Abstract

Satellite- and balloon-borne instruments may be used to measure the total transmission of electromagnetic radiation along a slant-path trajectory traversing the Earth’s atmosphere. Inversions of this type of measurement data to determine the altitude- and wavelength-dependent atmospheric extinction require accurate ray-tracing techniques. The ray-tracing equations used in our inversion procedures are derived. These equations describe the refraction angle and generate the path-length matrix for such radiation traversing a refracting, spherically symmetric atmosphere. Furthermore, they are presented in forms suitable for numerical integration.

© 1982 Optical Society of America

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References

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  1. M. P. McCormick, P. Hamill, T. J. Pepin, W. P. Chu, T. J. Swis-sler, and L. R. McMaster, “Satellite studies of the stratospheric aerosol,” Bull. Am. Meteorol. Soc. 60, 1038–1046 (1979).
    [CrossRef]
  2. T. J. Pepin, M. P. McCormick, W. P. Chu, F. Simon, T. J. Swissler, R. R. Adams, K. H. Crumbly, and W. H. Fuller, “Stratospheric aerosol measurements,” in Apollo–Soyuz Test Project Summary Science Report, Vol. I, NASA Spec. Publ. SP-412 (National Technical information Service, Springfield, Va.,1977), pp. 127–136.
  3. T. J. Pepin, “Inversion of solar extinction data from the Apollo-Soyuz test project stratospheric aerosol measurement (ASTP/ SAM) experiment,” in Inversion Methods in Atmospheric Remote Sounding, A. Deepak, ed.(Academic, New York, 1977), pp. 529–554.
  4. D. A. Thompson, T. J. Pepin, R. W. Lane, J. VanBaalen, and H. R. Bauer, “Balloon-borne scanning spectrometer system for atmospheric extinction studies in the 350–1100-nm spectral region,” Rev. Sci. Instrum. 53, 314–319 (1982).
    [CrossRef]
  5. D. A. Thompson, “A model of atmospheric extinction based on atmospheric attenuation measurements for altitudes to 37 km and wavelengths of 350–1100 nm,” Ph.D. thesis (University of Wyoming, Laramie, Wyo., 1981).
  6. A. R. McLeod, “On terrestrial refraction,” Philos. Mag. 38, 546–568 (1919).
  7. H. Bateman, “The solution of the integral equation connecting the velocity of propagation of an earthquake wave in the interior of the Earth with the times which the disturbance takes to travel to the different stations on the Earth’s surface,” Philos. Mag. 19, 576–587 (1910).
  8. H. Jeffreys, The Earth: Its Origin, History and Physical Constitution, 6th ed. (Cambridge U. Press, Cambridge, 1976), pp. 49–56.
  9. L. M. Jones, F. F. Fishbach, and J. W. Peterson, “Satellite measurements of atmospheric structure by refraction,” Planet. Space Sci. 9, 351–352 (1962).
    [CrossRef]
  10. R. Goody, “A note on the refraction of a twilight ray,” J. Atmos. Sci. 20, 502–505 (1963).
    [CrossRef]
  11. B. Edlén, “The dispersion of standard air,” J. Opt. Soc. Am. 43, 339–345 (1953).
    [CrossRef]
  12. R. Penndorf, “Tables of the refractive index for standard air and the Rayleigh scattering coefficient for the spectral region between 0.2 and 20.0 μ, and their application to atmospheric optics,” J. Opt. Soc. Am. 47, 176–182 (1957).
    [CrossRef]
  13. B. Edlén, “The refractive index of air,” Metrologia 2, 71–80 (1966).
    [CrossRef]
  14. J. C. Owens, “Optical refractive index of air: dependence on pressure, temperature and composition,” Appl. Opt. 6, 51–59 (1967).
    [CrossRef] [PubMed]
  15. W. A. Pearce, “Optical dispersion through a spherically symmetric atmosphere,” Tech. Rep. TR G960-0006 (EG&G Washington Analytical Services Center, Riverdale, Md., 1980).
  16. P. B. Hays and R. G. Roble, “Stellar spectra and atmospheric composition,” J. Atmos.Sci. 25, 1141–1153 (1968).
    [CrossRef]

1982 (1)

D. A. Thompson, T. J. Pepin, R. W. Lane, J. VanBaalen, and H. R. Bauer, “Balloon-borne scanning spectrometer system for atmospheric extinction studies in the 350–1100-nm spectral region,” Rev. Sci. Instrum. 53, 314–319 (1982).
[CrossRef]

1979 (1)

