Abstract

We discuss a maritime surveillance and detection concept based on Raman scattering of water molecules. Using a range-gated scanning lidar that detects Raman scattered photons from water, the absence or change of signal indicates the presence of a non-water object. With sufficient spatial resolution, a two-dimensional outline of the object can be generated by the scanning lidar. Because Raman scattering is an inelastic process with a relatively large wavelength shift for water, this concept avoids the often problematic elastic scattering for objects at or very close to the water surface or from the bottom surface for shallow waters. The maximum detection depth for this concept is limited by the attenuation of the excitation and return Raman light in water. If excitation in the UV is used, fluorescence can be used for discrimination between organic and non-organic objects. In this paper, we present a lidar model for this concept and discuss results of proof-of-concept measurements. Using published cross section values, the model and measurements are in reasonable agreement and show that a sufficient number of Raman photons can be generated for modest lidar parameters to make this concept useful for near-surface detection.

© 2018 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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2014 (1)

2013 (1)

2004 (1)

2003 (1)

M. E. Kushina, G. Heberle, M. Hope, D. Hall, M. Bethel, and L. K. Calmes, “ALMDS laser system,” Proc. SPIE 4968, 163–168 (2003).
[Crossref]

1998 (1)

1997 (1)

1995 (1)

D. P. Morris, H. Zagarese, C. E. Williamson, E. G. Balseiro, B. R. Hargreaves, B. Modenutti, R. Moeller, and C. Queimalinos, “The attenuation of solar UV radiation in lakes and the role of dissolved organic carbon,” Limnol. Oceanogr. 40, 1381–1391 (1995).
[Crossref]

1993 (1)

1992 (1)

1990 (1)

1984 (1)

S. Sugihara, M. Kishino, and M. Okami, “Contribution of Raman scattering to upward irradiance in the sea,” J. Oceanogr. Soc. Japan 40, 397–404 (1984).
[Crossref]

1983 (1)

W. K. Bischel and G. Black, “Wavelength dependence of Raman scattering cross sections from 200–600  nm,” AIP Conf. Proc. 100, 181–187 (1983).
[Crossref]

1981 (1)

1978 (1)

I. I. Kondilenko, P. A. Korotkov, V. A. Klimenko, and O. P. Demyanenko, “Transverse cross section of the Raman scattering of the u1 vibration of the water molecule in the liquid and gaseous states,” Opt. Spectrosc. 43, 384–386 (1978).

1975 (2)

N. P. Romanov and V. S. Shuklin, “Raman scattering cross section of liquid water,” Opt. Spectrosc. 38, 646–648 (1975).

R. B. Slusher and V. E. Derr, “Temperature dependence and cross sections of some Stokes and anti-Stokes Raman lines in ice Ih,” Appl. Opt. 14, 2116–2120 (1975).
[Crossref]

1973 (1)

Abreu, L. W.

L. W. Abreu and G. P. Anderson, “The MODTRAN® 2/3 report and LOWTRAN 7 model,” Ontar Corporation for PL/GPOS (1996).

Anderson, G. P.

L. W. Abreu and G. P. Anderson, “The MODTRAN® 2/3 report and LOWTRAN 7 model,” Ontar Corporation for PL/GPOS (1996).

Ashlock, T. A.

R. L. Schmitt, K. L. Schroder, M. W. Smith, L. J. Krumel, P. J. Hargis, I. R. Shokair, T. A. Ashlock, J. W. Daniels, J. R. Klarkowski, M. S. Johnson, C. M. Boney, P. J. Claassen, G. I. Magee, M. L. Pedroncelli, J. T. Spooner, and A. R. Lang, “Ares UV LIF standoff system development and testing,” in Proceedings of the 6th Joint Conference on Standoff Detection for Chemical and Biological Defense, Williamsburg, Virginia, October 2004.

Baker, K. S.

Balseiro, E. G.

D. P. Morris, H. Zagarese, C. E. Williamson, E. G. Balseiro, B. R. Hargreaves, B. Modenutti, R. Moeller, and C. Queimalinos, “The attenuation of solar UV radiation in lakes and the role of dissolved organic carbon,” Limnol. Oceanogr. 40, 1381–1391 (1995).
[Crossref]

Bartlett, J. S.

Bethel, M.

M. E. Kushina, G. Heberle, M. Hope, D. Hall, M. Bethel, and L. K. Calmes, “ALMDS laser system,” Proc. SPIE 4968, 163–168 (2003).
[Crossref]

Bischel, W. K.

