## Calibrated near-forward volume scattering function obtained from the LISST particle sizer

Optics Express, Vol. 14, Issue 8, pp. 3602-3615 (2006)

http://dx.doi.org/10.1364/OE.14.003602

Acrobat PDF (236 KB)

### Abstract

The physical nature of particles, such as size, shape, and composition govern their angular light scattering, which is described by the volume scattering function (VSF). Despite the fact that the VSF is one of the most important inherent optical properties, it has rarely been measured in aquatic environments since no commercial instrument exists
to measure the full VSF in the field. The commonly used LISST (Laser In Situ Scattering and Transmissometry) particle sizer (Sequoia Scientific,

© 2006 Optical Society of America

## 1. Introduction

*a*(λ) [m

^{-1}] and the volume scattering function (VSF)

*β*(λ,

*Ψ*,

*ϕ*) [m

^{-1}sr

^{-1}], where λ is the wavelength of light in a vacuum, and (Ψ,ϕ) is the polar direction of scattering with respect to the incident beam traveling in the direction of the

*z*-axis, shown in Fig. 1. For a collimated incident beam of spectral radiant power Φ

_{i}(λ) [W nm

^{-1}], a fraction Φ

_{S}(λ,Ψ,ϕ) of the beam is scattered at angle (Ψ,ϕ) into a solid angle ΔΩ. The VSF is then defined as

*Β*(λ,Ψ,ϕ) =

*Β*(λ,Ψ) [1]. The total volume scattering function,

*Β*(Ψ), can be separated into a summation of individual scattering components, usually the sum of pure water (w), salts (s), and particles (p):

*Β*(Ψ) = β

_{w}(Ψ)+β

_{s}(Ψ)+β

_{p}(Ψ). Additionally, the scattering due to turbulence [2

2. D.J. Bogucki, J.A. Domaradzki, R.E. Ecke, and C.R. Truman, “Light scattering on oceanic turbulence,” Appl. Opt. **43**, 5662–5668 (2004). [CrossRef] [PubMed]

3. X. Zhang, M. Lewis, M. Lee, B. Johnson, and G. Korotaev, “The volume scattering function of natural bubble
populations,” Limnol. Oceanogr. **47**, 1273–1282 (2002). [CrossRef]

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

8. 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. Tech. **20(4)**, 563–571 (2003). [CrossRef]

8. 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. Tech. **20(4)**, 563–571 (2003). [CrossRef]

10. WETLabs Inc. (http://www.wetlabs.com), PO Box 518, Philomath, OR 97370.

*b*(λ) [m

^{-1}] is routinely measured in situ. It can be computed indirectly by instruments such as the ac-9 [10

10. WETLabs Inc. (http://www.wetlabs.com), PO Box 518, Philomath, OR 97370.

*β*(Ψ) is highly peaked in the forward direction, with the majority (69-83% based on VSF observations of Petzold [9]) of particle scatter (

*b*

_{p}(λ)) contained in Ψ < 15°. In addition, the near-forward VSF can be combined with measurements from supplementary sensors that measure the VSF at other angles (such as the ECO VSF) to estimate the VSF across a wider angle range by fitting an analytic model (e.g. Fournier-Forand [11

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

12. Sequoia Scientific Inc. (http://www.sequoiasci.com), 2700 Richards Road, Suite 107, Bellevue, WA 98005.

13. P. Traykovski, R. Latter, and J.D. Irish, “A laboratory evaluation of the LISST instrument using natural sediments,” Mar. Geol. **159**, 355–367 (1999). [CrossRef]

14. Y.C. Agrawal and C. Pottsmith, “Instruments for particle size and settling velocity observations in sediment transport,” Mar. Geol. **168(1-4)**, 89–114 (2000). [CrossRef]

15. Y.C. Agrawal and P. Traykovski, “Particles in the bottom boundary layer: concentration and size dynamics through events,” J. Geophys. Res. **106(C5)**, 9533–9542 (2001). [CrossRef]

*shape*of the VSF [16

16. Y.C. Agrawal, “The optical volume scattering function: Temporal and vertical variability in the water column off the New Jersey coast,” Limnol. Oceanogr. **50(6)**, 1787–1794 (2005). [CrossRef]

## 2. Methods

### 2.1. LISST measurements

12. Sequoia Scientific Inc. (http://www.sequoiasci.com), 2700 Richards Road, Suite 107, Bellevue, WA 98005.

*zscat*

_{i}, where

*i*is the ring detector number, plus clean water laser transmitted power,

*tr*

_{0}, and reference power,

*ref*

_{0}. The raw LISST measurements (32 ring outputs,

*scat*

_{i}, laser transmitted power,

*tr*, and laser reference power,

*ref*, for each sample at approximately 1 Hz) were post processed to subtract the pure water ring outputs and for attenuation within the sample volume,

