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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 4 — Feb. 16, 2009
  • pp: 2132–2142
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Effects of bulk particle characteristics on backscattering and optical closure

Grace Chang and Amanda L. Whitmire  »View Author Affiliations


Optics Express, Vol. 17, Issue 4, pp. 2132-2142 (2009)
http://dx.doi.org/10.1364/OE.17.002132


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Abstract

Optical closure is essential for the determination of biogeochemical properties from ocean color remote sensing information. Mie scattering theory, a radiative transfer model, and a semi-analytical inversion algorithm were used to investigate the influence of particles and their properties on optical closure. Closure results were generally poor. Absorption coefficient (at) inversions were more accurate for moderate particle size distribution slopes (3.50 ≤ ξ ≤ 3.75). The degree of success in the derivation of the backscattering coefficient (bbp) was highest at moderate indices of refraction (1.15 ≤ np ≤ 1.20) and high values of ξ (> 3.75). Marked improvements in the estimates of bbp were enabled by a priori knowledge of bbp at one wavelength. At moderate values of np, derivations of at and bbp were within 25% of Mie-modeled values when Gershun’s relationship was used in combination with the semi-analytical algorithm.

© 2009 Optical Society of America

1. Introduction

Optical closure involves solutions to the forward and inverse problems in ocean optics. Both problems are twofold: Forward - the determination of the inherent optical properties (IOPs) from characteristics of the particulate and dissolved material and the prediction of the apparent optical properties (AOPs) from the IOPs; Inverse - the derivation of the IOPs from the AOPs, and the determination of biogeochemical properties from the IOPs. The latter component of the inverse problem is especially important for interpretation of ocean color remote sensing data to synoptically observe and monitor interdisciplinary processes such as biogeochemical cycling, particle transport, and ecosystem dynamics (e.g., see [1

1. Z. -P. Lee (ed.), “Remote sensing of inherent optical properties: Fundamentals, tests of algorithms, and applications,” in Reports of the International Ocean-Colour Coordinating Group, No. 5, (IOCCG, 2006).

]). Other than the computation of the AOPs from the IOPs, i.e. component #2 of the forward problem [2

2. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic Press, 1993).

], optical closure has proven problematic - particularly for global scale applications.

Optical theory suggests that the optical properties of particles depend on characteristics such as particle size, shape, composition, and index of refraction [3

3. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

, 4

4. C. J. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley, 1983).

]. Mie scattering theory involves the estimation of the IOPs from characteristics for a single particle or a population of mono-dispersed or poly-dispersed particles. Briefly, Mie theory determines the scattering and extinction efficiency factors (and absorption by difference) for a homogeneous sphere at a given wavelength, index of refraction, and particle diameter. Ensembles of efficiency factors for spheres of different sizes can then be applied, using a specified particle size distribution (PSD) and number concentration to derive the respective IOPs, i.e. the absorption, scattering, and attenuation coefficients for a particle population.

rrs(λ)[f(λ)/Q(λ)]{bbt(λ)/[at(λ)+bbt(λ)]},
(1)

where bbt(λ) is total spectral backscattering, at(λ) is total spectral absorption, and the f/Q ratio (wavelength notation hereafter suppressed) is a parameter that depends on the shape of the upwelling light field and the volume scattering function (VSF) [5

5. J. R. V. Zaneveld, “Remotely sensed reflectance and its dependence on vertical structure: A theoretical derivation,” Appl. Opt. 21, 4146–4150 (1982). [CrossRef] [PubMed]

]. The parameter Q(λ) is a measurable quantity: the ratio of upwelling irradiance to upwelling radiance. However the value, f(λ), is often approximated for a single VSF [6

6. J. T. O. Kirk, “Dependence of relationship between inherent and apparent optical properties of water on solar altitude,” Limnol. Oceanogr. 29, 350–356 (1984). [CrossRef]

] or using assumed scattering properties and sky conditions [7

7. H. R. Gordon, “Dependence of diffuse reflectance of natural waters on the Sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989). [CrossRef]

, 8

8. A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on Sun angles as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991). [CrossRef] [PubMed]

], which oftentimes leads to erroneous computations of rrs(λ) from the IOPs and vice versa when inverting this relationship (i.e. the inverse problem).

Empirical and semi-analytical algorithms have been developed to invert remote sensing reflectance signals to obtain the IOPs and biogeochemical constituents (e.g., chlorophyll concentration; [9

9. J. E. O’Reilly, S. Maritorena, B. G. Mitchell, D. A. Siegel, K. L. Carder, S. A. Garver, M. Kahru, and C. McClain, “Ocean color chlorophyll algorithms for SeaWiFS,” J. Geophys. Res. 103, 24,937–24,953 (1998). [CrossRef]

]). Empirical algorithms have proven challenging for Case II waters, whose optical properties are not dominated by chlorophyll-containing particles but rather by inorganic particles, colored dissolved organic matter (CDOM), or both quantities. Semi-analytical algorithms based on Eq. (1) (e.g., see [10

10. Z. -P. Lee, K. L. Carder, and R. A. Arnone, “Deriving inherent optical properties from water color: A multi-band quasi-analytical algorithm for optically deep waters,” Appl. Opt. 41, 5755–5772 (2002). [CrossRef] [PubMed]

]) have been more successful when applied to widely varying optical water types, however, assumptions about the spectral and angular scattering properties and the upwelling light field can negatively impact inversion results.

The effects of particles and particle characteristics on the variability of remote sensing reflectance need to be understood in order to solve the forward and inverse problems in ocean optics and importantly, to better utilize ocean color data to obtain information about biogeochemistry and ecosystem dynamics. We present results from forward and backward approaches to optical closure with emphasis on the effects of particles and their characteristics on backscattering, the backscattering ratio, and ocean color. We build upon decades of theoretical, laboratory, and field results (e.g., see [3

3. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

, 4

4. C. J. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley, 1983).

, 11–20

11. O. Ulloa, S. Sathyendranath, and T. Platt, “Effect of the particle size-distribution on the backscattering ratio in seawater,” Appl. Opt. 33, 7070–7077 (1994). [CrossRef] [PubMed]

]).

