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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 1 — Jan. 2, 2012
  • pp: 201–219
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Mass-specific scattering cross sections of suspended sediments and aggregates: theoretical limits and applications

Robert H. Stavn  »View Author Affiliations


Optics Express, Vol. 20, Issue 1, pp. 201-219 (2012)
http://dx.doi.org/10.1364/OE.20.000201


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Abstract

The spectral mass-specific scattering cross section σ[PIM](λ) is most important for the remote sensing inversion of the concentration of suspended mineral matter in the coastal ocean. This optical parameter is also important in optical theory and therefore the theoretical limits of this parameter are important. There are differing reports in the literature on the magnitude of σ[PIM](λ) and its spectral slope in different coastal ocean systems. To account for and predict these differences, I have applied a model of the size distribution of primary suspended mineral particles and aggregates of these particles to theoretical calculations of σ[PIM](λ). I utilized a model of mineral particle aggregates by Khelifa and Hill [Khelifa, A. and P.S. Hill, J. Hydraul. Res. 44, 390 (2006)] and Latimer's optical model of aggregates [Latimer, P., Appl. Opt. 24, 3231, (1985)]. I have been able to account for the variations in magnitude and spectral slope of σ[PIM](λ). This analysis will apply to not only inverting the concentration of suspended mineral matter but also provides the basis for inverting the processes of coagulation and aggregation of primary mineral particles in determining sedimentation rates, budgets, etc.

© 2011 OSA

1. Introduction

The scattering coefficient of the marine hydrosol is the source of remote sensing information about the suspended components in the ocean and the basis for mechanistic algorithms to invert the remote sensing reflectance and retrieve concentrations of materials of interest: suspended sediments, organic particulate matter, dissolved organic matter, chlorophyll concentration, etc [1

1. R. W. Gould Jr and R. A. Arnone, “Three-dimensional modelling of inherent optical properties in a coastal environment: coupling ocean colour imagery and in situ measurements,” Int. J. Remote Sens. 19(11), 2141–2159 (1998). [CrossRef]

3

3. R. W. Gould, Jr., R.H, Stavn, M. S. Twardowski, and G.M. Lamela. “Partitioning optical properties into organic and inorganic components from ocean color imagery,” in Ocean Optics XVI, Santa Fe, New Mexico, USA, S. Ackleson and C. Trees, eds. (Office of Naval Research, 2002) CDROM.

]. The absorption coefficient of the marine hydrosol is routinely coupled with the scattering coefficient to generate remote sensing algorithms [4

4. M. Sydor, R. W. Gould, R. A. Arnone, V. I. Haltrin, and W. Goode, “Uniqueness in remote sensing of the inherent optical properties of ocean water,” Appl. Opt. 43(10), 2156–2162 (2004). [CrossRef] [PubMed]

]. Furthermore the absorption coefficient is routinely partitioned by various methods into absorption due to phytoplankton, yellow substance, organic detritus, etc. to give more information about the materials suspended in the marine hydrosol [1

1. R. W. Gould Jr and R. A. Arnone, “Three-dimensional modelling of inherent optical properties in a coastal environment: coupling ocean colour imagery and in situ measurements,” Int. J. Remote Sens. 19(11), 2141–2159 (1998). [CrossRef]

,5

5. J. S. Cleveland, “Regional models for phytoplankton absorption as a function of chlorophyll a concentration,” J. Geophys. Res. 100(C7), 13,333–13,344 (1995). [CrossRef]

]). However, the scattering coefficient is not routinely partitioned except to attribute scattering to molecular water and “particles.” Suspended particulate matter can be broadly classified into inorganic, often terrigenous, matter and organic matter, some terrigenous but usually autochthonous. The two broad categories of suspended particulates have different indices of refraction and therefore rather different scattering coefficients. These differing scattering coefficients generate different effects on the remote sensing reflectance. This factor is most pronounced in coastal ocean waters and our program in biogeo-optics is now making contributions to coastal ocean remote sensing algorithms by the accurate partitioning of the particulate scattering coefficient [6

6. R. H. Stavn and S. J. Richter, “Biogeo-optics: particle optical properties and the partitioning of the spectral scattering coefficient of ocean waters,” Appl. Opt. 47(14), 2660–2679 (2008). [CrossRef] [PubMed]

].

The method of Stavn and Richter [6

6. R. H. Stavn and S. J. Richter, “Biogeo-optics: particle optical properties and the partitioning of the spectral scattering coefficient of ocean waters,” Appl. Opt. 47(14), 2660–2679 (2008). [CrossRef] [PubMed]

] is a direct mechanism for partitioning the particulate scattering coefficient from the determination of the concentrations of suspended mineral and organic matter in the marine hydrosol. Various attempts to partition the particulate scattering coefficient utilized more indirect information to accomplish this. Stavn and Keen [7

7. R. H. Stavn and T. R. Keen, “Suspended minerogenic particle distributions in high-energy coastal environments: Optical implications,” J. Geophys. Res. 109(C5), C05005 (2004), doi: (Oceans). [CrossRef]

] did this on the basis of the biogeo-optical model in which the majority of hydrosol absorption was ascribed to chlorophyll and/or CDOM while total scattering was ascribed primarily to mineral matter. Complex nonlinear optimizations have been performed utilizing information on the remote sensing reflectance and concentrations of various modes of suspended matter [8

8. R. P. Bukata, J. H. Jerome, K. Ya. Kondratyev, and D. V. Pozdnayakov, Optical Properties and Remote Sensing of Inland and Coastal Waters (CRC Press 1995).

