## Fast and accurate model of underwater scalar irradiance for stratified Case 2 waters

Optics Express, Vol. 14, Issue 5, pp. 1703-1719 (2006)

http://dx.doi.org/10.1364/OE.14.001703

Acrobat PDF (437 KB)

### Abstract

This paper is devoted to the derivation of a fast and accurate model of scalar irradiance for stratified Case 2 waters. Five strategies are formulated and employed in the new model, including (1) reallocating the sky radiance, (2) approximating the influence of the air-water interface, (3) constructing a look-up table of average cosine based on the single-scattering albedo and the backscatter fraction, (4) calculating the phase function of surrogate particles in Case 2 waters, and (5) using the average cosine as an index to cope with stratified waters. A comprehensive model-to-model comparison shows that the new model runs more than 1,400 times faster than the commercially-available Hydrolight model, while it limits the percentage error to 2.03% and the maximum error to less than 6.81%. This new model can be used interactively in models of the oceanic system, such as biogeochemical models or the heat budget part of global circulation models.

© 2006 Optical Society of America

## 1. Introduction

*E*

_{0}(

*z*) is a prerequisite for modeling hydrologic systems, such as the photosynthesis process in biogeochemical models or the heat budget part of ocean circulation models [2

2. C.-C. Liu, K. L. Carder, R. L. Miller, and J. E. Ivey, “Fast and accurate model of underwater scalar irradiance,” Appl. Opt. **41**, 4962–4974 (2002). [CrossRef] [PubMed]

3. A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. **22**, 709–722 (1977). [CrossRef]

7. A. Albert and C. D. Mobley, “An analytical model for subsurface irradiance and remote sensing reflectance in deep and shallow case-2 waters,” Opt. Express **11**, 2873–2890 (2003). [CrossRef] [PubMed]

2. C.-C. Liu, K. L. Carder, R. L. Miller, and J. E. Ivey, “Fast and accurate model of underwater scalar irradiance,” Appl. Opt. **41**, 4962–4974 (2002). [CrossRef] [PubMed]

## 2. Radiative transfer in aquatic environments

11. K. Stamnes, “The theory of multiple scattering of radiation in plane parallel atmospheres,” Rev. Geophys. **24**, 299–310 (1986). [CrossRef]

*L*through a distance

*r*along the direction

*cL*and gain due to scattering

*L*

_{*}and internal emission

*c*is the beam attenuation coefficient. The radiative energy transfers within the water body at a constant speed

*v*that equals the speed of light divided by the index of refraction. Therefore, the total rate of change along

*r*is

*v*is the unit vector along the path

*r*. Note that the time scale for reaching the steady state in studies of hydrologic optics is generally much shorter than the sampling time, therefore, the time-independent assumption is valid and the term of partial derivative with

*t*is dropped.

*L*

_{*}can be further divided into inelastic scattering

*β*describes the probability of a photon scattered to the direction

*S*. Liu

*et al*. [2

2. C.-C. Liu, K. L. Carder, R. L. Miller, and J. E. Ivey, “Fast and accurate model of underwater scalar irradiance,” Appl. Opt. **41**, 4962–4974 (2002). [CrossRef] [PubMed]

12. G. N. Plass and G. W. Kattawar, “Monte Carlo calculations of radiative transfer in the Earth’s atmosphere-ocean system. I. Flux in the atmosphere and ocean,” J. Phys. Oceanogr. **2**, 139–145 (1972). [CrossRef]

13. C.-C. Liu and J. Woods, “Prediction of ocean colour: Monte-Carlo simulation applied to a virtual ecosystem based on the Lagrangian Ensemble method,” Int. J. Remote Sens. **25**, 921–936 (2004). [CrossRef]

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

*z*. The general form of RTE, therefore, can be simplified as an integro-differential equation

## 3. Theory of approximation

*et al*. [2

**41**, 4962–4974 (2002). [CrossRef] [PubMed]

*E*

_{0}(

*z*) without losing accuracy compared with Hydrolight. Their model runs more than fourteen thousand times faster than the full Hydrolight code, while limiting the percentage error to 2.20% and the maximum error to less than 4.78%. However, their model was limited to (1) vertically homogeneous water columns, (2) Case 1 waters or Case 2 waters that happen to be gelbstoff rich, (3) spectral ranges from 400 to 700 nm, and (4) chlorophyll concentration (

*Chl*) from 0 to 10.0 mg∙m

^{-3}. This paper modifies one strategy described in Liu

*et al*. [2

**41**, 4962–4974 (2002). [CrossRef] [PubMed]

*E*

_{0}(

*z*) that can be applied to the stratified Case 2 waters. Details of these strategies are given as follows.