W. K. Bischel and G. Black, “Wavelength dependence of Raman scattering cross sections from 200–600  nm,” AIP Conf. Proc. 100, 181–187 (1983).
[Crossref]

Black, G.

W. K. Bischel and G. Black, “Wavelength dependence of Raman scattering cross sections from 200–600  nm,” AIP Conf. Proc. 100, 181–187 (1983).
[Crossref]

Boney, C. M.

R. L. Schmitt, K. L. Schroder, M. W. Smith, L. J. Krumel, P. J. Hargis, I. R. Shokair, T. A. Ashlock, J. W. Daniels, J. R. Klarkowski, M. S. Johnson, C. M. Boney, P. J. Claassen, G. I. Magee, M. L. Pedroncelli, J. T. Spooner, and A. R. Lang, “Ares UV LIF standoff system development and testing,” in Proceedings of the 6th Joint Conference on Standoff Detection for Chemical and Biological Defense, Williamsburg, Virginia, October 2004.

Bunkin, A. F.

Calmes, L. K.

M. E. Kushina, G. Heberle, M. Hope, D. Hall, M. Bethel, and L. K. Calmes, “ALMDS laser system,” Proc. SPIE 4968, 163–168 (2003).
[Crossref]

Chang, C. H.

C. H. Chang and L. A. Young, “Seawater temperature measurements from Raman spectra,” (Advanced Research Projects Agency, 1972).

Claassen, P. J.

R. L. Schmitt, K. L. Schroder, M. W. Smith, L. J. Krumel, P. J. Hargis, I. R. Shokair, T. A. Ashlock, J. W. Daniels, J. R. Klarkowski, M. S. Johnson, C. M. Boney, P. J. Claassen, G. I. Magee, M. L. Pedroncelli, J. T. Spooner, and A. R. Lang, “Ares UV LIF standoff system development and testing,” in Proceedings of the 6th Joint Conference on Standoff Detection for Chemical and Biological Defense, Williamsburg, Virginia, October 2004.

Cooper, K.

Copeland, R. A.

Daniels, J. W.

R. L. Schmitt, K. L. Schroder, M. W. Smith, L. J. Krumel, P. J. Hargis, I. R. Shokair, T. A. Ashlock, J. W. Daniels, J. R. Klarkowski, M. S. Johnson, C. M. Boney, P. J. Claassen, G. I. Magee, M. L. Pedroncelli, J. T. Spooner, and A. R. Lang, “Ares UV LIF standoff system development and testing,” in Proceedings of the 6th Joint Conference on Standoff Detection for Chemical and Biological Defense, Williamsburg, Virginia, October 2004.

Demyanenko, O. P.

I. I. Kondilenko, P. A. Korotkov, V. A. Klimenko, and O. P. Demyanenko, “Transverse cross section of the Raman scattering of the u1 vibration of the water molecule in the liquid and gaseous states,” Opt. Spectrosc. 43, 384–386 (1978).

Derr, V. E.

Faris, G. W.

Garvis, D.

Hale, G. M.

Hall, D.

M. E. Kushina, G. Heberle, M. Hope, D. Hall, M. Bethel, and L. K. Calmes, “ALMDS laser system,” Proc. SPIE 4968, 163–168 (2003).
[Crossref]

Hargis, P. J.

R. L. Schmitt, K. L. Schroder, M. W. Smith, L. J. Krumel, P. J. Hargis, I. R. Shokair, T. A. Ashlock, J. W. Daniels, J. R. Klarkowski, M. S. Johnson, C. M. Boney, P. J. Claassen, G. I. Magee, M. L. Pedroncelli, J. T. Spooner, and A. R. Lang, “Ares UV LIF standoff system development and testing,” in Proceedings of the 6th Joint Conference on Standoff Detection for Chemical and Biological Defense, Williamsburg, Virginia, October 2004.

Hargreaves, B. R.

D. P. Morris, H. Zagarese, C. E. Williamson, E. G. Balseiro, B. R. Hargreaves, B. Modenutti, R. Moeller, and C. Queimalinos, “The attenuation of solar UV radiation in lakes and the role of dissolved organic carbon,” Limnol. Oceanogr. 40, 1381–1391 (1995).
[Crossref]

Heberle, G.