]. Stavn and Richter [6

6. R. H. Stavn and S. J. Richter, “Biogeo-optics: particle optical properties and the partitioning of the spectral scattering coefficient of ocean waters,” Appl. Opt. 47(14), 2660–2679 (2008). [CrossRef] [PubMed]

] perform linear multiple regression of the particulate scattering coefficient against the concentrations of particulate inorganic matter (PIM) and particulate organic matter (POM), determined by loss-on-ignition analysis. The linear multiple regression applied is a new Model II type multiple regression that is valid where all the variables contain error and the independent variables are not controlled [6

6. R. H. Stavn and S. J. Richter, “Biogeo-optics: particle optical properties and the partitioning of the spectral scattering coefficient of ocean waters,” Appl. Opt. 47(14), 2660–2679 (2008). [CrossRef] [PubMed]

,9

9. E. A. Laws and J. W. Archie, “Appropriate use of regression analysis in marine biology,” Mar. Biol. 65(1), 13–16 (1981). [CrossRef]

]. This new multiple regression yields partial regression coefficients of PIM and POM against the particulate scattering coefficient which can be interpreted as mass-specific optical scattering cross sections [6

6. R. H. Stavn and S. J. Richter, “Biogeo-optics: particle optical properties and the partitioning of the spectral scattering coefficient of ocean waters,” Appl. Opt. 47(14), 2660–2679 (2008). [CrossRef] [PubMed]

]. These optical scattering cross sections plus the mass concentrations of mineral and organic matter can then be used to partition the particulate scattering coefficient into its major components, i.e. the scattering coefficient of suspended mineral matter and the scattering coefficient of suspended organic matter. PIM and POM can in principle be further partitioned to allow any number of functional relations between known inorganic and organic constituents and the particulate scattering coefficient.

Theoretical optical relations of suspended matter proposed by Stavn and Richter [6

6. R. H. Stavn and S. J. Richter, “Biogeo-optics: particle optical properties and the partitioning of the spectral scattering coefficient of ocean waters,” Appl. Opt. 47(14), 2660–2679 (2008). [CrossRef] [PubMed]

] illustrate the interrelations of a particle's size, density, refractive index, and scattering cross section that generate σ[PIM](λ). They propose that σ[PIM](λ), determined from field measurements of the particle scattering coefficient and mass concentration of mineral matter, is an optical property of a “mean mineral particle,” averaged over all mineral species in suspension. This mean mineral particle can then be characterized by the known optical properties of minerals, density and refractive index, to yield a true optical scattering cross section of the mean particle. An advantage of analysis of suspended minerals is their optical properties (density and refractive index) tend to be similar and thus easy to characterize (Table 2

Table 2. Relative Refractive Indices of Clay Minerals. Real Component n and Imaginary Component n’

table-icon
View This Table
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). An analysis of mineral optical cross sections was also applied by Green et al. [10

10. R. A. Green, H. M. Sosik, and R. J. Olson, “Contributions of phytoplankton and other particles to inherent optical properties in New England continental shelf waters,” Limnol. Oceanogr. 48(6), 2377–2391 (2003). [CrossRef]

] for characterizing mineral particulates determined from flow cytometry counts of total suspended particulates in the New England continental shelf waters. Our theoretical determinations for σ[PIM](λ) can be checked by quantitatively analyzing the mineral species in suspension through such techniques as X-ray diffraction and Energy Dispersive Spectra from the samples filtered from the coastal marine hydrosol. These data allow the analysis of the changes in σ[PIM](λ) attributable to changes in the mineral component. Additionally, field determinations of particle size spectra will allow further analysis of the mass-specific scattering cross section against predictions made from standard particle size distribution (PSD) models. The information we have at present indicates that laser-based particle sizing appears to have the best chance of not disturbing delicate structures of aggregates [12

12. E. Boss, W. Slade, and P. Hill, “Effect of particulate aggregation in aquatic environments on the beam attenuation and its utility as a proxy for particulate mass,” Opt. Express 17(11), 9408–9420 (2009). [CrossRef] [PubMed]

,13

13. R. A. Reynolds, D. Stramski, V. M. Wright, and S. B. Woźniak, “Measurements and characterization of particle size distributions in coastal waters,” J. Geophys. Res. 115(C8C08024), C08024 (2010), doi:. [CrossRef] [PubMed]

].

Risović [24

24. D. Risović, “Two component model of sea particle size distribution,” Deep Sea Res. Part I Oceanogr. Res. Pap. 40(7), 1459–1473 (1993). [CrossRef]

], in an extensive analysis of published PSD's, has proposed a model of particle size distribution in the sea made up of two components: component A (small sized) and component B (large sized), each described by 3 parameters of a generalized or extended gamma distribution. The extended gamma distribution has been well described by Deirmendjian [28

28. D. Deirmendjian, “Scattering and polarizaton properites of water clouds and hazes in the visible and infrared,” Appl. Opt. 3(2), 187–196 (1964). [CrossRef]

,29

29. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elesevier 1969).

] and Zdunkowski et al. [30

30. W. Zdunkowski, T. Trautman, and A. Bott, Radiation in the Atmosphere: A Course in Theoretical Meterology. (Cambridge 2007).

]. The parameters of the two-component model published by Risović [24

24. D. Risović, “Two component model of sea particle size distribution,” Deep Sea Res. Part I Oceanogr. Res. Pap. 40(7), 1459–1473 (1993). [CrossRef]

] describe a summation of all the gamma-distributed particle components in the ocean. They provide a useful starting point for analyzing the distribution of subsets of the total array of materials suspended in the ocean. Peng, et al. [17

17. F. Peng, S. W. Effler, D. O’Donnell, M. G. Perkins, and A. Weidemann, “Role of minerogenic particles in light scattering in lakes and a river in central New York,” Appl. Opt. 46(26), 6577–6594 (2007). [CrossRef] [PubMed]

,18

18. F. Peng, S. Effler, D. O'Donnell, A. Weidemann, and M. T. Auer, “Characterization of minerogenic particles in support of modeling light scattering in Lake Superior through a two-component approach,” Limnol. Oceanogr. 54(4), 1369–1381 (2009). [CrossRef]

] have successfully applied extended gamma distribution parameters for the B component to the PSD of suspended minerals in freshwater lakes and rivers. An advantage of the gamma distribution is the distribution shape and location parameters associated with it that help in the description and visualization of a particle size distribution. For this study of the optical properties of suspended minerals in the coastal ocean I have chosen the power and flexibility of Risović's approach for the PSD derived for this analysis.

