## Anomalous diffraction approximation to the light scattering coefficient spectra of marine particles with power-law size distribution |

Optics Express, Vol. 20, Issue 25, pp. 27603-27611 (2012)

http://dx.doi.org/10.1364/OE.20.027603

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### Abstract

Based on anomalous diffraction approximation, analytical expressions for the scattering coefficient of marine particles with power-law size distribution in the infinite domain of sizes (0, ∞) were derived. Comparison with the exact Mie solution for the light scattering by spheres indicated that the obtained expressions can describe the relative spectral variability of the scattering coefficient well. This is demonstrated and discussed for the scattering spectra of main types of marine particulates characterized by different optical properties.

© 2012 OSA

## 1. Introduction

8. D. Doxaran, K. Ruddick, D. McKee, B. Gentili, D. Tailliez, M. Chami, and M. Babin, “Spectral variations of light scattering by marine particles in coastal waters from the visible to the near infrared,” Limnol. Oceanogr. **54**(4), 1257–1271 (2009). [CrossRef]

12. 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**(2), 843–859 (2003). [CrossRef]

*k*varying within a narrower range, i.e.

*k*ranging between 3 and 5 [4

4. C. J. Buonassissi and H. M. Dierssen, “A regional comparison of particle size distributions and the power law approximation in oceanic and estuarine surface waters,” J. Geophys. Res. **115**(C10), C10028 (2010). [CrossRef]

5. R. A. Reynolds, D. Stramski, V. M. Wright, and S. B. Woźniak, “Measurements and characterization particle size distributions in coastal waters,” J. Geophys. Res. **115**(C8), C08024 (2010). [CrossRef]

12. 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**(2), 843–859 (2003). [CrossRef]

15. D. Stramski and D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr. **28**(4), 343–383 (1991). [CrossRef]

## 2. ADA scattering coefficients

*C*) and absorption (

_{e}*C*) cross sections by means of the surface integrals [16

_{a}16. F. D. Bryant and P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci. **30**(3), 291–304 (1969). [CrossRef]

*λ*is the light wavelength in the vacuum,

*r*is the light ray path through the particle,

*n - in*' is the wavelength dependent particle’s complex refractive index relative to water, and

*dP*is an element of the particle projected area (

*P*) on the plane perpendicular to the direction of the incident light. The ray paths are line segments within a particle and their direction is the same as the direction of the incident light. Neglecting the light refraction and reflection on a particle's surface results from the ADA assumption that the particle complex index of refraction is close to that of water that is, its real part is close to 1 and the imaginary part is close to 0. In this approach the light extinction is caused by the absorption of the light rays passing through a particle and by interference of those ones which were transmitted through a particle and those passing around the particle [14]. The difference between the cross sections expressed by Eqs. (2) and (3) gives the scattering cross-section.

*b*) can be determined as follows [2]:where the unit size

*D*was omitted for simplicity and

_{o}*D*. Accounting for the fact that the infinitely large size (

*Z*can equal

*A*. The integral

*k*must be lower than 4. Therefore, this range of the PSD slope values i.e.

*x*, which in this case is equal to 4 -

*k*, it is easy to find the limit:If we introduce this result into Eq. (5), the scattering coefficient accounting for all possible particle sizes equals:The power function

*A*and

*B*parameters by means of Eqs. (6) and (7) the final formula for the scattering coefficient is given by:whereThis non-dimensional integral is the only factor in Eq. (15) which depends on the particles' shape. The scattering coefficient must be a continuous function of

*k*. Therefore, to obtain an expression for the scattering coefficient when

*k*= 4, it is sufficient to determine the limit in Eq. (15) when

*k*approaches 4 through values less than 4. Then the factor in the brace approaches 0 whereas

*Mathematica*software:

*S*is very simple. It is the volume of a particle divided by its characteristic size raised to the third power. When 4 <

*k*< 5 the integral in Eq. (9) is integrated by parts once again to obtain:

*- k*) by (4

*- k*) Γ(4

*- k*) and accounting for the above limit Eq. (5) finally gives the same result as Eq. (15). Note that in this case Γ(4

*- k*) and the factor in the brace in Eq. (15) are both negative. Letting

*k*< 4 or 4 <

*k*< 5, andwhen

*k*= 4.

## 3. Comparison between ADA and Mie visible scattering coefficients

### 3.1 Calculation assumptions

12. 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**(2), 843–859 (2003). [CrossRef]

*Prorocentrum micans*(Dinophyceae). The data on the refractive index in the visible band (0.4

*-*0.75 µm) were obtained by means of digitalization of proper graphs (Figs. 3(a) and 3(b)) in Ahn et al. [19

19. Y. H. Ahn, A. Bricaud, and A. Morel, “Light backscattering efficiency and related properties of some phytoplankters,” Deep-Sea Res. **39**(11-12), 1835–1855 (1992). [CrossRef]

*Prorocentrum micans*refractive index shows only small fluctuations around the value of 1.05.

