Abstract

Analytic solutions to the refraction and extinction integrals are presented for the case of a horizontally or spherically stratified medium. These solutions are not only useful for the calculation of the images that would be seen through a lens with a continuously varying index of refraction, such as the atmosphere, they also provide a solution to the inverse problem of determining the refractive structure from measurements of the image. A remarkably simple inversion scheme is presented for determining the refractive (temperature) structure of the earth’s atmosphere by observations of the setting sun. The same scheme works for determination of vertical profiles of the extinction coefficient.

© 1977 Optical Society of America

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References

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  1. A. B. Fraser, W. H. Mach, Sci. Am. 234 (1), 102 (1976).
    [CrossRef]
  2. R. White, J. Opt. Soc. Am. 65, 676 (1975).
    [CrossRef]
  3. G. E. Backus, J. F. Gilbert, Philos. Trans. R. Soc. London Ser. A: 266, 123 (1970).
    [CrossRef]
  4. H. Bateman, Philos. Mag. S. 6, 19 (112), 576 (1910).
    [CrossRef]
  5. F. F. Fischbach, Bull. Am. Meteorol. Soc. 46, 9, 528 (1965).
  6. N. K. Johnson, O. F. T. Roberts, Q. J. R. Meteorol. Soc. 51, 131 (1925).
    [CrossRef]
  7. K. Brocks, Veroef. Meteorol. Inst. Univ. Berlin 3 (4), 1 (1939).
  8. R. G. Fleagle, J. Meteorol. 31, 51 (1950).
  9. R. G. Fleagle, Geophys. Res. Pap. 59, 128 (1958).
  10. J. K. Sparkman, “A remote sensing technique using terrestrial refraction, for the study of low-level lapse rate,” Ph.D. Thesis, University of Wisconsin (1971).
  11. A. B. Fraser, W. H. Mach, J. Opt. Soc. Am. 64, 1391A (1974).
  12. P. G. Tait, Trans. R. Soc.Edinburgh30, 551 (1881).
  13. J. M. Pernter, F. M. Exner, Meteorologisch Optik (Wilhelm Braumüller, Wien, 1922).
  14. G. H. Liljequist, Norwegian-British-Swedish Antarctic Expedition, 1949–52, Scientific Results (Oslo U. P., Oslo, 1964), Vol. 11.
  15. A. B. Fraser, R. Hemler, J. Opt. Soc. Am. 65, 1174A (1975).
  16. A. Fletcher, Mon. Not. R. Astron. Soc. 91, 559 (1931).
  17. A. B. Fraser, Atmosphere 13, 1 (1975).

1976 (1)

A. B. Fraser, W. H. Mach, Sci. Am. 234 (1), 102 (1976).
[CrossRef]

1975 (3)

R. White, J. Opt. Soc. Am. 65, 676 (1975).
[CrossRef]

A. B. Fraser, R. Hemler, J. Opt. Soc. Am. 65, 1174A (1975).

A. B. Fraser, Atmosphere 13, 1 (1975).

1974 (1)

A. B. Fraser, W. H. Mach, J. Opt. Soc. Am. 64, 1391A (1974).

1970 (1)

G. E. Backus, J. F. Gilbert, Philos. Trans. R. Soc. London Ser. A: 266, 123 (1970).
[CrossRef]

1965 (1)

F. F. Fischbach, Bull. Am. Meteorol. Soc. 46, 9, 528 (1965).

1958 (1)

R. G. Fleagle, Geophys. Res. Pap. 59, 128 (1958).

1950 (1)

R. G. Fleagle, J. Meteorol. 31, 51 (1950).

1939 (1)

K. Brocks, Veroef. Meteorol. Inst. Univ. Berlin 3 (4), 1 (1939).

1931 (1)

A. Fletcher, Mon. Not. R. Astron. Soc. 91, 559 (1931).

1925 (1)

N. K. Johnson, O. F. T. Roberts, Q. J. R. Meteorol. Soc. 51, 131 (1925).
[CrossRef]

1910 (1)

H. Bateman, Philos. Mag. S. 6, 19 (112), 576 (1910).
[CrossRef]

Backus, G. E.

