## 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, http://www.sequoiasci.com) measures near-forward angular scattering of a laser source (λ = 670 nm) at 32 logarithmically-spaced photodetectors arranged between 0.08 and 15 degrees and inverts the data to obtain particle size distribution (PSD). In order to calibrate the LISST to provide the near-forward VSF of unknown particle suspensions, we analyzed the scattering of light by polystyrene bead suspensions of known size distributions and composition, and empirically compared it with the results of Mie theory. This (1) allowed us to obtain a set of instrument specific scaling factors needed to retrieve the magnitude of the VSF and (2) provided validation that the shape of the VSF was appropriately obtained.

© 2006 Optical Society of America

## 1. Introduction

The fundamental properties that regulate radiative transfer within a body of water are the spectral absorption coefficient *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

For unpolarized light with randomly oriented particles, the scattering is assumed to be azimuthally symmetric about the axis of the incident beam, such that*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] and bubbles [3] may be important terms in some cases. In most natural waters, only the scattering of water, salts, and particles are considered. For the laboratory studies presented here, we are concerned only with pure water and particulate scattering.

The angular distribution of light scattered by particles is governed by the physical properties of the particles themselves, such as size, shape, and composition. The dependence has been demonstrated based on data obtained in situ, in the lab, and modeled using Mie theory and T-matrix methods [4, 5, 6]. As useful as the VSF is for modeling the light field in radiative transfer applications [7] or determining the properties of particles in sea water, it has seldom been measured in the field. The primary reason for the paucity of measurements has been the lack of commercially-available instruments for measuring the VSF in situ due to engineering difficulties [8] and lack of sufficient demand. Only limited measurements of the VSF have been made in the past, with custom-built instruments such as with the General Angle Scattering Meter (GASM) [9]. Lee and Lewis [8] presented a prototype instrument for measuring the complete VSF and reported significant environmental variability in the near-forward VSF.More recently, the use of instruments that measure the VSF at a few angles in the backward direction (such as 100°, 125°, and 150° for the WETLabs ECO-VSF) [10] has become routine.

The scattering coefficient is defined as the integral of the VSF over all angles, assuming azimuthal symmetry,

In contrast to the full VSF, the scattering coefficient *b*(λ) [m^{-1}] is routinely measured in situ. It can be computed indirectly by instruments such as the ac-9 [10] as the difference between total light attenuation and absorption. The VSF can be used to estimate the scattering coefficient, since *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])to the measured VSF data using a least squares approach.

The LISST [12] was designed for use in sedimentology to measure the PSD in the field [13, 14, 15]. In the LISST, light from a laser source (λ= 670 nm) scatters towards a lens on the receiving side of the sample volume, as in Fig. 2. On the other side of the receiving lens is a set of logarithmically spaced concentric ring photodetectors. The lens allows light scattered at a given angle to be collected by a given ring. The 32 photodetector rings are arranged such that their edge radii increase logarithmically, covering a total scattering angle range of approximately 0.08 to 15° in water (note that Sequoia reports the angles in air). There is also a detector at Ψ = 0 (acceptance angle of 0.0269° in water for the type-B instrument discussed here) used to measure beam attenuation.

Recently, the LISST has been used for quantification of natural variability in the *shape* of the VSF [16]. In the current paper, we present a method to obtain both the shape and magnitude of the near-forward VSF, using Mie theory and lab measurements of polystyrene bead PSD standards to calibrate the LISST-measured angular scattered power to the absolute VSF at near-forward angles.

## 2. Methods

#### 2.1. LISST measurements

The scattering response of the LISST was related to theoretical calculations of the VSF by analyzing microspheres of a known size distribution and composition. The microspheres were polystyrene particle size standards [17], with diameters ranging from approximately 2 to 100 μm, as summarized in Table 1. The EZY-CAL standards were contained in single-use bottles, suspended in water (with trace amounts of dispersant) at an approximate concentration of 2000 particles mL^{-1}. The other standards were contained in dropper-tipped bottles in concentrations on the order of 1% solids by weight. The mean diameters, μ_{D}, were traceable to the National Institute of Standards and Technology (NIST), evaluated by photon correlation spectroscopy or microscopy using a NIST calibrated stage micrometer. The reported uncertainty in mean diameter, δ_{D}, for each standard, was the sum of the calibration transfer uncertainty and measurement error. The standard deviation of the size distribution for each standard, σ_{D}, is certified by Duke Scientific through microscopy or electrical impedance analysis.

We made no independent attempt to validate the manufacturer-provided NIST-traceable particle size distribution. Given the agreement between the theoretical Mie results (which assumes a Gaussian PSD) and the LISST-measured VSF, we must suppose that the single Gaussian PSD is a reasonable assumption. However, users should use caution with such particle size standards, since the dispersant solution used may degrade over time, leading to destabilization of the suspension and aggregation of the beads. Bacterial contamination of the suspension is also possible.

