Abstract

Analytical solutions to the refraction integrals appropriate for ray trajectories along slant paths through the atmosphere are derived in this paper. This type of geometry is commonly encountered in remote-sensing applications utilizing an occultation technique. The solutions are obtained by evaluating higher-order terms from expansion of the refraction integral and are dependent on the vertical temperature distributions. Refraction parameters such as total refraction angles, air masses, and path lengths can be accurately computed. It is also shown that the method can be used for computing refraction parameters in astronomical refraction geometry for large zenith angles.

© 1983 Optical Society of America

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References

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  1. R. Goody, J. Atmos. Sci. 20, 502 (1963).
    [CrossRef]
  2. D. W. Goldsmith, Icarus 2, 341 (1963).
    [CrossRef]
  3. O. K. Garriott, J. Opt. Soc. Am. 69, 1064 (1979).
    [CrossRef]
  4. D. A. Graham, T. Ichikawa, J. S. Kim, Ann. Geophys. 25, 855 (1969).
  5. D. E. Snider, J. Atmos. Sci. 32, 2178 (1975).
    [CrossRef]
  6. D. W. Schuerman, F. Giovane, J. M. Greenberg, J. Appl. Meteorol. 14, 1182 (1975).
    [CrossRef]
  7. J. E. A. Selby, R. A. McClatchey, “Atmospheric Transmittance from 0.25 to 28.5 μm: Computer Code lowtran 2,” AFCRL-TR-72-0745, AD 763 721, 1972.
  8. D. A. Thompson, T. J. Pepin, F. W. Simon, J. Opt. Soc. Am. 72, 1498 (1982).
    [CrossRef]
  9. S. Weisbrod, L. J. Anderson, Proc. IRE 4, 1770 (1959).
    [CrossRef]
  10. R. S. Longhurst, Geometrical and Physical Optics (Longmans, London, 1964), p. 417.
  11. B. Edlen, J. Opt. Soc. Am. 43, 339 (1953).
    [CrossRef]
  12. R. Penndorf, J. Opt. Soc. Am. 47, 176 (1957).
    [CrossRef]
  13. B. Edlen, Metrologia 2, 71 (1966).
    [CrossRef]
  14. J. C. Owen, Appl. Opt. 6, 51 (1967).
    [CrossRef]
  15. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 123.
  16. U.S. Standard Atmospheric Supplements (U.S. GPO, Washington, D.C., 1966).
  17. W. A. Heiskanen, H. Moritz, Physical Geodesy (Freeman, San Francisco, 1967), p. 181.
  18. M. V. Klein, Optics (Wiley, New York, 1970), p. 31.

1982 (1)

1979 (1)

1975 (2)

D. E. Snider, J. Atmos. Sci. 32, 2178 (1975).
[CrossRef]

D. W. Schuerman, F. Giovane, J. M. Greenberg, J. Appl. Meteorol. 14, 1182 (1975).
[CrossRef]

1969 (1)

D. A. Graham, T. Ichikawa, J. S. Kim, Ann. Geophys. 25, 855 (1969).

1967 (1)

1966 (1)

B. Edlen, Metrologia 2, 71 (1966).
[CrossRef]

1963 (2)

R. Goody, J. Atmos. Sci. 20, 502 (1963).
[CrossRef]

D. W. Goldsmith, Icarus 2, 341 (1963).
[CrossRef]

1959 (1)

S. Weisbrod, L. J. Anderson, Proc. IRE 4, 1770 (1959).
[CrossRef]

1957 (1)

1953 (1)

Anderson, L. J.

S. Weisbrod, L. J. Anderson, Proc. IRE 4, 1770 (1959).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 123.

Edlen, B.

Garriott, O. K.

Giovane, F.

D. W. Schuerman, F. Giovane, J. M. Greenberg, J. Appl. Meteorol. 14, 1182 (1975).
[CrossRef]

Goldsmith, D. W.

D. W. Goldsmith, Icarus 2, 341 (1963).
[CrossRef]

Goody, R.

R. Goody, J. Atmos. Sci. 20, 502 (1963).
[CrossRef]

Graham, D. A.

D. A. Graham, T. Ichikawa, J. S. Kim, Ann. Geophys. 25, 855 (1969).

Greenberg, J. M.

D. W. Schuerman, F. Giovane, J. M. Greenberg, J. Appl. Meteorol. 14, 1182 (1975).
[CrossRef]

Heiskanen, W. A.

W. A. Heiskanen, H. Moritz, Physical Geodesy (Freeman, San Francisco, 1967), p. 181.

Ichikawa, T.

D. A. Graham, T. Ichikawa, J. S. Kim, Ann. Geophys. 25, 855 (1969).

Kim, J. S.