M. P. McCormick, P. Hamill, T. J. Pepin, W. P. Chu, T. J. Swis-sler, and L. R. McMaster, “Satellite studies of the stratospheric aerosol,” Bull. Am. Meteorol. Soc. 60, 1038–1046 (1979).
[CrossRef]

1968 (1)

P. B. Hays and R. G. Roble, “Stellar spectra and atmospheric composition,” J. Atmos.Sci. 25, 1141–1153 (1968).
[CrossRef]

1967 (1)

1966 (1)

B. Edlén, “The refractive index of air,” Metrologia 2, 71–80 (1966).
[CrossRef]

1963 (1)

R. Goody, “A note on the refraction of a twilight ray,” J. Atmos. Sci. 20, 502–505 (1963).
[CrossRef]

1962 (1)

L. M. Jones, F. F. Fishbach, and J. W. Peterson, “Satellite measurements of atmospheric structure by refraction,” Planet. Space Sci. 9, 351–352 (1962).
[CrossRef]

1957 (1)

1953 (1)

1919 (1)

A. R. McLeod, “On terrestrial refraction,” Philos. Mag. 38, 546–568 (1919).

1910 (1)

H. Bateman, “The solution of the integral equation connecting the velocity of propagation of an earthquake wave in the interior of the Earth with the times which the disturbance takes to travel to the different stations on the Earth’s surface,” Philos. Mag. 19, 576–587 (1910).

Adams, R. R.

T. J. Pepin, M. P. McCormick, W. P. Chu, F. Simon, T. J. Swissler, R. R. Adams, K. H. Crumbly, and W. H. Fuller, “Stratospheric aerosol measurements,” in Apollo–Soyuz Test Project Summary Science Report, Vol. I, NASA Spec. Publ. SP-412 (National Technical information Service, Springfield, Va.,1977), pp. 127–136.

Bateman, H.

H. Bateman, “The solution of the integral equation connecting the velocity of propagation of an earthquake wave in the interior of the Earth with the times which the disturbance takes to travel to the different stations on the Earth’s surface,” Philos. Mag. 19, 576–587 (1910).

Bauer, H. R.

D. A. Thompson, T. J. Pepin, R. W. Lane, J. VanBaalen, and H. R. Bauer, “Balloon-borne scanning spectrometer system for atmospheric extinction studies in the 350–1100-nm spectral region,” Rev. Sci. Instrum. 53, 314–319 (1982).
[CrossRef]

Chu, W. P.

M. P. McCormick, P. Hamill, T. J. Pepin, W. P. Chu, T. J. Swis-sler, and L. R. McMaster, “Satellite studies of the stratospheric aerosol,” Bull. Am. Meteorol. Soc. 60, 1038–1046 (1979).
[CrossRef]

T. J. Pepin, M. P. McCormick, W. P. Chu, F. Simon, T. J. Swissler, R. R. Adams, K. H. Crumbly, and W. H. Fuller, “Stratospheric aerosol measurements,” in Apollo–Soyuz Test Project Summary Science Report, Vol. I, NASA Spec. Publ. SP-412 (National Technical information Service, Springfield, Va.,1977), pp. 127–136.

Crumbly, K. H.

T. J. Pepin, M. P. McCormick, W. P. Chu, F. Simon, T. J. Swissler, R. R. Adams, K. H. Crumbly, and W. H. Fuller, “Stratospheric aerosol measurements,” in Apollo–Soyuz Test Project Summary Science Report, Vol. I, NASA Spec. Publ. SP-412 (National Technical information Service, Springfield, Va.,1977), pp. 127–136.

Edlén, B.

B. Edlén, “The refractive index of air,” Metrologia 2, 71–80 (1966).
[CrossRef]

B. Edlén, “The dispersion of standard air,” J. Opt. Soc. Am. 43, 339–345 (1953).
[CrossRef]

Fishbach, F. F.

L. M. Jones, F. F. Fishbach, and J. W. Peterson, “Satellite measurements of atmospheric structure by refraction,” Planet. Space Sci. 9, 351–352 (1962).
[CrossRef]

Fuller, W. H.

T. J. Pepin, M. P. McCormick, W. P. Chu, F. Simon, T. J. Swissler, R. R. Adams, K. H. Crumbly, and W. H. Fuller, “Stratospheric aerosol measurements,” in Apollo–Soyuz Test Project Summary Science Report, Vol. I, NASA Spec. Publ. SP-412 (National Technical information Service, Springfield, Va.,1977), pp. 127–136.

Goody, R.

R. Goody, “A note on the refraction of a twilight ray,” J. Atmos. Sci. 20, 502–505 (1963).
[CrossRef]

Hamill, P.

