Field observations and theoretical studies have shown that shapes of the volume scattering functions (VSFs) of oceanic particles in the backward directions, i.e., VSFs normalized by the total backscattering coefficient, exhibit a surprisingly low variability at angles near 120 degree, which is also confirmed by measurements of VSFs in coastal waters around the US. To investigate what this minimum variability angle (θ*) represents, we estimated mean values of the VSFs in the backward angles using four mean value theorems: mean value for integral, weighted mean value for integral, classic mean value for differentiation and Cauchy’s mean value. We also estimated the angles corresponding to the minimum values of the VSFs. We found θ* to be very close to the angles representing the classic mean values for differentiation of the VSFs. The low variability is due to the fact that the classic mean values vary little with the composition and sizes of particles.
© 2014 Optical Society of America
The volume scattering function (β(θ), m−1 sr−1) describes the angular distribution of scattered light by an assemblage of particles as a function of scattering angle, θ. It is one of the key determinants of the light field in the ocean . The backward portion of the VSF (i.e., 90 ≤ θ ≤180°) is of particular importance for applications of ocean color because it largely dictates the magnitude and shape of reflected solar radiation from the ocean  that is amenable to remote observation, say, from a satellite . The backscattering coefficient (bb, m−1) is often used to quantify the overall backward scattering and is defined as
Both particles and seawater contribute to the volume scattering additively. Molecular scattering by seawater has been known relatively well for several decades  and recent improvements [5–8] allow its prediction to within 2% of laboratory measurements. In the remaining part of the paper, VSFs or β(θ) refers to the volume scattering function due to particles, which can be easily estimated by subtracting the seawater contribution from the measured VSFs. Normalizing β(θ) with bb or alternatively the χ factor, defined as , describes the shape of β(θ) or 1/β(θ) in the backward directions. In the past, because of instrumentation difficulty, field measurements of the VSFs in the ocean were scant [9, 10]. Analyzing all the field data available up to his time, Oishi  found χ varies the least at 120° with a mean value of 1.14 and a standard deviation of ~5%. Let θ* denote the angle at which χ exhibits minimum variability. Recent technological advancement has allowed VSFs to be measured more frequently and extensively [12, 13], which led to a renewed interest in the behavior of the χ factor [13–17]. Sullivan and Twardowski  found θ* was between 110 – 120° based on an extensive data set collected by MASCOT, a prototype instrument measuring VSFs between 10 and 170° at 10° interval. Boss and Pegau  analyzed data collected near the LEO-15 site off the coast of New Jersey, using another prototype instrument, VSM  measuring VSFs at a finer angular resolution of 0.25°, and found θ* = 118° and χ(118°) = 1.10 ± 4%. We have measured VSFs using an improved version of VSM at three coastal waters around U.S., Monterey Bay (MT), Chesapeake Bay (CB) and Mobile Bay (MB) . Combining our data with the data measured at the LEO-15 site we found that while VSFs at backward angles exhibit over 2 orders of magnitude variation (Fig. 1-(a)), the χ factor shows a minimum variability at 122° and χ(θ*) = 1.10 ± 1.45% (Fig. 1-(b)).
The behavior of χ(θ) has also been simulated for spherical particles following the power-law size distribution. Under this simplified assumption, χ(θ) were modeled for different power-law exponents and different refractive indices [11, 17, 19]. Additional modeling was also done for different cut-off sizes at the small size end of power-law function and for coated particles . All of simulated χ(θ) intersect around 112 - 120°, indicating very limited variability in this angular range.
The χ represents the form of the VSFs in the backward angles, which is very sensitive to the shape , composition , and size distributions of particles and expected to vary significantly for different aquatic environment with differing particle assemblages . The existence of a consistent angle at which χ values estimated from different waters exhibit little variability is intriguing. This behavior was also found for mono-specific phytoplankton cultures . To further investigate this phenomenon, we examine particles of different sizes using an optical inversion technique that we recently developed  with results that have been validated in different studies [18, 25–28].
2. Small and large particles
The technical details of the inversion method have already been reported [18, 24]. Briefly, a measured VSF is partitioned using an optimization technique into contributions by different particle species, each of which is represented by a unique combination of log-normal size distribution and composition (we use refractive index as a proxy for composition). The Log-normal function closely represents size distribution of particles resulting from natural processes of breakage , coagulation , or cell division . Particles are simulated using asymmetrical hexagonal shape instead of spheres (an extreme shape having the smallest surface-area/volume). The inversion method has been tested in studies of natural bubble population [25, 26], phytoplankton and non-algal particles , particulate organic and inorganic matter  and bulk particulate size distributions  that are consistent with independent estimates. In this study, we will further group different particle species into two distinctive groups; small particles, representing those with sizes < 0.2 µm and large particles. There are two reasons for this approach: (1) marine particles are often operationally partitioned into dissolved (small) and particulate (large) particles via filtration; and (2) small and large particles exhibit distinctive shapes in VSFs. With this approach, we haveFig. 2. Note that VSFs for large particles have strong forward scattering as compared to small particles.
