Abstract

The theory of special relativity is used to analyze some of the physical phenomena associated with space-based coherent Doppler lidars aimed at Earth and the atmosphere. Two important cases of diffuse scattering and retroreflection by lidar targets are treated. For the case of diffuse scattering, we show that for a coaligned transmitter and receiver on the moving satellite, there is no angle between transmitted and returned radiation. However, the ray that enters the receiver does not correspond to a retroreflected ray by the target. For the retroreflection case there is misalignment between the transmitted ray and the received ray. In addition, the Doppler shift in the frequency and the amount of tip for the receiver aperture when needed are calculated. The error in estimating wind because of the Doppler shift in the frequency due to special relativity effects is examined. The results are then applied to a proposed space-based pulsed coherent Doppler lidar at NASA’s Marshall Space Flight Center for wind and aerosol backscatter measurements. The lidar uses an orbiting spacecraft with a pulsed laser source and measures the Doppler shift between the transmitted and the received frequencies to determine the atmospheric wind velocities. We show that the special relativity effects are small for the proposed system.

© 1999 Optical Society of America

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  1. W. E. Baker, G. D. Emmitt, F. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar-measured winds from space: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
    [CrossRef]
  2. M. J. Kavaya, G. D. Spiers, Elena S. Love, J. Rothermel, V. W. Keller, “Direct global measurements of tropospheric winds employing a simplified coherent laser radar using fully scalable technology and technique,” in Space Instrumentation and Dual-Use Technologies, F. A. Allahdadi, M. P. Chrisp, L. R. Giuliano, W. P. Lantham, F. Shanley, eds., Proc. SPIE2214, 237–249 (1994).
    [CrossRef]
  3. D. M. Winker, R. H. Couch, M. P. McCormick, “An overview of LITE: NASA’s lidar in-space technology experiment,” Proc. IEEE 84, 164–187 (1996).
    [CrossRef]
  4. J. M. Vaughan, K. O. Steinwall, C. Werner, P. H. Flamant, “Coherent laser radar in Europe,” Proc. IEEE 84, 205–226 (1996) and Refs. 114–117 therein.
    [CrossRef]
  5. R. G. Frehlich, “Heterodyne efficiency for a coherent laser radar with diffuse or aerosol targets,” J. Mod. Opt. 41, 2115–2129 (1994).
    [CrossRef]
  6. R. Frehlich, M. J. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30, 5325–5352 (1991).
    [CrossRef] [PubMed]
  7. R. T. Menzies, R. M. Hardesty, “Coherent Doppler lidar for measurements of wind fields,” Proc. IEEE 77, 449–462 (1989).
    [CrossRef]
  8. R. G. Beranek, J. W. Bilbro, D. E. Fitzjarrald, W. D. Jones, V. W. Keller, B. S. Perrine, “Laser atmospheric wind sounder (LAWS),” in Laser Applications in Meteorology and Earth Atmospheric Remote Sensing, M. M. Sokoloski, ed., Proc. SPIE1062, 234–248 (1989).
    [CrossRef]
  9. R. M. Huffaker, T. R. Lawrence, R. J. Keeler, M. J. Post, J. T. Priestly, J. A. Korrell, “Feasibility study of satellite-borne lidar global wind monitoring system Part II,” (Environmental Research Laboratories, Wave Propagation Laboratory, Boulder, Colo., August1980).
  10. J. W. Bilbro, R. Beranek, D. Fitzjarrald, J. Mabry, eds., “Shuttle coherent atmospheric lidar experiment: Final report,” (Marshall Space Flight Center, Huntsville, Ala., June1987).
  11. R. T. Menzies, “Doppler lidar atmospheric wind sensors: A comparative performance evaluation for global measurement applications from Earth orbit,” Appl. Opt. 25, 2546–2653 (1986).
    [CrossRef] [PubMed]
  12. M. J. Kavaya, R. T. Menzies, “Lidar aerosol backscatter measurements: systematic, modeling, and calibration error considerations,” Appl. Opt. 24, 3444–3453 (1985).
    [CrossRef] [PubMed]
  13. J. Fred Holmes, F. Amzajerdian, V. S. Rao Gudimetla, J. M. Hunt, “Remote sensing of atmospheric winds by utilizing speckle–turbulence interaction, a CO2 and optical heterodyne detection,” Appl. Opt. 27, 2532–2538 (1988).
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  14. M. J. Kavaya, G. D. Emmitt, “The Space Readiness Coherent Lidar Experiment (SPARCLE) Space Shuttle Mission,” in Laser Radar Technology and Applications III, G. W. Kamerman, ed., Proc. SPIE3380, 2–11 (1998).
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  15. J. Van Bladel, Relativity and Engineering (Springer-Verlag, New York, 1984).
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  16. D. De Zutter, “Doppler effect from a transmitter in translational motion,” Microwaves Opt. Acoust. 3, 85–92 (1979).
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  17. C. Yeh, “Reflection and transmission of electromagnetic waves by a moving dielectric medium,” J. Appl. Phys. 36, 3513–3517 (1965).
    [CrossRef]
  18. T. Shiozawa, K. Hazama, N. Kumagai, “Reflection and transmission of electromagnetic waves by a dielectric half-space moving perpendicular to the plane of incidence,” J. Appl. Phys. 38, 4459–4461 (1967).
    [CrossRef]
  19. H. Fujita, T. Yanese, S. Uchida, “Effect of a moving boundary on the reflections and the transmission of a plane electromagnetic wave,” Electron. Commun. Jpn. (English transl.) 53-B No. 10, 87–90 (1970).
  20. M. Saca, “Brewster angle in a semi-infinite dielectric moving perpendicularly to the interface,” Am. J. Phys. 48, 237–239 (1980).
    [CrossRef]
  21. C. Yeh, “Brewster angle for a dielectric medium moving at a relativistic speed,” J. Appl. Phys. 38, 5194–5200 (1967).
    [CrossRef]
  22. T. Shiozawa, N. Kumagai, “Total reflection at the interface between relatively moving media,” Proc. IEEE 55, 1243–1244 (1967).
    [CrossRef]
  23. T. Hosono, T. Hinata, K. Yuda, “Reflection of electromagnetic waves on the boundary surface between media moving relative to each other and parallel to the boundary and plane of incidence,” Electron. Commun. Jpn. (English transl.) 51-B No. 7, 66–72 (1968).
  24. T. Hosono, T. Hinata, K. Yuda, “Reflection of electromagnetic waves on the boundary surface between media in relative motion perpendicular to the plane of incidence,” Electron. Commun. Jpn. (English transl.) 52-B No. 7, 27–35 (1969).
  25. T. Shiozawa, K. Hazama, “General solution to the problem of reflection and transmission by a moving dielectric medium,” Radio Sci. 3, 569–576 (1968).
  26. V. P. Pyati, “Reflection and refraction of electromagnetic waves by a moving dielectric medium,” J. Appl. Phys. 38, 652–655 (1967).
    [CrossRef]
  27. J. M. Saca, “Snell’s law for light rays in moving isotropic dielectrics,” Proc. IEEE 68, 409–410 (1980).
    [CrossRef]
  28. D. Censor, “Scattering of a plane wave at a plane interface separating two moving media,” Radio Sci. 4, 1079–1088 (1969).
    [CrossRef]
  29. M. Ohkubo, “The surface impedance of a moving medium,” Electron. Commun. Jpn. (English transl.) 52-B No. 11, 125–128 (1969).
  30. C. Yeh, “Reflection from a dielectric-coated moving mirror,” J. Opt. Soc. Am. 57, 657–661 (1967).
    [CrossRef]
  31. K. Tanaka, K. Hazama, “Reflection and transmission of electromagnetic waves by a moving inhomogeneous medium,” Radio Sci. 7, 973–978 (1972).
    [CrossRef]
  32. K. Yasakawa, E. Ogawa, H. Fujioka, “Total reflection of Gaussian beam at the interface of moving media,” Electron. Commun. Jpn. (English transl.) 56-B No. 8, 65–71 (1973).
  33. D. G. Ashworth, P. A. Davies, “The Doppler effect in reflecting system,” Proc. IEEE 64, 280–281 (1976).
    [CrossRef]
  34. D. G. Ashworth, P. A. Davies, “Tests of special relativity using the Doppler effect,” Proc. IEEE 64, 281–283 (1976).
    [CrossRef]
  35. J. F. Holmes, A. Ishimaru, “Electric dipole radiation in the presence of a moving, dispersive dielectric half-space,” Radio Sci. 5, 61–72 (1970).
    [CrossRef]
  36. E. W. Young, “Doppler lidar on a polar orbiting platform—relativistic correction required,” in Laser Radar V, R. J. Becherer, ed., Proc. SPIE1222, 154–161 (1990).
    [CrossRef]
  37. E. C. Jordan, K. G. Balman, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood-Cliffs, N.J., 1968), Chap. 18.