M. P. McCormick, P. Hamill, T. J. Pepin, W. P. Chu, T. J. Swis-sler, and L. R. McMaster, “Satellite studies of the stratospheric aerosol,” Bull. Am. Meteorol. Soc. 60, 1038–1046 (1979).
[CrossRef]

Hays, P. B.

P. B. Hays and R. G. Roble, “Stellar spectra and atmospheric composition,” J. Atmos.Sci. 25, 1141–1153 (1968).
[CrossRef]

Jeffreys, H.

H. Jeffreys, The Earth: Its Origin, History and Physical Constitution, 6th ed. (Cambridge U. Press, Cambridge, 1976), pp. 49–56.

Jones, L. M.

L. M. Jones, F. F. Fishbach, and J. W. Peterson, “Satellite measurements of atmospheric structure by refraction,” Planet. Space Sci. 9, 351–352 (1962).
[CrossRef]

Lane, R. W.

D. A. Thompson, T. J. Pepin, R. W. Lane, J. VanBaalen, and H. R. Bauer, “Balloon-borne scanning spectrometer system for atmospheric extinction studies in the 350–1100-nm spectral region,” Rev. Sci. Instrum. 53, 314–319 (1982).
[CrossRef]

McCormick, M. P.

M. P. McCormick, P. Hamill, T. J. Pepin, W. P. Chu, T. J. Swis-sler, and L. R. McMaster, “Satellite studies of the stratospheric aerosol,” Bull. Am. Meteorol. Soc. 60, 1038–1046 (1979).
[CrossRef]

T. J. Pepin, M. P. McCormick, W. P. Chu, F. Simon, T. J. Swissler, R. R. Adams, K. H. Crumbly, and W. H. Fuller, “Stratospheric aerosol measurements,” in Apollo–Soyuz Test Project Summary Science Report, Vol. I, NASA Spec. Publ. SP-412 (National Technical information Service, Springfield, Va.,1977), pp. 127–136.

McLeod, A. R.

A. R. McLeod, “On terrestrial refraction,” Philos. Mag. 38, 546–568 (1919).

McMaster, L. R.

M. P. McCormick, P. Hamill, T. J. Pepin, W. P. Chu, T. J. Swis-sler, and L. R. McMaster, “Satellite studies of the stratospheric aerosol,” Bull. Am. Meteorol. Soc. 60, 1038–1046 (1979).
[CrossRef]

Owens, J. C.

Pearce, W. A.

W. A. Pearce, “Optical dispersion through a spherically symmetric atmosphere,” Tech. Rep. TR G960-0006 (EG&G Washington Analytical Services Center, Riverdale, Md., 1980).

Penndorf, R.

Pepin, T. J.

D. A. Thompson, T. J. Pepin, R. W. Lane, J. VanBaalen, and H. R. Bauer, “Balloon-borne scanning spectrometer system for atmospheric extinction studies in the 350–1100-nm spectral region,” Rev. Sci. Instrum. 53, 314–319 (1982).
[CrossRef]

M. P. McCormick, P. Hamill, T. J. Pepin, W. P. Chu, T. J. Swis-sler, and L. R. McMaster, “Satellite studies of the stratospheric aerosol,” Bull. Am. Meteorol. Soc. 60, 1038–1046 (1979).
[CrossRef]

T. J. Pepin, “Inversion of solar extinction data from the Apollo-Soyuz test project stratospheric aerosol measurement (ASTP/ SAM) experiment,” in Inversion Methods in Atmospheric Remote Sounding, A. Deepak, ed.(Academic, New York, 1977), pp. 529–554.

T. J. Pepin, M. P. McCormick, W. P. Chu, F. Simon, T. J. Swissler, R. R. Adams, K. H. Crumbly, and W. H. Fuller, “Stratospheric aerosol measurements,” in Apollo–Soyuz Test Project Summary Science Report, Vol. I, NASA Spec. Publ. SP-412 (National Technical information Service, Springfield, Va.,1977), pp. 127–136.

Peterson, J. W.

L. M. Jones, F. F. Fishbach, and J. W. Peterson, “Satellite measurements of atmospheric structure by refraction,” Planet. Space Sci. 9, 351–352 (1962).
[CrossRef]

Roble, R. G.

P. B. Hays and R. G. Roble, “Stellar spectra and atmospheric composition,” J. Atmos.Sci. 25, 1141–1153 (1968).
[CrossRef]

Simon, F.

T. J. Pepin, M. P. McCormick, W. P. Chu, F. Simon, T. J. Swissler, R. R. Adams, K. H. Crumbly, and W. H. Fuller, “Stratospheric aerosol measurements,” in Apollo–Soyuz Test Project Summary Science Report, Vol. I, NASA Spec. Publ. SP-412 (National Technical information Service, Springfield, Va.,1977), pp. 127–136.