### 3.1 Reallocation of sky radiance

*θ;ϕ*) and interact with the bodies of water. Because the quantity of concern,

*E*

_{0}(

*z*), is computed from integrals of the radiance over the entire sphere Ξ,

*z*= 0

^{+}) from all possible direction (

*θ;ϕ*) can be reallocated to the plane of the Sun (

*θ;ϕ*

_{0}) and the same integrated result of

*E*

_{0}(

*z*) still can be obtained. In other words, the planar irradiance incident on the surface can be expressed as

*E*

_{0}(

*z*) can be quickly calculated by summing the contribution from each light source in the plane of the Sun (

*θ;ϕ*

_{0}) by

*E̅*

_{0}(

*z;θ;ϕ*

_{0}) is the profile of

*E*

_{0}obtained by placing in a black sky a unit radiance in (

*θ;ϕ*

_{0}). As long as the distribution of incident radiance

*L*(0

^{+};

*θ;ϕ*) is given and

*E*

_{0}(

*z;θ;ϕ*

_{0}) is calculated and stored in advance, solving

*E*

_{0}(

*z*) is simply a procedure of multiplication and summation that can be done with very high efficiency [2

**41**, 4962–4974 (2002). [CrossRef] [PubMed]

### 3.2 Approximation of the influence of the air-water interface

*E*

_{0}(

*z*) can be regarded as redistributing the radiance field after the beam of light penetrates the interface. To consider this effect of redistribution, the profile of

*Ē*

_{0}(

*z;θ;ϕ*

_{0}) can be scaled by multiplying a corrective factor for wind speed

*CW*that is defined as

*Ē*

_{0}(0

^{-};

*θ;ϕ*

_{0})|

_{Vwind}and

*E*

_{0}(0

^{-};

*θ;ϕ*

_{0})|

_{V'wind}are the subsurface values of

*E*

_{0}obtained by placing in a black sky a unit light source in (

*θ;ϕ*

_{0}) under the condition of surface wind speed

*V*

_{wind}and

*V́*

_{wind}, respectively. Given

*CW*(

*θ;ϕ*

_{0})|

_{Vwind→V'wind}, the vertical profile of

*E*

_{0}(

*z*) calculated at wind speed

*V*

_{wind}(Eq. 8) can be easily extended to

*E*

_{0}(

*z*) at wind speed

*V′*

_{wind}by

*CW*is influenced by water constituents as well. Liu

*et al*. [2

**41**, 4962–4974 (2002). [CrossRef] [PubMed]

*CW*is only a weak function of water constituents. It is therefore feasible to construct a look-up table (LUT) based on the situation of clear water for quick reference to the

*CW*. Liu

*et al*. [2

**41**, 4962–4974 (2002). [CrossRef] [PubMed]

*CW*at 31 fixed wavelength from 400 nm to 700 nm at steps of 10 nm. To have a flexibility on the wavelength, the main variables of the new LUT are selected as

*ω*

_{0}(0.01 – 0.99),

*V*

_{wind}(0.0 – 15.0) and

*θ*(0.0 – 90.0). A standard volume scattering phase function for water molecules is obtained from the analytic Fournier-Forand phase function [20] by giving the backscatter fraction (

*BF*) a value of 0.5. A section of the LUT is denoted as LUT

_{CW}and given in Table 1. Note that the table is for pure water with an assumed absorption thus that

*ω*

_{0}can be varied from 0.01 to 0.99.

### 3.3 Look-up table of average cosine based on *ω*_{0} and BF

*E*

_{0}(

*z*) is the existence of a proper mathematical model that is able to approximate the vertical profile of the average cosine

*μ̄*(

*z*) with a set of only five parameters (

*B*

_{0},

*B*

_{1},

*P*,

*B*

_{2},

*Q*)

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

*ζ*=

*cz*and

*z*) is

*et al*. [2

**41**, 4962–4974 (2002). [CrossRef] [PubMed]

*B*

_{0},

*B*

_{1},

*P*,

*B*

_{2},

*Q*) from the given IOPs. With this LUT, the net diffuse attenuation coefficient

*K*

_{NET}(

*z*) can be calculated from

*μ̄*(

*z*) by use of Gershun’s equation [22]

*K*

_{NET}(

*z*) is

*E*

_{d}and

*E*

_{u}are the downwelling and upwelling planar irradiance, respectively. Therefore,

*K*

_{NET}(

*z*) gives us the profile of

*E*(

*z*) that can be converted to

*E*

_{0}(

*z*) using the definition of

*μ̄*(

*z*) (Eq. 12).