*λ*in [μm], and the real parts are wavelength-independent and equal 1.04 and 1.18 respectively (Fig. 1). It is believed that such optical properties are characteristic for nonliving particles [20

20. D. Stramski, A. Bricaud, and A. Morel, “Modeling the inherent optical properties of the ocean based on the detailed composition of the planktonic community,” Appl. Opt. **40**(18), 2929–2945 (2001). [CrossRef] [PubMed]

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

*S*equals 0.5

*π*/(

*k -*1). The value of

*m*in the power-law PSD (Eq. (1)) was taken to be 10

^{5}cm

^{−3}μm

^{−1}. It is a typical order of magnitude of this parameter [2]. The values of the PSD slope coefficient ranged between 3 and 5. This analytical approach was examined against the exact Mie solution to the light attenuation cross-sections. In this case the integral in Eq. (4) was evaluated numerically by means of a Matlab script [22

22. W. Slade and E. Boss, “Translation from Fortran to Matlab of the Bohren and Huffman Mie Code,” http://misclab.umeoce.maine.edu/software.php.

*-*µm step.

### 3.2 Results and discussion

*-*squared correlation coefficient) describes the similarity of the shapes of obtained spectra.

*k*ADA overestimated the magnitude of the scattering coefficients in relation to the Mie results. The overestimation got worse with increasing value of

*k*. For example, when

*n*value. Such a result is caused by two reasons. When the PSD slope coefficients becomes larger, the contribution of smallest particles in the light scattering gets more significant and their scattering cross-section according to ADA decreases slower (~

*k*= 3.1 the scattering coefficients calculated with the use of ADA were also distinctly higher than these obtained basing on the Mie solution (Table 1). Moreover, low values of the determination coefficients indicated significant differences between the shape of the scattering spectra. In this case, particles that are very large in comparison to the light wavelength, can contribute significantly into the light scattering. Therefore, when the upper limit of the particles' diameters was assumed to be 500 μm some of them were not taken into account in the Mie calculations. This resulted in such an unsatisfactory fit between obtained spectra. For example, after changing this upper limit to 5000 μm the recalculated value of

*α*parameter decreased to 1.288 (previously

*b*are products of two spectral functions. The first one is

*n*'/(

*n -*1) and

*k -*3. The ratio

*n*'/(

*n -*1) is the indicator characterizing the absorption-to-scattering properties of particles [14]. The higher it becomes, the more significant is the role of the absorption in the light attenuation. In such a case the absolute values of the latter function decrease, but the decrease is lower when

*k*gets higher. The values of

*n*'/(

*n -*1) are usually close to zero, even in the case of phytoplankton which strongly absorbs the light. Its maximum for

*Prorocentrum micans*is 0.055 in the blue part of the visible band. Therefore generally the function of

*n*'/(

*n -*1) does not change much and when

*k*is sufficiently large it can be expected that the

*ε*) were estimated by means of the least squares method after an appropriate logarithmic transformation of variables:where the constant on the right hand side of the equation is the logarithm of proportionality coefficient between

*k*was close to 3 and the determination coefficients were the lowest. In such a case the function

*n*'/(

*n -*1) cause greater effects.

**48**(2), 843–859 (2003). [CrossRef]

*k*increased the phytoplankton scattering spectra got smoother and more similar to the power function. On the other hand in the case of non-living particles (Figs. 2(b), 2(c))

*n*'/(

*n -*1) decreases exponentially because of the assumed spectrum of the imaginary part of the refractive index (Eq. (23)). Then the absorption reduces the magnitude of scattering in the short-wave part of the spectrum, and its influence gradually decreases with increasing light wavelength. For the lowest

*k*value (3.1), it can cause a weak power-law increase (

*ε*is negative) in the scattering coefficient with increasing wavelength. Beside this case the decreasing power function fitted excellently to the scattering spectra of non-living particles (Table 2). Generally, in such an approximation, the values of the

*ε*exponent were lower than

*k -*3. This difference became smaller for particles with lower values of

*n*'/(

*n -*1) ratio and when

*k*was higher.

## Acknowledgments

## References and links

1. | J. T. O. Kirk, |

2. | M. Jonasz and G. R. Fournier, |

3. | H. Bader, “The hyperbolic distribution of particles sizes,” J. Geophys. Res. |

4. | C. J. Buonassissi and H. M. Dierssen, “A regional comparison of particle size distributions and the power law approximation in oceanic and estuarine surface waters,” J. Geophys. Res. |

5. | R. A. Reynolds, D. Stramski, V. M. Wright, and S. B. Woźniak, “Measurements and characterization particle size distributions in coastal waters,” J. Geophys. Res. |

6. | 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. |

7. | G. Chang, A. Barnard, and J. R. V. Zaneveld, “Optical closure in a complex coastal environment: particle effects,” Appl. Opt. |

8. | D. Doxaran, K. Ruddick, D. McKee, B. Gentili, D. Tailliez, M. Chami, and M. Babin, “Spectral variations of light scattering by marine particles in coastal waters from the visible to the near infrared,” Limnol. Oceanogr. |

9. | G. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Ocean Optics XII, J.S. Jaffe, ed., Proc. SPIE 2258, 194–201 (1994). |