Swissler, T. J.

T. J. Pepin, M. P. McCormick, W. P. Chu, F. Simon, T. J. Swissler, R. R. Adams, K. H. Crumbly, and W. H. Fuller, “Stratospheric aerosol measurements,” in Apollo–Soyuz Test Project Summary Science Report, Vol. I, NASA Spec. Publ. SP-412 (National Technical information Service, Springfield, Va.,1977), pp. 127–136.

Swis-sler, T. J.

M. P. McCormick, P. Hamill, T. J. Pepin, W. P. Chu, T. J. Swis-sler, and L. R. McMaster, “Satellite studies of the stratospheric aerosol,” Bull. Am. Meteorol. Soc. 60, 1038–1046 (1979).
[CrossRef]

Thompson, D. A.

D. A. Thompson, T. J. Pepin, R. W. Lane, J. VanBaalen, and H. R. Bauer, “Balloon-borne scanning spectrometer system for atmospheric extinction studies in the 350–1100-nm spectral region,” Rev. Sci. Instrum. 53, 314–319 (1982).
[CrossRef]

D. A. Thompson, “A model of atmospheric extinction based on atmospheric attenuation measurements for altitudes to 37 km and wavelengths of 350–1100 nm,” Ph.D. thesis (University of Wyoming, Laramie, Wyo., 1981).

VanBaalen, J.

D. A. Thompson, T. J. Pepin, R. W. Lane, J. VanBaalen, and H. R. Bauer, “Balloon-borne scanning spectrometer system for atmospheric extinction studies in the 350–1100-nm spectral region,” Rev. Sci. Instrum. 53, 314–319 (1982).
[CrossRef]

Appl. Opt. (1)

Bull. Am. Meteorol. Soc. (1)

M. P. McCormick, P. Hamill, T. J. Pepin, W. P. Chu, T. J. Swis-sler, and L. R. McMaster, “Satellite studies of the stratospheric aerosol,” Bull. Am. Meteorol. Soc. 60, 1038–1046 (1979).
[CrossRef]

J. Atmos. Sci. (1)

R. Goody, “A note on the refraction of a twilight ray,” J. Atmos. Sci. 20, 502–505 (1963).
[CrossRef]

J. Atmos.Sci. (1)

P. B. Hays and R. G. Roble, “Stellar spectra and atmospheric composition,” J. Atmos.Sci. 25, 1141–1153 (1968).
[CrossRef]

J. Opt. Soc. Am. (2)

Metrologia (1)

B. Edlén, “The refractive index of air,” Metrologia 2, 71–80 (1966).
[CrossRef]

Philos. Mag. (2)

A. R. McLeod, “On terrestrial refraction,” Philos. Mag. 38, 546–568 (1919).

H. Bateman, “The solution of the integral equation connecting the velocity of propagation of an earthquake wave in the interior of the Earth with the times which the disturbance takes to travel to the different stations on the Earth’s surface,” Philos. Mag. 19, 576–587 (1910).

Planet. Space Sci. (1)

L. M. Jones, F. F. Fishbach, and J. W. Peterson, “Satellite measurements of atmospheric structure by refraction,” Planet. Space Sci. 9, 351–352 (1962).
[CrossRef]

Rev. Sci. Instrum. (1)

D. A. Thompson, T. J. Pepin, R. W. Lane, J. VanBaalen, and H. R. Bauer, “Balloon-borne scanning spectrometer system for atmospheric extinction studies in the 350–1100-nm spectral region,” Rev. Sci. Instrum. 53, 314–319 (1982).
[CrossRef]

Other (5)

D. A. Thompson, “A model of atmospheric extinction based on atmospheric attenuation measurements for altitudes to 37 km and wavelengths of 350–1100 nm,” Ph.D. thesis (University of Wyoming, Laramie, Wyo., 1981).

T. J. Pepin, M. P. McCormick, W. P. Chu, F. Simon, T. J. Swissler, R. R. Adams, K. H. Crumbly, and W. H. Fuller, “Stratospheric aerosol measurements,” in Apollo–Soyuz Test Project Summary Science Report, Vol. I, NASA Spec. Publ. SP-412 (National Technical information Service, Springfield, Va.,1977), pp. 127–136.

T. J. Pepin, “Inversion of solar extinction data from the Apollo-Soyuz test project stratospheric aerosol measurement (ASTP/ SAM) experiment,” in Inversion Methods in Atmospheric Remote Sounding, A. Deepak, ed.(Academic, New York, 1977), pp. 529–554.

