Abstract

A mirage is seen when atmospheric refraction distorts or displaces an image. We describe a mirage simulator that uses digital imaging equipment to generate mirage images from normal photographs. The simulation program relocates horizonal image lies into positions that they appear to occupy, according to rays traced from observer to object. Image-brightness adjustments are not required; we show that, while the atmosphere can change the size or shape of an object, it does not change its apparent brightness. The realistic quality of the computed images makes this simulator a useful tool in mirage analysis.

© 1992 Optical Society of America

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References

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  1. R. G. Fleagle, J. A. Businger, An Introduction to Atmospheric Physics, 2nd ed. (Academic, New York, 1980).
  2. W. H. Lehn, R. E. Wallace, “Continuous-tone mirage images computed from digitized source photographs,” in Digest of Topical Meeting on Meteorological Optics (Optical Society of America, Washington, D.C., 1986), of paper ThB4, pp. 36–38.
  3. M. Berger, T. Trout, N. Levit, “Ray tracing mirages,” IEEE Comput. Graphics Appl. 10, 36–41 (1990).
    [CrossRef]
  4. A. B. Fraser, W. H. Mach, “Mirages,” Sci. Am. 234, 102–111 (1976).
    [CrossRef]
  5. W. H. Lehn, M. B. El-Arini, “Computer-graphics analysis of atmospheric refraction,” Appl. Opt. 17, 3146–3151 (1978). The TC concept introduced in this reference plots zobj versus ray angle ϕ at the eye. For image calculations it is more convenient to interchange the axes and replace ϕ by zapp. The latter form previously appeared in 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, Wisconsin, 1971).
    [CrossRef] [PubMed]
  6. J. S. Morrish, “Inferior mirages and their corresponding temperature structures,” M.Sc. thesis (University of Manitoba, Winnipeg, Canada, 1985).
  7. 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]
  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. W. H. Lehn, “Atmospheric refraction and lake monsters,” Science, 205, 183–185 (1979).
    [CrossRef] [PubMed]
  10. W. H. Lehn, I. Schroeder, “The Norse merman as an optical phenomenon,” Nature (London) 289, 362–366 (1981).
    [CrossRef]
  11. J. B. Sweeney, A Pictorial History of Sea Monsters (Crown, New York, 1972).
  12. F. Bruemmer, Encounters with Arctic Animals (McGraw-Hill, Toronto, 1972), p. 73.
  13. L. M. Larson (translator), The King’s Mirror (Twayne, New York, 1917), pp. 135–136.
  14. W. J. Humphreys, Physics of the Air, 3rd ed. (Dover, New York, 1964).
  15. J. M. Pernter, F. Exner, Meteorologische Optik, 2nd ed. (Braumüller, Vienna, 1922).
  16. In the case of a flat earth, the ray invariant is n sin θ = k. Equation (A10) then becomes dα = dθ, but Eq. (A11) remains unchanged.
  17. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1986).
  18. An unnamed reviewer suggested an alternative approach to this result, based on the discussions of image brightness in Ref. 17: if the atmosphere is regarded as part of the (lossless) image-forming lens, and if the refractive indices of the medium are the same at object and image, the photometric brightness of the image will equal that of the object.

1990

M. Berger, T. Trout, N. Levit, “Ray tracing mirages,” IEEE Comput. Graphics Appl. 10, 36–41 (1990).
[CrossRef]

1985

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

1981

W. H. Lehn, I. Schroeder, “The Norse merman as an optical phenomenon,” Nature (London) 289, 362–366 (1981).
[CrossRef]

1979

W. H. Lehn, “Atmospheric refraction and lake monsters,” Science, 205, 183–185 (1979).
[CrossRef] [PubMed]

1978

1977

1976

A. B. Fraser, W. H. Mach, “Mirages,” Sci. Am. 234, 102–111 (1976).
[CrossRef]

Berger, M.

M. Berger, T. Trout, N. Levit, “Ray tracing mirages,” IEEE Comput. Graphics Appl. 10, 36–41 (1990).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1986).

Bruemmer, F.

F. Bruemmer, Encounters with Arctic Animals (McGraw-Hill, Toronto, 1972), p. 73.

Businger, J. A.

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

El-Arini, M. B.