2. Methods

Mie scattering theory and the radiative transfer model, Hydrolight, were used to compute the bulk IOPs and AOPs for a set of hypothetical water masses containing particle populations with variable bulk indices of refraction and PSDs. The influence of particles and their properties (e.g., bulk real index of refraction and PSD slope, np and ξ, respectively) on the IOPs and AOPs and on inversion algorithms to derive the IOPs from AOPs was investigated using a semi-analytical remote sensing inversion algorithm.

2.1 Forward approach

Mie scattering theory was used to generate bulk IOPs for a series of hypothetical water masses. Mie model input values were determined based on four years of time series measurements of optical properties collected on a mooring at 4 m water depth in shallow coastal waters of the Santa Barbara Channel (Fig. 1). Optical water types in this region of the Santa Barbara Channel are highly biogeochemically complex and can vary rapidly from relatively clear waters dominated by biogenic particles to very turbid and comprised of mostly inorganic particles [21

21. G. C. Chang, A. H. Barnard, S. McLean, P. J. Egli, C. Moore, J. R. V. Zaneveld, T. D. Dickey, and A. Hanson, “In situ optical variability and relationships in the Santa Barbara Channel: implications for remote sensing,” Appl. Opt. 45, 3593).3604 (2006). [CrossRef] [PubMed]

, 22

22. G. Chang, A. H. Barnard, and J. R. V. Zaneveld, “Optical closure in a complex coastal environment: Particle effects,” Appl. Opt. 46, 7679).7692 (2007). [CrossRef] [PubMed]

].

We used the Mie code provided by Bohren and Huffman [4

4. C. J. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley, 1983).

] for scattering by homogeneous spheres (routines BHMIE and CALLBH), translated from Fortran language into Matlab by E. Boss (U Maine). This code computes the elements and efficiencies of the amplitude scattering matrix. The modeled PSDs were restricted to a size range of diameters, D, ranging from 0.2 - 100 μm in 35 logarithmically spaced size bins. The PSDs followed a differential size distribution function where f(D) = N × D, where N is the particle concentration (held constant at 5 × 105 mL-1), i.e. we assumed Junge-type PSDs. We used np values of 1.01, 1.05, 1.10, 1.15 and 1.20 and ξ, values of 3.0, 3.25, 3.5, 3.75, 4.0 and 4.25 at nine wavelengths (λ = 412, 440, 488, 510, 532, 555, 650, 676, and 715 nm). The imaginary index of refraction of particles was held constant at n’ = 0.01 for all Mie theory calculations. Flatter PSD slopes, i.e. lower values of ξ, generally imply more large particles are present in a population, whereas steeper PSD slopes specify smaller particles. Lower values of np are indicative of biological particles (np phytoplankton = 1.02 - 1.07; [23

23. D. Stramski, A. Morel, and A. Bricaud, “Modeling the light attenuation and scattering by spherical phytoplankton cells: A retrieval of the bulk refractive index,” Appl. Opt. 27, 3954).3956 (1988). [CrossRef] [PubMed]

]) because of their high water content, and minerogenic particles are represented by higher values of np (np minerogenic = 1.14 to 1.26; [24

24. D. R. Lide, “Physical and optical properties of minerals,” in CRC Handbook of Chemistry and Physics, 77th ed., D. R. Lide, ed. (CRC Press, 1997), pp. 4130)–4136.

]). Two hundred and seventy sets of IOPs were determined from Mie modeling efforts. These Mie-computed IOPs were within the range of IOPs measured in the Santa Barbara Channel (Fig. 1).

Fig. 1. Measured IOPs (Santa Barbara Channel). Thick black lines indicate the spectral mean and dashed black lines denote one standard deviation from the mean. The lower right-hand panel shows a data series of bulk np (red) and ξ (blue) computed using measured IOPs and methods presented by Boss et al. [25] and Twardowski et al. [12]. [The IOP property subscript ‘p’ denotes particulate material and ‘g’ represents dissolved matter.]

Phase functions, together with Mie-computed scattering and attenuation (and absorption by difference) coefficients, were then inputted into the numerical radiative transfer model, Hydrolight, to compute the AOPs at the surface, just below the surface, and at 4 m geometric water depth in optically deep waters. Particle phase functions were generated from Mie-calculated VSFs using relevant phase function discretization operations in Hydrolight. Inelastic scattering processes were not included in computations and the following environmental conditions were assumed: wind speed = 5 m s-1, solar angle = 30°, and 0% cloud cover. Hydrolight calculations resulted in 270 arrays of AOPs representing five different bulk indices of refraction, six different PSD slopes, and nine different wavelengths.

2.2 Inverse approach

We employed a semi-analytical remote sensing inversion algorithm [10

10. Z. -P. Lee, K. L. Carder, and R. A. Arnone, “Deriving inherent optical properties from water color: A multi-band quasi-analytical algorithm for optically deep waters,” Appl. Opt. 41, 5755–5772 (2002). [CrossRef] [PubMed]

] in order to examine the influence of particles and their properties (np and ξ) on derivations of the IOPs from AOPs. This algorithm uses the relationship presented by Gordon et al. [26

26. H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988). [CrossRef]

]:

rrs(λ)=g0u(λ)+g1[u(λ)]2
(2a)

where

u(λ)=bbt(λ)/[at(λ)+bbt(λ)]
(2b)

and the g-constants are dependent on the particle phase function, oftentimes represented as g0 = 0.084 and g1 = 0.17 [10

10. Z. -P. Lee, K. L. Carder, and R. A. Arnone, “Deriving inherent optical properties from water color: A multi-band quasi-analytical algorithm for optically deep waters,” Appl. Opt. 41, 5755–5772 (2002). [CrossRef] [PubMed]

], which are the values we used in our computations. Due to the relatively large values of at(440) (> 0.3 m-1), at(650) was first parameterized as a function of Hydrolight-derived rrs(λ), then bbt(650) was derived from Eq. (2), as opposed to using the 555 nm wavelength method [10

10. Z. -P. Lee, K. L. Carder, and R. A. Arnone, “Deriving inherent optical properties from water color: A multi-band quasi-analytical algorithm for optically deep waters,” Appl. Opt. 41, 5755–5772 (2002). [CrossRef] [PubMed]

]. Next, bbt(λ) was computed; it was assumed to monotonically decrease with increasing wavelength:

bbt(λ)=bbw(λ)+bbp(λ0)(λ0/λ)η,
(3)

where λ 0 = 650 nm and η is a function of rrs(λ). Spectral at(λ) was calculated following Eq. (2). Details regarding empirical parameterizations of the IOPs and rrs(λ) can be found in Lee et al. [10

10. Z. -P. Lee, K. L. Carder, and R. A. Arnone, “Deriving inherent optical properties from water color: A multi-band quasi-analytical algorithm for optically deep waters,” Appl. Opt. 41, 5755–5772 (2002). [CrossRef] [PubMed]

].