**41**, 4962–4974 (2002). [CrossRef] [PubMed]

*Chl*,

*λ*and the backscatter fraction (

*BF*) as the main variables. As a result, the model of Liu

*et al*. [2

**41**, 4962–4974 (2002). [CrossRef] [PubMed]

*Chl*is limited to the range from 0 to 10.0 mg∙m

^{-3}, and

*λ*is limited to the range from 400 to 700 nm as well. Considering the fact that all IOPs can be derived from two principal IOPs: the absorption coefficient

*a*and the phase scattering function

*β*(

*ψ*) [23

23. V. I. Haltrin, ”Chlorophyll-Based Model of Seawater Optical Properties,“ Appl. Opt. **38**, 6826–6832 (1999). [CrossRef]

*ψ*is the scattering angle, the two principal IOPs should be selected as the main variables in the LUT. However,

*β*(

*ψ*) describes the probability of scattering in the

*ψ*direction, which varies from a symmetric distribution (molecular scale) to a heavily peak distribution in the forward direction (large particle). To consider a large range of the distribution of

*β*(

*ψ*), it is necessary to find an accurate and efficient way to parameterize

*β*(

*ψ*). Fournier and Forand (FF) [20] proposed an analytical phase function for ocean water, which parameterized

*β*(

*ψ*) as a function of the backscatter fraction (

*BF*), defined as the backscattering coefficient

*b*

_{b}divided by the scattering coefficient

*b*. A large range in

*β*(

*ψ*) can be obtained by varying the value of

*BF*. For example,

*BF*=0.0183 provides a very good fit to the Petzold’s average particle phase scattering function, while

*BF*=0.5 yields the pure water phase function. Mobley

*el al*. [24

24. C. D. Mobley, L. K. Sundman, and E. Boss, ”Phase function effects on oceanic light fields,“ Appl. Opt. **41**, 1035–1050 (2002). [CrossRef] [PubMed]

*BF*. Therefore,

*BF*is a good choice for parameterizing

*β*(

*ψ*).

*et al*. [2

**41**, 4962–4974 (2002). [CrossRef] [PubMed]

*BF*(0.0001 – 0.5) and

*ω*

_{0}(0.01 – 0.99) for quick reference to a set of parameters (

*B*

_{0},

*B*

_{1},

*P*,

*B*

_{2},

*Q*) used by the McCormick five-parameter model [21

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

*μ̄*(

*ζ*). The single-scattering albedo

*ω*

_{0}is defined as

*b*divided by

*c*. A section of the LUT is denoted as LUT

_{μ}and given in Table 2, where the Pearson correlation coefficient

*PCC*shows how good the McCormick five-parameter model fits the vertical profiles of the average cosine. Throughout the entire LUT, the average value of

*PCC*is 0.999577 and the lowest value is 0.980118.

*ω*

_{0}and

*BF*

_{P}can be calculated. The bulk property of

*BF*is given by

*BF*

_{w}=0.5. Referencing to the new LUT,

*ω*

_{0}and

*BF*

_{P}give us a profile of

*μ̄*

_{p}(

*ζ*), while

*ω*

_{0}and

*BF*give us another profile of

*μ̄*′(

*ζ*) that can be regarded as the value for a pure medium with a backscatter fraction

*BF*. However, neither

*μ̄*

_{p}(

*ζ*) nor

*μ̄*′(

*ζ*) represents the actual profile of

*μ̄*(

*ζ*), because both particles and water molecules contribute to the effect of scattering. The earlier work [2

**41**, 4962–4974 (2002). [CrossRef] [PubMed]

*μ̄*

_{p}(

*ζ*) and

*μ̄*(

*ζ*). Figure 1 illustrates that the assumption of a constant offset only held at depths where

*μ̄*(

*ζ*) has reached its asymptotic solution. To give a better approximation of

*μ̄*(

*ζ*) from

*μ̄*

_{p}(

*ζ*) and

*μ̄*′(

*ζ*), this research proposes a new approach to describe the vertical profile of offset Δ(

