## Improved method of Fournier-Forand marine phase function parameterization

Optics Express, Vol. 15, Issue 20, pp. 12763-12768 (2007)

http://dx.doi.org/10.1364/OE.15.012763

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### Abstract

Volume scattering functions (VSFs) and other optical seawater parameters were measured during a cruise in the Southern Baltic. Phase functions (PFs) calculated from VSFs were compared with Fournier-Forand phase functions parameterized with backscattering ratios. Due to significant divergences between experimental and modeled data a new method of Fournier-Forand phase function parameterization is proposed.

© 2007 Optical Society of America

## 1. Introduction

3. L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. **93**, 70–83 (1941). [CrossRef]

5. G. Fournier and J. L. Forand, “Analytic phase function for ocean water,” in *Ocean Optics XII*,
J. S. Jaffe, ed., Proc. SPIE **2258**, 194–201 (1994). [CrossRef]

6. M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmos. Ocean. Technol. **20**, 563–571 (2003). [CrossRef]

## 2. Materials and methods

5. G. Fournier and J. L. Forand, “Analytic phase function for ocean water,” in *Ocean Optics XII*,
J. S. Jaffe, ed., Proc. SPIE **2258**, 194–201 (1994). [CrossRef]

*n*and slope of the Junge particle size distribution μ. The form of the function is given by [8

8. G. Fournier and M. Jonasz, “Computer-based underwater imaging analysis,” in *Airborne and In-water Underwater Imaging*,
G. Gilbert, ed., Proc. SPIE **3761**, 62–77 (1999). [CrossRef]

*n*,

*μ*and the particle backscatter fraction

*B*:

_{p}*δ*

_{90}is

*δ*determined for scattering angle θ= 90 deg.

## 3. Results and discussion

*B*. The difference D between analytical and experimental functions was defined as:

_{p}*θ*, for which scattering intensities are known. This difference formula mimics the way an integral of the difference squared would be calculated if the phase functions were known for every scattering angle value.

*B*and the set of refractive indices

_{p}*n*, from 1.001 to 1.25 with step 0.001, a huge number of FF phase functions was calculated. The parameters

*n*and

*μ*were found for the FF function for which D was smallest. The algorithm was evaluated for each of over 30 measured PFs.

*n*,

*μ*) are presented in Fig. 1. (black circles). Contours of the backscattering ratios, obtained from Eq. (2) are also shown. The red dashed line which, according to Mobley et al., provides parameters

*n*and

*μ*is also marked. The red point describes the best fit of

*n*and

*μ*to the Petzold phase function. It is obvious that Mobley’s method only fits well for a small amount of measured data. This is why a better parameterization method is needed.

*n*and

*μ*are only weakly dependent on Bp. That is why we propose to draw a linear function

*n*(

*μ*), as Mobley et al. proposed, but with changing slope. We checked that the absorption coefficient a, measured by the mean of ac-9 values, would be a good characteristics of the linear slope of

*n*(

*μ*). Similarly to Mobley et al., we assumed that this linear function should intersect the point

*n*= 1 and

*μ*= 3. That is why the slope of line is given by (

*n*- 1)/(

*μ*- 3). In Fig. 2 factors (

*n*- 1)/(

*μ*- 3) are plotted with absorption coefficients a for all wavelengths for which VSFs were measured.

*n*and Junge slope

*μ*.

*B*in Eq. (2) are also required. The solution of the system of equations gives the real index of refraction

_{p}*n*and the slope of the Junge particle size distribution

*μ*, which are needed to plot Fournier Forand phase functions.

*B*in Eq. (2) is constant. We took the following average values of

_{p}*B*, for Southern Baltic waters, which were measured in the same cruise by mean of the VSM:

_{p}*B*,(443nm) = 0.0136,

_{p}*B*(490nm) = 0.0132,

_{p}*B*(555nm) = 0.012 and

_{p}*B*,(620nm) = 0.0119.

_{p}*a*are known. Averaged differences D calculated for all available stations are collected in table 1.

