Abstract

A hybrid method is presented by which Monte Carlo (MC) techniques are combined with an iterative relaxation algorithm to solve the radiative transfer equation in arbitrary one-, two-, or three-dimensional optical environments. The optical environments are first divided into contiguous subregions, or elements. MC techniques are employed to determine the optical response function of each type of element. The elements are combined, and relaxation techniques are used to determine simultaneously the radiance field on the boundary and throughout the interior of the modeled environment. One-dimensional results compare well with a standard radiative transfer model. The light field beneath and adjacent to a long barge is modeled in two dimensions and displayed. Ramifications for underwater video imaging are discussed. The hybrid model is currently capable of providing estimates of the underwater light field needed to expedite inspection of ship hulls and port facilities.

© 2004 Optical Society of America

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References

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  1. A. Tycho, T. M. Jorgensen, H. T. Yura, P. E. Andersen, “Derivation of a Monte Carlo method for modeling heterodyne detection in optical coherence tomography systems,” Appl. Opt. 41, 6676–6691 (2002).
    [CrossRef] [PubMed]
  2. R. J. Pahl, M. A. Shannon, “Analysis of Monte Carlo methods applied to blackbody and lower emissivity cavities,” Appl. Opt. 41, 691–699 (2002).
    [CrossRef] [PubMed]
  3. C. D. Mobley, Light and Water; Radiative Transfer in Natural Waters (Academic, San Diego, Calif., 1994).
  4. C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
    [CrossRef] [PubMed]
  5. P. N. Reinersman, K. L. Carder, “Monte Carlo simulation of the atmospheric point-spread function with an application to correction for the adjacency effect,” Appl. Opt. 34, 4453–4471 (1995).
    [CrossRef] [PubMed]
  6. P. N. Reinersman, K. L. Carder, F. R. Chen, “Satellite-sensor calibration verification with the cloud-shadow method,” Appl. Opt. 37, 5541–5549 (1998).
    [CrossRef]
  7. H. R. Gordon, “Ship perturbation of irradiance measurements at sea. 1: Monte Carlo simulations,” Appl. Opt. 24, 4172–4182 (1985).
    [CrossRef] [PubMed]
  8. K. L. Carder, C. C. Liu, Z. Lee, D. C. English, J. Patten, F. R. Chen, J. E. Ivey, “Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos,” Limnol. Oceanogr. 48, 355–363 (2003).
    [CrossRef]
  9. C. D. Mobley, L. K. Sundman, “Effects of optically shallow bottoms on upwelling radiances: inhomogeneous and sloping bottoms,” Limnol. Oceanogr. 48, 329–336 (2003).
    [CrossRef]
  10. D. S. Burnett, Finite Element Analysis (Addison-Wesley, Reading, Mass., 1987).
  11. C. D. Mobley, Hydrolight 4.0 User’s Guide, 2nd ed. (Sequoia Scientific, Redmond, Wash., 1998).

2003 (2)

K. L. Carder, C. C. Liu, Z. Lee, D. C. English, J. Patten, F. R. Chen, J. E. Ivey, “Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos,” Limnol. Oceanogr. 48, 355–363 (2003).
[CrossRef]

C. D. Mobley, L. K. Sundman, “Effects of optically shallow bottoms on upwelling radiances: inhomogeneous and sloping bottoms,” Limnol. Oceanogr. 48, 329–336 (2003).
[CrossRef]

2002 (2)

1998 (1)

1995 (1)

1993 (1)

1985 (1)

Andersen, P. E.

Burnett, D. S.

D. S. Burnett, Finite Element Analysis (Addison-Wesley, Reading, Mass., 1987).

Carder, K. L.

Chen, F. R.

K. L. Carder, C. C. Liu, Z. Lee, D. C. English, J. Patten, F. R. Chen, J. E. Ivey, “Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos,” Limnol. Oceanogr. 48, 355–363 (2003).
[CrossRef]

P. N. Reinersman, K. L. Carder, F. R. Chen, “Satellite-sensor calibration verification with the cloud-shadow method,” Appl. Opt. 37, 5541–5549 (1998).
[CrossRef]

English, D. C.

K. L. Carder, C. C. Liu, Z. Lee, D. C. English, J. Patten, F. R. Chen, J. E. Ivey, “Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos,” Limnol. Oceanogr. 48, 355–363 (2003).
[CrossRef]

Gentili, B.

Gordon, H. R.

Ivey, J. E.

K. L. Carder, C. C. Liu, Z. Lee, D. C. English, J. Patten, F. R. Chen, J. E. Ivey, “Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos,” Limnol. Oceanogr. 48, 355–363 (2003).
[CrossRef]

Jin, Z.

Jorgensen, T. M.

Kattawar, G. W.

Lee, Z.

K. L. Carder, C. C. Liu, Z. Lee, D. C. English, J. Patten, F. R. Chen, J. E. Ivey, “Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos,” Limnol. Oceanogr. 48, 355–363 (2003).
[CrossRef]

Liu, C. C.

K. L. Carder, C. C. Liu, Z. Lee, D. C. English, J. Patten, F. R. Chen, J. E. Ivey, “Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos,” Limnol. Oceanogr. 48, 355–363 (2003).
[CrossRef]

Mobley, C. D.

C. D. Mobley, L. K. Sundman, “Effects of optically shallow bottoms on upwelling radiances: inhomogeneous and sloping bottoms,” Limnol. Oceanogr. 48, 329–336 (2003).
[CrossRef]

C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
[CrossRef] [PubMed]

C. D. Mobley, Light and Water; Radiative Transfer in Natural Waters (Academic, San Diego, Calif., 1994).

C. D. Mobley, Hydrolight 4.0 User’s Guide, 2nd ed. (Sequoia Scientific, Redmond, Wash., 1998).

Morel, A.