*ζ*). The offset near the surface is given by

*μ̄*(

*ζ*) can be estimated from

*μ̄*

_{p}(

*ζ*) by adding the offset Δ(

*ζ*), i.e.,

*ζ*) is given by

### 3.4 Phase function of surrogate particles in Case 2 waters

*Chl*and scatter light like large particles), CDOM (assumed to absorb light but not scatter it) and small particles. Note the component of small particles represents minerals in waters that can both absorb and scatter light. The amount of each component does not need to be controlled by

*Chl*and the backscattering fraction for large particles

*BF*

_{l}and small particles

*BF*

_{s}do not need to be constant values either. To extend our approach to the general situation of Case 2 waters, it is equivalent to find a surrogate particle with backscattering fraction

*BF*

_{p}that is able to exhibit the same optical properties as those two types of particles presented. The normalized phase scattering function

*β̃*[1] for the four-component Case 2 waters is

*β̃*

_{l}(

*ψ,λ*) and

*β̃*

_{s}(

*ψ,λ*) can be obtained from the analytic Fournier-Forand phase function [20] by giving the values of

*BF*

_{l}and

*BF*

_{s}, respectively. Assuming a surrogate particle implies that

*BF*

_{p}is the value that the resulting phase function gives the best fit to

*β̃*

_{p}(

*ψ,λ*).

*BF*

_{p}based on the technique of optimization is illustrated in Fig. 2. A set of

*β̃*

_{p}(

*ψ,λ*) for ocean water is obtained by varying

*BF*

_{p}from 0.0001 to 0.5 (dashed lines). The thick solid line gives the corresponding

*β̃*

_{p}(

*ψ,λ*) of a four-component Case 2 waters, with which

*BF*

_{l}=0.001,

*BF*

_{s}= 0.08 and

*ratio*

_{l}= 0.31. Note that

*BF*

_{p}at a value of 0.053316 gives the best fit (thin solid line).

*BF*

_{p}consumes considerable computer time. To avoid repetitive calculation, the concept of constructing a LUT is again adopted. Three variables,

*ratio*

_{l},

*BF*

_{l}and

*BF*

_{s}are selected, with

*ratio*

_{l}ranging from 0 to 1,

*BF*

_{l}ranging from 0.0001 to 0.02 to represent large particles, and

*BF*

_{s}ranging from 0.018 to 0.3 to represent small particles. A section of the LUT is denoted as LUT

_{BF}and given in Table 3. Throughout the entire LUT

_{BF}, the average value of

*PCC*is 0.989577 and the lowest value is 0.930118. Note that the inelastic scattering effects were not included in the Hydrolight runs used to generate the three LUTs needed for the model of this paper.

### 3.5 Use average cosine as an index to cope with stratified waters

*μ̄*(

*ζ*) is a crude but useful one-parameter measure of the directional structures of the light field in spatially uniform waters. It can be used as an index to provide rough information about the radiance distribution at that depth, and hence the attenuation of light within that layer. This idea is implemented and examined in Fig. 3.

*et al*. [25

25. M. R. Lewis, J. J. Cullen, and T. Platt, ”Phytoplankton and thermal structure in the upper ocean: consequences of nonuniformity in chlorophyll profile,“ J. Geoph. Res. **88**, 2565–2570 (1983). [CrossRef]

## 4. Results

*θ*

_{S}, cloudiness, surface wind speed

*V*

_{wind}, the backscattering fraction for large particles

*BF*

_{l}, and the backscattering fraction for mineral particles

*BF*

_{S}(Table 4). The stratified layers were composed by randomly specifying values to four parameters of Eq. (23) to describe the vertical profiles of chlorophyll concentration

*Chl*(

*z*) and the mineral particle concentration

*M*(

*z*), respectively (Table 5). The CDOM absorption at a reference wavelength

*a*

_{g,440}(

*z*) was set to be varied linearly from the surface value

*γ*is randomly specified as well. Figure 4(a), 4(b) and 4(c) show the resulting profiles of

*Chl*(

*z*),

*a*

_{g,440}(

*z*) and

*M*(

*z*), respectively.