*tr*/

*ref*)(

*ref*

_{0}/

*tr*

_{0}), and

*dcal*

_{i}is a set of manufacturer-supplied detector responsivity correction factors. The

*zscat*corrected scatter,

*cscat*

_{i}is then corrected for the area of each ring [16

16. Y.C. Agrawal, “The optical volume scattering function: Temporal and vertical variability in the water column off the New Jersey coast,” Limnol. Oceanogr. **50(6)**, 1787–1794 (2005). [CrossRef]

*pscat*

_{i}:

*ϕ*is the fraction of a circle covered by the detector and

*psi*

_{i}are the angles (in water) corresponding to edges of the detector rings. Since the 32 detector rings are spaced logarithmically spanning a radius range of 200:1, the edge angles for each ring can be calculated by knowing the inner scattering angle (in air) of the first ring,

*psi*

_{min(air)}, and correcting for the index of refraction difference between water within the sample volume and air within the instrument,

*pscat*

_{i}(Fig. 3). In order to avoid negative

*pscat*

_{i}, for each run used for calibration, we discard rings whose scatter magnitude (median over calibration run) is not sufficiently greater than the variability (half the difference between 84th and 16th percentiles) in

*zscat*

_{i}, e.g. median(

*pscat*

_{i})/δ

*zscat*

_{i}< 10.

### 2.2. Theoretical Scattering Response

*x*= (2π/λ)

*r*, where

*r*and λ are the sphere radius and wavelength of light (in the medium surrounding the particle) and (2) the complex index of refraction of the spheres (

*m*

_{p}=

*m*′

_{p}+

*jm*′′

_{p}), relative to the surrounding medium (

*m*

_{w}, assumed non-absorbing),

*m*=

*m*

_{p}/

*m*

_{w}, where

*j*= (-1)

^{1/2}. The real index of refraction for polystyrene as a function of wavelength, λ, was calculated based on the results ofMa et al. [18

18. X. Ma, J.Q. Lu, R.S. Brock, K.M. Jacobs, P. Yang, and X. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. **48**, 4165–4172 (2003). [CrossRef]

*m*′′

_{p}~ O(10

^{-4})) was not included (see Section 4). The index of refraction of pure water at room temperature and λ = 670 nm is

*m*

_{w}= 1.3308 [1].

*x*and

*m*, the angular distribution of scattered polarized radiation,

*S*

_{1}(Ψ) and

*S*

_{2}(Ψ), as well as the scattering, extinction, and absorption cross sections (

*C*

_{sca},

*C*

_{ext}, and

*C*

_{abs}) were calculated [5].

*N*

_{0}is the particle concentration [

*m*

^{-3}] and Ñ(

*D*) is the size distribution normalized such that ∫Ñ(

*D*)

*dD*= 1. For each numerical calculation, the assumed distribution was discretized into 200 size bins spanning the range μ

_{D}±3σ

_{D}.

*c*

_{Mie}, is calculated by integration of the extinction cross section,

*C*

_{ext}, over size distribution,

^{2}] and the observed

*c*

_{LISST},

*N*

_{0}is estimated by assuming that

*c*

_{Mie}=

*c*

_{LISST}, hence

*N*

_{0}=

*c*

_{LISST}/

*c*=

*dΨ*. Based on this formulation, relative bias (Δ

*c*/

*c*

_{LISST}was found to be approximately 1.0% for the 100 μm beads, and less than 0.05% for 20 μm and smaller beads.

*Β̃*(Ψ), was calculated by combining the distributions of scattered power in each polarization,

*i*, depends on the integral of the VSF over the angle range of the detector, Ψ

_{i}to

*Ψ*

_{i+1}, so for the purpose of matching LISST-measured scattered power

*pscat*

_{i}for a particular ring to theoretical VSF, we calculated the ring average normalized theoretical VSF as

## 3. Results

_{D}. For each microsphere stock in Table 1,

*i*were generated for a population of 200 Gaussian size distributions with diameters uniformly varied within the bounds μ

_{D}± δ

_{D}. From the population, the median, 16th, and 84th percentiles (the difference of which is equivalent to twice the standard deviation for normally distributed data) were determined for

*i*calculated for each ring.

