford, 1933), Vol. 6, p. 487 (definition modified slightly).
  2. R. Greenler, Rainbows, Halos and Glories (Cambridge U. Press, Cambridge, 1980).
  3. W. G. Rees, “Polar mirages,” Polar Rec. 24, 193–198 (1988).
    [CrossRef]
  4. G. H. Liljequist, “Refraction phenomena in the polar atmosphere,” in Norwegian–British–Swedish Antarctic Expedition 1949–1952 Scientific Results II (Norsk Polarinstitutt, Oslo, 1956), Part 2B.
  5. F. K. Brunner, “Determination of line averages of sensible heat flux using an optical method,” Boundary-Layer Meteorol. 22, 193–207 (1982).
    [CrossRef]
  6. W. M. Porch, T. J. Green, “Remote lapse-rate sensing using atmospheric refraction over complex terrain,” Appl. Opt. 21, 981–982 (1982).
    [CrossRef] [PubMed]
  7. W. G. Rees, “Mirages with linear image diagrams,” J. Opt. Soc. Am. A 7, 1351–1354 (1990).
    [CrossRef]
  8. W. H. Lehn, “Inversion of superior mirage data to compute temperature profiles,”J. Opt. Soc. Am. 73, 1622–1625 (1983).
    [CrossRef]
  9. 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, pp. 1–266.
  10. F. J. W. Whipple, “Meteorological optics,” in Dictionary of Applied Physics, R. Glazebrook, ed. (Macmillan, London, 1921), Vol. 3, p. 158.
  11. W. H. Lehn, “A simple parabolic model for the optics of the atmospheric surface layer,” Appl. Math. Modelling 9, 447–453 (1985).
    [CrossRef]
  12. W. G. Rees, “Reconstruction of an atmospheric temperature profile from a 166-year old mirage,” Polar Rec. 24, 325–327 (1988).
    [CrossRef]
  13. W. H. Lehn, Department of Electrical Engineering, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada (personal communication).
  14. W. H. Mach, A. B. Fraser, “Inversion of optical data to obtain a micrometeorological temperature profile,” Appl. Opt. 18, 1715–1723 (1979).
    [CrossRef] [PubMed]
  15. W. H. Lehn, “Terrestrial images transmitted by the Novaya Zemlya effect,” in Meteorological Optics, 1986 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1986), p. 33.

1990 (1)

1988 (2)

W. G. Rees, “Polar mirages,” Polar Rec. 24, 193–198 (1988).
[CrossRef]

W. G. Rees, “Reconstruction of an atmospheric temperature profile from a 166-year old mirage,” Polar Rec. 24, 325–327 (1988).
[CrossRef]

1985 (1)

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

1983 (1)

1982 (2)

F. K. Brunner, “Determination of line averages of sensible heat flux using an optical method,” Boundary-Layer Meteorol. 22, 193–207 (1982).
[CrossRef]

W. M. Porch, T. J. Green, “Remote lapse-rate sensing using atmospheric refraction over complex terrain,” Appl. Opt. 21, 981–982 (1982).
[CrossRef] [PubMed]

1979 (1)

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, pp. 1–266.

Brunner, F. K.

F. K. Brunner, “Determination of line averages of sensible heat flux using an optical method,” Boundary-Layer Meteorol. 22, 193–207 (1982).
[CrossRef]

Fraser, A. B.

Green, T. J.

Greenler, R.

R. Greenler, Rainbows, Halos and Glories (Cambridge U. Press, Cambridge, 1980).

Lehn, W. H.

W. H. Lehn, “A simple parabolic model for the optics of the atmospheric surface layer,” Appl. Math. Modelling 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, “Terrestrial images transmitted by the Novaya Zemlya effect,” in Meteorological Optics, 1986 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1986), p. 33.

W. H. Lehn, Department of Electrical Engineering, University of Manitoba, Winnipeg, Manitoba R3T 2N2, 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 II (Norsk Polarinstitutt, Oslo, 1956), Part 2B.

Mach, W. H.

Porch, W. M.

Rees, W. G.

W. G. Rees, “Mirages with linear image diagrams,” J. Opt. Soc. Am. A 7, 1351–1354 (1990).
[CrossRef]

W. G. Rees, “Polar mirages,” Polar Rec. 24, 193–198 (1988).
[CrossRef]

W. G. Rees, “Reconstruction of an atmospheric temperature profile from a 166-year old mirage,” Polar Rec. 24, 325–327 (1988).
[CrossRef]

Whipple, F. J. W.

F. J. W. Whipple, “Meteorological optics,” in Dictionary of Applied Physics, R. Glazebrook, ed. (Macmillan, London, 1921), Vol. 3, p. 158.