For the calibration experiments, the LISST (LISST-100, type B, s/n 1102) was fitted with a small volume mixing chamber available from Sequoia Scientific [12]. Particle settling was counteracted by recirculating the sample through tubing connected to the sample volume and routed through a peristaltic pump. Turbulence generated due to the recirculation did not have a detectable effect on the optical scattering measured by the LISST, verified by comparing measured results on small beads with and without recirculation. EZY-CAL standards were added directly to the chamber in which pure water (Barnstead NANOpure Diamond) was recirculated. Test suspensions were prepared by filling the chamber with pure water and adding microsphere solution directly to the chamber in an addition series. Before additions to the LISST chamber, the sample bottles were agitated by hand or immersed in an ultrasonicated bath.

LISST measurements were logged using the standard LISST-SOP software provided by Sequoia Scientific and further processed in the MATLAB programming environment. Pure water
blanks were taken before readings on different microsphere stocks. Blank measurements consist of a set of 32 pure water ring detector outputs, *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,

where t is the transmission calculated as τ = (*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], and for laser power in the measurement relative to the clear water measurement, giving the uncalibrated scattered power in each ring, *pscat*_{i}
:

where *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,

For each calibration run, 1 Hz data were collected and processed according to Eqs. (3)-(4. In order to assess the uncertainty in LISST scattering measurements for a given calibration run, the median, 16th percentile, and 84th percentiles of scattered power were calculated for each ring. Median and percentile values were also calculated for the LISST-measured beam attenuation.

The LISST ring detectors are designed to increase in area geometrically with increasing angle in order to maintain an approximately equal photon flux per area of detector, accounting for the high peak in the near-forward scatter typically observed in natural particle assemblages. For the smallest microspheres, scattering in the most near-forward angles is very low, and due to the variability caused by experimental uncertainty and electronic noise, may lead to negative values of *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

Mie theory is an exact solution of Maxwell’s equations for the interaction of an electromagnetic plane wave with homogenous spheres. Detailed descriptions can be found in Bohren and Huffman [5] and Mishchenko [6]. Mie theory considers wavelength of light, particle size, and particle composition in terms of two fundamental parameters: (1) the size parameter, defined as *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], given in Eq. (6). Imaginary index of refraction for polystyrene (*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].

For a given *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].

The size distributions of microsphere calibration standards were assumed to be Gaussian in each case, with the mean diameters and standard deviations provided in Table 1. The general form of the Gaussian distribution is

where *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}.

From these quantities, the inherent optical properties can be determined for an assemblage of spheres with known size distribution. For example, particle beam attenuation, *c*_{Mie}
, is calculated by integration of the extinction cross section, *C*_{ext}
, over size distribution,

Since concentration varies for each calibration run, attenuation measured by the LISST is used to determine the particle number concentration by normalizing the Mie-calculated values to

Thus, based on the calculated ${c}_{\mathit{\text{Mie}}}^{\star}$ [m^{2}] and the observed *c*_{LISST}
, *N*
_{0} is estimated by assuming that *c*_{Mie}
= *c*_{LISST}
, hence *N*
_{0} = *c*_{LISST}
/${c}_{\mathit{\text{Mie}}}^{\star}$. To consider the possible bias introduced by the finite acceptance angle of the LISST transmissometer (0.0269°), the difference between theoretical and measured beam attenuation was determined using Δ*c* = ${\int}_{0}^{0.0269\xb0}$ ${\beta}_{\mathit{\text{Mie}}}^{\star}$(Ψ) sin (Ψ)*dΨ*. Based on this formulation, relative bias (Δ*c*/${c}_{\mathit{\text{Mie}}}^{\star}$) in *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.

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

and normalizing such that the integral over all angles is unity,

Similar to the case of beam attenuation, the VSF was then determined by integrating over the particle size distribution,

and normalized to particle concentration, as in Eq. (9),

Figure 3(a) shows the Mie-derived volume scattering functions, normalized to beam attenuation. Note that the VSF for smaller microspheres tend to be less highly peaked for smaller angles. Natural volume scattering functions tend to be more peaked in the forward direction [9], given that they are essentially a superposition of monospecific VSF including particles often larger (and hence their VSF are more forward-peaked) than the 100 μm VSF shown here.