1996 (2)

D. M. Winker, R. H. Couch, M. P. McCormick, “An overview of LITE: NASA’s lidar in-space technology experiment,” Proc. IEEE 84, 164–187 (1996).
[CrossRef]

J. M. Vaughan, K. O. Steinwall, C. Werner, P. H. Flamant, “Coherent laser radar in Europe,” Proc. IEEE 84, 205–226 (1996) and Refs. 114–117 therein.
[CrossRef]

1995 (1)

W. E. Baker, G. D. Emmitt, F. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar-measured winds from space: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

1994 (1)

R. G. Frehlich, “Heterodyne efficiency for a coherent laser radar with diffuse or aerosol targets,” J. Mod. Opt. 41, 2115–2129 (1994).
[CrossRef]

1991 (1)

1989 (1)

R. T. Menzies, R. M. Hardesty, “Coherent Doppler lidar for measurements of wind fields,” Proc. IEEE 77, 449–462 (1989).
[CrossRef]

1988 (1)

1986 (1)

1985 (1)

1980 (2)

M. Saca, “Brewster angle in a semi-infinite dielectric moving perpendicularly to the interface,” Am. J. Phys. 48, 237–239 (1980).
[CrossRef]

J. M. Saca, “Snell’s law for light rays in moving isotropic dielectrics,” Proc. IEEE 68, 409–410 (1980).
[CrossRef]

1979 (1)

D. De Zutter, “Doppler effect from a transmitter in translational motion,” Microwaves Opt. Acoust. 3, 85–92 (1979).
[CrossRef]

1976 (2)

D. G. Ashworth, P. A. Davies, “The Doppler effect in reflecting system,” Proc. IEEE 64, 280–281 (1976).
[CrossRef]

D. G. Ashworth, P. A. Davies, “Tests of special relativity using the Doppler effect,” Proc. IEEE 64, 281–283 (1976).
[CrossRef]

1973 (1)

K. Yasakawa, E. Ogawa, H. Fujioka, “Total reflection of Gaussian beam at the interface of moving media,” Electron. Commun. Jpn. (English transl.) 56-B No. 8, 65–71 (1973).

1972 (1)

K. Tanaka, K. Hazama, “Reflection and transmission of electromagnetic waves by a moving inhomogeneous medium,” Radio Sci. 7, 973–978 (1972).
[CrossRef]

1970 (2)

J. F. Holmes, A. Ishimaru, “Electric dipole radiation in the presence of a moving, dispersive dielectric half-space,” Radio Sci. 5, 61–72 (1970).
[CrossRef]

H. Fujita, T. Yanese, S. Uchida, “Effect of a moving boundary on the reflections and the transmission of a plane electromagnetic wave,” Electron. Commun. Jpn. (English transl.) 53-B No. 10, 87–90 (1970).

1969 (3)

D. Censor, “Scattering of a plane wave at a plane interface separating two moving media,” Radio Sci. 4, 1079–1088 (1969).
[CrossRef]

M. Ohkubo, “The surface impedance of a moving medium,” Electron. Commun. Jpn. (English transl.) 52-B No. 11, 125–128 (1969).