Appl. Math. Modelling (1)

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

Appl. Opt. (2)

Boundary-Layer Meteorol. (1)

F. K. Brunner, “Determination of line averages of sensible heat flux using an optical method,” Boundary-Layer Meteorol. 22, 193–207 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Polar Rec. (2)

W. G. Rees, “Polar mirages,” Polar Rec. 24, 193–198 (1988).
[CrossRef]

W. G. Rees, “Reconstruction of an atmospheric temperature profile from a 166-year old mirage,” Polar Rec. 24, 325–327 (1988).
[CrossRef]

Other (7)

W. H. Lehn, Department of Electrical Engineering, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada (personal communication).

W. H. Lehn, “Terrestrial images transmitted by the Novaya Zemlya effect,” in Meteorological Optics, 1986 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1986), p. 33.

G. H. Liljequist, “Refraction phenomena in the polar atmosphere,” in Norwegian–British–Swedish Antarctic Expedition 1949–1952 Scientific Results II (Norsk Polarinstitutt, Oslo, 1956), Part 2B.

The Oxford English Dictionary, 1st ed. (Oxford U. Press, Oxford, 1933), Vol. 6, p. 487 (definition modified slightly).

R. Greenler, Rainbows, Halos and Glories (Cambridge U. Press, Cambridge, 1980).

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, pp. 1–266.

F. J. W. Whipple, “Meteorological optics,” in Dictionary of Applied Physics, R. Glazebrook, ed. (Macmillan, London, 1921), Vol. 3, p. 158.

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Figures (9)

Fig. 1
Fig. 1

Geometry of a mirage in a spherically stratified medium such as the Earth’s atmosphere. The circles represent the Earth’s surface and a surface of constant radius r passing through the ray (irregular line) at the point B. The ray propagates from A to B. The angle θ measures its progress round the Earth’s surface so that r(θ) characterizes the trajectory. β is the angle between the ray and the local horizontal.

Fig. 2
Fig. 2

Model refractive-index profile (circles) and the fit produced by our algorithm (squares) for the quadratic refractive-index profile defined in the text. The figure shows height in the atmosphere plotted against 104(n − 1), where n is the refractive index.

Fig. 3
Fig. 3

The trajectory of the rays for the calculated refractive-index profile of Fig. 2. The rays are labeled with the value of β0, the angle of arrival at the eye, measured in milliradians. They travel from right to left.

Fig. 4
Fig. 4

Model (circles) and fitted (squares) refractive-index profile for the unphysically steep exponential profile defined in the text. The data have been plotted [as values of 104(n − 1)] only within the range over which the inversion algorithm provides information, i.e., the range of heights explored by the rays.

Fig. 5
Fig. 5

Temperature profile (circles) deduced from the image diagram of mirage 1. The figure also shows in situ temperature measurements made by Mach and Fraser,14 who report errors of ±0.04 K. The inverted data have been fitted to the temperature measured at a height of 0.25 m and are plotted only over the range of validity defined by the range of heights explored by the rays.

Fig. 6
Fig. 6

Like Fig. 5 but for mirage 2. The inverted data and the measurements have been constrained to match at 0.45 m, and the graph plots the excess over the temperature at that height, again in degrees Celsius. The errors in the observed temperatures are again ±0.04 K.

Fig. 7
Fig. 7

Like Fig. 6 but for mirage 3. The data are constrained to match at a height of 0.47 m.

Fig. 8
Fig. 8

Like Fig. 5 but for mirage 4. The errors in the temperature measurements made by Mach and Fraser14 are unknown.

Fig. 9
Fig. 9

Inversion of a mirage observation 5 reported by Lehn.8 The temperature profile, plotted in degrees Celsius (squares) is compared with the profile deduced by Lehn for the same mirage. There are no temperature observations except for a value of −2° C at 2.5 m, to which the inverted data are constrained.

Tables (1)

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Table 1 Parameters of the Real Mirages to Which the Inversion Method Was Applieda

Equations (15)

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d d s ( n r cos β ) = 0 ,
tan β = d ln r d θ .
d β = d θ n ( r d n d r + n ) ,
r ( θ , β 0 ) ,
β d z d x
d x R d θ ,
d 2 z d x 2 1 n d n d z + 1 R .
z = x 2 2 ( d n d z + 1 R ) + β 0 x + z 0 ,
D 2 2 d n d z
n ( z ) = 1.0003 + 10 - 6 z ,
n ( z ) = 1.000319 - 10 - 7 ( z - 20 ) 2 ,
n ( z ) = 1.0002 + 10 - 4 exp ( - z / 5 ) ,
- d n d z = n - 1 T ( d T d z + α ) ,
d T d z = T / 5 1 + 2 exp ( z / 5 ) - α ,
T = - α [ 2 z - 5 exp ( - z / 5 ) ] + 3 T 0 - 5 α 2 + exp ( - z / 5 ) .

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