Exner, F.

J. M. Pernter, F. Exner, Meteorologische Optik, 2nd ed. (Braumüller, Vienna, 1922).

Fleagle, R. G.

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

Fraser, A. B.

Humphreys, W. J.

W. J. Humphreys, Physics of the Air, 3rd ed. (Dover, New York, 1964).

Lehn, W. H.

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, I. Schroeder, “The Norse merman as an optical phenomenon,” Nature (London) 289, 362–366 (1981).
[CrossRef]

W. H. Lehn, “Atmospheric refraction and lake monsters,” Science, 205, 183–185 (1979).
[CrossRef] [PubMed]

W. H. Lehn, M. B. El-Arini, “Computer-graphics analysis of atmospheric refraction,” Appl. Opt. 17, 3146–3151 (1978). The TC concept introduced in this reference plots zobj versus ray angle ϕ at the eye. For image calculations it is more convenient to interchange the axes and replace ϕ by zapp. The latter form previously appeared in 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, Wisconsin, 1971).
[CrossRef] [PubMed]

W. H. Lehn, R. E. Wallace, “Continuous-tone mirage images computed from digitized source photographs,” in Digest of Topical Meeting on Meteorological Optics (Optical Society of America, Washington, D.C., 1986), of paper ThB4, pp. 36–38.

Levit, N.

M. Berger, T. Trout, N. Levit, “Ray tracing mirages,” IEEE Comput. Graphics Appl. 10, 36–41 (1990).
[CrossRef]

Mach, W. H.

A. B. Fraser, W. H. Mach, “Mirages,” Sci. Am. 234, 102–111 (1976).
[CrossRef]

Morrish, J. S.

J. S. Morrish, “Inferior mirages and their corresponding temperature structures,” M.Sc. thesis (University of Manitoba, Winnipeg, Canada, 1985).

Pernter, J. M.

J. M. Pernter, F. Exner, Meteorologische Optik, 2nd ed. (Braumüller, Vienna, 1922).

Schroeder, I.

W. H. Lehn, I. Schroeder, “The Norse merman as an optical phenomenon,” Nature (London) 289, 362–366 (1981).
[CrossRef]

Sweeney, J. B.

J. B. Sweeney, A Pictorial History of Sea Monsters (Crown, New York, 1972).

Trout, T.

M. Berger, T. Trout, N. Levit, “Ray tracing mirages,” IEEE Comput. Graphics Appl. 10, 36–41 (1990).
[CrossRef]

Wallace, R. E.

W. H. Lehn, R. E. Wallace, “Continuous-tone mirage images computed from digitized source photographs,” in Digest of Topical Meeting on Meteorological Optics (Optical Society of America, Washington, D.C., 1986), of paper ThB4, pp. 36–38.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1986).

Appl. Math. Model.

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.

IEEE Comput. Graphics Appl.

M. Berger, T. Trout, N. Levit, “Ray tracing mirages,” IEEE Comput. Graphics Appl. 10, 36–41 (1990).
[CrossRef]

Nature (London)

W. H. Lehn, I. Schroeder, “The Norse merman as an optical phenomenon,” Nature (London) 289, 362–366 (1981).
[CrossRef]

Sci. Am.

A. B. Fraser, W. H. Mach, “Mirages,” Sci. Am. 234, 102–111 (1976).
[CrossRef]

Science

W. H. Lehn, “Atmospheric refraction and lake monsters,” Science, 205, 183–185 (1979).
[CrossRef] [PubMed]

Other

J. S. Morrish, “Inferior mirages and their corresponding temperature structures,” M.Sc. thesis (University of Manitoba, Winnipeg, Canada, 1985).

J. B. Sweeney, A Pictorial History of Sea Monsters (Crown, New York, 1972).

F. Bruemmer, Encounters with Arctic Animals (McGraw-Hill, Toronto, 1972), p. 73.

L. M. Larson (translator), The King’s Mirror (Twayne, New York, 1917), pp. 135–136.

W. J. Humphreys, Physics of the Air, 3rd ed. (Dover, New York, 1964).

J. M. Pernter, F. Exner, Meteorologische Optik, 2nd ed. (Braumüller, Vienna, 1922).

In the case of a flat earth, the ray invariant is n sin θ = k. Equation (A10) then becomes dα = dθ, but Eq. (A11) remains unchanged.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1986).