T. Hosono, T. Hinata, K. Yuda, “Reflection of electromagnetic waves on the boundary surface between media in relative motion perpendicular to the plane of incidence,” Electron. Commun. Jpn. (English transl.) 52-B No. 7, 27–35 (1969).

1968 (2)

T. Shiozawa, K. Hazama, “General solution to the problem of reflection and transmission by a moving dielectric medium,” Radio Sci. 3, 569–576 (1968).

T. Hosono, T. Hinata, K. Yuda, “Reflection of electromagnetic waves on the boundary surface between media moving relative to each other and parallel to the boundary and plane of incidence,” Electron. Commun. Jpn. (English transl.) 51-B No. 7, 66–72 (1968).

1967 (5)

C. Yeh, “Brewster angle for a dielectric medium moving at a relativistic speed,” J. Appl. Phys. 38, 5194–5200 (1967).
[CrossRef]

T. Shiozawa, N. Kumagai, “Total reflection at the interface between relatively moving media,” Proc. IEEE 55, 1243–1244 (1967).
[CrossRef]

C. Yeh, “Reflection from a dielectric-coated moving mirror,” J. Opt. Soc. Am. 57, 657–661 (1967).
[CrossRef]

T. Shiozawa, K. Hazama, N. Kumagai, “Reflection and transmission of electromagnetic waves by a dielectric half-space moving perpendicular to the plane of incidence,” J. Appl. Phys. 38, 4459–4461 (1967).
[CrossRef]

V. P. Pyati, “Reflection and refraction of electromagnetic waves by a moving dielectric medium,” J. Appl. Phys. 38, 652–655 (1967).
[CrossRef]

1965 (1)

C. Yeh, “Reflection and transmission of electromagnetic waves by a moving dielectric medium,” J. Appl. Phys. 36, 3513–3517 (1965).
[CrossRef]

Amzajerdian, F.

Anderson, J. R.

W. E. Baker, G. D. Emmitt, F. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar-measured winds from space: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Ashworth, D. G.

D. G. Ashworth, P. A. Davies, “The Doppler effect in reflecting system,” Proc. IEEE 64, 280–281 (1976).
[CrossRef]

D. G. Ashworth, P. A. Davies, “Tests of special relativity using the Doppler effect,” Proc. IEEE 64, 281–283 (1976).
[CrossRef]

Atlas, R. M.

W. E. Baker, G. D. Emmitt, F. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar-measured winds from space: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Baker, W. E.

W. E. Baker, G. D. Emmitt, F. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar-measured winds from space: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Balman, K. G.

E. C. Jordan, K. G. Balman, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood-Cliffs, N.J., 1968), Chap. 18.

Beranek, R. G.

R. G. Beranek, J. W. Bilbro, D. E. Fitzjarrald, W. D. Jones, V. W. Keller, B. S. Perrine, “Laser atmospheric wind sounder (LAWS),” in Laser Applications in Meteorology and Earth Atmospheric Remote Sensing, M. M. Sokoloski, ed., Proc. SPIE1062, 234–248 (1989).
[CrossRef]

Bilbro, J. W.

R. G. Beranek, J. W. Bilbro, D. E. Fitzjarrald, W. D. Jones, V. W. Keller, B. S. Perrine, “Laser atmospheric wind sounder (LAWS),” in Laser Applications in Meteorology and Earth Atmospheric Remote Sensing, M. M. Sokoloski, ed., Proc. SPIE1062, 234–248 (1989).
[CrossRef]

Bowdle, D. A.

W. E. Baker, G. D. Emmitt, F. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar-measured winds from space: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Brown, R. A.

W. E. Baker, G. D. Emmitt, F. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar-measured winds from space: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Censor, D.

D. Censor, “Scattering of a plane wave at a plane interface separating two moving media,” Radio Sci. 4, 1079–1088 (1969).
[CrossRef]

Couch, R. H.

D. M. Winker, R. H. Couch, M. P. McCormick, “An overview of LITE: NASA’s lidar in-space technology experiment,” Proc. IEEE 84, 164–187 (1996).
[CrossRef]

Davies, P. A.

D. G. Ashworth, P. A. Davies, “Tests of special relativity using the Doppler effect,” Proc. IEEE 64, 281–283 (1976).
[CrossRef]

D. G. Ashworth, P. A. Davies, “The Doppler effect in reflecting system,” Proc. IEEE 64, 280–281 (1976).
[CrossRef]

De Zutter, D.

D. De Zutter, “Doppler effect from a transmitter in translational motion,” Microwaves Opt. Acoust. 3, 85–92 (1979).
[CrossRef]

Emmitt, G. D.

W. E. Baker, G. D. Emmitt, F. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar-measured winds from space: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

M. J. Kavaya, G. D. Emmitt, “The Space Readiness Coherent Lidar Experiment (SPARCLE) Space Shuttle Mission,” in Laser Radar Technology and Applications III, G. W. Kamerman, ed., Proc. SPIE3380, 2–11 (1998).
[CrossRef]

Fitzjarrald, D. E.

R. G. Beranek, J. W. Bilbro, D. E. Fitzjarrald, W. D. Jones, V. W. Keller, B. S. Perrine, “Laser atmospheric wind sounder (LAWS),” in Laser Applications in Meteorology and Earth Atmospheric Remote Sensing, M. M. Sokoloski, ed., Proc. SPIE1062, 234–248 (1989).
[CrossRef]

Flamant, P. H.

J. M. Vaughan, K. O. Steinwall, C. Werner, P. H. Flamant, “Coherent laser radar in Europe,” Proc. IEEE 84, 205–226 (1996) and Refs. 114–117 therein.
[CrossRef]

Frehlich, R.

Frehlich, R. G.

R. G. Frehlich, “Heterodyne efficiency for a coherent laser radar with diffuse or aerosol targets,” J. Mod. Opt. 41, 2115–2129 (1994).
[CrossRef]

Fujioka, H.

K. Yasakawa, E. Ogawa, H. Fujioka, “Total reflection of Gaussian beam at the interface of moving media,” Electron. Commun. Jpn. (English transl.) 56-B No. 8, 65–71 (1973).

