Abstract

Information derived from the superior mirage is used to compute the average vertical temperature profile in the atmosphere between the observer and a known object. The image is described by a plot of ray-elevation angle at the eye against elevation at which that ray intersects the object. The computational algorithm, based on the tracing of rays that have at most one vertex, iteratively adjusts the temperature profile until the observed image characteristics are reproduced. An example based on an observation made on the Beaufort Sea illustrates the process.

© 1983 Optical Society of America

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References

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  1. J. M. Pernter and F. Exner, Meteorologische Optik, 2nd ed. (Braumüller, Vienna, 1922).
  2. 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]
  3. W. H. Lehn and H. L. Sawatzky, “Image transmission under arctic mirage conditions,” Polarforschung 45, 120–128 (1975).
  4. W. H. Lehn and M. B. El-Arini, “Computer-graphics analysis of atmospheric refraction,” Appl. Opt. 17, 3146–3151 (1978).
    [Crossref] [PubMed]
  5. An extensive list of references to research before 1935 can be found in W. -E. Schiele, “Zur Theorie der Luftspiegelungen,” Veroeff. Geophys. Inst. Univ. Leipzig 7, 103–188 (1935).
  6. R. G. Fleagle, “The temperature distribution near a cold surface,” J. Meteorol. 13, 160–165 (1956).
    [Crossref]
  7. A. B. Fraser, “Simple solution for obtaining a temperature profile from the inferior mirage,” Appl. Opt. 18, 1724–1731 (1979).
    [Crossref] [PubMed]
  8. W. H. Mach, “Measurement of micrometeorological temperature profiles by the inversion of optical data,” Ph.D. Thesis (Pennsylvania State University, University Park, Pa., 1978).
  9. W. H. Mach and A. B. Fraser, “Inversion of optical data to obtain a micrometeorological temperature profile,” Appl. Opt. 18, 1715–1723 (1979).
    [Crossref] [PubMed]
  10. R. Meyer, “Die Entstehung optischer Bilder durch Brechung und Spiegelung in der Atmosphäre,” Meteorol. Z. 52, 405–408 (1935).
  11. In this context the concept of range is not absolute; it is linked to the nature of the image. Images of the requisite type may occur at object distances over 20 km, if the inversion has its steepest gradient 30 m above the observer’s eye, or at 1 km for inversions less than 1 m above the observer.
  12. The U.S. Standard Atmosphere is tabulated in Ref. 14. In the boundary layer, this atmosphere has a temperature gradient of −6.5 K/km.
  13. W. H. Lehn and B. A. German, “Novaya zemlya effect: analysis of an observation,” Appl. Opt. 20, 2043–2047 (1981).
    [Crossref] [PubMed]
  14. R. G. Fleagle and J. A. Businger, An Introduction to Atmospheric Physics, 2nd ed. (Academic, New York, 1980).

1981 (1)

1979 (2)

1978 (1)

1977 (1)

1975 (1)

W. H. Lehn and H. L. Sawatzky, “Image transmission under arctic mirage conditions,” Polarforschung 45, 120–128 (1975).

1956 (1)

R. G. Fleagle, “The temperature distribution near a cold surface,” J. Meteorol. 13, 160–165 (1956).
[Crossref]

1935 (2)

R. Meyer, “Die Entstehung optischer Bilder durch Brechung und Spiegelung in der Atmosphäre,” Meteorol. Z. 52, 405–408 (1935).

An extensive list of references to research before 1935 can be found in W. -E. Schiele, “Zur Theorie der Luftspiegelungen,” Veroeff. Geophys. Inst. Univ. Leipzig 7, 103–188 (1935).

Businger, J. A.

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

El-Arini, M. B.

Exner, F.

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

Fleagle, R. G.

R. G. Fleagle, “The temperature distribution near a cold surface,” J. Meteorol. 13, 160–165 (1956).
[Crossref]

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

Fraser, A. B.

German, B. A.

Lehn, W. H.

Mach, W. H.

W. H. Mach and A. B. Fraser, “Inversion of optical data to obtain a micrometeorological temperature profile,” Appl. Opt. 18, 1715–1723 (1979).
[Crossref] [PubMed]

W. H. Mach, “Measurement of micrometeorological temperature profiles by the inversion of optical data,” Ph.D. Thesis (Pennsylvania State University, University Park, Pa., 1978).