G. E. Backus, J. F. Gilbert, Philos. Trans. R. Soc. London Ser. A: 266, 123 (1970).
[CrossRef]

Bateman, H.

H. Bateman, Philos. Mag. S. 6, 19 (112), 576 (1910).
[CrossRef]

Brocks, K.

K. Brocks, Veroef. Meteorol. Inst. Univ. Berlin 3 (4), 1 (1939).

Exner, F. M.

J. M. Pernter, F. M. Exner, Meteorologisch Optik (Wilhelm Braumüller, Wien, 1922).

Fischbach, F. F.

F. F. Fischbach, Bull. Am. Meteorol. Soc. 46, 9, 528 (1965).

Fleagle, R. G.

R. G. Fleagle, Geophys. Res. Pap. 59, 128 (1958).

R. G. Fleagle, J. Meteorol. 31, 51 (1950).

Fletcher, A.

A. Fletcher, Mon. Not. R. Astron. Soc. 91, 559 (1931).

Fraser, A. B.

A. B. Fraser, W. H. Mach, Sci. Am. 234 (1), 102 (1976).
[CrossRef]

A. B. Fraser, Atmosphere 13, 1 (1975).

A. B. Fraser, R. Hemler, J. Opt. Soc. Am. 65, 1174A (1975).

A. B. Fraser, W. H. Mach, J. Opt. Soc. Am. 64, 1391A (1974).

Gilbert, J. F.

G. E. Backus, J. F. Gilbert, Philos. Trans. R. Soc. London Ser. A: 266, 123 (1970).
[CrossRef]

Hemler, R.

A. B. Fraser, R. Hemler, J. Opt. Soc. Am. 65, 1174A (1975).

Johnson, N. K.

N. K. Johnson, O. F. T. Roberts, Q. J. R. Meteorol. Soc. 51, 131 (1925).
[CrossRef]

Liljequist, G. H.

G. H. Liljequist, Norwegian-British-Swedish Antarctic Expedition, 1949–52, Scientific Results (Oslo U. P., Oslo, 1964), Vol. 11.

Mach, W. H.

A. B. Fraser, W. H. Mach, Sci. Am. 234 (1), 102 (1976).
[CrossRef]

A. B. Fraser, W. H. Mach, J. Opt. Soc. Am. 64, 1391A (1974).

Pernter, J. M.

J. M. Pernter, F. M. Exner, Meteorologisch Optik (Wilhelm Braumüller, Wien, 1922).

Roberts, O. F. T.

N. K. Johnson, O. F. T. Roberts, Q. J. R. Meteorol. Soc. 51, 131 (1925).
[CrossRef]

Sparkman, J. K.

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

Tait, P. G.

P. G. Tait, Trans. R. Soc.Edinburgh30, 551 (1881).

White, R.

Atmosphere (1)

A. B. Fraser, Atmosphere 13, 1 (1975).

Bull. Am. Meteorol. Soc. (1)

F. F. Fischbach, Bull. Am. Meteorol. Soc. 46, 9, 528 (1965).

Geophys. Res. Pap. (1)

R. G. Fleagle, Geophys. Res. Pap. 59, 128 (1958).

J. Meteorol. (1)

R. G. Fleagle, J. Meteorol. 31, 51 (1950).

J. Opt. Soc. Am. (3)

A. B. Fraser, R. Hemler, J. Opt. Soc. Am. 65, 1174A (1975).

A. B. Fraser, W. H. Mach, J. Opt. Soc. Am. 64, 1391A (1974).

R. White, J. Opt. Soc. Am. 65, 676 (1975).
[CrossRef]

Mon. Not. R. Astron. Soc. (1)

A. Fletcher, Mon. Not. R. Astron. Soc. 91, 559 (1931).

Philos. Mag. S. 6 (1)