Fujita, H.

H. Fujita, T. Yanese, S. Uchida, “Effect of a moving boundary on the reflections and the transmission of a plane electromagnetic wave,” Electron. Commun. Jpn. (English transl.) 53-B No. 10, 87–90 (1970).

Hardesty, R. M.

W. E. Baker, G. D. Emmitt, F. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar-measured winds from space: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

R. T. Menzies, R. M. Hardesty, “Coherent Doppler lidar for measurements of wind fields,” Proc. IEEE 77, 449–462 (1989).
[CrossRef]

Hazama, K.

K. Tanaka, K. Hazama, “Reflection and transmission of electromagnetic waves by a moving inhomogeneous medium,” Radio Sci. 7, 973–978 (1972).
[CrossRef]

T. Shiozawa, K. Hazama, “General solution to the problem of reflection and transmission by a moving dielectric medium,” Radio Sci. 3, 569–576 (1968).

T. Shiozawa, K. Hazama, N. Kumagai, “Reflection and transmission of electromagnetic waves by a dielectric half-space moving perpendicular to the plane of incidence,” J. Appl. Phys. 38, 4459–4461 (1967).
[CrossRef]

Hinata, T.

T. Hosono, T. Hinata, K. Yuda, “Reflection of electromagnetic waves on the boundary surface between media in relative motion perpendicular to the plane of incidence,” Electron. Commun. Jpn. (English transl.) 52-B No. 7, 27–35 (1969).

T. Hosono, T. Hinata, K. Yuda, “Reflection of electromagnetic waves on the boundary surface between media moving relative to each other and parallel to the boundary and plane of incidence,” Electron. Commun. Jpn. (English transl.) 51-B No. 7, 66–72 (1968).

Holmes, J. F.

J. F. Holmes, A. Ishimaru, “Electric dipole radiation in the presence of a moving, dispersive dielectric half-space,” Radio Sci. 5, 61–72 (1970).
[CrossRef]

Holmes, J. Fred

Hosono, T.

T. Hosono, T. Hinata, K. Yuda, “Reflection of electromagnetic waves on the boundary surface between media in relative motion perpendicular to the plane of incidence,” Electron. Commun. Jpn. (English transl.) 52-B No. 7, 27–35 (1969).

T. Hosono, T. Hinata, K. Yuda, “Reflection of electromagnetic waves on the boundary surface between media moving relative to each other and parallel to the boundary and plane of incidence,” Electron. Commun. Jpn. (English transl.) 51-B No. 7, 66–72 (1968).

Huffaker, R. M.

R. M. Huffaker, T. R. Lawrence, R. J. Keeler, M. J. Post, J. T. Priestly, J. A. Korrell, “Feasibility study of satellite-borne lidar global wind monitoring system Part II,” (Environmental Research Laboratories, Wave Propagation Laboratory, Boulder, Colo., August1980).

Hunt, J. M.

Ishimaru, A.

J. F. Holmes, A. Ishimaru, “Electric dipole radiation in the presence of a moving, dispersive dielectric half-space,” Radio Sci. 5, 61–72 (1970).
[CrossRef]

Jones, W. D.

R. G. Beranek, J. W. Bilbro, D. E. Fitzjarrald, W. D. Jones, V. W. Keller, B. S. Perrine, “Laser atmospheric wind sounder (LAWS),” in Laser Applications in Meteorology and Earth Atmospheric Remote Sensing, M. M. Sokoloski, ed., Proc. SPIE1062, 234–248 (1989).
[CrossRef]

Jordan, E. C.

E. C. Jordan, K. G. Balman, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood-Cliffs, N.J., 1968), Chap. 18.

Kavaya, M. J.

R. Frehlich, M. J. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30, 5325–5352 (1991).
[CrossRef] [PubMed]

M. J. Kavaya, R. T. Menzies, “Lidar aerosol backscatter measurements: systematic, modeling, and calibration error considerations,” Appl. Opt. 24, 3444–3453 (1985).
[CrossRef] [PubMed]

M. J. Kavaya, G. D. Emmitt, “The Space Readiness Coherent Lidar Experiment (SPARCLE) Space Shuttle Mission,” in Laser Radar Technology and Applications III, G. W. Kamerman, ed., Proc. SPIE3380, 2–11 (1998).
[CrossRef]

M. J. Kavaya, G. D. Spiers, Elena S. Love, J. Rothermel, V. W. Keller, “Direct global measurements of tropospheric winds employing a simplified coherent laser radar using fully scalable technology and technique,” in Space Instrumentation and Dual-Use Technologies, F. A. Allahdadi, M. P. Chrisp, L. R. Giuliano, W. P. Lantham, F. Shanley, eds., Proc. SPIE2214, 237–249 (1994).
[CrossRef]

Keeler, R. J.

R. M. Huffaker, T. R. Lawrence, R. J. Keeler, M. J. Post, J. T. Priestly, J. A. Korrell, “Feasibility study of satellite-borne lidar global wind monitoring system Part II,” (Environmental Research Laboratories, Wave Propagation Laboratory, Boulder, Colo., August1980).

Keller, V. W.

R. G. Beranek, J. W. Bilbro, D. E. Fitzjarrald, W. D. Jones, V. W. Keller, B. S. Perrine, “Laser atmospheric wind sounder (LAWS),” in Laser Applications in Meteorology and Earth Atmospheric Remote Sensing, M. M. Sokoloski, ed., Proc. SPIE1062, 234–248 (1989).
[CrossRef]

M. J. Kavaya, G. D. Spiers, Elena S. Love, J. Rothermel, V. W. Keller, “Direct global measurements of tropospheric winds employing a simplified coherent laser radar using fully scalable technology and technique,” in Space Instrumentation and Dual-Use Technologies, F. A. Allahdadi, M. P. Chrisp, L. R. Giuliano, W. P. Lantham, F. Shanley, eds., Proc. SPIE2214, 237–249 (1994).
[CrossRef]

Korrell, J. A.

