Abstract

An accurate method of analysis of underwater imaging systems using optical spread functions and their transforms, the modulation transfer functions, is presented. The overall system spread function is obtained from the water spread functions and the spread functions of the system components. The water point spread and beam spread functions are defined, and measurements of these are presented for clear coastal water for distances up to nine attenuation lengths. Relationships between the spread functions and their dependence on range are also given. In addition, their relationships to the conventional optical oceangraphic parameters of beam attenuation, absorption, and scattering are described. Combined optics-water spread functions and their transforms are developed for illuminator and receiver geometries typically used in underwater imaging. These are then used to determine the system (optics plus water) response to the target reflectance. An analytic technique for accurate computation of backscatter signals is developed. Computed and measured signals compare favorably. It is concluded that the use of spread functions is a convenient and viable technique for analytic computation of underwater images.

© 1977 Optical Society of America

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  1. S. Q. Duntley, J. Opt. Soc. Am. 53, 214 (1963).
  2. C. J. Funk, Appl. Opt. 12, 301 (1973).
  3. R. E. Morrison, Ph. D. thesis (Dept. of Meteorology and Oceanography, New York University, 1967).
  4. L. E. Mertens and D. L. Phillips, "Measurements of the Volume Scattering Function of Sea Water," Air Force Eastern Test Range, Tech. Rep. 334, Patrick AFB, Florida (1972), obtainable from author (unpublished).
  5. T. J. Petzold, "Volume Scattering Functions for Selected Ocean Waters," Scripps Inst. of Oceanography Visibility Laboratory, San Diego, Calif. (1972).
  6. W. H. Wells, AGARD Lecture Series No. 61, Sec. 4. 1, p. 1, obtainable from Report Distribution Unit, NASA, Langley, Va. (1973).
  7. Linear summation has been verified empirically for spatial frequencies below 10 000 cycles per rad—the region where salinity and thermal gradient structure in the water do not produce beam broadening (see Ref. 8 below).
  8. R. T. Hodgson and D. R. Caldwell, J. Opt. Soc. Am. 62, 1434 (1972).
  9. A receiver beam may be defined as the cylindrical volume whose umbral surface is bounded at the ends by the receiver aperture and by the image of the receiver field stop.
  10. W. H. Wells, Ref. 6, Sec. 3. 4, p. 1.
  11. E. C. Jordan, Electromagnetic Waves and Radiating Systems (Prentice-Hall, New York, 1950), p. 327.
  12. W. H. Wells, J. Opt. Soc. Am. 59, 686 (1969).
  13. W. H. Wells, Ref. 6, Sec. 3. 3, p. 1.
  14. This relationship has been verified up to a range of 5. 7 attenuation lengths. The maximum range of its validity is not known.
  15. H. Hodara, AGARD Lecture Series No. 61, Sec. 3. 4 (1973), see Ref. 6 for procurement.
  16. W. H. Wells, Ref. 6, Secs. 3. 3 and 4. 3.
  17. Equation (5) and others in the derivation assume that the spread functions are independent of position. This is true on spherical surfaces at a constant range. On plane surfaces, it represents an approximation useful over small fields.
  18. V. R. Muratov and R. L. Struzer, Optical Tech. Theory Exper. 39, 519 (1972).
  19. W. H. Wells, Ref. 6, Sec. 4. 3, p. 7.
  20. A. Gordon, D. Cozen, C. Funk, P. Heckman, Jr., "Design Study of Advanced Underwater Optical Imaging Systems, Appendix B," Naval Undersea Research and Development Center, San Diego, Tech. Publ. 275 (1972) (unpublished).
  21. By scattering calculations or from direct visual observations at a point outside a beam, it may be shown that the direction giving the maximum contribution to the illumination of a backscattering volume follows the direct line from the transmitter to the volume [dashed line of Fig. 8(b)], and the angular distribution of the flux falling on this backscattering volume is effectively only a few degrees wide. A similar observation applies to the flux backscattered from this volume and striking the receiver. Thus the dominant (back) scattering angle ηe for the incremental volume shown is approximately (π-G/r) rad over most of the scattering volume in a thin extended slab at range r, and the distribution of backscattering angles for this range is only a few degrees wide. Since the amplitude of the volume scattering function VSF varies only fractionally in a span of a few degrees when the mean scattering angle lies in the vicinity of π rad (Ref. 5), a similar accuracy will result if a single value of backscatter coefficient is used in the summation of the backscattering contribution from such a slab.
  22. W. H. Wells, Ref. 6, Sec. 4. 3, p. 2.