M. E. Kushina, G. Heberle, M. Hope, D. Hall, M. Bethel, and L. K. Calmes, “ALMDS laser system,” Proc. SPIE 4968, 163–168 (2003).
[Crossref]

Hope, M.

M. E. Kushina, G. Heberle, M. Hope, D. Hall, M. Bethel, and L. K. Calmes, “ALMDS laser system,” Proc. SPIE 4968, 163–168 (2003).
[Crossref]

Iarlori, M.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975).

Johnson, M. S.

I. R. Shokair, M. S. Johnson, R. L. Schmitt, and S. Sickafoose, “Concept for maritime near-surface surveillance using water Raman scattering,” (Sandia National Laboratories, 2016).

R. L. Schmitt, K. L. Schroder, M. W. Smith, L. J. Krumel, P. J. Hargis, I. R. Shokair, T. A. Ashlock, J. W. Daniels, J. R. Klarkowski, M. S. Johnson, C. M. Boney, P. J. Claassen, G. I. Magee, M. L. Pedroncelli, J. T. Spooner, and A. R. Lang, “Ares UV LIF standoff system development and testing,” in Proceedings of the 6th Joint Conference on Standoff Detection for Chemical and Biological Defense, Williamsburg, Virginia, October 2004.

Kattawar, G. W.

Kennedy, R.

Kishino, M.

S. Sugihara, M. Kishino, and M. Okami, “Contribution of Raman scattering to upward irradiance in the sea,” J. Oceanogr. Soc. Japan 40, 397–404 (1984).
[Crossref]

Klarkowski, J. R.

R. L. Schmitt, K. L. Schroder, M. W. Smith, L. J. Krumel, P. J. Hargis, I. R. Shokair, T. A. Ashlock, J. W. Daniels, J. R. Klarkowski, M. S. Johnson, C. M. Boney, P. J. Claassen, G. I. Magee, M. L. Pedroncelli, J. T. Spooner, and A. R. Lang, “Ares UV LIF standoff system development and testing,” in Proceedings of the 6th Joint Conference on Standoff Detection for Chemical and Biological Defense, Williamsburg, Virginia, October 2004.

Klimenko, V. A.

I. I. Kondilenko, P. A. Korotkov, V. A. Klimenko, and O. P. Demyanenko, “Transverse cross section of the Raman scattering of the u1 vibration of the water molecule in the liquid and gaseous states,” Opt. Spectrosc. 43, 384–386 (1978).

Klinkov, V. K.

Kondilenko, I. I.

I. I. Kondilenko, P. A. Korotkov, V. A. Klimenko, and O. P. Demyanenko, “Transverse cross section of the Raman scattering of the u1 vibration of the water molecule in the liquid and gaseous states,” Opt. Spectrosc. 43, 384–386 (1978).

Korotkov, P. A.

I. I. Kondilenko, P. A. Korotkov, V. A. Klimenko, and O. P. Demyanenko, “Transverse cross section of the Raman scattering of the u1 vibration of the water molecule in the liquid and gaseous states,” Opt. Spectrosc. 43, 384–386 (1978).

Krumel, L. J.

R. L. Schmitt, K. L. Schroder, M. W. Smith, L. J. Krumel, P. J. Hargis, I. R. Shokair, T. A. Ashlock, J. W. Daniels, J. R. Klarkowski, M. S. Johnson, C. M. Boney, P. J. Claassen, G. I. Magee, M. L. Pedroncelli, J. T. Spooner, and A. R. Lang, “Ares UV LIF standoff system development and testing,” in Proceedings of the 6th Joint Conference on Standoff Detection for Chemical and Biological Defense, Williamsburg, Virginia, October 2004.

Kulp, T. J.

Kushina, M. E.

M. E. Kushina, G. Heberle, M. Hope, D. Hall, M. Bethel, and L. K. Calmes, “ALMDS laser system,” Proc. SPIE 4968, 163–168 (2003).
[Crossref]

Lang, A. R.

R. L. Schmitt, K. L. Schroder, M. W. Smith, L. J. Krumel, P. J. Hargis, I. R. Shokair, T. A. Ashlock, J. W. Daniels, J. R. Klarkowski, M. S. Johnson, C. M. Boney, P. J. Claassen, G. I. Magee, M. L. Pedroncelli, J. T. Spooner, and A. R. Lang, “Ares UV LIF standoff system development and testing,” in Proceedings of the 6th Joint Conference on Standoff Detection for Chemical and Biological Defense, Williamsburg, Virginia, October 2004.

Lednev, V. N.

