Abstract

Plane-to-point transformations are used to develop a version of the Hydrolight computer program with which to compute the spatial dependence of the irradiance and the scalar irradiance of the light field away from an isotropic point source deep within a spatially uniform ocean. The transformations are also used to derive analytic approximations for determining the diffuse attenuation coefficient and the mean cosine of the radiance far from an isotropic point source. Approximations for determining the asymptotic diffuse attenuation coefficient from measurements at only two distances far from the source are derived and numerically tested with the modified version of the Hydrolight computer program. New spatial integrals of the outward irradiance are also derived that provide a different way for correlating the inherent optical properties of seawater.

© 2000 Optical Society of America

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References

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  1. L. E. Mertens, F. S. Replogle, “Use of the point spread function for analysis of imaging systems in water,” J. Opt. Soc. Am. 67, 1105–1117 (1977).
    [CrossRef]
  2. H. R. Gordon, “Equivalence of the point and beam spread functions of scattering media: a formal demonstration,” Appl. Opt. 33, 1120–1122 (1994).
    [CrossRef] [PubMed]
  3. R. A. Maffione, J. S. Jaffe, “The average cosine due to an isotropic light source in the ocean,” J. Geophys. Res. 100, 13179–13192 (1995).
    [CrossRef]
  4. R. A. Maffione, K. J. Voss, R. C. Honey, “Measurement of the spectral absorption coefficient in the ocean with an isotropic source,” Appl. Opt. 32, 3273–3279 (1993).
    [CrossRef] [PubMed]
  5. K. J. Voss, A. L. Chapin, “Measurement of the point spread function in the ocean,” Appl. Opt. 29, 3638–3642 (1990).
    [CrossRef] [PubMed]
  6. C. D. Mobley, Hydrolight, available from Sequoia Scientific, Inc., Sequoia Scientific, Inc., Westpark Technical Center, 15317 NE 90th St., Redmond, Wash. 98052.
  7. R. A. Leathers, C. S. Roesler, N. J. McCormick, “Ocean inherent optical property determination from in-water light field measurements,” Appl. Opt. 38, 5096–5103 (1999).
    [CrossRef]
  8. E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport (Wiley, New York, 1984), p. 31.
  9. W. S. Pegau, J. S. Cleveland, W. Doss, C. D. Kennedy, R. A. Maffione, J. L. Mueller, R. Stone, C. C. Trees, A. D. Weidemann, W. H. Wells, J. R. V. Zaneveld, “A comparison of methods for the measurement of the absorption coefficient in natural waters,” J. Geophys. Res. 100, 13201–13220 (1995).
    [CrossRef]
  10. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp. 87–91 and Appendices C and H.
  11. R. C. Erdmann, C. E. Siewert, “Green’s functions for the one-speed transport equation in spherical geometry,” J. Math. Phys. (N.Y.) 9, 81–89 (1968).
    [CrossRef]
  12. N. J. McCormick, “Mathematical models for the mean cosine of irradiance and the diffuse attenuation coefficient,” Limnol. Oceanogr. 40, 1013–1018 (1995).
    [CrossRef]
  13. C. C. Grosjean, “On a new approximate one-velocity theory of multiple scattering in infinite homogeneous media,” Nuovo Cimento X 4, 582–594 (1956).
    [CrossRef]
  14. T. J. Petzold, “Volume scattering functions for selected ocean waters,” (Scripps Institution of Oceanography, San Diego, Calif., 1972).
  15. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1970), p. 923.
  16. N. J. McCormick, I. Kuščer, “Singular eigenfunction expansions in neutron transport theory,” in Advances in Nuclear Science and Technology, E. J. Henley, J. Lewins, eds. (Academic, New York, 1973), Vol. 7, pp. 181–282.
  17. I. Kuščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transp. Theory Stat. Phys. 20, 351–381 (1991).
    [CrossRef]
  18. N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
    [CrossRef]
  19. P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
    [CrossRef]

1999 (1)

1995 (3)

W. S. Pegau, J. S. Cleveland, W. Doss, C. D. Kennedy, R. A. Maffione, J. L. Mueller, R. Stone, C. C. Trees, A. D. Weidemann, W. H. Wells, J. R. V. Zaneveld, “A comparison of methods for the measurement of the absorption coefficient in natural waters,” J. Geophys. Res. 100, 13201–13220 (1995).
[CrossRef]