Caldwell, D. R.

R. T. Hodgson and D. R. Caldwell, J. Opt. Soc. Am. 62, 1434 (1972).

Cozen, D.

A. Gordon, D. Cozen, C. Funk, P. Heckman, Jr., "Design Study of Advanced Underwater Optical Imaging Systems, Appendix B," Naval Undersea Research and Development Center, San Diego, Tech. Publ. 275 (1972) (unpublished).

Duntley, S. Q.

S. Q. Duntley, J. Opt. Soc. Am. 53, 214 (1963).

Funk, C.

A. Gordon, D. Cozen, C. Funk, P. Heckman, Jr., "Design Study of Advanced Underwater Optical Imaging Systems, Appendix B," Naval Undersea Research and Development Center, San Diego, Tech. Publ. 275 (1972) (unpublished).

Funk, C. J.

C. J. Funk, Appl. Opt. 12, 301 (1973).

Gordon, A.

A. Gordon, D. Cozen, C. Funk, P. Heckman, Jr., "Design Study of Advanced Underwater Optical Imaging Systems, Appendix B," Naval Undersea Research and Development Center, San Diego, Tech. Publ. 275 (1972) (unpublished).

Heckman, Jr., P.

A. Gordon, D. Cozen, C. Funk, P. Heckman, Jr., "Design Study of Advanced Underwater Optical Imaging Systems, Appendix B," Naval Undersea Research and Development Center, San Diego, Tech. Publ. 275 (1972) (unpublished).

Hodara, H.

H. Hodara, AGARD Lecture Series No. 61, Sec. 3. 4 (1973), see Ref. 6 for procurement.

Hodgson, R. T.

R. T. Hodgson and D. R. Caldwell, J. Opt. Soc. Am. 62, 1434 (1972).

Jordan, E. C.

E. C. Jordan, Electromagnetic Waves and Radiating Systems (Prentice-Hall, New York, 1950), p. 327.

Mertens, L. E.

L. E. Mertens and D. L. Phillips, "Measurements of the Volume Scattering Function of Sea Water," Air Force Eastern Test Range, Tech. Rep. 334, Patrick AFB, Florida (1972), obtainable from author (unpublished).

Morrison, R. E.

R. E. Morrison, Ph. D. thesis (Dept. of Meteorology and Oceanography, New York University, 1967).

Muratov, V. R.

V. R. Muratov and R. L. Struzer, Optical Tech. Theory Exper. 39, 519 (1972).

Petzold, T. J.

T. J. Petzold, "Volume Scattering Functions for Selected Ocean Waters," Scripps Inst. of Oceanography Visibility Laboratory, San Diego, Calif. (1972).

Phillips, D. L.

L. E. Mertens and D. L. Phillips, "Measurements of the Volume Scattering Function of Sea Water," Air Force Eastern Test Range, Tech. Rep. 334, Patrick AFB, Florida (1972), obtainable from author (unpublished).

Struzer, R. L.

V. R. Muratov and R. L. Struzer, Optical Tech. Theory Exper. 39, 519 (1972).

Wells, W. H.