D. A. Graham, T. Ichikawa, J. S. Kim, Ann. Geophys. 25, 855 (1969).

Klein, M. V.

M. V. Klein, Optics (Wiley, New York, 1970), p. 31.

Longhurst, R. S.

R. S. Longhurst, Geometrical and Physical Optics (Longmans, London, 1964), p. 417.

McClatchey, R. A.

J. E. A. Selby, R. A. McClatchey, “Atmospheric Transmittance from 0.25 to 28.5 μm: Computer Code lowtran 2,” AFCRL-TR-72-0745, AD 763 721, 1972.

Moritz, H.

W. A. Heiskanen, H. Moritz, Physical Geodesy (Freeman, San Francisco, 1967), p. 181.

Owen, J. C.

Penndorf, R.

Pepin, T. J.

Schuerman, D. W.

D. W. Schuerman, F. Giovane, J. M. Greenberg, J. Appl. Meteorol. 14, 1182 (1975).
[CrossRef]

Selby, J. E. A.

J. E. A. Selby, R. A. McClatchey, “Atmospheric Transmittance from 0.25 to 28.5 μm: Computer Code lowtran 2,” AFCRL-TR-72-0745, AD 763 721, 1972.

Simon, F. W.

Snider, D. E.

D. E. Snider, J. Atmos. Sci. 32, 2178 (1975).
[CrossRef]

Thompson, D. A.

Weisbrod, S.

S. Weisbrod, L. J. Anderson, Proc. IRE 4, 1770 (1959).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 123.

Ann. Geophys. (1)

D. A. Graham, T. Ichikawa, J. S. Kim, Ann. Geophys. 25, 855 (1969).

Appl. Opt. (1)

Icarus (1)

D. W. Goldsmith, Icarus 2, 341 (1963).
[CrossRef]

J. Appl. Meteorol. (1)

D. W. Schuerman, F. Giovane, J. M. Greenberg, J. Appl. Meteorol. 14, 1182 (1975).
[CrossRef]

J. Atmos. Sci. (2)

D. E. Snider, J. Atmos. Sci. 32, 2178 (1975).
[CrossRef]

R. Goody, J. Atmos. Sci. 20, 502 (1963).
[CrossRef]

J. Opt. Soc. Am. (4)

Metrologia (1)

B. Edlen, Metrologia 2, 71 (1966).
[CrossRef]

Proc. IRE (1)

S. Weisbrod, L. J. Anderson, Proc. IRE 4, 1770 (1959).
[CrossRef]

Other (6)

R. S. Longhurst, Geometrical and Physical Optics (Longmans, London, 1964), p. 417.

J. E. A. Selby, R. A. McClatchey, “Atmospheric Transmittance from 0.25 to 28.5 μm: Computer Code lowtran 2,” AFCRL-TR-72-0745, AD 763 721, 1972.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 123.

U.S. Standard Atmospheric Supplements (U.S. GPO, Washington, D.C., 1966).

W. A. Heiskanen, H. Moritz, Physical Geodesy (Freeman, San Francisco, 1967), p. 181.

M. V. Klein, Optics (Wiley, New York, 1970), p. 31.

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Tables (5)

Tables Icon

Table I Comparison of Calculated Total Refraction Angles in Arc Minutes: (A) From Eq. (19); (B) From a Ray Trace Method; and (C) Their Difference

Tables Icon

Table II Calculated Values of Refraction Angles in Arc Minutes for Standard Atmospheres: (A1) 15°N Annual; (A2) 30°N July; and (A3) 60°N July

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Table III Calculated Values of Air Masses (Normalized to One Air Mass for the 45°N Spring/Fall Standard Atmosphere) for Standard Atmospheres: (A) 45°N Spring/Fall; (A1) 15°N Annual; (A2) 30°N July; (A3) 60°N July

Tables Icon

Table IV Calculated Values of Ray Path in km for 45°N Spring/Fall Standard Atmosphere

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Table V Calculated Astronomical Refraction Parameters for an Observer Situated at Sea-Level Altitude for 45°N Spring/Fall Standard Atmosphere: (ϕ) Zenith Angle in Degrees; (A) Refraction Angle in Arc Minutes; and (B) Normalized Air Masses

Equations (34)