R. M. Huffaker, T. R. Lawrence, R. J. Keeler, M. J. Post, J. T. Priestly, J. A. Korrell, “Feasibility study of satellite-borne lidar global wind monitoring system Part II,” (Environmental Research Laboratories, Wave Propagation Laboratory, Boulder, Colo., August1980).

Krishnamurti, T. N.

W. E. Baker, G. D. Emmitt, F. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar-measured winds from space: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Kumagai, N.

T. Shiozawa, N. Kumagai, “Total reflection at the interface between relatively moving media,” Proc. IEEE 55, 1243–1244 (1967).
[CrossRef]

T. Shiozawa, K. Hazama, N. Kumagai, “Reflection and transmission of electromagnetic waves by a dielectric half-space moving perpendicular to the plane of incidence,” J. Appl. Phys. 38, 4459–4461 (1967).
[CrossRef]

Lawrence, T. R.

R. M. Huffaker, T. R. Lawrence, R. J. Keeler, M. J. Post, J. T. Priestly, J. A. Korrell, “Feasibility study of satellite-borne lidar global wind monitoring system Part II,” (Environmental Research Laboratories, Wave Propagation Laboratory, Boulder, Colo., August1980).

Lorenc, A. C.

W. E. Baker, G. D. Emmitt, F. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar-measured winds from space: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Love, Elena S.

M. J. Kavaya, G. D. Spiers, Elena S. Love, J. Rothermel, V. W. Keller, “Direct global measurements of tropospheric winds employing a simplified coherent laser radar using fully scalable technology and technique,” in Space Instrumentation and Dual-Use Technologies, F. A. Allahdadi, M. P. Chrisp, L. R. Giuliano, W. P. Lantham, F. Shanley, eds., Proc. SPIE2214, 237–249 (1994).
[CrossRef]

McCormick, M. P.

D. M. Winker, R. H. Couch, M. P. McCormick, “An overview of LITE: NASA’s lidar in-space technology experiment,” Proc. IEEE 84, 164–187 (1996).
[CrossRef]

McElroy, J.

W. E. Baker, G. D. Emmitt, F. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar-measured winds from space: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Menzies, R. T.

W. E. Baker, G. D. Emmitt, F. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar-measured winds from space: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

R. T. Menzies, R. M. Hardesty, “Coherent Doppler lidar for measurements of wind fields,” Proc. IEEE 77, 449–462 (1989).
[CrossRef]

R. T. Menzies, “Doppler lidar atmospheric wind sensors: A comparative performance evaluation for global measurement applications from Earth orbit,” Appl. Opt. 25, 2546–2653 (1986).
[CrossRef] [PubMed]

M. J. Kavaya, R. T. Menzies, “Lidar aerosol backscatter measurements: systematic, modeling, and calibration error considerations,” Appl. Opt. 24, 3444–3453 (1985).
[CrossRef] [PubMed]

Molinari, J. E.

W. E. Baker, G. D. Emmitt, F. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar-measured winds from space: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Ogawa, E.

K. Yasakawa, E. Ogawa, H. Fujioka, “Total reflection of Gaussian beam at the interface of moving media,” Electron. Commun. Jpn. (English transl.) 56-B No. 8, 65–71 (1973).

Ohkubo, M.

M. Ohkubo, “The surface impedance of a moving medium,” Electron. Commun. Jpn. (English transl.) 52-B No. 11, 125–128 (1969).

Paegle, J.

W. E. Baker, G. D. Emmitt, F. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar-measured winds from space: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Perrine, B. S.

R. G. Beranek, J. W. Bilbro, D. E. Fitzjarrald, W. D. Jones, V. W. Keller, B. S. Perrine, “Laser atmospheric wind sounder (LAWS),” in Laser Applications in Meteorology and Earth Atmospheric Remote Sensing, M. M. Sokoloski, ed., Proc. SPIE1062, 234–248 (1989).
[CrossRef]

Post, M. J.

W. E. Baker, G. D. Emmitt, F. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar-measured winds from space: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

R. M. Huffaker, T. R. Lawrence, R. J. Keeler, M. J. Post, J. T. Priestly, J. A. Korrell, “Feasibility study of satellite-borne lidar global wind monitoring system Part II,” (Environmental Research Laboratories, Wave Propagation Laboratory, Boulder, Colo., August1980).

Priestly, J. T.

R. M. Huffaker, T. R. Lawrence, R. J. Keeler, M. J. Post, J. T. Priestly, J. A. Korrell, “Feasibility study of satellite-borne lidar global wind monitoring system Part II,” (Environmental Research Laboratories, Wave Propagation Laboratory, Boulder, Colo., August1980).

Pyati, V. P.

V. P. Pyati, “Reflection and refraction of electromagnetic waves by a moving dielectric medium,” J. Appl. Phys. 38, 652–655 (1967).
[CrossRef]

Rao Gudimetla, V. S.

Robertson, F.

W. E. Baker, G. D. Emmitt, F. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar-measured winds from space: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Rothermel, J.

M. J. Kavaya, G. D. Spiers, Elena S. Love, J. Rothermel, V. W. Keller, “Direct global measurements of tropospheric winds employing a simplified coherent laser radar using fully scalable technology and technique,” in Space Instrumentation and Dual-Use Technologies, F. A. Allahdadi, M. P. Chrisp, L. R. Giuliano, W. P. Lantham, F. Shanley, eds., Proc. SPIE2214, 237–249 (1994).
[CrossRef]

Saca, J. M.

J. M. Saca, “Snell’s law for light rays in moving isotropic dielectrics,” Proc. IEEE 68, 409–410 (1980).
[CrossRef]

Saca, M.

M. Saca, “Brewster angle in a semi-infinite dielectric moving perpendicularly to the interface,” Am. J. Phys. 48, 237–239 (1980).
[CrossRef]

Shiozawa, T.

T. Shiozawa, K. Hazama, “General solution to the problem of reflection and transmission by a moving dielectric medium,” Radio Sci. 3, 569–576 (1968).