H. Jeffreys, The Earth: Its Origin, History and Physical Constitution, 6th ed. (Cambridge U. Press, Cambridge, 1976), pp. 49–56.

W. A. Pearce, “Optical dispersion through a spherically symmetric atmosphere,” Tech. Rep. TR G960-0006 (EG&G Washington Analytical Services Center, Riverdale, Md., 1980).

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Figures (2)

Fig. 1
Fig. 1

Onionskin model of a spherical atmosphere. Path lengths in each atmospheric layer are noted. The first subscript denotes the appropriate tangent layer and the second defines the layer traversed by the path segment under consideration. For simplicity, the refraction effects of the atmosphere are not specifically represented.

Fig. 2
Fig. 2

Geometry of a light ray traversing a refracting atmosphere. Notation is defined in the text.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

n r sin θ = k = n t r t .
tan θ = k [ ( n r ) 2 k 2 ] 1 / 2 .
d n n + cot θ d θ = 0 ,
ϕ = θ t θ i d θ = Δ θ ,
ϕ = n t 1 tan θ n d n
ϕ = 1 n t k n [ ( n r ) 2 k 2 ] 1 / 2 d n .
P t i = 2 r j r j 1 d r cos θ .
P t i = 2 r j r j 1 n r [ ( n r ) 2 k 2 ] 1 / 2 d r ,
n = 1 + μ ρ ,
δ = 1 n t k n [ k 2 ( n r t ) 2 ] 1 / 2 d n ,
δ = ln [ n t + ( n t 2 1 ) 1 / 2 ] .
ϕ δ = 1 n t k n { 1 [ ( n r ) 2 k 2 ] 1 / 2 1 [ k 2 ( n r t ) 2 ] 1 / 2 } d n .
ϕ = 0 ( n t 1 ) 1 / 2 2 k ( n t y 2 ) × { 1 r [ ( r r t r ) n t y 2 1 ] 1 / 2 [ ( r + r t r ) n t y 2 ] 1 / 2 1 r t ( 2 n t y 2 ) 1 / 2 } d y + ln [ n t + ( n t 2 1 ) 1 / 2 ] ,
ϕ = 1 ( n t 1 ) 1 / 2 2 n t ( n t y 2 ) D E × [ ( 1 + r t 2 r 2 ) ( 2 n t y 2 ) ( r + r t r ) ( r r t r ) n t 2 y 2 ] + ln [ n t + ( n t 2 1 ) 1 / 2 ] , d y
D = r t r ( 2 n t y 2 ) 1 / 2 + [ ( r r t r ) n t y 2 1 ] 1 / 2 [ ( r + r t r ) n t y 2 ] 1 / 2
E = ( 2 n t y 2 ) 1 / 2 [ ( r r t r ) n t y 2 1 ] 1 / 2 [ ( r + r t r ) n t y 2 ] 1 / 2
γ t j = 2 r j r j 1 n t r [ ( n t r ) 2 k 2 ] 1 / 2 d r ,
γ t j = 2 [ ( r j 1 2 r t 2 ) 1 / 2 ( r j 2 r t 2 ) 1 / 2 ] .
P t j γ t j = 2 r j r j 1 { n r [ ( n r ) 2 k 2 ] 1 / 2 n t r [ ( n t r ) 2 k 2 ] 1 / 2 } d r ,
P t j = 2 ( r j r t ) 1 / 2 ( r j 1 r t ) 1 / 2 2 ( x 2 + r t ) × { 1 [ 1 ( n n t n ) r t x 2 ] 1 / 2 [ x 2 ( n + n t n ) r t ] 1 / 2 1 ( x 2 + 2 r t ) 1 / 2 } d x + 2 [ ( r j 1 2 r t 2 ) 1 / 2 ( r j 2 r t 2 ) 1 / 2 ]
P t j = 2 ( r j r t ) 1 / 2 ( r j 1 r t ) 1 / 2 2 r t 2 ( n t + n n ) ( n t n n ) ( x 2 + r t ) x 2 F G d x + 2 [ ( r j 1 2 r t 2 ) 1 / 2 ( r j 2 r t 2 ) 1 / 2 ] ,
F = ( x 2 + 2 r t ) 1 / 2 + [ 1 + ( n n t n ) r t x 2 ] 1 / 2 × [ x 2 + ( n + n t n ) r t ] 1 / 2
G = ( x 2 + 2 r t ) 1 / 2 [ 1 + ( n n t n ) r t x 2 ] 1 / 2 [ x 2 + ( n + n t n ) r t ] 1 / 2 ,