26. L. Prieur and S. Sathyendranath, ”An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,“ Limnol. Oceanogr. **26**, 671–689 (1981). [CrossRef]

27. A. Morel, ”Light and marine photosynthesis: a spectral model with geochemical and climatological implications,“ Prog. Oceanogr. **26**, 263–306 (1991). [CrossRef]

27. A. Morel, ”Light and marine photosynthesis: a spectral model with geochemical and climatological implications,“ Prog. Oceanogr. **26**, 263–306 (1991). [CrossRef]

28. H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, and W. W. Broenkow, ”Phytoplankton pigment concentrations in the Middle Atlantic Bight: Comparison of ship determinations and CZCS estimates,“ Appl. Opt. **22**, 20–36 (1983). [CrossRef] [PubMed]

*λ*) is given in the paper of Morel [29

29. A. Morel, ”Optical modelling of the upper ocean in relation to its biogenous matter content (case 1 water),“ J. Geoph. Res. **93**, 10749–10768 (1988). [CrossRef]

*a*

_{w}(

*λ*) and scattering coefficient

*b*

_{w}(

*λ*) for pure water are given in the papers of Pope and Fry [30

30. R. M. Pope and E. S. Fry, ”Absorption spectrum (380-700 nm) of pure water. II. Integrating cavity measurements,“ Appl. Opt. **36**, 8710–8723 (1997). [CrossRef]

31. R. C. Smith and K. Baker, ”Optical properties of the clearest natural waters,” Appl. Opt. **20**, 177–184 (1981). [CrossRef] [PubMed]

*λ*) and scattering coefficient

*λ*) are taken from published data [32]. With the set of bio-optical models described from Eq. (24) to Eq. (28), the vertical profiles of

*Chl*(

*z*),

*a*

_{g,440}(

*z*) and

*M*(

*z*) can be converted to the vertical profiles of total absorption coefficient

*a*(

*z*,

*λ*) and total scattering coefficient

*b*(

*z*,

*λ*).

*b*

_{m}(

*z*,

*λ*) and

*b*

_{c}(

*z*,

*λ*), the vertical profile of

*ratio*

_{l}(

*z*,

*λ*) can be calculated from Eq. (20). Following the procedure described earlier, the backscattering fraction

*BF*

_{p}(

*z*,

*λ*) of surrogate particles can be quickly referred to in the LUT from the given values of

*ratio*

_{l}(

*z*,

*λ*),

*BF*

_{l}and

*BF*

_{S}. For illustration, Fig. 4(d), 4(e) and 4(f) show the vertical profiles of

*a*(

*z*),

*b*(

*z*) and

*BF*

_{p}(

*z*) respectively at wavelength 440 nm.

*E*

_{0,PAR}(

*z*) were simulated by use of the commercially available 4.3 version of the Hydrolight model and the new model developed in this work, respectively. A total of 35 wave bands were simulated, which ranged from 350 to 700 nm in steps of 10 nm. Figure 5 shows the comparison of

*E*

_{0,PAR}(

*z*) from 0 to 50 m with one meter intervals. To quantitate the comparison, the correlation coefficient

*r*, the maximum relative error

*ε*

_{max}, the average relative error

*ε*

_{average}and the percentage error ε% in the euphotic zone were calculated and given in Fig. 5. The definition of ε% is given by

_{log10}is defined by

*r*= 0.999924) as well as a large computational speed ratio (CSR) of 1402.8 is obtained for our model. The percentage error

*ε*% is 2.03% and the maximum relative error

*ε*

_{max}is not more than 6.81%. The average relative error

*ε*

_{average}is as low as 1.62%. Figure 5 demonstrates that the new model can indeed provide an accurate and fast simulation of the vertical profile of

*E*

_{0,PAR}(

*z*) for stratified Case 2 waters with a variety of compositions of the inherent optical properties.

## 5. Summary

## Acknowledgments

## References and links

1. | C D. Mobley, |

2. | C.-C. Liu, K. L. Carder, R. L. Miller, and J. E. Ivey, “Fast and accurate model of underwater scalar irradiance,” Appl. Opt. |

3. | A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. |

4. | H. R. Gordon and A. Y. Morel, |

5. | J. C. Pernetta and J. D. Milliman, “Land-ocean interactions in the coastal zone implementation plan,” IGBP Report 33 (Stockholm, 1995). |

6. | S. Sathyendranath, ed., |

7. | A. Albert and C. D. Mobley, “An analytical model for subsurface irradiance and remote sensing reflectance in deep and shallow case-2 waters,” Opt. Express |