An unnamed reviewer suggested an alternative approach to this result, based on the discussions of image brightness in Ref. 17: if the atmosphere is regarded as part of the (lossless) image-forming lens, and if the refractive indices of the medium are the same at object and image, the photometric brightness of the image will equal that of the object.

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

W. H. Lehn, R. E. Wallace, “Continuous-tone mirage images computed from digitized source photographs,” in Digest of Topical Meeting on Meteorological Optics (Optical Society of America, Washington, D.C., 1986), of paper ThB4, pp. 36–38.

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Figures (11)

Fig. 1
Fig. 1

Calculation of a transfer characteristic (TC). (a) Profile of a typical temperature inversion. (b) Corresponding ray paths entering an observer’s eye. (c) TC obtained from the ray paths at 20 km. The abscissa is the height at which a ray intersects the object at 20 km. The ordinate is the corresponding apparent height, calculated by projecting the ray tangent at the observer’s eye back onto Horizontal Distance (km) the object plane.

Fig. 2
Fig. 2

(a) Normal view of mountains on Somerset Island, Northwest Territories, Canada, on 31 May 1987. (b) Mirage of the same mountains photographed from Resolute Bay, Northwest Territories, at a distance of ~75 km on 7 June 1985.

Fig. 3
Fig. 3

TC deduced from the photographs of Fig. 2.

Fig. 4
Fig. 4

Mirage simulation obtained by applying the TC of Fig. 3 to the undistorted image of Fig. 2(a).

Fig. 5
Fig. 5

(a) Undistorted photograph of Naparotalik Spit, taken from Tuktoyaktuk, Northwest Territories, on 26 May 1979 (the distance is 13.2 km). The height of the domed hill at the left is 11.2 m. (b) Superior mirage of the same hills, observed on 16 May 1979. (c) Inferior mirage, 4 May 1983.

Fig. 6
Fig. 6

TC curves deduced from Fig. 5.

Fig. 7
Fig. 7

Mirage simulations obtained by applying the TC’s of Fig. 6 to Fig. 5(a): (a) superior mirage, (b) inferior mirage.

Fig. 8
Fig. 8

(a) Walruses in their natural environment, photographed by F. Bruemmer.12 (b) The walruses take on the aspect of sea monsters, much as described in the 13th Century book, King’s Mirror.13 (c) TC required to create the mirage.

Fig. 9
Fig. 9

Rays in a spherically symmetric atmosphere.

Fig. 10
Fig. 10

Straight-ray geometry for calculating image brightness.

Fig. 11
Fig. 11

Curved-ray geometry.

Equations (18)

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ϕ = ϕ A ϕ B d ϕ = r A r B f ( r , n , k ) d r ,
r A r B f ( r , n , k ) d r = r A r B + f ( r , n , k ) d r ,
f ( r , n , k ) = f ( r , n , k ) + f k d k ,
f k = n 2 r ( n 2 r 2 k 2 ) 3 / 2 .
r A r B f ( r , n , k ) d r = r A r B [ f ( r , n , k ) + f k d k ] d r + r A r B + [ f ( r , n , k ) + f k d k ] d r .
d θ = tan θ I r B ( n B 2 r B 2 k 2 ) 1 / 2 ,
I = r A r B n 2 r d r ( n 2 r 2 k 2 ) 3 / 2 .
d α = tan α I r A ( n A 2 r A 2 k 2 ) 1 / 2 .
tan α = sin α cos α = sin θ cos θ n A r A sin θ ( n B 2 r B 2 k 2 ) 1 / 2
( n A 2 r A 2 k 2 ) 1 / 2 = n A r A cos θ
d α = tan θ I r A ( n B 2 r B 2 k 2 ) 1 / 2 .
d α d θ = r B r A .
d α d θ = n B n A .
δ Ω δ A cos η = δω δ S cos ξ .
δ E = δ P δ S = B cos ξδ S δ Ω δ S .
δ E = B δ A f 2 cos 4 η .
δ Ω δ A cos η = δ ω δ S cos ξ .
δ E = B δ A f 2 cos 4 η .

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