When considering the effects of suspended mineral matter on the optics of coastal ocean waters, there is increasing attention being given to the coagulation of small mineral particles (approximately 1 μm diameter) into larger amorphous aggregates. Boss et al. [12

12. E. Boss, W. Slade, and P. Hill, “Effect of particulate aggregation in aquatic environments on the beam attenuation and its utility as a proxy for particulate mass,” Opt. Express 17(11), 9408–9420 (2009). [CrossRef] [PubMed]

] point out the significance of mineral particulate aggregations and discuss their effects on some optical properties of suspended minerals and their aggregates. The aggregation of smaller mineral particles is significant because in a mineral aggregate the optical properties can no longer be modeled with unmodified Mie scattering theory. For example, the porosity of the mineral aggregate means that optically the aggregate becomes a mixture of mineral properties and water properties. Latimer [31

31. P. Latimer, “Experimental tests of a theoretical method for predicting light scattering by aggregates,” Appl. Opt. 24(19), 3231–3239 (1985). [CrossRef] [PubMed]

], however, has proposed a simple optical theory for aggregate particles that involves Mie calculations for a coated sphere, the outer layer being the sum of the mineral mass of the aggregate and inner core being water. I have been able to apply the conclusions of Latimer to determine the theoretical optical properties of the mineral aggregate and its contribution to the mineral mass-specific scattering cross section σ[PIM](λ). I have chosen the model of Khelifa and Hill [32

32. A. Khelifa and P. S. Hill, “Models for effective density and settling velocity of flocs,” J. Hydraul. Res. 44(3), 390–401 (2006). [CrossRef]

], favored by Boss et al. [12

12. E. Boss, W. Slade, and P. Hill, “Effect of particulate aggregation in aquatic environments on the beam attenuation and its utility as a proxy for particulate mass,” Opt. Express 17(11), 9408–9420 (2009). [CrossRef] [PubMed]

], for modeling mineral particle aggregates. The Khelifa-Hill model describes the aggregate as a fractal entity composed of smaller primary particles, such as clay particles of about 1 μm diameter or so. Risović and Martinis [33

33. D. Risović and M. Martinis, “Fractal dimensions of suspended particles in seawater,” J. Colloid Interface Sci. 182(1), 199–203 (1996). [CrossRef]

] have explored the interrelations of the fractal nature of suspended aggregates and the two-component model [24

24. D. Risović, “Two component model of sea particle size distribution,” Deep Sea Res. Part I Oceanogr. Res. Pap. 40(7), 1459–1473 (1993). [CrossRef]

] of particle size distribution in the sea. The formation and breakup of aggregations is known to affect the particle scattering coefficient. The relative increase or decrease of total particles will correspondingly affect the magnitude of the particle scattering coefficient [12

12. E. Boss, W. Slade, and P. Hill, “Effect of particulate aggregation in aquatic environments on the beam attenuation and its utility as a proxy for particulate mass,” Opt. Express 17(11), 9408–9420 (2009). [CrossRef] [PubMed]

,34

34. A. Morel, “Optics of marine particles and marine optics,” in Particle Size Analysis, S. Demers, ed. (Springer Verlag, 1991), pp. 141–188.

,35

35. I. N. McCave, “Local and global aspects of the bottom nepheloid layers in the world ocean,” Neth. J. Sea Res. 20(2-3), 167–181 (1986). [CrossRef]

]. Furthermore, the tendency to scavenge either all particles equally or to remove some particle size classes preferentially will change the PSD. This will then affect the spectral scattering slope of the particle scattering coefficient [36

36. P. S. Hill, G. Voulgaris, and J. H. Trowbridge, “Controls on floc size in a continental shelf bottom boundary layer,” J. Geophys. Res. 106(C5), 9543–9549 (2001). [CrossRef]

]. The size range of suspended particles that will have the strongest effect on the particle scattering coefficient will be those of < 10 μm diameter [34

34. A. Morel, “Optics of marine particles and marine optics,” in Particle Size Analysis, S. Demers, ed. (Springer Verlag, 1991), pp. 141–188.

].

The purpose of this exercise is to explore the theoretical optical properties of the mineral mass-specific scattering cross section and how they are affected by suspended mineral aggregates. This is inspired in part by the divergence of mineral mass-specific scattering cross section results reported by Stavn and Richter [6

6. R. H. Stavn and S. J. Richter, “Biogeo-optics: particle optical properties and the partitioning of the spectral scattering coefficient of ocean waters,” Appl. Opt. 47(14), 2660–2679 (2008). [CrossRef] [PubMed]

] for Mobile Bay,Alabama, USA and Southwest Pass, mouth of the Mississippi River, USA and by Bowers and Binding [37

37. D. G. Bowers and C. E. Binding, “The optical properties of mineral suspended particles: A review and synthesis,” Estuar. Coast. Shelf Sci. 67(1-2), 219–230 (2006). [CrossRef]

] and McKee and Cunningham [38

38. D. M. McKee and A. Cunningham, “Identification and characterization of two optical water types in the Irish Sea from in situ inherent optical properties and seawater constituents,” Estuar. Coast. Shelf Sci. 68(1-2), 305–316 (2006). [CrossRef]

] for the Irish Sea, UK. The results will be compared with literature values on the optical properties of air dust suspensions in seawater [19

19. M. Stramska, D. Stramski, M. Cichocka, A. Cieplak, and S. B. Wozniak, “Effects of atmospheric particles from Southern California on the optical properties of seawater,” J. Geophys. Res. 113(C8), C08037 (2008), doi:. [CrossRef]

21

21. D. Stramski, M. Babin, and S. B. Wozniak, “Variations in the optical properties of terrigenous mineral-rich particulate matter suspended in seawater,” Limnol. Oceanogr. 52(6), 2418–2433 (2007). [CrossRef]

]. These results will also have a bearing on the issue of whether or not there is a spectral slope to the particle scattering coefficient in the ocean [39

39. R. W. Gould Jr, R. A. Arnone, and P. M. Martinolich, “Spectral dependence of the scattering coefficient in case 1 and case 2 waters,” Appl. Opt. 38(12), 2377–2383 (1999). [CrossRef] [PubMed]

].

2. Methods

I modeled mineral aggregates of both montmorillonite and illite primary particles. The mineral aggregates were modeled from the fractal-based model of Khelifa and Hill [32

32. A. Khelifa and P. S. Hill, “Models for effective density and settling velocity of flocs,” J. Hydraul. Res. 44(3), 390–401 (2006). [CrossRef]

]. The refractive indices of Montmorillonite and Illite were used to model the refractive index of the suspended mineral particles. Their refractive indices represent approximate limiting values of the optical properties of the typical clay minerals commonly found in the coastal ocean, with Montmorillonite often the dominant component [47

47. L. J. Doyle and T. N. Sparks, “Sediments of the Mississippi, Alabama, and Florida (MAFLA) continental shelf,” J. Sediment. Petrol. 50, 905–916 (1980).