*scat*

_{i}and beam attenuation. Mie-derived average VSF, 〈

*i*, was then calculated using

*c*

_{LISST}is the LISST-measured beam attenuation. The uncertainty in β

_{Mie}(Ψ) was addressed by propagating the uncertainties in

*c*

_{LISST},

*c*

_{LISST}, δ〈

*i*, and δ

*c*

_{LISST}, 〈

*i*, and

*i*was calculated according to Eq. (15). The resulting relationship between the theoretical 〈

*i*and LISST measurements was assumed linear, and a calibration coefficient [m

^{-1}sr

^{-1}count

^{-1}] relating the LISST measured

*pscat*

_{i}to 〈

*i*was calculated for each detector ring,

_{i}for each ring was determined using a robust regression [19

19. P.W. Holland and R.E. Welsch, “Robust regression using iteratively reweighted least-squares,” Comm. Stat.: Theory Meth. **A6**, 813–827 (1977). [CrossRef]

*i*values were small, since changes in VSF due to uncertainty in suspension mean diameter, μ

_{D}±δ

_{D}, and PSD standard deviation, σ

_{D}, were small. Note that accuracy in estimation of the magnitude of 〈

*i*is limited by the uncertainty in measuring beam attenuation (e.g. the suspension concentration). For the purposes of determining the ci, we used only runs where relative error δ

*c*

_{LISST}/

*c*

_{LISST}< 0.1. This does not imply that the instrument is not useful in the field when this is not the case, but only ensures relative accuracy in estimating

*N*

_{0}for the purpose of lab calibration. Smaller beads provide a much flatter scattering response over a wide angle range leading to less averaging error due to δβ (Ψ)/

*d*Ψ, compared with larger beads with highly variable VSF at larger angles (Fig. 3(a)). Our calibration procedure was to use the 2 μm runs, excluding the first six rings since their

*scat*

_{i}is not sufficiently greater than

*zscat*

_{i}(Fig. 3(b)). Similarly, we added data from 20 μm runs for intermediate rings, excluding rings 15 to 32 since the VSF falls off and fluctuates in this angle range. Data from 100 μm runs were also used since 100 μm beads scatter strongly in the most near-forward rings (Fig. 3(b)), excluding rings 10 to 32 due to fluctuations in the VSF. Our data and MATLAB source code are freely available on our website [20

20. W.H. Slade, “LISST Calibration Information,” (http://misclab.umeoce.maine.edu/code/lisstvsf.html).

_{i}was 3.1∙10

^{-6}[m

^{-1}sr

^{-1}count

^{-1}]. Deviation in the largest rings is likely due to increasing

*d*β(Ψ)/

*dΨ*, even for the small 2 μm beads. The error in the model-data fit for each ring (Fig. 4(e)) was calculated as the median relative absolute error between the expected Mie-derived ring average scattering and the LISST output for the calibration data,

*i*model fit for each ring averaged 6%. Similarly, we calculated the relative absolute error between 〈β

_{LISST}〉

_{i}estimated for the validation data (10 and 50 μm beads, which were not used in deriving the calibration coefficients) and the expected ring averaged VSF based on Mie theory. The validation error (Fig. 4(f)) ranged between approximately 10% and 80%, but was on average 30%. This error is likely an overestimate, since: (1) as in the case of each calibration run, our knowledge of the validation suspension concentration is limited by our ability to measure beam attenuation, (2) our knowledge of the shape of the calculated Mie VSF is limited by our inability to independently verify it (i.e. it is a function of the PSD and the bead properties that we did not verify independently), and (3) the uncertainty is calculated for rings even where our validation beads have complicated fluctuations in their VSF, which would likely not be the case for the VSF of natural waters.

## 4. Discussion and Conclusion

*χi*may indicate individual instrument characteristics. Determination of the calibration coefficients depended on knowing the concentration of microspheres within the sample chamber, which we determined using the LISST-measured beam attenuation. Therefore, we used a limited set of experimental runs where the relative error in

*c*

_{LISST}was low. We also selected data from particular detector rings for the different size classes of microspheres, since for some runs scatter in particular rings was very low (e.g. Fig 3(b)), and for some bead sizes the VSF was highly variable over particular rings (e.g. Fig 3(a)). A remaining source of uncertainty in the calibration was our assumption of the analytical PSD for the theoretical scattering calculations, which does not account for any destabilization or contamination of the measured suspensions. In the future this uncertainty may be reduced by using PSD measured at the time of calibration.