T. Shiozawa, K. Hazama, N. Kumagai, “Reflection and transmission of electromagnetic waves by a dielectric half-space moving perpendicular to the plane of incidence,” J. Appl. Phys. 38, 4459–4461 (1967).
[CrossRef]

T. Shiozawa, N. Kumagai, “Total reflection at the interface between relatively moving media,” Proc. IEEE 55, 1243–1244 (1967).
[CrossRef]

Spiers, G. D.

M. J. Kavaya, G. D. Spiers, Elena S. Love, J. Rothermel, V. W. Keller, “Direct global measurements of tropospheric winds employing a simplified coherent laser radar using fully scalable technology and technique,” in Space Instrumentation and Dual-Use Technologies, F. A. Allahdadi, M. P. Chrisp, L. R. Giuliano, W. P. Lantham, F. Shanley, eds., Proc. SPIE2214, 237–249 (1994).
[CrossRef]

Steinwall, K. O.

J. M. Vaughan, K. O. Steinwall, C. Werner, P. H. Flamant, “Coherent laser radar in Europe,” Proc. IEEE 84, 205–226 (1996) and Refs. 114–117 therein.
[CrossRef]

Tanaka, K.

K. Tanaka, K. Hazama, “Reflection and transmission of electromagnetic waves by a moving inhomogeneous medium,” Radio Sci. 7, 973–978 (1972).
[CrossRef]

Uchida, S.

H. Fujita, T. Yanese, S. Uchida, “Effect of a moving boundary on the reflections and the transmission of a plane electromagnetic wave,” Electron. Commun. Jpn. (English transl.) 53-B No. 10, 87–90 (1970).

Van Bladel, J.

J. Van Bladel, Relativity and Engineering (Springer-Verlag, New York, 1984).
[CrossRef]

Vaughan, J. M.

J. M. Vaughan, K. O. Steinwall, C. Werner, P. H. Flamant, “Coherent laser radar in Europe,” Proc. IEEE 84, 205–226 (1996) and Refs. 114–117 therein.
[CrossRef]

Werner, C.

J. M. Vaughan, K. O. Steinwall, C. Werner, P. H. Flamant, “Coherent laser radar in Europe,” Proc. IEEE 84, 205–226 (1996) and Refs. 114–117 therein.
[CrossRef]

Winker, D. M.

D. M. Winker, R. H. Couch, M. P. McCormick, “An overview of LITE: NASA’s lidar in-space technology experiment,” Proc. IEEE 84, 164–187 (1996).
[CrossRef]

Yanese, T.

H. Fujita, T. Yanese, S. Uchida, “Effect of a moving boundary on the reflections and the transmission of a plane electromagnetic wave,” Electron. Commun. Jpn. (English transl.) 53-B No. 10, 87–90 (1970).

Yasakawa, K.

K. Yasakawa, E. Ogawa, H. Fujioka, “Total reflection of Gaussian beam at the interface of moving media,” Electron. Commun. Jpn. (English transl.) 56-B No. 8, 65–71 (1973).

Yeh, C.

C. Yeh, “Reflection from a dielectric-coated moving mirror,” J. Opt. Soc. Am. 57, 657–661 (1967).
[CrossRef]

C. Yeh, “Brewster angle for a dielectric medium moving at a relativistic speed,” J. Appl. Phys. 38, 5194–5200 (1967).
[CrossRef]

C. Yeh, “Reflection and transmission of electromagnetic waves by a moving dielectric medium,” J. Appl. Phys. 36, 3513–3517 (1965).
[CrossRef]

Young, E. W.

E. W. Young, “Doppler lidar on a polar orbiting platform—relativistic correction required,” in Laser Radar V, R. J. Becherer, ed., Proc. SPIE1222, 154–161 (1990).
[CrossRef]

Yuda, K.

T. Hosono, T. Hinata, K. Yuda, “Reflection of electromagnetic waves on the boundary surface between media in relative motion perpendicular to the plane of incidence,” Electron. Commun. Jpn. (English transl.) 52-B No. 7, 27–35 (1969).

T. Hosono, T. Hinata, K. Yuda, “Reflection of electromagnetic waves on the boundary surface between media moving relative to each other and parallel to the boundary and plane of incidence,” Electron. Commun. Jpn. (English transl.) 51-B No. 7, 66–72 (1968).

Am. J. Phys. (1)

M. Saca, “Brewster angle in a semi-infinite dielectric moving perpendicularly to the interface,” Am. J. Phys. 48, 237–239 (1980).
[CrossRef]

Appl. Opt. (4)

Bull. Am. Meteorol. Soc. (1)

W. E. Baker, G. D. Emmitt, F. Robertson, R. M. Atlas, J. E. Molinari, D. A. Bowdle, J. Paegle, R. M. Hardesty, R. T. Menzies, T. N. Krishnamurti, R. A. Brown, M. J. Post, J. R. Anderson, A. C. Lorenc, J. McElroy, “Lidar-measured winds from space: a key component for weather and climate prediction,” Bull. Am. Meteorol. Soc. 76, 869–888 (1995).
[CrossRef]

Electron. Commun. Jpn. (English transl.) (5)

T. Hosono, T. Hinata, K. Yuda, “Reflection of electromagnetic waves on the boundary surface between media moving relative to each other and parallel to the boundary and plane of incidence,” Electron. Commun. Jpn. (English transl.) 51-B No. 7, 66–72 (1968).

T. Hosono, T. Hinata, K. Yuda, “Reflection of electromagnetic waves on the boundary surface between media in relative motion perpendicular to the plane of incidence,” Electron. Commun. Jpn. (English transl.) 52-B No. 7, 27–35 (1969).

M. Ohkubo, “The surface impedance of a moving medium,” Electron. Commun. Jpn. (English transl.) 52-B No. 11, 125–128 (1969).

K. Yasakawa, E. Ogawa, H. Fujioka, “Total reflection of Gaussian beam at the interface of moving media,” Electron. Commun. Jpn. (English transl.) 56-B No. 8, 65–71 (1973).