3. Results and discussion

To verify data and model integrity, we provide an independent parameterization of at(λ) derived from Gershun’s equation. The exact relationship (wavelength notation suppressed):

at=Kdμd[1+R(μd/μu)]1[1R+(Kd)1dR/dZ],
(4)

(where Kd is the diffuse attenuation coefficient for downwelling irradiance, μd and μu are the average cosines for downwelling and upwelling light, respectively, R is irradiance reflectance, and Z is depth) was applied to our Hydrolight-derived values for Kd, μd, μu, and R at Z = 4 m. Comparisons between Mie-modeled absorption coefficients, at Mie(λ), and absorption coefficients inverted using Gershun’s equation, at Ger(λ), at 4 m are shown in Fig. 2. The exact 1:1 correlation between at Mie(λ) and at Ger(λ) for all np, ξ, and λ indicates that no errors were made in Mie computed inputs to Hydrolight or Hydrolight modeling.

Fig. 2. Comparison between at Mie(λ) and at Ger(λ) [Eq. (4)] at nine wavelengths, five different values of np, and six different values of ξ.

3.1 Forward approach - IOPs

The effects of variable np and ξ on spectral IOPs are shown in Fig. 3. The absorption, attenuation, scattering, and backscattering coefficients (minus water) all increased with increasing np at all wavelengths. Harder particles (i.e. material with higher np-values) resulted in higher scattering and less transmission of light, which has been predicted from Mie theory (e.g., see [11

11. O. Ulloa, S. Sathyendranath, and T. Platt, “Effect of the particle size-distribution on the backscattering ratio in seawater,” Appl. Opt. 33, 7070–7077 (1994). [CrossRef] [PubMed]

, 12

12. M. S. Twardowski, E. Boss, J. B. MacDonald, W. S. Pegau, A. H. Barnard, and J. R. V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106, 14,129–14,142 (2001). [CrossRef]

]). The absorption coefficient was minimally affected by variations in np and decreased with increasing ξ; spectral variability was more or less constant across all np and ξ. PSD slope effects were similar between cpg(λ) and bp(λ), i.e. attenuation was controlled more by scattering processes. In general, cpg(λ) and bp(λ) decreased with increasing ξ but at np > 1.10, cpg(λ) and bp(λ) increased in the blue wavelengths as ξ exceeded 4.0 (Fig. 3). Wozniak and Stramski [17

17. S. B. Wozniak and D. Stramski, “Modeling the optical properties of mineral particles suspended in seawater and their influence on ocean reflectance and chlorophyll estimation from remote sensing algorithms,” Appl. Opt. 43, 3489–3503 (2004). [CrossRef] [PubMed]

] also showed higher blue-peaked spectra for mass-specific scattering coefficients at high values of ξ. Effects of np variability were more pronounced at higher ξ for cpg(λ) and bp(λ). This implies that variability in cpg(λ) and bp(λ) was more strongly affected by the presence of smaller, harder (i.e. minerogenic) particles.

Fig. 3. Spectral IOPs and IOP ratios as a function of np (legend in top left panel; colors correspond to plot lines and not fill colors) and ξ (x-axes).

Magnitudinal variability in bbp(λ) behaved as expected from Mie theory; bbp(λ) values increased with increasing np. Increasing ξ from 3.0 to 3.5 resulted in decreased bbp(λ) and then this trend reversed between ξ = 3.5 and 4.25. Mie-derived backscattering spectral shapes were not always represented by Eq. (3) and at np > 1.05, bbp(λ) exhibited spectral inflections, particularly at ξ = 3.0 (Fig. 3). The spectral shape of bbp(λ) became less steep with increasing np, with the mean power-law exponent changing from η = -2.601 to -0.1056, averaged over six different ξ. The mean of η over the 270 Mie-computed bbp(λ) spectra was calculated to be -0.8991 with a standard deviation of 1.0844. In comparison, Snyder et al. [20

20. W. A. Snyder, R. A. Arnone, C. O. Davis, W. Goode, R. W. Gould, S. Ladner, G. Lamela, W. J. Rhea, R. Stavn, M. Sydor, and A. Weidemann, “Optical scattering and backscattering by organic and inorganic particles in U.S. coastal waters,” Appl. Opt. 47, 666).677 (2008). [CrossRef] [PubMed]

] found that the mean and standard deviation of η for over 6000 field measurements of bbp(λ) in coastal waters was -0.942 ± 0.210. They stated that, “… the spread in this value is not random and represents small, but real, spectral changes in the backscattering spectral properties of the water” [20

20. W. A. Snyder, R. A. Arnone, C. O. Davis, W. Goode, R. W. Gould, S. Ladner, G. Lamela, W. J. Rhea, R. Stavn, M. Sydor, and A. Weidemann, “Optical scattering and backscattering by organic and inorganic particles in U.S. coastal waters,” Appl. Opt. 47, 666).677 (2008). [CrossRef] [PubMed]

]. Their results show that significant departures from the average power-law function occurred where biological particulates dominated. We also find that the spectral shape of bbp(λ) deviated from a power-law function at lower np (at np < 1.10, bbp(λ) exponentially decayed with increasing wavelength; not visible in Fig. 3), which is representative of particles with higher water content, i.e. biological particles. Spectral inflections in the particulate backscattering coefficient have been observed in modeling results that used input parameters based on measurements of phytoplankton cultures [27–30

27. A. Bricaud, A. Morel, and L. Prieur, “Optical efficiency factors of some phytoplankters,” Limnol. Oceanogr. 28, 816–832 (1983). [CrossRef]

]. Direct measurements of the spectral backscattering properties of marine phytoplankton cultures have also shown spectral inflections [31

31. A. L. Whitmire, The spectral backscattering properties of marine particles, Ph.D. dissertation, Oregon State University, 2008. ScholarsArchive@OSU, 28 Oct. 2008 <http://hdl.handle.net/1957/9088>.