*zscat*, in order to determine the total VSF, scatter due to pure water must be added back onto the derived VSF. Volume scattering function due to water can be calculated according to Morel [21], but must be integrated over the ring angle ranges.

*S*

_{11}element of the scattering matrix. However, the LISST source is a collimated laser diode (linear polarized), while the photodetector rings are nonpolarized receivers (neglecting any polarization imparted by the optical window and lens). Irradiances measured by such a configuration will incorporate the

*S*

_{12}and

*S*

_{13}elements of the Mueller scattering matrix [5]. In the case of spherical scatterers,

*S*

_{13}→ 0. Figure 7 shows the magnitude of polarization factor,

*P*(Ψ) = -

*S*

_{12}(Ψ)/

*S*

_{11}(Ψ), calculated using Mie theory for the polydispersions of calibration beads over the angle range of the type-B LISST. This ratio is indicative of the degree of linear polarization of the scattered light, where |

*P*(Ψ)|=1 for totally linear polarized light and

*P*(Ψ) = 0 for unpolarized light. Regardless of the composition or size of spherical scatterers,

*P*(0°) =

*P*(180°) = 0. For the suspensions used in this calibration, over the angle range of the type-B LISST, |

*P*(Ψ)| is in general less than 0.1, with maximum values in the larger-angle rings. Disagreement between the calculated unpolarized 〈β

*Mie*〉

_{i}and the polarized LISST-measured scatter could explain some of the deviation in the 〈β

*Mie*〉

_{i}vs. 〈β

_{LISST}〉

_{i}(or

*pscat*

_{i}) relationship for large-angle rings observed in Figs. 4 and 5. For oceanic suspensions the effect of polarization is expected to be even smaller, as |

*S*

_{12}(Ψ)/

*S*

_{11}(Ψ)| < 0.06±0.02 and |

*S*

_{13}(Ψ)/

*S*

_{11}(Ψ)| = 0±0.01 for the first 20 degrees based on the observation of Voss and Fry [22

22. K.J. Voss and E.S. Fry, “Measurement of the Mueller matrix for ocean water,” Appl. Opt. **23**, 4427–4439 (1984). [CrossRef] [PubMed]

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

*b*

_{p}. Knowledge of the full VSF will improve radiative transfer calculation and the associated understanding (through measurements-model comparison) of how ocean color varies in response to changes in the inherent optical properties (e.g. [7

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

## Acknowledgments

## References and links

1. | C.D. Mobley, |

2. | D.J. Bogucki, J.A. Domaradzki, R.E. Ecke, and C.R. Truman, “Light scattering on oceanic turbulence,” Appl. Opt. |

3. | X. Zhang, M. Lewis, M. Lee, B. Johnson, and G. Korotaev, “The volume scattering function of natural bubble
populations,” Limnol. Oceanogr. |

4. | 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. |

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

6. | M.I. Mishchenko, L.D. Travis, and A.A. Lacis, |

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

8. | 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. Tech. |

9. | T.J. Petzold, “Volume scattering functions for selected ocean waters,” Contract No. N62269-71-C-0676, UCSD, SIO Ref. 72–78 (1972). |

10. | WETLabs Inc. (http://www.wetlabs.com), PO Box 518, Philomath, OR 97370. |

11. | G.R. Fournier and J.L. Forand, “Analytical phase function for ocean water,” in |

12. | Sequoia Scientific Inc. (http://www.sequoiasci.com), 2700 Richards Road, Suite 107, Bellevue, WA 98005. |

13. | P. Traykovski, R. Latter, and J.D. Irish, “A laboratory evaluation of the LISST instrument using natural sediments,” Mar. Geol. |

14. | Y.C. Agrawal and C. Pottsmith, “Instruments for particle size and settling velocity observations in sediment transport,” Mar. Geol. |

15. | Y.C. Agrawal and P. Traykovski, “Particles in the bottom boundary layer: concentration and size dynamics through events,” J. Geophys. Res. |

16. | Y.C. Agrawal, “The optical volume scattering function: Temporal and vertical variability in the water column off the New Jersey coast,” Limnol. Oceanogr. |

17. | Duke Scientific Corporation (http://www.dukesci.com), 2463 Faber Place, Palo Alto, CA 94303. |

18. | X. Ma, J.Q. Lu, R.S. Brock, K.M. Jacobs, P. Yang, and X. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. |

19. | P.W. Holland and R.E. Welsch, “Robust regression using iteratively reweighted least-squares,” Comm. Stat.: Theory Meth. |

20. | W.H. Slade, “LISST Calibration Information,” (http://misclab.umeoce.maine.edu/code/lisstvsf.html). |