Pahl, R. J.

Patten, J.

K. L. Carder, C. C. Liu, Z. Lee, D. C. English, J. Patten, F. R. Chen, J. E. Ivey, “Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos,” Limnol. Oceanogr. 48, 355–363 (2003).
[CrossRef]

Reinersman, P.

Reinersman, P. N.

Shannon, M. A.

Stamnes, K.

Stavn, R. H.

Sundman, L. K.

C. D. Mobley, L. K. Sundman, “Effects of optically shallow bottoms on upwelling radiances: inhomogeneous and sloping bottoms,” Limnol. Oceanogr. 48, 329–336 (2003).
[CrossRef]

Tycho, A.

Yura, H. T.

Appl. Opt. (6)

Limnol. Oceanogr. (2)

K. L. Carder, C. C. Liu, Z. Lee, D. C. English, J. Patten, F. R. Chen, J. E. Ivey, “Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos,” Limnol. Oceanogr. 48, 355–363 (2003).
[CrossRef]

C. D. Mobley, L. K. Sundman, “Effects of optically shallow bottoms on upwelling radiances: inhomogeneous and sloping bottoms,” Limnol. Oceanogr. 48, 329–336 (2003).
[CrossRef]

Other (3)

D. S. Burnett, Finite Element Analysis (Addison-Wesley, Reading, Mass., 1987).

C. D. Mobley, Hydrolight 4.0 User’s Guide, 2nd ed. (Sequoia Scientific, Redmond, Wash., 1998).

C. D. Mobley, Light and Water; Radiative Transfer in Natural Waters (Academic, San Diego, Calif., 1994).

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Figures (14)

Fig. 1
Fig. 1

E 00, ρ0, ε0; E i i , ρ i , ε i )] is the part of the irradiance emitted from the region at wavelength λ0, point ρ0, and direction ε0 when the region is illuminated by irradiance incident at wavelength λ i , point ρ i , and direction ε i .

Fig. 2
Fig. 2

(a) Three-dimensional cube. Each face is subdivided into 16 grid cells. The directional resolution is 5° × 5°. The input at face 2, cell 4, Θ bin 24, and ϕ bin 68 is symbolized. (b) Two-dimensional plate. This plate represents a Fresnel interface. The reflected (E R ) and transmitted (E T ) irradiance results from input E i .

Fig. 3
Fig. 3

We can model one-dimensional environments using three-dimensional cubes by wrapping outputs from opposite faces of the cube as shown. In this case, the modeled light field varies only in the z direction; outputs in the x and y directions are wrapped.

Fig. 4
Fig. 4

We can model an infinitely long bar of square cross section by wrapping the MC model in the y direction. The spatial grid cells become long strips. An array of bars can be used to simulate two-dimensional problems. To solve one-dimensional problems, the bars can be stacked and the outputs wrapped in the x direction during the iterative phase of the model.

Fig. 5
Fig. 5

Model geometry for solution of the one-dimensional problem with long bars and strips. The output in the x direction (through the vertical sides of the bars) is wrapped in the iterative relaxation.

Fig. 6
Fig. 6

Model geometry for solution of the two-dimensional problem. The output through the vertical edges of the seawater region is wrapped, creating an infinite series of long, parallel barges.

Fig. 7
Fig. 7

Downward irradiance (E d ) versus depth as calculated by the hybrid model and Hydrolight. The difference between the two results, relative to the Hydrolight results, is 0.98% rms for the clear water and 2.36% rms for the turbid water. chl, chlorophyll.

Fig. 8
Fig. 8

Upward irradiance (E u ) versus depth as calculated by the hybrid model and Hydrolight. The difference between the two results, relative to the Hydrolight results, is 1.28% rms for the clear water and 3.63% rms for the turbid water. chl, chlorophyll.

Fig. 9
Fig. 9

Average cosines versus depth as calculated by the hybrid model and Hydrolight. The differences between the two models, from the top plot down, is 2.00%, 1.40%, 1.94%, and 2.48% rms. chl, chlorophyll.

Fig. 10
Fig. 10

Total upward radiance just above the surface in the half-plane perpendicular to the principal plane as calculated by the hybrid model and Hydrolight. The difference between the two models is 7.36% rms for the clear water and 8.94% rms for the turbid water. Chl, chlorophyll.

Fig. 11
Fig. 11

Normalized log10 E d (532 nm) and chlorophyll of 0.3 mg/m3 for downward irradiance as calculated by the hybrid model. Vertical and horizontal axes are not plotted to the same scale.

Fig. 12
Fig. 12

Normalized log10 E u (532 nm) and chlorophyll of 0.3 mg/m3 for upward irradiance as calculated by the hybrid model. Vertical and horizontal axes are not plotted to the same scale.

Fig. 13
Fig. 13

Normalized log10 E d (532 nm) and chlorophyll of 5.0 mg/m3 for downward irradiance as calculated by the hybrid model. Vertical and horizontal axes are not plotted to the same scale.

Fig. 14
Fig. 14

Normalized log10 E u (532 nm) and chlorophyll of 5.0 mg/m3 for upward irradiance as calculated by the hybrid model. Vertical and horizontal axes are not plotted to the same scale.

Tables (1)

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Table 1 Optical Parameters for Two Water Types, with a Wavelength of 532 nm

Equations (2)

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E0ρ0, ε0; Eiρi, εi=Rρ0, ε0; ρi, εiEiρi, εi.
E0ρ0, ε0dAρ0dΩε0= Rρ0, ε0; ρi, εi×Eiρi, εidAdΩ.

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