]. Reduced backscattering in spectral regions of strong absorption (e.g. peaks in absorption by Chlorophyll-a around 440 and 670 nm) is caused by the change in the imaginary index of refraction at these wavelengths, an effect known as anomalous dispersion [3

3. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

, 32

32. A. Morel and A. Bricaud, “Theoretical results concerning the optics of phytoplankton, with special Reference to remote sensing applications,” in Oceanography from Space, J. F. R. Gower, Ed. (Plenum, New York, 1981). [CrossRef]

]. However, spectral inflections present in our results are somewhat surprising, given that we kept the real and imaginary portions of the index of refraction constant across wavelengths in our model runs.

Based on theoretical studies [11

11. O. Ulloa, S. Sathyendranath, and T. Platt, “Effect of the particle size-distribution on the backscattering ratio in seawater,” Appl. Opt. 33, 7070–7077 (1994). [CrossRef] [PubMed]

, 12

12. M. S. Twardowski, E. Boss, J. B. MacDonald, W. S. Pegau, A. H. Barnard, and J. R. V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106, 14,129–14,142 (2001). [CrossRef]

], the backscattering ratio, bbp(λ)/bp(λ), has traditionally been assumed to be spectrally flat. However, Ulloa et al. [11

11. O. Ulloa, S. Sathyendranath, and T. Platt, “Effect of the particle size-distribution on the backscattering ratio in seawater,” Appl. Opt. 33, 7070–7077 (1994). [CrossRef] [PubMed]

] assert that in the case of monodispersions or in waters dominated by minerogenic particles, bbp(λ)/bp(λ) can vary strongly with wavelength depending on the size of the particles and their refractive index. We found that Mie-modeled bbp(λ)/bp(λ) was spectrally flat only at ξ ≤ 3.25 for all np, i.e. larger particles. During all other conditions, bbp(λ)/bp(λ) varied spectrally with increasing np and ξ. The spectral shape of bbp(λ)/bp(λ) exhibited increasing values with increasing wavelength, with the steepest spectral slopes observed at ξ = 4.25. The particulate backscattering ratios observed by Snyder et al. [20

20. W. A. Snyder, R. A. Arnone, C. O. Davis, W. Goode, R. W. Gould, S. Ladner, G. Lamela, W. J. Rhea, R. Stavn, M. Sydor, and A. Weidemann, “Optical scattering and backscattering by organic and inorganic particles in U.S. coastal waters,” Appl. Opt. 47, 666).677 (2008). [CrossRef] [PubMed]

] also exhibited a wide range in wavelength dependence and were highly variable in space and time. However, others have found little spectral variation in bbp(λ)/bp(λ). Boss et al. [15

15. E. Boss, W. S. Pegau, M. Lee, M. S. Twardowski, E. Shybanov, G. Korotaev, and F. Baratange, “The particulate backscattering ratio at LEO-15 and its use to study particle composition and distribution,” J. Geophys. Res. 109, C1, C0101410.1029/2002JC001514 (2004). [CrossRef]

] and Mobley et al. [33

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

] reported that bbp(λ)/bp(λ) spectra varied by less than 10% and 24% for all values of backscattering measured in coastal New Jersey and modeled using Hydrolight, respectively. Whitmire et al. [19

19. A. L. Whitmire, E. Boss, T. J. Cowles, and W. S. Pegau, “Spectral variability of the particulate backscattering ratio,” Opt. Express 15, 7019).7031 (2007). [CrossRef] [PubMed]

] found no significant spectral variability in a dataset of over 9000 observations that included data from diverse aquatic environments and particle populations, with rare exceptions in instances of monodispersions of large particles. Risović [13

13. D. Risović, “Effect of suspended particulate-size distribution on the backscattering ratio in the remote sensing of seawater,” Appl. Opt. 41, 7092–7101 (2002). [CrossRef] [PubMed]

] found that Mie-modeled bbp(λ)/bp(λ) was only weakly dependent on wavelength but suggested additional research on the spectral behavior of bbp(λ)/bp(λ).

The most striking results for the effects of particle characteristics on our set of hypothetical optical water types was the highly variable spectral shapes of the backscattering coefficient and the backscattering ratio. Contrary to assumptions made in the past, bbp(λ)/bp(λ) was not always spectrally flat and spectral bbp(λ) did not always follow the widely used expression shown in Eq. (3). This suggests that spectral variability, in addition to magnitudinal variability in bbp(λ)/bp(λ) and bbp(λ) could contain valuable information about the characteristics of particles. Importantly, variable spectral shapes (e.g., exponential versus power-law) at low np and spectral inflections at low ξ-values could have profound impacts on remote sensing and inversion algorithms for the derivation of the IOPs and biogeochemical properties.

3.2 Forward approach - AOPs

Hydrolight-computed AOP values were within the ranges of those reported in the literature (Fig. 4). For example, Morel et al. [35

35. A. Morel, K. J. Voss, and B. Gentili, “Bidirectional reflectance of oceanic waters: A comparison of modeled and measured upward radiance fields,” J. Geophys. Res. 100, 13,143–13,150 (1995). [CrossRef]

] reported field-measured and simulated values of Q(λ) between 0 and 8 sr and Loisel and Morel [36

36. H. Loisel and A. Morel, “Non-isotropy of the upward radiance field in typical coastal (Case 2) waters,” Int. J. Remote Sens. 22, 275–295 (2001). [CrossRef]

] found Q(λ) to vary between about 3 and 7 sr from numerical simulations of case II coastal waters with variable solar angles and scattering events. Loisel and Morel [36

36. H. Loisel and A. Morel, “Non-isotropy of the upward radiance field in typical coastal (Case 2) waters,” Int. J. Remote Sens. 22, 275–295 (2001). [CrossRef]

] also reported f/Q values between 0.06 and 0.20 sr-1 for the same simulations. Kd(λ) values and spectral shapes in Fig. 4 are typical of turbid coastal waters (e.g., Fig. 2 in [36

36. H. Loisel and A. Morel, “Non-isotropy of the upward radiance field in typical coastal (Case 2) waters,” Int. J. Remote Sens. 22, 275–295 (2001). [CrossRef]

]).