N. J. McCormick, “Mathematical models for the mean cosine of irradiance and the diffuse attenuation coefficient,” Limnol. Oceanogr. 40, 1013–1018 (1995).
[CrossRef]

R. A. Maffione, J. S. Jaffe, “The average cosine due to an isotropic light source in the ocean,” J. Geophys. Res. 100, 13179–13192 (1995).
[CrossRef]

1994 (2)

H. R. Gordon, “Equivalence of the point and beam spread functions of scattering media: a formal demonstration,” Appl. Opt. 33, 1120–1122 (1994).
[CrossRef] [PubMed]

P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
[CrossRef]

1993 (1)

1992 (1)

N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
[CrossRef]

1991 (1)

I. Kuščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transp. Theory Stat. Phys. 20, 351–381 (1991).
[CrossRef]

1990 (1)

1977 (1)

1968 (1)

R. C. Erdmann, C. E. Siewert, “Green’s functions for the one-speed transport equation in spherical geometry,” J. Math. Phys. (N.Y.) 9, 81–89 (1968).
[CrossRef]

1956 (1)

C. C. Grosjean, “On a new approximate one-velocity theory of multiple scattering in infinite homogeneous media,” Nuovo Cimento X 4, 582–594 (1956).
[CrossRef]

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp. 87–91 and Appendices C and H.

Chapin, A. L.

Cleveland, J. S.

W. S. Pegau, J. S. Cleveland, W. Doss, C. D. Kennedy, R. A. Maffione, J. L. Mueller, R. Stone, C. C. Trees, A. D. Weidemann, W. H. Wells, J. R. V. Zaneveld, “A comparison of methods for the measurement of the absorption coefficient in natural waters,” J. Geophys. Res. 100, 13201–13220 (1995).
[CrossRef]

Doss, W.

W. S. Pegau, J. S. Cleveland, W. Doss, C. D. Kennedy, R. A. Maffione, J. L. Mueller, R. Stone, C. C. Trees, A. D. Weidemann, W. H. Wells, J. R. V. Zaneveld, “A comparison of methods for the measurement of the absorption coefficient in natural waters,” J. Geophys. Res. 100, 13201–13220 (1995).
[CrossRef]

Erdmann, R. C.

R. C. Erdmann, C. E. Siewert, “Green’s functions for the one-speed transport equation in spherical geometry,” J. Math. Phys. (N.Y.) 9, 81–89 (1968).
[CrossRef]

Francisco, P. W.

P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
[CrossRef]

Gordon, H. R.

Grosjean, C. C.

C. C. Grosjean, “On a new approximate one-velocity theory of multiple scattering in infinite homogeneous media,” Nuovo Cimento X 4, 582–594 (1956).
[CrossRef]

Honey, R. C.

Jaffe, J. S.

R. A. Maffione, J. S. Jaffe, “The average cosine due to an isotropic light source in the ocean,” J. Geophys. Res. 100, 13179–13192 (1995).
[CrossRef]

Kennedy, C. D.

W. S. Pegau, J. S. Cleveland, W. Doss, C. D. Kennedy, R. A. Maffione, J. L. Mueller, R. Stone, C. C. Trees, A. D. Weidemann, W. H. Wells, J. R. V. Zaneveld, “A comparison of methods for the measurement of the absorption coefficient in natural waters,” J. Geophys. Res. 100, 13201–13220 (1995).
[CrossRef]

Kušcer, I.

I. Kuščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transp. Theory Stat. Phys. 20, 351–381 (1991).
[CrossRef]

N. J. McCormick, I. Kuščer, “Singular eigenfunction expansions in neutron transport theory,” in Advances in Nuclear Science and Technology, E. J. Henley, J. Lewins, eds. (Academic, New York, 1973), Vol. 7, pp. 181–282.

Leathers, R. A.

Lewis, E. E.

E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport (Wiley, New York, 1984), p. 31.

Maffione, R. A.