50

50. A. G. Johnson and J. T. Kelley, “Temporal, spatial, and textural variation in the mineralogy of Mississippi river suspended sediment,” J. Sediment. Petrol. 54, 67–72 (1984).

]. And, other suspended mineral components have refractive indices close to that of clay minerals [48

48. W.G. Egan and T.W. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic 1979).

]. Thus, I assumed that the optical properties of clay minerals would serve as a model of the optical properties of suspended minerals in general. The Khelifa–Hill model was then utilized to determine the relevant physical characteristics of the aggregate particles. Kehlifa-Hill predicts the porosity and density of clay mineral aggregates up to 2000 μm diameter, the aggregates composed of primary particles, 1 μm diameter. It describes an amorphous mineral aggregate with fractal geometry. The fundamental concept of fractal geometry is the fractal dimension that, in this case, summarizes the shape and relative porosity of a mineral aggregate. The fractal dimension is analogous to the dimensions of real objects [46

46. T. A. Witten and M. E. Cates, “Tenuous structures from disorderly growth processes,” Science 232(4758), 1607–1612 (1986). [CrossRef] [PubMed]

]. The fractal dimension of 3 indicates a 3-dimensional spherical object, tightly packed primary particles perhaps. The Fractal dimension of 2 indicates a 2-dimensional object, which could be considered a “sheet” composed of randomly branching primary particles. The fractal dimension of 1 indicates an object composed of randomly collected particles in a predominantly linear structure. Fractal dimensions in between these limits would then have intermediate structural properties. The fractal dimension F for the clay mineral flocs was determined with the equation
F=α(Dfd)β,
(6)
where α = 3.0, Df is the equivalent spherical diameter of the floc in μm, Dfc is the equivalent spherical diameter of a maximal characteristic floc size, here 2000 μm, Fc is the fractal dimension of the maximal characteristic floc size, d (primary particle diameter) = 1.0 μm, β = log(Fc/3)/log(Dfc/d). The Khelifa-Hill model [32

32. A. Khelifa and P. S. Hill, “Models for effective density and settling velocity of flocs,” J. Hydraul. Res. 44(3), 390–401 (2006). [CrossRef]

] is expressed as a “median” value and an upper and lower model limit that account for the majority of the experimental and observational values of the clay mineral aggregation studies in their database. These three versions of the model illustrated in Khelifa and Hill [32

32. A. Khelifa and P. S. Hill, “Models for effective density and settling velocity of flocs,” J. Hydraul. Res. 44(3), 390–401 (2006). [CrossRef]

] were utilized in this study. The model is based on a maximal size for a clay mineral aggregate, an assumed 2000 μm diameter, and a fractal dimension Fc of this maximal aggregate. The fractal dimension for a clay mineral aggregate decreases monotonically from 3.0 for an “aggregate” that is the diameter of the primary particle to the value of Fc for the maximal aggregate diameter. The three model versions are controlled by the value of Fc: 2.4, 2.0, and 1.6, for the maximal aggregate diameter. In turn, these values determine the value of β: – 0.02936,.− 0.0533, and − 0.08270 respectively. The clay mineral aggregate models were referred to as the Low, Standard, and High Khelifa-Hill models. We can see that the dimension of the low fractal limit model predicts an aggregate that is relatively loosely and randomly organized. The high dimension model predicts a more tightly packed clay mineral aggregate. The Khelifa-Hill model then predicts the density of an aggregate or floc based on the fractal modeling of the aggregate. As a floc gets bigger its density becomes lower, i.e. the fractal dimension decreases, and this is important for predicting the settling velocity of the floc and for estimating the optical properties of the floc [12

12. E. Boss, W. Slade, and P. Hill, “Effect of particulate aggregation in aquatic environments on the beam attenuation and its utility as a proxy for particulate mass,” Opt. Express 17(11), 9408–9420 (2009). [CrossRef] [PubMed]

]. Τhe effective density of the floc is termed the excess density and it was determined from
ρrρw=(ρsρw)(Dfd)F3,
(7)
where ρs = density of the primary particle (montmorillonite or illite), ρw = the density of water, ρr = the density of the floc (montmorillonite or illite + water). These 3 densities made it possible to construct an optical model of the aggregate proposed by Latimer [31

31. P. Latimer, “Experimental tests of a theoretical method for predicting light scattering by aggregates,” Appl. Opt. 24(19), 3231–3239 (1985). [CrossRef] [PubMed]

], a coated sphere, of the various sized montmorillonite or illite flocs. Latimer’s [31

31. P. Latimer, “Experimental tests of a theoretical method for predicting light scattering by aggregates,” Appl. Opt. 24(19), 3231–3239 (1985). [CrossRef] [PubMed]

] model consists of the mass of the mineral portion of the floc condensed into a shell of montmorillonite or illite surrounding a core of water. The index of refraction used was that for either montmorillonite or illite and the imaginary components of the refractive indices were determined for the wavelengths utilized, Table 2 [48

48. W.G. Egan and T.W. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic 1979).

]. As per Latimer's specifications, coated sphere Mie calculations were performed to determine the scattering efficiency and particle scatteringcross section of a floc (B component) of a particular diameter. Fortran code for the Mie calculation of both a solid and a coated sphere came from Bohren and Huffman [51

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

]. The particles of Montmorillonite and Illite, somewhat larger than 1 μm radius, representing the A component, were assumed to be essentially spheres of tightly packed primary particles. Ordinary Mie calculations of a solid homogeneous sphere were performed on the primary clay mineral particles (A component) to determine their scattering efficiencies and particle scattering cross sections. Risović and Martinis [33

33. D. Risović and M. Martinis, “Fractal dimensions of suspended particles in seawater,” J. Colloid Interface Sci. 182(1), 199–203 (1996). [CrossRef]

] demonstrated that the fractal dimension of A component particles in the coastal ocean varied from about 2.7 – 3.0, essentially tightly packed spherical particles or aggregates. The primary particles that contributed the most to the calculation of the mass-specific scattering cross section varied from 0.5 - 3.0 μm radius (1.0 - 6.0 μm diameter) while the most important equivalent spherical radii of the montmorillonite and illite aggregates varied from 2.0 - 8 μm (4.0 - 16 μm diameter). Any primary particles or aggregates larger than those mentioned above did not contribute significantly to the mass-specific scattering cross section. The wavelengths used in the calculations were: 410 nm, 440 nm, 488 nm, 510 nm, 532 nm, 550 nm, 650 nm, 676 nm, and 715 nm.