## 4. Conclusions

## Acknowledgments

## References and links

1. | T. J. Petzold, “Volume scattering functions for selected ocean waters,” SIO Ref. 72–78, Scripps Institute of Oceanography, U. California, San Diego (1972). |

2. | C. D. Mobley, |

3. | L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. |

4. | V. I. Haltrin, “Two-term Henyey-Greenstein light scattering phase function for seawater,” in |

5. | G. Fournier and J. L. Forand, “Analytic phase function for ocean water,” in |

6. | M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmos. Ocean. Technol. |

7. | E. B. Shybanov, O. V. Martynov, T. Król, W. Freda, and R. Hapter, “Scattering functions of Southern Baltic sea waters”, |

8. | G. Fournier and M. Jonasz, “Computer-based underwater imaging analysis,” in |

9. | C. D. Mobley, L. K. Sundman, and E. Boss, “Phase Function Effects on Oceanic Light Fields,” Appl. Opt. |

10. | M. Chami, D. McKee, E. Leymarie, and G. Khomenko, “Influence of the angular shape of the volume-scattering function and multiple scattering on remote sensing reflectance,” Appl. Opt. |

11. | D. McKee and A. Cunningham, “Evidence for wavelength dependence of the scattering phase function and its implication for modeling radiance transfer in shelf seas,” Appl. Opt. |

12. | A. H. Hakim and N. J. McCormick, “Ocean Optics Estimation for Absorption, Backscattering, and Phase Function Parameters,” Appl. Opt. |

**OCIS Codes**

(010.4450) Atmospheric and oceanic optics : Oceanic optics

(290.1350) Scattering : Backscattering

(290.5820) Scattering : Scattering measurements

**ToC Category:**

Atmospheric and oceanic optics

**History**

Original Manuscript: July 3, 2007

Revised Manuscript: August 8, 2007

Manuscript Accepted: August 17, 2007

Published: September 21, 2007

**Virtual Issues**

Vol. 2, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Wlodzimierz Freda and Jacek Piskozub, "Improved method of Fournier-Forand marine phase function parameterization," Opt. Express **15**, 12763-12768 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-20-12763

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### References

- T. J. Petzold, "Volume scattering functions for selected ocean waters," SIO Ref. 72-78, Scripps Institute of Oceanography, U. California, San Diego (1972).
- C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, San Diego, Calif., 1994).
- L. C. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93, 70-83 (1941). [CrossRef]
- V. I. Haltrin, "Two-term Henyey-Greenstein light scattering phase function for seawater," in IGARSS ’99: Proceedings of the International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 1423-1425.
- G. Fournier and J. L. Forand, "Analytic phase function for ocean water," Proc. SPIE 2258, 194-201 (1994). [CrossRef]
- M. E. Lee and M. R. Lewis, "A new method for the measurement of the optical volume scattering function in the upper ocean," J. Atmos. Ocean. Technol. 20, 563-571 (2003).Q1 [CrossRef]
- E. B. Shybanov, O. V. Martynov, T. Król, W. Freda, R. Hapter, "Scattering functions of Southern Baltic sea waters", in Physicochemical Problems of Natural Waters Ecology, T. Król, ed., (Gdynia Maritime University Publishing, 2007), Vol. 5, pp. 90-95.Q2
- G. Fournier and M. Jonasz, "Computer-based underwater imaging analysis," in Airborne and In-water Underwater Imaging, G. Gilbert, ed., Proc. SPIE 3761, 62-77 (1999). [CrossRef]
- C. D. Mobley, L. K. Sundman, and E. Boss, "Phase Function Effects on Oceanic Light Fields," Appl. Opt. 41,1035-1050 (2002), http://www.opticsinfobase.org/abstract.cfm?URI=ao-41-6-1035. [CrossRef] [PubMed]
- M. Chami, D. McKee, E. Leymarie, and G. Khomenko, "Influence of the angular shape of the volume-scattering function and multiple scattering on remote sensing reflectance," Appl. Opt. 45, 9210-9220 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=ao-45-36-9210. [CrossRef] [PubMed]
- D. McKee and A. Cunningham, "Evidence for wavelength dependence of the scattering phase function and its implication for modeling radiance transfer in shelf seas," Appl. Opt. 44, 126-135 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ao-44-1-126 [PubMed]
- A. H. Hakim and N. J. McCormick, "Ocean Optics Estimation for Absorption, Backscattering, and Phase Function Parameters," Appl. Opt. 42, 931-938 (2003). [CrossRef] [PubMed]

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