W. S. Pegau, J. S. Cleveland, W. Doss, C. D. Kennedy, R. A. Maffione, J. L. Mueller, R. Stone, C. C. Trees, A. D. Weidemann, W. H. Wells, J. R. V. Zaneveld, “A comparison of methods for the measurement of the absorption coefficient in natural waters,” J. Geophys. Res. 100, 13201–13220 (1995).
[CrossRef]

R. A. Maffione, J. S. Jaffe, “The average cosine due to an isotropic light source in the ocean,” J. Geophys. Res. 100, 13179–13192 (1995).
[CrossRef]

R. A. Maffione, K. J. Voss, R. C. Honey, “Measurement of the spectral absorption coefficient in the ocean with an isotropic source,” Appl. Opt. 32, 3273–3279 (1993).
[CrossRef] [PubMed]

McCormick, N. J.

R. A. Leathers, C. S. Roesler, N. J. McCormick, “Ocean inherent optical property determination from in-water light field measurements,” Appl. Opt. 38, 5096–5103 (1999).
[CrossRef]

N. J. McCormick, “Mathematical models for the mean cosine of irradiance and the diffuse attenuation coefficient,” Limnol. Oceanogr. 40, 1013–1018 (1995).
[CrossRef]

P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
[CrossRef]

N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
[CrossRef]

I. Kuščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transp. Theory Stat. Phys. 20, 351–381 (1991).
[CrossRef]

N. J. McCormick, I. Kuščer, “Singular eigenfunction expansions in neutron transport theory,” in Advances in Nuclear Science and Technology, E. J. Henley, J. Lewins, eds. (Academic, New York, 1973), Vol. 7, pp. 181–282.

Mertens, L. E.

Miller, W. F.

E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport (Wiley, New York, 1984), p. 31.

Mueller, J. L.

W. S. Pegau, J. S. Cleveland, W. Doss, C. D. Kennedy, R. A. Maffione, J. L. Mueller, R. Stone, C. C. Trees, A. D. Weidemann, W. H. Wells, J. R. V. Zaneveld, “A comparison of methods for the measurement of the absorption coefficient in natural waters,” J. Geophys. Res. 100, 13201–13220 (1995).
[CrossRef]

Pegau, W. S.

W. S. Pegau, J. S. Cleveland, W. Doss, C. D. Kennedy, R. A. Maffione, J. L. Mueller, R. Stone, C. C. Trees, A. D. Weidemann, W. H. Wells, J. R. V. Zaneveld, “A comparison of methods for the measurement of the absorption coefficient in natural waters,” J. Geophys. Res. 100, 13201–13220 (1995).
[CrossRef]

Petzold, T. J.

T. J. Petzold, “Volume scattering functions for selected ocean waters,” (Scripps Institution of Oceanography, San Diego, Calif., 1972).

Replogle, F. S.

Roesler, C. S.

Siewert, C. E.

R. C. Erdmann, C. E. Siewert, “Green’s functions for the one-speed transport equation in spherical geometry,” J. Math. Phys. (N.Y.) 9, 81–89 (1968).
[CrossRef]

Stone, R.

W. S. Pegau, J. S. Cleveland, W. Doss, C. D. Kennedy, R. A. Maffione, J. L. Mueller, R. Stone, C. C. Trees, A. D. Weidemann, W. H. Wells, J. R. V. Zaneveld, “A comparison of methods for the measurement of the absorption coefficient in natural waters,” J. Geophys. Res. 100, 13201–13220 (1995).
[CrossRef]

Trees, C. C.

W. S. Pegau, J. S. Cleveland, W. Doss, C. D. Kennedy, R. A. Maffione, J. L. Mueller, R. Stone, C. C. Trees, A. D. Weidemann, W. H. Wells, J. R. V. Zaneveld, “A comparison of methods for the measurement of the absorption coefficient in natural waters,” J. Geophys. Res. 100, 13201–13220 (1995).
[CrossRef]

Voss, K. J.

Weidemann, A. D.

W. S. Pegau, J. S. Cleveland, W. Doss, C. D. Kennedy, R. A. Maffione, J. L. Mueller, R. Stone, C. C. Trees, A. D. Weidemann, W. H. Wells, J. R. V. Zaneveld, “A comparison of methods for the measurement of the absorption coefficient in natural waters,” J. Geophys. Res. 100, 13201–13220 (1995).
[CrossRef]

Wells, W. H.

W. S. Pegau, J. S. Cleveland, W. Doss, C. D. Kennedy, R. A. Maffione, J. L. Mueller, R. Stone, C. C. Trees, A. D. Weidemann, W. H. Wells, J. R. V. Zaneveld, “A comparison of methods for the measurement of the absorption coefficient in natural waters,” J. Geophys. Res. 100, 13201–13220 (1995).
[CrossRef]

Zaneveld, J. R. V.