H. Fujita, T. Yanese, S. Uchida, “Effect of a moving boundary on the reflections and the transmission of a plane electromagnetic wave,” Electron. Commun. Jpn. (English transl.) 53-B No. 10, 87–90 (1970).

J. Appl. Phys. (4)

V. P. Pyati, “Reflection and refraction of electromagnetic waves by a moving dielectric medium,” J. Appl. Phys. 38, 652–655 (1967).
[CrossRef]

C. Yeh, “Brewster angle for a dielectric medium moving at a relativistic speed,” J. Appl. Phys. 38, 5194–5200 (1967).
[CrossRef]

C. Yeh, “Reflection and transmission of electromagnetic waves by a moving dielectric medium,” J. Appl. Phys. 36, 3513–3517 (1965).
[CrossRef]

T. Shiozawa, K. Hazama, N. Kumagai, “Reflection and transmission of electromagnetic waves by a dielectric half-space moving perpendicular to the plane of incidence,” J. Appl. Phys. 38, 4459–4461 (1967).
[CrossRef]

J. Mod. Opt. (1)

R. G. Frehlich, “Heterodyne efficiency for a coherent laser radar with diffuse or aerosol targets,” J. Mod. Opt. 41, 2115–2129 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

Microwaves Opt. Acoust. (1)

D. De Zutter, “Doppler effect from a transmitter in translational motion,” Microwaves Opt. Acoust. 3, 85–92 (1979).
[CrossRef]

Proc. IEEE (7)

D. M. Winker, R. H. Couch, M. P. McCormick, “An overview of LITE: NASA’s lidar in-space technology experiment,” Proc. IEEE 84, 164–187 (1996).
[CrossRef]

J. M. Vaughan, K. O. Steinwall, C. Werner, P. H. Flamant, “Coherent laser radar in Europe,” Proc. IEEE 84, 205–226 (1996) and Refs. 114–117 therein.
[CrossRef]

R. T. Menzies, R. M. Hardesty, “Coherent Doppler lidar for measurements of wind fields,” Proc. IEEE 77, 449–462 (1989).
[CrossRef]

J. M. Saca, “Snell’s law for light rays in moving isotropic dielectrics,” Proc. IEEE 68, 409–410 (1980).
[CrossRef]

T. Shiozawa, N. Kumagai, “Total reflection at the interface between relatively moving media,” Proc. IEEE 55, 1243–1244 (1967).
[CrossRef]

D. G. Ashworth, P. A. Davies, “The Doppler effect in reflecting system,” Proc. IEEE 64, 280–281 (1976).
[CrossRef]

D. G. Ashworth, P. A. Davies, “Tests of special relativity using the Doppler effect,” Proc. IEEE 64, 281–283 (1976).
[CrossRef]

Radio Sci. (4)

J. F. Holmes, A. Ishimaru, “Electric dipole radiation in the presence of a moving, dispersive dielectric half-space,” Radio Sci. 5, 61–72 (1970).
[CrossRef]

T. Shiozawa, K. Hazama, “General solution to the problem of reflection and transmission by a moving dielectric medium,” Radio Sci. 3, 569–576 (1968).

D. Censor, “Scattering of a plane wave at a plane interface separating two moving media,” Radio Sci. 4, 1079–1088 (1969).
[CrossRef]

K. Tanaka, K. Hazama, “Reflection and transmission of electromagnetic waves by a moving inhomogeneous medium,” Radio Sci. 7, 973–978 (1972).
[CrossRef]

Other (8)

E. W. Young, “Doppler lidar on a polar orbiting platform—relativistic correction required,” in Laser Radar V, R. J. Becherer, ed., Proc. SPIE1222, 154–161 (1990).
[CrossRef]

E. C. Jordan, K. G. Balman, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood-Cliffs, N.J., 1968), Chap. 18.

R. G. Beranek, J. W. Bilbro, D. E. Fitzjarrald, W. D. Jones, V. W. Keller, B. S. Perrine, “Laser atmospheric wind sounder (LAWS),” in Laser Applications in Meteorology and Earth Atmospheric Remote Sensing, M. M. Sokoloski, ed., Proc. SPIE1062, 234–248 (1989).
[CrossRef]

R. M. Huffaker, T. R. Lawrence, R. J. Keeler, M. J. Post, J. T. Priestly, J. A. Korrell, “Feasibility study of satellite-borne lidar global wind monitoring system Part II,” (Environmental Research Laboratories, Wave Propagation Laboratory, Boulder, Colo., August1980).

J. W. Bilbro, R. Beranek, D. Fitzjarrald, J. Mabry, eds., “Shuttle coherent atmospheric lidar experiment: Final report,” (Marshall Space Flight Center, Huntsville, Ala., June1987).

M. J. Kavaya, G. D. Spiers, Elena S. Love, J. Rothermel, V. W. Keller, “Direct global measurements of tropospheric winds employing a simplified coherent laser radar using fully scalable technology and technique,” in Space Instrumentation and Dual-Use Technologies, F. A. Allahdadi, M. P. Chrisp, L. R. Giuliano, W. P. Lantham, F. Shanley, eds., Proc. SPIE2214, 237–249 (1994).
[CrossRef]

M. J. Kavaya, G. D. Emmitt, “The Space Readiness Coherent Lidar Experiment (SPARCLE) Space Shuttle Mission,” in Laser Radar Technology and Applications III, G. W. Kamerman, ed., Proc. SPIE3380, 2–11 (1998).
[CrossRef]

J. Van Bladel, Relativity and Engineering (Springer-Verlag, New York, 1984).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Lidar and target geometry and coordinate systems.

Fig. 2
Fig. 2

Path geometry for the diffusely reflected ray.

Fig. 3
Fig. 3

Path geometry for the retroreflected ray.