From Eq. (2), it followsFig. 3. Several features can be discerned clearly. First, χ varies but the general angular shapes are similar within each particle group. Second, the general shapes of χ differ significantly between the two particle groups. Third, the χ factor shows minimum variability at angles about 120° for both groups, even though the exact values of χs(θ*) and χl(θ*) are different. And finally, there are insignificant differences in χs(θ) or χl(θ) between different experiment sites.
3. Theoretical interpretation
While both direct observation and optical inversion indicate the existence of this minimum variability angle θ*, the question remains as to what θ* represents? Following the weighted mean value theorem for integrals, there exists , for which it follows from Eq. (1) that: . Oishi  conjectured that θ* ≈θ1. To evaluate whether this is indeed the case, we merged β(θ) measured in the four field experiments into one data set and estimated θ1 for both small and large particle groups. Figure 4 compares the values estimated for θ1 with the minimum variability angle, θ*. Clearly, these two angles are not the same. Actually if the two angles were the same, then χ(θ*) = 1, differing considerably from either theoretical or experimental results that have shown χ(θ*) ≈1.05 – 1.20. Expanding on Oishi’s approach, we estimated three other angles (θ2, θ3, and θ4) , based on: the mean value theorem for integrals, ; the classic mean value theorem, ; and Cauchy’s mean value theorem, . The symbol ′ denotes differentiation. Note that we do not have VSF measurements at exactly 180°; the measurements were down to 177.3° in the LEO-15 experiment and down to 179° in the other three experiments. We used β(177.3) or β(179) directly in the above estimations. We tested by extrapolating the VSFs to 180°, with negligible differences in the estimations of these angles. Also, we estimated the minimum β(θ) angle (θ5), i.e., . The comparisons for these four additional angles are also shown in Fig. 4. For both small and large particle populations and for the bulk particle population, θ3, among all the four angles, is the closest to θ* in values and exhibits the smallest variability.
Figure 4 naturally leads to a question—why are the angles (θ3) estimated for different particle groups more similar to each other? In an attempt to answer this question, we look at two extreme cases, particles of sizes smaller than the wavelength in vaccum (λ), i.e. size, with m0 being the refractive index of the medium (m0 ≈1.33 for water) and particles of sizes larger than the wavelength, i.e. size . Assuming λ = 0.532 µm, particles smaller than the wavelength are roughly of sizes < ~0.07 µm and those larger than the wavelength of sizes > ~10 µm. For oceanic particles smaller than the wavelength, the scattering can be approximated as ,33] and “the scattering pattern caused by reflection on very large convex particles with random orientation is identical with the scattering pattern by reflection on a very large sphere of the same material and surface condition” . Following Fournier , the VSF due to reflection can be approximated as:Eq. (4) for particles smaller than the wavelength and they are 125°, 135°, 110° 145° and 90°, respectively. Similarly, we estimated the values from Eq. (5) for particles much larger than the wavelength. They vary slightly with the refractive index, and range approximately 111 – 112° for θ1, 116 – 117° for θ2, 117 – 119° for θ3, 120 – 122° for θ4, and 180° for θ5, respectively. Between the two sets of estimates, the difference for θ3 is the smallest, varying only 7 – 9° between extremely small and large particles, whereas the differences for other angles are at least 15° or larger. Two interesting notes regarding the minimum β(θ) angle (θ5): it shifts from 90° for particles smaller than the wavelength to 180° for particles larger than the wavelength and its value does not change with the refractive index for extremely large particles (geometric optics).
