Abstract

Differential geometric techniques are presented and used to model the optical properties of the atmosphere under conditions that produce superior mirages. Optical path length replaces the usual Euclidean metric as a distance-measuring function and is used to construct a surface on which the paths of light rays are geodesics. The geodesic equations are shown to be equivalent to the ray equation in the plane. A differential equation that relates the Gaussian curvature of the surface and the refractive index of the atmosphere is derived. This equation is solved for the cases in which the curvature vanishes or is constant. Illustrative examples based on observation demonstrate the use of geometric techniques in the analysis of mirage images.

© 1992 Optical Society of America

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References

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  1. J. M. Pernter, F. M. Exner, Meteorologische Optik, 2nd ed. (Braumüller, Vienna, 1922).
  2. S. Vince, “Observations upon an unusual horizontal refraction of the air; with remarks on the variations to which the lower parts of the atmosphere are sometimes subject,” Philos. Trans. R. Soc. London 89, 13–23 (1799).
    [Crossref]
  3. W. H. Lehn, I. I. Schroeder, “Polar mirages as aids to Norse navigation,” Polarforschung 49, 173–187 (1979).
  4. W. Kropla, “Obtaining temperature profiles from superior mirage data,” master’s thesis (University of Manitoba, Winnipeg, Manitoba, Canada, 1988).
  5. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).
  6. D. Lovelock, H. Rund, Tensors, Differential Forms, and Variational Principles (Wiley, New York, 1975).
  7. R. G. Fleagle, J. A. Businger, An Introduction to Atmospheric Physics, 2nd ed. (Academic, New York, 1980).
  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, M. B. El-Arini, “Computer-graphics analysis of atmospheric refraction,” Appl. Opt. 17, 3146–3151 (1978).
    [Crossref] [PubMed]
  10. W. Tape, “The topology of mirages,” Sci. Am. 252(6), 120–129 (1985).
    [Crossref]
  11. W. G. Rees, “Mirages with linear image diagrams,” J. Opt. Soc. Am. A 7, 1351–1354 (1990).
    [Crossref]
  12. W. H. Lehn, B. A. German, “Novaya Zemlya effect: analysis of an observation,” Appl. Opt. 20, 2043–2047 (1981).
    [Crossref] [PubMed]
  13. G. H. Liljequist, “Refraction phenomena in the polar atmosphere,” in Norwegian–British–Swedish Antarctic Expedition, 1949–52, Scientific Results (Oslo U. Press, Oslo, Norway, 1964), Vol. 2, Part 2.
  14. W. H. Lehn, “The Novaya Zemlya effect: an arctic mirage,”J. Opt. Soc. Am. 69, 776–781 (1979).
    [Crossref]
  15. The elevation had previously been reported as 18.7 m. A survey conducted in 1983 indicated that the true elevation of Whitefish Summit is 20.3 m above sea level.
  16. B. O’Neill, Elementary Differential Geometry (Academic, New York, 1966).

1990 (1)

1985 (2)

W. Tape, “The topology of mirages,” Sci. Am. 252(6), 120–129 (1985).
[Crossref]

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

1981 (1)

1979 (2)

W. H. Lehn, “The Novaya Zemlya effect: an arctic mirage,”J. Opt. Soc. Am. 69, 776–781 (1979).
[Crossref]

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

1978 (1)

1799 (1)

S. Vince, “Observations upon an unusual horizontal refraction of the air; with remarks on the variations to which the lower parts of the atmosphere are sometimes subject,” Philos. Trans. R. Soc. London 89, 13–23 (1799).
[Crossref]

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. M.

J. M. Pernter, F. M. 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).

German, B. A.

Kropla, W.

W. Kropla, “Obtaining temperature profiles from superior mirage data,” master’s thesis (University of Manitoba, Winnipeg, Manitoba, Canada, 1988).

Lehn, W. H.

Liljequist, G. H.

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

Lovelock, D.

D. Lovelock, H. Rund, Tensors, Differential Forms, and Variational Principles (Wiley, New York, 1975).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).

O’Neill, B.

B. O’Neill, Elementary Differential Geometry (Academic, New York, 1966).

Pernter, J. M.

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

Rees, W. G.

Rund, H.

D. Lovelock, H. Rund, Tensors, Differential Forms, and Variational Principles (Wiley, New York, 1975).

Schroeder, I. I.

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

Tape, W.

W. Tape, “The topology of mirages,” Sci. Am. 252(6), 120–129 (1985).
[Crossref]

Vince, S.

S. Vince, “Observations upon an unusual horizontal refraction of the air; with remarks on the variations to which the lower parts of the atmosphere are sometimes subject,” Philos. Trans. R. Soc. London 89, 13–23 (1799).
[Crossref]

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. (2)

J. Opt. Soc. Am. (1)

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

Philos. Trans. R. Soc. London (1)

S. Vince, “Observations upon an unusual horizontal refraction of the air; with remarks on the variations to which the lower parts of the atmosphere are sometimes subject,” Philos. Trans. R. Soc. London 89, 13–23 (1799).
[Crossref]

Polarforschung (1)

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

Sci. Am. (1)

W. Tape, “The topology of mirages,” Sci. Am. 252(6), 120–129 (1985).
[Crossref]

Other (8)

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

W. Kropla, “Obtaining temperature profiles from superior mirage data,” master’s thesis (University of Manitoba, Winnipeg, Manitoba, Canada, 1988).

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).

D. Lovelock, H. Rund, Tensors, Differential Forms, and Variational Principles (Wiley, New York, 1975).

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

The elevation had previously been reported as 18.7 m. A survey conducted in 1983 indicated that the true elevation of Whitefish Summit is 20.3 m above sea level.