H. Bateman, Philos. Mag. S. 6, 19 (112), 576 (1910).
[CrossRef]

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

G. E. Backus, J. F. Gilbert, Philos. Trans. R. Soc. London Ser. A: 266, 123 (1970).
[CrossRef]

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

N. K. Johnson, O. F. T. Roberts, Q. J. R. Meteorol. Soc. 51, 131 (1925).
[CrossRef]

Sci. Am. (1)

A. B. Fraser, W. H. Mach, Sci. Am. 234 (1), 102 (1976).
[CrossRef]

Veroef. Meteorol. Inst. Univ. Berlin (1)

K. Brocks, Veroef. Meteorol. Inst. Univ. Berlin 3 (4), 1 (1939).

Other (4)

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

P. G. Tait, Trans. R. Soc.Edinburgh30, 551 (1881).

J. M. Pernter, F. M. Exner, Meteorologisch Optik (Wilhelm Braumüller, Wien, 1922).

G. H. Liljequist, Norwegian-British-Swedish Antarctic Expedition, 1949–52, Scientific Results (Oslo U. P., Oslo, 1964), Vol. 11.

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Equations (57)

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d x = cot β d z ,
d ( n cos β ) / d s = o
n cos β = n v = n o cos β o ,
x t = n v z e z t ( n 2 - n v 2 ) - 1 / 2 d z .
τ = 1 - ( n / n o ) 2 , ϕ = tan β cos β o , ξ = x sec β o , ζ = z .
d ξ = ϕ - 1 d ζ
d ( ϕ 2 + τ ) / d s = 0 ,
τ = ϕ o 2 - ϕ 2 ,             τ v = ϕ o 2 , τ e = ϕ o 2 - ϕ e 2 ,             τ - τ e = ϕ e 2 - ϕ 2 .
d ξ = - 2 ζ d ϕ
ξ t = - 2 ϕ e ϕ t ζ d ϕ ,
ζ d ζ d τ ζ ( 1 ) ,
ζ = o τ ζ d τ ,
d ζ = - 2 ζ ϕ d ϕ
ζ t = - 2 ϕ e ϕ t ζ ϕ d ϕ .
ζ = m = 1 p a m ψ m ( τ ) = m = 1 p a m ψ m ( ϕ o 2 - ϕ 2 ) ,
- ξ t / 2 = m = 1 p a m ψ ^ m , - ζ t / 2 = m = 1 p a m ψ ˜ m ,
ψ ^ m = ψ ^ m ( ϕ e , ϕ t ) = ϕ e ϕ t ψ m ( ϕ o 2 - ϕ 2 ) d ϕ , ψ ˜ m = ψ ˜ m ( ϕ e , ϕ t ) = ϕ e ϕ t ψ m ( ϕ o 2 - ϕ 2 ) ϕ d ϕ .
ζ a = ζ e + ξ t ϕ e ,
ζ = m = 1 p ζ o ( m ) τ m / m ! ,
ξ t = m = 1 p ζ o ( m ) s = 1 m 2 2 s - 1 ( s - 1 ) ! ( m - s ) ! ( 2 s - 1 ) ! × [ ϕ e 2 s - 1 τ e m - s - ϕ t 2 s - 1 ( τ e + ϕ e 2 - ϕ t 2 ) m - s ] ; ζ t = m = 1 p ζ o ( m ) ( τ e + ϕ e 2 - ϕ t 2 ) m / m !
ξ t = m = 1 p ζ o ( m ) [ 2 2 m - 1 ( m - 1 ) ! ( 2 m - 1 ) ! ϕ o 2 m - 1 - s = 1 m 2 2 s - 1 ( s - 1 ) ! ( m - s ) ! ( 2 s - 1 ) ! ϕ t 2 s - 1 ( ϕ o 2 - ϕ t 2 ) m - s ] ζ t = m = 1 p ζ o ( m ) ( ϕ o 2 - ϕ t 2 ) m / m !