W. S. Pegau, J. S. Cleveland, W. Doss, C. D. Kennedy, R. A. Maffione, J. L. Mueller, R. Stone, C. C. Trees, A. D. Weidemann, W. H. Wells, J. R. V. Zaneveld, “A comparison of methods for the measurement of the absorption coefficient in natural waters,” J. Geophys. Res. 100, 13201–13220 (1995).
[CrossRef]

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp. 87–91 and Appendices C and H.

Appl. Opt. (4)

J. Geophys. Res. (2)

W. S. Pegau, J. S. Cleveland, W. Doss, C. D. Kennedy, R. A. Maffione, J. L. Mueller, R. Stone, C. C. Trees, A. D. Weidemann, W. H. Wells, J. R. V. Zaneveld, “A comparison of methods for the measurement of the absorption coefficient in natural waters,” J. Geophys. Res. 100, 13201–13220 (1995).
[CrossRef]

R. A. Maffione, J. S. Jaffe, “The average cosine due to an isotropic light source in the ocean,” J. Geophys. Res. 100, 13179–13192 (1995).
[CrossRef]

J. Math. Phys. (N.Y.) (1)

R. C. Erdmann, C. E. Siewert, “Green’s functions for the one-speed transport equation in spherical geometry,” J. Math. Phys. (N.Y.) 9, 81–89 (1968).
[CrossRef]

J. Opt. Soc. Am. (1)

Limnol. Oceanogr. (3)

N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
[CrossRef]

P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
[CrossRef]

N. J. McCormick, “Mathematical models for the mean cosine of irradiance and the diffuse attenuation coefficient,” Limnol. Oceanogr. 40, 1013–1018 (1995).
[CrossRef]

Nuovo Cimento X (1)

C. C. Grosjean, “On a new approximate one-velocity theory of multiple scattering in infinite homogeneous media,” Nuovo Cimento X 4, 582–594 (1956).
[CrossRef]

Transp. Theory Stat. Phys. (1)

I. Kuščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transp. Theory Stat. Phys. 20, 351–381 (1991).
[CrossRef]

Other (6)

T. J. Petzold, “Volume scattering functions for selected ocean waters,” (Scripps Institution of Oceanography, San Diego, Calif., 1972).

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1970), p. 923.

N. J. McCormick, I. Kuščer, “Singular eigenfunction expansions in neutron transport theory,” in Advances in Nuclear Science and Technology, E. J. Henley, J. Lewins, eds. (Academic, New York, 1973), Vol. 7, pp. 181–282.

C. D. Mobley, Hydrolight, available from Sequoia Scientific, Inc., Sequoia Scientific, Inc., Westpark Technical Center, 15317 NE 90th St., Redmond, Wash. 98052.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp. 87–91 and Appendices C and H.

E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport (Wiley, New York, 1984), p. 31.

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Figures (2)

Fig. 1
Fig. 1

Diffuse attenuation coefficient K(cr) versus dimensionless radial distance cr from Hydrolight (△) and from approximation (18) (●) for ω = 0.3 (upper set of curves), ω = 0.6, and ω = 0.9 (lower set).

Fig. 2
Fig. 2

Mean cosine μ̅(cr) versus dimensionless radial distance cr from Hydrolight (△) and from approximation (26) (●) for ω = 0.3 (μ̅ as = 0.8900), ω = 0.6 (μ̅ as = 0.7787), and ω = 0.9 (μ̅ as = 0.5217).

Tables (2)

Tables Icon

Table 1 Symmetry Test for the Point Source Version of Hydrolight

Tables Icon

Table 2 Comparison of K from Hydrolight to K T, the Transcendental Approximation (21), and K Q, the Quadratic Approximation (23), Obtained for RS ≤ r ≤ RL

Equations (51)