8. | R. W. Preisendorfer, |

9. | R. L Fante, “Relationship between radiative-transport theory and Maxwell’s equations in dielectric media,” J. Opt. Soc. Am. |

10. | R. M. Measures, |

11. | K. Stamnes, “The theory of multiple scattering of radiation in plane parallel atmospheres,” Rev. Geophys. |

12. | G. N. Plass and G. W. Kattawar, “Monte Carlo calculations of radiative transfer in the Earth’s atmosphere-ocean system. I. Flux in the atmosphere and ocean,” J. Phys. Oceanogr. |

13. | C.-C. Liu and J. Woods, “Prediction of ocean colour: Monte-Carlo simulation applied to a virtual ecosystem based on the Lagrangian Ensemble method,” Int. J. Remote Sens. |

14. | K. L. Carder, C.-C. Liu, Z. P. Lee, D. C. English, J. Patten, R. F. Chen, J. E. Ivey, and C. O. Davis, “Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos,” Limnol. Oceanogr. |

15. | R. W. Preisendorfer and C. D. Mobley, “Unpolarized irradiance reflectances and glitter patterns of random capillary waves on lakes and seas, by Monte Carlo simulation,” NOAA Technical Memorandum, ERL PMEL-63 (NOAA Pacific Marine Environmental Laboratory, Seattle, WA, 1985). |

16. | C. D. Mobley and R. W. Preisendorfer, “A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces,” NOAA Technical Memorandum, ERL PMEL-75 (NOAA Pacific Marine Environmental Laboratory, Seattle, WA, 1988). |

17. | C. D. Mobley, “A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces,” Limnol. Oceanogr. |

18. | C. D. Mobley, “Hydrolight 3.0 Users’ Guide,” (SRI International, Menlo Park, CA, 1995). |

19. | C.-C. Liu, J. D. Woods, and C. D. Mobley, ”Optical model for use in oceanic ecosystem models,“ Appl. Opt. |

20. | G. R. Fournier and J. L. Forand, ”Analytic phase function for ocean water,“ presented at the SPIE: Ocean Optics XII, 1994. |

21. | N. J. McCormick, ”Mathematical models for the mean cosine of irradiance and the diffuse attenuation coefficient,“ Limnol. Oceanogr. |

22. | A. Gershun, ”The light field,“ J. Math. Phys. |

23. | V. I. Haltrin, ”Chlorophyll-Based Model of Seawater Optical Properties,“ Appl. Opt. |

24. | C. D. Mobley, L. K. Sundman, and E. Boss, ”Phase function effects on oceanic light fields,“ Appl. Opt. |

25. | M. R. Lewis, J. J. Cullen, and T. Platt, ”Phytoplankton and thermal structure in the upper ocean: consequences of nonuniformity in chlorophyll profile,“ J. Geoph. Res. |

26. | L. Prieur and S. Sathyendranath, ”An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,“ Limnol. Oceanogr. |

27. | A. Morel, ”Light and marine photosynthesis: a spectral model with geochemical and climatological implications,“ Prog. Oceanogr. |

28. | H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, and W. W. Broenkow, ”Phytoplankton pigment concentrations in the Middle Atlantic Bight: Comparison of ship determinations and CZCS estimates,“ Appl. Opt. |

29. | A. Morel, ”Optical modelling of the upper ocean in relation to its biogenous matter content (case 1 water),“ J. Geoph. Res. |

30. | R. M. Pope and E. S. Fry, ”Absorption spectrum (380-700 nm) of pure water. II. Integrating cavity measurements,“ Appl. Opt. |

31. | R. C. Smith and K. Baker, ”Optical properties of the clearest natural waters,” Appl. Opt. |

32. | R. P. Bukata, J. H. Jerome, K. Y. Kondratyev, and D. V. Pozdnyakov, |

**OCIS Codes**

(010.4450) Atmospheric and oceanic optics : Oceanic optics

(010.7340) Atmospheric and oceanic optics : Water

(030.5620) Coherence and statistical optics : Radiative transfer

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: December 15, 2005

Revised Manuscript: February 19, 2006

Manuscript Accepted: February 23, 2006

Published: March 6, 2006

**Virtual Issues**

Vol. 1, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Cheng-Chien Liu, "Fast and accurate model of underwater scalar irradiance for stratified Case 2 waters," Opt. Express **14**, 1703-1719 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-5-1703

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### References

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