Equations (40)

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

xG=xV+VtV1-α21/2, yG=yV, zG=zV-H, ctG=ctV+αxV1-α21/2.
xV=xG-VtG1-α21/2, yV=yG, zV=zG+H, ctV=ctG-αxG1-α21/2.
cosψx=sinθcosϕ, cosψy=sinθsinϕ, cosψz=cosθ.
tanϕ=cosψycosψx,  cosθ=cosψz.
xV=ctV cosψVtx=ctV sinθVtcosϕVt, yV=ctV cosψVty=ctV sinθVtsinϕVt, zV=ctV cosψVtz=ctV cosθVt.
tVg=Hc cosψVtz=Hc cosθVt,
xVg=H cosψVtxcosψVtz=H tanθVtcosϕVt, yVg=H cosψVtycosψVtz=H tanθVtsinϕVt, zVg=H.
xGg=H α+sinθVtcosϕVt1-α21/2 cosθVt=H α+cosψVtx1-α21/2 cosψVtz, yGg=H tanθVtsinϕVt=H cosψVtycosψVtz, zGg=0,
tGg=Hc1+α sinθVtcosϕVt1-α21/2 cosθVt=Hc1+α cosψVtx1-α21/2 cosψVtz.
rG=ctGg-0=H1-α21/21+α cosψVtxcosψVtz=H1-α21/21+α sinθVtcosϕVtcosθVt.
cosψGgx=xGg-0r=α+cosψVtx1+α cosψVtx=α+sinθVtcosϕVt1+α sinθVtcosϕVt,cosψGgy=yGg-0r=1-α21/2 cosψVty1+α cosψVtx=1-α21/2 sinθVtsinϕVt1+α sinθVtcosϕVt,cosψGgz=zGg--Hr=1-α21/2 cosψVtz1+α cosψVtx=1-α21/2 cosθVt1+α sinθVtcosϕVt.
cosθGg=cosψGgz=1-α21/2 cosθVt1+α sinθVtcosϕVt, tanϕGg=cosψGgycosψGgx=1-α21/2 sinθVtsinϕVtα+sinθVtcosϕVt.
xG=ctG-tGgcosψGdx+xGg, yG=ctG-tGgcosψGdy+yGg, zG=ctG-tGgcosψGdz.
tGret = tGg - Hc cosψGdz=H 1+α cosψVtxc1-α21/2 cosψVtz-Hc cosψGdz.
xGret=-H cosψGdxcosψGdz+H α+cosψVtx1-α21/2 cosψVtz,
yGret=-H cosψGdycosψGdz+H cosψVtycosψVtz, zGret=-H.
xVret=H α-cosψGdx1-α21/2 cosψGdz+H cosψVtxcosψVtz, yVret=yGret=-H cosψGdycosψGdz+H cosψVtycosψVtz. zVret=0.
rV=|xVret-xVg|2+|yVret-yVg|2+|zVret-zVg|21/2=H -1+α sinθGdcosϕGd1-α21/2 cosθGd=H -1+α cosψGdx1-α21/2 cosψGdz.
cosψVretx=xVret-xVgrV=α-cosψGdx-1+α cosψGdx,cosψVrety=yVret-yVgrV=-1-α21/2 cosψGdy-1+α cosψGdx,cosψVretz=zVret-zVgrV=-1-α21/2 cosψGdz-1+α cosψGdx.
cosθVret=cosψVretz=1-α21/2 cosθGd1-α sinθGdcosϕGd,tanϕVret=cosψVretycosψVretx=-1-α21/2 sinθGdsinϕGdα-sinθGdcosϕGd.
xVret=0, yVret=0, zVret=0.
cosψGd1x=α-cosψVtx1-α cosψVtx,cosψGd1y=-1-α21/2 cosψVty1-α cosψVtx,cosψGd1z=-1-α21/2 cosψVtz1-α cosψVtx.
ψVretx=ψVtx-π, ψVrety=ψVty-π, ψVretz=π-ψVtz,
cosθGd1=-cosθVt1-α21/21-α sinθVtcosϕVt,tanϕGd1=-sinθVtsinϕVt1-α21/2α-sinθVtcosϕVt.
xVret=-2Hα 1+α sinθVtcosϕVt1-α2cosθVt, yVret=0, zVret=0.
xVret=-2HαcosθVt1+α sinθVtcosϕVt+,
cosθVret=-1-α2cosθVt1+α2+2α sinθVtcosϕVt, tanϕVret=1-α2sinθVtsinϕVt2α+1+α2sinθVtcosϕVt.
cosθVret=cosθVt1-2α sinθVtcosϕVt-2α1-2α sinθVtcosϕVt+,
βxG2+βyG2+βzG2=β2,
ωVttV-βxVxV-βyVyV-βzVzV=ωgtG-βxGxG-βyGyG-βzGzG.
ωVttV-βxVxV-βyVyV-βzVzV=ωVt1-α21/2tG-α xGc-βxV1-α21/2xG-vtG-βyVyG-βzVzG=t 11-α21/2ωVt+βxVV-xG1-α21/2βxV+αβt0-yGβyV-zGβzV.
ωg=ωVt+βxVV1-α21/2,  βxG=βxV+αβt01-α21/2, βyG=βyV,  βzG=βzV.
ωVt=ωg-βxGV1-α21/2,  βxV=βxG-αβg01-α21/2, βV1=βyG,  βzV=βzG,
βxV=βt0 cosψVtx=βt0 sinθVtcosϕVt, βyV=βt0 cosψVty=βt0 sinθVtsinϕVt, βzV=βt0 cosψVtz=βt0 cosθVt.
ωg=ωVt+βxVV1-α21/2=ωVt1+α sinθVtcosϕVt1-α21/2=ωVt1+αcosψVtx1-α21/2.
βGdx=βg0 sinθGdcosϕGd, βGdy=βg0 sinθGdsinϕGd, βGdz=βg0 cosθGd,
ωVret=γωg-βGdxV.
ωVret=ωVt1+α sinθVt cos ϕVt1-α sin θGd cos ϕGd1-α2.
ωVret=ωVt1+α sin θVt cos ϕVt1-α sin θVt cos ϕVt.
ωVret=ωVt1+α2+2α sin θVt cos ϕVt1-α2=wt1+2α sinθVtcosϕVt+2α2+2α3 sinθVtcosϕVt+.

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