B. O’Neill, Elementary Differential Geometry (Academic, New York, 1966).

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

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Figures (6)

Fig. 1
Fig. 1

Fitting of parabolic (constant K) temperature profile to two Novaya Zemlya observations; the circles mark the input temperatures, and the solid curves show the parabolic fit.

Fig. 2
Fig. 2

Calculated ray paths derived from profile 1, exhibiting strong focusing at the antipodal points of the spherical modeling surface.

Fig. 3
Fig. 3

Whitefish Summit photographed with a 560-mm lens from Tuktoyaktuk, Northwest Territories, Canada, over a range of 20 km. (a) Superior mirage: May 16, 1979, 03.12 mountain daylight time, camera elevation 2.5 m. (b) Normal view: camera elevation 5.8 m.

Fig. 4
Fig. 4

TC’s: heavy lines, observed; thin lines, calculated. Zones I–III are described in the text.

Fig. 5
Fig. 5

Temperature profile that generates the fitted TC of Fig. 4. Zones I–III are described in the text.

Fig. 6
Fig. 6

Calculated mirage image based on the calculated TC.

Tables (2)

Tables Icon

Table 1 Temperature (T) Profiles for Examples A and B

Tables Icon

Table 2 Parameter Values for Profiles 1 and 2

Equations (48)

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g i j = n 2 ( z ) δ i j , g i j = 1 n 2 δ i j ,
- Γ 1 2 1 = Γ 2 2 2 = Γ 1 1 2 = Γ 2 1 1 = 1 n d n d z .
d 2 x i d l 2 + Γ j i k d x j d l d x k d l = 0 ,
d 2 x d l 2 + 2 n d n d z d x d l d z d l = 0 , d 2 z d l 2 + 1 n d n d z [ ( d z d l ) 2 - ( d x d l ) 2 ] = 0.
d x d l = d x d s d s d l = 1 n d x d s , d z d l = d z d s d s d l = 1 n d z d s ,
d 2 x d l 2 = 1 n 2 d 2 x d s 2 + 1 n d d z ( 1 n d x d s d z d s ) , d 2 z d l 2 = 1 n 2 d 2 z d s 2 + 1 n d d z ( 1 n d z d s d z d s ) ,
d d s ( n d r d s ) = n ,
K = R 1212 g ,
d 2 n d z 2 - 1 n ( d n d z ) 2 = - n 3 K .
n = 1 + ρ = 1 + β p T ,
r T = β p ,
d ( r T ) d z = - β g ( ρ ) = - β g r ,
T ( z ) = 1 r ( z ) [ r 0 T 0 - β g 0 z r ( z ) d z ] .
T ( z ) = 1 a exp ( b z ) - 1 { r 0 T 0 - β g a b [ exp ( b z ) - 1 ] + β g z } .
a = n 0 = r 0 + 1 ,
b = 1 z e ln n e n 0 = 1 z e ln ( 1 + r e 1 + r e ) .
b r e - r 0 z e = n e - n 0 z e .
T ( z ) = T 0 + z ( - β g - a b T 0 r 0 ) + z 2 [ β g a b r 0 - a b 2 T 0 r 0 ( 1 2 - a r 0 ) ] ,
κ = - sin θ n d n d z ,
κ = - b sin θ - b ,
n = 2 a c K exp ( - a z ) exp ( - 2 a z ) + c 2 ,
n ( z ) = n 0 ( 1 + c 2 ) exp ( - a z ) exp ( - 2 a z ) + c 2 ,
a = n 0 K 1 + c 2 2 c .
r ( z ) = r 0 + n 0 2 K 1 - c 2 2 c z + n 0 3 K 1 - 6 c 2 + c 4 8 c 2 z 2 ,
r ( z ) = r 0 - n 0 2 γ K z - n 0 3 K 2 z 2 ,
a = n 0 K .
T ( z ) = T 0 + b 1 z + b 2 z 2 ,
b 1 = n 0 2 γ T 0 K r 0 - g β ,
b 2 = n 0 3 T 0 K 2 r 0
a = ( 2 b 2 r 0 n 0 T 0 ) 1 / 2 ,
γ = r 0 ( b 1 + g β ) a n 0 T 0 ,
K = a 2 n 0 2 .
T ( z ) = 271.12 - 0.0316 z + 0.0133 z 2 .
r ( z ) = 293.8 × 10 - 6 - ( 2.749 × 10 - 9 ) z - ( 14.40 × 10 - 9 ) z 2 .
T ( z ) = 240.96 - 0.01272 z + 0.001254 z 2 .
r ( z ) = 330.5 × 10 - 6 - ( 29.36 × 10 - 9 ) z - ( 1.719 × 10 - 9 ) z 2 .
T ( y ) = 273.825 + 0.075 y + 0.471 y 2 ,
g i j = [ 1 0 0 sin 2 θ ] .
Γ h l k = 1 2 ( g k l x h + g l h x k - g h k x l ) ,
Γ h j k = g l j Γ h l k .
D X j = d X j + Γ h j k X h d x k
D Y h = d Y h - Γ h l k Y l d x k .
D x ˙ j = d x ˙ j + Γ h j k x ˙ h d x k = 0
d 2 x j d t 2 + Γ h j k d x h d t d x k d t = 0.
X k p = X p x k + Γ h p k X h
Y h k = Y h x k - Γ h m k Y m .
R l j h k = Γ l j k x k - Γ l j k x h + Γ m j k Γ l m h - Γ m j h Γ l m k
k ( s ) = d t d s · n ,

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