ξ t = 2 ζ o ( ϕ o - ϕ t ) ,             ξ t = ζ o ( ϕ o 2 - ϕ t 2 ) .
ξ a - ξ t = ξ t 2 / 4 ξ o .
δ ζ x 2 10 - 6 2 d T d z ,
ξ t = m = 1 p ζ o ( m ) 2 2 m ( m - 1 ) ! ( 2 m - 1 ) ! ϕ o 2 m - 1 .
ξ v = m = 1 p ζ o ( m ) 2 2 m - 1 ( m - 1 ) ! ( 2 m - 1 ) ! ϕ o 2 m - 1 , ζ v = m = 1 p ζ o ( m ) ϕ o 2 m / m !
d s = csc β d z ,
d ln I / ( - d s ) = - b ,
ln I e / I t = - z e z t b n ( n 2 - n v 2 ) - 1 l / 2 d z .
τ = 1 - ( n / n o ) 2 , ϕ = tan β cos β o , ξ = ln I / I t , ζ = - z e z b n n o d z
ζ ( τ ) = - b ( 1 - τ ) 1 / 2 d z / d τ , = - b ( n / n o ) d z / d τ , = b ( n o / 2 ) d z / d n
d θ = cot β d ln r ,
- d R = cot β d ln n ,
d ( n r cos β ) / d s = 0 ,
θ t = n v r v r e r t ( n 2 r 2 - n v 2 r v 2 ) - 1 / 2 d ln r ,
- R t = n v r v n e n t ( n 2 r 2 - n v 2 r v 2 ) - 1 / 2 d ln n ,
θ t - R t = β t - β e .
τ = 1 - ( n r / n o r o ) 2 , ϕ = tan β cos β o , ξ = θ sec β o , ζ = ln ( r / r o ) ,
τ = 1 - ( n r / n o r o ) 2 , ϕ = tan β cos β o , ξ = - R sec β o , ζ = ln ( n / n o ) ,
ψ ( τ ) = exp ( - m τ / τ s ) ,
- ξ t = ( π ) 1 / 2 m = 1 p a m ( - τ s m ) 1 / 2 exp ( m ϕ o 2 - τ s ) erfc [ ( m - τ s ) 1 / 2 ϕ o ] .
- ξ t = ln n o ( π / - τ s ) 1 / 2 exp ( ϕ o 2 / - τ s ) erfc [ ϕ o / ( - τ s ) 1 / 2 ] .
τ s = 2 ( n o - 1 ) - 2 T o / r o ( g / R - γ o ) ,
- ½ ξ t = ϕ e ζ d ϕ
- ½ d ξ d ϕ o = 2 ϕ o ϕ o ζ ( 2 ) d ϕ - ζ o ,
( d ξ t d ϕ o ) ϕ o = 0 = 2 ζ o ,
- ½ d 2 ξ t d ϕ o 2 = 2 ϕ o ζ ( 2 ) d ϕ + 4 ϕ o 2 ϕ o ζ ( 3 ) d ϕ - 2 ϕ o ζ o ( 2 ) .
( d 2 m - 1 ξ t d ϕ o 2 m - 1 ) ϕ o = 0 = 2 2 m - 1 ( m - 1 ) ! ζ o ( m ) .
( d ζ t d ϕ o ) ϕ o = 0 - 1 + ( d h d β o ) β o = 0 .
d ξ t d ϕ o - 1 + c d t d β o ,
ζ o ( 1 ) = ½ [ c ( d t d β o ) β o = 0 - 1 ] , ζ o ( m ) = c 2 2 m - 1 ( m - 1 ) ! ( d 2 m - 1 t d β o 2 m - 1 ) β o = 0             m > 1.
( d β o d h ) β o = 0 = 1 - ( n o - 1 ) r o T o ( g R - γ )
d s = csc β d r ,
d ln I / ( - d s ) = - b ,
ln I e / I t = - r e r t b n r ( n 2 r 2 - n v 2 r v 2 ) - 1 / 2 d r .
τ = 1 - ( n r / n o r o ) 2 , ϕ = tan β cos β o , ξ = ln I / I t ζ = - r e r b ( n r / n o r o ) d r ,
b o = 1 r o ( d ln I o d β o ) β o = 0 .

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