Fig. 4. 3-D plots of spectral AOPs as a function of np (legend in top left panel; colors correspond to plot lines and not fill colors) and ξ (x-axes).

3.3 Inverse approach

Absorption and backscattering coefficients derived using a semi-analytical algorithm [10

10. Z. -P. Lee, K. L. Carder, and R. A. Arnone, “Deriving inherent optical properties from water color: A multi-band quasi-analytical algorithm for optically deep waters,” Appl. Opt. 41, 5755–5772 (2002). [CrossRef] [PubMed]

] were within 150% of “true” values, i.e. Mie-computed (Fig. 5). Lower values of at(λ) were estimated more accurately than higher values, likely because the algorithm was formulated based on at(λ) values of less than 0.3 m-1 whereas our Mie-computed at(λ) values were much higher (mean of 0.5 m-1 at 412 nm; Fig. 1). The degree of success of derivation of at(λ) was more influenced by ξ as compared to np or wavelength. For derivations of at(λ), the inversion algorithm performed best at moderate ξ-values (3.50 ≤ ξ ≤ 3.75) for all values of np and at all wavelengths. Absorption was slightly overestimated for high values of ξ (i.e. smaller particles) and underestimated for low ξ (i.e. larger particles). On the other hand, successful derivations of bbp(λ) were dependent on np and ξ but not necessarily on wavelength or the magnitude of bbp(λ). We show that more accurate bbp(λ) estimations were achievable at 1.15 ≤ np ≤ 1.20 and at higher values of ξ (> 4.0; i.e. smaller particles). Derivations were underestimated for lower ξ.

Fig. 5. Derived at QAA(λ) and bbp QAA(λ) compared to at Mie(λ) and bbp Mie(λ). Wavelengths are represented by different colors from blue (400 nm range) to red (600 nm range). Open symbols in the first column denote the different ξ-values (triangles = 3.0, circles = 3.25, squares = 3.50, pluses = 3.75, stars = 4.0, and asterisks = 4.25) and closed symbols in the second column represent the different np-values (triangles = 1.01, circles = 1.05, squares = 1.10, crosses = 1.15, and stars = 1.20).

The right two columns of Fig. 6 (e, f) show derivations of at(λ) using a simplified version of Eq. (4), where μu = 0.40, μd = 0.90, and dR/dZ is neglected [39

39. A. Morel and S. Maritorena, “Bio-optical properties of oceanic waters: A reappraisal,” J. Geophys. Res. 106, 7163–7180 (2001). [CrossRef]

]:

at(λ)=Kd(λ)0.90[1+2.25R(λ)]1[1R(λ)]
(5)

and by assuming R(λ) = 3.5 × rrs(λ). Eq. (3) was then used to estimate bbp(λ) (Fig. 6g, h). The use of a simplified Gershun’s equation yielded more accurate derivations of at(λ), mostly within 25% of Mie-computed values (Fig. 6e, f). The largest deviations in at(λ) and bbp(λ) were found at higher np values. Unsuccessful derivations of bbp(λ) at np = 1.01 in both cases are due to the fact that the spectral shape of bbp(λ) at the lowest np-value (Fig. 6d, h) is represented by an exponential, rather than a power-law function. This implies that for complex coastal waters, improvements need to be made to the empirical parameterization of the power-law exponent, η.

Wozniak and Stramski [17

17. S. B. Wozniak and D. Stramski, “Modeling the optical properties of mineral particles suspended in seawater and their influence on ocean reflectance and chlorophyll estimation from remote sensing algorithms,” Appl. Opt. 43, 3489–3503 (2004). [CrossRef] [PubMed]

] also show that mineral particles in seawater can result in significant errors in the derivation of, in their case chlorophyll concentration from ocean color data. Marked improvements in the inversion algorithm are found by assuming a priori knowledge of bbp at one wavelength or by applying a simplified Gershun’s equation to first derive at(λ). This shows that minimal in situ measurements can result in accurate assessments of the IOPs and hence biogeochemical properties from ocean color remote sensing data.

Fig. 6. (a–d) Same as Fig. 5 but assuming bbp(650) is known; (e–h) Same as Fig. 6 but for at(λ) derived using Eq. (5); bbp(λ) was derived using Eq. (3).

4. Summary and conclusions

Mie scattering theory and the radiative transfer model, Hydrolight, were used to compute the IOPs and AOPs for a set of hypothetical optical water types with variable bulk real indices of refraction and particle size distribution slopes. The influence of particles and their properties on inversion algorithms to derive the IOPs from modeled AOPs was investigated using a semi-analytical remote sensing inversion algorithm. Notable conclusions are highlighted here.

The spectral shape and magnitude of the absorption coefficient was minimally affected by variations in np, but its magnitude decreased significantly with increasing ξ. Variability in the attenuation and scattering coefficients was strongly influenced by the presence of smaller, harder particles. The spectral shape of the particulate backscattering coefficient was not always represented by a power-law function, and often exhibited spectral inflections. The particulate backscattering ratio was spectrally flat only in the presence of larger particles. During all other conditions, bbp(λ)/bp(λ) varied spectrally with increasing np and ξ. In evaluating the effect of particle properties on AOPs, we found that the f/Q ratio exhibited escalating spectral variability with increasing np and ξ. In tests of the performance of a semi-analytical algorithm in predicting IOPs, the degree of success in the derivation of at(λ) was more influenced by the magnitude of at(λ) and ξ as compared to np or wavelength. Accurate derivations of bbp(λ) were dependent on np and ξ but not necessarily on wavelength or the magnitude of bbp(λ). Marked improvements in estimates of bbp(λ) were enabled by a priori knowledge of bbp at one wavelength. At 1.05 < np < 1.15, derivations of at(λ) and bbp(λ) were well within 25% of true values when Gershun’s equation was used in combination with the semi-analytical algorithm.