Liu, F.

Magee, G. I.

R. L. Schmitt, K. L. Schroder, M. W. Smith, L. J. Krumel, P. J. Hargis, I. R. Shokair, T. A. Ashlock, J. W. Daniels, J. R. Klarkowski, M. S. Johnson, C. M. Boney, P. J. Claassen, G. I. Magee, M. L. Pedroncelli, J. T. Spooner, and A. R. Lang, “Ares UV LIF standoff system development and testing,” in Proceedings of the 6th Joint Conference on Standoff Detection for Chemical and Biological Defense, Williamsburg, Virginia, October 2004.

Marshall, B. R.

Mobley, C. D.

C. D. Mobley, Radiative Transfer in the Ocean (Academic, 2001).

Modenutti, B.

D. P. Morris, H. Zagarese, C. E. Williamson, E. G. Balseiro, B. R. Hargreaves, B. Modenutti, R. Moeller, and C. Queimalinos, “The attenuation of solar UV radiation in lakes and the role of dissolved organic carbon,” Limnol. Oceanogr. 40, 1381–1391 (1995).
[Crossref]

Moeller, R.

D. P. Morris, H. Zagarese, C. E. Williamson, E. G. Balseiro, B. R. Hargreaves, B. Modenutti, R. Moeller, and C. Queimalinos, “The attenuation of solar UV radiation in lakes and the role of dissolved organic carbon,” Limnol. Oceanogr. 40, 1381–1391 (1995).
[Crossref]

Morris, D. P.

D. P. Morris, H. Zagarese, C. E. Williamson, E. G. Balseiro, B. R. Hargreaves, B. Modenutti, R. Moeller, and C. Queimalinos, “The attenuation of solar UV radiation in lakes and the role of dissolved organic carbon,” Limnol. Oceanogr. 40, 1381–1391 (1995).
[Crossref]

Okami, M.

S. Sugihara, M. Kishino, and M. Okami, “Contribution of Raman scattering to upward irradiance in the sea,” J. Oceanogr. Soc. Japan 40, 397–404 (1984).
[Crossref]

Pedroncelli, M. L.

R. L. Schmitt, K. L. Schroder, M. W. Smith, L. J. Krumel, P. J. Hargis, I. R. Shokair, T. A. Ashlock, J. W. Daniels, J. R. Klarkowski, M. S. Johnson, C. M. Boney, P. J. Claassen, G. I. Magee, M. L. Pedroncelli, J. T. Spooner, and A. R. Lang, “Ares UV LIF standoff system development and testing,” in Proceedings of the 6th Joint Conference on Standoff Detection for Chemical and Biological Defense, Williamsburg, Virginia, October 2004.

Pershin, S. M.

Queimalinos, C.

D. P. Morris, H. Zagarese, C. E. Williamson, E. G. Balseiro, B. R. Hargreaves, B. Modenutti, R. Moeller, and C. Queimalinos, “The attenuation of solar UV radiation in lakes and the role of dissolved organic carbon,” Limnol. Oceanogr. 40, 1381–1391 (1995).
[Crossref]

Querry, M. R.

Rizi, V.

Rocci, G.

Romanov, N. P.

N. P. Romanov and V. S. Shuklin, “Raman scattering cross section of liquid water,” Opt. Spectrosc. 38, 646–648 (1975).

Salmon, T.

Sathyendranath, S.

Schmitt, R. L.

I. R. Shokair, M. S. Johnson, R. L. Schmitt, and S. Sickafoose, “Concept for maritime near-surface surveillance using water Raman scattering,” (Sandia National Laboratories, 2016).

R. L. Schmitt, K. L. Schroder, M. W. Smith, L. J. Krumel, P. J. Hargis, I. R. Shokair, T. A. Ashlock, J. W. Daniels, J. R. Klarkowski, M. S. Johnson, C. M. Boney, P. J. Claassen, G. I. Magee, M. L. Pedroncelli, J. T. Spooner, and A. R. Lang, “Ares UV LIF standoff system development and testing,” in Proceedings of the 6th Joint Conference on Standoff Detection for Chemical and Biological Defense, Williamsburg, Virginia, October 2004.

Schroder, K. L.