10. | C. D. Mobley, |

11. | A. Morel, “The scattering of light by sea water. Experimental results and theoretical approach,” AGARD lecture series 61 on Optics of the sea, Interface and in-water transmission and imaging, (Advisory Group for Aerospace Research and Development NATO, London, 1973). |

12. | 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. |

13. | C. F. Bohren and D. R. Huffman, |

14. | H. C. van de Hulst, 1981, |

15. | D. Stramski and D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr. |

16. | F. D. Bryant and P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci. |

17. | M. Abramowitz and I. A. Stegun, |

18. | D. Sarason, |

19. | Y. H. Ahn, A. Bricaud, and A. Morel, “Light backscattering efficiency and related properties of some phytoplankters,” Deep-Sea Res. |

20. | D. Stramski, A. Bricaud, and A. Morel, “Modeling the inherent optical properties of the ocean based on the detailed composition of the planktonic community,” Appl. Opt. |

21. | S. B. Woźniak and D. Stramski, “Modelling the optical properties of mineral particles suspended in seawater and their influence on ocean reflectance and chlorophyll estimation for remote sensing algorithm,” Appl. Opt. |

22. | W. Slade and E. Boss, “Translation from Fortran to Matlab of the Bohren and Huffman Mie Code,” http://misclab.umeoce.maine.edu/software.php. |

**OCIS Codes**

(010.4450) Atmospheric and oceanic optics : Oceanic optics

(290.5850) Scattering : Scattering, particles

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: October 10, 2012

Revised Manuscript: November 9, 2012

Manuscript Accepted: November 10, 2012

Published: November 28, 2012

**Virtual Issues**

Vol. 8, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Maciej Matciak, "Anomalous diffraction approximation to the light scattering coefficient spectra of marine particles with power-law size distribution," Opt. Express **20**, 27603-27611 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-25-27603

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### References

- J. T. O. Kirk, Light and Photosynthesis in Aquatic Ecosystems, (Cambridge University Press, 1994).
- M. Jonasz and G. R. Fournier, Light Scattering by Particles in Water, (Academic, Amsterdam, 2007).
- H. Bader, “The hyperbolic distribution of particles sizes,” J. Geophys. Res.75(15), 2822–2830 (1970). [CrossRef]
- C. J. Buonassissi and H. M. Dierssen, “A regional comparison of particle size distributions and the power law approximation in oceanic and estuarine surface waters,” J. Geophys. Res.115(C10), C10028 (2010). [CrossRef]
- R. A. Reynolds, D. Stramski, V. M. Wright, and S. B. Woźniak, “Measurements and characterization particle size distributions in coastal waters,” J. Geophys. Res.115(C8), C08024 (2010). [CrossRef]
- 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(27), 4885–4893 (2001). [CrossRef] [PubMed]
- G. Chang, A. Barnard, and J. R. V. Zaneveld, “Optical closure in a complex coastal environment: particle effects,” Appl. Opt.46(31), 7679–7692 (2007). [CrossRef] [PubMed]
- D. Doxaran, K. Ruddick, D. McKee, B. Gentili, D. Tailliez, M. Chami, and M. Babin, “Spectral variations of light scattering by marine particles in coastal waters from the visible to the near infrared,” Limnol. Oceanogr.54(4), 1257–1271 (2009). [CrossRef]
- G. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Ocean Optics XII, J.S. Jaffe, ed., Proc. SPIE 2258, 194–201 (1994).
- C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters, (Academic, San Diego, Calif.,1994).
- A. Morel, “The scattering of light by sea water. Experimental results and theoretical approach,” AGARD lecture series 61 on Optics of the sea, Interface and in-water transmission and imaging, (Advisory Group for Aerospace Research and Development NATO, London, 1973).
- 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(2), 843–859 (2003). [CrossRef]
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2004).
- H. C. van de Hulst, 1981, Light Scattering by Small Particles, (Dover Publications, New York, 1981).
- D. Stramski and D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr.28(4), 343–383 (1991). [CrossRef]
- F. D. Bryant and P. Latimer, “Optical efficiencies of large particles of arbitrary shape and orientation,” J. Colloid Interface Sci.30(3), 291–304 (1969). [CrossRef]
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, (National Bureau of Standards, Applied Mathematics Series 55, 1964).
- D. Sarason, Complex Function Theory, (American Mathematical Society, 2007).
- Y. H. Ahn, A. Bricaud, and A. Morel, “Light backscattering efficiency and related properties of some phytoplankters,” Deep-Sea Res.39(11-12), 1835–1855 (1992). [CrossRef]
- D. Stramski, A. Bricaud, and A. Morel, “Modeling the inherent optical properties of the ocean based on the detailed composition of the planktonic community,” Appl. Opt.40(18), 2929–2945 (2001). [CrossRef] [PubMed]
- S. B. Woźniak and D. Stramski, “Modelling the optical properties of mineral particles suspended in seawater and their influence on ocean reflectance and chlorophyll estimation for remote sensing algorithm,” Appl. Opt.43(17), 3489–3503 (2004). [CrossRef] [PubMed]
- W. Slade and E. Boss, “Translation from Fortran to Matlab of the Bohren and Huffman Mie Code,” http://misclab.umeoce.maine.edu/software.php .

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