Acknowledgments

GC was supported by the National Oceanographic Partnership Program as part of the MOSEAN program. AW was supported by a NOAA Sea Grant Industry Fellowship. The authors would like to acknowledge MOSEAN PIs Tommy Dickey, Casey Moore, Al Hanson, and Dave Karl. The authors also thank Tim Cowles, Andrew Barnard, Frank Spada, and an anonymous reviewer who provided an excellent, insightful and thorough review of an earlier version of this paper.

References and links

1.

Z. -P. Lee (ed.), “Remote sensing of inherent optical properties: Fundamentals, tests of algorithms, and applications,” in Reports of the International Ocean-Colour Coordinating Group, No. 5, (IOCCG, 2006).

2.

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic Press, 1993).

3.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

4.

C. J. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley, 1983).

5.

J. R. V. Zaneveld, “Remotely sensed reflectance and its dependence on vertical structure: A theoretical derivation,” Appl. Opt. 21, 4146–4150 (1982). [CrossRef] [PubMed]

6.

J. T. O. Kirk, “Dependence of relationship between inherent and apparent optical properties of water on solar altitude,” Limnol. Oceanogr. 29, 350–356 (1984). [CrossRef]

7.

H. R. Gordon, “Dependence of diffuse reflectance of natural waters on the Sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989). [CrossRef]

8.

A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on Sun angles as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991). [CrossRef] [PubMed]

9.

J. E. O’Reilly, S. Maritorena, B. G. Mitchell, D. A. Siegel, K. L. Carder, S. A. Garver, M. Kahru, and C. McClain, “Ocean color chlorophyll algorithms for SeaWiFS,” J. Geophys. Res. 103, 24,937–24,953 (1998). [CrossRef]

10.

Z. -P. Lee, K. L. Carder, and R. A. Arnone, “Deriving inherent optical properties from water color: A multi-band quasi-analytical algorithm for optically deep waters,” Appl. Opt. 41, 5755–5772 (2002). [CrossRef] [PubMed]

11.

O. Ulloa, S. Sathyendranath, and T. Platt, “Effect of the particle size-distribution on the backscattering ratio in seawater,” Appl. Opt. 33, 7070–7077 (1994). [CrossRef] [PubMed]

12.

M. S. Twardowski, E. Boss, J. B. MacDonald, W. S. Pegau, A. H. Barnard, and J. R. V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106, 14,129–14,142 (2001). [CrossRef]

13.

D. Risović, “Effect of suspended particulate-size distribution on the backscattering ratio in the remote sensing of seawater,” Appl. Opt. 41, 7092–7101 (2002). [CrossRef] [PubMed]

14.

M. Babin, A. Morel, V. Fournier-Sicre, F. Fell, and D. Stramski, “Light scattering properties of marine particles in coastal and open ocean waters as related to the particle mass concentration,” Limnol. Oceanogr. 48, 843–859 (2003). [CrossRef]

15.

E. Boss, W. S. Pegau, M. Lee, M. S. Twardowski, E. Shybanov, G. Korotaev, and F. Baratange, “The particulate backscattering ratio at LEO-15 and its use to study particle composition and distribution,” J. Geophys. Res. 109, C1, C0101410.1029/2002JC001514 (2004). [CrossRef]

16.

E. Boss, D. Stramski, T. Bergmann, W. S. Pegau, and M. Lewis, “Why Should We Measure the Optical Backscattering Coefficient?” Oceanography 17, 44–49 (2004).

17.

S. B. Wozniak and D. Stramski, “Modeling the optical properties of mineral particles suspended in seawater and their influence on ocean reflectance and chlorophyll estimation from remote sensing algorithms,” Appl. Opt. 43, 3489–3503 (2004). [CrossRef] [PubMed]

18.

J. M. Sullivan, M. S. Twardowski, P. L. Donaghay, and S. Freeman, “Use of optical scattering to discriminate particle types in coastal waters,” Appl. Opt. 44, 1667–1680 (2005). [CrossRef] [PubMed]

19.

A. L. Whitmire, E. Boss, T. J. Cowles, and W. S. Pegau, “Spectral variability of the particulate backscattering ratio,” Opt. Express 15, 7019).7031 (2007). [CrossRef] [PubMed]

20.

W. A. Snyder, R. A. Arnone, C. O. Davis, W. Goode, R. W. Gould, S. Ladner, G. Lamela, W. J. Rhea, R. Stavn, M. Sydor, and A. Weidemann, “Optical scattering and backscattering by organic and inorganic particles in U.S. coastal waters,” Appl. Opt. 47, 666).677 (2008). [CrossRef] [PubMed]

21.

G. C. Chang, A. H. Barnard, S. McLean, P. J. Egli, C. Moore, J. R. V. Zaneveld, T. D. Dickey, and A. Hanson, “In situ optical variability and relationships in the Santa Barbara Channel: implications for remote sensing,” Appl. Opt. 45, 3593).3604 (2006). [CrossRef] [PubMed]

22.

G. Chang, A. H. Barnard, and J. R. V. Zaneveld, “Optical closure in a complex coastal environment: Particle effects,” Appl. Opt. 46, 7679).7692 (2007). [CrossRef] [PubMed]

23.

D. Stramski, A. Morel, and A. Bricaud, “Modeling the light attenuation and scattering by spherical phytoplankton cells: A retrieval of the bulk refractive index,” Appl. Opt. 27, 3954).3956 (1988). [CrossRef] [PubMed]

24.

D. R. Lide, “Physical and optical properties of minerals,” in CRC Handbook of Chemistry and Physics, 77th ed., D. R. Lide, ed. (CRC Press, 1997), pp. 4130)–4136.

25.