Meyer, R.

R. Meyer, “Die Entstehung optischer Bilder durch Brechung und Spiegelung in der Atmosphäre,” Meteorol. Z. 52, 405–408 (1935).

Pernter, J. M.

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

Sawatzky, H. L.

W. H. Lehn and H. L. Sawatzky, “Image transmission under arctic mirage conditions,” Polarforschung 45, 120–128 (1975).

Schiele, W. -E.

An extensive list of references to research before 1935 can be found in W. -E. Schiele, “Zur Theorie der Luftspiegelungen,” Veroeff. Geophys. Inst. Univ. Leipzig 7, 103–188 (1935).

Appl. Opt. (5)

J. Meteorol. (1)

R. G. Fleagle, “The temperature distribution near a cold surface,” J. Meteorol. 13, 160–165 (1956).
[Crossref]

Meteorol. Z. (1)

R. Meyer, “Die Entstehung optischer Bilder durch Brechung und Spiegelung in der Atmosphäre,” Meteorol. Z. 52, 405–408 (1935).

Polarforschung (1)

W. H. Lehn and H. L. Sawatzky, “Image transmission under arctic mirage conditions,” Polarforschung 45, 120–128 (1975).

Veroeff. Geophys. Inst. Univ. Leipzig (1)

An extensive list of references to research before 1935 can be found in W. -E. Schiele, “Zur Theorie der Luftspiegelungen,” Veroeff. Geophys. Inst. Univ. Leipzig 7, 103–188 (1935).

Other (5)

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

W. H. Mach, “Measurement of micrometeorological temperature profiles by the inversion of optical data,” Ph.D. Thesis (Pennsylvania State University, University Park, Pa., 1978).

In this context the concept of range is not absolute; it is linked to the nature of the image. Images of the requisite type may occur at object distances over 20 km, if the inversion has its steepest gradient 30 m above the observer’s eye, or at 1 km for inversions less than 1 m above the observer.

The U.S. Standard Atmosphere is tabulated in Ref. 14. In the boundary layer, this atmosphere has a temperature gradient of −6.5 K/km.

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

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Figures (7)

Fig. 1
Fig. 1

a, A common TC that transforms b, an object, into c, two erect and one inverted image.

Fig. 2
Fig. 2

The process of inversion.

Fig. 3
Fig. 3

Transfer characteristic zones (I, II, III) and a typical vertex locus.

Fig. 4
Fig. 4

The layers below the pivot. The spherical layer boundaries are represented as straight lines.

Fig. 5
Fig. 5

a, Normal view of Whitefish Summit, May 20, 1979, 19.36 h MDT; b, superior mirage of Whitefish Summit, May 16, 1979, 03.22 h MDT; and c, calculated image based on the iteratively estimated temperature profile. The angular scale (in arc minutes) is the same for all figures.

Fig. 6
Fig. 6

Transfer characteristics: a, observed; b, calculated in iteration three; c, calculated in iteration eight and used to create Fig. 5c.

Fig. 7
Fig. 7

Temperature profiles: a, initial guess; b, iteration 3; c, iteration 8, the final approximation. The portion below the pivot, corresponding to zone I, is calculated only once.

Equations (14)

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2 [ ρ ( z ) - ρ ( z e ) ] + 2 R E ( z - z e ) + ϕ 2 = 0 ,
T v - T ( z e ) = T ( z e ) [ - g β T m ( z v - z e ) + T ( z e ) ( z v - z e ) β p 0 R E + T ( z e ) ϕ 2 2 β p 0 ] .
1 r = ρ ( 1 + ρ ) T ( d T d z + g β ) ,
z = - x 2 2 R E ,
z = - x 2 2 R E + h .
z = - x 2 2 r + x tan ϕ + z e ,
r = ( 1 R E - tan 2 ϕ h 2 z e ) - 1
x h = ( 1 r - 1 R E ) - 1 tan ϕ h .
ϕ = x R E - x r + ϕ .
ρ = β p / T .
p ( z ) = p 0 exp [ - g β 0 z d z T ( z ) ] ,
p ( z ) = p 0 ( 1 - g β z T m ) .
p ( 50 ) = p 0 ( 1 - 0.00625 ) ,
ρ ( z ) = β p 0 T ( z ) ( 1 - g β z T m ) ,