21. | A. Morel, “Optical properties of pure water and pure sea water,” in |

22. | K.J. Voss and E.S. Fry, “Measurement of the Mueller matrix for ocean water,” Appl. Opt. |

**OCIS Codes**

(000.2170) General : Equipment and techniques

(010.4450) Atmospheric and oceanic optics : Oceanic optics

(120.5820) Instrumentation, measurement, and metrology : Scattering measurements

(290.4020) Scattering : Mie theory

(290.5820) Scattering : Scattering measurements

(290.5850) Scattering : Scattering, particles

**ToC Category:**

Scattering

**History**

Original Manuscript: November 15, 2005

Revised Manuscript: March 30, 2006

Manuscript Accepted: April 11, 2006

Published: April 17, 2006

**Virtual Issues**

Vol. 1, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Wayne H. Slade and Emmanuel S. Boss, "Calibrated near-forward volume scattering function obtained from the LISST particle sizer," Opt. Express **14**, 3602-3615 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3602

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### References

- C.D. Mobley, Light and Water (Academic Press, San Diego, 1994).
- D.J. Bogucki, J.A. Domaradzki, R.E. Ecke and C.R. Truman, "Light scattering on oceanic turbulence," Appl. Opt. 43, 5662-5668 (2004). [CrossRef] [PubMed]
- X. Zhang, M. Lewis, M. Lee, B. Johnson and G. Korotaev, "The volume scattering function of natural bubble populations," Limnol. Oceanogr. 47, 1273-1282 (2002). [CrossRef]
- 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(C01014), doi:10.1029/2002JC001514 (2004).
- C.F Bohren and D.R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, New York, 1983).
- M.I. Mishchenko, L.D. Travis, and A.A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University Press, Cambridge, 2002).
- C.D. Mobley, L.K. Sundman, and E. Boss, "Phase function effects on oceanic light fields," Appl. Opt. 41, 1035-1050 (2002). [CrossRef] [PubMed]
- 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(4), 563-571 (2003). [CrossRef]
- T.J. Petzold, "Volume scattering functions for selected ocean waters," Contract No. N62269-71-C-0676, UCSD, SIO Ref. 72-78 (1972).
- WETLabs Inc. (http://www.wetlabs.com), PO Box 518, Philomath, OR 97370.
- G.R. Fournier and J.L. Forand, "Analytical phase function for ocean water," in Ocean Optics XII, J.S. Jaffe, ed., Proc. SPIE 2258, 194-201 (1994). [CrossRef]
- Sequoia Scientific Inc. (http://www.sequoiasci.com), 2700 Richards Road, Suite 107, Bellevue, WA 98005.
- P. Traykovski, R. Latter, and J.D. Irish, "A laboratory evaluation of the LISST instrument using natural sediments," Mar. Geol. 159, 355-367 (1999). [CrossRef]
- Y.C. Agrawal and C. Pottsmith, "Instruments for particle size and settling velocity observations in sediment transport," Mar. Geol. 168(1-4), 89-114 (2000). [CrossRef]
- Y.C. Agrawal and P. Traykovski, "Particles in the bottom boundary layer: concentration and size dynamics through events," J. Geophys. Res. 106(C5), 9533-9542 (2001). [CrossRef]
- Y.C. Agrawal, "The optical volume scattering function: Temporal and vertical variability in the water column off the New Jersey coast," Limnol. Oceanogr. 50(6), 1787-1794 (2005). [CrossRef]
- Duke Scientific Corporation (http://www.dukesci.com), 2463 Faber Place, Palo Alto, CA 94303.
- X. Ma, J.Q. Lu, R.S. Brock, K.M. Jacobs, P. Yang, and X. Hu, "Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm," Phys. Med. Biol. 48, 4165-4172 (2003). [CrossRef]
- P.W. Holland and R.E. Welsch, "Robust regression using iteratively reweighted least-squares," Comm. Stat.: Theory Meth. A6, 813-827 (1977). [CrossRef]
- W.H. Slade, "LISST Calibration Information," (http://misclab.umeoce.maine.edu/code/lisstvsf.html).
- A. Morel, "Optical properties of pure water and pure sea water," in Optical Aspects of Oceanography, N. G. Jerlov and E. S. Neilsen, eds. (Academic, New York, 1974), pp. 1-24.
- K.J. Voss and E.S. Fry, "Measurement of the Mueller matrix for ocean water," Appl. Opt. 23, 4427-4439 (1984). [CrossRef] [PubMed]

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