R. L. Schmitt, K. L. Schroder, M. W. Smith, L. J. Krumel, P. J. Hargis, I. R. Shokair, T. A. Ashlock, J. W. Daniels, J. R. Klarkowski, M. S. Johnson, C. M. Boney, P. J. Claassen, G. I. Magee, M. L. Pedroncelli, J. T. Spooner, and A. R. Lang, “Ares UV LIF standoff system development and testing,” in Proceedings of the 6th Joint Conference on Standoff Detection for Chemical and Biological Defense, Williamsburg, Virginia, October 2004.

Shokair, I. R.

R. L. Schmitt, K. L. Schroder, M. W. Smith, L. J. Krumel, P. J. Hargis, I. R. Shokair, T. A. Ashlock, J. W. Daniels, J. R. Klarkowski, M. S. Johnson, C. M. Boney, P. J. Claassen, G. I. Magee, M. L. Pedroncelli, J. T. Spooner, and A. R. Lang, “Ares UV LIF standoff system development and testing,” in Proceedings of the 6th Joint Conference on Standoff Detection for Chemical and Biological Defense, Williamsburg, Virginia, October 2004.

I. R. Shokair, M. S. Johnson, R. L. Schmitt, and S. Sickafoose, “Concept for maritime near-surface surveillance using water Raman scattering,” (Sandia National Laboratories, 2016).

Shuklin, V. S.

N. P. Romanov and V. S. Shuklin, “Raman scattering cross section of liquid water,” Opt. Spectrosc. 38, 646–648 (1975).

Sickafoose, S.

I. R. Shokair, M. S. Johnson, R. L. Schmitt, and S. Sickafoose, “Concept for maritime near-surface surveillance using water Raman scattering,” (Sandia National Laboratories, 2016).

Slusher, R. B.

Smith, M. W.

R. L. Schmitt, K. L. Schroder, M. W. Smith, L. J. Krumel, P. J. Hargis, I. R. Shokair, T. A. Ashlock, J. W. Daniels, J. R. Klarkowski, M. S. Johnson, C. M. Boney, P. J. Claassen, G. I. Magee, M. L. Pedroncelli, J. T. Spooner, and A. R. Lang, “Ares UV LIF standoff system development and testing,” in Proceedings of the 6th Joint Conference on Standoff Detection for Chemical and Biological Defense, Williamsburg, Virginia, October 2004.

Smith, R. C.

Spooner, J. T.

R. L. Schmitt, K. L. Schroder, M. W. Smith, L. J. Krumel, P. J. Hargis, I. R. Shokair, T. A. Ashlock, J. W. Daniels, J. R. Klarkowski, M. S. Johnson, C. M. Boney, P. J. Claassen, G. I. Magee, M. L. Pedroncelli, J. T. Spooner, and A. R. Lang, “Ares UV LIF standoff system development and testing,” in Proceedings of the 6th Joint Conference on Standoff Detection for Chemical and Biological Defense, Williamsburg, Virginia, October 2004.

Sugihara, S.

S. Sugihara, M. Kishino, and M. Okami, “Contribution of Raman scattering to upward irradiance in the sea,” J. Oceanogr. Soc. Japan 40, 397–404 (1984).
[Crossref]

Visconti, G.

Vodacek, A.

Voss, K. J.

Williamson, C. E.

D. P. Morris, H. Zagarese, C. E. Williamson, E. G. Balseiro, B. R. Hargreaves, B. Modenutti, R. Moeller, and C. Queimalinos, “The attenuation of solar UV radiation in lakes and the role of dissolved organic carbon,” Limnol. Oceanogr. 40, 1381–1391 (1995).
[Crossref]

Xu, X.

Yi, F.

Young, L. A.

C. H. Chang and L. A. Young, “Seawater temperature measurements from Raman spectra,” (Advanced Research Projects Agency, 1972).

Yulmetov, R. N.

Zagarese, H.

D. P. Morris, H. Zagarese, C. E. Williamson, E. G. Balseiro, B. R. Hargreaves, B. Modenutti, R. Moeller, and C. Queimalinos, “The attenuation of solar UV radiation in lakes and the role of dissolved organic carbon,” Limnol. Oceanogr. 40, 1381–1391 (1995).
[Crossref]

AIP Conf. Proc. (1)

W. K. Bischel and G. Black, “Wavelength dependence of Raman scattering cross sections from 200–600  nm,” AIP Conf. Proc. 100, 181–187 (1983).
[Crossref]

Appl. Opt. (10)

G. W. Faris and R. A. Copeland, “Wavelength dependence of the Raman cross section for liquid water,” Appl. Opt. 36, 2686–2688 (1997).
[Crossref]