E. Boss, M. S. Twardowski, and S. Herring, “Shape of the particulate beam attenuation spectrum and its inversion to obtain the shape of the particulate size distribution,” Appl. Opt. 40, 4885–4893 (2001). [CrossRef]

26.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988). [CrossRef]

27.

A. Bricaud, A. Morel, and L. Prieur, “Optical efficiency factors of some phytoplankters,” Limnol. Oceanogr. 28, 816–832 (1983). [CrossRef]

28.

A. Bricaud and A. Morel, “Light attenuation and scattering by phytoplanktonic cells: a theoretical modeling,” Appl. Opt. 25, 571–580 (1986). [CrossRef] [PubMed]

29.

Y.-H. Ahn, A. Bricaud, and A. Morel, “Light backscattering efficiency and related properties of some phytoplankters,” Deep Sea Res. 39, 1835–1855 (1992). [CrossRef]

30.

D. Stramski, A. Bricaud, and A. Morel, “Modeling the inherent optical properties of the ocean based on the detailed composition of planktonic community,” Appl. Opt. 40, 2929–2945 (2001). [CrossRef]

31.

A. L. Whitmire, The spectral backscattering properties of marine particles, Ph.D. dissertation, Oregon State University, 2008. ScholarsArchive@OSU, 28 Oct. 2008 <http://hdl.handle.net/1957/9088>.

32.

A. Morel and A. Bricaud, “Theoretical results concerning the optics of phytoplankton, with special Reference to remote sensing applications,” in Oceanography from Space, J. F. R. Gower, Ed. (Plenum, New York, 1981). [CrossRef]

33.

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

34.

H. Loisel, X. Mériaux, J-F. Berthon, and A. Poteau, “Investigation of the optical backscattering to scattering ratio of marine particles in relation to their biogeochemical composition in the eastern English Channel and southern North Sea,” Limnol. Oceanogr. 52(2), 739–752 (2007). [CrossRef]

35.

A. Morel, K. J. Voss, and B. Gentili, “Bidirectional reflectance of oceanic waters: A comparison of modeled and measured upward radiance fields,” J. Geophys. Res. 100, 13,143–13,150 (1995). [CrossRef]

36.

H. Loisel and A. Morel, “Non-isotropy of the upward radiance field in typical coastal (Case 2) waters,” Int. J. Remote Sens. 22, 275–295 (2001). [CrossRef]

37.

J. R. V. Zaneveld, “A theoretical derivation of the dependence of the remotely sensed reflectance of the ocean on the inherent optical properties,” J. Geophys. Res. 100, 13,135–13,142 (1995). [CrossRef]

38.

Y. Park and K. Ruddick, “Model of remote-sensing reflectance including bi-directional effects for case 1 and case 2 waters,” Appl. Opt. 44, 1236–1249 (2005). [CrossRef] [PubMed]

39.

A. Morel and S. Maritorena, “Bio-optical properties of oceanic waters: A reappraisal,” J. Geophys. Res. 106, 7163–7180 (2001). [CrossRef]

OCIS Codes
(010.4450) Atmospheric and oceanic optics : Oceanic optics
(010.0280) Atmospheric and oceanic optics : Remote sensing and sensors

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: November 3, 2008
Revised Manuscript: December 3, 2008
Manuscript Accepted: January 23, 2009
Published: February 2, 2009

Virtual Issues
Vol. 4, Iss. 4 Virtual Journal for Biomedical Optics

Citation
Grace Chang and Amanda L. Whitmire, "Effects of bulk particle characteristics on backscattering and optical closure," Opt. Express 17, 2132-2142 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2132