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

μ Lr, μr+1-μ2rLr, μμ+cLr, μ=b -11 β˜cos ΘLr, μdμ+Φsδr8π2r2 δμ-1, =b2n=02n+1fnPnμ-11 PnμLr, μdμ+Φsδr8π2r2 δμ-1,  r0,
μr2r2Lr, μr+μ1-μ2Lr, μr=-cLr, μ+b2n=02n+1fnPnμ×-11 PnμLr, μdμ+Φsδr8π2r2 δμ-1,  r0.
E0r=2π -11 Lr, μdμ
Er=2π -11 μLr, μdμ
limr0 4πr2E0r=limr0 4πr2Er=Φs.
Kr=-1ErdErdr.
μ¯r=Er/E0r,
a=μ¯rKr-2r
K=limz-1EplzdEplzdz,
K˜r-1r2Erdr2Erdr=Kr-2r.
a=lnΦs/4π-lnR2ER0Rμ¯r-1dr-1.
alnΦs/4π-lnR2ERR-1
E0r=-12πrdE0,plzdzz=r,
Er=-12πddzEplzzz=r.
E0r=Φsc4πrσ+exp-cr/ννNνdν,  r>0,
Er=Φsa4πrσ+1+νcrexp-cr/νNνdν,  r>0,
K=c/ν1.
KrK+1r2+Kr1+Kr+Oexp-αcr,  cr  1,
K˜rK2r1+Kr+Oexp-αcr,  cr  1.
E+r=2π 01 μLr, μdμ
KRL-RS-ln1+KRL1+KRS-lnRS2E+RSRL2E+RL0, cRS  1.
1K1+1Kr-1r2E+rdr2E+rdr1,  cr  1,
1K2+RS+RL21K+RL2-RS24RL2E+RL+RS2E+RSRL2E+RL-RS2E+RS0, cRS  1.
μ¯r=ErE0r=acσ+1+νcrexp-cr/νNνdνσ+exp-cr/ννNνdν-1.
μ¯rac1+ν1crexp-cr/ν1Nν1+1+ν2crexp-cr/ν2Nν2×exp-cr/ν1ν1Nν1+exp-cr/ν2ν2Nν2-1.
μ¯ra/K-m1 exp-αcr+,  large r,
m1=aν1Nν1cNν2ν1ν2-1>0.
μ¯rk0+k1 exp-k2r,  small r,
RHS=-c-bfkLn,k+Φsδn,0,
Ln,k=8π20-11 rn+2PkμLr, μdrdμ
μPkμ=2k+1-1k+1Pk+1μ+kPk-1μ,
1-μ2Pkμμ=kk+12k+1Pk-1μ-Pk+1μ.
k+1k-nLn-1,k+1-kn+k+1Ln-1,k-1+2k+1c-bfkLn,k=2k+1Φsδn,0.
0 E0r4πr2dr=L0,0=Φsa-1,
0 rEr4πr2dr=L1,1=L0,0c-bf1-1=Φsac-bf1-1.
0 r2E0r4πr2dr=L2,0=2L1,1/a=2Φsa2c-bf1-1.
r2=0 r2E0r4πr2dr0 E0r4πr2dr=2ac-bf1.
0 r3Er4πr2dr=L3,1=4L2,2+5L2,03c-bf1=2Φs4a+5c-bf23a2c-bf12c-bf2.
0 r3Er4πr2dr0 rEr4πr2dr=24a+5c-bf23ac-bf1c-bf2.
0 rEr4πr2dr0 rE+r4πr2dr, 0 r3Er4πr2dr0 r3E+r4πr2dr,
0 r3E+r4πr2dr0 rE+r4πr2dr24a+5c-bf23ac-bf1c-bf2,
Luncr, μ=Φs8π2r2exp-crδμ-1.
Lplz, μ=j=1J Cνjϕνj, μexp-cz/νj+01 Cνϕν, μexp-cz/νdν+j=1J C-νjϕ-νj, μexpcz/νj+01 C-νϕ-ν, μexpcz/νdν.
-11 ϕν, μdμ=1,  all ν,
-11 μϕν, μdμ=ν1-ω,  all ν,
Lplz, μ=j=1J Cνjϕνj, μexp-cz/νj+01 Cνϕν, μexp-cz/νdν,  z0,
Cνj=4πNνj-1,  Cν=4πNν-1.
E0,plz=j=1J2Nνj-1 exp-cz/νj+012Nν-1 exp-cz/νdν,  z>0.
Eplz=1-ωj=1J2Nνj-1νj exp-cz/νj+012Nν-1 ν exp-cz/νdν,  z>0.
E0,plz=σ+2Nν-1 exp-cz/νdν,  z>0,
Eplz=1-ωσ+2Nν-1 ν exp-cz/νdν, z>0.

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