J. S. Bartlett, K. J. Voss, S. Sathyendranath, and A. Vodacek, “Raman scattering by pure water and seawater,” Appl. Opt. 37, 3324–3332 (1998).
[Crossref]

V. Rizi, M. Iarlori, G. Rocci, and G. Visconti, “Raman lidar observations of cloud liquid water,” Appl. Opt. 43, 6440–6453 (2004).
[Crossref]

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Figures (10)

Fig. 1.
Fig. 1. Illustration of lidar beam incidence, refraction at the air–water interface, and return Raman light. The range is R and s is the path length along the propagation direction in the water.
Fig. 2.
Fig. 2. MODTRAN calculation of solar spectral irradiance. MODTRAN options: “Direct solar Irradiance,” zero zenith angle, zero altitude above sea level, “Mid-latitude Winter” atmospheric model, and 23 km visibility aerosol model. The spectral resolution or width per spectral bin is 50  cm1.
Fig. 3.
Fig. 3. Attenuation coefficient of pure water based on [21]. Also shown are the attenuation coefficient at the corresponding Raman wavelength and the summed coefficients.
Fig. 4.
Fig. 4. Schematic diagram of the Ares UV-LIF lidar system at 355 nm.
Fig. 5.
Fig. 5. Left: The Ares lidar mounted in a trailer with beam aligned for horizontal propagation. Right: horizontal water pipe used for the Raman measurements. The pipe is 152 cm long and 14.0 cm in diameter with fused silica windows on both ends. The ports in the pipe were used for placing a target to set the water depth. The pipe front window is located 70  m from the lidar.
Fig. 6.
Fig. 6. Spectrally integrated N2 Raman signal and range-corrected signal versus range for the Ares lidar. The measured signal is normalized to maximum of unity. Also shown is the ratio of measured counts to range-corrected counts, which is the overlap factor of the lidar LIF channel. These measurements are for horizontal beam propagation in air with a gate delay of 0.05  μs relative to the lidar range and a width of 0.1 μs, corresponding to a width of 15 m of air.
Fig. 7.
Fig. 7. Pure water Raman spectra with target at various locations along the water column. The spectra are normalized to 500 pulses. For all measurements the gate delay is 50  ns with respect to the front window, the width is 100 ns, and the detector gain setting is at 50. The manually measured pulse energy is 30  mJ.
Fig. 8.
Fig. 8. Summed and corrected counts of the spectra in Fig. 7 versus target location. The last location at 152 cm. has no target but is the end of the tube. The dashed curve is the model fit based on an overall attenuation coefficient of β0.26  m1.
Fig. 9.
Fig. 9. Spectra for the ocean water sample at several target locations. All spectra are normalized to 500 pulses. The gate delay is 16  ns, the gate width is 100 ns, and the gain setting is 100. At this gain setting, the detector calibration factor is 4.9 relative to the gain setting of 50 used in Fig. 7 (counts are divided by this factor to get equivalent counts at gain setting of 50). The manually measured laser energy is 7.5  mJ/pulse. After partitioning of the Raman and fluorescence spectra, the summed Raman signal at the maximum target location is 1.7×107 counts for 500 pulses.
Fig. 10.
Fig. 10. Spectra for the San Francisco Bay water sample at several target locations and normalized to 500 pulses. All measurement parameters are the same as in Fig. 9. After partitioning of the Raman and fluorescence spectra, the summed Raman signal is 4.5×106 counts for 500 pulses.

Equations (8)

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dσ(θ,ϕ)dΩ=dσ(θ=90°,ϕ=90°)dΩ{2ρ+(1ρ)(cos2θcos2ϕ+sin2ϕ)1+ρ},
dσ(θ=90°,ϕ=90°)dΩ=Aνs4(νi2νp2)2,
dNR=Ib(t,s)ϵbdσ(θ=180°)dΩNWdsΔΩ,
dER(τ,s)=Pb(t)dtexp[(μ+μR)s]ds(dσdΩNWλbλR)L,
ERTotal=L(dσdΩNWλbλR)1β0ΔTdtPb(t)[exp(βsmin)exp(βsmax)],
smin=[D(2R/c+t)]c/2,smax=[D+W(2R/c+t)]c/2,
ER=(dσdΩNWλbλR)ΔΩηLidarEBeamST,
ESolar=(ΔkdkS(k))ΔΩ2πρSurfaceAFOVT(kR)WηLidar,

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