W. H. Wells, Ref. 6, Sec. 4. 3, p. 2.

W. H. Wells, Ref. 6, Sec. 3. 3, p. 1.

W. H. Wells, Ref. 6, Sec. 4. 3, p. 7.

W. H. Wells, Ref. 6, Secs. 3. 3 and 4. 3.

W. H. Wells, Ref. 6, Sec. 3. 4, p. 1.

W. H. Wells, AGARD Lecture Series No. 61, Sec. 4. 1, p. 1, obtainable from Report Distribution Unit, NASA, Langley, Va. (1973).

W. H. Wells, J. Opt. Soc. Am. 59, 686 (1969).

Other (22)

S. Q. Duntley, J. Opt. Soc. Am. 53, 214 (1963).

C. J. Funk, Appl. Opt. 12, 301 (1973).

R. E. Morrison, Ph. D. thesis (Dept. of Meteorology and Oceanography, New York University, 1967).

L. E. Mertens and D. L. Phillips, "Measurements of the Volume Scattering Function of Sea Water," Air Force Eastern Test Range, Tech. Rep. 334, Patrick AFB, Florida (1972), obtainable from author (unpublished).

T. J. Petzold, "Volume Scattering Functions for Selected Ocean Waters," Scripps Inst. of Oceanography Visibility Laboratory, San Diego, Calif. (1972).

W. H. Wells, AGARD Lecture Series No. 61, Sec. 4. 1, p. 1, obtainable from Report Distribution Unit, NASA, Langley, Va. (1973).

Linear summation has been verified empirically for spatial frequencies below 10 000 cycles per rad—the region where salinity and thermal gradient structure in the water do not produce beam broadening (see Ref. 8 below).

R. T. Hodgson and D. R. Caldwell, J. Opt. Soc. Am. 62, 1434 (1972).

A receiver beam may be defined as the cylindrical volume whose umbral surface is bounded at the ends by the receiver aperture and by the image of the receiver field stop.

W. H. Wells, Ref. 6, Sec. 3. 4, p. 1.

E. C. Jordan, Electromagnetic Waves and Radiating Systems (Prentice-Hall, New York, 1950), p. 327.

W. H. Wells, J. Opt. Soc. Am. 59, 686 (1969).

W. H. Wells, Ref. 6, Sec. 3. 3, p. 1.

This relationship has been verified up to a range of 5. 7 attenuation lengths. The maximum range of its validity is not known.

H. Hodara, AGARD Lecture Series No. 61, Sec. 3. 4 (1973), see Ref. 6 for procurement.

W. H. Wells, Ref. 6, Secs. 3. 3 and 4. 3.

Equation (5) and others in the derivation assume that the spread functions are independent of position. This is true on spherical surfaces at a constant range. On plane surfaces, it represents an approximation useful over small fields.

V. R. Muratov and R. L. Struzer, Optical Tech. Theory Exper. 39, 519 (1972).

W. H. Wells, Ref. 6, Sec. 4. 3, p. 7.

A. Gordon, D. Cozen, C. Funk, P. Heckman, Jr., "Design Study of Advanced Underwater Optical Imaging Systems, Appendix B," Naval Undersea Research and Development Center, San Diego, Tech. Publ. 275 (1972) (unpublished).

By scattering calculations or from direct visual observations at a point outside a beam, it may be shown that the direction giving the maximum contribution to the illumination of a backscattering volume follows the direct line from the transmitter to the volume [dashed line of Fig. 8(b)], and the angular distribution of the flux falling on this backscattering volume is effectively only a few degrees wide. A similar observation applies to the flux backscattered from this volume and striking the receiver. Thus the dominant (back) scattering angle ηe for the incremental volume shown is approximately (π-G/r) rad over most of the scattering volume in a thin extended slab at range r, and the distribution of backscattering angles for this range is only a few degrees wide. Since the amplitude of the volume scattering function VSF varies only fractionally in a span of a few degrees when the mean scattering angle lies in the vicinity of π rad (Ref. 5), a similar accuracy will result if a single value of backscatter coefficient is used in the summation of the backscattering contribution from such a slab.

W. H. Wells, Ref. 6, Sec. 4. 3, p. 2.

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