With all of the information supplied above, we are now ready to calculate theoretically the average mineral mass-specific scattering cross section [52

52. J. B. Austin, “Methods of representing distribution of particle size,” Ind. Eng. Chem. Anal. Ed. 11(6), 334–339 (1939). [CrossRef]

] of a hydrosol σ[PIM](λ) and determine its limits. Stavn and Richter [6

6. R. H. Stavn and S. J. Richter, “Biogeo-optics: particle optical properties and the partitioning of the spectral scattering coefficient of ocean waters,” Appl. Opt. 47(14), 2660–2679 (2008). [CrossRef] [PubMed]

] derived the theoretical value of the mineral mass-specific scattering cross section determined in the field
σ[PIM](λ)=ij[σm(λ)ρmvm]ij,
(8)
where σm(λ) is the particle (primary or aggregate/floc) scattering cross section, here determined from the Mie calculations, λ denotes light wavelength in a vacuum, ρm is the mineral density, either as primary particle or as a floc, vm is the volume of either a primary mineral particle or of a mineral aggregate, i subscript indicates summation over particle species, j subscript indicates summation over the weighted size classes of a particular mineral species. I utilized mineral mass-specific scattering cross sections from Mobile Bay, Alabama, USA, Southwest Pass, mouth of the Mississippi, USA [6

6. R. H. Stavn and S. J. Richter, “Biogeo-optics: particle optical properties and the partitioning of the spectral scattering coefficient of ocean waters,” Appl. Opt. 47(14), 2660–2679 (2008). [CrossRef] [PubMed]

], and the Irish Sea, UK [37

37. D. G. Bowers and C. E. Binding, “The optical properties of mineral suspended particles: A review and synthesis,” Estuar. Coast. Shelf Sci. 67(1-2), 219–230 (2006). [CrossRef]

,38

38. D. M. McKee and A. Cunningham, “Identification and characterization of two optical water types in the Irish Sea from in situ inherent optical properties and seawater constituents,” Estuar. Coast. Shelf Sci. 68(1-2), 305–316 (2006). [CrossRef]

] to compare with the theory.

3. Results

The two-component mineral model particle size distribution dominated by primary particles isplotted in Fig. 1
Fig. 1 Particle size distribution for primary mineral particle dominated PSD. The ratio of small particles to large in this model is NA/NB = 1.55. The parameters of the Two-component Mineral Model applied here are: CA = 6.0 x 1023, CB = 1.55 x 1012, μA = 3.2, μB = 5.0, γA = 0.12, γB = 2.35
. The dominance by primary particles is indicated by the ratio NA/NB = 1.55. The distribution of the mineral aggregates of component B occurs at least one order of magnitude lower than the peaked portion of the distribution of primary particles, component A. The particle size distribution in which the primary particles and the aggregate particles are of about the same order of magnitude concentration is plotted in Fig. 2
Fig. 2 Particle size distribution for equivalent primary and aggregated mineral PSD. The ratio of small particles to large in this model is NA/NB = 0.42. The parameters of the Two-component Model applied here are: CA = 6.0 x 1022, CB = 1.75 x 1012, μA = 3.2, μB = 5.0, γA = 0.15, γB = 2.35
. The near equivalenceof primary particles and aggregated particles is indicated by the ratio NA/NB = 0.42. The particle size distribution that is dominated by mineral aggregates is plotted in Fig. 3
Fig. 3 Particle size distribution for mineral aggregate dominated PSD. The ratio of small particles to large in this model is NA/NB = 0.135. The parameters of the distribution applied here are: CA = 6.0 x 1023, CB = 1.75 x 1013, μA = 3.2, μB = 5.0, γA = 0.12, γB = 2.35
. The mineral aggregates occur at significantly greater concentration than the primary mineral particles, indicated by the ratio NA/NB = 0.135, nearly an order of magnitude higher. The essential difference between these distributions is the CA/CB ratio (Table 2) which serves to simply move the distribution of the mineral aggregates, component B, up or down relative to the distribution of primary particles, component A, and redistribute the particle concentrations. It is possible to see that the log plot of total particles with radius, in some cases, could be interpreted as a constant slope in some sections of the distribution, especially the larger radii (Figs. 1-3).

Mineral mass-specific scattering cross sections σ[PIM](λ) calculated from the aggregate models utilized here (PSD's delineated in Figs. 1-3) and field data [6

6. R. H. Stavn and S. J. Richter, “Biogeo-optics: particle optical properties and the partitioning of the spectral scattering coefficient of ocean waters,” Appl. Opt. 47(14), 2660–2679 (2008). [CrossRef] [PubMed]

] with new correction factors are plotted in Figs. 4
Fig. 4 Mass-specific scattering cross sections for primary particle dominated PSD compared to Mobile Bay, Alabama, USA results (Stavn and Richter, 2008). Results from Montmorillonite and Illite aggregate models determined at low, median, and high fractal dimensions.
and 5
Fig. 5 Mass-specific scattering cross sections for primary and aggregate co-dominant PSD and compared to Southwest Pass, mouth of the Mississippi, USA data (Stavn and Richter, 2008). Results from Montmorillonite and Illite aggregate models determined at low, median, and high fractal dimensions.
and the comparisons with Irish Sea data [37

37. D. G. Bowers and C. E. Binding, “The optical properties of mineral suspended particles: A review and synthesis,” Estuar. Coast. Shelf Sci. 67(1-2), 219–230 (2006). [CrossRef]

,38

38. D. M. McKee and A. Cunningham, “Identification and characterization of two optical water types in the Irish Sea from in situ inherent optical properties and seawater constituents,” Estuar. Coast. Shelf Sci. 68(1-2), 305–316 (2006). [CrossRef]