Next, we examine if these two extreme cases would form a bounding constraint. In nature, particles form a continuum in sizes, ranging from molecules to particulates of sizes millimeters or larger and all contribute to the volume scattering functions. The β(θ) by oceanic particles is often approximated using Fournier and Forand  phase function [see also 33], developed based on two simplified assumptions: particles follow a power law (or Junge) size distribution and have the same refractive index, with the values of the power-law exponent and index of refraction relative to water varying −5.0 – −3.5 and 1.02 – 1.20, respectively . Figure 5 shows the variation of θ3 estimated for the Fournier and Forand phase functions as a function of the power-law exponent (or Junge slope) and the refractive index of particles. Relatively, θ3 is found to be more sensitive to the size distribution than the composition of particles. For size distributions favoring the smaller particles with a steeper slope, the values of θ3 are close to the theoretical value of 110° estimated for particles of sizes less than the wavelength, and θ3 approaches 120° as large particles become increasingly dominant. This indicates that the variability of θ3 is well constrained, at least from this simplified theoretical analysis. Our result is consistent with the early finding by Boss and Pegau  showing that χ(117) based on Fournier and Forand phase functions have roughly the same value as χ(117) by seawater, for which β(θ) is the same as Eq. (4) for particles smaller than the wavelength.
We now examine bubbles, which are found in copious amounts in both rough [36, 37] and quiescent seas [38–40]. With m = 0.75, a (large) bubble behaves optically different from a typical particle, whose m > 1. Since Eq. (4) does not depend on the particle type, the five angles (θ1, θ2, θ3, θ4 and θ5 = 125°, 135°, 110°, 145° and 90°, respectively) are the same for a bubble and a particle both of sizes smaller than the wavelength. But these five angles are quite different at sizes much larger than the wavelength between a bubble and a particle; for a bubble these five angles are 108°, 112°, 110°, 114°and 180°, respectively, as compared to 111 – 112°, 116 – 117°, 117 – 119°, 120 – 122°, and 180°, respectively, for particles of different refractive indices. Interestingly, values of θ3 for bubbles are nearly the same ( = 110°) between the two extreme sizes, whereas the values for other angles differ significantly. Since the Fournier and Forand phase functions do not apply to bubble populations, we estimated β(θ) using Mie theory (bubbles are basically spherical) and five angles assuming the same power-law size distributions. For the exponent from −5 to −3.5, θ3 varies from 114° to 108°, tightly around 110° which is found for both extremely small and large bubbles.
We have shown that the angle θ* at which minimum variability has been observed in the backward shape of β(θ) represents the location of the classic mean value of β(θ) (we denote this angle as θ3 in this study). While β(θ) in the backward angles is expected to vary strongly with the size, composition and shape of particles, the values of θ3 are surprisingly invariant. In terms of particles sizes, we have examined variations of θ3 in two general size fractions of dissolved and particulate (Fig. 4), in two end size members (Eqs. (4) and (5)), and in varying proportions of small and large particles. In terms of the effect of particle shape, our inversion method (Fig. 2) uses asymmetrical hexahedra to represent particles  and the reflective contribution to β(θ) was estimated using spheres. The former is almost a diametrical opposite to the latter: asymmetry vs symmetry and sharp edges vs smooth curvature. In terms of composition, the refractive indices used in the inversion and simulation cover the typical range for oceanic particles  and bubbles. While not exhaustive, our analyses do cover a wide range of possible representations of oceanic particles over a variety of water types. The values of θ3 estimated over all these analyses varied within a tight range of 110 – 125°.
In remote observation of the aquatic environment (i.e., ocean color), the magnitude of the reflected signal is directly proportional to bb, which is very difficult to measure accurately (see Eq. (1)). The existence of a minimum variability angle θ* for β(θ) in the backward angular range has led to practical applications, because it is possible to infer bb relatively accurately by measuring scattering at the angle of θ* (~120°). For example, commercial sensors are available measuring scattering at fixed angles [19, 43], from which bb are estimated.
Boss et al.  showed that scattering measured around 120° performs better as a predictor of particulate matter concentration (or total suspended solids) than measurements of side-scattering at 90° or the attenuation coefficient at coastal sites. Generally, VSFs vary strongly with the concentration, size distribution and type of particulate matter in the water . However, taking measurement of VSFs around 120°, approximately representing the classic mean value of VSFs in the backward angular range that we have shown is not very sensitive to the size distribution and composition of particles, leaves the dominant sensitivity attributable to the concentration.
Even though θ3 is most consistent with θ* and this numerical consistency remain relatively unchanged for different particle assemblages either by simulations or observations, we do not know its physical root. It remains to be investigated as to what physical connotation the classic mean value of β(θ) in the backward angles carries that leads to such a constrained variability?
XZ acknowledges the funding support from National Aeronautics and Space Administration (NASA) (NNX13AB20A, NNX13AN72G). EB acknowledges funding support from the Office of Naval Research (00014-13-1-0146). We thank two anonymous reviewers for their valuable comments.
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