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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  1. M. P. McCormick, P. Hamill, T. J. Pepin, W. P. Chu, T. J. Swissler, and L. R. McMaster, "Satellite studies of the stratospheric aerosol," Bull. Am. Meteorol. Soc. 60, 1038–1046 (1979).
  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, Jr., "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 Re-mote 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 III, "Balloon-borne scanning spectrometer system for atmospheric extinction studies in the 350–1100-nm spectral region," Rev. Sci. Instrum. 53, 314–319 (1982).
  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 mea-surements of atmospheric structure by refraction," Planet. Space Sci. 9, 351–352 (1962).
  10. R. Goody, "A note on the refraction of a twilight ray," J. Atmos. Sci. 20, 502–505 (1963).
  11. B. Edlén, "The dispersion of standard air," J. Opt. Soc. Am. 43, 339–345 (1953).
  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).
  13. B. Edlén, "The refractive index of air," Metrologia 2, 71–80 (1966).
  14. J. C. Owens, "Optical refractive index of air: dependence on pressure, temperature and composition," Appl. Opt. 6, 51–59 (1967).
  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).

1982 (1)

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

1979 (1)

M. P. McCormick, P. Hamill, T. J. Pepin, W. P. Chu, T. J. Swissler, and L. R. McMaster, "Satellite studies of the stratospheric aerosol," Bull. Am. Meteorol. Soc. 60, 1038–1046 (1979).

1968 (1)

P. B. Hays and R. G. Roble, "Stellar spectra and atmospheric composition," J. Atmos. Sci. 25, 1141–1153 (1968).

1967 (1)

1966 (1)

B. Edlén, "The refractive index of air," Metrologia 2, 71–80 (1966).

1963 (1)

R. Goody, "A note on the refraction of a twilight ray," J. Atmos. Sci. 20, 502–505 (1963).

1962 (1)

L. M. Jones, F. F. Fishbach, and J. W. Peterson, "Satellite mea-surements of atmospheric structure by refraction," Planet. Space Sci. 9, 351–352 (1962).

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, Jr., "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 III, H. R.

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

Chu, W. P.

M. P. McCormick, P. Hamill, T. J. Pepin, W. P. Chu, T. J. Swissler, and L. R. McMaster, "Satellite studies of the stratospheric aerosol," Bull. Am. Meteorol. Soc. 60, 1038–1046 (1979).

T. J. Pepin, M. P. McCormick, W. P. Chu, F. Simon, T. J. Swissler, R. R. Adams, K. H. Crumbly, and W. H. Fuller, Jr., "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, Jr., "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).

B. Edlén, "The dispersion of standard air," J. Opt. Soc. Am. 43, 339–345 (1953).

Fishbach, F. F.

L. M. Jones, F. F. Fishbach, and J. W. Peterson, "Satellite mea-surements of atmospheric structure by refraction," Planet. Space Sci. 9, 351–352 (1962).

Fuller, Jr., 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, Jr., "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).

Hamill, P.

M. P. McCormick, P. Hamill, T. J. Pepin, W. P. Chu, T. J. Swissler, and L. R. McMaster, "Satellite studies of the stratospheric aerosol," Bull. Am. Meteorol. Soc. 60, 1038–1046 (1979).

Hays, P. B.

P. B. Hays and R. G. Roble, "Stellar spectra and atmospheric composition," J. Atmos. Sci. 25, 1141–1153 (1968).

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 mea-surements of atmospheric structure by refraction," Planet. Space Sci. 9, 351–352 (1962).

Lane, R. W.

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

McCormick, M. P.

M. P. McCormick, P. Hamill, T. J. Pepin, W. P. Chu, T. J. Swissler, and L. R. McMaster, "Satellite studies of the stratospheric aerosol," Bull. Am. Meteorol. Soc. 60, 1038–1046 (1979).

T. J. Pepin, M. P. McCormick, W. P. Chu, F. Simon, T. J. Swissler, R. R. Adams, K. H. Crumbly, and W. H. Fuller, Jr., "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. Swissler, and L. R. McMaster, "Satellite studies of the stratospheric aerosol," Bull. Am. Meteorol. Soc. 60, 1038–1046 (1979).

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 III, "Balloon-borne scanning spectrometer system for atmospheric extinction studies in the 350–1100-nm spectral region," Rev. Sci. Instrum. 53, 314–319 (1982).

M. P. McCormick, P. Hamill, T. J. Pepin, W. P. Chu, T. J. Swissler, and L. R. McMaster, "Satellite studies of the stratospheric aerosol," Bull. Am. Meteorol. Soc. 60, 1038–1046 (1979).

T. J. Pepin, M. P. McCormick, W. P. Chu, F. Simon, T. J. Swissler, R. R. Adams, K. H. Crumbly, and W. H. Fuller, Jr., "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 Re-mote Sounding, A. Deepak, ed. (Academic, New York, 1977), pp. 529–554.

Peterson, J. W.

L. M. Jones, F. F. Fishbach, and J. W. Peterson, "Satellite mea-surements of atmospheric structure by refraction," Planet. Space Sci. 9, 351–352 (1962).

Roble, R. G.

P. B. Hays and R. G. Roble, "Stellar spectra and atmospheric composition," J. Atmos. Sci. 25, 1141–1153 (1968).

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, Jr., "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.

M. P. McCormick, P. Hamill, T. J. Pepin, W. P. Chu, T. J. Swissler, and L. R. McMaster, "Satellite studies of the stratospheric aerosol," Bull. Am. Meteorol. Soc. 60, 1038–1046 (1979).

T. J. Pepin, M. P. McCormick, W. P. Chu, F. Simon, T. J. Swissler, R. R. Adams, K. H. Crumbly, and W. H. Fuller, Jr., "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.

Thompson, D. A.

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

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 III, "Balloon-borne scanning spectrometer system for atmospheric extinction studies in the 350–1100-nm spectral region," Rev. Sci. Instrum. 53, 314–319 (1982).

Appl. Opt. (1)

Bull. Am. Meteorol. Soc. (1)

M. P. McCormick, P. Hamill, T. J. Pepin, W. P. Chu, T. J. Swissler, and L. R. McMaster, "Satellite studies of the stratospheric aerosol," Bull. Am. Meteorol. Soc. 60, 1038–1046 (1979).

J. Atmos. Sci. (2)

R. Goody, "A note on the refraction of a twilight ray," J. Atmos. Sci. 20, 502–505 (1963).

P. B. Hays and R. G. Roble, "Stellar spectra and atmospheric composition," J. Atmos. Sci. 25, 1141–1153 (1968).

J. Opt. Soc. Am. (2)

Metrologia (1)

B. Edlén, "The refractive index of air," Metrologia 2, 71–80 (1966).

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 mea-surements of atmospheric structure by refraction," Planet. Space Sci. 9, 351–352 (1962).

Rev. Sci. Instrum. (1)

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

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, Jr., "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 Re-mote 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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