] are in Fig. 6
Fig. 6 Mass-specific scattering cross sections for aggregate dominated PSD compared with results from the Irish Sea (Bowers and Binding, 2006, McKee and Cunningham, 2006). Results from Montmorillonite and Illite aggregate models determined at low, median, and high fractal dimensions.
. In Fig. 4 we compare the calculated plots of σ[PIM](λ) for the primary particle dominated PSD with the σ[PIM](λ) plot for Mobile Bay, Alabama, USA. The values of σ[PIM](λ) in Mobile Bay varied from 0.84 – 0.64 m2 g−1, from the blue to the red end of the spectrum, andthese values were the highest of any field data recorded. The plots from all the models demonstrate values of σ[PIM](λ) of 1.03–0.59 m2 g−1 from the blue to the red end of the spectrum. Most of the model values tend to fall above the values of the field data. All the model plots fall within the standard errors for the field data at the wavelengths 650 nm – 715 nm. The Montmorillonite Low Model falls above the standard error of the field data at all shorter wavelengths. All models fall above the standard error of the field value at 412 nm. All models except the Montmorilllonite Low Model just equal or fall inside the upper standard error value at the wavelengths of 488 nm – 550 nm. The spectral slopes for models and field data appear to be comparable, are negative from the blue to the red end of the spectrum, and appear to be the largest slopes recorded from the field. The extreme model values are represented by the Montmorillonite Low Model and the Illite High Model. The closest fits appear to be Illite in either the Standard or High Models.

In Fig. 5, we compare the plots of σ[PIM](λ) from the intermediate particle PSD to the plot from Southwest Pass, Mississippi River, USA. The values of σ[PIM](λ) at the Southwest Pass varied from 0.63 – 0.51 m2 g−1, from the blue to the red end of the spectrum, and these valueswere in the mid-range of any recorded from the field. The plots from all the models demonstrate values of σ[PIM](λ) of 0.69 – 0.37 m2 g−1 from the blue to the red end of the spectrum. All of the model plots fall within the standard errors determined for the field data with the exception of the Illite High and Montmorillonite High Models at 676 nm and 715 nm. Most of the model values tend to fall below the values of the field data. The spectral slopes for models and the field data appear to be comparable with the Illite Low Model and the Montmorillonite Low Model having slopes closest to the field data. The spectral slopes are negative from the blue to the red end of the spectrum and they appear to be less than the spectral slopes for the primary particle dominant PSD in Fig. 4. The extreme model values are represented by the Montmorillonite Low model and the Illite High model. The closest fit appears to be the Illite Low model.

In Fig. 6, we compare the plots of calculated σ[PIM](λ) values from the aggregate dominated PSD models with two plots of field data from the Irish Sea, UK [37

37. D. G. Bowers and C. E. Binding, “The optical properties of mineral suspended particles: A review and synthesis,” Estuar. Coast. Shelf Sci. 67(1-2), 219–230 (2006). [CrossRef]

,38

38. D. M. McKee and A. Cunningham, “Identification and characterization of two optical water types in the Irish Sea from in situ inherent optical properties and seawater constituents,” Estuar. Coast. Shelf Sci. 68(1-2), 305–316 (2006). [CrossRef]

]. The values of σ[PIM](λ) from the Irish Sea vary from approximately 0.23 to 0.34 m2 g−1 at the blueend of the spectrum to 0.29 to 0.36 m2 g−1 at the yellow/red end of the spectrum and these are the lowest recorded field values. The values from aggregate dominated PSD models for σ[PIM](λ) cluster reasonably closely around the Irish Sea data but no model settles completely within the standard deviation limits of the field data. The field data indicate either no significant spectral slope to the Irish Sea data or perhaps a positive spectral slope from the blue to the red end of the spectrum. In addition, there is a possible spectral peak in the region around 550 nm in the field data. The model calculations tend to fall equally above and below the field data. The extreme values from the model calculations occur with the Montmorillonite Low Model to the Illite High Model. The Illite Low Model appears have a spectral slope comparable to the data from Mckee and Cunningham [38

38. D. M. McKee and A. Cunningham, “Identification and characterization of two optical water types in the Irish Sea from in situ inherent optical properties and seawater constituents,” Estuar. Coast. Shelf Sci. 68(1-2), 305–316 (2006). [CrossRef]

] and for the most part hovers just above the standard deviation limits of the data. The Illite Low Model calculations fit within the standard deviation limits of the middle portion of the Bowers and Binding [37

37. D. G. Bowers and C. E. Binding, “The optical properties of mineral suspended particles: A review and synthesis,” Estuar. Coast. Shelf Sci. 67(1-2), 219–230 (2006). [CrossRef]

] data. The calculations from the Montmorillonite Low Model demonstrate a comparable spectral slope. There is an interesting “hinge point” with the model calculations in which the spectral slopes are either small to 0 or varying between being negative or positive The field data reported from Bowers and Binding [37

37. D. G. Bowers and C. E. Binding, “The optical properties of mineral suspended particles: A review and synthesis,” Estuar. Coast. Shelf Sci. 67(1-2), 219–230 (2006). [CrossRef]

] are comparable to much of the data of McKee and Cunningham [38

38. D. M. McKee and A. Cunningham, “Identification and characterization of two optical water types in the Irish Sea from in situ inherent optical properties and seawater constituents,” Estuar. Coast. Shelf Sci. 68(1-2), 305–316 (2006). [CrossRef]

] but diverge somewhat at at the blue and yellow/red ends. There is a positive spectral slope to the Bowers and Binding [37

37. D. G. Bowers and C. E. Binding, “The optical properties of mineral suspended particles: A review and synthesis,” Estuar. Coast. Shelf Sci. 67(1-2), 219–230 (2006). [CrossRef]

] data if we discount the data point at 650 nm

4. Discussion

We have demonstrated theoretical characteristics of the mass-specific scattering cross section of suspended mineral matter that are verified in part by experimental observations from field studies and this will have practical applications. I have proposed a two-component mineral PSD model based on Risović’s two-component model for the total particle PSD. The smaller (A) component of the two-component mineral model is assumed to be primary mineral particles. The larger (B) component of the PSD model is assumed to be aggregates composed of the primary particles. The results presented here will potentially apply to all studies of suspended mineral matter in the coastal ocean and inland waters.