The power sensed by each photodetector ring, *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

In addition to determination of suspension concentration using measured beam attenuation, two further issues were addressed in the calibration: manufacturer-specified uncertainty in the bead standards and experimental uncertainty in the LISST measurements. A Monte Carlo technique was used to address the reported uncertainty in nominal diameter, δ_{D}. For each microsphere stock in Table 1, ${c}_{\mathit{\text{Mie}}}^{\star}$
and 〈${\beta}_{\mathit{\text{Mie}}}^{\star}$〉*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 ${c}_{\mathit{\text{Mie}}}^{\star}$ and for 〈${\beta}_{\mathit{\text{Mie}}}^{\star}$
〉*i* calculated for each ring.

As discussed in Section 2.1, median values and percentiles were calculated for the LISST p*scat*_{i}
and beam attenuation. Mie-derived average VSF, 〈${\beta}_{\mathit{\text{Mie}}}$
〉*i*, was then calculated using

where *c*_{LISST}
is the LISST-measured beam attenuation. The uncertainty in β_{Mie}(Ψ) was addressed by propagating the uncertainties in *c*_{LISST}
, ${\beta}_{\mathit{\text{Mie}}}^{\star}$, and ${c}_{\mathit{\text{Mie}}}^{\star}$,

where δ*c*_{LISST}
, δ〈${\beta}_{\mathit{\text{Mie}}}^{\star}$
〉*i*, and δ${c}_{\mathit{\text{Mie}}}^{\star}$, are the half the difference between the 84th and 16th percentiles of *c*_{LISST}
, 〈${\beta}_{\mathit{\text{Mie}}}^{\star}$
〉*i*, and ${c}_{\mathit{\text{Mie}}}^{\star}$, respectively. For each run used in the calibration (a summary of all runs is presented in Table 2), 〈${\beta}_{\mathit{\text{Mie}}}$
〉*i* was calculated according to Eq. (15). The
resulting relationship between the theoretical 〈${\beta}_{\mathit{\text{Mie}}}$
〉*i* and LISST measurements was assumed
linear, and a calibration coefficient [m^{-1} sr^{-1} count^{-1}] relating the LISST measured *pscat*_{i}
to 〈${\beta}_{\mathit{\text{Mie}}}$
〉*i* was calculated for each detector ring,

The parameter χ_{i} for each ring was determined using a robust regression [19] (Fig. 4(a)–(e)). In general, δ〈${\beta}_{\mathit{\text{Mie}}}$
〉*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 〈${\beta}_{\mathit{\text{Mie}}}$
〉*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].

The median of calibration coefficients χ_{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,

Relative error in the χ*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

Our results support the use of the LISST to measure the near-forward volume scattering function of suspended particles. Independent trials using several different sizes of microspheres yielded a set of calibration constants for the detector rings of the LISST, and additional tests on microsphere standards provided an independent consistency check on the calibration. We are unsure of the applicability of the calibration constants derived for the LISST used in this study to other LISST instruments. Variability in the ring-specific *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.

In the field, the VSF obtained using the derived calibration coefficients includes scattering due to all substances in the sample volume besides that of pure water. This will likely include particles of a variety of different shapes, both single and aggregated, compact and fluffy. The scattering may also include that of colloidal material, salts, turbulence, and bubbles. For a LISST instrument that has been calibrated using pure water *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.

The effects of absorption on the Mie-derived VSF in larger angles for larger beads was not negligible, as can be seen in Fig. 6. We avoid these problematic regions of the VSF for the larger, e.g. 20 and 100 μm beads entirely, since (1) errors in calculating the VSF for absorbing spheres may bias calibration data, (2) the VSF is highly variable in these regions leading to possible errors in averaging over ring angles, such as due to error in knowing the absolute ring edges or due to slight misalignment of optical components, and (3) these regions are possibly affected by the polarization of the LISST laser source (discussed below). Coefficients for rings derived using data selected based on these three criteria will possess less uncertainty due to our inability to independently verify the VSF and suspension PSD.

The VSF, by definition, describes the scattering distribution of unpolarized light (unpolarized source and detector), namely the *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]. Thus it appears that the errors introduced to the VSF due to the use of a polarized source are small.

This is an exciting time in the field of ocean optics, as for the first time a commercial instrument is available to measure in situ and understand environmental variability in the VSF in near-forward angles. Combining the LISST measurements with other commercially-available sensors measuring the VSF at angles in the backwards direction can provide a means to infer the full VSF, using theoretical VSF models (e.g. Fournier-Forand [11]) to fit measured data, and directly estimate *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]). Comparison with available in-water and bench-top instrumentation will provide the necessary optical closure test for this approach.

## Acknowledgments

We thank Laura Azevedo for assistance with the measurements. We also thank A. Weidemann and two anonymous reviewers for thoughtful reviews of an earlier version of this manuscript. This work was supported by the Optical & Biological Oceanography program at the Office of Naval Research.

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