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d P ( z ) / d z = ( g / R ) P ( z ) / T ( z ) ,
T ( z ) = T i + β i ( z z i ) z i z z i + 1 ,
P ( z ) = P 0 j = 0 i 1 f ( T j , z j + 1 , z j ) f ( T i , z , z i ) ,
f ( T i , z , z i ) = [ 1 + β i ( z z i ) / T i ] g / ( R β i ) .
f ( T i , z , z i ) = exp [ g ( z z i ) / ( R T i ) ] .
n ( z ) = 1 + a λ ρ ( z ) / ρ 0 ,
n ( z ) = 1 + a λ T 0 T ( z ) j = 0 i 1 f ( T j , z j + 1 , z j ) f ( T i , z , z i ) for z i z z i + 1 ,
θ = r t n t r t d r r [ n 2 ( r ) r 2 n t 2 r t 2 ] 1 / 2 ,
θ = 1 d x x { x 2 [ 1 + ε G ( x ) ] 2 1 } 1 / 2 .
G i ( x ) = [ σ i g i ( x ) 1 ] for x i x x i + 1 ,
σ i = a λ T 0 ( n t 1 ) T i j = 0 i 1 f ( T j , x j + 1 , x j ) , g i ( x ) = { C i γ i [ 1 + b i ( x 1 ) ] γ i for β i 0 d i exp [ ( x 1 ) / H i ] for β i = 0 ,
γ i = g / R β i + 1 , C i = 1 + β i ( z t z i ) / T i , b i = β i r t / T i C i , d i = exp [ ( 1 x i ) / H i ] , and H i = R T i / g r t .
G i ( x ) = { [ 1 + r t β i ( x 1 ) / T i ] γ i 1 for β i 0 exp [ ( x 1 ) / H i ] 1 for β i = 0.
β c = [ n t T t ( n t 1 ) r t q R ] .
θ = i θ i ,
θ i = x i x i + 1 d x x { x 2 [ 1 + ε G ( x ) ] 2 1 } 1 / 2 .
θ i = I i + 0 ε I i + 1 ε 2 2 ( I i 2 I i + 2 ) + ε 3 3 ( 3 I i 3 5 I i + 3 ) ,
I i ± p = x i x + 1 G i p ( x ) x ( 2 P 2 ) ± 1 d x ( x 2 1 ) P ± 1 / 2 .
δ T = i δ i ,
δ i = 2 ε σ i J i 11 ε 2 [ x ( G i 2 ) ( x 2 1 ) 1 / 2 | x i x i + 1 σ i 2 J i 22 + 2 σ i J i 21 ] + ε 3 3 [ x 3 ( G i 3 ) ( x 2 1 ) 3 / 2 + x 2 ( G i 2 ) ( x 2 1 ) 1 / 2 | x i x i + 1 σ i 3 J i 33 + 3 σ i 2 J i 32 3 σ i J i 31 ] ,
J i p q = ( q H i ) P ( d i ) q π H i 2 q erf [ q ( x 1 ) H i ] | x i x i + 1 ,
J i p q = D i { F ( a p ) B z ( 1 2 , | γ i q | p + 1 ) for β i < 0 F ( a + p ) B ω ( 1 2 , | γ i q | + p 1 2 ) for β i > 0 ,
τ 2 = τ 1 n ( r 1 ) / n ( r 2 ) n ( r ) Δ S / n ( r 2 ) ,
δ T = 2 π H ( ε + 0.41 H ε 2 + 0.27 H 2 ε 3 ) ,
m ( r t ) = 2 r t n ( r ) ρ ( r ) r d r [ n 2 ( r ) r 2 n t 2 r t 2 ] 1 / 2 .
m ( r t ) = A 1 x [ 1 + G ( x ) ] [ 1 + ε G ( x ) ] d x { x 2 [ 1 + ε G ( x ) ] 2 1 } 1 / 2 ,
m ( r t ) = i m i ,
m i = A ( σ i J i 01 ε ( σ i 2 J i 12 σ i J i 11 ) 1 2 ε 2 [ x 3 d d x { [ 1 + G ( x ) ] G 2 ( x ) } ( x 2 1 ) 1 / 2 | x i x i + 1 σ i 3 J i 23 + 2 σ i 2 J i 22 σ i J i 21 ] ) ,
S 12 = r 1 r 2 n ( r ) r d r [ n 2 ( r ) r 2 n t 2 r t 2 ] 1 / 2 .
S 12 = r t x 1 x 2 x [ 1 + ε G ( x ) ] d x { x 2 [ 1 + ε G ( x ) ] 2 1 } 1 / 2 .
S 12 = i S i ,
S i = r t { ( x 2 1 ) 1 / 2 | x i x i + 1 + ε x 2 G i ( x ) ( x 2 1 ) 1 / 2 | x i x i + 1 ε σ i J i 11 } ε 2 [ 2 x 4 G 2 ( x ) ( x 2 1 ) 3 / 2 + 3 2 x 3 G ( x ) G ( x ) ( x 2 1 ) 1 / 2 | x i x i + 1 3 4 σ i 2 J i 22 + 3 2 σ i J i 21 ] } ,
n t r t = n 0 r 0 sin ϕ 0 ,
θ ( ϕ 0 ) = x 0 d x x 2 { x 2 [ 1 + ε 0 G ( x ) ] 2 1 } 1 / 2 ,

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