There is a series of experimental studies that have a bearing on what we have reported here [19

19. M. Stramska, D. Stramski, M. Cichocka, A. Cieplak, and S. B. Wozniak, “Effects of atmospheric particles from Southern California on the optical properties of seawater,” J. Geophys. Res. 113(C8), C08037 (2008), doi:. [CrossRef]

21

21. D. Stramski, M. Babin, and S. B. Wozniak, “Variations in the optical properties of terrigenous mineral-rich particulate matter suspended in seawater,” Limnol. Oceanogr. 52(6), 2418–2433 (2007). [CrossRef]

]. These studies report primarily on the mass-specific scattering coefficient, an empirical ratio of the total particle scattering coefficient and the total concentration of suspended matter, both mineral and organic. Thus the empirical ratio is not exactly defined optically, as is done in Eq. (8) and Stavn and Richter [6

6. R. H. Stavn and S. J. Richter, “Biogeo-optics: particle optical properties and the partitioning of the spectral scattering coefficient of ocean waters,” Appl. Opt. 47(14), 2660–2679 (2008). [CrossRef] [PubMed]

], but this empirical ratio can approach the exactly defined values of Eq. (8). Stramski et al. [21

21. D. Stramski, M. Babin, and S. B. Wozniak, “Variations in the optical properties of terrigenous mineral-rich particulate matter suspended in seawater,” Limnol. Oceanogr. 52(6), 2418–2433 (2007). [CrossRef]

] performed experiments on clay minerals suspended in sea water and these values are a close approximation to an exactly defined and investigated mass-specific scattering cross section. These suspensions were not corrected for the presence of organic matter in suspension with the clay minerals. The majority of the experiments are performed on air dust samples that have not been analyzed as to the type of minerals in the dust and the amount of organic materials in the dust such as black carbon. Another factor that renders the experimental ratios less than comparable with what is reported here is that suspended minerals, being dense, are notoriously difficult to work with and maintain in a stable, repeatable state in suspension [44

44. C. L. Gallegos and R. G. Menzel, “Submicron size distribution of inorganic suspended solids in turbid waters by photon correlation spectroscopy,” Water Resour. Res. 23(4), 596–602 (1987). [CrossRef]

]. Therefore, many of the reported experimental results may simply be a function of how a particular suspension of minerals or dust was prepared and handled. The particle size distributions for the suspensions were determined with a Coulter counter which has unknown effects on the aggregation state of suspended particles, a major topic of investigation here.

Mass-specific scattering coefficients were reported in Stramski et al. [21

21. D. Stramski, M. Babin, and S. B. Wozniak, “Variations in the optical properties of terrigenous mineral-rich particulate matter suspended in seawater,” Limnol. Oceanogr. 52(6), 2418–2433 (2007). [CrossRef]

] for montmorillonite, illite, and calcite that closely approached σ[PIM](λ) values as reported in this study, defined in Eq. (8). The values approximating σ[PIM](λ) for calcite in the range 400 nm to 700 nm were 1.6 – 1.2 m2 g−1 for a small particle dominated PSD and 1.18 – 0.92 m2 g−1 for a large particle dominated PSD, both of which had a negative spectral slope. These values tend to bracket the σ[PIM](λ) for calcite liths of approximately 1.0 μm equivalent spherical diameter reported in Stavn and Richter [6

6. R. H. Stavn and S. J. Richter, “Biogeo-optics: particle optical properties and the partitioning of the spectral scattering coefficient of ocean waters,” Appl. Opt. 47(14), 2660–2679 (2008). [CrossRef] [PubMed]

]. A negative spectral slope was recorded for both montmorillonite and illite suspensions [21

21. D. Stramski, M. Babin, and S. B. Wozniak, “Variations in the optical properties of terrigenous mineral-rich particulate matter suspended in seawater,” Limnol. Oceanogr. 52(6), 2418–2433 (2007). [CrossRef]

] with a mass-specific scattering coefficient of 0.84 – 0.63 m2 g−1 for 412 nm – 715 nm which accords well with the σ[PIM](λ) values reported for Mobile Bay [6

6. R. H. Stavn and S. J. Richter, “Biogeo-optics: particle optical properties and the partitioning of the spectral scattering coefficient of ocean waters,” Appl. Opt. 47(14), 2660–2679 (2008). [CrossRef] [PubMed]

] and the theoretical calculations in Fig. 4. These results of Stramski et al. [21

21. D. Stramski, M. Babin, and S. B. Wozniak, “Variations in the optical properties of terrigenous mineral-rich particulate matter suspended in seawater,” Limnol. Oceanogr. 52(6), 2418–2433 (2007). [CrossRef]

] were recorded for a large particle PSD. There were no flat spectral slopes for the mass-specific scattering coefficients of mineral suspensions but there were flat spectral slopes reported for the mass-specific scattering coefficients of air dust suspensions. The approximate value for Sahara Dust across the visible spectrum was 0.75 m2 g−1 which is higher than the nearly flat spectral values for σ[PIM](λ) reported from the Irish Sea and our theoretical calculations in Fig. 6. From the results of this report it appears that a large value of a flat spectrum could be attributed to very loosely packed aggregate particles of the air dust. Stramski et al. [21

21. D. Stramski, M. Babin, and S. B. Wozniak, “Variations in the optical properties of terrigenous mineral-rich particulate matter suspended in seawater,” Limnol. Oceanogr. 52(6), 2418–2433 (2007). [CrossRef]

] also state the possibility that scattering by relatively small mineral particles might have been coupled with strong absorption in the blue region of the spectrum by organic matter to yield a relatively flat spectrum for the mass-specific scattering coefficient. Also, lower values of the mass-specific scattering coefficient that accord with the results of Fig. 6 were reported by Stramska et al. [19

19. M. Stramska, D. Stramski, M. Cichocka, A. Cieplak, and S. B. Wozniak, “Effects of atmospheric particles from Southern California on the optical properties of seawater,” J. Geophys. Res. 113(C8), C08037 (2008), doi:. [CrossRef]

] for air dust samples that were not allowed to settle as had been the case for the earlier reports [20

20. D. Stramski, S. B. Wozniak, and P. J. Flatau, “Optical properties of Asian mineral dust suspended in seawater,” Limnol. Oceanogr. 49(3), 749–755 (2004). [CrossRef]

,21

21. D. Stramski, M. Babin, and S. B. Wozniak, “Variations in the optical properties of terrigenous mineral-rich particulate matter suspended in seawater,” Limnol. Oceanogr. 52(6), 2418–2433 (2007). [CrossRef]

]. This sample treatment retained larger particles but the difficulties of maintaining stable mineral suspensions were indicated in the erratic, highly variable values of the optical coefficients for these suspensions. Reversal of the spectral slope of the mass-specific scattering coefficients to a positive spectral slope was not reported. However, the spectral slope of the mass-specific scattering coefficient for Oahu Air Dust reported in Stramski et al. [21

21. D. Stramski, M. Babin, and S. B. Wozniak, “Variations in the optical properties of terrigenous mineral-rich particulate matter suspended in seawater,” Limnol. Oceanogr. 52(6), 2418–2433 (2007). [CrossRef]

] indicates a possible positive spectral slope from 400 nm – 650 nm with a tendency to flatten at longer wavelengths. In this study we have reported a reversal of spectral slope in the field data of the Irish Sea and in our theoretical calculations of σ[PIM](λ).