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References

  1. Z. -P. Lee (ed.), "Remote sensing of inherent optical properties: Fundamentals, tests of algorithms, and applications," in Reports of the International Ocean-Colour Coordinating Group, No. 5, (IOCCG, 2006).
  2. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic Press, 1993).
  3. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
  4. C. J. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley, 1983).
  5. J. R. V. Zaneveld, "Remotely sensed reflectance and its dependence on vertical structure: A theoretical derivation," Appl. Opt. 21, 4146-4150 (1982). [CrossRef] [PubMed]
  6. J. T. O. Kirk, "Dependence of relationship between inherent and apparent optical properties of water on solar altitude, "Limnol. Oceanogr. 29, 350-356 (1984). [CrossRef]
  7. H. R. Gordon, "Dependence of diffuse reflectance of natural waters on the Sun angle," Limnol. Oceanogr. 34, 1484-1489 (1989). [CrossRef]
  8. A. Morel and B. Gentili, "Diffuse reflectance of oceanic waters: its dependence on Sun angles as influenced by the molecular scattering contribution," Appl. Opt. 30, 4427-4438 (1991). [CrossRef] [PubMed]
  9. J. E. O’Reilly, S. Maritorena, B. G. Mitchell, D. A. Siegel, K. L. Carder, S. A. Garver, M. Kahru, and C. McClain, "Ocean color chlorophyll algorithms for SeaWiFS," J. Geophys. Res. 103, 24,937-24,953 (1998). [CrossRef]
  10. Z. -P. Lee, K. L. Carder, and R. A. Arnone, "Deriving inherent optical properties from water color: A multi-band quasi-analytical algorithm for optically deep waters," Appl. Opt. 41,5755-5772 (2002). [CrossRef] [PubMed]
  11. O. Ulloa, S. Sathyendranath, and T. Platt, "Effect of the particle size-distribution on the backscattering ratio in seawater," Appl. Opt. 33, 7070-7077 (1994). [CrossRef] [PubMed]
  12. M. S. Twardowski, E. Boss, J. B. MacDonald, W. S. Pegau, A. H. Barnard, and J. R. V. Zaneveld, "A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters," J. Geophys. Res. 106, 14,129-14,142 (2001). [CrossRef]
  13. D. Risović, "Effect of suspended particulate-size distribution on the backscattering ratio in the remote sensing of seawater," Appl. Opt. 41, 7092-7101 (2002). [CrossRef] [PubMed]
  14. M. Babin, A. Morel, V. Fournier-Sicre, F. Fell, and D. Stramski, "Light scattering properties of marine particles in coastal and open ocean waters as related to the particle mass concentration," Limnol. Oceanogr. 48, 843-859 (2003). [CrossRef]
  15. E. Boss, W. S. Pegau, M. Lee, M. S. Twardowski, E. Shybanov, G. Korotaev, and F. Baratange, "The particulate backscattering ratio at LEO-15 and its use to study particle composition and distribution," J. Geophys. Res. 109, C1, C0101410.1029/2002JC001514 (2004). [CrossRef]
  16. E. Boss, D. Stramski, T. Bergmann, W. S. Pegau, and M. Lewis, "Why Should We Measure the Optical Backscattering Coefficient?" Oceanography 17, 44-49 (2004).
  17. S. B. Wozniak and D. Stramski, "Modeling the optical properties of mineral particles suspended in seawater and their influence on ocean reflectance and chlorophyll estimation from remote sensing algorithms," Appl. Opt. 43, 3489-3503 (2004). [CrossRef] [PubMed]
  18. J. M. Sullivan, M. S. Twardowski, P. L. Donaghay, and S. Freeman, "Use of optical scattering to discriminate particle types in coastal waters," Appl. Opt. 44, 1667-1680 (2005). [CrossRef] [PubMed]
  19. A. L. Whitmire, E. Boss, T. J. Cowles, and W. S. Pegau, "Spectral variability of the particulate backscattering ratio," Opt. Express 15, 7019-7031 (2007). [CrossRef] [PubMed]
  20. W. A. Snyder, R. A. Arnone, C. O. Davis, W. Goode, R. W. Gould, S. Ladner, G. Lamela, W. J. Rhea, R. Stavn, M. Sydor, and A. Weidemann, "Optical scattering and backscattering by organic and inorganic particles in U.S. coastal waters," Appl. Opt. 47, 666-677 (2008). [CrossRef] [PubMed]
  21. G. C. Chang, A. H. Barnard, S. McLean, P. J. Egli, C. Moore, J. R. V. Zaneveld, T. D. Dickey, and A. Hanson, "In situ optical variability and relationships in the Santa Barbara Channel: implications for remote sensing," Appl. Opt. 45,3593-3604 (2006). [CrossRef] [PubMed]
  22. G. Chang, A. H. Barnard, and J. R. V. Zaneveld, "Optical closure in a complex coastal environment: Particle effects," Appl. Opt. 46, 7679-7692 (2007). [CrossRef] [PubMed]
  23. D. Stramski, A. Morel, and A. Bricaud, "Modeling the light attenuation and scattering by spherical phytoplankton cells: A retrieval of the bulk refractive index," Appl. Opt. 27, 3954-3956 (1988). [CrossRef] [PubMed]
  24. D. R. Lide, "Physical and optical properties of minerals," in CRC Handbook of Chemistry and Physics, 77th ed., D. R. Lide, ed. (CRC Press, 1997), pp. 4130-4136.
  25. E. Boss, M. S. Twardowski, and S. Herring, "Shape of the particulate beam attenuation spectrum and its inversion to obtain the shape of the particulate size distribution," Appl. Opt. 40, 4885-4893 (2001). [CrossRef]
  26. H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, "A semianalytic radiance model of ocean color," J. Geophys. Res. 93, 10,909-10,924 (1988). [CrossRef]
  27. A. Bricaud, A. Morel and L. Prieur, "Optical efficiency factors of some phytoplankters," Limnol. Oceanogr. 28, 816-832 (1983). [CrossRef]
  28. A. Bricaud. and A. Morel, "Light attenuation and scattering by phytoplanktonic cells: a theoretical modeling," Appl. Opt. 25, 571-580 (1986). [CrossRef] [PubMed]
  29. Y.-H. Ahn, A. Bricaud, and A. Morel, "Light backscattering efficiency and related properties of some phytoplankters," Deep Sea Res. 39, 1835-1855 (1992). [CrossRef]
  30. D. Stramski, A. Bricaud, and A. Morel, "Modeling the inherent optical properties of the ocean based on the detailed composition of planktonic community," Appl. Opt. 40, 2929-2945 (2001). [CrossRef]
  31. A. L. Whitmire, The spectral backscattering properties of marine particles, Ph.D. dissertation, Oregon State University, 2008. ScholarsArchive@OSU, 28 Oct. 2008 <http://hdl.handle.net/1957/9088>.
  32. A. Morel and A. Bricaud, "Theoretical results concerning the optics of phytoplankton, with special Reference to remote sensing applications," in Oceanography from Space, J. F. R. Gower, Ed. (Plenum, New York, 1981). [CrossRef]
  33. C. D. Mobley, L. K. Sundman, and E. Boss, "Phase function effects on oceanic light fields," Appl. Opt. 41, 1035-1050 (2002). [CrossRef] [PubMed]
  34. H. Loisel, X. Mériaux, J-F. Berthon, and A. Poteau, "Investigation of the optical backscattering to scattering ratio of marine particles in relation to their biogeochemical composition in the eastern English Channel and southern North Sea," Limnol. Oceanogr. 52(2), 739-752 (2007). [CrossRef]
  35. A. Morel, K. J. Voss, and B. Gentili, "Bidirectional reflectance of oceanic waters: A comparison of modeled and measured upward radiance fields," J. Geophys. Res. 100, 13,143-13,150 (1995). [CrossRef]
  36. H. Loisel and A. Morel, "Non-isotropy of the upward radiance field in typical coastal (Case 2) waters," Int. J. Remote Sens. 22, 275-295 (2001). [CrossRef]
  37. J. R. V. Zaneveld, "A theoretical derivation of the dependence of the remotely sensed reflectance of the ocean on the inherent optical properties," J. Geophys. Res. 100, 13,135-13,142 (1995). [CrossRef]
  38. Y. Park and K. Ruddick, "Model of remote-sensing reflectance including bi-directional effects for case 1 and case 2 waters," Appl. Opt. 44, 1236-1249 (2005). [CrossRef] [PubMed]
  39. A. Morel and S. Maritorena, "Bio-optical properties of oceanic waters: A reappraisal," J. Geophys. Res. 106, 7163-7180 (2001). [CrossRef]

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