The results of this study may have a bearing on a discussion in the literature concerning the spectral slope of the marine hydrosol scattering coefficient, which is primarily affected by the suspended particulates of the hydrosol. Gould et al. [39

39. R. W. Gould Jr, R. A. Arnone, and P. M. Martinolich, “Spectral dependence of the scattering coefficient in case 1 and case 2 waters,” Appl. Opt. 38(12), 2377–2383 (1999). [CrossRef] [PubMed]

] report that the hydrosol scattering coefficient of the coastal ocean has a negative spectral slope, from the blue to the red end of the spectrum. This spectral slope decreases as one goes from the coastal ocean to the clear water of the open ocean. Others have reported difficulty in reproducing this result [55

55. A. H. Barnard, W. S. Pegau, and J. R. V. Zaneveld, “Global relationships of the inherent optical properties of the oceans,” J. Geophys. Res. 103(C11), 24955–24968 (1998). [CrossRef]

] even though Gould et al. [39

39. R. W. Gould Jr, R. A. Arnone, and P. M. Martinolich, “Spectral dependence of the scattering coefficient in case 1 and case 2 waters,” Appl. Opt. 38(12), 2377–2383 (1999). [CrossRef] [PubMed]

] quote similar studies from sediment-laden river mouths, etc. The results of Gould et al. [39

39. R. W. Gould Jr, R. A. Arnone, and P. M. Martinolich, “Spectral dependence of the scattering coefficient in case 1 and case 2 waters,” Appl. Opt. 38(12), 2377–2383 (1999). [CrossRef] [PubMed]

] were from readings taken in the upper 1.5 meters while others, Barnard et al. [55

55. A. H. Barnard, W. S. Pegau, and J. R. V. Zaneveld, “Global relationships of the inherent optical properties of the oceans,” J. Geophys. Res. 103(C11), 24955–24968 (1998). [CrossRef]

], took optical readings from the surface to the bottom. It is entirely possible, in the case of the coastal ocean, that near surface samples such as from Gould et al. [39

39. R. W. Gould Jr, R. A. Arnone, and P. M. Martinolich, “Spectral dependence of the scattering coefficient in case 1 and case 2 waters,” Appl. Opt. 38(12), 2377–2383 (1999). [CrossRef] [PubMed]

] would be dominated by smaller primary mineral particles that would sink slowly compared to larger aggregates. We have demonstrated that in a mineral dominated system of mostly smaller primary mineral particles (Fig. 4) one would expect a negative spectral slope for the scattering coefficient. Furthermore, sinking aggregates can grow in size in the process of aggregation by differential sedimentation which would increase the relative contribution of the B component (larger particles) of the PSD.to the scattering coefficient [22

22. D. Risović and M. Martinis, “The role of coagulation and sedimentation mechanisms in the two-component model of sea-particle size distribution,” Fizika B: J. Exp. Theoret. Phys. (Zagreb, Croatia) 2, 103–118 (1994).

]. These mineral aggregates, if dominant, would generate a smaller spectral slope to none to possibly a positive spectral slope (Figs. 5,6). This might explain the discrepancies reported in the above studies.

5. Conclusions

Both montmorillonite and illite appear to provide usable models of the refractive index of suspended mineral particles and mineral aggregates. More field data may allow the determination of a reasonable “average” value for the refractive index of suspended mineral matter for the purposes of biogeo-optical modeling of the coastal ocean and providing new, accurate remote sensing algorithms for the coastal ocean. The clay mineral aggregate model of Khelifa and Hill [32

32. A. Khelifa and P. S. Hill, “Models for effective density and settling velocity of flocs,” J. Hydraul. Res. 44(3), 390–401 (2006). [CrossRef]

] has provided valuable insights into when the degree of “packing” of an aggregate, expressed as the fractal dimension, may have a bearing on the magnitude and the slope of the mass-specific scattering cross section. In addition, the two-component mineral model, inspired by Risović’s two-component total particle formulation, utilizing the extended gamma distribution for the PSD of the suspended particles, gives us great flexibility in investigating the dynamics of particle aggregation and their effects on hydrosol optical properties [22

22. D. Risović and M. Martinis, “The role of coagulation and sedimentation mechanisms in the two-component model of sea-particle size distribution,” Fizika B: J. Exp. Theoret. Phys. (Zagreb, Croatia) 2, 103–118 (1994).

].

Acknowledgments

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C. E. Binding, D. G. Bowers, and E. G. Mitchelson-Jacob, “An algorithm for the retrieval of suspended sediment concentrations in the Irish Sea from SeaWiFS ocean colour satellite imagery,” Int. J. Remote Sens. 24(19), 3791–3806 (2003). [CrossRef]

55.

A. H. Barnard, W. S. Pegau, and J. R. V. Zaneveld, “Global relationships of the inherent optical properties of the oceans,” J. Geophys. Res. 103(C11), 24955–24968 (1998). [CrossRef]

56.

H. R. Gordon and T. Du, “Light scattering by nonspherical particles: application to coccoliths detached from Emiliania huxleyi,” Limnol. Oceanogr. 46(6), 1438–1454 (2001). [CrossRef]

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(010.4450) Atmospheric and oceanic optics : Oceanic optics
(290.4020) Scattering : Mie theory
(290.5850) Scattering : Scattering, particles
(010.4458) Atmospheric and oceanic optics : Oceanic scattering

ToC Category:
Scattering

History
Original Manuscript: September 20, 2011
Revised Manuscript: November 18, 2011
Manuscript Accepted: December 1, 2011
Published: December 20, 2011

Virtual Issues
Vol. 7, Iss. 3 Virtual Journal for Biomedical Optics

Citation
Robert H. Stavn, "Mass-specific scattering cross sections of suspended sediments and aggregates: theoretical limits and applications," Opt. Express 20, 201-219 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-1-201


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